diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznvja" "b/data_all_eng_slimpj/shuffled/split2/finalzznvja" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznvja" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{intro}\nDeep learning has become a promising way to model the complexity of stock movements. It enables us to capture non-linear movements, to associate large data, and to reduce noise without an assumption of a pre-specified underlying structure. At the same time, it leaves us with a difficulty in selecting numerous hyperparameters, which critically affects the performance of the resulting models.\nMost studies dealing with a financial time series typically choose pre-specified hyperparameters and check the robustness of the model based on small changes in the parameters. This approach requires experts to put a lot of effort into tuning numerous parameters simultaneously, which often results in a suboptimal model. \n\n Hyperparameter optimization (HPO) can be used to mitigate this problem by automatically searching\nfor the most optimal hyperparameters in machine learning learners, and has been widely used to identify good configurations more quickly, such as through the use of a sequential model-based algorithm configuration (SMAC), tree-structure Parzen estimator (TPE), and Sprearmint \\cite{feurer2014using}.\nHPO has also been demonstrated to be an extremely powerful approach for automatic image and speech recognition, and offers advantages for dealing with machine learning in a systematic manner. \nFirst, it reduces the human effort necessary in tuning the hyperparameters and opens up the possibility of improving the performance of machine learning \\cite{melis2018state}\\cite{snoek2012practical}. Second, it improves the reproducibility and fairness of scientific studies because an automated HPO is more reproducible than a hand-tuned approach using trial-and-error\nsearches to produce a desired behavior, thereby allowing us to compare different methods more fairly through the same level of tuning \\cite{bergstra2013making}\\cite{sculley2018winner}.\n\n\nDespite such advantages, financial studies have generally not considered this method. HPO requires a large data scale to avoid an overfitting occurring in both the training and validation data. Stock-related data are obtained only over a relatively short time span, typically from the year 1950 to the present. As shown in Fig. \\ref{fig:log_return}, a random evolution of a stock return, such as time-varying volatility and occasional jumps related to crashes or sudden upsurges, causes a time dependency of the model parameter set to specific periods. Furthermore, cross-validation and shuffling, which are crucial techniques for preventing an overfitting, cannot be used because stock-related data are time-ordered, and a modeling process requires preserving the time ordering. \nFor these reasons, the use of HPO has\nrarely been assessed and there is a poor understanding of its efficiency in financial data modeling. As a result, practitioners need to pay more attention to hyperparameter tuning and the resulting models largely depending on their experience. \n\\begin{figure}[t]\n\\centering\n \\scalebox{0.5}\n {\n\t\\includegraphics{Fig1.pdf}\n\n }\n\\caption{S$\\&$P 500 index and its returns from Jan. 1, 1950 to Dec. 31, 2017.}\n\\label{fig:log_return}\n\\end{figure}\n\n\nIn this study, we evaluate the viability of HPO in terms of the stock return predictability problem. We examined the HPO performance across different conditions, the input features of the fundamentals and technical indicators, and the regularization of a dropout and batch normalization.\nOur key findings are as follows:\n\\begin{itemize}\n\\item[\u2022] We show that, whereas the prediction models with an input of fundamentals are likely to overfit the in-sample data, \nmodels with the input feature of the technical indicators achieves a strong predictability throughout the in- and out-of-sample periods. A dropout is more effective for a positive predictability in an out-of-sample than a batch normalization.\n\\item[\u2022] We show that the model with good predictability in both an in- and out-of-sample is less sensitive to the time evolution, which reveals that it is a general model for adapting to the changes in the economic and business conditions.\n\\end{itemize}\nWe believe this study provides insight into the application of machine learning for investment purposes or risk management.\n\\\\\n\\\\\n\\noindent {\\bf Related work} \n In financial economics, there is a long-standing debate whether (excess) stock market returns are predictable. \nThe conventional framework for analyzing equity premium predictability is a `linear predictive regression' model taking the following form:\n\\begin{equation}\nr_{t+1}=\\alpha+\\bm \\beta^{'} \\bm x_{t}+\\varepsilon_{t+1},\n\\end{equation}\nwhere $r_{t+1}$ is the return on the stock market index in excess of the risk-free interest rate, $\\alpha$ is an intercept term, $\\bm \\beta$ is a $p\\times 1$ dimensional vector of the slope parameters, $\\bm x_{t}$ is a $p\\times 1$ dimensional vector of the predictor variables observed at time $t$, and $\\varepsilon_{t+1}$ is a zero-mean disturbance term. \nThe most commonly followed approaches are the use of individual bivariate regressions using one variable at a time from the Goyal and Welch (GW)\npredictor variables \\cite{welch2007comprehensive}, or a multivariate regression, which includes the full set of GW predictors in (1) (see \\cite{goyal2003predicting}\\cite{welch2007comprehensive}\\cite{campbell2007predicting} for a bivariate regression and \\cite{rapach2010out}\\cite{neely2014forecasting}\\cite{buncic2017macroeconomic} for a multivariate regression).\n\n\nDeep learning models are on the rise, showing impressive results in modeling the complex behavior of financial data. Examples include stock prediction based on long short-term memory (LSTM) networks \\cite{fischer2018deep},\ndeep portfolios based on deep autoencoders \\cite{heaton2017deep},\nthreshold-based approaches using recurrent neural networks \\cite{lee2018threshold}, and\ndeep factor models involving deep feed-forward networks \\cite{nakagawa2018deep}, LSTM networks \\cite{nakagawa2019deep}, and fundamentals \\cite{alberg2017improving}. These studies apply hand-tuned hyper-parameters.\n\nIn section \\ref{sec:2}, we provide the data used in this study and the preprocessing methods.\nIn section \\ref{sec:3}, we describe the experimental setting and its implementation.\nIn section \\ref{sec:4}, we provide the experimental results and make comparisons between models.\nFinally, some concluding remarks are given in section \\ref{sec:5}.\n\n\n\n\\section{Data and preprocessing}\n\\label{sec:2}\nWe used sets of fundamentals and technical indicators that have traditionally been used for studying stock predictability. \n\\\\\n\\\\\n\\noindent {\\bf Technical indicators} \nTechnical analysis is a method for forecasting price movements using past prices and volume and includes a variety of forecasting techniques such as a chart\nanalysis, cycle analysis, and computerized technical trading systems. \n\nTechnical analysis has a long history of widespread use by participants in\nspeculative markets \n\\cite{smidt1965amateur}\n\\cite{billingsley1996benefits}\n\\cite{fung1997information}\n\\cite{menkhoff1997examining}\n\\cite{cheung2001currency}\n\\cite{gehring2003technical}, and\nthere is a large body of academic evidence\ndemonstrating\nthe usefulness of a technical analysis, including theoretical support \n\\cite{brown1989technical} and empirical evidence \n\\cite{lo2000foundations}\\cite{blume1994market}, as well as their role in out-of-sample equity premium predictability \n\\cite{baetje2016equity}\n\\cite{rapach2010out}\n\\cite{neely2014forecasting}.\n\n\nThe monthly market data for the S$\\&$P500 were obtained from Yahoo Finance and contain daily trading data, i.e., the opening prices, high prices, low prices, adjusted closing prices, and end-of-day volumes. The data are from the period between January 1, 1950 and December 31, 2017 (Fig. \\ref{fig:log_return}).\nWe used a full set of 14 technical indicators based on 3 types of popular technical strategies, moving average crossover rules, momentum rules, and volume rules:\n\\begin{itemize}\n \\setlength\\itemsep{1em}\n\\item\tThe time-series momentum indicator, MOM($m$), is the generation of a buy signal when the price is higher than the historical price. Its validation is supported by the observation that the ``trend'' effect persists for approximately 1 year and then partially reverses over a longer timeframe. \nHere, $\\textrm{MOM}_{t}(m)$ at time $t$ is\ndefined as follows:\n\\begin{equation}\n \\textrm{MOM}_{t}(m)=\\begin{cases}\n 1 \\textrm{ (Buy signal) }, & \\text{if} \\quad P_{t} \\geq P_{t-m}\\\\\n -1 \\textrm{ (Sell signal) }, & \\text{otherwise}.\n \\end{cases}\n\\end{equation}\nwhere $P_{t}$ is the index value at time $t$, and $m$ is the\nlook-back period. \nWe use $m = 1,3,6,9$ and $12$, which are respectively labeled as\n$\\textrm{MOM}_{t}$(1M), \n$\\textrm{MOM}_{t}$(3M),\n$\\textrm{MOM}_{t}$(6M),\n$\\textrm{MOM}_{t}$(9M), and\n$\\textrm{MOM}_{t}$(12M).\n\t \n\\item The moving average indicator, MA$(s,l)$, \nprovides a signal for an upward or downward trend.\nA buy signal is generated when the short-term moving average crosses above the long-term moving average because this represents the beginning of an upward trend. A sell signal is generated when the short-term moving average \ncrosses below the long-term moving average because this represents the beginning of a downward trend.\n\nLet us define\na simple moving average of the index as follows:\n\\begin{equation}\n\\textrm{MA}_{j,t}^{P}=(1\/j)\\sum_{i=0}^{j-1}P_{t-m} \n\\textrm{ for } j=s \\textrm{ or }l,\n\\end{equation}\t\nwhere $s$ and $l$ are the look-back periods for short and long moving averages. \nThe moving average indicator $\\textrm{MA}_{t}(s,l)$\nis then designed as follows:\n\\begin{equation}\n \\textrm{MA}_{t}(s,l)=\\begin{cases}\n 1 \\textrm{ (Buy signal) }, & \\text{if} \\quad \\textrm{MA}_{s,t}^{P} \\geq \\textrm{MA}_{l,t}^{P}\\\\\n -1 \\textrm{ (Sell signal) }, & \\text{otherwise}.\n \\end{cases}\n \\end{equation}\n\tThe six moving average indicators are constructed for $s=1$, $2$, $3$, and \n$l=9$, $12$, which are symbolized as \n\tMA(1M-9M), MA(1M-12M), MA(2M-9M), MA(2M-12M), MA(3M-9M), and MA(3M-12M).\n\\item The volume indicator, VOL($s,l$), indicates a strong market\ntrend if the recent stock market volume and stock price increase.\nLet us define the on-balance volume (OBV) as follows:\n\\begin{equation}\n\\textrm{OBV}_{t}=\\sum_{k=1}^{t}VOL_{k}D_{k},\n\\end{equation}\nwhere $VOL_{k}$ is a measure of the trading volume (i.e., number of shares traded) during period $k$, and $D_{k}$ is a binary variable: \n\\begin{equation}\n D_{k}=\\begin{cases}\n 1, & \\text{if} \\quad P_{k}\\geq P_{k-1}\\\\\n -1, & \\text{otherwise}.\n \\end{cases}\n\\end{equation}\nThe value of $\\textrm{OBV}_{t}$ conceptionally measures both positive and negative \nvolume based on the belief that changes in volume can predict a stock movement. The volume-based indicator is then defined as the difference between\nthe moving averages with a $s$-period and $l$-period:\n\\begin{equation}\n \\textrm{VOL}(s,l)=\\begin{cases}\n 1 \\textrm{ (Buy signal) }, & \\text{if} \\quad \\textrm{MA}_{s,t}^{\\textrm{OBV}} \\geq \\textrm{MA}_{l,t}^{\\textrm{OBV}}\\\\\n -1 \\textrm{ (Sell signal) }, & \\text{otherwise}.\n \\end{cases}\n\\end{equation}\nHere,\n$\n\\textrm{MA}_{j,t}^{\\textrm{OBV}}=(1\/j)\\sum_{i=0}^{j-1}\\textrm{OBV}_{t-i}\n$ is the moving average of $\\textrm{OBV}_{t}$ for $j=s$ or $l$. \nThe six moving average indicators are constructed for $s=1$, $2$, $3$ and \n$l=9$, $12$, which are symbolized as \nVOL(1M-9M), VOL(1M-12M),\nVOL(2M-9M), VOL(1M-12M), VOL(3M-9M) and VOL(3M-12M).\n\\end{itemize}\n\\noindent {\\bf Fundamental indicators}\nWe use the financial indicators employed by \n\\cite{welch2007comprehensive} for the\nU.S. stock market, which is available from Amit Goyal's web site. \nWe use updated data consisting of 14 popular fundamental variables\nspanning from January 1950 to December 2017. We provide a short definition of\nthese variables as follows.\n\\begin{itemize}\n \\setlength\\itemsep{1em}\n\\item Dividend-price ratio, DP: Log of a 12-month moving sum of dividends paid on the S\\&P 500 index minus the log of the stock prices. \n\\item Dividend yield, DY: Log of a 12-month moving sum of dividends minus the log of 1-month lagging stock prices.\n\\item Earning-price ratio, EP: Log of a 12-month moving sum of earnings on the S\\&P 500 index minus the log of the stock prices.\n\\item Dividend-payout ratio, DE: Log of a 12-month moving sum of dividends minus the log of a 12-month moving sum of earnings.\n\\item Stock variance, SVAR: Sum of squared daily returns on the S$\\&$P500. \n\\item Book-to-market ratio, BM: Ratio of book value to market value for the Dow Jones Industrial Average. \n\\item Net equity expansion, NTIS: Ratio of 12-month moving sum of net issues by NYSE listed stocks divided by their total market capitalization.\n\\item Treasury Bill rate, TBL: Interest rate on a 3-month treasury bill from the secondary market. \n\\item Long-term yield, LTY: Long-term government bond yields. \n\\item Long-term rate of return, LTR: Long-term government bond returns \n\\item Term spread, TMS: Difference between the long and term yield on government bonds and T-bills. \n\\item Default yield spread, DFY: Difference between BAA- and AAA-rated corporate bonds and returns on long-term government bonds. \n\\item Default return spread, DFR: Difference between the return on long-term corporate bonds and returns on the long-term government bonds.\n\\item Inflation, INFL: Consumer Price Index (CPI) for all urban consumers.\\\\ \n\\end{itemize} \n\\section{Experiments}\n\\label{sec:3}\n\\noindent {\\bf Data Splits:}\nAs mentioned earlier, the predictability found in traditional studies is not uniform over time and is concentrated within certain periods \\cite{neely2014forecasting}. To check the robustness, we investigated the predictability over four different periods, the entire period of $1950-2017$ (Exp. 1) and its sub-periods of $1950-2015$ (Exp. 2), $1950-2007$ (Exp. 3), and $1950-2002$ (Exp. 4).\nFor each experiment, we split the\ndata into in-sample and out-of-sample periods.\nThe in-sample data were divided into a training dataset (50$\\%$) for developing the prediction models and a validation set (50$\\%$) for evaluating its predictive ability.\n\\\\\n \\\\\n\\noindent{\\bf Training:} \nDeep feedforward neural networks (DNNs) were used in this study. We applied TPE for automated hyperparameter tuning with\nadditional tests using simulated annealing and a random search to further confirm our results. \nThe hyperparameters and their\nprior distributions are summarized in Table \\ref{params}.\nFor hyperparameter selection, we trained DNNs on an in-sample training set and selected the model with the lowest validation error.\nWe limited the number of function evaluations for finding optimal hyper-parameters to $50$. \nEach evaluation comprised training the DNN\nmodels for 200 epochs and selecting the model with the lowest validation error.\n\\\\\n\\\\\n\\noindent{\\bf Regularizer:} \nWe are particularly interested in regularization methods for model generalization\nbecause the time-dependent behavior of financial data is likely to cause a parameter instability over an out-of-sample. \nWe examined the effectiveness of the most popular regularization methods, namely, a dropout and batch normalization (BN). \nA dropout\\cite{srivastava2014dropout} \nis a simple way to prevent co-adaptation among\nhidden nodes of deep feed-forward neural networks by randomly dropping out selected hidden nodes.\nIn recent years, batch normalization \\cite{IoffeS2015batch} has replaced a dropout\nin modern neural network architectures. \nIt uses the distribution of\nthe summed input to a neuron over a mini-batch of training cases to compute the\nmean and variance, which are then used to normalize the summed input to the\nneuron for each training case.\nDropout and BN layers are employed for all hidden layers.\n\\\\\n\\\\\n\\begin{table}[htbp]\n\\centering\n\\caption{List of parameters and their corresponding range of\nvalues used in the grid search.}\n\\label{table:meanerrorbaseline}\n\\small\n\\begin{tabular}{lll}\n\\toprule\n Hyperparamter &\\quad \\quad \\quad & Considered values\/functions \\\\\n\\midrule\n Number of Hidden Layers && \\{2, 3\\} \\\\\n Number of Hidden Units && \\{2, 4, 8, 16\\} \\\\\n \\makecell[l]{ Standard deviation} &&\\{0.025,0.05,0.075\\}\\\\\n Dropout && \\{0.25, 0.5, 0.75\\} \\\\\n Batch Size && \\{28, 64, 128\\} \\\\\n Optimizer && \\{RMSProp, ADAM, SGD (no momentum)\\} \\\\\n Activation Function&& Hidden layer: \\{tanh, ReLU, sigmoid\\}, Output layer: Linear \\\\\n Learning Rate && \\{0.001\\} \\\\\n Number of Epochs && \\{100\\} \\\\ \n\\bottomrule\n\\end{tabular}\n\\parbox{\\textwidth}{\\small%\n\\vspace{1eX}\n{\\bf Number of Layers}: number of the layers of the neural network.\n{\\bf Number of Hidden Units}: number of units in the hidden layers\nof the neural network.\n{\\bf Standard Deviation}: standard deviation of a random normal initializer. \n{\\bf Dropout}: dropout rates. \n{\\bf Batch Size}: number of samples per \nbatch. \n{\\bf Activation}: sigmoid function $\\sigma(z)=1\/(1+e^{-z})$, hyperbolic \ntangent function $\\textrm{tanh}(z)=(e^{z}-e^{-z})\/(e^{z}-e^{-z})$,\nand rectified linear unit (ReLU) function $\\textrm{ReLU}(z)=\\textrm{max}(0,z)$.\n{\\bf Learning Rate}: learning rate of the back-propagation algorithm.\n{\\bf Number of Epochs}: number of iterations for all of the training data.\n{\\bf Optimizer}: stochastic gradient descent (SGD) \\cite{kingma2014adam}, RMSProp \\cite{tieleman2012lecture}, and ADAM \\cite{kingma2014adam}}\n \\label{params}%\n\\end{table}\n\n\\noindent {\\bf Out-of-sample $R^{2}$ statistic:} \nWe measured the out-of-sample $R^{2}$ statistics ($R_{\\textrm{OS}}$) \\cite{campbell2007predicting} for a comparison with the in-sample $R^{2}$ statistics ($R^{2}_{\\textrm{IS}}$)\nand evaluated the forecasting power of the models. \nThe $R_{\\textrm{OS}}^{2}$ statistic measures the improvement in the mean square\nforecast error (MSFE) for the return forecast relative to the simple historical\naverage (or constant expected return) forecast, which ignores information contained in the predictors. This is computed as follows:\n\\begin{equation}\nR^{2}_{\\textrm{OS}}=1-\\frac{\\sum_{t=1}^{T}(r_{t}-\\hat{r}_{t})^{2}}{\\sum_{t=1}^{T}(r_{t}-\\bar{r}_{t})^{2}},\n\\end{equation} \nwhere $\\hat{r}_{t}$ is the fitted value from a predictive regression estimated through period $t-1$, and $\\bar{r}_{t}$ is the historical average return estimated through period $t-1$.\n\\\\\n\\\\\n\\noindent {\\bf Model stability:} We analyzed the model stability over time in terms of the feature importance. \nStock price dynamics is so complex with complicated interactions among changing micro\nbehavior, varying product cycles, interdependent industrial structures, and cyclic macro environment, thus it leads to gradual or sudden shifts in the model parameters. For example, traditional univariate models are highly exposed to\nthe model instability in the in-sample, which demonstrates the time-dependency of \nthe statistical significance and the coefficient of the predictor variables \\cite{neely2014forecasting}. To overcome this problem, a multivariate regression model is proposed through which the changes to the parameters at breaks are estimated \\cite{paye2006instability}.\n\nWe examined the stability of the trained model over time by \ncomputing the SHapley Additive exPlanation (SHAP) values of the features \\cite{lundberg2018consistent}\nto find the contribution of the features in the prediction and determine the change in ranking of the features over time. \n\\section{Results}\n\\label{sec:4}\n\n\n\n\n\\subsection{Technical Indicators}\n\n\\subsubsection{Dropout versus batch normalization}\nWe compared a DNN with a dropout and a DNN with batch normalization for the four experiments. The following observations can be made regarding the results reported in Table \\ref{tab:dropout}.\n\\begin{itemize}\n\\item Both DNNs show a good in-sample predictive power of a positive $R^{2}_{\\textrm{IS}}$ for all experiments. The in-sample predictive power of the BN ranging over 1.740 to 2.968 is stronger than that of the dropout ranging over 0.424 to 0.748. \n\\item The DNN with a dropout achieves a good out-of-sample predictive power, showing positive $R^{2}_{OS}$ values for all experiments, which means that it outperforms the historical mean return over the training and validation periods.\nHowever, the BM model achieves a poor out-of-sample predictive power, with negative $R^{2}_{OS}$ values for all experiments. A dropout is more effective at preventing a model instability.\n\\item The instability of the BN model is derived from an overfitting to the in-sample set based on the observation that, although $\\textrm{MSE}_{\\textrm{train}}$ and $\\textrm{MSE}_{\\textrm{val}}$ of the BN model are lower than those of the dropout model (except for only $\\textrm{MSE}_{\\textrm{train}}$ in Exp. 2), $\\textrm{MSE}_{\\textrm{test}}$ of the BN model is higher than that of the dropout model. Figure \\ref{MSE_TPE_BN} graphically shows the overfitting occurring during the training in Exp. 1.\n\\item The results indicate that an in-sample predictive content does not necessarily translate into an out-of-sample predictive ability, nor ensure the stability of the predictive relation over time.\n\\item The degree of predictability varies according to the experimental period, showing that Exp. 2 and 3 show a strong predictability of $1.889$ and $1.670$, and Exp. 1 and 4 show a relatively weak predictability of $0.569$ and $0.319$, respectively.\n\\item Figure \\ref{MSE_forecast_pattern} graphically shows how to beat the historical average in Exp. 1. The dropout model forecasts returns around the mean of the out-of-sample, whereas the historical average showed a greater deviation. This means the model can be adjusted better to a new market environment than the historical average. \n\\item The DNN with a dropout achieves an average predictability of 0.53$\\%$ in-sample and 1.11$\\%$ out-of-sample. The DNN with a dropout has an average predictability of 2.312$\\%$ in-sample \nand $-2.8545\\%$ out-of-sample.\n\\end{itemize}\n\n\\begin{table}[htbp]\n\\small\n \\centering\n \\caption{Comparison of models based on average prediction performance ($\\pm$1 s.d. in parentheses) over 5 runnings with different random initial seeds for each experiment.}\n \\begin{tabular}{L{3.cm} C{2.cm} C{2.cm} C{2.cm} C{2.cm} C{2.cm} }\n \\toprule\n Model & \\multicolumn{1}{c}{$\\textrm{MSE}_{\\textrm{train}}$ } & \\multicolumn{1}{c}{$\\textrm{MSE}_{\\textrm{val}}$ } & \\multicolumn{1}{c}{$\\textrm{MSE}_{\\textrm{test}}$ } &\n \\multicolumn{1}{c}{$R^{2}_{IS}$ } & \\multicolumn{1}{c}{$R^{2}_{OS}$ } \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 1} \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{0.129 \\\\($\\pm 3.236 $)}& \\makecell{0.197 \\\\ ($\\pm 0.171 $)} & \\makecell{ $\\bm{0.186}$ \\\\ ($\\pm\\bm{ 1.506 }$)}& \\makecell{0.748 \\\\ ($\\pm$1.040)}& \\makecell{$\\bm{0.569}$ \\\\($\\bm{\\pm 0.621}$)} \\\\\n DNN w. BN & \\makecell{ $\\bm{0.128}$\\\\($\\bm{\\pm 0.646}$)}& \\makecell{$\\bm{0.193}$\\\\($\\bm{\\pm 1.333 }$)} & \\makecell{0.194\\\\($\\pm 1.713$)} & \\makecell{$\\bm{1.740}$ \\\\($\\bm{\\pm 0.247$})} &\\makecell{$-3.74$ \\\\($\\pm$0.804)} \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 2} \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{ $\\bm{0.126}$ \\\\ ($\\bm{\\pm 0.062}$)} & \\makecell{ 0.206 \\\\ ($\\pm 0.130$)} &\\makecell{$\\bm{0.201}$ \\\\ ($\\bm{\\pm 0.540}$)}& \\makecell{0.451 \\\\($\\pm$0.028)}& \\makecell{$\\bm{1.889}$ \\\\ ($\\bm{\\pm 0.242)}$} \\\\\n DNN w. BN & \\makecell{0.127\\\\($\\pm 1.739$)}& \\makecell{ $\\bm{0.201}$\\\\($\\bm{ \\pm 0.298}$)} &\\makecell{0.209\\\\($\\pm 3.240$)} & \\makecell{$\\bm{1.890}$\\\\ ($\\bm{\\pm 0.293}$)} &\\makecell{$ -2.70$\\\\ ($\\pm$0.906)} \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 3 } \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{0.130\\\\($\\pm 0.189 $)} & \\makecell{0.216 \\\\($\\pm 0.038 $)} & \\makecell{$\\bm{0.147}$ \\\\($\\bm{\\pm 0.174}$)}&\\makecell{0.507 \\\\($\\pm$0.056)} & \\makecell{$\\bm{1.670}$ \\\\($\\bm{\\pm 0.143}$)} \\\\\n DNN w. BN & \\makecell{$\\bm{0.125}$\\\\($\\bm{\\pm 1.618 }$)}& \\makecell{$\\bm{0.213}$\\\\($\\bm{\\pm 0.768 }$)} & \\makecell{0.153\\\\($\\pm 2.781 $)} &\\makecell{$\\bm{2.650}$ \\\\($\\bm{\\pm 0.543}$)} & \\makecell{$-3.518$ \\\\($\\pm$2.650)} \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 4} \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{0.1193 \\\\($\\pm 0.697$)}& \\makecell{0.197 \\\\($\\pm 0.716$)} & \\makecell{$\\bm{0.218}$ \\\\($\\bm{\\pm 0.356}$) }& \\makecell{0.424 \\\\($\\pm$0.181)} & \\makecell{$\\bm{0.319}$ \\\\($\\bm{\\pm 0.139}$)}\\\\\n DNN w. BN & \\makecell{$\\bm{0.115}$\\\\($\\bm{\\pm 3.998 }$)}& \\makecell{$\\bm{0.196}$\\\\($\\bm{\\pm 0.702}$)} & \\makecell{0.219\\\\($\\pm 2.160$)} & \\makecell{$\\bm{2.968}$ \\\\($\\bm{\\pm 0.959}$)} & \\makecell{$-1.460$ \\\\($\\pm$0.989)}\\\\\n \\bottomrule\n \\end{tabular}%\n\\begin{tablenotes}[flushleft]\\footnotesize\n \\item[]\n Note: All the $\\textrm{MSE}$ and $R^{2}$ values have been multiplied by a factor of $10^{-2}$ and all the s.d. values has been multiplied by a factor of $10^{-5}$.\n \\end{tablenotes}\n \\label{tab:dropout}%\n \\end{table}%\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n \\scalebox{0.5}\n {\n\t\\includegraphics{Fig2.pdf}\n\n }\n\\caption{Validation and testing errors of the DNNs with dropout and with BN with regards to $50$ function evaluations and 200 epochs for each function evaluation. \nThe (dashed) lines are the average score over\nfive random initializations and \nthe shaded regions correspond to one standard deviation.\n\\iffalse\nThe (dashed) lines represent the\naverage over five random initializations.\n\\fi\n\\iffalse\nof five repetitions with different training and validation splits, and the shaded\nareas represent the standard deviation over those repetitions.\nThe results are the average testing score\nover five trials where the shaded regions correspond to the\nstandard deviation.\n\\fi}\n\\label{MSE_TPE_BN}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centering\n \\scalebox{0.5}\n {\n\t\\includegraphics{Fig3.png}\n }\n\\caption{Comparison of actual and predicted values over the out-of-sample period. The actual return is drawn by the thin solid black line.\nForecasted values from the DNN with dropout,\nin-sample mean, and\nout-of-sample mean \nare drawn by the solid green, blue and yellow lines, respectively.\n}\n\\label{MSE_forecast_pattern}\n\\end{figure}\n\n\\subsubsection{Effect of optimizer choice}\nTo further check the robustness of a dropout with respect to the dependency on the selected optimization algorithm, we repeated the experiments using a random search and simulated annealing. As shown in\nFig. \\ref{MSE_three_opt}, a comparable performance is shown for both the validation and test set, without an overfitting to the former.\nOur observations on different optimizers consistently suggest that a dropout helps improve the generalization. This indicates that the benefits of the HPO are general, without\ndepending on a specific optimizer, thereby demonstrating its robustness.\n\n\n\\begin{figure}[t]\n\\centering\n \\scalebox{0.55}\n {\n\t\\includegraphics{Fig4.pdf}\n }\n \n\\caption{Comparison of the simulated annealing (SA), TPE, and random search (RS) performances on the validation and test sets for the first 100 observations. \nThe (dashed) lines are the average score over\nfive random initializations and \nthe shaded regions correspond to one standard deviation.\n}\n\\label{MSE_three_opt}\n\\end{figure}\n\n\n\n\\subsubsection{Model stability over time}\nFigure \\ref{fig_SHAP} shows the importance of features arranged in decreasing order for the dropout and batch normalization models. They were calculated by summing the absolute values of the SHAP values. Table \\ref{SHAP_rank} shows the rank of the features over time from Exp. 4 to Exp. 1. The following observation was made based on the results.\n \\begin{itemize}\n \\item The feature importance is sensitive to the selected experimental periods for both models. This implies that the selection of a small number of features based on their importance can prevent a model generalization for unseen (new) data. \n \n\\item Overall, we observed that a DNN with a BN \nachieves a greater variability than a DNN with a dropout. In the experiments with a dropout, the five variables $\\{$MA112, MA212, MA39, MA29, MOM6M$\\}$ and\nthe six variables $\\{$MOM12M, VOL29, VOL212, VOL312, MOM3M, MOM1M $\\}$\nremain in the top half (from 1st to 8th) and bottom half across the experiments, respectively.\nBy contrast, in the experiments using a BN, only $\\{$MA19$\\}$ remains in the top half and $\\{$MA29, MA312$\\}$ remain in the bottom half. This indicates that a DNN with a dropout is more generalized against a time change and explains the outperformance of $R^{2}_{OS}$ in a more fundamental manner.\n \\end{itemize}\n\n \n\n\\begin{figure}[h]\n \\advance\\leftskip-1cm\n \\centering\n \\subfigure[DNN with dropout. From left, Exp. 1\u20134.]{%\n \\includegraphics[width=0.25\\linewidth]{Fig5.pdf}%\n \\includegraphics[width=0.25\\linewidth]{Fig6.pdf}%\n \\includegraphics[width=0.25\\linewidth]{Fig7.pdf}%\n \\includegraphics[width=0.25\\linewidth]{Fig8.pdf}%\n }\\\\\n \\subfigure[DNN with BN. From left, Exp. 1\u20134.]{%\n \\includegraphics[width=0.25\\linewidth]{Fig9.pdf}\n \\includegraphics[width=0.25\\linewidth]{Fig10.pdf}\n \\includegraphics[width=0.25\\linewidth]{Fig11.pdf}\n \\includegraphics[width=0.25\\linewidth]{Fig12.pdf}\n }\n \\caption{Mean absolute value of SHAP values for each features for Exp. 1--4.}\n \\label{fig_SHAP}\n\\end{figure}\n\n\n\n \\begin{table}[!htb]\n \\caption{\\label{SHAP_rank}\nFeature ranking results of DNNs with dropout (left) and with BN (right).}\n \\small\n \\begin{minipage}{.55\\textwidth}\n \\centering\n \\begin{tabular}{lccccc}\n \\toprule\n & \\multicolumn{1}{c}{Exp. 4} & \\multicolumn{1}{c}{Exp. 3} & \\multicolumn{1}{c}{Exp. 2} & \\multicolumn{1}{c}{Exp. 1} \\\\\n \\hline\n MA112 & 1 & 4 & 3 & 2 \\\\\n MA212 & 2 & 3 & 4 & 7 \\\\\n MA39 & 3 & 6 & 5 & 1 \\\\\n MA19 & 4 & 1 & 1 & 10 \\\\\n MA29 & 5 & 2 & 2 & 4 \\\\\n MOM6M & 6 & 5 & 6 & 5 \\\\\n VOL19 & 7 & 8 & 8 & 13 \\\\\n MOM9M & 8 & 9 & 9 & 9 \\\\\n MOM12M & 9 & 13 & 11 & 11 \\\\\n VOL29 & 10 & 10 & 10 & 16 \\\\\n VOL212& 11 & 12 & 14 & 17 \\\\\n VOL312 & 12 & 11 & 13 & 12 \\\\\n MA312 & 13 & 7 & 7 & 14 \\\\\n VOL112& 14 & 14 & 15 & 8 \\\\\n VOL39 & 15 & 16 & 16 & 3 \\\\\n MOM3M & 16 & 15 & 12 & 16 \\\\\n MOM1M & 17 & 17 & 17 & 15 \\\\\n \\bottomrule\n \\end{tabular}%\n \\end{minipage}\n \\begin{minipage}{.3\\textwidth}\n \\centering\n \\begin{tabular}{lcccc}\n \\toprule\n & \\multicolumn{1}{c}{Exp. 4} & \\multicolumn{1}{c}{Exp. 3} & \\multicolumn{1}{c}{Exp. 2} & \\multicolumn{1}{c}{Exp. 1} \\\\\n \\hline\n MA212 & 1 & 1 & 17 & 8 \\\\\n VOL19 & 2 & 9 & 8 & 4 \\\\\n MA112 & 3 & 15 & 14 & 17 \\\\\n MA19 & 4 & 3 & 5 & 6 \\\\\n VOL39 & 5 & 16 & 13 & 15 \\\\\n VOL312& 6 & 5 & 3 & 14 \\\\\n VOL29 & 7 & 11 & 7 & 13 \\\\\n MOM3M & 8 & 6 & 4 & 12 \\\\\n MA39 & 9 & 4 & 1 & 9 \\\\\n MOM9M & 10 & 12 & 6 & 2 \\\\\n MOM6M & 11 & 2 & 16 & 1 \\\\\n MA29 & 12 & 17 & 11 & 16 \\\\\n VOL112& 13 & 14 & 12 & 5 \\\\\n MOM12M& 14 & 8 & 9 & 7 \\\\\n MOM1M & 15 & 10 & 2 & 3 \\\\\n MA312 & 16 & 13 & 10 & 11 \\\\\n VOL212& 17 & 7 & 15 & 10 \\\\\n \\bottomrule\n \\end{tabular}%\n \\end{minipage}\n \\iffalse\n \\begin{tablenotes}[flushleft]\\footnotesize\n \\item[]\n Note: Superscripts $t$ and $b$ denote the features remaining\n in the top and bottom halves of the features during all experiments.\n \\end{tablenotes}\n \\fi\n \\end{table}\n\n\n\\subsection{Fundamentals}\n\\subsubsection{Predictability and model stability}\nTable \\ref{tab:fundamentals} shows the results produced through the\nsame procedure as used in the previous experiments applying fundamentals. \nThe following observations can be made regarding the results.\n\\begin{itemize}\n\\item For both models, fundamental data are prone to an overfitting to the in-sample data as shown in\nthe positive $R_{IS}^{2}$ and negative $R_{OS}^{2}$ values. \n\\item A DNN with a dropout outperforms a DNN with a BN in terms of better values of $R_{IS}^{2}$ and $R_{OS}^{2}$ except for only $R_{IS}^{2}$ in Exp. 1. \n\\end{itemize}\n\n\\begin{table}[htbp]\n\\small\n \\centering\n \\caption{Comparison of models based on average prediction performance ($\\pm$1 s.d. in parentheses) over 5 runnings with different random initial seeds for each experiment.}\n \\begin{tabular}{L{3.cm} C{2.cm} C{2.cm} C{2.cm} C{2.cm} C{2.cm} }\n \\toprule\n Regularizer & \\multicolumn{1}{c}{$\\textrm{MSE}_{\\textrm{train}}$ } & \\multicolumn{1}{c}{$\\textrm{MSE}_{\\textrm{val}}$ } & \\multicolumn{1}{c}{$\\textrm{MSE}_{\\textrm{test}}$ } &\n \\multicolumn{1}{c}{$R^{2}_{IS}$ } & \\multicolumn{1}{c}{$R^{2}_{OS}$ } \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 1} \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{0.129 \\\\($\\pm 0.580 $)}& \\makecell{0.198 \\\\ ($\\pm 0.329 $)} & \\makecell{ $\\bm{0.189}$ \\\\ ($\\pm\\bm{ 0.904 }$)}& \\makecell{$-0.179$ \\\\ ($\\pm$0.177)}& \\makecell{$\\bm{-0.341}$ \\\\($\\bm{\\pm 0.620}$)} \\\\\n DNN w. BN & \\makecell{ $\\bm{0.127}$\\\\($\\bm{\\pm 2.778}$)}& \\makecell{$\\bm{0.193}$\\\\($\\bm{\\pm 2.412 }$)} & \\makecell{0.205\\\\($\\pm 18.817$)} & \\makecell{$\\bm{2.700}$ \\\\($\\bm{\\pm 0.544$})} &\\makecell{$-10.462$ \\\\($\\pm$9.105)} \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 2} \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{ $\\bm{0.119}$ \\\\ ($\\bm{\\pm 1.797}$)} & \\makecell{ 0.202 \\\\ ($\\pm 0.687$)} &\\makecell{$2.334$ \\\\ ($\\pm 15.298$)}& \\makecell{$\\bm{3.841}$ \\\\($\\bm{\\pm 0.645}$)}& \\makecell{$\\bm{-13.794}$ \\\\ ($\\bm{\\pm 7.457)}$} \\\\\n DNN w. BN & \\makecell{0.156\\\\($\\pm 9.133$)}& \\makecell{ $\\bm{0.196}$\\\\($\\bm{ \\pm 0.823}$)} &\\makecell{$\\bm{ 0.598}$ \\\\($\\bm{\\pm 89.937$})} & \\makecell{$-5.098$\\\\ ($\\pm 2.558$)} &\\makecell{$ -191.78$\\\\ ($\\pm$43.838)} \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 3 } \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{$\\bm{0.122}$\\\\($\\bm{\\pm 2.632}$)} & \\makecell{2.122 \\\\($\\pm 1.185 $)} & \\makecell{$\\bm{0.170}$ \\\\($\\bm{\\pm 5.285}$)}&\\makecell{$\\bm{4.248}$ \\\\($\\bm{\\pm0.736}$)} & \\makecell{$\\bm{-13.670}$ \\\\($\\bm{\\pm 3.527}$)} \\\\\n DNN w. BN & \\makecell{$0.138$\\\\($\\pm 18.870$)}& \\makecell{$\\bm{0.206}$\\\\($\\bm{\\pm 0.715}$)} & \\makecell{0.271\\\\($\\pm 134.062$)} &\\makecell{$1.160$ \\\\($\\pm 5.526$)} & \\makecell{$-81.173$ \\\\($\\pm$89.469)} \\\\\n \\hline\n & \\multicolumn{5}{c}{Exp. 4} \\\\\n \\cline{2-6}\n DNN w. dropout & \\makecell{$\\bm{0.113}$ \\\\($\\bm{\\pm 1.315}$)}& \\makecell{0.194 \\\\($\\pm 1.385$)} & \\makecell{$\\bm{0.232}$ \\\\($\\bm{\\pm 8.753}$) }& \\makecell{$\\bm{3.443}$ \\\\($\\bm{\\pm 0.764}$)} & \\makecell{$\\bm{-6.058}$ \\\\($\\bm{\\pm 3.992}$)}\\\\\n DNN w. BN & \\makecell{$0.129$\\\\($\\pm 5.911 $)}& \\makecell{$\\bm{0.185}$\\\\($\\bm{\\pm 0.121}$)} & \\makecell{0.395\\\\($\\pm 49.916$)} & \\makecell{$1.003$ \\\\($\\pm 1.816$)} & \\makecell{$-80.260$ \\\\($\\pm$22.767)}\\\\\n \\bottomrule\n \\end{tabular}%\n\\begin{tablenotes}[flushleft]\\footnotesize\n \\item[]\n Note: All the $\\textrm{MSE}$ and $R^{2}$ values have been multiplied by a factor of $10^{-2}$ and all the s.d. values has been multiplied by a factor of $10^{-5}$.\n \\end{tablenotes}\n \\label{tab:fundamentals}%\n \\end{table}%\n \n \n \n \n\\section{Conclusion}\n\\label{sec:5}\nIn this study, we explored hyperparameter optimization techniques used in\nA stock return prediction by applying DNN-based predictors. The experiment was validated\nby considering different settings for the datasets, periods, and regularization.\nWe found that technical indicators are robust to an overfitting\nduring the HPO procedure, showing positive $R_{IS}$ and $R_{OS}$ values over different time periods, whereas the fundamental indicators are prone to an overfitting to the in-sample data. To summarize, dropout layers can efficiently decrease the risk of an overfitting and increase the model generalizability. \n\nThis system can be seen as a first step toward a better and\nmore fruitful integration of the recent developments in HPO techniques. Future efforts for improving\nthe current solution will be devoted to the design of a neural architecture for the fundamental data, which are robust to an overfitting. Fundamental data evidently reflect the fundamental values, which can\nserve as useful predictors or provide complementary information for a stock return prediction. \nWe expect the development to improve the prediction accuracy by\ncombining fundamental and technical indicators.\n\n\n\n\\bibliographystyle{unsrt}\n\\input{HyperparamterOptimizationForStock.bbl}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\\label{sec:intro}\n\\IEEEPARstart{V}{isual} sensor networks are used in a diverse set of applications such as surveillance \\cite{wang2006surveillance}, traffic monitoring and control \\cite{rachmadi2011trafficControl}, parking lot management \\cite{baroffio2015visual} and indoors patient monitoring \\cite{8361445}.\nRecently, integrating such sensor networks to traffic infrastructure has been suggested as promising means to support autonomous driving functionality in complex urban zones to enable cooperative perception \\cite{wang2018deployment,arnold2019cooperative}.\nIn such setting, the infrastructure-based sensors, which may include cameras and lidars, augment vehicles' on-board sensor data using emerging V2X technologies.\nThe usage of these networks is expected to grow further as high resolution sensors become more affordable and new generations of highly reliable wireless communication systems become widely deployed \\cite{5G}.\nWhen designing sensor networks, the choice of the number and pose of the sensors, \\textit{i.e.} their location and rotation angles, is critical in determining their coverage.\nThis directly impacts the performance of object detection, classification, and tracking applications that use the data from these sensor networks.\n\nThe problem of optimising sensor poses for a network of sensors has been explored in the literature.\nA major category of the existing studies formulate this problem as a discrete optimisation problem where a finite set of possible sensor poses is considered and the target objects' visibility is described by a set of binary variables \\cite{Chakrabarty2002,horster2006optimal,gonzales2009optimalIP,zhao2009optimal}.\nThe problem is then solved by using various forms of Integer Programming (IP) solvers or heuristic methods \\cite{zhao2013approximate} to either maximise the number of visible target objects (coverage) with a fixed number of sensors; or to minimise the number of sensors required to achieve a given coverage constraint.\nHowever, the majority of the applications that may use such sensor networks, \\textit{e.g.} object detection \\cite{arnold2019cooperative} and tracking \\cite{granstrom2017extended}, require a minimum level of visibility over the target objects which cannot be encoded by single binary variables.\nFor example, an object may have different degrees of visibility due to occlusions and due to its position \\textit{w.r.t.} the sensors, which causes ambiguity in the assignment of a binary visibility variable.\n\nAnother category of the existing studies consider the optimisation of continuous sensor pose variables using simulated annealing \\cite{1345252}, Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno (BFGS) \\cite{akbarzadeh2013probabilistic}, particle swarm \\cite{nguyen2015lineofsight}, evolutionary algorithms \\cite{saad2020realistic} and gradient-based optimisation \\cite{akbarzadeh2014efficient}.\nMost of the studies in this category focus on maximising the coverage (visible ground area) of extensive 3D environments described by digital elevation maps.\nHowever, such formulation does not consider the distribution of objects in the environment, and instead, assume an object would be visible if it is within a region covered by the sensors.\nAs a result, these studies fail to detect and prevent occlusions between objects since they do not explicitly model the visibility of the target objects.\nThis becomes a limiting factor when considering cluttered environments such as traffic junctions with a significant number of vehicles and pedestrians.\n\nThe visibility models that have been used in the literature usually consider simplifying assumptions which hinder the applicability of such methods in many practical settings.\nExamples of such simplifications include the use of a 2D visibility model that does not take into account the sensors' pitch and yaw angles \\cite{Chakrabarty2002,horster2006optimal,gonzales2009optimalIP} or the assumption of cameras focusing on a single target object without occlusions \\cite{ercan2006optimal}.\nIn the cases where occlusions are considered, \\textit{e.g.} \\cite{zhao2009optimal}, the visibility model only takes into account the centroid of objects to determine if the whole object is occluded.\nAs a result, partial occlusions, which are common in practice, are not considered.\nFurthermore, the use of Line-of-Sight (LoS) visibility models based on the Bresenham's algorithm \\cite{akbarzadeh2013probabilistic,saad2020realistic} and derived methods \\cite{nguyen2015lineofsight} is only applicable to environments represented by an elevation map.\n\nDue to the aforementioned limitations, the problem of determining the optimal poses for a network of sensors that avoid occlusions and guarantee a minimum degree of visibility for all target objects remains unsolved.\nTo this end, we propose an occlusion-aware visibility model based on a differentiable rendering framework and develop two novel approaches for object-centric sensor pose optimisation based on gradient-ascent and Integer Programming, respectively.\nDifferent from the existing approaches in the literature that aim to cover ground areas on elevation maps, we explicitly model the visibility of the target objects by considering a set of object configurations, defined as frames.\nIn this definition, each frame contains a number of target objects with specific sizes, positions and orientations within the environment.\nThe objective of the proposed method is to maximise the visibility of all target objects across the frames.\nThe distribution of frames is considered to be application specific and can be obtained by empirical evaluations or simulation.\nWe perform a comprehensive evaluation of the proposed methods in a challenging traffic junction environment and compare them with previous methods in the literature.\nThe results of this evaluation indicate that explicitly modelling the visibility of objects is critical to avoid occlusions in cluttered scenarios.\nFurthermore, the results show that both of the proposed methods outperform existing methods in the literature by a significant margin in terms of object visibility.\nIn summary, the contributions of this paper are:\n\\begin{itemize}\n\t\\item A realistic visibility model, created using a rendering process, that produces pixel-level visibility information and is capable of detecting occlusions between objects;\n\t\\item A novel gradient-based sensor pose optimisation method based on the aforementioned visibility model;\n\t\\item A novel IP sensor pose optimisation method that guarantees minimum object visibility based on aforementioned rendering process;\n\t\\item The performance comparison between both methods and existing works in the literature in a simulated traffic junction environment.\n\\end{itemize}\n\nThe rest of this paper is organised as follows.\nSection \\ref{sec:relatedworks} provides a review of related methods and highlights the distinguishing aspects of our work.\nSection \\ref{sec:problem} defines the system model and the formal underlying optimisation problem, including a novel sensor pose parametrisation.\nSections \\ref{sec:gradopt} and \\ref{sec:ipopt} describe the proposed gradient-based and Integer Programming solutions of the underlying optimisation problem, respectively.\nFinally, Section \\ref{sec:experiments} presents the evaluation results and Section \\ref{sec:conclusion} presents the concluding remarks.\n\n\\section{RELATED WORKS}\n\\label{sec:relatedworks}\nThis section provides a review of existing works related to sensor pose optimisation and differentiable rendering, and articulates how our work in this paper compares to these existing works in the literature.\n\n\\subsection{Sensor pose optimisation}\n\tThe problem of sensor pose optimisation has its historical origin in the field of computational geometry with the art-gallery problem \\cite{orourke1987art}, where the aim is to place a minimal number of sensors within a polygon environment in such a way that all points within the polygon are visible.\n\tAlthough further work extends the art-gallery problem to a 3D environment considering finite field-of-view and image quality metrics \\cite{fleishman2000automatic}, it still falls short of providing realistic sensor and environmental models.\n\tFurther efforts treat the pose optimisation problem as an extension of the maximum coverage problem, however use very simplistic sensor assumptions, such as radial sensor coverage in 2D environments \\cite{agarwal2009efficient}.\n\t\n\tOne category of methods consider sensor poses as continuous variables which are optimised according to some objective function.\t\t\n\tAkbarzadeh \\textit{et al.} \\cite{akbarzadeh2013probabilistic} propose a probabilistic visibility model using logistic functions conditioned on the distance and vertical\/horizontal angles between the sensor and a target point.\n\tThe authors then optimise the aggregated coverage over an environment described by a digital elevation map using simulated annealing and Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno (BFGS) optimisation.\n\tIn further work \\cite{akbarzadeh2014efficient}, the same authors propose a gradient-ascent optimisation to maximise the aggregated coverage using their previous visibility model.\n\tThis method requires obtaining the analytical forms of the derivatives of sensor parameters.\n\tIn contrast, we propose a gradient-ascent method that uses Automatic Differentiation (AD) \\cite{paszke2019pytorch} allowing efficient sensor pose optimisation without specifying the analytical forms of the derivatives.\n\t\n\tRecent work by Saad \\textit{et al.} \\cite{saad2020realistic} uses a visibility model similar to \\cite{akbarzadeh2013probabilistic} with a LoS formulation. \n\tThe authors introduce constraints over sensors' locations and detection requirements, which are application-specific, and optimise the sensor pose to achieve the detection requirements using a genetic algorithm.\n\tTemel \\textit{et al.} \\cite{temel2014} uses a LoS binary visibility model and a stochastic Cat Swarm Optimisation (CSO) to maximise the coverage of a set of sensors.\n\tThe aforementioned methods aim to maximise the coverage of extensive 3D environments represented by digital elevation maps and use LoS algorithms \\cite{nguyen2015lineofsight,akbarzadeh2013probabilistic} to detect occlusions in these elevation maps.\n\tHowever, digital elevation maps are not ideal to represent target objects due to their coarse spatial resolution, which may conceal the objects' shapes.\n\tIn this paper, we propose a novel visibility model, based on the depth buffer of a rendering framework \\cite{ravi2020pytorch3d}, which allows to accurately and efficiently detect any occlusions using arbitrarily shaped environments and objects.\n\tFurthermore, we explicitly model the visibility of target objects using a differentiable visibility score which is based on a realistic perspective camera model.\n\t\n\tGiven the difficulty of optimising the sensors' pose as continuous variables, another category of methods consider a discrete approach, where a subset of candidate sensor poses must be chosen to maximise the binary visibility of target points \\cite{zhao2013approximate,1345252,gonzalez2009optimal,yao2009can}.\n\tThis formulation allows to solve the problem using various forms of Integer Programming (IP) solvers \\cite{gonzalez2009optimal}, including Branch-and-Bound methods \\cite{schrijver1998theory}.\n\tIn some cases, solving the IP problem can be computationally infeasible, particularly when the set of candidate sensors is large, and thus, approximated methods, such as Simulated Annealing \\cite{zhao2013approximate,1345252} and Markov-Chain Monte Carlo (MCMC) sampling strategies \\cite{zhao2013approximate,yao2009can}, can be used.\n\tThe drawback of the aforementioned approximated methods is that they cannot guarantee the optimality of the solution found.\n\t\n\tThe methods in the IP category consider the visibility of a target object as a binary variable (\\textit{i.e.} visible or invisible), which cannot represent different levels of visibility and may result in sub-optimal sensor poses.\n\tConsider, for example, two sensor poses that can observe a given target object; one of the poses is closer to the object and provides more information than the other; yet, both poses obtain the same binary visibility result, \\textit{i.e.} the object is visible.\n\tIn contrast to methods in this category that assign a binary visibility for target objects, we propose a novel IP formulation that considers the number of points (or pixels) that each sensor cast over each object, obtained using a rendering framework.\n\tOur proposed IP formulation takes into account the effect of partial occlusions and guarantees a minimum visibility across all target objects.\n\n\\subsection{Differentiable Rendering}\n\tA general renderer is a process that generates images, or pixel values, given 3D scene parameters, which include objects' meshes and textures, camera and lighting parameters.\n\tIn contrast, differentiable renderers are a subclass capable of providing the derivative of pixel values with respect to any of the aforementioned scene parameters \\cite{loper2014opendr}.\n\tSuch formulation bridges the gap between 3D scene parameters and their 2D projections \\cite{kato2020differentiable}, allowing efficient gradient-based optimisation solutions to inverse-graphics problems such as 3D object reconstruction \\cite{kato2019learning}, object\/camera pose estimation \\cite{rhodin2015versatile} and adversarial examples generation \\cite{liu2018beyond}.\n\tHowever, the potential of differentiable renderers is yet to be explored in the context of sensor pose optimisation for visual sensor networks.\n\tIn this paper, we use PyTorch3D's \\cite{ravi2020pytorch3d} perspective camera models to create an end-to-end differentiable pipeline that can be optimised using gradient-ascent.\n\tThis pipeline allows to directly optimise the sensors' pose to maximise the visibility of multiple target objects, as described in Section \\ref{sec:gradopt}.\n\tFor a detailed review of differentiable renderers and applications the reader may refer to \\cite{kato2020differentiable}.\n\n\\section{PROBLEM FORMULATION}\n\\label{sec:problem}\n\tThis section firstly presents the formulation of the sensor pose optimisation problem upon which we base our gradient-based and Integer Programming (IP) methods in Sections \\ref{sec:gradopt} and \\ref{sec:ipopt}, respectively.\n\tNext, a novel sensor pose parametrisation is introduced to constrain the sensor poses to feasible regions which are pre-defined according to the environment where the sensors are deployed.\n\t\n\t\\begin{figure*}[htp]\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{smodel-visibility}\t\t\n\t\t\\caption{Illustration of the problem formulation for an exemplar driving environment with $N=3$ sensors and $M=5$ target objects. (a) Physical representation of target objects and sensor poses. (b) Objects and environment representation under the sensor pose problem formulation. (c) re-projected point cloud $P(S)$ and objects' visibility metric. Note that the visibility metric of a target object is obtained by counting the number of points of $P(S)$ on the surface the respective object, as defined in Equation \\ref{eq:vis}.}\n\t\t\\label{fig:sysmodel-visibility}\n\t\\end{figure*}\n\t\n\tThe sensor network, depicted in Figure \\ref{fig:sysmodel-visibility}a, consists of a set of fixed infrastructure sensors $S$ that collectively observe a set of target objects, denoted by $O$, in a driving environment.\n\tEach target object is represented using a three-dimensional cuboid encoded by $o=(x,y,z,w,h,l,\\theta) \\in O$, where $x,y,z$ correspond to the 3D centroid of the box, while $w,h,l$ represent the box size and $\\theta$ corresponds to the pitch angle (rotation around the vertical axis), as depicted in Figure \\ref{fig:sysmodel-visibility}b.\n\tThe visibility of object $o$ by the sensor set $S$, denoted by $\\vis(o,S)$, is defined as the number of pixels, or points, that the set of sensors $S$ project onto the object's surfaces.\n\tVisibility in this sense intuitively quantifies the information that sensors capture about each object and has shown to be correlated with the performance of perception tasks such as 3D object detection \\cite{arnold2019cooperative} and tracking \\cite{granstrom2017extended}.\n\tThis visibility metric is computed in two steps.\n\tFirst, the frame containing objects $O$ is rendered.\n\tThen, the depth-buffer from each sensor in $S$ is re-projected into 3D space, creating an aggregated point cloud $P(S)$, as described in Section \\ref{sec:gradopt:occlusion} and illustrated in Figure \\ref{fig:sysmodel-visibility}c.\n\tFinally, the visibility of each object $o \\in O$ is obtained by counting the number of points of $P(S)$ that lie on the surface of each respective object:\n\t\\begin{equation}\n\t\t\\label{eq:vis}\n\t\t\\vis(o, S) = \\sum_{\\bm{p} \\in P(S)} \n\t\t\\begin{cases}\n\t\t1,& \\text{if } \\bm{p} \\text{ on } o \\text{'s surface} \\\\\n\t\t0,& \\text{otherwise}.\n\t\t\\end{cases}\n\t\\end{equation}\n\tThis visibility metric provides pixel-level resolution which successfully captures the effects of total or partial occlusions caused by other target objects and by the environment.\n\tThe environment model, denoted by $E$, can also be modified according to the application requirements.\n\tFor example, it is possible to include static scene objects, such as buildings, lamp posts and trees, that may affect the visibility of target objects.\n\t\n\tThe formulation proposed so far considered a single, static configuration of target objects, denoted by $O$.\n\tHowever, driving environments are dynamic and typically contain moving vehicles and pedestrians.\n\tWe account for dynamic environments by considering a set of $L$ static frames.\n\tEach frame contains a number of target objects with specific sizes, positions and orientations within the environment.\n\tThe number of frames, denoted by $L$, must be chosen such that the distribution of objects over the collection of frames approximates the distribution of target objects' in the application environment.\n\tFor example, one can obtain a set of frames for driving environments using microscopic scale traffic simulation tools, such as SUMO \\cite{SUMO2018} or through the empirical observation of the driving environment.\n\t\n\tThe underlying optimisation problem is to find the optimum poses for $N$ sensors, denoted by $S=\\{s_1,\\dots,s_N\\}$ that maximise the visibility of target objects across the $L$ frames.\n\tFormally, the optimal set of sensor poses is defined as\n\t\\begin{equation}\n\t\t\\label{eq:optimalP}\n\t\t\\hat{S} \\triangleq \\argmax_S \\min_{o \\in \\mathbb{O}} \\vis(o, S),\n\t\\end{equation}\n\twhere $\\mathbb{O}$ is the set of objects across $L$ frames.\n\tIn practice, each frame is rendered independently so that objects from different frames do not occlude one another, but the optimisation is still performed across all frames.\n\t\n\tIt should be noted that one can alternatively maximise the mean visibility of the target objects, which can be formulated as $\\argmax_S \\frac{1}{M}\\sum_{o \\in \\mathbb{O}} \\vis(o,S)$.\n\tHowever, this may result in some of the objects having very low or zero visibility in the favour of others having un-necessarily large visibility.\n\tBut maximising the minimum visibility biases the optimisation algorithm towards sensor poses that guarantee the visibility of all target objects.\n\t\n\t\\subsection{Sensor Pose Parametrisation}\n\t\\label{sec:poseparam}\n\t\tGenerally speaking, the pose of a sensor in a 3D environment can be described by the canonical six degrees-of-freedom parametrisation $s=(x,y,z,\\varphi,\\theta,\\phi)$, where the $(x,y,z)$ represent the sensor position and $(\\varphi,\\theta,\\phi)$ its viewing angles.\n\t\tHowever, unconstrained optimisation under such parametrisation is seldom useful in practice as most environments have restrictions regarding sensors' location, \\textit{e.g.} sensors must be mounted close to a wall, on lamp posts, and clear from a road, etc.\n\t\tTo this end, we propose a continuous sensor pose parametrisation called virtual rail which imposes constraints over the sensors' location without adding any penalty term to the optimisation objective function or requiring any changes to the optimisation process, such as gradient projection.\n\t\t\n\t\tA virtual rail is defined by a line segment between two points in 3D space.\n\t\tThe sensors can be placed at any point within this line segment, as illustrated in Figure \\ref{fig:virtualRails}.\n\t\tThe viewing angles are described by the rotations along the X and Y axis, as we assume no rotation along the camera axis (Z). \n\t\tThe pose of a sensor on a virtual rail between points $\\bm{p_1,p_2} \\in \\mathbb{R}^3$ has its pose fully determined by the parameters $s=(t,\\alpha,\\beta)$ through the parametrisation\n\t\t\\begin{equation}\n\t\t\\label{eq:railparam}\n\t\t\t\\begin{aligned}\n\t\t\t\t(x,y,z) &= \\bm{p_1} + \\sigma(t)(\\bm{p_2}-\\bm{p_1}), \\\\\n\t\t\t\t\\varphi &= 2\\pi\\sigma(\\alpha), \\\\\n\t\t\t\t\\theta &= \\pi\\sigma(\\beta), \\\\\n\t\t\t\t\\phi &= 0, \n\t\t\t\\end{aligned}\n\t\t\\end{equation}\n\t\twhere\n\t\t\\begin{equation}\n\t\t\\label{eq:sigmoid}\n\t\t\t\\sigma(z) = \\frac{1}{1+e^{-z}},\n\t\t\\end{equation}\n\t\tis the sigmoid function.\n\t\tThis function enforces the bounds of position within the rail, \\textit{i.e.} $(x,y,z)$ on the line segment between $\\bm{p_1,p_2}$, and viewing angles $\\varphi \\in [0,2\\pi]$, $\\theta \\in [0,\\pi]$ for unbounded variables $t,\\alpha,\\beta \\in \\mathbb{R}$.\n\t\t\n\t\tThis parametrisation allows the use of unbounded gradient optimisation with guaranteed constraints over the sensors' poses.\n\t\tNote that the choice of the number and position of virtual rails are hyper-parameters defined to fit the needs of the application according to the complexity of the environment\/task.\n\n\\section{GRADIENT-BASED SENSOR POSE OPTIMISATION}\n\\label{sec:gradopt}\n\tThis section describes the proposed gradient-based sensor pose optimisation for multi-object visibility maximisation.\n\t\t\n\tThe objective function proposed in Equation \\ref{eq:optimalP} is not differentiable \\textit{w.r.t.} the sensor pose parameters due to the non-continuity introduced by the threshold operation in $\\vis(\\cdotp)$.\n\tThus, gradient-based solutions cannot be applied to solve this optimisation problem.\n\tWe, therefore, propose a processing pipeline featuring a differentiable objective function that approximates the objective function in Equation \\ref{eq:optimalP}.\n\tA crucial element of this approximation is the visibility score, a continuous variable in the interval $[0,1]$ that measures the visibility of a given 3D point \\textit{w.r.t.} a sensor.\n\tThe processing pipeline considers the continuous visibility score of multiple points over each target object, which ensures the objects' visibility and implicitly approximates the original problem in Section \\ref{sec:problem}.\n\tIt shall be noted that the visibility score is different from the visibility metric (Equation \\ref{eq:vis}) in two ways: 1) the former is differentiable while the latter is not; 2) the former indicates the degree of visibility of a single point on a target object while the latter is the number of points on the surface of a target object.\n\tThe proposed processing pipeline for the computation and optimisation of the objective function is depicted in Figure \\ref{fig:diagramGD}.\n\t\n\tThe processing pipeline consists of five stages.\n\t\\begin{enumerate}\n\t\t\\item a set of target points, denoted by $T \\in \\mathbb{R}^{MF\\times 3}$, is created by sampling $F$ points from each of the $M$ target objects.\n\t\tThe points are randomly distributed along the objects' surfaces proportionally to each surface area.\n\t\t\n\t\t\\item the points $T$ are projected onto the image plane of each sensor and a visibility score is computed for each target point according to their position \\textit{w.r.t.} the visible frustum of the respective sensor, as described in Section \\ref{sec:gradopt:visibility}.\n\t\t\n\t\t\\item an occlusion-aware visibility model, described in Section \\ref{sec:gradopt:occlusion}, is used to update the visibility score created in the previous stage.\n\t\tThis is required since some projected points will be in the visible frustum of a given sensor but occluding objects prevent direct line-of-sight between the point and the sensor.\n\t\t\n\t\t\\item the objective function is computed as the mean visibility score of all points $T$ on target objects, as described in Section \\ref{sec:gradopt:objective}. \n\t\t\n\t\t\\item gradient-ascent is used to maximise the objective computed in the previous step, as described in Section \\ref{sec:gradopt:optim}.\n\t\\end{enumerate}\n\n\tThe proposed processing pipeline can work for any continuous sensor pose parametrisation.\n\tIn this paper, we consider the parametrisation proposed in Section \\ref{sec:poseparam}, which constrains the sensor position to a line segment and allows for unconstrained gradient-based optimisation.\n\t\n\t\\begin{figure*}[htp]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{diagramGD}\t\t\n\t\\caption{Processing pipeline of the proposed Gradient-based sensor pose optimisation method. (a) an exemplar frame with two objects and a set $S$ of $N$ sensors, including an environmental model with an occluding block (in green). (b) the optimisation pipeline.}\n\t\\label{fig:diagramGD}\n\t\\end{figure*}\n\t\t\n\t\\subsection{Visibility Model}\n\t\\label{sec:gradopt:visibility}\n\t\tIn this section we propose a realistic visibility model based on the perspective camera model provided by PyTorch3D \\cite{ravi2020pytorch3d}.\n\t\tBuilt on top of PyTorch, this camera model provides differentiable transformations from the global coordinate system to the camera image plane which is fundamental for a fully differentiable pipeline.\n\t\tThe cameras' extrinsic matrix is determined by the pose of the sensors, specified by the set of parameters $S$, being optimised.\n\t\tIt shall be noted that all cameras intrinsic properties are identical: 90-degree horizontal field-of-view, $D_\\text{near}=1$m, $D_\\text{far}=100$m near and far clipping planes, respectively, and resolution of $W=200$ x $H=200$ pixels.\n\t\tThe resolution is kept relatively small in order to reduce the computational complexity of the rendering process, described in Section \\ref{sec:gradopt:occlusion}.\n\t\tIncreasing the image resolution directly increases the visibility of the target objects as there will be a higher number of pixels\/points per object.\n\t\tIn practice, the sensor poses resulting from the optimisation process can be used for cameras with higher resolution, as long as they have the same aspect ratio and field-of-view.\n\t\t\n\t\tThe projection of a point $p = [x \\; y \\; z]^T \\in \\mathbb{R}^3$ in the global coordinate system into the image plane of sensor $s$ is given by\n\t\t\\begin{equation}\n\t\t\\label{eq:imagePlaneProjection}\n\t\t\\begin{bmatrix}\n\t\t\tu_sd_s \\\\\n\t\t\tv_sd_s \\\\\n\t\t\td_s\n\t\t\\end{bmatrix} =\n\t\tM_i M_e(s) \n\t\t\\begin{bmatrix}\n\t\tx \\\\\n\t\ty \\\\\n\t\tz\n\t\t\\end{bmatrix},\n\t\t\\end{equation}\n\t\twhere $[ud \\; vd \\; d]^T$ are homogeneous coordinates that can be divided by $d$ to obtain the canonical form $[u \\; v \\; 1]^T$.\n\t\tHere, $u,v$ are the image plane coordinates in pixels, $d$ is the depth of the point in the view frustum and $M_i, M_e$ are the intrinsic and extrinsic camera matrices of sensor $s$, respectively.\n\t\tThe point $p$ is within the visible frustum if and only if $W \\geq u \\geq 0$, $H \\geq v \\geq 0$ and $D_\\text{far} \\geq d \\geq D_\\text{near}$ where $W$ and $H$ are the image width and height in pixels.\n\t\t$D_\\text{near},D_\\text{far}$ are the camera near and far clipping planes in meters, respectively.\n\t\t\n\t\tThe visibility of a given point from the perspective of a sensor $s$ is determined by verifying that the image plane projection of this point, given by Equation \\ref{eq:imagePlaneProjection}, satisfies the bounds described in the previous paragraph.\n\t\tIf the bounds are satisfied, the point is considered visible, otherwise it is not.\n\t\tSince the threshold operations used to identify the visibility of a point are not differentiable, we opt to use the sigmoid function (Equation \\ref{eq:sigmoid}) as a differentiable approximation of the binary visibility.\n\t\tThis continuous visibility score can be interpreted as a probabilistic visibility measure \\cite{akbarzadeh2013probabilistic} of a point, ranging from 0 (completely invisible) to 1 (completely visible).\n\t\tThis is formulated by a \\textit{window} function as follows:\n\t\t\\begin{equation}\n\t\tw(z,\\gamma,z_0,z_1) = \\sigma(\\gamma(z-z_0))-\\sigma(\\gamma(z-z_1)),\n\t\t\\end{equation}\n\t\twhere $\\gamma \\in \\mathbb{R}$ controls the rate of transition on the limits of the interval $[z_0,z_1]$, as illustrated in Figure \\ref{fig:windowf}.\n\t\tAs $\\gamma$ increases the window function tends to a binary threshold operation.\n\t\tHowever, this reduces the intervals with non-zero gradients, and consequently inhibits parameters updates through gradient optimisation.\n\t\tEmpirical tests revealed that $\\gamma=1$ was the best out of the three tested values (0.1, 1, 10) for this hyper-parameter.\n\t\t\n\t\tThe visibility score of a point $p$ with image plane projection $[u_sd_s \\ v_sd_s \\ d_s]^T$ is given by\t\t\n\t\t\\begin{equation}\n\t\t\\label{eq:visScore}\n\t\t\t\\Psi(p,s) = w(u_s,\\gamma,0,W) \\cdot w(v_s,\\gamma,0,H) \\cdot w(d_s,\\gamma,D_\\text{near},D_\\text{far}).\n\t\t\\end{equation}\n\t\t\n\t\t\\begin{figure}[htp]\n\t\t\t\\centering\t\n\t\t\t\\includegraphics[width=\\linewidth]{windowfunction}\n\t\t\t\\caption{Window function $w(z,\\gamma,z_0,z_1)$ plotted for $z_0=0,z_1=200$ and varying values of $\\gamma$.}\n\t\t\t\\label{fig:windowf}\n\t\t\\end{figure}\t\t\n\t\t\n\t\tThis visibility model does not take into account occlusions caused by other objects or the environment since a point being within the visible frustum of a sensor is a required but not sufficient condition to guarantee direct line-of-sight visibility from the sensor to the point.\n\t\tWe account for occlusion by proposing an occlusion aware visibility model in Section \\ref{sec:gradopt:occlusion}.\t\t\t\n\t\t\n\t\\subsection{Occlusion Awareness}\n\t\\label{sec:gradopt:occlusion}\n\t\tWe verify line-of-sight visibility using the depth buffer generated by PyTorch3D's \\cite{ravi2020pytorch3d} rasteriser. \n\t\tThis rasteriser transforms the meshes representing the environment and the target objects into a raster image with a corresponding depth buffer.\n\t\tWhen an object is projected to the image plane, the orthogonal distance between the object and the sensor is stored in the corresponding pixel position of the depth buffer.\n\t\tIf another object is projected to the same pixel, the depth buffer keeps the smallest depth distance among the two.\n\t\tThis solves the hidden surface problem in computer graphics, where some objects overlap over the sensor's field-of-view and the closest objects occlude\/hide other objects in the background.\n\t\tWe use the same approach in our processing pipeline to determine if a given sensor has line-of-sight visibility of a point in 3D space.\n\t\t\n\t\tGiven a target point $p \\in T$ and a sensor $s$, the point is considered to be occluded from the point of view of sensor $s$ if\n\t\t\\begin{equation}\n\t\t\\label{eq:occlusionCriteria}\n\t\t|d_s - Z_s(u_s,v_s)| > \\kappa,\n\t\t\\end{equation}\n\t\twhere $[u_sd_s \\ v_sd_s \\ d_s]^T$ is the projection of $p$ on the image plane of $s$ according to Equation \\ref{eq:imagePlaneProjection}.\n\t\tHere, $Z_s(u_s,v_s)$ is the depth buffer of sensor $s$ at the pixel position $(u_s,v_s)$ and $\\kappa$ is a threshold for the maximum disparity between the projection depth value and the depth buffer.\n\t\tIn our experiments, we consider $\\kappa=0.5$m, which allows to accurately detect occlusions.\n\t\tFigure \\ref{fig:occlusion-aware} illustrates this occlusion-aware visibility model for a visible and an occluded point.\n\t\tIn this figure, the depth of a point projected on the image plane matches the depth buffer measurement at the corresponding pixel if the point is visible; if the point is occluded the depth buffer value will be smaller since there is another object closer to the sensor.\n\t\t\n\t\tThis occlusion-aware visibility model leads to an enhanced version of the visibility score of a point $p$ observed by sensor $s$, given by:\n\t\t\\begin{equation}\n\t\t\\label{eq:visScoreOcc}\n\t\t\t\\Psi(p,s) = \\left\\{\n\t\t\t\\begin{aligned}\n\t\t\t& w(u_s,\\gamma,0,W) \\cdot \\\\ \n\t\t\t& \\quad w(v_s,\\gamma,0,H) \\cdot \\\\\n\t\t\t& \\quad \\quad w(d_s,\\gamma,D_\\text{near},D_\\text{far}), && \\text{if } |d_s - Z_s(u_s,v_s)| \\leq \\kappa \\\\\n\t\t\t& 0, && \\text{otherwise.}\n\t\t\t\\end{aligned}\n\t\t\t\\right.\n\t\t\\end{equation}\n\t\tIn the case where $p$ is out of the visible frustum of sensor $s$, the visibility score is given by Equation \\ref{eq:visScore}.\n\t\tNote that if the point is occluded, there is no gradient signal to change the pose of the sensor in which the point is occluded.\n\t\tYet, the occluded point can be targeted by other sensors in the network.\n\t\t\n\t\tThe depth buffer from each sensor is re-projected into 3D space, using the inverse of Equation \\ref{eq:imagePlaneProjection}, to create a 3D point cloud representing all points observed by the respective sensor.\n\t\tEffectively, the depth buffers from each sensor $s \\in S$ are re-projected and aggregated into a fused point cloud $P(S)$, shown in Figure \\ref{fig:simpleMesh}.\n\t\tThis fused point cloud is used to compute the visibility metric $\\vis(o,S)$, used by the IP method and during the system performance evaluation.\n\t\t\n\t\t\\begin{figure}[htp]\n\t\t\t\\centering\t\n\t\t\t\\includegraphics[width=\\linewidth]{diagramOcc}\n\t\t\t\\caption{Illustration of the occlusion-aware visibility model: a point is considered to be visible by sensor $s$ if it lies within the visible frustum of $s$ and the Z component of the projection in the image plane closely matches the Z component obtained from the depth buffer (red point on $o_1$). If the disparity between these distances is above a threshold ($\\kappa=0.5$m) the point is considered occluded (blue point on $o_2$).}\n\t\t\t\\label{fig:occlusion-aware}\n\t\t\\end{figure}\t\t\n\n\t\\subsection{Objective Function}\n\t\\label{sec:gradopt:objective}\n\t\tA target point $p$ may be observed by multiple sensors, thus, the overall visibility of a point by a set of sensors $S$ is computed as\n\t\t\\begin{equation}\n\t\t\\label{eq:visScoreAll}\n\t\t\\Psi(p,S) = 1-\\prod_{s \\in S} (1-\\Psi(p,s)).\n\t\t\\end{equation}\n\t\tAccording to Equation \\ref{eq:visScoreAll}, a point's overall visibility score is forced to be 1 if at least one sensor has a visibility score of one.\n\t\tConversely, sensors that cannot observe a point (zero visibility score) do not affect the overall visibility score.\n\t\tFurthermore, when multiple sensors observe the same point, the combined visibility score improves.\n\n\t\tThe proposed sensor pose optimisation model in this paper aims to maximise the mean visibility score across all objects $O$ for a given set of sensors $S$.\n\t\tHence, the following objective function is maximised in our gradient-based formulation:\n\t\t\\begin{equation}\n\t\t\\label{eq:objective-func}\n\t\t\\mathcal{L} = \\frac{1}{|T|}\\sum_{p \\in T} \\Psi(p,S),\n\t\t\\end{equation}\n\t\twhere $T$ is a set of randomly sampled target points from target objects' surfaces, and $\\Psi(p,S)$ is the overall visibility score of point $p$ across all sensors $S$ according to Equation \\ref{eq:visScoreAll} considering the enhanced occlusion-aware visibility model, described by Equation \\ref{eq:visScoreOcc}.\t\n\t\t\n\t\\subsection{Optimisation}\n\t\\label{sec:gradopt:optim}\t\n\t\tWe adopt the Adam optimiser \\cite{kingma2014adam} to allow per-parameter learning rate and adaptive gradient-scaling, which has been shown to stabilise and shorten the optimisation process.\n\t\tThe optimiser uses a global learning rate of 0.1, and is executed for 20 iterations over the whole collection of frames.\n\t\tThese optimisation hyper-parameters were determined empirically through experiments.\n\t\tAlgorithm \\ref{alg:gd} describes the optimisation process for a set of frames and Table \\ref{tab:gdalgodesc} specifies the input variables used in the algorithm.\n\t\t\n\t\tThe objective function in Equation \\ref{eq:objective-func} is maximised \\textit{w.r.t.} the continuous sensor pose parameters $(t,\\alpha,\\beta)$ described in Section \\ref{sec:poseparam}.\n\t\tThese parameters specify the pose of a sensor within a virtual rail.\n\t\tIn an environment containing multiple virtual-rails, there must be an assignment between each sensor and the virtual-rail it belongs to.\n\t\tThis assignment is represented by a discrete variable that maps each sensor to one of the virtual-rails and is also subject to optimisation.\n\t\tHowever, since it is a discrete variable, it cannot be part of the gradient-based optimisation process.\n\t\tWe overcome this problem by performing multiple runs of the optimisation process, each with a random virtual-rail assignment, and reporting the best results across all runs in terms of the objective function.\n\t\t\t\t\n\t\tThe sensor poses are initialised using a uniform distribution on the interval $[-2,2]$ over the parameter $t$, which controls the sensor position $(x,y,z)$ along the virtual-rail according to Equation \\ref{eq:railparam}.\n\t\tThe limits of the uniform distribution are chosen such that the sensors initial position within the rail can be anywhere from $10\\%$ to $90\\%$ of the length of the rail.\n\t\tThe viewing angles can be randomly initialised in the same fashion.\n\t\tHowever, there may be some prior information of the environment that can guide this decision.\n\t\tFor example, in a traffic junction objects are likely to traverse the central area of the junction, thus, sensors could benefit by focusing towards the junction centre.\n\t\tAlthough this step is not strictly required, it introduces prior information into the problem which reduces the amount of time required to achieve satisfactory results in the optimisation process.\n\t\t\n\t\t\\begin{algorithm}\n\t\t\\caption{Gradient-Ascent Sensor Pose Optimisation}\n\t\t\\label{alg:gd}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\REQUIRE $N,O_1,O_2,O_3,\\dots,O_L,F,E,\\text{virtualRails},\\text{epochs}$\n\t\t\t\\ENSURE $\\hat{S}$, minVisibility\n\t\t\t\\\\ \\textit{Initialisation} :\n\t\t\t\\STATE $S \\gets \\emptyset$\n\t\t\t\\FOR{$i \\gets 1$ to $N$}\n\t\t\t\\STATE $p_1,p_2 \\gets $ random virtual rail from \\text{virtualRails}\n\t\t\t\\STATE Draw sample $t$ from $\\text{Uniform}(-2,2)$\n\t\t\t\\STATE Set $\\alpha,\\beta$ such that sensor focus on the centre of the junction \\COMMENT{Alternatively, sample them from the uniform distribution.}\n\t\t\t\\STATE Set $s=f(p_1,p_2,t,\\alpha,\\beta)$ \\COMMENT{$f$ is the sensor pose parametrisation given by Equation \\ref{eq:railparam}}\n\t\t\t\\STATE $S \\gets S \\cup s$\n\t\t\t\\ENDFOR\n\t\t\t\\\\ \\textit{Optimisation loop}\n\t\t\t\\FOR{$e \\gets 1$ to epochs}\n\t\t\t\\STATE $\\mathcal{L} \\gets 0$\n\t\t\t\\FOR{$O \\in \\{O_1,\\dots,O_L\\}$}\n\t\t\t\\STATE $T \\gets$ sample $F$ points from each target objects $o \\in O$ surfaces \n\t\t\t\\STATE $T' \\gets$ image plane projection of $p \\in T$ for each sensor $s \\in S$ according to Equation \\ref{eq:imagePlaneProjection}\n\t\t\t\\STATE $Z,P \\gets$ depth-buffer and reconstructed point-cloud from rasteriser as a function of $O,S,E$\n\t\t\t\\STATE $\\Psi \\gets $ visibility score for each $p \\in T'$ according to Equation \\ref{eq:visScoreOcc}\n\t\t\t\\STATE $\\Psi_S \\gets $ overall visibility score over all sensors according to Equation \\ref{eq:visScoreAll}\n\t\t\t\\STATE $\\mathcal{L} \\gets \\mathcal{L} + \\text{mean}(\\Psi_S)$\n\t\t\t\\ENDFOR\n\t\t\t\\STATE minVisibilityMetric $\\gets \\min_o \\vis(o, S) \\forall o \\in O_1 \\cup \\dots \\cup O_L$ \\COMMENT{Computes the visibility metric using the reconstructed point-cloud $P$}\n\t\t\t\\IF{minVisibilityMetric improved since last epoch}\n\t\t\t\\STATE $\\hat{S} \\gets S$\n\t\t\t\\ENDIF\n\t\t\t\\STATE Compute $\\frac{\\partial \\mathcal{L}}{\\partial S}$ using automatic differentiation\n\t\t\t\\STATE Update $S$ based on gradient-ascent update rule\n\t\t\t\\ENDFOR\n\t\t\t\\RETURN $\\hat{S}$, minVisibility\n\t\t\\end{algorithmic} \n\t\\end{algorithm}\n\n\t\\begin{table}[]\n\t\t\\caption{Description of variables in Algorithm \\ref{alg:gd}}\n\t\t\\label{tab:gdalgodesc}\n\t\t\\resizebox{\\linewidth}{!}{%\n\t\t\\begin{tabular}{@{}lll@{}}\n\t\t\t\\toprule\n\t\t\t\\textbf{Variable} & \\textbf{Description} & \\textbf{Value} \\\\ \\midrule\n\t\t\tN & Number of sensors & 1-6 \\\\\n\t\t\t$O_1,\\dots,O_L$ & Sets of objects for each of the L frames & \\\\\n\t\t\tL & Number of frames in the dataset & 1000 \\\\\n\t\t\tF & Number of target points sampled per object & 400 \\\\\n\t\t\tE & Environmental model & \\\\\n\t\t\tvirtualRails & The set of virtual rails described by two end-points in $\\mathbb{R}^3$ & \\\\\n\t\t\tepochs & Number of optimisation iterations & 20 \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t\t}\n\t\\end{table}\n\n\t\t\n\\section{INTEGER PROGRAMMING-BASED SENSOR POSE OPTIMISATION}\n\\label{sec:ipopt}\nInteger Programming (IP) is an effective approach for solving optimisation problems where some or all of the variables are integers and may be subject to other constraints \\cite{schrijver1998theory}.\nApplied to sensor pose optimisation, this formulation assumes that the optimal set of sensors are chosen from a finite set of sensor poses, called candidate poses.\nThe problem is a combinatorial search to find the optimal subset of candidate poses that maximise an objective function.\nThis objective function typically models the visibility of an area or objects.\nAdditional constraints, such as the maximum number of sensors in the optimal set can be added to the problem formulation.\nThis section describes how IP can be applied to solve the sensor pose optimisation problem formulated in Section \\ref{sec:problem}.\nThe objective is to find the subset of candidate sensor poses that maximises the minimum visibility metric of target objects.\nWe firstly introduce a method for the discretisation of the sensor pose parameter space into a finite set of candidate poses.\nWe then cast the base optimisation problem in Eq. \\ref{eq:optimalP} into an IP optimisation problem and present three approaches to solve it: a heuristic off-the-shelf solver and two approximate methods based on sampling strategies.\n\n\t\\subsection{Discretising Pose Parameters}\n\t\tTo apply Integer Programming to the sensor placement problem we need to discretise the sensor pose parameter space into a finite set of candidate sensor poses.\n\t\tWe use the concept of virtual rails, described in Section \\ref{sec:poseparam}, to create the set of candidate sensor poses by dividing each virtual rail into 10 equally spaced sensor positions.\n\t\tThe horizontal viewing angles at each position is also divided into 10 feasible angles, between 0 and 360 degrees.\n\t\tThe vertical viewing angles at each position is divided into 3 feasible angles.\n\t\tTo this end, the set of candidate sensor poses for a given virtual rail is $S' = \\{(t, \\varphi, \\theta) : \\sigma(t) \\in \\{0.1,0.2,\\dots,1\\}, \\varphi \\in \\{36,72,\\dots,360\\}, \\theta \\in \\{18,36,54\\} \\}$.\n\t\tFor simplicity, in the rest of this paper we assume that $S'$ represents the union of candidate poses from all virtual rails and the number of candidate poses is given by $|S'|=N'$.\n\t\tFigure \\ref{fig:virtualRails} illustrates the set of candidate poses $S'$ for a T-junction scenario.\n\t\t\n\t\\subsection{IP Objective}\t\t\n\t\tThe general sensor pose optimisation problem can be formulated as the following IP problem\n\t\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\max_{b_1,\\dots,b_{N'}} & f(b_1,\\dots,b_{N'},o_1,\\dots,o_M) \\\\\n\t\t\t\\textrm{s.t.} \\quad & \\sum_{i=1}^{N'} b_i \\leq N, \n\t\t\\end{aligned}\n\t\t\\end{equation}\n\t\twhere $b_i$ is a binary variable indicating if the $i$-th sensor in the candidate set, denoted by $s_i \\in S'$, is part of the optimal set.\n\t\tIn other words, the sensor $s_i$ is part of the optimal set if $b_i$ is 1 and the optimal set of sensors is given by \n\t\t$\\hat{S} = \\{s_i \\in S' : b_i = 1 \\quad \\forall i \\in \\{1,\\dots,N'\\} \\}$.\n\t\tThe constraint guarantees that the maximum number of chosen sensors do not exceed $N$.\n\t\tThe objective function $f(\\cdot)$ represents the targets' visibility, which depends on the choice of sensors $b_1,\\dots,b_{N'}$ and the targets $o_1,\\dots,o_M$.\n\t\tPrevious works \\cite{gonzales2009optimalIP,zhao2013approximate} define $f(\\cdot)$ as the sum of binary visibilities of environment points.\n\t\tThis is a poor estimate of target objects' visibility since there are varying degrees of visibility which cannot be encoded as a binary variable.\n\t\tTo address this problem, we propose a novel IP formulation that takes into account the visibility metric of a target object $o$ observed by a sensor $s$, $\\vis(o, \\{s\\})$, defined in Equation \\ref{eq:vis}.\t\t\n\t\tThe motivation is to to find the sensor set that maximise the minimum visibility metric among target objects.\n\t\tHence, the equivalent IP problem is described by\n\t\t\\begin{equation}\n\t\t\\label{eq:ipoptim}\n\t\t\\begin{aligned}\n\t\t\t\\max_{z,b_1,\\dots,b_{N'}} \\quad & z \\\\\n\t\t\t\\textrm{s.t.} \\quad & \\sum_{i=1}^{N'} b_i \\vis(o, \\{s_i\\}) \\geq z \\quad \\forall o \\in O, \\\\\n\t\t\t & \\sum_{i=1}^{N'} b_i \\leq N,\n\t\t\\end{aligned}\n\t\t\\end{equation}\n\t\twhere $z \\in \\mathbb{Z}_{\\geq 0}$ is the minimum visibility metric among target objects.\n\t\tThe first constraint guarantees that $z$ is the minimum visibility metric among all objects.\n\t\tNote that the effect of multiple sensors observing a given object is cumulative \\textit{w.r.t.} the visibility metric, \\textit{i.e.} $\\vis(o, \\{s_1,s_2\\})=\\vis(o, \\{s_1\\})+\\vis(o, \\{s_2\\})$.\n\t\t\n\t\tThe formulation proposed so far considers a single frame, denoted by $O$, containing the target objects.\n\t\tThis is extended to $L$ frames by rendering each frame individually, including the objects and the environmental model, for all candidate sensors.\n\t\tThe visibility of an object $o$, as observed by sensor $s_i$, denoted by $\\vis(o, \\{s_i\\})$, is obtained by counting the number of points in the re-projected point cloud generated by sensor $s_i$ that are on the surface of the object $o$, as described in Section \\ref{sec:problem} and illustrated in Figure \\ref{fig:diagramIP}.\n\t\tIn practice, the visibility of all objects are computed frame by frame, for each candidate sensor, prior to the optimisation and stored in a visibility matrix $V$.\n\t\tThis allows to solve the IP problem in Eq. \\ref{eq:ipoptim} for any number of sensors without recomputing the objects' visibilities.\n\t\t\n\t\t\\begin{figure*}[htp]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{diagramIP}\t\t\n\t\t\t\\caption{Illustration of the process of computing the visibility metric of object $o_i \\in O$ by each candidate sensor $s_i \\in S'$. The rendered point cloud naturally handles any occlusion caused by the environment model $E$ and other target objects in the frame. The visibility of a given object is obtained by counting all points (represented by the blue dots) from the respective sensor that are on the surface of the respective object. The output is a visibility matrix $V$ that depicts how many points each candidate sensor cast on each object in the frame, \\textit{i.e.} the object's visibility. This process is repeated for all frames and the matrices computed for each frame are concatenated horizontally.}\n\t\t\t\\label{fig:diagramIP}\n\t\t\\end{figure*}\n\t\n\t\t\\begin{figure}[htp]\n\t\t\t\\centering\t\n\t\t\t\\includegraphics[width=\\linewidth]{virtualRails}\n\t\t\t\\caption{Candidate set $S'$ over 5 virtual rails (red line segments) in a T-junction environment. Each yellow wireframe represent a sensor's viewing pose. To ease visualisation, only 10 candidates sensors are represented for each virtual rail, but the entire set consider 10 rotations along the Y axis and 3 rotations along the X axis, resulting in a total of 300 candidates per rail, or 1500 candidates overall.}\n\t\t\t\\label{fig:virtualRails}\n\t\t\\end{figure}\n\t\n\t\\subsection{Heuristic Solution}\n\t\tIP problems are NP-complete \\cite{schrijver1998theory}, thus, finding the solution using exhaustive search is computationally expensive or even unfeasible when the search space is large.\n\t\tParticularly, the size of the search space of the IP problem in Eq. \\ref{eq:ipoptim} is $\\binom{N'}{N}$.\n\t\tFor example, for a candidate set with $N'=1500$ poses and a given number of sensors $N$, \\textit{e.g.} 6, the size of the search space is $\\binom{1500}{6} \\approx 3^{17}$. \n\t\tFor this reason, there are multiple algorithms that attempt to solve the problem using heuristic methods such as cutting plane and branch-and-cut methods \\cite{schrijver1998theory}.\n\t\t\n\t\tIn this paper, we use the \\textit{Coin-or Branch and Cut (CBC)} open-source IP solver \\cite{cbcsolver} and the \\textit{python-mip} wrapper \\cite{python-mip} to solve the problem.\n\t\tThis solver uses Linear Programming (LP) relaxation for continuous variables and applies branching and cutting plane methods where the integrality constraint does not hold.\n\t\tThe solver cannot always guarantee the optimality of the solution, specially when exhaustive search is infeasible.\n\t\tThus, the problem in Equation \\ref{eq:ipoptim} is solved using the default optimisation settings until the optimal solution is found or the time since an improvement in the objective function exceeds a limit.\n\t\n\t\\subsection{Approximate Solutions}\n\t\tApproximate solutions to the IP problem are often used for the camera placement problem when exact solutions cannot be obtained in feasible time \\cite{zhao2013approximate}.\n\t\tAs described in the previous section, exhaustive search is unfeasible for the IP problem in Eq. \\ref{eq:ipoptim} due to the size of the search space.\n\t\tFor this reason, we implement two approximate methods: Na\\\"ive sampling and Markov Chain Monte Carlo (MCMC) sampling.\n\t\t\n\t\tThe Na\\\"ive sampling method assumes that all sensors in the candidate set $S'$ are equally likely to be part of the optimal set. \n\t\tThis method explores the search space by uniformly sampling $N$ sensors from the candidate set $S'$ without replacement.\n\t\tThe algorithm, described in Algorithm \\ref{alg:naiveIP}, runs until time since the last improvement in the objective function exceeds a limit.\n\t\t\n\t\tThe MCMC method uses the Metropolis-Hastings sampling algorithm \\cite{zhao2013approximate} to select sensors that are likely to maximise the objective function.\n\t\tAlgorithm \\ref{alg:mcmcIP} describes the full and detailed execution steps of the proposed sampling scheme.\n\t\tThe process starts with an initial sample of $N$ random sensors from $S'$, denoted by $S_0$.\n\t\tAt each subsequent iteration, a new sample set is computed as follows.\n\t\tAt iteration $i$, a random and uniformly selected element of $S_{i-1}$ is exchanged with a random and uniformly selected element of $S'$, generating an intermediate set $S_{i}^*$. \n\t\tThe ratio $r=\\frac{f(S_{i}^*)}{f(S_{i-1})}$, is computed, where $f(S) = \\min_{o \\in O} \\vis(o, S)$.\n\t\tThe solution set at iteration $i$ is then set according to\n\t\t\\begin{equation}\n\t\t\tS{i} = \n\t\t\t\\begin{cases}\n\t\t\tS_{i-1}, & \\text{if } u \\leq r \\\\\n\t\t\tS_{i}^*,& \\text{otherwise},\n\t\t\t\\end{cases}\n\t\t\\end{equation}\n\t\twhere $u$ is a sample from the uniform distribution $U[0,1]$.\n\t\tThe algorithm is executed until the time since the last improvement in the objective function exceeds a limit.\n\t\t\n\t\t\\begin{algorithm}\n\t\t\t\\caption{Na\\\"ive Sampling Approximate IP Solution}\n\t\t\t\\label{alg:naiveIP}\n\t\t\t\\begin{algorithmic}[1]\n\t\t\t\t\\REQUIRE $S',N,O,\\text{maxTime}$\n\t\t\t\t\\ENSURE $\\hat{S}$\n\t\t\t\t\\\\ \\textit{Initialisation} :\n\t\t\t\t\\STATE z\\_best = 0\n\t\t\t\t\\STATE $\\hat{S} = \\emptyset$\n\t\t\t\t\\\\ \\textit{Sampling loop}\n\t\t\t\t\\WHILE {timeSinceLastImprovement $\\leq$ maxTime}\n\t\t\t\t\t\\STATE $S$ = $N$ samples from $S'$ without replacement;\n\t\t\t\t\t\\STATE z = $\\min_{o \\in O} \\vis(o,S)$\n\t\t\t\t\t\\IF {(z $\\geq$ z\\_best)}\n\t\t\t\t\t\t\\STATE z\\_best = z\n\t\t\t\t\t\t\\STATE $\\hat{S} = S$\n\t\t\t\t\t\t\\STATE reset timeSinceLastImprovement\n\t\t\t\t\t\\ENDIF\n\t\t\t\t\\ENDWHILE\n\t\t\t\t\\RETURN $\\hat{S}$ \n\t\t\t\\end{algorithmic} \n\t\t\\end{algorithm}\n\t\n\t\t\\begin{algorithm}\n\t\t\\caption{MCMC Metropolis-Hastings Sampling Approximate IP Solution}\n\t\t\\label{alg:mcmcIP}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\REQUIRE $S',N,O,\\text{maxTime}$\n\t\t\t\\ENSURE $\\hat{S}$\n\t\t\t\\\\ \\textit{Initialisation} :\n\t\t\t\\STATE z\\_best = 0\n\t\t\t\\STATE $\\hat{S} = \\emptyset$\n\t\t\t\\STATE $S$ = $N$ samples from $S'$\n\t\t\t\\\\ \\textit{MCMC loop}\n\t\t\t\\WHILE {timeSinceLastImprovement $\\leq$ maxTime}\n\t\t\t\\STATE $S^* = S$\n\t\t\t\\STATE replace one random element of $S^*$ with a random element from $S'\\setminus S^*$ \n\t\t\t\\STATE $z = \\min_{o \\in O} \\vis(o,S)$\n\t\t\t\\STATE $z^* = \\min_{o \\in O} \\vis(o,S^*)$\n\t\t\t\\STATE $r = \\frac{z^*}{z + \\epsilon}$ \\COMMENT{$\\epsilon$ is a small value to avoid division by zero}\n\t\t\t\\STATE $u \\gets \\text{sample from Uniform}(0,1)$\n\t\t\t\\IF {($u \\leq r$)}\n\t\t\t\t\\STATE $S = S^*$\n\t\t\t\t\\STATE $z = z^*$\n\t\t\t\t\\IF {(z $\\geq$ z\\_best)}\n\t\t\t\t\\STATE z\\_best = z\n\t\t\t\t\\STATE $\\hat{S} = S$\n\t\t\t\t\\STATE reset timeSinceLastImprovement\t\t\t\t\t\t\n\t\t\t\t\\ENDIF\t\t\t\t\n\t\t\t\\ENDIF\n\t\t\t\\ENDWHILE\n\t\t\t\\RETURN $\\hat{S}$ \n\t\t\\end{algorithmic} \n\t\\end{algorithm}\n\n\\section{EVALUATION}\n\\label{sec:experiments}\n\tThis section describes the evaluation of the proposed sensor pose optimisation methods.\n\tFirst, the evaluation metrics are defined in Section \\ref{sec:experiments:metrics}.\n\tNext, the experiment setup is described, including details of the simulation scenario in Section \\ref{sec:experiments:setup}.\n\tThen, a comparative evaluation between the methods proposed in this paper is presented in Section \\ref{sec:experiments:cproposed}.\n\tFinally, a comparison of the proposed methods with existing works in the literature and a comparison of different visibility models are reported in Section \\ref{sec:experiments:cprev} and \\ref{sec:experiments:vismodels}, respectively.\n\t\n\t\\subsection{Evaluation Metrics}\n\t\\label{sec:experiments:metrics}\n\t\tExisting studies in the literature assess sensor pose optimisation methods using the number of visible targets \\cite{zhao2013approximate} or the mean ground area coverage \\cite{akbarzadeh2014efficient,akbarzadeh2013probabilistic,saad2020realistic}, where coverage is defined as the probability that an area is visible to a sensor.\n\t\tHowever, such metrics are unsuitable for object-centric visibility for two reasons.\n\t\tFirst, adopting a binary visibility for an object is a coarse measure, since an object can be visible to different degrees due to its distance from the sensors, due to occlusions and limited sensor field-of-view.\n\t\tSecondly, the coverage of a ground area does not guarantee that an object placed within this area will be visible, as occlusions may limit the object's visibility.\n\t\tFor the aforementioned reasons, in our analysis we evaluate a set of sensor poses $S$ based on the minimum visibility metric across all objects, denoted by $\\min_{o \\in \\mathbb{O}} \\vis(o,S)$.\n\t\tRecalling from Equation \\ref{eq:vis}, the visibility metric is defined as number of pixels that the set of sensors $S$ observe on the surface of a given target object.\n\t\tIn addition to the minimum visibility metric, we compute the Empirical Cumulative Distribution Function (ECDF) of the visibility metric for all objects across frames, which provides broader insight into the visibility patterns across objects.\n\t\t\n\t\\subsection{Evaluation Setup}\n\t\\label{sec:experiments:setup}\n\t\tThe performance evaluation of the proposed sensor pose optimisation methods is carried out by simulating traffic on a T-junction environment.\n\t\tThis is motivated by the challenging conditions faced in such environments.\n\t\tFor safety reasons, it is critical to guarantee that all vehicles, \\textit{i.e.} target objects, are visible to the sensors.\n\t\tYet, vehicles are subject to occlusions from other vehicles and buildings.\n\t\t\n\t\tThe driving environment is simulated using the CARLA open-source simulator \\cite{Dosovitskiy17carla}.\n\t\tA typical urban T-junction is chosen from one of the existing maps in the simulation tool.\n\t\tIt has an area of 80 x 40 meters with several tall buildings and road-side objects, such as trees, bus shelters and lamp-posts.\n\t\tWithin this environment, a dataset consisting of 1000 frames is generated.\n\t\tEach frame is a snapshot of the environment at a particular time, containing the number of vehicles and their representation.\n\t\tThe objects' representation, as described in Section \\ref{sec:problem}, defines their position, size and orientation in the environment.\n\t\t\n\t\tThe environment model, available through CARLA open-source assets, contains a high number of complex meshes that slow down the rendering process.\n\t\tFor this reason, we opt to create a simplified version of the environment.\n\t\tTo this end, we create cuboid meshes for the buildings near the junction, and represent vehicles as cuboids using the same dimensions of the original objects' bounding boxes.\n\t\tThis approximation significantly speed up the rendering process without detrimental impact to the measurement of objects' visibility metric.\n\t\tFigures \\ref{fig:fullMesh} and \\ref{fig:simpleMesh} illustrate the original and simplified environment models, respectively.\t\t\t\n\t\t\n\t\tSensors placed in such driving environments must be placed by the road-side and clear from the road.\n\t\tThis constraint is addressed by creating five virtual rails alongside the junction, each aligned with the curb over a segment of the junction, as illustrated in Figure \\ref{fig:virtualRails}.\n\t\tThe parametrisation of the rails is application dependent and may need adjustment.\n\t\tIn this application, the virtual rail configuration allows sensors to be positioned on existing road-side infrastructure, such as traffic lights.\n\t\tThe virtual rails are positioned on a height of 5.2m above the ground, following the standards of public light infrastructure in the UK \\cite{durhamLighting}.\n\t\tHowever, the height of each sensor could also be included in the optimisation process.\n\t\t\n\t\t\\begin{figure*}[htp]\n\t\t\t\\centering\n\t\t\t\n\t\t\t\\subfloat[\\label{fig:fullMesh}]{\\includegraphics[width=0.46\\textwidth]{tjuncPCLFullMesh}}\\hfill\n\t\t\t\\subfloat[\\label{fig:simpleMesh}]{\\includegraphics[width=0.48\\textwidth]{tjuncPCLSimpleMesh}}\t\t\n\t\t\t\n\t\t\t\\caption{T-junction environment models described by re-projected point clouds created using \\protect\\subref{fig:fullMesh} the original environment model representation from CARLA and \\protect\\subref{fig:simpleMesh} the simplified environment model proposed in this paper.}\n\t\t\t\\label{fig:meshEnvironments}\n\t\t\\end{figure*}\t\n\n\t\\subsection{Comparative evaluation of the proposed methods}\n\t\\label{sec:experiments:cproposed}\n\t\tTable \\ref{tab:results} shows the results comparing the gradient-based and IP optimisation methods in terms of the minimum object visibility metric and duration of the optimisation process for a varying number of sensors, denoted by $N$.\n\t\tThe runtime performance of the IP methods does not include the time required to compute the visibility matrix, i.e. rendering 1000 frames for each of the 1500 candidate sensors poses, which took 28 hours.\n\t\tHowever, this process is only done once and the resulting visibility matrix is used by all IP methods for any number of sensors.\n\t\tNone of the methods could find a pose for a single sensor that can observe all objects, thus, the results are reported for $N > 1$.\n\t\tThe gradient-based method results are reported for the best out of 10 runs for each number of sensors.\n\t\tEach run has a random sensor-rail assignment and random sensor position initialisation, as described in Section \\ref{sec:gradopt:optim}.\n\t\tThe best minimum visibility metric observed in each run is reported in Figure \\ref{fig:runsGD}.\n\t\t\n\t\tThe evaluation shows that the IP method consistently outperform the gradient-based method, which we believe is explained by two factors.\n\t\tFirst, the loss function being maximised in the gradient method is non-convex and presents local-maxima, which may result in sub-optimal results.\n\t\tSecondly, the gradient-based method does not optimise the sensor-rail assignment.\n\t\tWe circumvent the latter by performing multiple optimisation runs for the gradient-based method, each with a random sample of sensor-rail assignment.\n\t\tHowever, the variance of the visibility metric obtained across runs, observed in Figure \\ref{fig:runsGD}, suggests that ten samples may not be enough to explore the sensor-rail assignment space.\n\t\tIncluding more samples of sensor-rail assignments requires more optimisation runs, which becomes time costly.\n\t\tOn the other hand, the IP method handles the sensor-rail assignment naturally as the candidate sensor pose set includes sensor poses in all virtual rails.\n\t\t\n\t\tFigure \\ref{fig:poseECDF} shows the resulting sensor poses found by each method for different numbers of sensors and the associated ECDF of the visibility metric of the target objects for each set of sensor poses.\n\t\tThe visibility metric distributions obtained with IP solutions show similar visibility patterns, except for $N=6$ where the heuristic IP approach has a significant advantage over its counterparts. \n\t\tThe distribution of visibilities for gradient-based solutions is significantly skewed towards smaller visibilities if compared to IP solutions.\n\t\tParticularly, for $N=5$, approximately 80\\% of the objects have less than 1000 points when observed by the gradient-based solution, while only 40\\% of objects have less than 1000 points for the IP solutions.\n\t\t\n\t\t\\begin{figure}[htp]\n\t\t\t\\centering\t\n\t\t\t\\includegraphics[width=\\linewidth]{runsGD}\n\t\t\t\\caption{Best Minimum visibility for each out of the ten runs of the Gradient-ascent optimisation method for varying number of sensors $N$.}\n\t\t\t\\label{fig:runsGD}\n\t\t\\end{figure}\n\t\t\n\t\t\\begin{table}[]\n\t\t\t\\caption{Comparison of optimisation results for different number of sensors across methods}\n\t\t\t\\label{tab:results}\n\t\t\t\\resizebox{\\linewidth}{!}{%\n\t\t\t\t\\begin{tabular}{@{}lllll@{}}\n\t\t\t\t\t\\toprule\n\t\t\t\t\t\\textbf{Method} & \\textbf{N} & \\textbf{Min Visibility} & \\textbf{Runtime till Best (min)} & \\textbf{Overall Runtime (min)} \\\\ \\midrule\n\t\t\t\t\t\\multirow{5}{*}{Gradient-based*}& 2 & 17 & 25 & 27 \\\\\n\t\t\t\t\t& 3 & 55 & 32 & 32 \\\\\n\t\t\t\t\t& 4 & 102 & 39 & 39 \\\\\n\t\t\t\t\t& 5 & 123 & 30 & 46 \\\\\n\t\t\t\t\t& 6 & 178 & 16 & 53 \\\\ \\midrule\n\t\t\t\t\t\\multirow{5}{*}{IP CBC} & 2 & \\textbf{26} & 325 & 565 \\\\\n\t\t\t\t\t& 3 & 67 & 416 & 656 \\\\\n\t\t\t\t\t& 4 & 213 & 286 & 526 \\\\\n\t\t\t\t\t& 5 & \\textbf{447} & 175 & 415 \\\\\n\t\t\t\t\t& 6 & \\textbf{590} & 354 & 594 \\\\ \\midrule\n\t\t\t\t\t\\multirow{5}{*}{IP Na\\\"ive} & 2 & 26 & 0.2 & 240 \\\\\n\t\t\t\t\t& 3 & \\textbf{114} & 163 & 406 \\\\\n\t\t\t\t\t& 4 & 201 & 44 & 284 \\\\\n\t\t\t\t\t& 5 & 354 & 179 & 419 \\\\\n\t\t\t\t\t& 6 & 405 & 97 & 337 \\\\ \\midrule\n\t\t\t\t\t\\multirow{5}{*}{IP MCMC} & 2 & 26 & 0.2 & 240 \\\\\n\t\t\t\t\t& 3 & 107 & 190 & 430 \\\\\n\t\t\t\t\t& 4 & \\textbf{220} & 423 & 663 \\\\\n\t\t\t\t\t& 5 & 321 & 87 & 327 \\\\\n\t\t\t\t\t& 6 & 411 & 20 & 260 \\\\ \\bottomrule\n\t\t\t\t\\end{tabular}\n\t\t\t}\n\t\t\t\\footnotesize *Best results out of 10 runs with random initialisation. Overall Runtime reported for the single best run.\n\t\t\\end{table}\n\t\n\t\t\\begin{figure*}[htp]\n\t\t\\centering\n\t\t\t\\subfloat[$\\hat{S}$ for $N=2$ ]{\\includegraphics[width=0.45\\textwidth]{pose-2}}\\hfill\n\t\t\t\\subfloat[ECDF of $\\vis(o,\\hat{S})$]{\\includegraphics[width=0.45\\textwidth]{ecdf-2}}\n\t\t\t\n\t\t\t\\subfloat[$\\hat{S}$ for $N=3$ ]{\\includegraphics[width=0.45\\textwidth]{pose-3}}\\hfill\n\t\t\t\\subfloat[ECDF of $\\vis(o,\\hat{S})$]{\\includegraphics[width=0.45\\textwidth]{ecdf-3}}\n\t\t\t\n\t\t\t\\subfloat[$\\hat{S}$ for $N=4$ ]{\\includegraphics[width=0.45\\textwidth]{pose-4}}\\hfill\n\t\t\t\\subfloat[ECDF of $\\vis(o,\\hat{S})$]{\\includegraphics[width=0.45\\textwidth]{ecdf-4}}\n\t\t\t\n\t\t\t\\subfloat[$\\hat{S}$ for $N=5$ ]{\\includegraphics[width=0.45\\textwidth]{pose-5}}\\hfill\n\t\t\t\\subfloat[ECDF of $v\\vis(o,\\hat{S})$]{\\includegraphics[width=0.45\\textwidth]{ecdf-5}}\n\t\t\t\n\t\t\t\\subfloat[$\\hat{S}$ for $N=6$ ]{\\includegraphics[width=0.45\\textwidth]{pose-6}}\\hfill\n\t\t\t\\subfloat[ECDF of $\\vis(o,\\hat{S})$]{\\includegraphics[width=0.45\\textwidth]{ecdf-6}}\n\t\t\t\n\t\t\t\\caption{Resulting sensor poses and visibility distributions. The left column represents the perspective view of the junction showing the pose of the resulting set $\\hat{S}$. The right column shows the ECDF of object's visibility for the optimal set of sensors found by different methods. The colour of the sensors in the perspective view follows the legend of the ECDF plot. Each row describes the results for a given number of sensors, denoted by $N$. Note that some of the camera poses are the same across methods and may appear as a single one, particularly for $N=2$. }\n\t\t\\label{fig:poseECDF}\n\t\t\\end{figure*}\n\t\n\t\\subsection{Comparison with existing works}\n\t\\label{sec:experiments:cprev}\n\t\tWe compare our sensor pose optimisation methods with two existing works.\n\t\tAkbarzadeh \\textit{et al.} \\cite{akbarzadeh2014efficient} maximise the coverage of a ground area using gradient-ascent and Zhao \\textit{et al.} \\cite{zhao2013approximate} uses Integer Programming to maximise the number of target points visible in an environment.\n\t\tWe reproduce these methods in the simulated T-junction environment considering the coverage of uniformly distributed points over the T-junction ground area.\n\t\tNote that these methods do not explicitly model the visibility of the target objects, instead they maximise the coverage of the ground area.\n\t\tThe evaluation considers the ground surface coverage, \\textit{i.e.} the ratio of ground points that are visible to the sensors, and the minimum visibility of objects placed over this area.\n\t\tThe results are reported in Table \\ref{tab:results-other-methods}.\n\t\tThese results show that the previous methods are successful in maximising the coverage of the T-junction's ground area.\n\t\tHowever, this does not guarantee the visibility of target objects since occlusions between objects are a key factor in determining the visibility of objects in cluttered environments.\n\t\tThis underpins the importance of explicitly considering the visibility of target objects in contrast to the coverage of ground areas.\n\t\t\n\t\t\\begin{table}[]\n\t\t\t\\caption{Performance comparison with existing works in terms of ground area coverage and minimum object visibility for different number of sensors}\n\t\t\t\\label{tab:results-other-methods}\n\t\t\t\\resizebox{\\linewidth}{!}{%\n\t\t\t\t\\begin{tabular}{@{}lllll@{}}\n\t\t\t\t\t\\toprule\n\t\t\t\t\t\\textbf{Method} & \\textbf{N} & \\textbf{Ground Area Coverage (\\%)} & \\textbf{Min Visibility} & \\textbf{Overall Runtime (min)} \\\\ \\midrule\n\t\t\t\t\t\\multirow{5}{*}{Akbarzadeh et al. \\cite{akbarzadeh2014efficient}} & 2 & 51 & 0 & 2 \\\\\n\t\t\t\t\t& 3 & 69 & 0 & 2 \\\\\n\t\t\t\t\t& 4 & 73 & 0 & 2 \\\\\n\t\t\t\t\t& 5 & 78 & 1 & 2 \\\\\n\t\t\t\t\t& 6 & 91 & 41 & 2 \\\\ \\midrule\n\t\t\t\t\t\\multirow{5}{*}{Zhao et al. \\cite{zhao2013approximate}} & 2 & 79 & 0 & 3 \\\\\n\t\t\t\t\t& 3 & 88 & 0 & 3 \\\\\n\t\t\t\t\t& 4 & 91 & 0 & 3 \\\\\n\t\t\t\t\t& 5 & 92 & 0 & 3 \\\\\n\t\t\t\t\t& 6 & 92 & 0 & 3 \\\\ \\bottomrule\n\t\t\t\t\\end{tabular}\n\t\t\t}\n\t\t\\end{table}\n\t\n\t\\subsection{Comparison between visibility models}\n\t\\label{sec:experiments:vismodels}\n\t\tWe perform a study comparing the performance of the gradient-based method considering three different visibility models: our visibility model with and without occlusion awareness (Eq. \\ref{eq:visScore} and \\ref{eq:visScoreOcc}, respectively) and the visibility model from Akbarzadeh \\textit{et al.} \\cite{akbarzadeh2014efficient}.\n\t\tIn this study, we consider $N=6$ sensors and explicitly model the visibility of the target objects using the three aforementioned visibility models.\n\t\tTable \\ref{tab:ablation} reports the results of this study.\n\t\tOur occlusion-aware visibility model achieves the best performance as it can realistically determine which points are visible and accordingly change the sensors' pose to account for potential occlusions.\n\t\tThis is highlighted in Figure \\ref{fig:visibilityAblation}, depicting the point clouds of target points, where the colour of each point encodes its visibility score, ranging from blue (invisible) to red (visible).\n\t\tNote that our occlusion-aware visibility model correctly identify non-visible parts of the objects due to occlusion (blue) or only partially visible (yellow).\n\t\tIn contrast, the two other visibility models fail to identify areas of occlusion, mistakenly determining that all points are visible (red).\n\t\tAs a result, the optimisation process cannot improve the visibility of such areas.\n\t\t\t\t\n\t\t\\begin{table}[]\n\t\t\t\\caption{Comparison between visibility models used in the gradient-based method}\n\t\t\t\\label{tab:ablation}\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{@{}ll@{}}\n\t\t\t\t\\toprule\n\t\t\t\t\\textbf{Visibility Model} & \\textbf{Min Visibility} \\\\ \\midrule\n\t\t\t\tOurs (without Occlusion-Aware model, Eq. \\ref{eq:visScore}) & 82 \\\\\n\t\t\t\tOurs (with Occlusion-Aware model, Eq. \\ref{eq:visScoreOcc}) & \\textbf{178} \\\\\n\t\t\t\tAkbarzadeh et al. \\cite{akbarzadeh2014efficient} & 115 \\\\ \\bottomrule\n\t\t\t\\end{tabular}\n\t\t\\end{table}\n\t\n\t\t\\begin{figure*}[htp]\n\t\t\t\\centering\n\t\t\t\n\t\t\t\\subfloat[\\label{fig:visibilityAblation:our}]{\\includegraphics[width=0.3\\textwidth]{visOurs}}\\hfill\n\t\t\t\\subfloat[\\label{fig:visibilityAblation:ourNoOcc}]{\\includegraphics[width=0.3\\textwidth]{visOursNoOcc}}\\hfill\n\t\t\t\\subfloat[\\label{fig:visibilityAblation:vahab}]{\\includegraphics[width=0.3\\textwidth]{visVahab}}\t\t\n\t\t\t\n\t\t\t\\caption{Point clouds showing the target points over all objects in all the frames for three visibility models. \\protect\\subref{fig:visibilityAblation:our} our visibility model including occlusion awareness, \\protect\\subref{fig:visibilityAblation:ourNoOcc} our visibility model without occlusion awareness and \\protect\\subref{fig:visibilityAblation:vahab} Akbarzadeh et al. \\cite{akbarzadeh2014efficient}. The point colors indicate the visibility score $\\Psi$, ranging from blue ($\\Psi = 0$, invisible) to red ($\\Psi = 1$, visible). The white vertical pointer marks the position of the object with least visibility. Sensors poses are indicated by XYZ axis within coloured spheres.}\n\t\t\t\\label{fig:visibilityAblation}\n\t\t\\end{figure*}\n\t\t\n\n\\section{CONCLUSION}\n\\label{sec:conclusion}\n\tSensor pose optimisation methods such as the ones proposed in this paper can guide the cost-effective deployment of visual sensor networks in traffic infrastructure to maximise the visibility of objects of interest.\n\tSuch sensor network infrastructures can be used to increase the safety and efficiency of traffic monitoring systems and aid the automation of driving in complex road segments, particularly, in areas where accidents are more likely to happen.\n\t\n\tOur systematic study, in addition to the proposition of novel approaches for sensor pose optimisation, reveals a number of key insights that can be useful for researchers and system designers.\n\tFirstly, explicit modelling of the visibility of the target objects is critical when optimising the poses of sensors, particularly in cluttered environments where sensors are prone to severe occlusions.\n\tSecondly, rendering-based visibility models can realistically determine the visibility of target objects at the pixel level and, thus, improve the pose optimisation process.\n\tThirdly, the IP optimisation method seems to outperform the gradient-ascent method in terms of minimum object visibility, at the cost of increased computational time.\n\tThe sensor pose optimisation methods proposed in this paper can guide the deployment of sensor networks in traffic infrastructure to maximise the visibility of objects of interest.\n\t\n\tAs a follow-on study, we believe that it can be interesting to investigate how to reduce the search space of the IP formulation, for example, by using heuristics to remove candidate sensor poses that have limited observability.\n\tAdditionally, strategies to incorporate the discrete rail assignment variables directly into the gradient optimisation should be investigated, \\textit{e.g.} considering differentiable discrete distribution sampling via Gumbel-Softmax \\cite{45822}.\n\tOther global-optimisation strategies could be used to circumvent the impact of local-minima in the gradient-ascent method.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe electric field produced as a consequence of the separation of electrical charges inside clouds is the origin of lightning in the troposphere. However, as originally proposed by \\cite{Wilson1925\/PPhSocLon} and later detected by \\cite{Franz1990\/Sci}, atmospheric electrical discharges can also take place in upper regions of the atmosphere. These types of electrical phenonema are known as Transient Luminous Events (TLEs).\n\nThe first detected TLE was a sprite \\citep{Franz1990\/Sci}, an upper atmospheric discharge formed by a complex structure of thousands of streamers and a diffuse non-streamer zone that can extend from 40~km up to 85~km of altitude \\citep{Sentman1994\/VIDEO, Lyons1994\/GeoRL}. Other types of TLEs, known as halos and elves, can also be produced in the upper atmosphere at altitudes greater than 70~km and 80~km. Both halos and sprites can have a duration of several milliseconds \\citep{Lyons2000\/ETAGU, Barrington-Leigh2001\/JGR,Wescott2001\/JGR\/1, Bering2002\/AdSpR,Moudry2003\/JASTP,Bering2004\/AdSpR,Bering2004\/GeoRL,Frey2007\/GeoRL,Sentman2008\/JGRD\/1, Gordillo-Vazquez2008\/JPhD, Luque2011\/NatGe}, while elves have a duration of less than 1~ms \\citep{Inan1991\/GRL, Inan1997\/GeoRL, Taranenko1993\/GRL, Moudry2003\/JASTP, Kuo2007\/JGRA, Marshall2010\/JGRA\/2, gordillo2016upper, van_der_Velde2016\/GRL, perezmodeling, perezspectroscopic}.\n\nIn 1995, \\cite{Wescott1995\/GeoRL, Wescott1996\/GeoRL\/1} discovered the existence of upward propagating conical-shaped jets in the ranges of altitudes between 15~km and 25~km. Later in year 2002, \\cite{Pasko2002\/Natur} reported another type of upward propagating jets that reached the ionosphere. These upward propagating discharges were later called Blue Jets (BJ) and Gigantic Jets (GJ), two types of TLEs that can propagate in the range of altitudes between 15~km and 40~km in the case of Blue Jets, and between 15~km and 90~km in the case of GJs. The upper altitude reached by Blue Jets (about 40~km) corresponds to the level where propagation time equals the relaxation timescale of about 0.2~s \\citep{Sukhorukov1996\/GeoRL\/1}. Blue Jets and Gigantic Jets are different events triggered right above the cloud layer \\citep{Pasko2002\/Natur, van_der_Velde2010\/JGR, Pasko2012\/SSR, Chanrion2017\/GRL}. According to some evidences \\citep{Krehbiel2008\/NatGe, Riousset2010\/JGRA, Pasko2012\/SSR}, Gigantic Jets could be initiated as a cloud lightning discharge propagating upward, while Blue Jets are triggered as a consequence of the electrical breakdown produced between the storm upper charge layer and the screening charge attracted to the cloud top \\citep{Krehbiel2008\/NatGe, Riousset2010\/JGRA, Pasko2012\/SSR}. \n\n\nSince their discovery in 1989, TLEs have been observed from planes, balloons, ground-based detectors and space-based instrumentation. Several campaigns have recorded the spectra of sprites \\citep{Hampton1996\/GeoRL, Kanmae2007\/GeoRL, Passas2016\/APO, Gordillo-Vazquez2018\/JGR}. Some space-based missions, such as the Space Shuttle \\citep{Boeck1992\/GRL}, the Imager of Sprites and Upper Atmospheric Lightning (ISUAL) of the National Space Organization of Taiwan (NSPO) \\citep{Chern2003\/JASTP, Chen2008\/JGRA, hsu2017\/TAOC} and the Global Lightning and sprIte MeasurementS (GLIMS) of the Japan Aerospace Exploration Agency (JAXA) \\citep{sato2015overview,Adachi2016\/JASTP} have reported TLE observations from space. Last April 2, 2018 the Atmosphere-Space Interactions Monitor (ASIM) \\citep{Neubert2006\/ILWS} of the European Space Agency (ESA) was successfully launched. ASIM is equipped with the Modular Multi-Imaging Assembly (MMIA), devoted to the study of TLEs from space. In addition, the Tool for the Analysis of RAdiations from lightNIng and Sprites (TARANIS) \\citep{Blanc2007\/AdSpR} of the Centre National d'\\'Etudes Spatiales (CNES), will also be devoted to the observation of these events after its expected launch in 2019 or 2020.\n\nSeveral authors have investigated the local chemical impact of TLEs \\citep{Gordillo-Vazquez2008\/JPhD, Sentman2008\/JGRD\/1, Gordillo-Vazquez2009\/PSST, Gordillo-Vazquez2010\/JGRA, Pasko2012\/SSR, Parra-Rojas\/JGR, Parra-Rojas\/JGR2015, Winkler2015\/JASTP, PerezInvernon2016\/GRL, Hoder2016\/IOP, perezmodeling}. Recently, \\cite{Winkler2015\/JASTP} developed a local chemical model of Blue Jets obtaining an important local enhancement of NO$_x$, N$_2$O and O. The global chemical influence of TLEs has been investigated by previous studies. According to previous local models of halos and elves \\citep{perezmodeling}, their global chemical impact would be negligible. \\cite{Arnone2014\/JGR} estimated the global production of NO$_x$ by sprites using the Whole Atmosphere Community Climate Model version 4 (WACCM4). \\cite{Arnone2014\/JGR} found that a perturbation in the tropical concentration of nitrogen oxide by sprites could lie between 0.015~ppbv and 0.15~ppbv. These quantities correspond to a perturbation of the background concentration of NO$_x$ between less than 1 \\% and up to 20 \\% at different altitudes. Some observational studies have attempted to measure sprite-NO$_x$ through satellite observations \\citep{Arnone2008\/GeoRL, Rodger2008\/GRL, Arnone2009\/PSST, arnone2016chimtea}. However, according to these studies sprite-NO$_x$ is at the edge of current detectability.\nThe predicted significant local chemical influence \\citep{Winkler2015\/JASTP} suggest that Blue Jets could have a non-negligible influence in the chemistry of the atmosphere.\n\nIn this work, we have developed the first global parameterization of Blue Jets. We have used the WACCM4 model in order to study the global occurrence rate of Blue Jets and their global chemical impact by developing three different Blue Jet parameterizations. WACCM4 includes a lightning parameterization developed by \\cite{Price1992\/JGR} based on the cloud top height (CTH). Here we also use other lightning parameterizations based on, respectively, the amount of convective precipitation (CP) \\citep{Allenp2002\/JGR}; the upward mass flux (MFLUX) \\citep{Allenp2002\/JGR}; the precipitation rate and the Convective Available Potential Energy (CPCAPE) \\citep{Romps2014\/SCI}; and on the upward cloud ice flux (ICEFLUX) \\citep{Finney2014\/ACP}.The combined use of lightning and Blue Jet parameterizations allow us to predict the geographical and seasonal chemical impact of Blue Jets.\n\n\n\\subsection{Physics and chemistry of Blue Jets} \n\\label{sec:bjstatistics}\n\n\\begin{figure}\n\\includegraphics[width=0.6\\columnwidth]{BJ_wescott.png}\n\\footnotesize\n\\caption{\\label{fig:BJ_wescott}\nLeft panel: Inverted black and white photography of a Blue Jet. The spatial scales of the leader and streamer regions can be appreciated. Image adapted from \\cite{Wescott2001\/JGR}. Right panel: Blue Jet simulated by \\cite{Krehbiel2008\/NatGe} illustrating the charge structure of clouds. Blue and red lines correspond to positive and negative charges, respectively. Image adapted from \\cite{Krehbiel2008\/NatGe}.\n\\normalsize\n}\n\\end{figure}\n\nBlue Jets are formed by a leader channel surrounded by a large number of streamers. The leader is a highly conductive plasma channel that can heat the air up to thousands of Kelvin. The first interpretations of Blue Jets by a streamer corona of a leader were made by \\cite{Sukhorukov1998\/JASTP} and \\cite{Petrov1999\/JTePh}. \\cite{Raizer2006\/GeoRL, Raizer2007\/JASTP} proposed the development of Blue Jets as a bi-leader channel that propagates upward from the streamer zone of a positive leader. According to \\cite{Raizer2006\/GeoRL, Raizer2007\/JASTP}, this leader can transfer the energy contained in the clouds to upper regions of the atmosphere, where the low density allows the development of a streamer corona. Figure~\\ref{fig:BJ_wescott} shows a photography of a real Blue Jet from\\cite{Wescott2001\/JGR} and the structure of charges in clouds that trigger the inception of Blue Jets simulated by \\cite{Krehbiel2008\/NatGe}.\n \\cite{Krehbiel2008\/NatGe} developed a model based on quasielectrostatic fields, formed as an imbalance of the electric charge in the cloud tops, to predict Blue Jet and Gigantic Jet inception. After lightning occurs, a charged layer can remain near the storm top layer creating a local electric field. \\cite{Krehbiel2008\/NatGe} found that conventional electric breakdown near this charged layer could trigger an upward propagating leader, forming a Blue Jet. \\cite{Riousset2010\/JGRA} upgraded the model proposed by \\cite{Krehbiel2008\/NatGe}, confirming the obtained results. Observations by \\cite{Lu2011\/GRL} supported some of the predictions by \\cite{Krehbiel2008\/NatGe} and \\cite{Riousset2010\/JGRA}. As hypothesized by \\cite{Krehbiel2008\/NatGe}, there would exist a competition between intra-cloud discharges and Blue Jets in the process of discharging the cloud. The result of this competition would depend on the capability of the convective fluxes to mix the oppositely charged layers located in the cloud top before the inception of a Blue Jet.\n\n\\cite{Chanrion2017\/GRL} have recently described the observation of a Blue Jet from the International Space Station (ISS). The top height of the thundercloud that initiated the Blue Jet reached the tropopause, an atmospheric region where convection is weak. The reported Blue Jet was preceded in 1.16~s by a strong negative CG lightning with a peak current of -167.5~kA. This CG discharge could possibly be the parent lightning of the Blue Jet.\nA Blue Jet would then be formed by an upward propagating leader traveling to upper regions of the atmosphere with lower pressure, reaching its maximum altitude between 30~km and 40~km \\citep{van_der_Velde2010\/JGR, Pasko2012\/SSR, daSilva2013\/GRL, Milikh2014\/JGR, Chanrion2017\/GRL}. Streamers could then emerge from the leader as it passes through the low pressure regions of the atmosphere \\citep{Raizer2007\/JASTP}.\n\n\\cite{Mishin1997\/GeoRL} and \\cite{smirnova2003\/IJGA} developed the first models to estimate the local chemical impact of Blue Jets. However, these models do not include the latest results on the electrodynamical mechanisms of Blue Jets \\citep{Raizer2007\/JASTP, Krehbiel2008\/NatGe, Riousset2010\/JGRA}. \\cite{Winkler2015\/JASTP} developed the most detailed model to date to study the local chemical impact of a Blue Jet including 88~species interacting through more than 1000~reactions. They used their model to estimate the local chemical impact of the leader and streamers of a Blue Jet at several altitudes.\nAccording to their estimations, the high-temperature reactions taking place in the Blue Jet leader can enhance by several orders of magnitude the local background concentrations of stratospheric N$_2$O and NO (due to the high temperature reactions collected in Table~3 of \\citep{Winkler2015\/JASTP}) and produce a significant depletion of ozone \\citep{Winkler2015\/JASTP}. In addition, the high electric field in the streamer phase would produce an enhancement in the concentration of N$_2$O by the chemical reactions\n\n\\begin{linenomath*}\n\\begin{equation}\ne + N_2 \\rightarrow e + N_2(A^3\\Sigma_u^+) \\label{reactionN2A}\n\\end{equation}\n\\end{linenomath*}\nand\n\\begin{linenomath*}\n\\begin{equation}\nN_2(A^3\\Sigma_u^+) + O_2 \\rightarrow N_2O + O. \\label{reactionN2A}\n\\end{equation}\n\\end{linenomath*}\n\n\nThe injection of NO$_x$ into the stratosphere could also influence the concentration of other species. According to investigations about the chemical influence of lightning-produced NO in the atmosphere, atmospheric electricity phenomena can also contribute to the concentration of OH, HO$_2$ and CO \\citep{rohrer2006strong, murray2013interannual, siingh2015lightning}. In particular, NO interacts with HO$_2$ producing OH. The production of OH molecules can influence the acidity of rainwater \\citep{seinfeld2016atmospheric}, as they can react with NO$_2$ molecules producing HNO$_3$ following the chemical reaction NO$_2$ + OH + M $\\rightarrow$ HNO$_3$ + M \\citep{labrador2005effects}. The formation of OH contributes to the loss of CO by the process CO + OH $\\rightarrow$ HO$_2$ + C \\citep{murray2013interannual}. OH molecules can also contribute to the oxidation of SO$_2$, leading to the production of H$_2$SO$_4$. In addition, NO$_2$ molecules contribute to the production of N$_2$O$_5$. The oxidation of N$_2$O$_5$ followed by a heterogeneous hydrolysis reaction on aerosol particles contributes to the enhancement of HNO$_3$.\n\n\n\n\\subsection{Global budgets of N$_2$O and NO$_x$ and their relation with atmospheric electricity} \n\\label{sec:budjet}\n\nAccording to \\cite{Winkler2015\/JASTP}, Blue Jets could inject an important amount of nitrous oxide (N$_2$O), nitric oxide (NO) and nitrogen dioxide NO$_2$ at stratospheric altitudes. These gases play important roles in the chemical balance of stratospheric ozone. In addition, N$_2$O is one of the most important greenhouse gases. \n\nNatural N$_2$O sources are estimated to inject about 10.2~Tg~N$_2$O-N~yr$^{-1}$ in the atmosphere, while anthropogenic sources could produce around 6.3~Tg N$_2$O-N~yr$^{-1}$ \\citep{davidson2009\/nat, Prather2015\/JGR}, where 1~Tg = 10$^{12}$~g and N$_2$O-N stands for the mass of nitrogen atoms in N$_2$O molecules \\citep{davidson2009\/nat}. The major natural and anthropogenic sources of N$_2$O are basically due to nitrification and denitrification produced by microbes at ground level \\citep{davidson2009\/nat}. However, \\cite{Sheese2015\/GRL} have recently proposed an atmospheric source of N$_2$O based on observations from the satellite instrument ``Atmospheric Chemistry Experiment-Fourier Transform Spectrometer\" (ACE-FTS) consisting of the chemical reaction described in equation~(\\ref{reactionN2A}) \\citep{arnone2012stratosphere, Sheese2015\/GRL}. N$_2$O is the major source of NO in the stratosphere. 90 \\% of the stratospheric destruction of N$_2$O is by photolysis (N$_2$O + h$\\nu$ $\\rightarrow$ N$_2$ + O) and 10 \\% is by reaction with O($^1$D) producing NO, N$_2$ and O$_2$ molecules \\citep{seinfeld2016atmospheric}. \n\n\n\n\\cite{Plieninger2016\/ACP} compared global-average vertical profiles of N$_2$O obtained by different instruments. In particular, \\cite{Plieninger2016\/ACP} showed the vertical stratospheric concentration of N$_2$O obtained by the ``Michelson Interferometer for Passive Atmospheric Sounding\" (MIPAS), the ``Atmospheric Chemistry Experiment-Fourier Transform Spectrometer (ACE-FTS), the ``Microwave Limb Sounder onboard Aura\" (Aura-MLS) and the ``Sub-Milimetre Radiometer onboard Odin\" (Odin-SMR). It is worth noting that the global-average concentration of N$_2$O estimated by each of the above mentioned instruments between 20~km and 40~km indicates that the observational uncertainty in the global amount of N$_2$O is about 10~\\% \\citep{Plieninger2016\/ACP}. \n\nLightning is not considered an important source of atmospheric N$_2$O, as shown in Table~11 of \\cite{SchumannHuntrieser2007\/SCP} where results from different studies and campaigns conclude that the global lightning-produced emission rate of N$_2$O is below 5~$\\times$10$^{-4}$ ~Tg~N$_2$O-N~yr$^{-1}$.\n\nLet us now turn to the global budget of NO and NO$_2$, which together make up NO$_x$. \\cite{SchumannHuntrieser2007\/SCP} presented an extensive study about the global production of NO$_x$ by lightning (or LNO$_x$) based on satellite and aircraft measurements, laboratory experiments and theoretical studies. \nLightning is considered one of the major natural sources of atmospheric NO$_x$ emissions. Different studies estimate the global production of NO$_x$ in thunderstorms in a wide range between 1 and 20~Tg~NO-N~yr$^{-1}$ \\citep{SchumannHuntrieser2007\/SCP, huntrieser2016injection}. However, the most likely range is 5$\\pm$3~Tg~N~yr$^{-1}$. Lightning would then contribute up to $\\sim$10-15$\\%$ of the total global emissions of NO$_x$. It is probably something greater than 10\\% now that anthropogenic emissions have decreased substantially in North America and Europe.\n\nThe uncertainties in the contribution of lightning to the global concentration of NO$_x$ is based on theoretical and empirical challenges. Laboratory results of NO$_x$ produced by electrical discharges are difficult to extrapolate to real lightning discharges, as both Cloud-to-Ground (CG) and Intra-Cloud (IC) lightning discharges are different from each other \\citep{Price1997\/JGR} and cannot be accurately reproduced in the laboratory. Space-based instruments measure NO$_2$ and cannot accurately measure the concentration of tropospheric NO$_x$ \\citep{SchumannHuntrieser2007\/SCP, beirle2010direct, bucsela2010lightning, pickering2016estimates}. This concentration has to be usually deduced from the concentration of other species that can react with NO$_x$ molecules. However, some uncertainties in the atmospheric chemical kinetics of NO$_x$ lead to imprecisions in the estimation of NO$_x$ from measurements. These estimations are often based on an assumed upper tropospheric (UT) chemical lifetime of NO$_x$ in a range between 2 and 8~days. \nBased on reanalysis of the measurements taken by the Deep Convective Clouds and Chemistry (DC3) atmospheric experiment, \\cite{Nault2017\/JGR} recently revised the interaction of atmospheric CH$_3$O$_2$NO$_2$ and HNO$_3$ with NO$_x$ molecules, estimating a new UT NO$_x$ lifetime of about 3 hours in the first few hours downwind of a thunderstorm instead of the previous scale of days. Using this new analysis, \\cite{Nault2017\/JGR} estimated a global lightning production of NO$_x$ of about 9~Tg~NO-N~yr$^{-1}$. \\cite{Nault2017\/JGR} results indicate higher LNO$_x$ in the mid-latides than in the tropical regions, in agreement with \\cite{SchumannHuntrieser2007\/SCP}. However, the latest estimations of the global lightning NO$_x$ emissions by new cloud-sliced observations of UT NO$_2$ in the 6~km - 9~km range from the Ozone Monitoring Instrument (OMI) of the Aura mission combined with the GEOS-Chem model point to a global lightning NO$_x$ source of 5.5~Tg~N yr$^{-1}$ \\citep{maraisnitrogen}. \\cite{maraisnitrogen} reports no significant difference in LNO$_x$ production per flash between the tropics and mid-latitudes.\nStratospheric NO can cause ozone depletion through the processes \\citep{crutzen1979\/ARPS} \n\n\\begin{linenomath*}\n\\begin{equation}\nNO + O_3 \\rightarrow NO_2 + O_2,\n\\end{equation}\n\\begin{equation}\nNO_2 + O \\rightarrow NO + O_2.\n\\end{equation}\n\\label{ozonedep}\n\\end{linenomath*}\n\nMoreover, the oxidation of N$_2$O is the major source of stratospheric NO, producing 1~Tg~N~yr$^{-1}$ of NO$_x$ \\citep{crutzen1979\/ARPS}. Thus, the introduction of Blue Jets in global models as a new possible atmospheric source of N$_2$O and NO$_x$ could have a non negligible effect in the global budget of ozone.\n\n\n\n\n\n\n\n\\section{Model} \n\\label{sec:models}\n\n\n\\subsection{WACCM4}\n\\label{sec:models}\n\nThe Whole Atmosphere Community Climate Model version 4 (WACCM4) \\citep{Marsh2013\/JC} is a global circulation model included in the Community Earth Climate System Model version 1 (CESM1). WACCM4 is an extension of the Community Atmosphere Model (CAM4) \\citep{Marsh2013\/JC, tilmes2015description, tilmes2016representation}. CAM4 couples the troposphere and the stratosphere chemistry, while WACCM4 extends up to the thermosphere. The chemistry of WACCM4 is based on version 4 of the Model for OZone And Related chemical Tracers (MOZART4) \\citep{kinnison2007sensitivity, emmons2010description, lamarque2012\/GMD, tilmes2015description}, including 183~species and 472~chemical reactions including gas-phase chemistry of neutrals and ions, photolysis and heterogenous chemistry.\n\n\nWe set the model domain extending from the surface to 140~km of altitude (5.96$\\times$10$^{-6}$~hPa). We divide the vertical domain in 88 levels and set a horizontal resolution of 1.9$^{\\circ}$ in latitude and 2.5$^{\\circ}$ in longitude. We start the numerical experiment with WACCM4 running a complete year (from January 1999 to January 2000) without Blue Jets allowing free dynamics for each considered lightning parameterization. Then, we run the same period of time in the specified dynamics mode (SD-WACCM4) \\citep{lamarque2012\/GMD, smith2017\/JAS}. In this study, we use the facility of SD but, instead of nudging to reanalysis fields, we nudge to the meteorological fields from a previous (free-running) WACCM simulation. The reason for using SD is to ensure that the basic dynamics in the lower and middle atmosphere is identical in simulations in which other changes are made. In this second run, temperature fields and horizontal winds in the troposphere and stratosphere are nudged at each model time step using the output of the first run. We apply the nudging from ground level to 80~km. The nudging is then tapered off in the ranges of altitude between 80 and 90~km, and finally removed at 90~km of altitude \\citep{smith2017\/JAS}.\n\nAfterward, we use the same specified dynamics in order to run a complete year including all the combinations of lightning and Blue Jet parameterizations. This approach allows us to compare the simulations with and without Blue Jets in order to estimate their global chemical impact in the atmosphere.\n\nAs the lifetime of N$_2$O in the atmosphere is of the order of a century \\citep{Prather2015\/JGR}, we select the most realistic cases and repeat the process for a period of one decade. Following this approach, the obtained results would be closer to the chemical equilibrium. We discuss these cases in section~\\ref{sec:results}.\n \n\n\n\n\n\\subsection{Lightning parameterizations}\n\\label{sec:lightning}\n\nThe temporal and geographical occurrence of Blue Jets obtained with WACCM4 will strongly depend on the global occurrence of lightning. In this section, we briefly highlight the particularities of each considered lightning parameterization.\n\nThe characteristic size of lightning is some orders of magnitude smaller than the WACCM4 grid size. Therefore, lightning are considered as sub-grid events in the model. WACCM4 includes a lightning parameterization based on the cloud top heights (CTH) \\citep{Price1992\/JGR} that estimates the density of lightning (or flashes) in each domain cell for every time step of 30~minutes. The regional and seasonal flash frequency produced by this parameterization roughly agrees with the observations recorded by the Lightning Imaging Sensor (LIS) and the Optical Transient Detector (OTD) \\citep{Christian2003\/JGR, cecil2014gridded} over a period of two decades. However, the implementation of this lightning parameterization in WACCM4 overestimates the total flashes per second taking place in the globe over a period of one year. According to OTD\/LIS, the global lightning occurrence over a year is around 44~flashes per second, while this parameterization produces around 65~flashes per second. In addition, the lightning parameterization by \\cite{Price1992\/JGR} implemented in WACCM4 also underestimates the lightning occurrence over the oceans. For these reasons, the use of a parameterization for Blue Jets together with the CTH lightning parameterization by \\cite{Price1992\/JGR} would probably underestimate the occurrence of Blue Jets over the oceans and would overestimate the global occurrence of Blue Jets. However, the spatial correlation between the flash frequency reported by OTD\/LIS and the flash frequency estimated by CTH is 0.7602. This is the highest spatial correlation obtained by the use of different lightning parameterizations. Therefore, we choose the CTH lightning parameterization to show the primary results.\n\n\\cite{Allenp2002\/JGR} developed a lightning parameterization based on the amount of convective precipitation (CP) over USA. We have tested this lightning parameterization in WACCM4, obtaining a good agreement between the predicted flash frequency (51~flashes per second) and the lightning occurrence reported by OTD\/LIS. We obtain a spatial correlation between the flash frequency derived by CP and OTD\/LIS of 0.5760. However, the implementation of this parameterization in WACCM4 produces a lightning occurrence that remains almost constant over the four seasons, in disagreement with observations. \\cite{Allenp2002\/JGR} also derived a lightning parameterization based on the upward mass flux (MFLUX) at 440~hPa. This parameterization produces again a good agreement between the predicted flash frequency (43~flashes per second) and the lightning occurrence reported by OTD\/LIS. However, it slightly overestimates the occurrence of lightning in the oceans and in South America, while underestimates the flash density in some regions of Africa. In addition, MFLUX produces a low spatial correlation (0.4963) with the flash frequency reported by OTD\/LIS.\n\nApart from these three ``classical\" lightning parameterizations by \\cite{Price1992\/JGR} and \\cite{Allenp2002\/JGR}, we also implement Blue Jet parameterizations in WACCM4 together with two, more recent, lightning parameterizations developed by \\cite{Romps2014\/SCI} (CPCAPE) and \\cite{Finney2014\/ACP} (ICEFLUX), respectively. The parameterization of \\cite{Romps2014\/SCI} is based on the precipitation rate and on the Convective Available Potential Energy (CAPE), while the parameterization by \\cite{Finney2014\/ACP} is based on the upward cloud ice flux at 440~hPa. \nThe parameterization by \\cite{Romps2014\/SCI} produces global (52~flashes per second), regional and seasonal lightning frequencies that agree with the observation by OTD\/LIS but it overestimates the flash occurrence over the oceans. The spatial correlation between the flash frequency derived by CPCAPE and the observations of OTD\/LIS is 0.4540. The implementation of the parameterization developed by \\cite{Finney2014\/ACP} underestimates by a factor of~2 the global lightning occurrence rate and results in a spatial correlation with the flash frequency reported by OTD\/LIS of 0.6739. \n\n\n\\subsection{Blue Jet parameterizations in WACCM4}\n\\label{sec:bj}\n\n\\subsubsection{Blue Jet frequency}\n\\label{subsec:BJfrequency}\nThe characteristic size of Blue Jets is some orders of magnitude smaller than the horizontal size of WACCM4 grids. As in the case of lightning, Blue Jets have to be treated as sub-grid phenomena in WACCM4. Following the basic idea of the previously described global lightning parameterizations, we have developed two different Blue Jet parameterizations to be considered in global models. The first developed parameterization prescribes the estimation of global Blue Jets per minute to predict their chemical influence in the atmosphere. The second proposed parameterization is based on physical assumptions and does not impose the rate of occurrence of Blue Jets.\n\n\\subsubsection*{Parameterization based on ISUAL and the altitude of the tropopause (IS-TROP LOW \/ IS-TROP UP)}\n\n\n\n\\cite{Ignaccolo2006\/GeoRL} proposed a formula to estimate the global rate of sprites based on reports of sprite detections. \\cite{Ignaccolo2006\/GeoRL} obtained a global occurrence rate of sprites about 2.8 per minute. According to ISUAL, the global occurrence rate of TLEs is around 4.13~per minute \\cite{Chen2008\/JGRA}, among which 3.23 are elves, 0.50 are sprites, 0.39 are halos and 0.01 are gigantic jets. As optical emissions from Blue Jets and lightning are difficult to separate, the global global occurrence rate of Blue Jets was not derived from ISUAL data. However, we assume that Blue Jets are less frequent than sprites and more frequent than Gigantic Jets. Therefore, as a first approximation we assume that the global occurrence rate of Blue Jet is in the range between 0.01 and 1.0 events per minute. Given that Blue Jets are triggered as a consequence of the remaining imbalance of charge in thunderclouds after lightning occurs \\citep{Krehbiel2008\/NatGe}, Blue Jet parameterizations must be spatially and temporally connected with any considered parameterization of lightning. Following these considerations, the total occurrence of Blue Jets at a given time would be the total occurrence of lightning flashes given multiplied by 3.6 $\\times$ 10$^{-4}$ (UP) or 3.6 $\\times$ 10$^{-6}$ (LOW), respectively. \n\nAs we discussed in section~\\ref{sec:bjstatistics}, the model proposed by \\cite{Krehbiel2008\/NatGe} indicates that the inception of Blue Jets is favored when the two oppositely charged layers located in the upper part of thunderclouds do not mix. Hence, it is reasonable to expect the inception of Blue Jets when the convection near the cloud top is weak. In this regard, the Blue Jet reported by \\cite{Chanrion2017\/GRL} was triggered in a thundercloud whose top height was near the tropopause, where the lack of convection keeps the temperature relatively constant. WACCM4 and the most of Global Circulation Models can calculate the altitude of the tropopause and the cloud top height, two atmospheric variables that can be related with the possibility of Blue Jet inception \\citep{Krehbiel2008\/NatGe, Chanrion2017\/GRL}. We can then distribute the predicted Blue Jets exclusively in the domain cells where the cloud top height is above the beginning of the tropopause or below tropopause by no more than one kilometer and there is lightning. We also impose as a condition to the existence of Blue Jets that the flash frequency is greater than zero in that domain cell. We restrict the locations where Blue Jets can be distributed to the range of latitudes between 60$^{\\circ}$ S and 60$^{\\circ}$ N. We name this Blue Jet parameterization as ``IS-TROP LOW\" and ``IS-TROP UP\", depending on whether the global occurrence rate of BJ is set to 3.6 $\\times$ 10$^{-6}$ (LOW) or 3.6 $\\times$ 10$^{-4}$ (UP) Blue Jets per lightning and where IS and TROP refer to ISUAL and to the height of the tropopause, respectively. Although this parameterization could produce a realistic geographical occurrence of Blue Jets, the global occurrence rate is somehow imposed. \n\n\n\n\n\\subsubsection*{Parameterization based on lightning peak currents and the altitude of the tropopause (LPC-TROP LOW \/ LPC-TROP UP)}\n\nAccording to the model developed by \\cite{Krehbiel2008\/NatGe}, a strong lightning discharge or a set of small amplitude CG lightning discharges occurring within a short time distance would probably precede the inception of a Blue Jet, since Blue Jets are produced by a large imbalance of charge. \\cite{Chanrion2017\/GRL} reported the observation of a Blue Jet preceded by a strong lightning discharge. Let us now use this observation to derive a more realistic Blue Jet parameterization based on the peak current value of the lightning possibly preceding a Blue Jet.\nThe Blue Jet reported by \\cite{Chanrion2017\/GRL} was preceded by a negative Cloud-to-Ground (CG) lightning discharge with a peak current of -167.5~kA. Elves, the less energetic TLEs, seem to be triggered by lightning discharges with peak currents whose absolute value is above 60~kA \\citep{Barrington-Leigh1999\/GeoRL\/1}. As an approximation, we can assume that the peak current threshold of the lightning preceding Blue Jets is between 60~kA and 167.5~kA. We choose two representative values in this range to be the threshold of Blue Jets, such as 100~kA and 150~kA. According to the distribution of global lightning peak current reported by \\cite{Said2013\/JGR} using the Vaisala global lightning data set GLD360, approximately 1 \\% of lightning have a peak current above 100~kA and only 0.1 \\% have a peak current above 150~kA. We can then develop a Blue Jet parameterization in WACCM4 where the spatial occurrence of Blue Jets is again restricted to domain cells where the cloud top height is higher than one kilometer below the tropopause. However, instead of imposing the global occurrence of Blue Jets, we can now assume that the Blue Jet frequency in such domain cells is given by the amount of lightning in the domain cell with peak currents above 100~kA or 150~kA. Hence, we define the Blue Jet frequency in each cell where the cloud top height is near the tropopause as 0.01 (UP) or 0.001 (LOW) times the frequency of lightning. We refer to these two Blue Jet parameterizations as ``LPC-TROP UP\" and ``LPC-TROP LOW\", where LPC and TROP recall to lightning peak current and to the height of the tropopause, respectively. The maximum peak current of lightning is not homogeneously distributed over land and ocean \\citep{Said2013\/JGR}. However, we do not include in this parameterization any parameter to take into account this inhomogeneity. This simplification is justified because the scope of this paper is to describe the global chemical influence of Blue Jets rather that the regional influence.\n\n\n\\subsubsection{Chemical impact of Blue Jets}\n\\label{subsec:chemicalimpact}\n\\cite{Winkler2015\/JASTP} developed the most detailed zero-dimensional model until now to predict the local chemical impact of a Blue Jet in the center of Blue Jet leader and streamers. \\cite{Winkler2015\/JASTP} estimated the chemical impact of an upward propagating Blue Jet at different altitudes between 18~km and 38~km, obtaining a significant enhancement in the densities of some chemical species such as N$_2$O, NO or O and a decrease in the density of O$_3$ in the center of Blue Jets. We use the local chemical impact of a single Blue Jet obtained by \\cite{Winkler2015\/JASTP} together with the previously derived Blue Jet parameterizations to estimate the global chemical impact of Blue Jets using WACCM4. We assume that each Blue Jet would produce an enhancement in the concentrations of N$_2$O, NO or O. However, as Blue Jets are considered as sub-grid phenomena in WACCM4, we take into account the following considerations:\n\n\\begin{enumerate}\n\\item As the area of a WACCM4 cell is larger than the horizontal cross-section of the Blue Jet, we have to estimate the total number of species produced by all Blue Jets at each altitude and distribute them over the area of the grid at each altitude level. Hence, we need to estimate the electrodynamical radius of the Blue Jet where the chemical reactions are produced. \\cite{Winkler2015\/JASTP} noted that while the optical radius of a Blue Jet is a few hundreds of meters \\citep{Wescott2001\/JGR}, the electrodynamical radius could be between 20 and 100~times smaller \\citep{Shneider2012\/PhysPlas, Milikh2014\/JGR}. Based on optical observations by \\cite{Wescott2001\/JGR}, we have assumed that Blue Jets have an optical radius of 250~m at their base. Hence, the electrodynamical radius $R_e$ would be in the range between $R_1$ = 2.5~m and $R_2$ = 12.5~m. According to \\cite{Winkler2015\/JASTP}, the production of N$_2$O and NO is dominated by the leader for altitudes ranging between 18~km and 28~km and by streamers between 28~km and 38~km. We have then assumed that the electrodynamical radius of the Blue Jet is completely filled by a leader between 18~km and 28~km of altitude and by streamers between 28~km and 38~km of altitude.\n\nThe global production of N$_2$O by Blue Jets can then be estimated using the density variations reported by \\cite{Winkler2015\/JASTP} (figure~19 of \\cite{Winkler2015\/JASTP}). As a first approximation, we can consider a Blue Jet as a 20~km long cylinder with a constant radii of 2.5~m or 12.5~m formed by a leader and a streamer region. A single Blue Jet would produce between 2 $\\times$ 10$^{28}$ and 6 $\\times$ 10$^{29}$ molecules of N$_2$O for electrodynamical radii of 2.5~m and 12.5~m, respectively. Assuming that the global occurrence rate of Blue Jets is between 0.01 and 1~per minute \\citep{Ignaccolo2006\/GeoRL, Chen2008\/JGRA}, we find that Blue Jets with a radii of 2.5~m would produce between 6 $\\times$ 10$^{-3}$~Tg~N$_2$O-N~yr$^{-1}$ and 0.6~Tg~N$_2$O-N~yr$^{-1}$, while Blue Jets with a radii of 12.5~m would produce between 0.15~Tg~N$_2$O-N~yr$^{-1}$ and 15 ~Tg~N$_2$O-N~yr$^{-1}$\n\n\n\\item We assume that the production of species decays parabolically across the radial coordinate $r$ from the center of the Blue Jet up to the limit of the electrodynamical radius as \n\n\\begin{linenomath*}\n\\begin{equation}\nN(r) = N_{max} \\left(1 - \\frac{r^2}{R^2_e} \\right),\n\\label{NR}\n\\end{equation}\n\\end{linenomath*}\n\nwhere $R_e$ is the electrodynamical radius and $N_{max}$ is the enhancement in the density of species in the symmetry axis of the Blue Jet as predicted by \\cite{Winkler2015\/JASTP}.\n\n\n\\item Recorded optical emissions from Blue Jets indicate an increase of its radius with altitude \\citep{Wescott2001\/JGR}. We assume that the Blue Jet radius increases with altitude following the simple scale law\n\n\\begin{linenomath*}\n\\begin{equation}\nP_i R_i = P_j R_j,\n\\label{scalelay}\n\\end{equation}\n\\end{linenomath*}\n\nwhere $P$ and $R$ are the atmospheric pressure and Blue Jet radius at two different altitude levels denoted as $i$ and $j$.\n\n\n\n\\end{enumerate}\n\nFollowing these considerations, the Blue Jet region at 30~km of altitude filled by streamers would have a radius between 16~m and 80~m. Given the assumed altitude-dependence of the leader and streamer-region radius and the production profile by \\cite{Winkler2015\/JASTP}, the total production of N$_2$O and NO by a Blue Jet is dominated by the leader phase.\n\n\n\n\n\\section{Results} \n\\label{sec:results}\n\nWe have implemented in WACCM4 the Blue Jet parameterizations derived in section~\\ref{sec:bj} using five different lightning parameterizations. We present the obtained global occurrence rate of Blue Jets in subsection~\\ref{sec:occurrence}. We have coupled these Blue Jet frequencies with the chemical impact of a single Blue Jet predicted by \\cite{Winkler2015\/JASTP}. In subsection~\\ref{sec:gchemical} we present the predicted global impact of Blue Jets for each global parameterization.\n\n\\subsection{Global occurrence and seasonal cycle of Blue Jets}\n\\label{sec:occurrence}\n\nLet us firstly focus on the global occurrence of Blue Jets derived for each combination of lightning and Blue Jet parameterizations. The first column of figure~\\ref{fig:bj_cases_1_2} shows the obtained lightning flash frequency using different lightning parameterizations in WACCM4. The second column of figure~\\ref{fig:bj_cases_1_2} shows the Blue Jet frequency for the Blue Jet parameterization denoted as ``IS-TROP LOW\" using different lightning parameterizations. As we have previously detailed, this Blue Jet parameterization take as input the global occurrence rate of Blue Jet estimated from the TLE occurrence reported by ISUAL. The Blue Jet frequency obtained with the BJ parameterizations ``LPC-TROP LOW\" and ``LPC-TROP UP\" using different lightning parameterizations is plotted in figure~\\ref{fig:bj_cases_3}. In the ``LPC-TROP\" parameterization, the Blue Jet occurrence rate is not fixed to any given value. Instead, it is based on the physical assumptions previously described in section~\\ref{sec:bj}.\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{bj_cases_1_2.pdf}\n\\caption{\\label{fig:bj_cases_1_2}\n(First column) annual average lightning flash frequencies in flashes km$^{-2}$day$^{-1}$ and (second column) annual average Blue Jet frequencies in BJ km$^{-2}$day$^{-1}$ using the Blue Jet parameterization denoted as ``IS-TROP LOW\" and different lightning parameterizations. We have used different lightning parameterizations denoted as CTH \\citep{Price1992\/JGR} based on the cloud top height, CP \\citep{Allenp2002\/JGR} based on the precipitation rate, CPCAPE \\citep{Romps2014\/SCI} based on the precipitation rate and convective available potential energy (CAPE), MFLUX \\cite{Allenp2002\/JGR} based on the updraft mass flux and ICEFLUX \\citep{Finney2014\/ACP} based on the upward cloud ice flux. We annotate in boxes the total annual Blue Jets per minute.\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{bj_cases_3.pdf}\n\\footnotesize\n\\caption{\\label{fig:bj_cases_3}\nAnnual average Blue Jet frequencies BJ km$^{-2}$day$^{-1}$ using the parameterizations denoted as ``LPC-TROP LOW\" and ``LPC-TROP UP\". The shown Blue Jet frequencies have been calculated using different lightning parameterizations denoted as CTH \\citep{Price1992\/JGR} based on the cloud top height , CP \\citep{Allenp2002\/JGR} based on the precipitation rate, CPCAPE \\citep{Romps2014\/SCI} based on the precipitation rate and convective available potential energy (CAPE), MFLUX \\citep{Allenp2002\/JGR} based on the updraft mass flux and ICEFLUX \\citep{Finney2014\/ACP} based on the upward cloud ice flux. We annotate in boxes the total annual Blue Jets per minute.\n\\normalsize\n}\n\\end{figure}\n\nThe Blue Jet frequencies obtained with all the considered Blue Jet parameterizations and with the CTH lightning parameterization are collected in table~\\ref{tab:results}. The results obtained with the other tested lightning parameterizations are shown in the supplementary material.\n\nThe Blue Jet parameretizations ``IS-TROP\" and the most of the Blue Jet parameterizations ``LPC-TROP LOW\" produce a global occurrence rate of Blue Jets lower than 1 BJ per minute, as estimated from the TLE frequency reported by ISUAL. However, the Blue Jet parameterizations ``LPC-TROP UP\" and ``CP LPC-TROP LOW\" significantly overestimate the Blue Jet frequency. \n\nThe comparison of the spatial distribution of lightning flashes and Blue Jets indicates that the relative occurrence of Blue Jets in Asia with respect to other regions is larger than the relative occurrence of lightning flashes. In addition, most of the considered parameterizations produce a maximum in the lightning flash frequency and in the Blue Jet frequency over Africa and in the north of South America. \n\nThe monthly global average occurrences of Blue Jets obtained with different lightning parameterization schemes are presented in figure~\\ref{fig:bj_monthly}. The maximum occurrence of Blue Jets takes place between June and August, coinciding with the maximum occurrence of lightning. As the occurrence of Blue Jets is related with a high lightning activity on clouds \\citep{Krehbiel2008\/NatGe} (see figure~\\ref{fig:BJ_wescott}), we conclude that the obtained coincidence of the seasonal cycle of lightning and Blue Jets can be considered as realistic.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{bj_monthly.pdf}\n\\caption{\\label{fig:bj_monthly}\nMonthly global average Blue Jet frequencies BJ km$^{-2}$day$^{-1}$ using the developed Blue Jets parameterization. The shown Blue Jet frequencies have been calculated using different lightning parameterizations denoted as CTH \\citep{Price1992\/JGR} based on the cloud top height, CP \\citep{Allenp2002\/JGR} based on the precipitation rate, CPCAPE \\citep{Romps2014\/SCI} based on the precipitation rate and convective available potential energy (CAPE), MFLUX \\citep{Allenp2002\/JGR} based on the updraft mass flux and ICEFLUX \\citep{Finney2014\/ACP} based on the upward cloud ice flux.\n}\n\\end{figure}\n\n\n\\subsection{Global chemical impact of Blue Jets}\n\\label{sec:gchemical}\n\nWe follow the simulation scheme proposed in section~\\ref{sec:models} in order to predict the global chemical impact of Blue Jets in the atmosphere. We use the same specified dynamics to simulate the atmosphere with and without Blue Jets. \n\nFirst, we run simulations of one year for all the considered Blue Jet parameterizations (subsection~\\ref{sec:chem1}). Then, we choose some of the most representative cases and extend the simulations up to ten years in order to obtain the chemical influence of Blue Jets when equilibrium is reached (subsection~\\ref{sec:chem10}). We do not include in this discussion the chemical impact of Blue Jets using the parameterizations that overestimate the Blue Jet frequency (``LPC-TROP UP\"). \n\n\\subsubsection{Transient response}\n\\label{sec:chem1}\nIn this section we present and discuss the global impact of Blue Jets over one year for all the considered cases. It is important to note that simulations of one year including the chemical perturbation of Blue Jets do not allow the atmosphere to reach the equilibrium. However, these short simulations are useful to choose the most realistic cases before extending the simulation to five and ten years. \nWe collect in table~\\ref{tab:results} the total annual production, i.e., the total number of NO, N$_2$O and O molecules injected in the atmosphere by Blue Jets for all the considered cases of the CTH lightning parameterization, while we show in the supplementary material the results corresponding to other lightning parameterizations. The cases in which the production of N$_2$O is larger than the natural source of atmospheric N$_2$O (10.2~Tg~N$_2$O-N~yr$^{-1}$) \\citep{davidson2009\/nat, Prather2015\/JGR} and the total occurrence rate of Blue Jets is higher than 1 BJ per minute will be considered as unrealistic scenarios. Therefore, the realistic scenarios would be most of the``IS-TROP UP\", all ``IS-TROP LOW\" and most of ``LPC-TROP LOW\". The lower realistic scenario (3.6 $\\times$ 10$^{-6}$ BJ per lightning flash and R$_1$ = 2.5~m) produces a Blue Jet frequency of 1.4 $\\times$ 10$^{-3}$ BJ per minute and 6.6 $\\times$ 10$^{-4}$ Tg~N$_2$O-N~yr$^{-1}$ (ICEFLUX IS-TROP LOW R$_1$), while the higher realistic case (3.6 $\\times$ 10$^{-4}$ BJ per lightning flash and R$_2$ = 12.5~m) produces a Blue Jet frequency of 0.72~BJ per minute and 7.6~Tg~N$_2$O-N~yr$^{-1}$ (CTH LPC-TROP LOW R$_2$). \nThe predicted production of NO in the so-called realistic cases is about two orders of magnitude lower that the production of NO by lightning. The global production of NO by Blue Jets is then negligible. The global production of O is also negligible.\n\n\nLet us now estimate the transient chemical impact of Blue Jets in the atmosphere over one year. For this purpose, we calculate the global annual average vertical profile of some chemical species obtained from the simulations of Blue Jets and compare them with the profiles produced in the simulations without Blue Jets. We plot the obtained results with the cases whose predicted Blue Jet frequency is close to the maximum value estimated from the TLE occurrence reported by ISUAL (IS-TROP UP R$_1$ and R$_2$; and LPC-TROP LOW R$_1$ and R$_2$, respectively) in figure~\\ref{fig:bj_price}, together with the percentage of change at each altitude between simulations with and without Blue Jets (relative enhancement). The last three plots of figure~\\ref{fig:bj_price} correspond to the vertical production rate of NO, N$_2$O and O by both Blue Jets and lightning. Figure~\\ref{fig:bj_price} shows results for the CTH lightning parameterization, while the supplementary material shows figures collecting results with other lightning parameterizations (CP, CPCAPE, MFLUX, ICEFLUX).\n\n\nIt is worth analyzing the obtained chemical impact for each considered species. The most remarkable chemical impact of Blue Jets are the enhancements in the densities of NO$_x$ and N$_2$O at altitudes between 10~km and 30~km. Most of the simulations producing a realistic Blue Jet frequency and imposing R$_2$ = 12.5~m (IS-TROP UP R$_2$ and LPC-TROP LOW R$_2$) predict maximum density increases of NO$_x$ and N$_2$O of 30~\\% and 5~\\%, respectively. Simulations that produce the lowest possible Blue Jet frequency (IS-TROP LOW R$_1$ and R$_2$) and simulations producing a realistic Blue Jet frequency (IS-TROP UP R$_1$ and LPC-TROP LOW R$_1$) and imposing R$_1$ = 2.5~m predict a negligible influence of Blue Jets in the global amount of NO$_x$ and N$_2$O. \n\nVertical profiles of other species can also be influenced by the inclusion of Blue Jets in the global atmospheric chemistry. Blue Jets could produce a decrease in the upper tropospheric density of OH and HO$_2$ of about 5~\\% and 20~\\%, respectively. The injected NO molecules would led to an increase in OH and a reduction of HO$_2$ by the process NO + HO$_2$ $\\rightarrow$ NO$_2$ + OH \\citep{murray2013interannual}. However, the conversion of NO into NO$_2$ can also contribute to a decrease in the concentration of OH, specially at lower altitudes by the process NO$_2$ + OH + M $\\rightarrow$ HNO$_3$ + M, where M represents air molecules (N$_2$ + O$_2$). According to our results, the concentration of HNO$_3$ and SO$_2$ could increase about 20~\\%. HNO$_3$ and SO$_2$ molecules can directly contribute to the production of acid rain \\citep{seinfeld2016atmospheric}. The density profile of CO can exhibit both relative increases and decreases at different altitudes, as its gains and losses mechanisms depend on the concentration of OH according to the process CO + OH $\\rightarrow$ CO$_2$ + H \\citep{murray2013interannual}.\nThe global density profiles of other species, such as O$_3$ and O, are not significantly influenced by Blue Jets.\n\n\nThe first panel of figure~\\ref{fig:bj_column_chem_enhancement_cases} shows the annual average total column density of N$_2$O after a WACCM4 simulation of 1~year using the lightning parameterization by \\cite{Price1992\/JGR} without Blue Jets. The rest of the panels in figure~\\ref{fig:bj_column_chem_enhancement_cases} show the annual average total column density difference of N$_2$O between two simulations of 1~year with and without Blue Jets using different lightning parameterizations. The geographical distribution of N$_2$O changes is directly linked to the adopted lightning parameterization (see figure~\\ref{fig:bj_cases_3}), with strong increases in N$_2$O in the tropics and\/or mid-latitudes in relation to a local stronger Blue Jet occurrence in those regions. Interestingly, all lightning parameterizations produce an enhancement in the concentration of N$_2$O near the northern high latitude and polar regions. Given the limited amount of Blue Jet simulated to occur at high latitude, N$_2$O is likely increased by wave-driven transport and mixing from lower latitudes in the extratropical upper troposphere-lowermost stratosphere on relatively fast timescales (see e.g. \\cite{holton1995stratosphere}). On longer timescales (see e.g. the 5 and 10 years cases presented below), increases in N$_2$O at high latitude can occur through poleward and downward adiabatic transport of tropical air. Similar effects are produced also in simulations of LNO$_x$ by \\cite{grewe2009impact} with high impact of LNO$_x$ to changes at high latitude.\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{bj_price.pdf}\n\\caption{\\label{fig:bj_price}\nSolid lines correspond to annual global average density of some species after a WACCM4 simulation of 1~year including Blue Jets and using the lightning parameterization CTH \\citep{Price1992\/JGR}. Triangles correspond to the same simulation with lightning but without Blue Jets. Dashed lines represent the percentage difference when Blue Jets are included. The last three subplots in the lower row show the total production rate of NO, N$_2$O and O, respectively by Blue Jets and lightning.\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{bj_column_chem_enhancement_cases.pdf}\n\\caption{\\label{fig:bj_column_chem_enhancement_cases}\nThe top left panel shows the annual average total column density of N$_2$O after a WACCM4 simulation of 1~year using the lightning parameterization by \\cite{Price1992\/JGR} without Blue Jets. The other panels correspond to the variation in the annual average total column density of N$_2$O between two simulations of 1~year with and without Blue Jets using different lightning parameterizations and the realistic Blue Jet parameterization ``LPC-TROP UP R$_1$\" . \n}\n\\end{figure}\n\n\\subsubsection{Response close to equilibrium}\n\\label{sec:chem10}\n\nThe analysis of the global chemical impact of Blue Jets presented in the previous section is based on a simulation of one year. As we pointed out in section~\\ref{sec:models}, the lifetime of N$_2$O is of the order of a century, while the time scale of the overturning circulation is about 5~years.\n\nWe select two of the previously identified realistic cases in terms of Blue Jet frequency (LPC-TROP LOW R$_1$ and R$_2$) and extend the CTH-based simulations up to five and ten years. We also extend the control case without Blue Jets up to five and ten years. This approach allows us to see the global distribution after the injected species have been transported. It is important to emphasize that a simulation of more than 100~years would be necessary to reach the complete chemical equilibrium. However, such a long simulation is out of the scope of this paper. We plot on figures~\\ref{fig:bj_price_2004} and~\\ref{fig:bj_price_2009} the atmospheric chemical influence of Blue Jets annual averaged for the fifth and the tenth year of simulation, respectively. The chemical influence of Blue Jets is a factor of two larger in the five year simulation (see figure~\\ref{fig:bj_price_2004}) than in the simulation of one year (figure~\\ref{fig:bj_price}). Hence, we cannot assume that the atmosphere has already reached an equilibrium after including Blue Jets. However, the chemical influence of Blue Jets as shown in the 10 year simulation (see figure~\\ref{fig:bj_price_2009}) is quite similar to the one obtained in the simulation of five years, indicating that a simulation of ten years may be sufficient to estimate the global chemical impact of Blue Jets despite the 100~year lifetime of N$_2$O. After a simulation of ten years, the density enhancements and decreases obtained in the previous section (one year simulations) are increased by a factor of two. The increase in the tropospheric density of HNO$_3$ suggests that Blue Jets could also have a direct influence in the acidity of rainwater.\n\nLet us now investigate the geographical chemical impact of Blue Jets resulting from a 10 year simulation. We plot in figure~\\ref{fig:bj_column_chem_absolute} the annual average total column density of some species after simulating a decade using the realistic BJ parameterization ``LPC-TROP LOW R$_2$\". The differences with respect to a simulation without Blue Jets are plotted in figure~\\ref{fig:bj_column_chem_enhancement} (total column density) and in figure~\\ref{fig:lat} (longitudinally averaged vertical profile of N$_2$O and O$_3$). The maximum influence in the density of N$_2$O and HNO$_3$ is concentrated near the North Pole, as can be seen in figures~\\ref{fig:bj_column_chem_enhancement} and~\\ref{fig:lat}. \nFigure~\\ref{fig:lat} also shows a slight depletion of about 5 \\% in the column density of O$_3$ above 30~km near the Equator. Although the total column density of O$_3$ in polar regions is not significantly affected by Blue Jets (see figure~\\ref{fig:bj_column_chem_enhancement}). Figure~\\ref{fig:lat} shows that there is an increase of O$_3$ below 18~km of altitude at all latitudes and a decrease above 20~km of altitude of about~5 \\%. \nSome other species show differences that are distributed through mid latitudes, especially around points of maximum Blue Jet occurrence rates. \n\n\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{bj_price_2004.pdf}\n\\caption{\\label{fig:bj_price_2004}\nSolid lines correspond to annual global average density of some species after a WACCM4 simulation of 5~years including Blue Jets and using the lightning parameterization CTH \\citep{Price1992\/JGR}. Triangles correspond to the same simulation with lightning but without Blue Jets. Dashed lines represent the percentage variation with respect to a similar simulation without Blue Jets. The last three subplots in the lower row show the total production rate of NO, N$_2$O and O by lightning and Blue Jets. Note that the horizontal upper scale of figures~\\ref{fig:bj_price}, \\ref{fig:bj_price_2004} and \\ref{fig:bj_price_2009} are different.\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{bj_price_2009.pdf}\n\\caption{\\label{fig:bj_price_2009}\nSolid lines correspond to annual global average density of some species after a WACCM4 simulation of 10~year including Blue Jets and using the lightning parameterization CTH \\citep{Price1992\/JGR}. Triangles correspond to the same simulation with lightning but without Blue Jets. Dashed lines represent the percentage variation with respect to a similar simulation without Blue Jets. The last three subplots in the lower row show the total production rate of NO, N$_2$O and O by lightning and Blue Jets. Note that the horizontal upper scale of figures~\\ref{fig:bj_price}, \\ref{fig:bj_price_2004} and \\ref{fig:bj_price_2009} are different.\n}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{bj_column_chem_absolute.pdf}\n\\caption{\\label{fig:bj_column_chem_absolute}\nAnnual average total column density of some chemical species after a WACCM4 simulation of 10~years including Blue Jets. These subplots have been calculated using the lightning parameterization based on the cloud-top height CTH \\citep{Price1992\/JGR} and the Blue Jets parameterization denoted as ``LPC-TROP LOW R$_2$\".\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{bj_column_chem_enhancement}\n\\caption{\\label{fig:bj_column_chem_enhancement}\nDifferences in the annual average total column density of some chemical species between two simulations of 10~years with (as in figure~\\ref{fig:bj_column_chem_absolute}) and without Blue Jets. Positive values correspond to enhancement in densities due to Blue Jets, while negative variations represent density decrease produced by Blue Jets. \n}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{lat.pdf}\n\\caption{\\label{fig:lat}\nLatitude-altitude distribution of the differences in the annual average density profile of N$_2$O and O$_3$ between two simulations of 10~years with (as in figure~\\ref{fig:bj_column_chem_absolute}) and without Blue Jets. These variations are longitudinally average.\n}\n\\end{figure}\n\n\nThe obtained density profile of N$_2$O can be compared with measurements of Aura-MLS and MIPAS \\citep{Plieninger2016\/ACP} to determine how well the obtained response corresponds with observations. As detailed by \\cite{Plieninger2016\/ACP}, Aura-MLS and MIPAS (operating at reduced resolution mode) measured the density profile of N$_2$O over a wide range of latitudes. Figure~7 of \\cite{Plieninger2016\/ACP} shows the reported N$_2$O and their error bars from Aura-MLS and MIPAS. We plot these profiles together with the N$_2$O obtained after 10 year simulations with Blue Jets in figure~\\ref{fig:bj_price_2009_N2O}. It can be seen that the equilibrium response of WACCM4 including Blue Jets produce a global average N$_2$O profile that falls in the range reported by Aura-MLS and MIPAS for the cases ``LPC-TROP LOW R$_1$ and R$_2$\".\n\n\\begin{figure}\n\\includegraphics[width=0.6\\columnwidth]{bj_price_2009_N2O.pdf}\n\\caption{\\label{fig:bj_price_2009_N2O}\nComparison between the global average profile of N$_2$O obtained by Aura-MLS and MIPAS \\citep{Plieninger2016\/ACP} and the global average profile of N$_2$O after 10 year simulation with and without Blue Jets in WACCM4 (second subplot of the first row of figure~\\ref{fig:bj_price_2009}) shown here with red, purple and superimposed green solid lines (absolute values) and dashed lines (percentage of change). Blue and yellow solid lines correspond to the lower and upper total N$_2$O density reported by Aura-MLS and MIPAS according to the error bars shown in figure 7 of \\cite{Plieninger2016\/ACP}. Blue and yellow dashed lines are the percentage of difference between the lower and upper total N$_2$O density reported by Aura-MLS and MIPAS and the N$_2$O profile of a WACCM4 simulation without Blue Jets.\n}\n\\end{figure}\n\n\n\n\\normalsize\n\n\\section{Summary and conclusions}\n\n\nWe have introduced for the first time Blue Jets in an atmospheric global circulation model. The Blue Jet parameterization presented here is a step further in the coupling between local and global models of atmospheric electricity phenomena. \nPrevious local models of Blue Jets predicted an important local enhancement of N$_2$O and NO$_x$ molecules between 18~km and 38~km of altitude, as well as a depletion of O$_3$ \\citep{Winkler2015\/JASTP}. The significant local chemical influence of Blue Jets suggests that their global chemical influence could be non-negligible. \n\nIn this work, we have developed two different global parameterizations of Blue Jets. The first parameterizations (IS-TROP LOW and UP) is based on the ratio between lightning and TLE occurrence rate as reported by ISUAL and introduces a physical-based geographical dependence for the occurrence of Blue Jets. These parameterizations link the occurrence of TLEs with the altitude of the cloud top. It imposes the condition that the top of the thunderclouds must be near the tropopause in order to favor the inception of Blue Jets. Finally, the second Blue Jet parameterizations (LCP-TROP LOW and UP) are based on the observational evidences pointing to a close relationship between strong lightning discharges and Blue Jets in thunderstorms. We have obtained a good agreement between the TLE occurrence rate reported by ISUAL and the predicted ones by the Blue Jet parameterizations introduced in WACCM4 except with LCP-TROP UP. \n\nThe implementation of these Blue Jet parameterizations in WACCM4 has allowed us to estimate their global chemical influence in the atmosphere. We have made several assumptions about the geometry of single Blue Jets in order to couple the local chemical model of \\cite{Winkler2015\/JASTP} with the global chemistry implemented in WACCM4. Depending on the differences between the obtained N$_2$O profile and the profiles reported by Aura-MLS and MIPAS \\citep{Plieninger2016\/ACP}, we have distinguished between realistic and extreme cases. According to the most realistic cases, Blue Jets would inject between 6.6 $\\times$ 10$^{-4}$~Tg~N$_2$O-N~yr$^{-1}$ and 7.6~Tg~N$_2$O-N~yr$^{-1}$ near 20~km of altitude. The average value 3.8~Tg~N$_2$O-N~yr$^{-1}$ corresponds about 38~\\% of natural N$_2$O sources. In addition, we have obtained that the global production of NO$_x$ by Blue Jets is between 10$^{-5}$~Tg~NO-N~yr$^{-1}$ and 0.14~Tg~NO-N~yr$^{-1}$. The average value 0.07~Tg~NO-N~yr$^{-1}$ is about 1~\\% of natural NO sources, two orders of magnitude below the production of NO$_x$ by lightning on the troposphere. WACCM4 has allowed us to estimate the influence of Blue Jets in other chemical species apart from NO$_x$ and N$_2$O. In particular, we have found that the stratospheric (between 20~km and 40~km) concentration of some species such as OH, HO$_2$, SO$_2$ and HNO$_3$ could also be influenced by Blue Jets. Finally, we have also found that the inclusion of Blue Jets in WACCM4 can account for a maximum decrease of O$_3$ by about 5 \\% between 20~km and 40~km of altitude. \n\n\nThere are several reasons behind the high uncertainty in what we call realistic results. First, there is not a clear convenient global parameterization of lightning to be combined with the proposed Blue Jet parameterizations \\citep{Tost2007\/ACP}. Second, the detailed mechanisms behind the production of Blue Jets are still poorly described, which makes it difficult to build global parameterization for Blue Jets. Finally, the complex chemistry taking place in the high temperature leader-phase of Blue Jets together with the uncertainties in their electrodynamical radius imply an important uncertainty of the local chemical influence of Blue Jets. All in all, we consider this work as a first approximation to the understanding of the influence of Blue Jets on the global atmospheric chemistry.\n\n\\clearpage\n\n\\small\n\n\\begin{longtable}{|c|c|c|c|c|}\n\n\\hline \\multicolumn{1}{|c}{\\textbf{L-BJ parameterizations}} & \\multicolumn{1}{|c}{\\textbf{BJ frequency [min$^{-1}$]}} & \\multicolumn{1}{|c}{\\textbf{Tg~NO-N~yr$^{-1}$}} & \\multicolumn{1}{|c}{\\textbf{Tg~N$_2$O-N~yr$^{-1}$}} & \\multicolumn{1}{|c|}{\\textbf{Tg~O~yr$^{-1}$}} \\\\\n\n\\endfirsthead\n\n\\multicolumn{1}{|c}{\\textbf{L-BJ parameterization}} & \\multicolumn{1}{|c}{\\textbf{BJ frequency [min$^{-1}$]}} & \\multicolumn{1}{|c}{\\textbf{Tg~NO-N~yr$^{-1}$}} & \\multicolumn{1}{|c}{\\textbf{Tg~N$_2$O-N~yr$^{-1}$}} & \\multicolumn{1}{|c|}{\\textbf{Tg~O~yr$^{-1}$}} \\\\\n\\hline\n\\endhead\n\n\n\n\\hline\n\nCTH IS-TROP UP R$_1$ & 0.9 & 6 $\\times$ 10$^{-3}$ & 0.42 & 3 $\\times$ 10$^{-7}$ \\\\ \n\\hline\nCTH IS-TROP UP R$_2$ & 0.9 & 0.16 & 10.4 & 7 $\\times$ 10$^{-6}$ \\\\ \n\\hline\nCTH IS-TROP LOW R$_1$ & 9 $\\times$ 10$^{-3}$ & 6 $\\times$ 10$^{-6}$ & 4 $\\times$ 10$^{-3}$ & 3 $\\times$ 10$^{-9}$ \\\\ \n\\hline\nCTH IS-TROP LOW R$_2$ & 9 $\\times$ 10$^{-3}$ & 1.5 $\\times$ 10$^{-3}$ & 0.1 & 8 $\\times$ 10$^{-8}$ \\\\ \n\\hline\nCTH LPC-TROP UP R$_1$ & 7.2 & 5 $\\times$ 10$^{-2}$ & 3.0 & 3 $\\times$ 10$^{-6}$ \\\\ \n\\hline\nCTH LPC-TROP UP R$_2$ & 7.2 & 1.36 & 76.0 & 7 $\\times$ 10$^{-5}$ \\\\ \n\\hline\nCTH LPC-TROP LOW R$_1$ & 0.72 & 5 $\\times$ 10$^{-3}$ & 0.3 & 3 $\\times$ 10$^{-7}$ \\\\ \n\\hline\nCTH LPC-TROP LOW R$_2$ & 0.72 & 0.14 & 7.6 & 7 $\\times$ 10$^{-6}$ \\\\ \n\\hline\n\n\\caption{BJ frequency and production of NO, N$_2$O and O obtained for different one year simulations using BJ parameterizations and the CTH lightning parameterization.} \\label{tab:results} \\\\\n\n\\end{longtable}\n\n\n\n\\section*{Acknowledgement}\n\nThe authors acknowledge helpful discussions with Rolando Garcia, Daniel Marsh, Michael Mills, Charles Bardeen, Douglas Kinnison, Andrew Gettelman, Simone Tilmes, Louisa Emmons and Heidi Huntrieser. This work was supported by the Spanish Ministry of Science and Innovation, MINECO under projects ESP2015-69909-C5-2-R and ESP2017-86263-C4-4-R and by the EU through the H2020 Science and Innovation with Thunderstorms (SAINT) project (Ref. 722337) and the FEDER program. The National Center for Atmospheric Research is sponsored by the National Science Foundation. FJPI acknowledges a PhD research contract, code BES-2014-069567. FJGV acknowledges support from the Spanish Ministry of Education and Culture under the Salvador de Madariaga program PRX17\/00078. Data and codes presented here are available from figshare repository at bit.ly\/WACCMBJ.\n\n\\newcommand{\\pra}{Phys. Rev. A} \n\\newcommand{\\jgr}{J. Geoph. Res. } \n\\newcommand{\\jcp}{J. Chem. Phys. } \n\\newcommand{\\ssr}{Space Sci. Rev.} \n\\newcommand{\\planss}{Plan. Spac. Sci.} \n\\newcommand{\\pre}{Phys. Rev. E} \n\\newcommand{\\nat}{Nature} \n\\newcommand{\\icarus}{Icarus} \n\\newcommand{\\ndash}{-} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCellularity is a concept due to Graham and Lehrer ~\\cite{Graham-Lehrer-cellular} that is useful for studying non--semisimple specializations of certain algebras such as Hecke algebras, $q$--Schur algebras, etc. A number of important examples of cellular algebras, including the Hecke algebras of type $A$ and the Birman--Wenzl--Murakami (BMW) algebras, actually occur in towers $A_0 \\subseteq A_1 \\subseteq A_2 \\subseteq \\dots$ with coherent cellular structures. Coherence means that the cellular structures are well--behaved with respect to induction and restriction.\n\nThis paper establishes a framework for proving cellularity of towers of algebras $(A_n)_{n \\ge 0}$ that are obtained by repeated Jones basic constructions from a coherent tower of cellular algebras $(Q_n)_{n \\ge 0}$. \n\nExamples that fit in our framework include: Temperley-Lieb algebras, Brauer algebras, walled Brauer algebras, Birman--Wenzl--Murakami (BMW) algebras, cyclotomic BMW algebras, partition algebras, and contour algebras.\nWe give a uniform proof of cellularity for all of these algebras. \n\nWe should alert the reader that we use a definition of cellular algebras that is slightly weaker than the original definition of Graham and Lehrer. The two definitions are equivalent in case $2$ is invertible in the ground ring, and we know of no consequence of cellularity that would not also hold with the weaker definition; in particular, all results of Graham and Lehrer ~\\cite{Graham-Lehrer-cellular} go through with the modified definition. See Section \\ref{subsection: cellularity} for details. Our contention is that the relaxed definition is in fact superior, as it allows one to deal more naturally with extensions of cellular algebras. \nFor this reason, we have retained the terminology ``cellularity\" for our weaker definition, rather than inventing some new terminology such as ``weak cellularity.\" \n\n Once we have proved our abstract result (Theorem \\ref\n{main theorem}), it is generally very easy to check that each example fits our framework, and thus that the tower $(A_n)_{n\\ge 0}$ in the example is a coherent tower of cellular algebras. What we need is, for the most part, already in the literature, or completely elementary. The application of our method to the cyclotomic BMW algebras depends on a very recent result of Mathas regarding induced modules of cyclotomic Hecke algebras ~\\cite{mathas-2009}. \n\n\n\nFor most of our examples, cellularity has been established previously (but coherence of the cellular structures is a new result). Many of the existing proofs of cellularity for these algebras follow the pattern made explicit by Xi in his paper on cellularity of the partition algebras ~\\cite{Xi-Partition}. The cellular bases obtained are pieced together from cellular bases of the (quotient) algebras $Q_k$ and bases of certain $R$--modules $V_k$ of tangles or diagrams, where $R$ is the ground ring for $A_n$; a formal method for piecing the parts together is K\\\"onig and Xi's method of ``inflation\" \n~\\cite{KX-Morita}. It is not evident that the resulting ``tangle bases'' yield coherent cellular structures.\nBy contrast, the cellular bases that we produce are indexed by paths on the branching diagram (Bratteli diagram) for the generic semisimple representation theory of the tower $(A_n)_{n \\ge 0}$ over a field, and coherence is built into the construction. \n\nFor example, for the Brauer algebras, the BMW algebras, and the cyclotomic BMW algebras, \nour cellular basis of the $n$--th algebra is indexed by up--down tableaux of length $n$, and may be regarded as\nan analogue of Murphy's cellular basis ~\\cite{murphy-hecke95} for the Hecke algebra, or the basis of Dipper, James and Mathas ~\\cite{dipper-james-mathas} for the cyclotomic Hecke algebras.\n A Murphy type basis for the BMW and Brauer algebras has been constructed by Enyang ~\\cite{Enyang2}, but such a basis for the cyclotomic BMW algebras has not been obtained previously. It would be fairly involved to extend Enyang's method to the cyclotomic case, but our method applies to this case without difficulty. \n\n\nLet us remark on the role played by the generic ground ring for our examples. For each of our examples $(A_n)_{n \\ge 0}$, there is a generic ground ring $R$ such that any specialization $A_n^S$ to a ground ring $S$ is obtained as $A_n^S = A_n^R \\otimes_R S$. Moreover, $R$ is an integral domain, and if $F$ denotes the field of fractions of $R$, then the algebras $(A_n^F)_{n \\ge 0}$ are split semisimple with a known representation theory and branching diagram. It suffices for us to prove that the sequence of algebras defined over the generic ground ring $R$ is a coherent cellular tower, and we find that we can use the structure of the algebras defined over $F$ as a tool to accomplish this.\n\n\n \n \nOur approach is influenced by the work of K\\\"onig and Xi ~\\cite{KX-Morita} as well as by the work of Cox et.~al. on ``towers of recollement\" ~\\cite{cox-towers}. In fact, the idea behind our approach is roughly the following: Each algebra $A_n$ (over the generic ground ring $R$) contains an essential idempotent $e_{n-1}$ with the properties that $e_{n-1} A_n e_{n-1} \\cong A_{n-2}$ and \\break $A_n\/(A_n e_{n-1} A_n) \\cong Q_n$, where $Q_n$ is a cellular algebra. Assuming that $A_{n-2}$ and $A_{n-1}$ are cellular, we show that the\n(generally non--unital) ideal $I_n = A_n e_{n-1} A_n$ is a ``cellular ideal\" in $A_n$ by relating ideals of \n$A_{n-2}$ to ideals of $A_n$ contained in $I_n$. This proof involves a new basis--free characterization of cellularity and also involves showing that $I_n \\cong A_{n-1} \\otimes_{A_{n-2}} A_{n-1}$ as\n$A_{n-1}$ bimodules; thus $I_n$ is a sort of Jones basic construction for the pair $A_{n-2} \\subseteq A_{n-1}$. Since our version of cellularity behaves well under extensions, we can conclude that\n$A_n$ is cellular. Our method is related to ideas introduced by K\\\"onig and Xi in their treatment of \ncellularity and Morita equivalence ~\\cite{KX-Morita}. \n\n\n\n\n\n\n\n Following Cox et.~al.~\\cite{cox-towers}, our approach employs the interaction between induction and restriction functors relating $A_{n-1}$--mod and $A_{n}$--mod, on the one hand, and localization and globalization functions relating $A_n$--mod and $A_{n-2}$--mod, on the other hand. (Write $e = e_{n-1} \\in A_n$. The localization functor $F: A_n\\text{--mod} \\rightarrow e A_n e\\text{--mod} \\cong A_{n-2}\\text{--mod}$ is $F: M \\mapsto e M$. The globalization function $G: A_{n-2}\\text{--mod} \\cong e A_n e\\text{--mod} \\rightarrow A_n\\text{--mod}$ is $G: N \\mapsto A_n e \\otimes_{e A_n e} N$.) \n \n Our framework and that of Cox et.~al. dovetail nicely; in fact, our main result (Theorem ~\\ref{main theorem}) says that if \n $(A_n)$, $(Q_n)$ are two sequences of algebras satisfying our framework axioms, then $(A_n)$ satisfies a cellular version of the axioms for towers of recollement; see ~\\cite{cox-walled-brauer} for a discussion of cellularity and towers of recollement.\n\n\nAlthough our techniques do not seem to be adaptable to proving ``strict\" cellularity in the sense of \n~\\cite{Graham-Lehrer-cellular}, by combining our results with previous proofs of ``strict\" cellularity for our examples, we can show the existence of ``strictly\" cellular Murphy type bases, i.e. bases indexed by\npaths on the generic branching diagram for the sequence of algebras $(A_n)_{n \\ge 0}$. We will indicate how this can be done for the cyclotomic BMW algebras; other examples are similar. \n\nSeveral other general frameworks have been proposed for cellularity which also successfully encompass many of our examples; see ~\\cite{KX-Morita, green-martin-tabular, wilcox-cellular}.\n\nIn a companion paper ~\\cite{GG2}, we refine the framework of this paper to take into account the role played by Jucys--Murphy elements. At the same time, we modify Andrew Mathas's theory \\cite{mathas-seminormal} of cellular algebras with Jucys--Murphy elements to take into account coherent sequences of such algebras.\n\n\\medskip\n\\noindent{\\bf Acknowledgement.} Part of this work was done while both authors were visiting MSRI in 2008. We are grateful to the organizers of the program in Combinatorial Representation Theory and to the staff at MSRI for a pleasant and stimulating visit. We thank the referees for helpful suggestions which resulted in several improvements. \n\n\\hfill \\newpage\n\n\\section{Preliminaries}\n\n \n\\subsection{Algebras with involution}\nLet $R$ be a commutative ring with identity. In the following, assume $A$ is an $R$--algebra with an involution $i$ (that is, an $R$--linear algebra anti--automorphism of $A$ with \\def\\id{{\\rm id}} $i^2 = \\id$). \n\nIf $M$ is a left $A$--module, we define a right $A$--module $i(M)$ as follows. As a set, $i(M)$ is a copy of $M$, with elements marked with the symbol $i$, $i(M) = \\{i(m) : m \\in M\\}$. The $R$--module structure of $i(M)$ is given by $i(m_1) + i(m_2) = i(m_1 + m_2)$, and $r i(m) = i(r m)$. Finally, the right $A$--module structure is defined by\n$i(m) a = i( (i(a) m)$. If $\\alpha : M \\to N$ is a homomorphism of left $A$--modules, define $i(\\alpha) : i(M) \\to i(N)$ by $i(\\alpha)(i(m)) = i(\\alpha(m))$. Then $i : A\\text{--mod} \\to \\text{mod--}A$ is a functor. \nFor any fixed $M$, \n$i : M \\to i(M)$ given by $m \\mapsto i(m)$ is, by definition, an isomorphism of $R$--modules.\n\nIf $\\Delta$ is a left ideal in $A$, we have two possible meanings for $i : \\Delta \\to i(\\Delta)$, namely the restriction to $\\Delta$ of the involution $i$, whose image is a right ideal in $A$, or the application of the functor $i$. However, there is no problem with this, as the right $A$--module obtained by applying the functor $i$ can be identified with the\nright ideal $i(\\Delta)$.\n \n\n\n The same construction gives a map from right $A$--modules to left $A$--modules. Moreover, if \n$A$ and $B$ are $R$--algebras with involutions $i_A$ and $i_B$, and $M$ is an $A$--$B$--bimodule, then\n$i(M)$, defined as above as an $R$--module has the structure of a $B$--$A$--bimodule with\n$b\\, i(m) a =i( i_A(a) m \\, i_B(b))$. \nNote that $i\\circ i(M)$ is naturally isomorphic to $M$, so $i$ is an equivalence between the categories of \n$A$--$B$--bimodules and the category of $B$--$A$--bimodules.\n\n\\begin{lemma} \\label{lemma; involutions and tensor products of bimodules}\n Suppose $A$, $B$, and $C$ are $R$--algebras with involutions $i_A$, $i_B$, and $i_C$. Let\n$_B P_A$ and $_A Q_C$ be bimodules. Then\n$$\ni(P \\otimes_A Q) \\cong i(Q) \\otimes_A i(P),\n$$\nas $C$--$B$--bimodules.\n\\end{lemma}\n\n\\begin{proof} It is straightforward to check that there is a well defined $R$--linear isomorphism\n$f_0 : P \\otimes_A Q \\to i(Q) \\otimes_A i(P)$ such that $f_0(p \\otimes q) = i(q) \\otimes i(p)$. Then\n$$f = f_0 \\circ i^{-1} : i(P\\otimes_A Q) \\to i(Q) \\otimes_A i(P)$$ is an $R$--linear isomorphism. Finally, one can check that $f$ is a $C$--$B$--bimodule map.\n\\end{proof}\n \n\n\n\\begin{remark} \\label{remark: i applied to tensor product}\nNote that if we identify $i(P \\otimes_A Q)$ with $i(Q) \\otimes_A i(P)$ via $f$, then we have the \nformula $i(p \\otimes q) = i(q) \\otimes i(p)$. In particular, let $M$ be a $B$--$A$--bimodule, and identify\n$i\\circ i(M)$ with $M$, and $i(M\\otimes_A i(M))$ with $i\\circ i(M) \\otimes_A i(M) = M\\otimes_A i(M)$.\nThen we have the formula $i(x \\otimes i(y)) = y \\otimes i(x)$. We will use these identifications throughout the paper.\n\\end{remark}\n \n \n\\subsection{Cellularity} \\label{subsection: cellularity}\n\nWe recall the definition of {\\em cellularity} from ~\\cite{Graham-Lehrer-cellular}; see also\n~\\cite{Mathas-book}. The version of the definition given here is slightly weaker than the original definition in ~\\cite{Graham-Lehrer-cellular}; we justify this below.\n\n\n\\begin{definition} \\label{gl cell} Let $R$ be an integral domain and $A$ a unital $R$--algebra. A {\\em cell datum} for $A$ consists of an algebra involution $i$ of $A$; a partially ordered set $(\\Lambda, \\ge)$ and \nfor each $\\lambda \\in \\Lambda$ a set $\\mathcal T(\\lambda)$; and a subset $\n\\mathcal C = \\{ c_{s, t}^\\lambda : \\lambda \\in \\Lambda \\text{ and } s, t \\in \\mathcal T(\\lambda)\\} \\subseteq A$; \nwith the following properties:\n\\begin{enumerate}\n\\item $\\mathcal C$ is an $R$--basis of $A$.\n\\item \\label{mult rule} For each $\\lambda \\in \\Lambda$, let $\\breve A^\\lambda$ be the span of the $c_{s, t}^\\mu$ with\n$\\mu > \\lambda$. Given $\\lambda \\in \\Lambda$, $s \\in \\mathcal T(\\lambda)$, and $a \\in A$, there exist coefficients \n$r_v^s( a) \\in R$ such that for all $t \\in \\mathcal T(\\lambda)$:\n$$\na c_{s, t}^\\lambda \\equiv \\sum_v r_v^s(a) c_{v, t}^\\lambda \\mod \\breve A^\\lambda.\n$$\n\\item $i(c_{s, t}^\\lambda) \\equiv c_{t, s}^\\lambda \\mod \\breve A^\\lambda$ for all $\\lambda\\in \\Lambda$ and, $s, t \\in \\mathcal T(\\lambda)$.\n\n\\end{enumerate}\n$A$ is said to be a {\\em cellular algebra} if it has a cell datum. \n\\end{definition}\n\n\n\nFor brevity, we will write that $(\\mathcal C, \\Lambda)$ is a cellular basis of $A$. \n\n\n\n\n\n\\begin{remark} \\mbox{} \\label{remark: on definition of cellularity}\n\\begin{enumerate}\n\\item The original definition in ~\\cite{Graham-Lehrer-cellular} requires that $i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda $ for all $\\lambda, s, t$. However, one can check that the results of \\cite{Graham-Lehrer-cellular} remain valid with our weaker axiom.\nIn fact, we are not aware of any consequence of cellularity that would not also hold with our weaker definition. \n\\item In case $2 \\in R$ is invertible, our definition is equivalent to the original. Here is the proof: Suppose that $2$ is invertible in the ground ring and that\n$\\{c_{s, t}^\\lambda\\}$ is a cellular basis in the sense of Definition \\ref{gl cell}. We want to produce a new cellular basis $\\{a_{s, t}^\\lambda\\}$ satisfying the strict equality $i(a_{s, t}^\\lambda) = a_{t, s}^\\lambda $ for all $\\lambda, s, t$. By hypothesis, for each $\\lambda, s, t$ there is a unique $f(\\lambda, s, t) \\in \\breve A^\\lambda$ such that $i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda + f(\\lambda, s, t)$. One easily checks that $i(f(\\lambda, s, t)) = -f(\\lambda, t, s)$. Declare\n$a_{s, t}^\\lambda = c_{s, t}^\\lambda + (1\/2) f(\\lambda, t, s)$ for all $\\lambda, s, t$. \nThen $\\{a_{s, t}^\\lambda\\}$ has the desired properties. \n\\end{enumerate}\n\\end{remark}\n\nWe recall some basic structures related to cellularity, see ~\\cite{Graham-Lehrer-cellular}.\nGiven $\\lambda\\in\\Lambda$. Let $A^\\lambda$ denote the span of the $c_{s,t}^{\\mu}$ with $\\mu \\geq \\lambda$. It follows that both $A^\\lambda$ and $\\breve A^\\lambda$ (defined above) are $i$--invariant two sided ideals of $A$.\nIf $t \\in \\mathcal T(\\lambda)$, define $C_t^\\lambda$ to be the $R$-submodule of $A^\\lambda\/\\breve A^\\la$ with basis $\\{ c_{s,t}^\\lambda + \\breve A^\\la : s \\in \\mathcal T(\\lambda) \\}$. Then $C_t^\\lambda$ is a left $A$-module by Definition \\ref{gl cell} (\\ref{mult rule}). Furthermore, the action of $A$ on $C_t^\\lambda$ is independent of $t$, i.e $C_u^{\\lambda}\\cong C_t^{\\lambda}$ for any $u,t \\in \\mathcal T(\\lambda)$. The {\\em left cell module} $\\Delta^\\lambda$ \nis defined as follows: as an $R$--module, $\\Delta^\\lambda$ is free with basis $\\{c_s^\\lambda$ : $s \\in \\mathcal T(\\lambda)\\}$; for each $a \\in A$, the action of $a$ on $\\Delta^\\lambda$ is defined by $ ac_s^\\lambda=\\sum_v r_v^s(a) c_v^\\lambda$ where $r_v^s(a)$ is as in Definition \\ref{gl cell} (\\ref{mult rule}). Then $\\Delta^\\lambda \\cong C_t^\\lambda$, for any $t \\in \\mathcal T(\\lambda)$.\n For all $s,t \\in \\mathcal T(\\lambda)$, we have a canonical $A-A$--bimodule isomorphism $\\alpha : A^\\lambda\/\\breve A^\\la \\rightarrow \\Delta^\\lambda \\otimes_R i(\\Delta^\\lambda)$ defined by $\\alpha(c_{s,t}^{\\lambda}+\\breve A^\\la)=c_s^\\lambda \\otimes_R i(c_t^\\lambda)$. Moreover, we have\n $i \\circ \\alpha = \\alpha \\circ i$, using Remark \\ref{remark: i applied to tensor product} and point (3) of Definition \\ref{gl cell}.\n\n\n\\begin{definition} Suppose $A$ is a unital $R$--algebra with involution $i$, and $J$ is an $i$--invariant ideal; then we have an induced algebra involution $i$ on $A\/J$. \nLet us say that $J$ is a {\\em cellular ideal} in $A$ if it satisfies the axioms for a cellular algebra (except for being unital) with cellular basis \n$$ \\{ c_{s, t}^\\lambda : \\lambda \\in \\Lambda_J \\text{ and } s, t \\in \\mathcal T(\\lambda)\\} \\subseteq J$$\nand we have, as in point (2) of the definition of cellularity, \n$$\na c_{s, t}^\\lambda \\equiv \\sum_v r_v^s(a) c_{v, t}^\\lambda \\mod \\breve J^\\lambda\n$$\nnot only for $a \\in J$ but also for $a \\in A$.\n\\end{definition}\n\n\n\\begin{remark} \\label{remark on extensions of cellular algebras} (On extensions of cellular algebras.) \nIf $J$ is a cellular ideal in $A$, and \n$H = A\/J$ is cellular (with respect to the involution induced from the involution on $A$), then $A$ is cellular. In fact, let $(\\Lambda_J, \\ge)$ be the partially ordered set in the cell datum for $J$ and $\\mathcal C_J$ the cellular basis.\nLet $(\\Lambda_H, \\ge)$ be the partially ordered set in the cell datum for $H$ and $\\{\\bar h_{u, v}^\\mu\\}$ the cellular basis. Then $A$ has a cell datum with partially ordered set $\\Lambda = \\Lambda_J \\cup \\Lambda_H$, with partial order agreeing with the original partial orders on $\\Lambda_J$ and on $\\Lambda_H$ and with $\\lambda > \\mu$ if $\\lambda \\in \\Lambda_J$ and $\\mu \\in \\Lambda_H$.\nA cellular basis of $A$ is $\\mathcal C_J \\cup \\{h_{s, t}^\\mu\\}$, where $h_{s, t}^\\mu$ is any lift of $\\bar h_{s, t}^\\mu$.\n\n\nWith the original definition of ~\\cite{Graham-Lehrer-cellular}, the assertions of this remark would be valid only if the ideal\n$J$ has an $i$--invariant $R$--module complement in $A$.\nThe ease of handling extensions is our motivation for using the weaker definition of cellularity.\n\\end{remark}\n\n\n\\subsection{Basis--free formulations of cellularity}\nK\\\"onig and Xi have given a basis--free definition of cellularity ~\\cite{KX-Morita}. We describe a slight weakening of their definition, which corresponds exactly to our weaker form of Graham--Lehrer cellularity\n\n\\begin{definition}[K\\\"onig and Xi] \\label{defKX} Let $R$ be an integral domain and $A$ a unital $R$-algebra\nwith involution $i$. An $i$--invariant two sided ideal $J$ in $A$ is called a {\\em split ideal} if, and only if,\nthere exists a left ideal $\\Delta$ of $A$ contained in $J$, with $\\Delta$ finitely generated and free over $R$, and \nthere is an isomorphism of $A$--$A$--bimodules $\\alpha : J \\rightarrow \\Delta \\otimes_R i(\\Delta)$ making the following diagram commute:\n\\begin{diagram}\nJ\t&\\rTo^{\\alpha}\t&&\\Delta \\otimes_R i(\\Delta)\\\\\n\\dTo_{i} &&& \\dTo_{i}\\\\\nJ\t&\\rTo^{\\alpha}\t&&\\Delta \\otimes_R i(\\Delta)\\\\\n\\end{diagram}\n\n\\ignore{\n$$\\begin{CD}\nJ\t@>\\alpha>>\t\\Delta \\otimes_R i(\\Delta)\\\\\n@VViV\t\t\t@VViV\\\\\nJ\t@>\\alpha>>\t\t\\Delta \\otimes_R i(\\Delta).\n\\end{CD}$$\n}\nA finite chain of $i$--invariant two sided ideals\n$$0=J_0 \\subset J_1 \\subset J_2 \\subset \\cdots \\subset J_n = A$$\nis called a {\\em cell chain} if for each $j$ ($1 \\leq j \\leq n$), the quotient $J_j\/J_{j-1}$ is a split ideal of $A\/J_{j-1}$ (with respect to the involution induced by $i$ on $A\/J$). \n\\end{definition} \n\n\\begin{remark} \\label{remark on Konig Xi definition}\n\\mbox{}\n\\begin{enumerate}\n\\item K\\\"onig and Xi call a split ideal a ``cell ideal.\" We changed the terminology to avoid confusion with other concepts. \n\\item The definition of a cell chain differs from the one given by Konig and Xi in that we dropped the requirement that $J_{j-1}$ have an $i$-invariant $R$-module complement in $J_j$.\n\\end{enumerate}\n\\end{remark}\n\n\n \n\\begin{lemma} Let $R$ be an integral domain and let $A$ be a unital $R$--algebra with involution $i$.\nAn ideal $J$ of $A$ is split if, and only if, there exists a left $A$--module $M$ that is finitely generated and free as an $R$--module, and there exists an\nisomorphism of $A$--$A$--bimodules $\\gamma : J \\rightarrow M \\otimes_R i(M)$ making the following diagram commute:\n\\begin{diagram}\nJ &\\rTo^{\\gamma} &&M \\otimes_R i(M)\\\\\n\\dTo_{i} &&& \\dTo_{i}\\\\\nJ &\\rTo^{\\gamma} &&M \\otimes_R i(M)\\\\\n\\end{diagram}\n\\end{lemma}\n\n\n\n\\begin{proof} If $J$ is split, it clearly satisfies the condition of the lemma. Conversely, suppose the condition of the lemma is satisfied. Fix some element $b_0$ of the basis of $M$ over $R$ and define a left $A$--module map\n$\\beta : M \\to A$ by $\\beta(m) = \\gamma^{-1}(m \\otimes b_0)$. Then $\\beta$ is an isomorphism of $M$ onto a left\nideal $\\Delta$ of $A$ contained in $J$.\n\n\\ignore{\nRegard $M$ as an $A$--$R$--bimodule.\nAs in Remark \\ref{remark: i applied to tensor product}, identify $i\\circ i(M)$ with $M$ and\n$i(M \\otimes_R i(M))$ with $M \\otimes_R i(M)$. Then the map $x \\otimes i(y) \\mapsto y \\otimes i(x)$ is just\n$i : M \\otimes_R i(M) \\to M \\otimes_R i(M)$. The same considerations apply to $\\Delta$ as well.\n}\n\n\nNow we have $\\beta \\otimes i(\\beta) : M \\otimes_R i(M) \\to \\Delta \\otimes_R i(\\Delta)$\nis an isomorphism satisfying $(\\beta \\otimes i(\\beta)) \\circ i = i \\circ (\\beta \\otimes i(\\beta))$. It follows that\n$\\alpha = (\\beta \\otimes i(\\beta)) \\circ \\gamma : J \\to \\Delta \\otimes_R i(\\Delta)$ is an isomorphism of \n$A$--$A$--bimodules satisfying the requirement for a split ideal, namely, $ \\alpha \\circ i = i \\circ \\alpha$.\n\\end{proof}\n \n\n\\begin{lemma}[K\\\"onig and Xi] \\label{lemma: equivalence of GL and KX} Let $A$ be an $R$--algebra with involution.\n$A$ is {cellular} if, and only if, $A$ has a finite cell chain.\n\\end{lemma}\n\n\\begin{proof} We sketch the proof from ~\\cite{KX-structure}, p.\\ 372.\n\nSuppose $A$ has a cell datum with partially ordered set $(\\Lambda, \\ge)$ and cell basis $\\{ c_{s, t}^\\lambda\\}$. \nWrite $\\Lambda$ as a sequence $(\\lambda_1, \\lambda_2, \\dots, \\lambda_n)$, where $\\lambda_1$ is maximal in $\\Lambda$, and, for $1 \\le j < n$, \n$\\lambda_{j+1}$ is maximal in $\\Lambda \\setminus \\{\\lambda_1, \\dots, \\lambda_j\\}$. Then for each $j\\ge 1$, $\\Gamma_j = \\{\\lambda_1, \\dots, \\lambda_j\\}$ is an order ideal in $\\Lambda$. Set $\\Gamma_0 = \\emptyset$.\n Define $A(\\Gamma_j)$ to be the $R$-submodule of $A$ \nspanned by the basis elements $c_{u, v}^\\lambda$, with $\\lambda \\in \\Gamma_j$.\nThen $A(\\Gamma_j)$ is an $i$--invariant two sided ideal in $A$, and\n$$0=A(\\Gamma_0)\\subset A(\\Gamma_1)\\subset\\cdots\\subset A(\\Gamma_{n})=A.$$\nMoreover (see ~\\cite{Graham-Lehrer-cellular}, p. 6), \n$$A(\\Gamma_j)\/A(\\Gamma_{j-1})\\cong\nA^{\\lambda_j}\/\\breve A^{\\lambda_j} \\cong\n\\Delta^{\\lambda_j}\\otimes_Ri(\\Delta^{\\lambda_j}),$$\nand the isomorphism $\\alpha : A(\\Gamma_j)\/A(\\Gamma_{j-1}) \\to \\Delta^{\\lambda_j}\\otimes_Ri(\\Delta^{\\lambda_j})$ satisfies\n$\\alpha \\circ i = i \\circ \\alpha$. Thus $(A(\\Gamma_j))_{1 \\le j \\le n}$ is a cell chain.\n\nConversely, suppose $(J_j)_{0 \\le j \\le n}$ is a cell chain in $A$. Then for each $j \\ge 1$, we have an \n$A$--module $\\Delta_j$ that is finitely generated and free as an $R$--module, and an isomorphism of $A$--$A$--bimodules\n$\\alpha_j : J_j\/J_{j-1} \\to \\Delta_j \\otimes_R i(\\Delta_j)$ satisfying $i \\circ \\alpha_j = \\alpha_j \\circ i$.\n Let $\\{b^j_s : s\\in \\mathcal T(j)\\}$ be an $R$--basis of $\\Delta_j$ and let\n$c_{s, t}^{\\lambda_j}$ be any lift in $J_j$ of $\\alpha_j^{-1}(b^j_s \\otimes i(b^j_t))$. Now take $\\Lambda'$ to be $\\Lambda$ with the order $\\lambda_1 > \\lambda_2 > \\cdots > \\lambda_n$. Let $\\mathcal C = \\{c_{s, t}^{\\lambda_j} : 1\\le j \\le n; \\ s, t \\in \\mathcal T(j)\\}$. Then \n$(\\mathcal C, \\Lambda')$ is a cellular basis of $A$.\n\\end{proof}\n\n\\begin{remark} In the Lemma, $A$ has a cellular basis $\\{c_{s, t}^\\lambda\\}$ with\n$i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda$ if, and only if, $A$ has a finite cell chain $(J_j)$ such that\nfor each $j\\ge 1$, $J_{j-1}$ has an $i$--invariant $R$--module complement in $J_j$. \n\\end{remark}\n\nNote that if we follow the procedure of the proof, starting with a cell datum on $A$ with partially ordered set\n$(\\Lambda, \\ge)$, then the only information that we retain about $\\Lambda$ is that $\\lambda_{j+1}$ is maximal in \n$\\Lambda \\setminus \\Gamma_j$; we cannot recover the partial order on $\\Lambda$ from this. Moreover, if we continue to\nproduce a cellular basis $\\{c_{s, t}^j\\}$ from the cell chain $(A(\\Gamma_j))_{0 \\le j \\le n}$, the result will not necessarily have the properties of a cellular basis with respect to the original partially ordered set $(\\Lambda, \\ge)$.\n\nIn order to prove our main results, we will need a different basis--free formulation of cellularity that allows us to pass back and forth between\nthe formulation of Definition \\ref{gl cell} and the basis--free formulation without losing information about the partially ordered set.\n\n\n\\begin{definition} \\label{definition: cell net}\nLet $A$ be an $R$--algebra with involution $i$. Let $(\\Lambda, \\ge)$ be a finite partially ordered set. For $\\lambda \\in \\Lambda$, let\n $\\Gamma_{\\ge \\lambda}$ denote the order ideal $\\{\\mu : \\mu \\ge \\lambda\\}$ and $\\Gamma_{> \\lambda}$ the order ideal $\\{\\mu : \\mu > \\lambda\\}$.\n \nA {\\em $\\Lambda$--cell net} is a map from the set of order ideals of $\\Lambda$ to the set of $i$--invariant two sided ideals of $A$, \n$\\Gamma \\mapsto A_\\Gamma$, with the following properties:\n\\begin{enumerate}\n\\item $A_\\emptyset = \\{0\\}$. If $\\Gamma_1 \\subseteq \\Gamma_2$, then $A_{\\Gamma_1} \\subseteq A_{\\Gamma_2} $.\n\\item For $\\lambda \\in \\Lambda$, write $A_{\\ge \\lambda} = A_{\\Gamma_{\\ge \\lambda}}$ and $A_{> \\lambda} = A_{\\Gamma_{> \\lambda}}$. Then\n$$A = {\\rm span}\\{A_{\\ge \\mu} : \\mu \\in \\Lambda \\},$$ and for all $\\lambda \\in \\Lambda$, $$A_{> \\lambda} ={\\rm span} \\{ A_{\\ge \\mu} : \\mu > \\lambda\\}.$$\n\\ignore{\n\\item If $\\mu$ is not comparable with $\\lambda_i$ ($1 \\le i \\le s$), then\n$$A_{\\ge \\mu} \\cap {\\rm span}\\{A_{\\ge \\lambda_i} : 1 \\le i \\le s\\} \\subseteq A_{> \\mu}.$$\n}\n\\item For each $\\lambda \\in \\Lambda$, there is an $A$--module $M^\\lambda$, finitely generated and free as an $R$--module, such that whenever $\\Gamma \\subseteq \\Gamma'$ are order ideals of $\\Lambda$, with $\\Gamma' \\setminus \\Gamma = \\{\\lambda\\}$, then there exists an\nisomorphism of $A$--$A$--bimodules $$\\alpha : A_{\\Gamma'}\/A_{\\Gamma} \\to M^\\lambda \\otimes_R i(M^\\lambda),$$ satisfying\n$i \\circ \\alpha = \\alpha \\circ i$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition} \\label{lemma: cell net characterization of cellularity}\n Let $A$ be an $R$--algebra with involution, and let $(\\Lambda, \\ge)$ be a finite partially ordered set.\n Then $A$ has a cell datum with partially ordered set $\\Lambda$ if, and only if, $A$ has a $\\Lambda$--cell net.\n\\end{proposition}\n\n\\begin{proof} Suppose that $A$ has a cell datum with partially ordered set $\\Lambda$ and cell basis\n$\\{c_{s, t}^\\lambda\\}$. For each order ideal $\\Gamma$ of $\\Lambda$, let $A(\\Gamma)$ denote the span of those $c_{s, t}^\\lambda$ with\n$\\lambda \\in \\Gamma$. Then $\\Gamma \\mapsto A(\\Gamma)$ is a $\\Lambda$--cell net.\n\nConversely, suppose that $A$ has a $\\Lambda$--cell net, $\\Gamma \\mapsto A_\\Gamma$. For each $\\lambda \\in \\Lambda$, we have\nan isomorphism of $A$--$A$--bimodules $\\alpha_\\lambda : A_{\\ge \\lambda}\/A_{ > \\lambda} \\to M^\\lambda \\otimes_R i(M^\\lambda)$.\nLet $\\{b_s^\\lambda : s \\in \\mathcal T(\\lambda)\\}$ be an $R$--basis of $M^\\lambda$ and let $c_{s, t}^\\lambda$ be any lift of\n$\\alpha_\\lambda^{-1}(b_s^\\lambda \\otimes i(b_t^\\lambda))$ to $A_{\\ge \\lambda}$. We claim that $$\\mathcal C = \\{c_{s, t}^\\lambda : \\lambda \\in \\Lambda; s, t \\in \\mathcal T(\\lambda)\\}$$ is an $R$--basis of $A$. \n\nLet $A^\\lambda$ be the span of those $c_{s, t}^\\mu$ with $\\mu \\ge \\lambda$ and $\\breve A^\\lambda$ the span of those $c_{s, t}^\\mu$ with $\\mu > \\lambda$. If $\\mu \\ge \\lambda$, then for all $s, t \\in \\mathcal T(\\mu)$, $c_{s, t}^\\mu \\in A_{\\ge \\mu} \\subseteq A_{\\ge \\lambda}$, using point (1) of Definition \\ref{definition: cell net}.\nHence $A^\\lambda \\subseteq A_{\\ge \\lambda}$. Similarly, $\\breve A^\\lambda \\subseteq A_{> \\lambda} $.\n\n\n \nWe claim that \n\\begin{equation} \\label{equation: equality of ideals related to order ideals}\n\\text{for all} \\ \\lambda \\in \\Lambda, \\quad A_{\\ge \\lambda} = A^\\lambda.\n\\end{equation}\nThis is clear if $\\lambda$ is a maximal element of $\\Lambda$. (Note that $A_{> \\lambda} = A_\\emptyset = \\{0\\}$.)\n Now suppose that $\\lambda$ is not maximal and that for all\n $\\mu > \\lambda$, $A_{\\ge \\mu} = A^\\mu$. Then $$A_{> \\lambda} = {\\rm span}\\{A_{\\ge \\mu} : \\mu > \\lambda\\}\n = {\\rm span}\\{A^\\mu : \\mu > \\lambda\\} = \\breve A^\\lambda,$$\n where the first equality comes from (2) of Definition \\ref{definition: cell net} and the second from the induction hypothesis. By definition of $\\{c_{s, t}^\\lambda\\}$, we have $$A_{\\ge \\lambda} = {\\rm span}\\{c_{s, t}^\\lambda\\} + A_{> \\lambda} = {\\rm span}\\{c_{s, t}^\\lambda\\} + \\breve A^\\lambda = A^\\lambda.$$\n Assertion (\\ref{equation: equality of ideals related to order ideals}) now follows by induction. Point (2) of Definition \\ref{definition: cell net} and (\\ref{equation: equality of ideals related to order ideals}) imply that\n $A_{> \\lambda} = \\breve A^\\lambda$ for all $\\lambda \\in \\Lambda$, and that $A = {\\rm span}(\\mathcal C)$.\n \nWe now proceed to establish linear independence of $\\mathcal C$. \nWrite $\\Lambda$ as a sequence $(\\lambda_1, \\lambda_2, \\dots, \\lambda_K)$ with \n$\\lambda_1$ maximal and $\\lambda_{j+1}$ maximal in $\\Lambda \\setminus \\{\\lambda_1, \\dots, \\lambda_j\\}$ for $1 \\le j < K$. Put \n$\\Gamma_j = \\{\\lambda_1, \\dots, \\lambda_j\\}$ for $j \\ge 1$ and $\\Gamma_0 = \\emptyset$. Then $(\\Gamma_j)_{0 \\le j \\le K}$ is a maximal chain of order ideals. Since $\\Gamma_j \\setminus \\Gamma_{j-1} = \\{\\lambda_j\\}$, we have an isomorphism $\\gamma_j : A_{\\Gamma_j}\/A_{\\Gamma_{j-1}} \\to M^{\\lambda_j} \\otimes_R \ni(M^{\\lambda_j})$ with $i \\circ \\gamma_j = \\gamma_j \\circ i$. Thus $(A_{\\Gamma_j})_{0 \\le j \\le K}$ is a cell chain\nin $A$. So by the proof of Lemma \\ref{lemma: equivalence of GL and KX}, $A$ has a cellular basis\n$$\\mathcal B = \\{ b_{s, t}^\\lambda : \\lambda \\in \\Lambda;\\ s, t, \\in \\mathcal T(\\lambda)\\},$$ but with respect to the ``wrong\" partial order on $\\Lambda$. Since $\\mathcal C$ is a spanning set of the same cardinality as the basis\n$\\mathcal B$, it follows that $\\mathcal C$ is linearly independent over $R$, and thus an $R$--basis of\n$A$. \n\n\\ignore{\nWe now proceed to establish linear independence of $\\mathcal C$, making use of condition (3) of Definition \\ref{definition: cell net}.\n Suppose we have a non--trivial linear relation $\\sum_{\\lambda, s, t} r(\\lambda, s, t) c_{s, t}^\\lambda = 0$ with coefficients in $R$. Let $\\mu$ be minimal among those $\\lambda$ such that some $r(\\lambda, s, t)$ is non--zero. Rewrite the linear relation as\n$\\Sigma' + \\Sigma'' + \\Sigma''' = 0$, where $\\Sigma'$ is the sum of those terms with $\\lambda$ not comparable to $\\mu$,\n\\ $\\Sigma''$ is the sum of those terms with $\\lambda > \\mu$, and $\\Sigma'''$ is the sum of those terms with $\\lambda = \\mu$.\nThen $\\Sigma'' + \\Sigma''' \\in A^{\\mu} = A_{\\ge \\mu}$. Hence also $\\Sigma' \\in A_{\\ge \\mu}$. But\n $\\Sigma' $ is also in the span of those $A_{\\ge \\lambda}$ with $\\lambda$ not comparable to $\\mu$. By point (3) of \n Definition \\ref{definition: cell net}, we have $\\Sigma' \\in A_{> \\mu}$. But then $\\Sigma' + \\Sigma'' \\in A_{> \\mu}$. \n Taking the quotient by $A_{> \\mu}$, we get $\\sum_{s, t} r(\\mu, s, t) ( c_{s, t}^\\mu + A_{> \\mu}) = 0$.\n Since the set of $( c_{s, t}^\\mu + A_{> \\mu})$ is a basis of $A_{\\ge \\mu}\/A_{ > \\mu} $, it follows that all the \n coefficients $r(\\mu, s, t)$ are zero, a contradiction. \n }\n \n Because $A_{> \\lambda} = \\breve A^\\lambda$ for all $\\lambda \\in \\Lambda$, it is now easy to see that properties (2) and (3) of Definition \\ref{gl cell} are satisfied by $\\mathcal C$.\n\\end{proof}\n\n\\begin{remark} \\label{remark: conditions for cell net to give strict cellular basis}\n In the Proposition, the following are equivalent:\n\\begin{enumerate}\n\\item $A$ has a cellular basis $\\{c_{s, t}^\\lambda\\}$ with\n$i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda$.\n\\item $A$ has a $\\Lambda$ cell net $\\Gamma \\to A_\\Gamma$ such that for each pair $\\Gamma \\subseteq \\Gamma'$, $A_\\Gamma$ has an $i$--invariant $R$--module complement in $A_{\\Gamma'}$. \n\\item $A$ has a $\\Lambda$ cell net $\\Gamma \\to A_\\Gamma$ such that for each $\\lambda \\in \\Lambda$, \n$A_{> \\lambda}$ has an $i$--invariant $R$--module complement in $A_{\\ge \\lambda}$. \n\\end{enumerate}\nThe implications (1) $\\implies$ (2) $\\implies$ (3) are evident. For (3) $\\implies$ (1), let $B_\\lambda$ denote the $i$--invariant $R$--module complement of $A_{> \\lambda}$ in $A_{\\ge \\lambda}$, and, \nin the 2nd\nparagraph of the proof of the Proposition, let $c_{s, t}^\\lambda$ be the unique lift of\n$\\alpha_\\lambda^{-1}(b_s^\\lambda \\otimes i(b_t^\\lambda))$ in $B_\\lambda$. \n\\end{remark}\n\n\\subsection{Coherent towers of cellular algebras}\n\\begin{definition}\nLet $H_0 \\subseteq H_1 \\subseteq H_2 \\subseteq \\cdots$ be an increasing sequence of cellular algebras, with a common multiplicative identity element, over an integral domain $R$. Let $\\Lambda_n$ denote the partially ordered set in the cell datum for $H_n$. We say that $(H_n)_{n \\ge 0}$ is a {\\em coherent tower of cellular algebras} if the following conditions are satisfied:\n\\begin{enumerate}\n\\item The involutions are consistent; that is, the involution on $H_{n+1}$, restricted to $H_n$, agrees with the involution on $H_n$.\n\\item For each $n\\ge 0$ and for each $\\lambda \\in \\Lambda_n$, the induced module ${\\rm Ind}_{H_n}^{H_{n+1}} (\\Delta^\\lambda)$\nhas a filtration by cell modules of $H_{n+1}$. That is, there is a filtration\n$$\n{\\rm Ind}_{H_n}^{H_{n+1}} (\\Delta^\\lambda) = M_t \\supseteq M_{t-1} \\supseteq \\cdots \\supseteq M_0 = (0)\n$$\nsuch that for each $j\\ge1$, there is a $\\mu_j \\in \\Lambda_{n+1}$ with $M_j\/M_{j-1} \\cong \\Delta^{\\mu_j}$.\n\\item For each $n\\ge 0$ and for each $\\mu \\in \\Lambda_{n+1}$, the restriction ${\\rm Res}_{H_n}^{H_{n+1}} (\\Delta^\\mu)$\nhas a filtration by cell modules of $H_{n}$. That is, there is a filtration\n$$\n{\\rm Res}_{H_n}^{H_{n+1}} (\\Delta^\\mu) = N_s \\supseteq N_{s-1} \\supseteq \\cdots \\supseteq N_0 = (0)\n$$\nsuch that for each $i\\ge1$, there is a $\\lambda_i \\in \\Lambda_{n}$ with $N_j\/N_{j-1} \\cong \\Delta^{\\lambda_i}$.\n\n\\end{enumerate}\n\\end{definition}\n\nThe modification of the definition for a {\\em finite} tower of cellular algebras is obvious.\n\nWe call a filtration as in (2) and (3) a {\\em cell filtration}.\nIn the examples that we study, we will also have {\\em uniqueness of the multiplicities} of the cell modules appearing as subquotients of the cell filtrations, and {\\em Frobenius reciprocity} connecting the multiplicities in the two types of filtrations. We did not include uniqueness of multiplicities and Frobenius reciprocity as requirements in the definition, as they will follow from additional assumptions that we will impose later; see Lemma \\ref{lemma: multiplicities in cell filtrations}.\\footnote{Hemmer and Nakano ~\\cite{Hemmer-Nakano} have obtained remarkable general results about uniqueness of multiplicities in Specht filtrations of modules over Hecke algebras of type A. Hartmann and Paget \\cite{Hartmann-Paget} obtained analogous results for modules over Brauer algebras. The assertions that we require here are much more special, applying only to induced modules of cell modules and restrictions of cell modules. }\n\n\n\\begin{example} \\label{example: Hn coherent tower} {\\em The tower of Hecke algebras of type $A$ is a coherent tower of cellular algebras.}\nLet $R$ be an integral domain and $q$ an invertible element of $R$. Let $H_n(R, q)$ denote the Hecke algebra of type $A$ generated by elements $T_1, \\dots, T_{n-1}$ satisfying the braid relations and the quadratic relations $(T_j-q)(T_j + 1) = 0$ for $1 \\le j \\le n-1$. When $q = 1$, $H_n(R, q)$ is the group algebra $R \\mathfrak S_n$ of the symmetric group $\\mathfrak S_n$. \nAs is well known, $H_n(R, q)$ has a basis $T_w$ ($w \\in \\mathfrak S_n$) given by \n$T_w = T_{j_1} \\dots T_{j_\\ell}$ for any reduced expression $w = s_{j_1} \\dots s_{j_\\ell}$.\nThe map defined by $i(T_w) = T_{w^{-1}}$ is an algebra involution. The map defined by $(T_w)^\\# = (-q)^{\\ell(w)}(T_{w^{-1}})^{-1}$ is an algebra automorphism. The assignment $T_w \\mapsto T_w$ is an embedding of $H_n(R, q)$ into\n$H_{n+1}(R, q)$. The algebra involutions are consistent on $(H_n)_{n \\ge 0}$.\n\nDipper and James ~\\cite{dipper-james1, dipper-james2} studied the representation theory of the Hecke algebras, defining Specht modules $S^\\lambda$ which generalize Specht modules for symmetric groups. \nThey showed that induced modules of Specht modules have a filtration by Specht modules\n~\\cite{dipper-james1}. Jost ~\\cite{jost} showed that restrictions of Specht modules have Specht filtrations.\n \n Murphy ~\\cite{murphy-hecke95} showed that the Hecke algebras are cellular (before the formalization of the notion of cellularity in ~\\cite{Graham-Lehrer-cellular}). Murphy shows that his cell modules $\\Delta^\\lambda$ satisfy\n $\\Delta^\\lambda \\cong ({S^{\\lambda'}})^\\#$, where $\\lambda'$ is the transpose of $\\lambda$ and the superscript $\\#$ means that the module is twisted by the automorphism $\\#$. Thus it follows from the results of Dipper, James, and Jost cited above that restricted modules and induced modules of Murphy's cell modules have cell filtrations.\n\\end{example}\n\n\n\\subsection{Inclusions of split semisimple algebras and branching diagrams}\n A general \\break source for the material in this section is ~\\cite{GHJ}. \n \n A finite dimensional split semisimple algebra over a field $F$ is one which is isomorphic to a finite direct sum of full matrix algebras over $F$.\n \nSuppose\n$A \\subseteq B$ are finite dimensional split semisimple algebras over $F$ (with the same identity element). Let $A(i)$, $i \\in I$, be the minimal ideals of $A$ and $B(j)$, $j \\in J$, the minimal ideals of $B$. \nWe associate a $J \\times I$ {\\em inclusion matrix}\n$\\Omega$ to the inclusion $A \\subseteq B$, as follows. Let $W_j$ be a simple $B(j)$--module.\nThen $W_j$ becomes an $A$--module via the inclusion, and $\\Omega(j, i)$ is the multiplicity of a simple $A_i$--module \n in the decomposition of $W_j$ as an $A$--module. \n An equivalent characterization of the inclusion matrix is the following. Let $q_i$ be a minimal idempotent in $A(i)$ and let $z_j$ be the identity of \n$B(j)$ (a minimal central idempotent in $B$). Then $q_i z_j$ is the sum of\n $\\Omega(j, i)$ minimal idempotents in $B(j)$.\n\nIt is convenient to encode an inclusion matrix by a bipartite graph, called the {\\em branching diagram}; the branching diagram has vertices labeled by $I$ arranged on one horizontal line, vertices labeled by $J$ arranged along a second (higher) horizontal line, and $\\Omega(j, i)$ edges connecting\n$j \\in J$ to $i \\in I$.\n\nIf $A_1 \\subseteq A_2 \\subseteq A_3 \\cdots$ is a (finite or infinite) sequence of inclusions of finite dimensional split semisimple algebras over $F$, then the branching diagram for the sequence is obtained by stacking the branching diagrams for each inclusion, with the \nupper vertices of the diagram for $A_i \\subseteq A_{i+1}$ being identified with the lower vertices of the diagram for $A_{i+1} \\subseteq A_{i+2}$.\n\nFor our purposes, it suffices to restrict our attention to the case that $A_0 \\cong F$. In most of our examples,\n the entries in each inclusion matrix are all $0$ or $1$; thus in the branching diagram there are no multiple edges between vertices.\n\n\n\n\n\\begin{definition} \\label{def of branching} An (infinite) abstract branching diagram $\\mathfrak{B}$ is an infinite graph with vertex set\n$V = \\coprod_{i \\ge 0} V_i$, with the following properties\n\\begin{enumerate}\n\\item $V_0$ is a singleton and $V_i$ is finite for all $i$.\n\\item Two vertices $v \\in V_i$ and $w \\in V_j$ are adjacent only if $|i - j|=1$. Multiple edges are allowed between adjacent vertices. \n\\item If $i \\ge 1$ and $v \\in V_i$, then $v$ is adjacent to at least one vertex in $V_{i-1}$ and to at least one vertex in $V_{i+1}$.\n\\end{enumerate}\n\\end{definition}\n\nThe definition can be modified in the obvious way for a {\\em finite} abstract branching diagram. \nWhen we treat the walled Brauer algebra in Section \\ref{subsection: walled Brauer algebras}, we will loosen the definition by dropping the requirement that $V_0$ is a singleton. \n\n\nThe branching diagram for a sequence of finite dimensional split semisimple algebras (with the restrictions mentioned above) is an abstract branching diagram, and conversely, given an abstract branching diagram $\\mathfrak{B}$, one can construct a sequence of finite dimensional split semisimple algebras (over any given field) whose branching diagram is (isomorphic to) $\\mathfrak{B}$.\n\nLet $\\mathfrak{B}$ be an abstract branching diagram with vertex set $V = \\coprod_{i \\ge 0} V_i$. We usually denote the unique element of $V_0$ by $\\emptyset$. We picture $\\mathfrak{B}$ with the elements of $V_i$ arranged on the horizontal line $y = i$ in the plane, and we call $V_i$ the $i$--th {\\em row} of vertices in $\\mathfrak{B}$. If $v \\in V_i$ and $w \\in V_{i+1}$ are adjacent, we write $v \\nearrow w$. The subgraph of $\\mathfrak{B}$ consisting of $V_i$ and $V_{i+1}$ and the edges connecting them is called the $i$--th {\\em level} of $\\mathfrak{B}$.\n\n\nNow suppose we are given an abstract branching diagram $\\mathfrak{B}_0$ with vertex set\n\\def\\spp #1{^{(#1)}}\n$V\\spp 0 = \\coprod_{i \\ge 0} V_i\\spp 0$. We construct a new abstract branching diagram $\\mathfrak{B}$ as follows:\nThe vertex set of $\\mathfrak{B}$ is $V = \\coprod_{k \\ge 0} V_k$, where\n$$\nV_k = \\coprod_{\\substack{i\\leq k\\\\k-i\\text{ even}}} V_i\\spp 0 \\times \\{k\\}.\n$$\nThus the $k$--th row of vertices of $\\mathfrak{B}$ consists of copies of rows $k$, $k-2$, $k-4$, \\dots of vertices of\n$\\mathfrak{B}_0$. Now if $(\\lambda, k) \\in V_k$ and $(\\mu, k+1) \\in V_{k+1}$, there exist $i \\le k$ with $k -i$ even such that\n$\\lambda \\in V_i \\spp 0$, and $j \\le k+1$ with $k+ 1 - j$ even such that $\\mu \\in V_j \\spp 0$.\nWe declare $(\\lambda, k) \\nearrow (\\mu, k+1)$ if, and only if, $|i - j| = 1$ and $\\lambda$ and $\\mu$ are adjacent\nin $\\mathfrak{B}_0$. The number of edges connecting $(\\lambda, k) $ and $(\\mu, k+1)$ is the same as the number of edges connecting $\\lambda$ and $\\mu$ in $\\mathfrak{B}_0$. \n\nThe first few levels of $\\mathfrak{B}$ is picture schematically in Figure \\ref{figure: branching diagram}, where each diagonal line represents\nall the edges connecting vertices in $V_i^{(0)}$ with vertices in $V_{i \\pm 1}^{(0)}$. \n\\begin{figure}\n$$ \\inlinegraphic[scale=.5]{branching1}$$\n\\caption{Branching diagram obtained by reflections}\n\\label{figure: branching diagram}\n\\end{figure}\nNote that the $k$--th level of $\\mathfrak{B}$ is a folded copy of the first $k$ levels of $\\mathfrak{B}_0$.\nWe call $\\mathfrak{B}$ the {\\em branching diagram obtained by reflections from $\\mathfrak{B}_0$}.\n\n\\begin{example} Take $\\mathfrak{B}_0$ to be Young's lattice. Thus $V_k\\spp 0$ consists of Young diagrams of size\n$k$, and $\\lambda \\nearrow \\mu$ in $\\mathfrak{B}_0$ if $\\mu$ is obtained from $\\lambda$ by adding one box.\nThen the $k$--th row of vertices in the abstract branching diagram $\\mathfrak{B}$ obtained from $\\mathfrak{B}_0$ by reflections consists of all pairs $(\\lambda, k)$, where $\\lambda$ is a Young diagram of size $i \\le k$, with $k - i$ even.\n Moreover, $(\\lambda, k) \\nearrow (\\mu, k+1)$ in $\\mathfrak{B}$ if, and only if, \n$\\mu$ is obtained from $\\lambda$ either by adding one box or by removing one box.\n\\end{example}\n\n\\subsection{The Jones basic construction} This paper could be written without ever mentioning the Jones basic construction.\n Nevertheless, in our view, the basic construction plays an essential role behind the scenes.\n\nThe Jones basic construction was introduced ~\\cite{Jones-index} in the theory of von Neumann algebras and is crucial in the analysis of von Neumann subfactors. Translated to the context of finite dimensional split semisimple algebras over a field, the basic construction was a fundamental ingredient in Wenzl's analysis of the generic structure of the Brauer algebras and the BMW algebras ~\\cite{Wenzl-Brauer, Birman-Wenzl, Wenzl-BCD} .\n\nThe basic construction for finite dimensional split semisimple algebras can be described as follows (see ~\\cite{GHJ}): let $ A \\subseteq B $ be finite dimensional split semisimple algebras over field $ F$, with the same multiplicative identity element.\nThe basic construction for the pair $ A \\subseteq B $ is the algebra ${\\rm End}(B_A)$.\nThis algebra is also split semisimple and the inclusion matrix for the pair $B \\subseteq {\\rm End}(B_A)$ is a transpose of that for the pair $ A \\subseteq B $. Suppose now that $ B$ has a faithful $F$-- valued trace $ \\varepsilon$ with faithful restriction to $ A$. Here faithful means that the bilinear form $(x, y) \\mapsto \\varepsilon(x y)$ is non--degenerate. In this case there is a unique trace preserving conditional expectation\n$ \\varepsilon_A : B \\to A$, i.e. a unital $A$--$A$--bimodule map satisfying $\\varepsilon\\circ \\varepsilon_A = \\varepsilon$. \nIdentify $ B$ with its image in ${\\rm End}_F(B)$ under the left regular representation.\nThe basic construction $ {\\rm End}(B_A)$ is equal to $ B \\varepsilon_A B =\n\\{ \\sum_{i = 1}^n b_i' \\varepsilon_A b_i'' : n \\ge 1, b_i', b_i'' \\in B\\}$. Moreover, $ B \\varepsilon_A B \\cong\nB \\otimes_A B$, where the latter is given the algebra structure determined by\n$(b_1 \\otimes b_2)(b_3 \\otimes b_4) = b_1 \\otimes \\varepsilon_A(b_2 b_3) b_4$. Note that we have three realizations for the basic construction,\n$$\n{\\rm End}(B_A) \\cong B \\varepsilon_A B \\cong\nB \\otimes_A B,\n$$\nany of which could serve as a potential definition of the basic construction in a more general setting.\n\nSuppose in addition that we are given an algebra $C$ with $B \\subseteq C$ and that $ C$ contains an idempotent $ e$ such that $ exe = \\varepsilon_{A}(x) e$ for $ x \\in B$, and $x \\mapsto x e$ is injective from $B$ to $Be \\subseteq C$. Note that\n $ BeB$ is a possibly non--unital subalgebra of $C$.\nBy ~\\cite{Wenzl-Brauer}, Theorem 1.3, $ BeB \\cong B\\varepsilon_{A}B \\cong {\\rm End}(B_{A})$, and, in particular, $ BeB$ is unital and semisimple.\n\nLet's now describe how Wenzl used these ideas to show the generic semisimplicity of the \nBrauer algebras. We refer the reader to Section \\ref{subsection: Brauer algebras} for the definition of the Brauer algebras.\nConsider the Brauer algebras $B_{n} = B_n(F, \\delta)$ over $F= {\\mathbb C}$ or $F = {\\mathbb Q}(\\delta)$, in the first case with parameter\n$\\delta $ a non-integer complex number, and in the second case with parameter $\\delta $\nan indeterminant over ${\\mathbb Q}$. The Brauer algebras have a canonical $F$--valued trace $\\varepsilon$ and conditional expectations $\\varepsilon_{n}: B_{n} \\to B_{n-1}$ preserving the trace. \n Each Brauer algebra $B_{n}$ contains an essential idempotent $e_{n-1}$ with\n $e_{n-1}^{2} = \\delta e_{n-1}$ and $e_{n-1} xe_{n-1} = \\delta \\varepsilon_{n-1}(x) e_{n-1}$ for\n $x \\in B_{n-1}$. Moreover, $x \\mapsto x e_{n-1}$ is injective from $B_{n-1}$ to $B_{n}$ and one has $B_{n}\/ B_{n} e_{n-1} B_{n} \\cong F \\mathfrak S_{n}$,\n which is semisimple, since $F$ has characteristic 0.\n Let $f_{n-1} = \\delta^{-1} e_{n-1}$; then $f_{n-1}$ is an idempotent with$f_{n-1} xf_{n-1} = \\varepsilon_{n-1}(x) f_{n-1}$ for\n $x \\in B_{n-1}$. We have $B_{0} \\cong B_{1} \\cong F$. \n \n Suppose it is known for some $n$ that $B_{k}$ is split semisimple and that\n the trace $\\varepsilon$ is faithful on $B_{k}$ for $k \\le n$. By Wenzl's observation applied to\n $B_{n-1} \\subseteq B_{n} \\subseteq B_{n+1}$ and the idempotent $f_{n} \\in B_{n+1}$, we have\n $B_{n} e_{n} B_{n} = B_{n} f_{n} B_{n} \\cong B_{n} \\varepsilon_{n} B_{n} \\cong {\\rm End}((B_{n})_{B_{n-1}})$. But it is elementary to check that $B_{n} e_{n} B_{n} = \n B_{n+1} e_{n} B_{n+1}$. Thus we have that the ideal $B_{n+1} e_{n} B_{n+1} \\subseteq \n B_{n+1}$ is split semisimple, and the quotient of $B_{n+1}$ by this ideal ($\\cong F \\mathfrak S_{n+1}$) is also split semisimple, so $B_{n+1}$ is split semisimple. To continue the inductive argument, it is necessary to verify that the trace $\\varepsilon$ \nis faithful on $B_{n+1}$. Wenzl uses a Lie theory argument for this.\n\n In this paper, we develop a cellular analog of this argument. Let's continue to use the example of the Brauer algebras to illustrate this. Cellularity is a property that is preserved under specializations, so it suffices to consider the Brauer algebras over the generic ring\n $R = {\\mathbb Z}[ \\delta ] $. Let $F$ denote the field of fractions of $R$, $F = {\\mathbb Q}(\\delta)$.\t\n Write $B_{n}$ for $B_{n}(R, \\delta)$ and $B_{n}^{F}$ for $B_{n}(F, \\delta)$. By Wenzl's theorem, $B_{n}^{F}$ is split semisimple. \n We have $B_{0} \\cong B_{1} \\cong R$. \n \n Suppose it is known for some $n$ that $B_{k}$ is cellular for $k \\le n$. We want to show that\n$B_{n+1} e_{n} B_{n+1} = B_{n} e_{n} B_{n}$ is a cellular ideal in $B_{n+1}$. It will then follow that $B_{n+1}$ is cellular, because the quotient $B_{n+1}\/B_{n+1} e_{n} B_{n+1} \\cong R \\mathfrak S_{n+1}$ is cellular. Let $\\Lambda_{n-1}$ denote the partially ordered set in the \ncell datum for $B_{n-1}$. For each order ideal $\\Gamma$ of $\\Lambda_{n-1}$, write $J(\\Gamma)$ for the span in $B_{n-1}$ of all $c_{s, t}^\\lambda$ with $\\lambda \\in \\Gamma$. The crucial point is to show that\n$\\Gamma \\mapsto B_n e_n J(\\Gamma) B_n = B_{n+1} e_n J(\\Gamma)B_{n+1}$ is a \n$\\Lambda_{n-1}$--cell net in $B_{n+1} e_n B_{n+1}$. Along the way to doing this, we show that\n \\begin{equation}\\label{equation: jones bc 1}\nJ'(\\Gamma) := B_{n} \\otimes_{B_{n-1}} J(\\Gamma) \\otimes_{B_{n-1}} B_{n} \\cong B_{n}e_{n}J(\\Gamma)B_{n} \n \\end{equation} \nvia $b' \\otimes x \\otimes b'' \\mapsto b' e_{n} x b''$; consequently, if $\\Gamma_{1} \\subseteq \\Gamma_{2}$, then $J'(\\Gamma_{1})$ imbeds in $J'(\\Gamma_{2}) $. In particular,\n \\begin{equation} \\label{equation: jones bc 2}\n B_{n} \\otimes_{B_{n-1}} B_{n} \\cong B_{n}e_{n}B_{n} = B_{n+1}e_{n}B_{n+1}, \n \\end{equation}\nand $J'(\\Gamma)$ imbeds as an ideal in the (non--unital) algebra $\n B_{n} \\otimes_{B_{n-1}} B_{n}$. Essentially, what we show is that \n $B_{n+1}e_{n}B_{n+1} = B_{n}e_{n} B_{n}$ is isomorphic to the basic construction \n $\n B_{n} \\otimes_{B_{n-1}} B_{n}$, and that\n $\\Gamma \\mapsto J'(\\Gamma)$ is a $\\Lambda_{n-1}$--cell net in $\n B_{n} \\otimes_{B_{n-1}} B_{n}$. \n \n We note that $B_{n}$ is {\\em not} a projective $B_{n-1}$--module, but the isomorphisms\n (\\ref{equation: jones bc 1}) and the embeddings $J'(\\Gamma_{1}) \\hookrightarrow J'(\\Gamma_{2}) $\n reflect the projectivity of $B_{n}^{F}$ over $B_{n-1}^{F}$.\n \n \n\\subsection{Coherent cellular towers and extension of the ground ring}\n\nLet $R$ be an integral domain and let $F$ denote the field of fractions of $R$. We will be interested in \ncoherent towers $(H_n)_{n \\ge 0}$ of cellular algebras over $R$ such that for all $n$, the $F$--algebra $H_n^F := H_n \\otimes_R F$ is (split) semisimple. We will see that in this situation we have uniqueness of multiplicities in the\nfiltrations of induced and restricted modules by cell modules, and Frobenius reciprocity connecting these multiplicities.\n\nFor any algebra $A$ over $R$, write $A^F$ for the $F$--algebra $A\\otimes_R F$. Moreover, for a left (or right)\n$A$--module $M$, write $M^F$ for the left (or right) $A^F$ module $M\\otimes_R F$.\n\n\\begin{lemma} \\label{first tensor iso}\nLet $R$ be an integral domain and $F$ its field of fractions. Let $A$ and $B$ be $R$-algebras. For modules $M_{A}$ and \n$_A N$, we have\n\\begin{equation} \\label{Fisomorphism}\nM\\otimes_A N\\otimes_R F\\cong M^F\\otimes_{A^F}N^F\n\\end{equation}\n\\noindent\nas $F$-vector spaces. The isomorphism\n$$M\\otimes_A N\\otimes_RF\\rightarrow M^F\\otimes_{A^F}N^F$$\nis determined by $(x\\otimes_Ay\\otimes_R f)\\mapsto(x\\otimes_R\\bm 1_F)\\otimes_{A^F}(y\\otimes_R f)$. If ${}_{A}{N}_B$ is a bimodule, then the isomorphism in (\\ref{Fisomorphism}) is an isomorphism of right $B^F$--modules, and similarly, if \n$_B M_A$ is a bimodule, then the isomorphism is an isomorphism of left $B^F$--modules. \n\\end{lemma}\n\n\\begin{proof}\nNote that\n\\begin{align}\nM&\\otimes_A(N\\otimes_R F)\\cong M\\otimes_A A^F\\otimes_{A^F}(N\\otimes_R F) \\notag \\\\\n\t&=(M\\otimes_A A\\otimes_R F)\\otimes_{A^F}(N\\otimes_R F) \\notag \\\\\n\t&\\cong (M\\otimes_R F)\\otimes_{A^F}(N\\otimes_R F) \\notag \\\\\n\t&=M^F\\otimes_{A^F}N^F. \\notag\n\\end{align}\n\n\\noindent\nIf we track a simple tensor through these equalities and isomorphisms, we see that\n\\begin{align}\n&x\\otimes_A y\\otimes_R f\\mapsto x\\otimes_A\\bm 1_{A^F}\\otimes_{A^F}(y\\otimes_R f) \\notag \\\\\n\t&\\quad=x\\otimes_A\\bm 1_A\\otimes_R\\bm 1_F\\otimes_{A^F}(y\\otimes_R f)\\mapsto(x\\otimes_R\\bm 1_F)\\otimes_{A^F}(y\\otimes_R f). \\notag\n\\end{align}\nThe final statement follows from this.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemma injectivity of x to x tensor 1} Let $R$ be an integral domain and $F$ its field of fractions. If $M$ is a free $R$--module, then the map $M \\to M \\otimes_R F$ determined by \n$x \\mapsto x \\otimes 1_F$ is injective.\n\\end{lemma}\n\n\\begin{proof} It follows from ~\\cite{Jacobson}, Propositions 3.2 and 3.3 that the map $x \\mapsto x \\otimes 1$ takes an $R$--basis of $M$ to an $F$--basis of $M \\otimes_R F$. In particular, the map is injective.\n\\end{proof}\n\n\\begin{lemma} \\label{injectivity of iota tensor id(F) with free R modules}\nLet $R$ be an integral domain and $F$ its field of fractions. Let $N_1 \\subseteq N_2$ be $R$--modules with $N_2$ free. Let $\\iota : N_1 \\to N_2$ denote the injection. Then\n$\\iota \\otimes \\id_F : N_1 \\otimes_R F \\to N_2 \\otimes_R F$ is injective.\n\\end{lemma}\n\n\\begin{proof}\n Any element of $N_1 \\otimes_R F $ can be written as $y = (1\/q) (x \\otimes 1_F)$,\nwith $q \\in R^\\times$ and $x \\in N_1$. Then $\\iota \\otimes \\id_F(y) = (1\/q)(\\iota(x) \\otimes 1_F) =\n(1\/q)\\, \\gamma \\circ \\iota (x)$, where $\\gamma : N_2 \\to N_2 \\otimes_R F$ is determined\nby $z \\mapsto z \\otimes 1_F$. Because $N_2$ is a free $R$--module, $\\gamma$ is injective, by Lemma \\ref{lemma injectivity of x to x tensor 1}, and it follows that $\\iota \\otimes \\id_F$ is injective.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemma: multiplicities in cell filtrations}\nLet $R$ be an integral domain with field of fractions $F$.\n Suppose that $(H_n)_{n \\ge 0}$ is a coherent tower of cellular algebras over $R$ and that\n $H_n^F$ is split semisimple for all $n$. Let $\\Lambda_n$ denote the partially ordered set in the cell datum for\n $H_n$. \n Then\n \\begin{enumerate}\n \\item \n $\\{(\\Delta^\\lambda)^F : \\lambda \\in \\Lambda_n\\}$ is a complete family of simple $H_n^F$--modules.\n \\item Let $[\\omega(\\mu, \\lambda)]_{\\mu \\in \\Lambda_{n+1}, \\, \\lambda \\in \\Lambda_n}$ denote the inclusion matrix for\n $H_n^F \\subseteq H_{n+1}^F$. Then for any $\\lambda \\in \\Lambda_n$ and $\\mu \\in \\Lambda_{n+1}$, \n and any cell filtration of ${\\rm Res}_{H_n}^{H_{n+1}}(\\Delta^\\mu)$, the number of subquotients of the filtration isomorphic to $\\Delta^\\lambda$ is $\\omega(\\mu, \\lambda)$.\n \n \\item Likewise, for any $\\lambda \\in \\Lambda_n$ and $\\mu \\in \\Lambda_{n+1}$, \n and any cell filtration of ${\\rm Ind}_{H_n}^{H_{n+1}}(\\Delta^\\lambda)$, the number of subquotients of the filtration isomorphic to $\\Delta^\\mu$ is $\\omega(\\mu, \\lambda)$.\n \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} For point (1), $(\\Delta^\\lambda)^F$ is a cell module for $H_n^F$, and, for a semisimple cellular algebra, the cell modules are precisely the simple modules.\n\nWe have \n\\begin{equation} \\label{first direct sum decomp of Res Delta}\n({\\rm Res}_{H_n}^{H_{n+1}}(\\Delta^\\mu))^F = {\\rm Res}_{H_n^F}^{H_{n+1}^F}((\\Delta^\\mu)^F) \\cong\n \\bigoplus_{\\lambda \\in \\Lambda_n} \\omega(\\mu ,\\lambda) (\\Delta^\\lambda)^F,\n\\end{equation}\nby definition of the inclusion matrix. On the other hand, if \n$$\n{\\rm Res}_{H_n}^{H_{n+1}} (\\Delta^\\mu) = N_s \\supseteq N_{s-1} \\supseteq \\cdots \\supseteq N_0 = (0)\n$$\nis a cell filtration, with $N_j\/N_{j-1} \\cong \\Delta^{\\lambda_j}$, then\n$$\n({\\rm Res}_{H_n}^{H_{n+1}}(\\Delta^\\mu))^F = N_s^F \\supseteq N_{s-1}^F \\supseteq \\cdots \\supseteq N_0^F = (0),\n$$\nby Lemma \\ref{injectivity of iota tensor id(F) with free R modules}, because all the modules\n$N_j$ are free as $R$--modules. Moreover, \n$N_j^F\/N_{j-1}^F \\cong (N_j\/N_{j-1})^F \\cong (\\Delta^{\\lambda_j})^F$ by right exactness of tensor products. Since $H_n^F$ modules are semisimple,\n\\begin{equation}\\label{second direct sum decomp of Res Delta}\n({\\rm Res}_{H_n}^{H_{n+1}}(\\Delta^\\mu))^F \\cong \\bigoplus_{j = 1}^s (\\Delta^{\\lambda_j})^F.\n\\end{equation}\nComparing (\\ref{first direct sum decomp of Res Delta}) and (\\ref{second direct sum decomp of Res Delta}) and taking into account that $\\Delta^\\lambda \\mapsto (\\Delta^\\lambda)^F$ is injective, we obtain conclusion (2).\n\nLikewise,\n$$\n({\\rm Ind}_{H_n}^{H_{n+1}}(\\Delta^\\lambda))^F = H_{n+1} \\otimes_{H_n} \\Delta^\\lambda \\otimes_R F \\cong \n H_{n+1}^F \\otimes_{H_n^F} (\\Delta^\\lambda)^F,\n$$\nby Lemma \\ref{first tensor iso}. \nBut\n$$\nH_{n+1}^F \\otimes_{H_n^F} (\\Delta^\\lambda)^F = {\\rm Ind}_{H_n^F}^{H_{n+1}^F}((\\Delta^\\lambda)^F) \\cong \\bigoplus_{\\mu \\in \\Lambda_{n+1}} \\omega(\\mu ,\\lambda) (\\Delta^\\mu)^F,\n$$\nusing (\\ref{first direct sum decomp of Res Delta}) and Frobenius reciprocity. The rest of the argument for point (3) is similar to that for point (2).\n\\end{proof}\n\n\n \n\\begin{lemma} \\label{lemma: cell basis indexed by paths}\nAdopt the assumptions and notation of Lemma \\ref{lemma: multiplicities in cell filtrations}.\nAssume in addition that the branching diagram $\\mathfrak{B}$ for $(H_n^F)_{n\\ge 0}$ has no multiple edges and that\n$H_0^F = F$. It follows that each $H_n$ has a cell datum (perhaps different from the one initially given)\nwith the same partially ordered set $\\Lambda_n$ but with $\\mathcal T(\\lambda)$ equal to the set of paths\non $\\mathfrak{B}$ from $\\emptyset$ to $\\lambda$. \n\\end{lemma}\n\n\\begin{proof}\nReferring to the proof of Proposition \\ref{lemma: cell net characterization of cellularity}, it suffices to show that, for each $n$ and for each $\\lambda \\in \\Lambda_n$, the cell module $\\Delta^\\lambda$ has\n an $R$--basis indexed by the set $\\mathcal P(\\lambda)$ of paths in $\\mathfrak{B}$ from $\\emptyset$ to $\\lambda$. \nBut this says only that the rank of $\\Delta^\\lambda$ over $R$ is $|\\mathcal P(\\lambda)|$, and this is true because ${\\rm rank}_R(\\Delta^\\lambda) = {\\rm dim}_F(\\Delta^\\lambda \\otimes_R F) = |\\mathcal P(\\lambda)|$. See also the following remark.\n\\end{proof}\n\n\\begin{remark} In principle, in the situation of Lemma \\ref{lemma: cell basis indexed by paths}, we can recursively build bases of cell modules, using the cell filtrations of restrictions. Suppose we have bases of $\\Delta^\\lambda$ for all $\\lambda \\in \\Lambda_n$ for some $n$. \nLet $\\mu \\in \\Lambda_{n+1}$. Then $\\Delta^\\mu$, regarded as an $H_n$--module, has a filtration by cell modules of $H_n$, \n$$\n\\Delta^\\mu = N_s \\supseteq N_{s-1} \\supseteq \\cdots \\supseteq N_0 = (0),\n$$\n with $N_j\/N_{j-1} \\cong \\Delta^{\\lambda_j}$; and $\\lambda \\in \\Lambda_n$ appears (exactly once) in the list \n of $\\lambda_j$, if, and only if, $\\lambda \\nearrow \\mu$. Now we inductively build bases of the $N_j$ to obtain a basis of $N_s = \\Delta^\\mu$. The isomorphism $N_1 \\cong \\Delta^{\\lambda_1}$ provides a basis of $N_1$. For $j \\ge 2$, if we have a basis of $N_{j-1}$, then that basis together with any lift of a basis of $N_j\/N_{j-1} \\cong \\Delta^{\\lambda_j}$ gives a basis of $N_j$.\n\n\n\n\n\\end{remark}\n \n\n\n\n\n\n\\section{A framework for cellularity}\nIn this section we describe our framework for cellularity of algebras related to the Jones basic construction.\n\n \n\\subsection{Framework Axioms} \\label{subsection: framework axioms}\nLet $R$ be an integral domain with field of fractions $F$. We consider two sequences of $R$--algebras\n$$\nA_0 \\subseteq A_1 \\subseteq A_2 \\subseteq \\cdots, \\quad\\text{and} \\quad Q_0 \\subseteq Q_1 \\subseteq Q_2 \\subseteq \\cdots,\n$$\neach with a common multiplicative identity element. \nWe assume the following axioms:\n\\begin{enumerate}\n\\item \\label{axiom Hn coherent} $(Q_n)_{n \\ge 0}$ is a coherent tower of cellular algebras.\n\\item \\label{axiom: involution on An} There is an algebra involution $i$ on $\\cup_n A_n$ such that $i(A_n) = A_n$.\n\\item \\label{axiom: A0 and A1} $A_0 = Q_0 = R$, and $A_1 = Q_1$ (as algebras with involution).\n\\item \\label{axiom: semisimplicity}\nFor all $n$, $A_n^F : = A_n \\otimes_R F$ is split semisimple. \n\\item \\label{axiom: idempotent and Hn as quotient of An}\n For $n \\ge 2$, $A_n$ contains an essential idempotent $e_{n-1}$ such that $i(e_{n-1}) = e_{n-1}$ and\n$A_n\/(A_n e_{n-1} A_n) \\cong Q_n$, as algebras with involution.\n\n\\item \\label{axiom: en An en} For $n \\ge 1$, $e_{n}$ commutes with $A_{n-1}$ and $e_{n} A_{n} e_{n} \\subseteq A_{n-1} e_{n}$.\n\\item \\label{axiom: An en}\nFor $n \\ge 1$, $A_{n+1} \te_{n} = A_{n} e_{n}$, and the map $x \\mapsto x e_{n}$ is injective from\n$A_{n}$ to $A_{n} e_{n}$.\n\\item \\label{axiom: e(n-1) in An en An} For $n \\ge 2$, $e_{n-1} \\in A_{n+1} e_n A_{n+1}$.\n\\end{enumerate}\n\n\n\\begin{remark} \\mbox{}\n\\begin{enumerate}\n\\item\nLet $\\Lambda_n \\spp 0$ denote the partially ordered set in the cell datum for $Q_n$. It follows from axioms (\\ref{axiom Hn coherent}) and (\\ref{axiom: semisimplicity}) and Lemma \\ref{lemma: multiplicities in cell filtrations} that $\\Lambda_n \\spp 0$ can be identified with the $n$--th row of vertices of the branching diagram for $(Q_n^F)_{n \\ge 0}$.\n\n\\item Applying the involution in axiom (\\ref{axiom: An en}), we also have $e_{n} A_{n+1} \t = e_{n} A_n $, and the map $x \\mapsto e_{n} x $ is injective from\n$A_{n}$ to $e_{n} A_{n}$. \n\\item Since $e_n$ is an essential idempotent, there is a non--zero $\\delta_n \\in R$ with $e_n^2 = \\delta_n e_n$. Thus we have $e_n A_n e_n \\supseteq e_n A_{n-1} e_n = A_{n-1} e_n^2 =\n\\delta_n A_{n-1} e_n$. Combining this with axiom (\\ref{axiom: en An en}), we have\n$\\delta_n A_{n-1} e_n \\subseteq e_n A_n e_n \\subseteq A_{n-1} e_n$. Hence\n$e_n A_n^F e_n = A_{n-1}^F e_n$. \n\n\n\\item From axiom (\\ref{axiom: en An en}), we have for every\n$x \\in A_{n}$, there is a $y \\in A_{n-1}$ such that $e_{n} x e_{n} = y e_{n}$; but by axiom\n(\\ref{axiom: An en}), $y$ is uniquely determined, so we have a map ${\\rm cl}_{n} : A_{n} \\rightarrow A_{n-1}$ \nwith $e_{n} x e_{n} = {\\rm cl}_{n}(x) e_{n}$. It is easy to check that ${\\rm cl}_{n}$ is an $A_{n-1}$--$A_{n-1}$--bimodule map, but it is not unital in general; if $e_{n-1}^2 = \\delta_n e_{n-1}$, then \n${\\rm cl}_{n}(\\bm 1) = \\delta_n \\bm 1$. If $\\delta_n$ is invertible in $R$, then $\\varepsilon_{n} = (1\/\\delta_n) {\\rm cl}_{n}$ is a conditional expectation, i.e., a unital $A_{n-1}$--$A_{n-1}$--bimodule map.\n\\item From axioms (\\ref{axiom: semisimplicity}) and (\\ref{axiom: idempotent and Hn as quotient of An}), we have $Q_n^F : = Q_n \\otimes_R F$ is split semisimple. \n\\item In our examples, there is a single non--zero $\\delta$ with $e_n^2 = \\delta e_n$ for all $n$. \n\\end{enumerate}\n\\end{remark}\n\n \n\n\n \n\n\\subsection{The main theorem}\n\n\n\\begin{theorem} \\label{main theorem}\n Let $R$ be an integral domain with field of fractions $F$. Let $(Q_k)_{k\\ge 0}$ and\n$(A_k)_{k\\ge 0}$ be two towers of $R$--algebras satisfying the framework axioms of Section \\ref{subsection: framework axioms}. Then\n\\begin{enumerate}\n\\item $(A_k)_{k\\ge 0}$ is a coherent tower of cellular algebras.\n\\item For all $k$, the partially ordered set in the cell datum for $A_k$ can be realized as\n$$\n\\Lambda_k = \\coprod_{\\substack{i\\leq k\\\\k-i\\text{ even}}} \\Lambda_i\\spp 0 \\times \\{k\\},\n$$\nwith the following partial order: Let $\\lambda \\in \\Lambda_i\\spp 0$ and $\\mu \\in \\Lambda_j\\spp 0$, with\n$i$, $j$, and $k$ all of the same parity. Then\n $(\\lambda, k) > (\\mu, k)$ if, and only if, $i < j$, or $i = j$ and $\\lambda > \\mu$ in $\\Lambda_i\\spp 0$.\n \\item Suppose $k \\ge 2$ and $(\\lambda, k) \\in \\Lambda_i\\spp 0 \\times \\{k\\} \\subseteq \\Lambda_k$. Let \n $\\Delta^{(\\lambda, k)}$ be the corresponding cell module. \n If $i < k$, \n then\n $(A_k e_{k-1} A_k \\ \\Delta^{(\\lambda, k)})\\otimes_R F = \\Delta^{(\\lambda, k)}\\otimes_R F $, while if $i = k$ then \\break $A_k e_{k-1} A_k \\ \\Delta^{(\\lambda, k)} = 0$.\n\\item The branching diagram $\\mathfrak{B}$ for $(A_k^F)_{k \\ge 0}$ is that obtained by reflections from the branching diagram\n$\\mathfrak{B}_0$ for $(Q_k^F)_{n \\ge 0}$.\n\n\\end{enumerate}\n\\end{theorem}\n\n \\begin{remark} \\label{remark: point 5 comes from other statements}\n In most of our examples, the branching diagrams have no multiple edges. In this case,\n for all $k$ and for all $(\\lambda, k) \\in \\Lambda_k$, the index set $\\mathcal T((\\lambda, k))$ in the cell datum for $A_k$ can be taken to be the set of paths on $\\mathfrak{B}$ from $\\emptyset$ to $(\\lambda, k)$.\nThis follows from (1) and (4), using Lemma \\ref{lemma: cell basis indexed by paths}.\n \\end{remark}\n \n \n\\section{Proof of the main theorem} \\label{section: basic construction preserves cellularity}\n \n \n We will prove Theorem \\ref{main theorem} in this section. Our strategy is to prove \n the following statement by induction on $n$:\n\n\\smallskip\n\\noindent {\\bf Claim:}\\quad\n{\\em For all $n \\ge 0$, the statements (1) --(4) of Theorem \\ref{main theorem} hold for the finite tower $(A_k)_{0 \\le k \\le n}$.}\n\\smallskip\n\nOf course, by statement (4) for the finite tower, we mean that the branching diagram for the\nfinite tower $(A_k^F)_{0 \\le k \\le n}$ is that obtained by reflections from the branching diagram\nof the finite tower $(Q_k^F)_{0 \\le k \\le n}$. \n\nThe claim holds trivially for $n = 0$ and $n = 1$. We assume that the claim holds \nfor some $n \\ge 1$ and prove that it also holds for $n + 1$. \n\n\\subsection{$A_{n+1}$ is cellular}\nWe will show that $A_{n+1}$ is a cellular algebra.\n\nSince $A_{n+1}\/A_{n+1} e_n A_{n+1} \\cong Q_{n+1}$ is cellular, to prove that\n $A_{n+1}$ is cellular, it suffices to show that $A_{n+1} e_n A_{n+1}$ is a cellular ideal in $A_{n+1}$; see Remark\n\\ref{remark on extensions of cellular algebras}.\n\nRecall that $\\Lambda_k$ denotes the partially ordered set in the cell datum for $A_k$ for each $k$, $0 \\le k \\le n$.\nDenote the elements of the cellular basis of $A_k$ by $c_{u, v}^\\lambda$ for $\\lambda \\in \\Lambda_k$ and\n$u, v \\in \\mathcal T(\\lambda)$.\n\n\nFor each order ideal $\\Gamma$ of $\\Lambda_{n-1}$, recall that $A_{n-1}(\\Gamma)$ is the span in $A_{n-1}$ of all $c_{s, t}^\\lambda$ with $\\lambda \\in \\Gamma$.\n$A_{n-1}(\\Gamma)$ is an $i$--invariant two sided ideal of $A_{n-1}$. \n\\ignore{\nIf $\\Gamma \\subseteq \\Gamma'$ are two order ideals with\n$\\Gamma' \\setminus \\Gamma = \\{\\lambda\\}$, then\n$$A_{n-1}(\\Gamma')\/A_{n-1}(\\Gamma)\\cong\nA_{n-1}^{\\lambda}\/\\breve A_{n-1}^{\\lambda} \\cong\n\\Delta^{\\lambda}\\otimes_Ri(\\Delta^{\\lambda}),$$\nand the isomorphism $\\alpha : A_{n-1}(\\Gamma')\/A_{n-1}(\\Gamma) \\to \\Delta^{\\lambda}\\otimes_Ri(\\Delta^{\\lambda})$ satisfies\n$\\alpha \\circ i = i \\circ \\alpha$.\n}\nIn the following, we will write $J(\\Gamma) = A_{n-1}(\\Gamma)$\nand $$\\hat J(\\Gamma)= A_n e_n J(\\Gamma) A_n = A_{n+1} e_n J(\\Gamma)A_{n+1}, $$ which is a two sided ideal in $A_{n+1}$. \nOur goal is to show that $\\Gamma \\mapsto \\hat J(\\Gamma)$ is a $\\Lambda_{n-1}$--cell net in $A_{n+1} e_n A_{n+1}$.\n \n \n\\begin{lemma} \\label{second tensor iso}\nLet $R$ be an integral domain and $F$ its field of fractions. \nSuppose that $A$ and $B$ are $R$-algebras. Let\n$P_{A}$, $_A M_A$ and $_A Q$\n be modules. Then\n$$P\\otimes_AM\\otimes_AQ\\otimes_R F\\cong P^F\\otimes_{A^F}M^F\\otimes_{A^F}Q^F$$\nas $F$-vector spaces. The isomorphism \n$$P\\otimes_A M\\otimes_A Q\\otimes_R F\\rightarrow P^F\\otimes_{A^F} M^F\\otimes_{A^F}Q^F$$\nis determined by \n$$x\\otimes_Ay\\otimes_Az\\otimes_R f \\mapsto (x\\otimes_R\\bm 1_F)\\otimes_{A^F}(y\\otimes_R\\bm 1_F)\\otimes_{A^F}(z\\otimes_R f).$$\nIf $_B P_A$ and $_A Q_B$ are bimodules, then the isomorphism is an isomorphism of $B^F$--$B^F$--bimodules.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{first tensor iso},\n\\begin{equation} \\label{equation: 2nd tensor iso 1}\n(P\\otimes_AM)\\otimes_AQ\\otimes_R F\\cong(P\\otimes_AM)^F\\otimes_{A^F}Q^F.\\end{equation}\nApplying Lemma \\ref{first tensor iso} again, we have that \n\\begin{equation} \\label{equation: 2nd tensor iso 2}\n(P\\otimes_AM)^F\\cong P^F\\otimes_{A^F}M^F\n\\end{equation}\nas right $A^F$--modules. Combining the two isomorphisms we have\n\\begin{equation} \\label{equation: 2nd tensor iso 3}\nP\\otimes_AM\\otimes_AQ\\otimes_R F\\cong P^F\\otimes_{A^F}M^F\\otimes_{A^F}Q^F.\n\\end{equation}\nIf we track a simple tensor through these isomorphisms, we see that\n\\begin{align}\nx\\otimes_A&y\\otimes_Az\\otimes_R f\\mapsto(x\\otimes_Ay\\otimes_R\\bm 1_F)\\otimes_{A^F}(z\\otimes_R f) \\notag \\\\\n\t&\\mapsto(x\\otimes_R\\bm 1_F)\\otimes_{A^F}(y\\otimes_R\\bm 1_F)\\otimes_{A^F}(z\\otimes_R f). \\notag\n\\end{align}\nIf $_B P_A$ and $_A Q_B$ are bimodules, then the isomorphism in (\\ref{equation: 2nd tensor iso 1}) is an isomorphism of $B^F$--$B^F$--bimodules, and the isomorphism in (\\ref{equation: 2nd tensor iso 2}) is an isomorphism of \n$B^F$--$A^F$-- bimodules. Hence the final isomorphism (\\ref{equation: 2nd tensor iso 3}) is an isomorphism of\n$B^F$--$B^F$--bimodules.\n\\end{proof}\n\n\n\\begin{lemma} \\label{beta inj}\nLet $K$ be a field and $A$ a semisimple $K$-algebra. Suppose that $I \\subseteq A$ is a two-sided ideal and $M_A$, $_A N$ are modules. Then the homomorphism $M\\otimes_{A} I\\otimes_{A} N\\rightarrow M\\otimes_{A} N$ defined by $x\\otimes y\\otimes z\\mapsto x\\otimes yz$ is injective.\n\\end{lemma}\n\n\\begin{proof}\nThe semisimplicity of $A$ implies that all $A$-modules are projective. Thus $N\\otimes_{A}-$ and $-\\otimes_{A} M$ are exact, and \n$$N\\otimes_AI\\otimes_AM\\rightarrow N\\otimes_AA\\otimes_AM\\cong N\\otimes_AM$$\nis injective.\n\\end{proof}\n\n\\ignore{\n\\begin{remark} \\label{special case of beta inj}\nIf $J$ is a two-sided ideal such that $I\\trianglelefteq J\\trianglelefteq A$, then the proof of Lemma \\ref{beta inj} implies that the map $N\\otimes_AI\\otimes_AM\\rightarrow N\\otimes_AJ\\otimes_AM$ is injective.\n\\end{remark}\n}\n\n\n\n\\vbox{\n\\begin{proposition} \\label{proposition: Phi isomorphism} For all order ideals $\\Gamma$ of $\\Lambda_{n-1}$:\n\\begin{enumerate}\n\\item \n The map\n$$\\Phi_\\Gamma : \\ A_n e_n \\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n\\rightarrow A_n e_n J(\\Gamma) A_n$$\ndetermined by $$\\Phi_\\Gamma(a_1 e_n \\otimes x\\otimes e_n a_2)=a_1 e_n xa_2$$ is an isomorphism of \n$A_{n+1}$--$A_{n+1}$--bimodules.\n\\item $ A_n e_n \\otimes_{A_{n-1}}J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module.\n\\item Let $ \\Gamma'$ be another order ideal containing $\\Gamma$, such that $\\Gamma' \\setminus \\Gamma$ is a singleton.\n Let $\\iota$ denote the injection $J(\\Gamma) \\ \\to J(\\Gamma')$. Then\n$$\n\\begin{aligned}\n\\beta_{\\Gamma,\\Gamma'} := \\id \\otimes \\iota \\otimes \\id : \\ &A_n e_n \\otimes_{A_{n-1}}J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n \\to \\\\\n&A_n e_n \\otimes_{A_{n-1}}J(\\Gamma') \\otimes_{A_{n-1}} e_n A_n\n\\end{aligned}\n$$\nis injective.\n\\end{enumerate}\n\\end{proposition}\n}\n\nWe provide two lemmas on the way to proving Proposition \\ref{proposition: Phi isomorphism}.\n\n\\begin{lemma} \\label{lemma 1 for induction on j}\n Let \\ $\\Gamma \\subseteq \\Gamma'$ be two order ideals in $\\Lambda_{n-1}$ such that $\\Gamma' \\setminus \\Gamma$ is a singleton.\nSuppose that \\ $\\Phi_{\\Gamma}$ is an isomorphism and that $ A_n e_n \\otimes_{A_{n-1}}J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module. Then $\\beta_{\\Gamma, \\Gamma'}$ is injective and \n$ A_n e_n \\otimes_{A_{n-1}}J(\\Gamma') \\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module.\n\\end{lemma}\n\n\\begin{proof} Let $\\{\\lambda\\} = \\Gamma' \\setminus \\Gamma$.\nSince $\\Phi_{\\Gamma}$ is assumed injective, it follows from considering the commutative diagram below that\n$\\beta_{\\Gamma,\\Gamma'}$ is also injective:\n\\begin{diagram}\nA_n e_n \\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n &&\\rTo^{\\Phi_\\Gamma} &&A_n e_n J(\\Gamma) A_n \\\\\n\\dTo_{ \\beta_{\\Gamma, \\Gamma'} } & &&& \\dTo\\\\\n A_n e_n \\otimes_{A_{n-1}} J(\\Gamma')\\otimes_{A_{n-1}} e_n A_n &&\\rTo^{\\Phi_{\\Gamma'}} &&A_n e_n J(\\Gamma') A_n\n\\end{diagram}\n\\ignore{\n$$\\begin{CD}\nA_n e_n \\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n @>\\Phi_\\Gamma>> A_n e_n J(\\Gamma) A_n \\\\\n@VV \\beta_{\\Gamma, \\Gamma'} V @VV V \\\\\nA_n e_n \\otimes_{A_{n-1}} J(\\Gamma')\\otimes_{A_n} e_n A_n @> \\Phi_{\\Gamma'} >> A_n e_n J(\\Gamma') A_n\n\\end{CD}\n$$\n}\n\nBy the right exactness of tensor products, we have\n\\begin{equation} \\label{equation 1 for lemma 1 of induction on j}\n\\begin{aligned}\n&(A_n e_n\\otimes_{A_{n-1}} J(\\Gamma')\\otimes_{A_{n-1}} e_{n }A_n) \/ \\beta_{\\Gamma,\\Gamma'}(A_n e_{n}\\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n ) \\\\\n&\\cong A_n e_n\\otimes_{A_{n-1}} (J(\\Gamma')\/J(\\Gamma)) \\otimes_{A_{n-1}} e_n A_n \\\\\n&\\cong A_n e_n\\otimes_{A_{n-1}} \\Delta^{\\lambda} \\otimes_R i( \\Delta^{\\lambda}) \\otimes_{A_{n-1}} e_n A_n\n\\end{aligned}\n\\end{equation}\nConsider $A_n e_n = A_{n+1} e_n$ (because of framework axiom (\\ref{axiom: An en})) as an $A_{n+1}$--$A_{n-1}$--bimodule. One can easily check that\n$i(A_n e_n) \\cong e_n A_n $ as $A_{n- 1}$--$A_{n+1}$--bimodules. Therefore,\n\\begin{equation} \\label{equation 2 for lemma 1 of induction on j}\ni( \\Delta^{\\lambda}) \\otimes_{A_{n-1}} e_n A_n \\cong i( \\Delta^{\\lambda}) \\otimes_{A_{n-1}} i( A_n e_n)\n\\cong i(A_n e_n\\otimes_{A_{n-1}} \\Delta^{\\lambda}),\n\\end{equation}\nusing Lemma \\ref{lemma; involutions and tensor products of bimodules}. By framework axioms \n(\\ref {axiom: en An en}) and (\\ref{axiom: An en}), $A_n e_n \\cong A_n$ as $A_n$--$A_{n-1}$--bimodules. Hence,\n\\begin{equation} \\label{equation 3 for lemma 1 of induction on j}\nA_n e_n\\otimes_{A_{n-1}} \\Delta^{\\lambda} \\cong A_n \\otimes_{A_{n-1}} \\Delta^{\\lambda} = {\\rm Ind}_{A_{n-1}}^{A_n}( \\Delta^{\\lambda}),\n\\end{equation}\nas $A_n$ modules. Combining (\\ref{equation 1 for lemma 1 of induction on j}), \n(\\ref{equation 2 for lemma 1 of induction on j}), and (\\ref{equation 3 for lemma 1 of induction on j}), we have\n\\begin{equation} \\label{equation 4 for lemma 1 of induction on j}\n\\begin{aligned}\n&(A_n e_n\\otimes_{A_{n-1}} J(\\Gamma')\\otimes_{A_{n-1}} e_{n}A_n) \/ \\beta_{\\Gamma,\\Gamma'}(A_n e_{n}\\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n ) \\\\\n&\\cong {\\rm Ind}_{A_{n-1}}^{A_n}( \\Delta^{\\lambda}) \\otimes_R i( {\\rm Ind}_{A_{n-1}}^{A_n}( \\Delta^{\\lambda})),\n\\end{aligned}\n\\end{equation}\nas $A_n$--$A_n$--bimodules.\n\n\nBy the induction assumption on $n$, ${\\rm Ind}_{A_{n-1}}^{A_n}( \\Delta^{\\lambda})$ has a filtration\nwith subquotients isomorphic to cell modules for $A_n$, and in particular ${\\rm Ind}_{A_{n-1}}^{A_n}( \\Delta^{\\lambda})$ is a free $R$--module. By (\\ref{equation 4 for lemma 1 of induction on j}), \n$$\n(A_n e_n\\otimes_{A_{n-1}} J(\\Gamma')\\otimes_{A_n} e_{n}A_n) \/ \\beta_{\\Gamma,\\Gamma'}(A_n e_{n}\\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n )\n$$\nis a free $R$--module. Since $A_n e_{n }\\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n$ is free by hypothesis, and $\\beta_{\\Gamma,\\Gamma'}$ is injective, \n $$\\beta_{\\Gamma,\\Gamma'}(A_n e_{n}\\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n )$$ is a free $R$--module. Hence \n$$\nA_n e_n\\otimes_{A_{n-1}} J(\\Gamma')\\otimes_{A_{n-1}} e_{n}A_n\n$$\nis also a free $R$--module.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemma 2 for induction on j} Let $\\Gamma$ be an order ideal in $\\Lambda_{n-1}$.\n If $A_n e_n\\otimes_{A_{n-1}} J(\\Gamma) \\otimes_{A_{n-1}} e_{n}A_n$ is a free $R$--module, then $\\Phi_\\Gamma$ is an isomorphism.\n\\end{lemma}\n\n\n\n\n\\begin{proof} $\\Phi_\\Gamma$ is surjective, so we only have to prove $\\Phi_\\Gamma$ is injective.\nDefine\n$$\\alpha_1~:~ A_{n} e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}\\rightarrow A_{n}e_n\\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}\\otimes_R F$$\nand\n$$\\alpha_2~:~ A_{n} e_n J(\\Gamma) A_{n}\\rightarrow A_{n} e_n J(\\Gamma) A_{n}\\otimes_R F$$\nby $x \\mapsto x \\otimes \\bm 1_F$.\n Since $A_{n}e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}$ is a free $R$--module, by assumption, \n $\\alpha_1$ is injective, according to Lemma \\ref{lemma injectivity of x to x tensor 1}. Let \n $$\n \\tau: A_{n} e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}\\otimes_R F \\rightarrow\n A_n^F e_n \\otimes_{ A_{n-1}^F}J(\\Gamma)^F \\otimes_{ A_{n-1}^F} e_n A_n^F\n $$\n be the isomorphism from Lemma \\ref{second tensor iso}. (We are writing $e_n$ for $e_n \\otimes \\bm 1_F$.)\n Let\n $$\n \\Phi_\\Gamma^F: A_n^F e_n \\otimes_{ A_{n-1}^F}J(\\Gamma)^F \\otimes_{ A_{n-1}^F} e_n A_n^F\t\\rightarrow\t A_n^F e_nJ(\\Gamma)^F A_n^F\n $$\n be defined by $xe_n \\otimes a \\otimes e_n y \\mapsto x e_n a y$. \n \n Consider the following diagram\n \\newarrow{Equals} =====\n \\begin{diagram}\n A_n^F e_n \\otimes_{ A_{n-1}^F}J(\\Gamma)^F \\otimes_{ A_{n-1}^F} e_n A_n^F && \\rTo^{\\Phi_\\Gamma^F} && A_n^F e_nJ(\\Gamma)^F A_n^F \\\\\n \\uTo^{\\tau}&&&& \\uEquals\\\\\n A_{n}e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}\\otimes_R F &&\\rTo^{\\Phi_\\Gamma\\otimes id_F}&&\n A_{n} e_n J(\\Gamma) A_{n}\\otimes_R F\\\\\n \\uTo^{\\alpha_{1}}&&&&\\uTo_{\\alpha_{2}}\\\\\n A_{n}e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}&&\\rTo^{\\Phi_\\Gamma}&&A_{n} e_n J(\\Gamma) A_{n}.\\\\\n \\end{diagram}\n \\ignore{\n$$\\begin{CD}\n A_n^F e_n \\otimes_{ A_{n-1}^F}J(\\Gamma)^F \\otimes_{ A_{n-1}^F} e_n A_n^F\t@>\\Phi_\\Gamma^F>>\t A_n^F e_nJ(\\Gamma)^F A_n^F \\\\\n@A\\tau AA\t@|\\\\\n A_{n}e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n}\\otimes_R F @>\\Phi_\\Gamma\\otimes id_F>> A_{n} e_n J(\\Gamma) A_{n}\\otimes_R F\\\\\n@A\\alpha_1AA\t@AA\\alpha_2A\\\\\n A_{n}e_n \\otimes_{A_{n-1}} J(\\Gamma)\\otimes_{A_{n-1}} e_n A_{n} @>\\Phi_\\Gamma>> A_{n} e_n J(\\Gamma) A_{n}.\\\\\n\\end{CD}$$\n}\nIt is straightforward to check that $\\Phi_\\Gamma^F \\circ \\tau\\circ \\alpha_1 = \\alpha_2 \\circ \\Phi_\\Gamma$. Thus, to prove that $\\Phi_\\Gamma$ is injective, it suffices to show that $\\Phi_\\Gamma^F$ is injective.\n\n Define \n $$\\beta~:~ A_n^F e_n \\otimes_{A_{n-1}^F}J(\\Gamma)^F\\otimes_{A_{n-1}^F} e_n A_n^F\\rightarrow A_n^F e_n\\otimes_{A_{n-1}^F} e_n A_n^F$$\n by $\\beta(x\\otimes y\\otimes z)=x\\otimes yz$. Observe that $\\beta$ is injective by\n Lemma \\ref{beta inj}. Define $$\\phi^F : A_n^F e_n\\otimes_{A_{n-1}^F} e_n A_n^F \\to A_n^F e_n A_n^F$$ by\n $\\phi^F( x e_n \\otimes e_n y) = x e_n y$. Observe that $\\phi^F \\circ \\beta = \\Phi_\\Gamma^F$, so to prove that $\\Phi_\\Gamma^F$ is injective, it suffices to show that $\\phi^F$ is injective.\n \n \nSince $A_{n+1}^F$ is split semisimple (by framework axiom (\\ref{axiom: semisimplicity})), the ideal $A_{n+1}^F e_n A_{n+1}^F$\n(which equals $ A_n^F e_n A_n^F$ by framework axiom (\\ref{axiom: An en})) is a unital algebra in its own right, and Morita equivalent to $e_n A_{n+1}^F e_n = e_n A_n^F e_n \\cong A_{n-1}^F$.\nIn fact, let \n$$\\psi^F : e_n A_n \\otimes_{A_n^F e_n A_n^F} A_n^F e_n \\to e_n A_n^F e_n$$ be given by \n$e_n x \\otimes y e_n \\mapsto (1\/\\delta_n) e_n xy e_n$, where $e_n^2 = \\delta_n e_n$. Then\n$$(e_n A_{n}^F e_n, A_n^F e_n A_n^F, A_n^F e_n, e_n A_n^F, \\psi^F, \\phi^F)$$\n is a Morita context, in the sense of ~\\cite{Jacobson}, Section 3.12, with surjective\nbimodule maps $\\psi^F$ and $\\phi^F$. It follows from Morita theory, for example ~\\cite{Jacobson}, Morita Theorem I, page\n167, that $\\psi^F$ and $\\phi^F$ are isomorphisms.\n\\end{proof}\n\n\n\\noindent\n{\\em Proof of Proposition \\ref{proposition: Phi isomorphism}}: \\quad \nLet $\\Gamma$ be an order ideal of $\\Lambda_{n-1}$. There exists a chain of order ideals\n$$\n\\emptyset = \\Gamma_0 \\subseteq \\Gamma_1 \\subseteq \\cdots \\subseteq \\Gamma_s = \\Gamma,\n$$\nsuch that the difference between any two successive order ideals is a singleton. Write $\\beta_j$ for\n$\\beta_{\\Gamma_j, \\Gamma_{j+1}}$, for $0 \\le j < s$.\n\nWe prove by induction that for $0 \\le j \\le s$, \n$\\Phi_{\\Gamma_j}$ is an isomorphism and $A_n e_n \\otimes_{A_{n-1}}J(\\Gamma_j) \\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module; and that for $0 \\le j < s$, $\\beta_j$ is injective.\nFor $j = 0$, these statements are trivial since $J(\\emptyset) = 0$.\n\nFix $j$ ($0 \\le j < s$) and suppose that \n$A_n e_n \\otimes_{A_{n-1}}J(\\Gamma_j)\\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module, that\n$\\Phi_{\\Gamma_j}$ is an isomorphism. Then it follows from Lemma \\ref{lemma 1 for induction on j} that \n$ A_n e_n \\otimes_{A_{n-1}}J(\\Gamma_{j+1})\\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module. Next, it follows from Lemma \\ref{lemma 2 for induction on j} that $\\Phi_{\\Gamma_{j+1}}$ is an isomorphism. \n\nWe conclude that $A_n e_n \\otimes_{A_{n-1}}J(\\Gamma)\\otimes_{A_{n-1}} e_n A_n$ is a free $R$--module and that\n $\\Phi_\\Gamma$ is an isomorphism. Applying Lemma \n\\ref{lemma 1 for induction on j} again gives statement (3) of the Proposition.\n \\qed\n \n\n\\medskip\n\nWe continue to work with the following assumptions: \n $R$ is an integral domain with field of fractions $F$. $(Q_k)_{k\\ge 0}$ and\n$(A_k)_{k\\ge 0}$ are two towers of $R$--algebras satisfying the framework axioms of Section \\ref{subsection: framework axioms}. \nThe following induction assumption is in force:\nFor some fixed $n \\ge 1$, the conclusions (1) --(4) of Theorem \\ref{main theorem} hold for the finite tower $(A_k)_{0 \\le k \\le n}$.\nWe use the notation of the discussion preceding Lemma \\ref{second tensor iso}.\n\nThe following is a corollary of Proposition \\ref{proposition: Phi isomorphism}.\n\n\\begin{corollary} \\label{corollary: a basic construction isomorphism}\n$A_n e_n \\otimes_{A_{n-1}} e_n A_n \\cong A_n e_n A_n$, as $A_{n+1}$--$A_{n+1}$ bimodules, with the isomorphism determined by $x e_n \\otimes e_n y \\mapsto x e_n y$. \n\\end{corollary}\n\n\\begin{proof} In Proposition \\ref{proposition: Phi isomorphism}, take $\\Gamma = \\Lambda_{n-1}$, so \n$J(\\Gamma) = A_{n-1}$. \n\\end{proof}\n\n\\begin{proposition} \\mbox{} \\label{lemma: cellularity induction step}\n\\begin{enumerate}\n\\item $\\Gamma \\mapsto \\hat J(\\Gamma)$ is a $\\Lambda_{n-1}$--cell net in $A_n e_n A_n$.\n\\item\n $A_n e_n A_n$ is a cellular ideal in $A_{n+1}$.\n\\item $A_{n+1}$ is a cellular algebra. The partially ordered set in the cell datum for $A_{n+1}$ can be realized as\n$\\Lambda_{n+1} = \\Lambda_{n-1} \\cup \\Lambda_{n+1}\\spp 0$, where $ \\Lambda_{n+1}\\spp 0$ is the partially ordered set in the cell datum for\n$Q_{n+1}$; moreover the partial order on $\\Lambda_{n+1}$ agrees with the original partial orders on $\\Lambda_{n-1} $ and \n$\\Lambda_{n+1}\\spp 0$, and satisfies $\\lambda > \\mu$ if $\\lambda \\in \\Lambda_{n-1}$ and $\\mu \\in \\Lambda_{n+1}\\spp 0$.\n\\item Let $\\lambda \\in \\Lambda_{n-1}$, and let $ \\Delta^\\lambda$ denote the corresponding cell module of $A_{n-1}$. The cell module of $A_{n+1}$ corresponding to $\\lambda$ is isomorphic to \n$A_n e_n \\otimes_{A_{n-1}} \\Delta^\\lambda$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} It is evident that $\\hat J(\\emptyset) = \\{0\\}$, and that $\\Gamma_1 \\subseteq \\Gamma_2$ implies $\\hat J(\\Gamma_1) \\subseteq \\hat J(\\Gamma_2)$.\nNote that $J(\\Gamma_{\\ge \\lambda}) = A_{n-1}^\\lambda$, so $\\hat J(\\Gamma_{\\ge \\lambda}) = A_n e_n A_{n-1}^\\lambda A_n$.\nSimilarly, $\\hat J(\\Gamma_{> \\lambda}) = A_n e_n \\breve A_{n-1}^\\lambda A_n$. It follows that\n$A_n e_n A_n = {\\rm span}\\{\\hat J(\\Gamma_{\\ge \\lambda}) : \\lambda \\in \\Lambda_{n-1}\\}$ and that\nfor all $\\lambda \\in \\Lambda_{n-1}$, $\\hat J(\\Gamma_{> \\lambda}) = {\\rm span}\\{ \\hat J(\\Gamma_{\\ge \\mu}) : \\mu > \\lambda\\}$. We have shown that\n$\\Gamma \\mapsto \\hat J(\\Gamma)$ satisfies conditions (1) and (2) of Definition \\ref{definition: cell net}.\n \n Next we show that $\\Gamma \\mapsto \\hat J(\\Gamma)$ satisfies condition (3) of Definition \\ref{definition: cell net}.\nLet $\\Gamma \\subseteq \\Gamma'$ be two order ideals of $\\Lambda_{n-1}$, with $\\Gamma' \\setminus \\Gamma = \\{\\lambda\\}$.\nFrom the proof of Proposition \\ref{proposition: Phi isomorphism}, we already have $\\hat J(\\Gamma')\/\\hat J(\\Gamma) \\cong M^\\lambda \\otimes_R i(M^\\lambda)$, with\n$M^\\lambda = A_n e_n\\otimes_{A_{n-1}} \\Delta^{\\lambda} $. Let $\\chi : \\hat J(\\Gamma')\/\\hat J(\\Gamma) \\to M^\\lambda \\otimes_R i(M^\\lambda)$ denote the isomorphism. We have to check that $\\chi \\circ i = i \\circ \\chi$. The isomorphism\n$\\Phi_\\Gamma$ of Proposition \\ref{proposition: Phi isomorphism} satisfies $i \\circ \\Phi_\\Gamma = \\Phi_\\Gamma\\circ i$. Moreover,\n$$\\beta_{\\Gamma, \\Gamma'}( A_n e_n \\otimes_{A_{n-1}} J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n) \\subseteq A_n e_n \\otimes_{A_{n-1}} J(\\Gamma') \\otimes_{A_{n-1}} e_n A_n$$\nand $\\hat J(\\Gamma) \\subseteq \\hat J(\\Gamma')$ are $i$--invariant, so the induced isomorphism\n$$\n\\begin{aligned}\n\\tilde \\Phi_\\Gamma : A_n e_n \\otimes_{A_{n-1}} J(\\Gamma')& \\otimes_{A_{n-1}} e_n A_n\/ \\beta_{\\Gamma, \\Gamma'}( A_n e_n \\otimes_{A_{n-1}} J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n) \\\\ &\\to \\hat J(\\Gamma')\/\\hat J(\\Gamma)\n\\end{aligned}\n$$\nsatisfies $i \\circ \\tilde\\Phi_\\Gamma = \\tilde\\Phi_\\Gamma\\circ i$. Next, the map $$\\pi : A_n e_n \\otimes_{A_{n-1}} J(\\Gamma') \\otimes_{A_{n-1}} e_n A_n \\to\nA_n e_n \\otimes_{A_{n-1}} J(\\Gamma')\/J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n$$\nsatisfies $i \\circ \\pi = \\pi \\circ i$, \nso the induced isomorphism\n$$\n\\begin{aligned}\n\\tilde \\pi: A_n e_n \\otimes_{A_{n-1}} J(\\Gamma') &\\otimes_{A_{n-1}} e_n A_n\/ \\beta_{\\Gamma, \\Gamma'}( A_n e_n \\otimes_{A_{n-1}} J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n) \\\\ &\\to A_n e_n \\otimes_{A_{n-1}} J(\\Gamma')\/J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n\n\\end{aligned}\n$$\nsatisfies $i \\circ \\tilde \\pi = \\tilde \\pi \\circ i$. Finally, we have an isomorphism $\\alpha: J(\\Gamma')\/J(\\Gamma) \\to \\Delta^{\\lambda} \\otimes_R i(\\Delta^{\\lambda})$ satisfying $i \\circ \\alpha = \\alpha \\circ i$, so the map\n$$\\begin{aligned}\n\\bar\\alpha = \\id \\otimes \\alpha \\otimes \\id : &A_n e_n \\otimes_{A_{n-1}} J(\\Gamma')\/J(\\Gamma) \\otimes_{A_{n-1}} e_n A_n \\to \\\\\n&A_n e_n \\otimes_{A_{n-1}} \\Delta^{\\lambda} \\otimes_R i(\\Delta^{\\lambda}) \\otimes_{A_{n-1}} e_n A_n\n\\end{aligned}$$\nsatisfies $i \\circ \\bar\\alpha = \\bar\\alpha \\circ i$.\nThe map $\\chi$ is $ \\bar\\alpha \\circ \\tilde \\pi \\circ \\tilde \\Phi_\\Gamma^{-1}$, so we have $i \\circ \\chi = \\chi \\circ i$.\n\nThis completes the proof that $\\Gamma \\mapsto \\hat J(\\Gamma)$ is a $\\Lambda_{n-1}$--cell net in $A_n e_n A_n$. By\nProposition \\ref{lemma: cell net characterization of cellularity}, $A_n e_n A_n$ has a cell datum with partially ordered set equal to $\\Lambda_{n-1}$. Moreover, since\nthe isomorphisms $\\hat J(\\Gamma')\/\\hat J(\\Gamma) \\cong M^\\lambda \\otimes_R i(M^\\lambda)$ are actually \nisomorphisms of $A_{n+1}$--$A_{n+1}$--bimodules, the cellular basis $\\tilde{\\mathcal C}$ of $A_n e_n A_n$\nsatisfies the property (2) of Definition \\ref{gl cell} not only for $a \\in A_n e_n A_n$ but also for $a \\in A_{n+1}$;\nthat is $A_n e_n A_n$ is a cellular ideal in $A_{n+1}$.\n\nStatement (3) of the Lemma follows from applying Remark \\ref{remark on extensions of cellular algebras}.\nStatement (4) follows from the isomorphism $\\hat J(\\Gamma')\/\\hat J(\\Gamma) \\cong M^\\lambda \\otimes_R i(M^\\lambda)$.\n\\end{proof}\n\n\\begin{corollary} \\label{corollary: p.o. set for cellular structure}\n\n The description of the partially ordered set given in Theorem \\ref{main theorem}, point (2), is valid for $k = n+1$.\n\\end{corollary}\n\n\\begin{proof}\n Combining point (3) of Proposition \\ref{lemma: cellularity induction step} with the induction assumption (specifically the description of $\\Lambda_{n-1}$ as the union of copies of \n$\\Lambda_{n-1}\\spp 0$, $\\Lambda_{n-3}\\spp 0$, etc.), we see that $\\Lambda_{n+1}$ is the union of copies of \n$\\Lambda_{n+1}\\spp 0$, $\\Lambda_{n-1}\\spp 0$, $\\Lambda_{n-3}\\spp 0$, etc., with the following partial order:\nthe partial order agrees with the original partial order on each $\\Lambda_i \\spp 0$, and \n$\\lambda > \\mu$ if $\\lambda \\in \\Lambda_i \\spp 0$, $\\mu \\in \\Lambda_j \\spp 0$, and $i < j$. \n\\end{proof}\n\n\n\nFor the remainder of Section \\ref{section: basic construction preserves cellularity}, we denote elements of $\\Lambda_k$ ($0 \\le k \\le n+1$) by ordered pairs $(\\lambda, k)$, where it is understood that\n$\\lambda \\in \\Lambda_i\\spp 0$ for some $i \\le k$ with $k -i$ even.\n\n\\begin{corollary} \\label{corollary: cell modules and An en An}\n Point (3) of Theorem \\ref{main theorem} holds for $k = n+1$.\n\\end{corollary}\n\n\\begin{proof}\n The cell modules of $A_{n+1}$ are of two types: There are the cell modules\n$\\Delta^{(\\lambda, n+1)}$ with $\\lambda \\in \\Lambda_{n+1}\\spp 0$, which are actually cell modules of\n$A_{n+1}\/(A_n e_n A_n) \\cong Q_{n+1}$. These satisfy $$A_n e_n A_n \\ \\Delta^{(\\lambda, n+1)} = 0.$$\nOn the other hand, there are the cell modules of the cellular ideal $A_n e_n A_n$, namely\n$\\Delta^{(\\lambda, n+1)} = A_n e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda, n-1)}$, with\n$\\lambda \\in \\Lambda_i\\spp 0$ for some $i < n+1$ with $n+1 -i$ even. These satisfy\n$$A_n e_n A_n \\ \\Delta^{(\\lambda, n+1)} = \nA_n e_n A_n e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda, n-1)}.\n$$\nBut \n$$\nA_n e_n A_n e_n \\otimes_R F = A_n^F A_{n-1}^F e_n = A_n^F e_n, \n$$ using framework axiom (\\ref{axiom: en An en}), so we have $$A_n e_n A_n \\ \\Delta^{(\\lambda, n+1)} \\otimes_R F = \\Delta^{(\\lambda, n+1)} \\otimes_R F ,$$\nby application of Lemma \\ref{first tensor iso}..\n\\end{proof}\n\n\n\\subsection{Cell filtrations of restrictions and induced modules}\n\nNext we show that the restriction of a cell module from $A_{n+1}$ to $A_n$, and the induction of a cell module from\n$A_n$ to $A_{n+1}$, have cell filtrations.\n\n\\begin{proposition} \\label{lemma: cell filtration of restrictions}\nLet $(\\lambda, n+1) \\in \\Lambda_{n+1}$, and let $\\Delta = \\Delta^{(\\lambda, n+1)}$ be the corresponding\ncell module of $A_{n+1}$. Then the restriction of $\\Delta$ to $A_n$ has a cell filtration.\n\\end{proposition}\n\n\\begin{proof} Write ${\\rm Res}(\\Delta)$ for the restriction to $A_n$.\n\nIf $A_{n+1} e_n A_{n+1} \\ \\Delta = 0$, then $\\Delta$ is an $Q_{n+1}$--module; moreover,\nby framework axiom (\\ref{axiom: e(n-1) in An en An}) from Section \\ref{subsection: framework axioms}, $A_n e_{n-1} A_n\\ {\\rm Res}(\\Delta) = 0$ as well, so ${\\rm Res}(\\Delta)$ is a $Q_{n}$--module.\nThen it follows from the assumption of coherence of $(Q_k)_{k \\ge 0}$ that ${\\rm Res}(\\Delta)$ has a cell filtration as an $Q_n$--module, hence as an $A_n$--module.\n\nIf $A_{n+1} e_n A_{n+1}\\ \\Delta \\ne 0$, then $\\lambda \\in \\Lambda_i\\spp 0$ for some $i < n$, and \n$$\\Delta \\cong A_n e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda, n-1)}.$$ \nSince $A_n e_n \\cong A_n$ as $A_n$--$A_{n-1}$ bimodules, \n ${\\rm Res}(\\Delta) \\cong \n{\\rm Ind}_{A_{n-1}}^{A_{n}}( \\Delta^{(\\lambda, n-1)})$, which has a cell filtration by the induction assumption.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma} \\label{lemma: sort of flatness}\n Let $R$ be an integral domain with field of fractions $F$. Let $A$ be a unital $R$--algebra, $P$ a right $A$--module, and $N_1 \\subseteq N_2$ left $A$--modules, such that\n\\begin{enumerate}\n\\item $A^F = A \\otimes_R F$ is semisimple, and \n\\item $N_2$ and $P \\otimes_A N_1$ are free $R$--modules.\n\\end{enumerate}\nLet $\\iota : N_1 \\to N_2$ denote the injection. Then $$\\id_P \\otimes \\iota : P \\otimes_A N_1 \\to P \\otimes_A N_2$$ is injective.\n\\end{lemma}\n\n\\begin{proof} First, $\\iota \\otimes \\id_F : N_1 \\otimes_R F \\to N_2 \\otimes_R F$ is injective by Lemma \\ref{injectivity of iota tensor id(F) with free R modules}.\nWrite $\\beta = \\id_P \\otimes \\iota$, and let \n$$\\beta^F = \\id_{P^F} \\otimes (\\iota \\otimes \\id_F) : P^F \\otimes_{A^F} N_1^F \\to \nP^F \\otimes_{A^F} N_2^F. \n$$\nSince $A^F$ is semisimple, $P^F$ is projective; hence $\\beta^F$ is injective.\n\nConsider the following diagram: \n\\begin{diagram}\n P^F \\otimes_{A^F} N^F_{1} \t&&\\rTo^{\\beta^F} &&\t P^F \\otimes_{A^F} N^F_{2} \t\\\\\n \\uTo^{\\tau_{1}} &&&& \\uTo_{\\tau_{2}}\\\\\n P \\otimes_{A} N_1 \\otimes_R F &&\\rTo^{\\beta\\otimes id_F} && P \\otimes_{A} N_2 \\otimes_R F\\\\\n\\uTo^{\\alpha_1}&&&&\t\\uTo_{\\alpha_2}\\\\\n P \\otimes_{A} N_1 &&\\rTo^{\\beta}&& P \\otimes_{A} N_2,\\\\\n\\end{diagram}\n\\ignore{\n$$\\begin{CD}\n P^F \\otimes_{A^F} N^F_{1} \t@>\\beta^F>>\t P^F \\otimes_{A^F} N^F_{2} \t\\\\\n@A\\tau_1 AA\t@A\\tau_2 AA\\\\\nP \\otimes_{A} N_1 \\otimes_R F @>\\beta\\otimes id_F>> P \\otimes_{A} N_2 \\otimes_R F\\\\\n@A\\alpha_1AA\t@AA\\alpha_2A\\\\\n P \\otimes_{A} N_1 @>\\beta>> P \\otimes_{A} N_2,\\\\\n\\end{CD}$$\n}\nwhere $\\alpha_i$ is determined by $x \\mapsto x \\otimes 1_F$ and $\\tau_i$ is the isomorphism of Lemma \\ref{first tensor iso} ($i = 1, 2$). Note that \n$\\alpha_1$ is injective\nby Lemma \\ref{lemma injectivity of x to x tensor 1}, since $ P \\otimes_{A} N_1$ is assumed to be free over $R$. One can check that\n$\\beta^F \\circ \\tau_1 \\circ \\alpha_1 = \\tau_2 \\circ \\alpha_2 \\circ \\beta$. It follows that $\\beta$ is injective.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemma: globalization preserves cell filtrations}\n Let $M$ be an $A_{n-1}$ module with a cell filtration:\n$$\n(0) = M_0 \\subseteq M_1 \\subseteq \\cdots \\subseteq M_t = M,\n$$\nwith $M_j\/M_{j-1} \\cong \\Delta^{(\\lambda_j, n-1)}$ for $1 \\le j \\le t$. Then for $1 \\le j \\le t$,\n\\begin{enumerate}\n\\item $A_n e_n \\otimes_{A_{n-1}} M_{j}$ is a free $R$--module,\n\\item $A_n e_n \\otimes_{A_{n-1}} M_{j-1}$ imbeds in $A_n e_n \\otimes_{A_{n-1}} M_{j}$, and\n\\item $(A_n e_n \\otimes_{A_{n-1}} M_{j})\/(A_n e_n \\otimes_{A_{n-1}} M_{j-1}) \\cong \nA_n e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda_j, n-1)}$.\n\\end{enumerate}\nThus, the $A_{n+1}$--module $A_n e_n \\otimes_{A_{n-1}} M$ has a cell filtration with subquotients\n$\\Delta^{(\\lambda_j, n+1)} = A_n e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda_j, n-1)}$ ($1 \\le j \\le t$).\n\\end{lemma}\n\n\\begin{proof} We have $M_1 \\cong \\Delta^{(\\lambda_1, n-1)}$, so $A_n e_n \\otimes_{A_{n-1}} M_1$ is a free $R$--module.\nFix $j \\ge 2$ and suppose that $A_n e_n \\otimes_{A_{n-1}} M_{j-1}$ is a free $R$--module.\nLet $\\iota: M_{j-1} \\to M_j$ denote the injection and let $$\\beta = \\id_{ A_n e_n} \\otimes \\iota : A_n e_n \\otimes_{A_{n-1}} M_{j-1} \\to A_n e_n \\otimes_{A_{n-1}} M_{j}.$$\nThen $\\beta$ is injective by an application of Lemma \\ref{lemma: sort of flatness}, with \n$A = A_{n-1}$, $P = A_n e_n$, $N_1 = M_{j-1}$, and $N_2 = M_j$.\nThe quotient $$( A_{n}e_n \\otimes_{A_{n-1}} M_{j},)\/\\beta( A_{n}e_n \\otimes_{A_{n-1}} M_{j-1})$$ is free over $R$,\nbecause\n$$\n\\begin{aligned}\n( A_{n}e_n \\otimes_{A_{n-1}} &M_{j})\/\\beta( A_{n}e_n \\otimes_{A_{n-1}} M_{j-1}) \\\\\n&\\cong\nA_{n}e_n \\otimes_{A_{n-1}} (M_j\/ M_{j-1}) \\\\\n&\\cong A_{n}e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda_j, n-1)}.\n\\end{aligned}\n$$\nConsequently, $ A_{n}e_n \\otimes_{A_{n-1}} M_{j}$ is free over $R$. All the assertions of the lemma now follow by induction on $j$.\n\\end{proof}\n\n\\begin{lemma} \\label{induction to An en An from An}\n Let $M$ be an $A_n$--module, \nand let ${\\rm Res}(M)$ denote the restriction of $M$ to $A_{n-1}$. We have\n$$A_n e_n A_n \\otimes_{A_n} M \\cong A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(M),\n$$\nas $A_{n+1}$ modules.\n\\end{lemma}\n\n\\begin{proof} \n\\ignore{The map from $A_n e_n A_n \\times M$ to \n$A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(M)$ determined by\n$$\n(\\sum_i x'_i e_n x''_i, m) \\mapsto \\sum_i (x'_i e_n \\otimes x''_i m)\n$$\nis $R$--bilinear and $A_n$--balanced, so yields a linear map $\\psi: A_n e_n A_n \\otimes_{A_n} M \\to A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(M)$. Likewise the map from $A_n e_n \\times M$ to\n$A_n e_n A_n \\otimes_{A_n} M$ determined by $(x e_n, m) \\mapsto x e_n \\otimes m$ is\n$R$--bilinear and $A_{n-1}$--balanced, so gives a linear map\n$\\phi : A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(M) \\to A_n e_n A_n \\otimes_{A_n} M $. It is straightforward to check that $\\psi$ and $\\phi$ are inverses.\n}\nBy Corollary \\ref{corollary: a basic construction isomorphism}, we have $A_n e_n A_n \\cong A_n e_n \\otimes_{A_{n-1}} e_n A_n \\cong\n A_n e_n \\otimes_{A_{n-1}} A_n$ as $A_{n+1}$--$A_n$ bimodules. Thus\n $$A_n e_n A_n \\otimes_{A_n} M \\cong A_n e_n \\otimes_{A_{n-1}} A_n \\otimes_{A_n} M \\cong\n A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(M).$$\n\\end{proof}\n\n\\begin{proposition} \\label{proposition: cell filtration of induced modules}\n Let ${(\\mu, n)} \\in \\Lambda_n$ and let $\\Delta^{(\\mu, n)}$ be the corresponding cell module of \n$A_n$. \n\\begin{enumerate}\n\\item\n $A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)}$ has cell filtration (as an\n$A_{n+1}$--module). In particular, $A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)}$ is free as an $R$--module.\n\\item $A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)}$ imbeds in ${\\rm Ind}_{A_n}^{A_{n+1}}(\\Delta^{(\\mu, n)})$,\nand $${\\rm Ind}_{A_n}^{A_{n+1}}(\\Delta^{(\\mu, n)})\/(A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)}) \\cong\nQ_{n+1} \\otimes_{A_n} \\Delta^{(\\mu, n)}.$$\n\\item $Q_{n+1} \\otimes_{A_n} \\Delta^{(\\mu, n)}$ has cell filtration (as a $Q_{n+1}$--module, hence as an \n$A_{n+1}$--module). \n\\item ${\\rm Ind}_{A_n}^{A_{n+1}}(\\Delta^{(\\mu, n)})$ has a cell filtration.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} For point (1), \nlet ${\\rm Res}(\\Delta^{(\\mu, n)})$ denote the restriction to $A_{n-1}$.\nBy Lemma \\ref{induction to An en An from An}, we have\n$A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)} \\cong A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(\\Delta^{(\\mu, n)})$, as\n$A_{n+1}$ modules. By the induction assumption stated at the beginning of Section \n\\ref{section: basic construction preserves cellularity}, $ {\\rm Res}(\\Delta^{(\\mu, n)})$ has cell filtration,\n$$\n(0) = M_0 \\subseteq M_1 \\subseteq \\cdots \\subseteq M_t = {\\rm Res}(\\Delta^{(\\mu, n)}),\n$$\nwith $M_j\/M_{j-1} \\cong \\Delta^{(\\lambda_j, n-1)}$ for some $(\\lambda_j, n-1) \\in \\Lambda_{n-1}$. By Lemma \\ref {lemma: globalization preserves cell filtrations},\n$A_n e_n \\otimes_{A_{n-1}} {\\rm Res}(\\Delta^{(\\mu, n)})$ has a cell filtration with subquotients\n$\\Delta^{(\\lambda_j, n+1)} = A_n e_n \\otimes_{A_{n-1}} \\Delta^{(\\lambda_j, n-1)}$.\n\nPoint (2) follows from Lemma \\ref{lemma: sort of flatness} (with left and right modules interchanged), taking\n$A = A_n$, $P = \\Delta^{(\\mu, n)}$, $N_1 = A_n e_n A_n$, and $N_2 = A_{n+1}$. Note that\n$A_{n+1}$ is a free $R$--module by Proposition \\ref{lemma: cellularity induction step}, and $A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)} $ is a free $R$--module by point (1). The statement regarding the quotient follows from the right exactness of tensor products.\n\nFor $n=1$, $A_1 = Q_1$, and $\\Delta^{(\\mu, n)}$ is an $Q_1$--cell module; statement (3) follows from\nthe assumption of coherence of $(Q_k)_{k \\ge 0}$. If $n \\ge 2$, then by the induction assumption,\neither $A_n e_{n-1} A_n\\ \\Delta^{(\\mu, n)} = \\Delta^{(\\mu, n)}$, or $A_n e_{n-1} A_n \\ \\Delta^{(\\mu, n)} = (0)$. In the former case,\n$$\n\\begin{aligned}\nQ_{n+1} \\otimes_{A_n} \\Delta^{(\\mu, n)} &= Q_{n+1} \\otimes_{A_n}A_n e_{n-1} A_n\\ \\Delta^{(\\mu, n)} \\\\\n&= Q_{n+1}A_n e_{n-1} A_n \\otimes_{A_n} \\Delta^{(\\mu, n)} = 0,\\\\\n\\end{aligned}\n$$\nbecause $e_{n-1} \\in A_{n+1} e_n A_{n+1}$, by the framework axiom (\\ref{axiom: e(n-1) in An en An}). In the latter case, $A_n e_{n-1} A_n$ annihilates both $Q_{n+1}$ and $\\Delta^{(\\mu, n)}$, so both are $A_n\/(A_n e_{n-1} A_n) \\cong Q_n$--modules.\nThus $Q_{n+1} \\otimes_{A_n} \\Delta^{(\\mu, n)} = Q_{n+1} \\otimes_{Q_n} \\Delta^{(\\mu, n)}$, which has\nan $Q_{n+1}$--cell filtration by the assumption of coherence of $(Q_k)_{k \\ge 0}$. This proves point (3).\n\nFinally, we have an exact sequence\n$$\n0 \\to A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)} \\to {\\rm Ind}_{A_n}^{A_{n+1}}(\\Delta^{(\\mu, n)}) \\to\nQ_{n+1} \\otimes_{A_n} \\Delta^{(\\mu, n)} \\to 0,\n$$\nwhere both $A_n e_n A_n \\otimes_{A_n} \\Delta^{(\\mu, n)}$ and $Q_{n+1} \\otimes_{A_n} \\Delta^{(\\mu, n)} $ have $A_{n+1}$--cell filtrations. Hence $ {\\rm Ind}_{A_n}^{A_{n+1}}(\\Delta^{(\\mu, n)})$ has an $A_{n+1}$--cell filtration.\n\\end{proof}\n\n\\begin{corollary} \\label{corollary: Ak coherent tower} \nThe finite tower $(A_k)_{0 \\le k \\le n+1}$ is a coherent tower of cellular algebras.\n\\end{corollary}\n\n\\begin{proof} Combine the induction hypothesis, Proposition \\ref{lemma: cellularity induction step}, Proposition \\ref{lemma: cell filtration of restrictions}, and Proposition \\ref{proposition: cell filtration of induced modules}.\n\\end{proof}\n\n\\begin{corollary} \\label{corollary: branching diagram obtained by reflections}\n The branching diagram for the finite tower \n $(A_k^F)_{0 \\le k \\le n + 1}$ is that obtained by reflections from the branching diagram\nof the finite tower $(Q_k^F)_{0 \\le k \\le n + 1}$.\n\\end{corollary}\n\n\\begin{proof} From the induction hypothesis, we already know that the branching diagram\nfor $(A_k^F)_{0 \\le k \\le n }$ is obtained by reflections from the branching diagram\nof the finite tower $(Q_k^F)_{0 \\le k \\le n}$. So we have only to consider the branching diagram\nfor $A_{n-1}^F \\subseteq A_n^F \\subseteq A_{n+1}^F$; specifically, we need to show that\nif $\\lambda \\in \\Lambda_i \\spp 0$ with $i < n+1$ and $n+1 -i $ even, and $(\\mu, n) \\in \\Lambda_n$ is arbitrary, \nthen $$(\\mu, n) \\nearrow (\\lambda, n+1) \\text{\\quad if, and only if \\quad } (\\lambda, n-1) \\nearrow (\\mu, n),$$\nin the branching diagram for $A_{n-1}^F \\subseteq A_n^F \\subseteq A_{n+1}^F$, and the number of \nedges connecting $(\\mu, n)$ and $(\\lambda, n+1)$ is the same as the number of edges connecting \n$(\\lambda, n-1)$ and $ (\\mu, n)$.\nBut this follows from Lemma \\ref{lemma: multiplicities in cell filtrations} and the proof of either Proposition \\ref{lemma: cell filtration of restrictions}, or Proposition \\ref{proposition: cell filtration of induced modules}, point (1).\n\\end{proof}\n\n\\medskip\n\\noindent\n{\\em Conclusion of the proof of Theorem \\ref{main theorem}.} \\ \\ Under the assumption that statements (1)--(4) of the theorem are valid for the finite tower $(A_k)_{0 \\le k \\le n}$, for some fixed $n$, we had to show that they are also valid for the tower $(A_k)_{0 \\le k \\le n + 1}$. This was verified in Corollary\n\\ref{corollary: Ak coherent tower}, Corollary \\ref{corollary: p.o. set for cellular structure}, Corollary \n\\ref{corollary: cell modules and An en An}, and Corollary \\ref{corollary: branching diagram obtained by reflections}. \n\n\n\\section{Examples}\n\n\\subsection{Preliminaries on tangle diagrams} \\label{subsection: preliminaries on tangle diagrams}\nSeveral of our examples involve {\\em tangle diagrams} in the rectangle $\\mathcal R = [0, 1] \\times [0, 1]$.\nFix points $a_i \\in [0, 1]$, $i \\ge 1$, with $0 < a_1 < a_2 < \\cdots$. Write\n$\\p i = (a_i, 1)$ and $\\overline{ \\p i} = (a_i, 0)$.\n\n\nRecall that a {\\em knot diagram} means a collection of piecewise smooth closed curves in the plane\nwhich may have intersections and self-intersections, but only simple\ntransverse intersections. At each intersection or crossing, one of the\ntwo strands (curves) which intersect is indicated as crossing\nover the other. \n\nAn {\\em $(n,n)$--tangle diagram} is a piece of a\nknot diagram in $\\mathcal R$ consisting of exactly $n$ topological intervals and possibly some number of closed curves, such that: (1) the endpoints of the intervals are the points $\\p 1, \\dots \\p n, \\pbar 1, \\dots, \\pbar n$, and these are the only points of intersection of the family of curves with the boundary of the rectangle, and (2) each interval intersects the boundary of the rectangle transversally. \n\n\nAn {\\em $(n,n)$--Brauer diagram} is a ``tangle\" diagram containing no closed curves, \nin which information about over and under crossings is ignored. Two Brauer diagrams are identified if the pairs of boundary points joined by curves is the same in the two diagrams.\nBy convention, there is a unique $(0, 0)$--Brauer diagram, the empty diagram with no curves.\nFor $n \\ge 1$, the number of $(n,n)$--Brauer diagrams is $(2n-1)!! = (2n-1)(2n-3)\\cdots (3)(1)$.\n\nA {\\em Temperley--Lieb} diagram is a Brauer diagram without crossings. For $n \\ge 0$, the number of $(n, n)$--Temperley--Lieb diagrams is the Catalan number $\\frac{1}{n+1} {2n \\choose n}$.\n\nFor any of these types of diagrams, we call $P = \\{\\p 1, \\dots, \\p n, \\pbar 1,\\dots, \\pbar n\\}$ the set of {\\em vertices} of the diagram, $P^+ = \\{\\p 1, \\dots, \\p n\\}$ the set of {\\em top vertices}, and\n$P^- = \\{\\pbar 1,\\dots, \\pbar n\\}$ the set of {\\em bottom vertices}. A curve or {\\em strand} in the diagram is called a {\\em vertical} or {\\em through} strand if it connects a top vertex and a bottom vertex, and a {\\em horizontal} strand if it connects two top vertices or two bottom vertices.\n\n\n\\subsection{The Brauer algebras}\n\\subsubsection{Definition of the Brauer algebras} \\label{subsection: Brauer algebras}\n\nLet $S$ be a commutative ring with identity, with a distinguished element $\\delta$.\nThe Brauer algebra $B_n(S, \\delta)$ is the free $S$--module with basis the set of $(n, n)$--Brauer diagrams, and with multiplication defined as follows.\nThe product of two Brauer diagrams is defined\nto be a certain multiple of another Brauer diagram. Namely, given two\nBrauer diagrams $a, b$, first ``stack\" $b$ over $a$; the result is a planar tangle that may contain some number of closed curves. Let $r$ denote the number of closed curves, and let $c$ be the Brauer\ndiagram obtained by removing all the closed curves. Then\n$\na b = \\delta^r c.\n$\n\n\\begin{definition}\\r\nFor $n \\ge 1$, the {\\em Brauer algebra} $B_n(S, \\delta)$ over $S$ with parameter $\\delta$ is the free $S$-module with basis the set of \n$(n,n)$-Brauer diagrams, with the bilinear product determined by the\nmultiplication of Brauer diagrams. In particular, $B_0(S, \\delta) = S$.\n\\end{definition}\n\n\nNote that the Brauer diagrams with only vertical strands are in\nbijection with permutations of $\\{1, \\dots, n\\}$, and that the\nmultiplication of two such diagrams coincides with the multiplication of\npermutations. Thus the Brauer algebra contains the group algebra $S\\mathfrak S_n$ of\nthe permutation group $\\mathfrak S_n$. The identity element of the Brauer algebra is the diagram corresponding to the trivial permutation.\n\n\n\n\\subsubsection{Brief history of the Brauer algebras}\nThe Brauer algebras were introduced by Brauer~\\cite{Brauer} as a device\nfor studying the invariant theory of orthogonal and symplectic groups.\nWenzl ~\\cite{Wenzl-Brauer} observed that generically, the sequence of Brauer algebras (over a field) is obtained\nby repeated Jones basic constructions from the symmetric group algebras; he used this to show that\n$B_n(k, \\delta)$ is semisimple, when $k$ is a field of characteristic zero and $\\delta$ is not an integer.\nGraham and Lehrer ~\\cite{Graham-Lehrer-cellular} showed that the Brauer algebras are cellular, and classified the simple modules of $B_n(k, \\delta)$ when $k$ is a field and $\\delta$ is arbitrary. Another illuminating proof of cellularity of\nthe Brauer algebras was given by K\\\"onig and Xi ~\\cite{KX-Brauer}. Enyang's two proofs of cellularity for \nBirman--Wenzl algebras ~\\cite{Enyang1, Enyang2} also apply to the Brauer algebras.\n\n\\subsubsection{Some properties of the Brauer algebras} In this section, write $B_n$ for $B_n(S, \\delta)$. \nFor $n \\ge 1$, let $\\iota$ denote the map from $(n,n)$--Brauer diagrams to\n$(n+1, n+1)$--Brauer diagrams that adds an additional strand to a diagram, connecting $\\p {n+1}$ to\n$\\pbar {n+1}$.\n$$\n\\iota: \\quad \\inlinegraphic{tangle_box2} \\quad \\mapsto \\quad \n\\inlinegraphic{iota}\n$$\nThe linear extension of $\\iota$ to $B_n$ is an injective unital homomorphism into\n$B_{n+1}$. Using $\\iota$, we identify $B_n$ with its image in $B_{n+1}$.\n\nFor $n \\ge 1$ define a map ${\\rm cl}$ from $(n,n)$--Brauer diagrams into $B_{n-1}$ as follows. First ``partially close\" a given $(n,n)$--Brauer diagram by adding an additional smooth curve connecting $\\p n$ to $\\pbar n$,\n$$\n \\inlinegraphic{tangle_box2} \\quad \\mapsto \\quad \n\\inlinegraphic{partial_closure}.\n$$\nIn case the resulting ``tangle\" contains a closed curve (which happens precisely when the original diagram already had a strand connecting $\\p n$ to $\\pbar n$), remove this loop and replace it with a factor of $\\delta$. The linear extension of ${\\rm cl}$ to $B_n$ is a (non-unital)\n$B_{n-1}$--$B_{n-1}$ bimodule map, and ${\\rm cl}\\circ\\iota(x) = \\delta\\ x$ for $x \\in \nB_n$.\n\nIf $\\delta$ is invertible in $S$, we can define $\\varepsilon_n = (1\/\\delta) {\\rm cl}$, which is a \nconditional expectation, that is, a unital $B_{n-1}$--$B_{n-1}$ bimodule map. We have\n${\\varepsilon_{n+1}}\\circ\\iota(x) = x$ for $x \\in \nB_n$. The map $\\varepsilon = \\varepsilon_1\\circ \\cdots \\circ \\varepsilon_n : B_n \\to B_0 \\cong S$ is a normalized trace; that is, $\\varepsilon(\\bm 1) = 1$ and $\\varepsilon(a b) = \\varepsilon(b a)$ for all $a, b$. The value of $\\varepsilon$ on a Brauer diagram $d$ is obtained as follows: first close all the strands of $d$ by introducing new curves joining $\\p j$ to $\\pbar j$ for all $j$; let $c$ be the number of components (closed loops) in the resulting\n$(0,0)$--tangle; then $\\varepsilon(d) = \\delta^{c - n}$ if $d \\in B_n$. The trace and condition expectation play an essential role in Wenzl's treatment of the structure of the Brauer algebra over ${\\mathbb Q}({\\mathbold \\delta})$\n~\\cite{Wenzl-Brauer}, and thus implicitly in our verification of the framework axioms in Proposition\n\\ref{proposition: framework axioms for Brauer}.\n\nThe involution $i$ on $(n, n)$--Brauer diagrams which reflects a diagram in the axis $y = 1\/2$\nextends linearly to an algebra involution of $B_n$. We have $\\iota \\circ i = i \\circ \\iota$ and ${\\rm cl}\\circ i = i \\circ {\\rm cl}$.\n\nThe products $a b$ and $b a$ of two Brauer diagrams have at most as many through strands as $a$. Consequently, the span of diagrams with at most $r$ through strands ($r \\le n$ and $n-r$ even) is a two--sided ideal $J_r$ in $B_n$. $J_r$ is $i$--invariant.\n\nLet $e_j$ and $s_j$ denote the $(n, n)$--Brauer diagrams:\n$$\ne_j = \\inlinegraphic[scale=.7]{ordinary_E_j}\\qquad\ns_j = \\inlinegraphic[scale= .7]{ordinary_s_j} \n$$\nNote that $e_j^2 = \\delta e_j$, so $e_j$ is an essential idempotent if $\\delta \\ne 0$, and nilpotent if $\\delta = 0$.\nWe have $i(e_j) = e_j$ and $i(s_j) = s_j$. \nIt is easy to see that $e_1, \\dots, e_{n-1}$ and $s_1, \\dots, s_{n-1}$ generate $B_n$ as an algebra. \n\nLet $r \\le n$ with $n - r$ even, and let $f_r = e_{r+1} e_{r+3} \\cdots e_{n-1}$.\nAny Brauer diagram with exactly $r$ through strands can be factored as $\\pi_1 f_r \\pi_2$, where $\\pi_i$ are permutation diagrams. Consequently, $J_r$ is generated by $f_r$. In particular the\nideal $J = J_{n-2}$ spanned by diagrams with fewer than $n$ through strands is generated by $e_{n-1}$. We have $B_n\/J \\cong S\\mathfrak S_n$, as algebras with involutions.\n\n\\begin{lemma} \\label{lemma: Brauer axiom 6} Write $B_n$ for $B_n(S, \\delta)$.\n\\begin{enumerate}\n\\item \nFor $n \\ge 2$, $e_{n} B_{n} e_{n} = B_{n-1} e_{n}$.\n\\item $e_1 B_1 e_1 = \\delta B_0 e_1$\n\\item For $n \\ge 2$, \n$e_{n}$ commutes with $ B_{n-1} $. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} For $n \\ge 2$, if $x$ is an $(n, n)$--Brauer diagram, then $e_{n} x e_{n} \\in B_{n-1}\\, e_{n}$. Thus, $e_{n} B_{n} \\,e_{n} \\break \\subseteq B_{n-1} \\,e_{n}$. On the other hand, for $x \\in B_{n-1}$, we \nhave $e_{n} x e_{n-1} e_{n} = x e_{n}$. Hence, \n $e_{n} B_{n} e_{n} \\supseteq B_{n-1} \\,e_{n}$. This proves (1). Points (2) and (3) are obvious.\n\\end{proof}\n\n\\begin{lemma} \\label{B(n+1) e(n) = B(n) e(n) for Brauer algebras} Write $B_n$ for $B_n(S, \\delta)$.\nFor $n \\ge 1$, \n $B_{n+1}\\, e_{n} = B_{n} \\, e_{n}$. Moreover, \n $x \\mapsto x e_{n}$ is injective from $ B_{n}$ to $ B_{n+1}$.\n\\end{lemma}\n\n\\begin{proof} \nBy ~\\cite{Wenzl-Brauer}, Proposition 2.1, any $(n+1, n+1)$--Brauer diagram is either already in \n$B_{n}$, or can be written in the form $a \\chi_{n} b$, with $a, b \\in B_{n}$ and\n$\\chi_{n} \\in \\{e_{n}, s_{n}\\}$. Applying this again to $b$, either $b \\in B_{n-1}$, or $b$ can be factored as $b_1 \\chi_{n-1} b_2$, with $b_i \\in B_{n-1}$ and $\\chi_{n-1} \\in \\{e_{n-1}, s_{n-1}\\}$. Since $e_{n}^2 = \\delta e_{n}$ and $s_{n} e_{n} = e_{n}$, it follows that if\n$b \\in B_{n-1}$, then $a \\chi_{n} b e_{n} = a b \\chi_{n} e_{n} \\in B_{n} e_{n}$.\nIf $b = b_1 \\chi_{n-1} b_2$, then $a \\chi_{n} b e_{n} = a b_1 \\chi_{n} \\chi_{n-1} e_{n} b_2$.\nNow we can apply the following identities: $e_{n} \\chi_{n-1} e_{n} = e_{n}$ for \n$\\chi_{n-1} \\in \\{e_{n-1}, s_{n-1}\\}$, $s_{n} e_{n-1} e_{n} = s_{n-1} e_{n}$, and\n $s_{n} s_{n-1} e_{n} = e_{n-1} e_{n}$ to conclude that $a \\chi_{n} b e_{n} \\in B_{n} e_{n}$. This shows that $B_{n+1} e_{n} = B_{n} e_{n}$. \n\nFor $x \\in B_{n}$, we have\n${\\rm cl}( x e_{n}) = x$, so the map $x \\mapsto x e_{n}$ is injective from\n$B_{n}$ to $B_{n}e_{n}$. \\end{proof}\n\n\n\\subsubsection{Verification of framework axioms for the Brauer algebras}\n\nWe take $R = {\\mathbb Z}[{\\mathbold \\delta}]$, where ${\\mathbold \\delta}$ is an indeterminant. Then $R$ is the universal ground ring for the Brauer algebras; for any commutative ring $S$ with distinguished element $\\delta$, we have\n$B_n(S, \\delta) \\cong B_n(R, {\\mathbold \\delta})\\otimes_R S$. Let $F = {\\mathbb Q}({\\mathbold \\delta})$ denote the field of fractions of $R$. Write $B_n = B_n(R, {\\mathbold \\delta})$.\n\n\\begin{proposition} \\label{proposition: framework axioms for Brauer} The two sequence of $R$--algebras $(B_n)_{n \\ge 0}$ and $(R \\mathfrak S_n)_{n \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}.\n\\end{proposition}\n\n\\begin{proof} According to Example \\ref{example: Hn coherent tower}, $(R\\mathfrak S_n)_{n \\ge 0}$ is a coherent tower of cellular algebras, so axiom (\\ref{axiom Hn coherent}) holds.\nFramework axioms (\\ref{axiom: involution on An}) and (\\ref{axiom: A0 and A1}) are evident.\n$B_n^F$ is split semisimple by ~\\cite{Wenzl-Brauer}, Theorem 3.2, so axiom (\\ref{axiom: semisimplicity}) holds.\n\nWe take $e_{n-1} \\in B_n$ to be the element defined in the previous section. Let us verify the axioms (\\ref{axiom: idempotent and Hn as quotient of An})--(\\ref{axiom: e(n-1) in An en An}) involving $e_{n-1}$. As observed above, $e_{n-1}$ is $i$--invariant, $J = B_n e_{n-1} B_n$ is the ideal spanned by diagrams with fewer than $n$ through strands, and $B_n\/J \\cong R\\mathfrak S_n$ as algebras with involution. This verifies axiom (\\ref{axiom: idempotent and Hn as quotient of An}).\nAxiom (\\ref{axiom: en An en}) follows from Lemma \\ref{lemma: Brauer axiom 6} and axiom (\\ref{axiom: An en}) from \n Lemma \\ref{B(n+1) e(n) = B(n) e(n) for Brauer algebras}. Axiom (\\ref{axiom: e(n-1) in An en An}) holds because $e_{n-1} e_n e_{n-1} = e_{n-1}$.\n\\end{proof}\n\n\n\\begin{corollary} For any commutative ring $S$ and for any $\\delta \\in S$, \nthe sequence of Brauer algebras $(B_n(S, \\delta))_{n \\ge 0}$ is a coherent tower of cellular algebras.\n$B_n(S, \\delta)$ has cell modules indexed by all Young diagrams of size $n$, $n-2$, $n-4, \\dots$. The cell module labeled by\na Young diagram $\\lambda$ has a basis labeled by up--down tableaux of length $n$ and shape $\\lambda$.\n\\end{corollary}\n\n\\subsection{The Jones--Temperley--Lieb algebras}\n\n\n\n\\subsubsection{Definition of the Jones--Temperley--Lieb algebras} \nLet $S$ be a commutative ring with identity, with distinguished element $\\delta$. The Jones--Temperley--Lieb algebra $T_n(S, \\delta)$ is the unital $S$--algebra with generators $e_1, \\dots, e_{n-1}$ satisfying the relation:\n\\begin{enumerate}\n\\item $e_j^2 = \\delta e_j$,\n\\item $e_j e_{j \\pm 1} e_j = e_j$,\n\\item $e_j e_k = e_k e_j$, if $|j - k| \\ge 2$,\n\\end{enumerate}\nwhenever all indices involved are in the range from $1$ to $n-1$.\n\n\\subsubsection{Diagramatic realization of the Jones--Temperley-Lieb algebras}\n\nThe $S$--span \\break $\\tilde T_n(S, \\delta)$ of Temperley--Lieb diagrams is a subalgebra of the Brauer algebra. We have an algebra map $\\varphi$ from $T_n(S, \\delta)$ to $\\tilde T_n(S, \\delta)$, determined by $e_j \\mapsto e_j$ for $1 \\le j \\le n-1$.\nKauffman shows (\\cite{Kauffman}, Theorem 4.3) that the map is an isomorphism. In fact, to show that $\\varphi$ is surjective, it suffices to show that any Temperley--Lieb diagram can be written as a product of $e_j$'s. Kauffman indicates by example how this is to be done, and it is not difficult to invent a measure of complexity of Temperley--Lieb diagrams and to show this formally, by induction on complexity. For injectivity, Jones shows (\\cite{Jones-index}, p.\\ 14) that $T_n(S, \\delta)$ is spanned by a family $\\mathbb B$ of $\\frac{1}{n+1}{2n \\choose n}$ reduced words in the $e_j$'s. \nSince $\\varphi$ is surjective and $\\tilde T_n(S, \\delta)$ is a free $S$--module of rank $\\frac{1}{n+1}{2n \\choose n}$, it follows easily that $\\mathbb B$ is a basis and $\\varphi$ is an isomorphism. Because of this, we will no longer distinguish between $T_n(S, \\delta)$ and $\\tilde T_n(S, \\delta)$.\n\n\n\n\\subsubsection{Brief history of the Jones--Temperley--Lieb algebras} The Jones-Temperley-Lieb algebras were introduced by Jones in his study of subfactors ~\\cite{Jones-index} and then employed by him to define the Jones link invariant ~\\cite{jones-invariant}. The name derives from the appearance of specific representations of the algebras in statistical mechanics that had been found some years earlier. By now, there is a huge literature related to these algebras because of their multiple roles in subfactor theory, invariants of links and 3-manifolds, statistical mechanics and quantum field theory. The Jones--Temperley--Lieb algebras were shown to be cellular in ~\\cite{Graham-Lehrer-cellular}. Several other proofs of cellularity are known, for example ~\\cite{wilcox-cellular, green-martin-tabular}.\n\n\n\\subsubsection{Some properties of the Jones--Temperley--Lieb algebra} The Brauer algebra maps\n$\\iota$, ${\\rm cl}$, $\\varepsilon_n$ (when $\\delta$ is invertible), and $i$ restrict to maps\nof the Jones--Temperley--Lieb algebras having similar properties. For example, $i$ is an algebra involution on each\n$T_n(S, \\delta)$ and $i \\circ \\iota = \\iota\\circ i$.\n\nThe span of Temperley--Lieb diagrams having at least one horizontal strand is an ideal $J$ in $T_n(S, \\delta)$, and\n$T_n(S, \\delta)\/J \\cong S$.\nThe proof of surjectivity of $\\varphi$ sketched above shows that any Temperley--Lieb diagram with at least one horizontal edge\ncan be written as a non-trivial product of $e_j$'s; so $J$ is equal to the ideal generated by all of the $e_j$'s.\nHowever, the identities $e_j e_{j+1} e_j = e_j$ imply that $J$ is the ideal generated by $e_{n-1}$. \n\n\n\\subsubsection{Verification of the framework axioms for the Jones--Temperley--Lieb algebras}\nWe take $R = {\\mathbb Z}[{\\mathbold \\delta}]$, where ${\\mathbold \\delta}$ is an indeterminant. Then $R$ is the universal ground ring for the Jones--Temperley--Lieb algebras; for any integral domain $S$ with distinguished element $\\delta$, we have\n$T_n(S, \\delta) \\cong T_n(R, {\\mathbold \\delta})\\otimes_{R} S$. Let $F = {\\mathbb Q}({\\mathbold \\delta})$ denote the field of fractions of $R$. Write $T_n = T_n(R, {\\mathbold \\delta})$.\n\n\\begin{proposition} \\label{proposition: framework axioms for TL} The two sequences of $R$--algebras $(T_n)_{n \\ge 0}$ and $(R)_{n \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}.\n\\end{proposition}\n\n\\begin{proof} Axioms (\\ref{axiom Hn coherent}), (\\ref{axiom: involution on An}), and (\\ref{axiom: A0 and A1}) are obvious. \nFor semisimplicity of $T_n^F$, see ~\\cite{GHJ}, \nTheorem 2.8.5. This gives axiom (\\ref{axiom: semisimplicity}).\nWe checked axiom (\\ref{axiom: idempotent and Hn as quotient of An}) in the previous section. The proof for\naxiom (\\ref{axiom: en An en}) is the same as for the Brauer algebras.\n\nAccording to ~\\cite{Jones-index}, Lemma 4.1.2, any \n $(n+1, n+1)$--Temperley--Lieb diagram is either already in \n$T_{n}$, or can be written in the form $a e_{n} b$, with $a, b \\in T_{n}$. Given this, the verification of\naxiom (\\ref{axiom: An en}) is the same as for the Brauer algebras; we have to use only the identity\n$e_{n} e_{n-1} e_{n} = e_{n}$ in place of several similar identities for the Brauer algebras. \n\nAs for the Brauer algebras, axiom \n(\\ref{axiom: e(n-1) in An en An}) follows from the identity $e_{n-1} e_n e_{n-1} = e_{n-1}$.\n \\end{proof}\n \n \\begin{corollary} For any ring $S$ and $\\delta \\in S$, the sequence of Jones--Temperley--Lieb algebras $(T_n(S, \\delta))_{n \\ge 0}$ is a coherent tower of cellular algebras. The cell modules of $T_n(S, \\delta)$ can be labeled by Young diagrams with one or two rows and size $n$, and the basis of the cell module labeled by $\\lambda$ by standard tableaux of shape $\\lambda$.\n \\end{corollary}\n \n \\begin{proof} We only have to remark that the vertices on the $n$-th row of the branching diagram for\n $(T_k^F)_{k\\ge 0}$ (see ~\\cite{GHJ}, Lemma 2.8.4) can be labeled by Young diagrams of size $n$ with no more than 2 rows, and the paths on the branching diagram by standard tableaux. \n (Alternatively, the vertices on the $n$-th row of the branching diagram can be labeled by Young diagrams with one row and size $n$, $n-2$, $n-4, \\dots$, and the paths on the branching diagram by up--down tableaux.)\n \\end{proof}\n\n\n\\subsection{The Birman--Wenzl--Murakami (BMW) algebras}\n\n\\subsubsection{Definition of the BMW algebras}\nThe BMW algebras were first introduced by Birman and Wenzl ~\\cite{Birman-Wenzl} and independently by Murakami ~\\cite{Murakami-BMW} as abstract algebras defined by generators and relations. The version of the presentation given here follows ~\\cite{Morton-Wassermann}.\n\n\\def\\unskip\\kern.55em\\ignorespaces{\\unskip\\kern.55em\\ignorespaces}\n\\def\\vskip-\\lastskip\\vskip4pt{\\vskip-\\lastskip\\vskip4pt}\n\\def\\vskip-\\lastskip\\vskip4pt plus2pt{\\vskip-\\lastskip\\vskip4pt plus2pt}\n\\def\\vskip-\\lastskip\\vskip12pt plus2pt minus2pt{\\vskip-\\lastskip\\vskip12pt plus2pt minus2pt}\n\n\n\n\\begin{definition} \\label{definition: BMW algebra}\nLet $S$ be a commutative unital ring with invertible elements $\\rho$ and $q$ and an element $\\delta$ satisfying $\\rho^{-1} - \\rho = (q^{-1} -q)(\\delta -1)$. The {\\em Birman--Wenzl--Murakami algebra}\n$\\bmw n(S; \\rho, q, \\delta)$ is the unital $S$--algebra \n with generators $g_i^{\\pm 1}$ and\n$e_i$ ($1 \\le i \\le n-1$) and relations:\n\\begin{enumerate}\n\\item (Inverses) \\unskip\\kern.55em\\ignorespaces $g_i g_i^{-1} = g_i^{-1} g_i = 1$.\n\\item (Essential idempotent relation)\\unskip\\kern.55em\\ignorespaces $e_i^2 = \\delta e_i$.\n\\item (Braid relations) \\unskip\\kern.55em\\ignorespaces $g_i g_{i+1} g_i = g_{i+1} g_i g_{i+1}$ \nand $g_i g_j = g_j g_i$ if $|i-j| \\ge 2$.\n\\item (Commutation relations) \\unskip\\kern.55em\\ignorespaces $g_i e_j = e_j g_i$ and\n$e_i e_j = e_j e_i$ if $|i-j|\\ge 2$. \n\\item (Tangle relations)\\unskip\\kern.55em\\ignorespaces $e_i e_{i\\pm 1} e_i = e_i$, $g_i\ng_{i\\pm 1} e_i = e_{i\\pm 1} e_i$, and $ e_i g_{i\\pm 1} g_i= e_ie_{i\\pm 1}$.\n\\item (Kauffman skein relation)\\unskip\\kern.55em\\ignorespaces $g_i - g_i^{-1} = (q - q^{-1}) (1 - e_i)$.\n\\item (Untwisting relations)\\unskip\\kern.55em\\ignorespaces $g_i e_i = e_i g_i = \\rho^{-1} e_i$,\nand $e_i g_{i \\pm 1} e_i = \\rho e_i$.\n\\end{enumerate}\n\\end{definition}\n\n\\subsubsection{Geometric realization of the BMW algebras}\nA geometric realization of the BMW algebra is as the algebra of framed $(n, n)$--tangles in the disc cross the interval, modulo certain skein relations. It is more convenient, at least for our purposes, to describe this geometric version in terms of tangle diagrams.\n\nFirst, tangle diagrams can be multiplied by stacking, as for Brauer or Temperley--Lieb diagrams (but closed loops are allowed, and there is no reduction by removing closed loops after stacking). Recall that our convention is that the product $ab$ of tangle diagrams is given by stacking $b$ over $a$. This makes $(n, n)$--tangle diagrams into a monoid, the identity being the tangle diagram in which each top vertex $\\p j$ is connected to the bottom vertex $\\pbar j$ by a vertical line segment, when $n \\ge 1$.\n(The identity for the monoid of $(0, 0)$--tangle diagrams is the empty tangle.)\n\n\\medskip\n\\vbox{\n\\begin{eqnarray*}\n\\text{I}&\\quad\\ &\\inlinegraphic{right_twist} \\quad \\longleftrightarrow\n \\quad \\inlinegraphic{vertical_line} \\quad\n \\longleftrightarrow \\quad \\inlinegraphic{left_twist}\\\\\n\\text{II}&\\quad\\ &\\inlinegraphic[scale =.5] {ReidemeisterII} \\quad \n\\longleftrightarrow \n\\quad \\inlinegraphic[scale=1.75]{id_smoothing} \\\\\n \\text{III}&\\quad\\ &\\inlinegraphic{ReidIIIleft} \\quad \\longleftrightarrow \\quad \n\\inlinegraphic{ReidIIIright} \n\\end{eqnarray*} \n\n\\centerline{Reidemeister moves}\n}\n\n\\medskip\nTwo tangle diagrams are said to be {\\em regularly isotopic} if they are related by a sequence of Reidemeister moves of types II and III, followed by an isotopy of $\\mathcal R$ fixing the boundary.\n(Reidemeister moves of type I are not allowed.) See the figure above for the Reidemeister moves.\n\nStacking of tangle diagrams respects regular isotopy; thus one obtains a monoid structure on the regular isotopy classes of $(n, n)$--tangle diagrams. Let us denote this monoid by $\\mathcal U_n$.\nLet $S$ be a ring with elements $\\rho$, $q$ and $\\delta$ as in the definition of the BMW algebras.\nThe {\\em Kauffman tangle algebra} $\\kt n(S; \\rho, q, \\delta)$ is the monoid algebra $S\\ \\mathcal U_n$ modulo the following skein relations:\n\\begin{enumerate}\n\\item Crossing relation:\n$\n\\quad \\inlinegraphic[scale=.6]{pos_crossing} - \\inlinegraphic[scale=.3]{neg_crossing} \n\\quad = \n\\quad\n(q^{-1} - q)\\,\\left( \\inlinegraphic[scale=1.2]{e_smoothing} - \n\\inlinegraphic[scale=1.2]{id_smoothing}\\right).\n$\n\\item Untwisting relation:\n$\\quad \n\\inlinegraphic{right_twist} \\quad = \\quad \\rho \\quad\n\\inlinegraphic{vertical_line} \\quad\\ \\text{and} \\quad\\ \n\\inlinegraphic{left_twist} \\quad = \\quad \\rho^{-1} \\quad\n\\inlinegraphic{vertical_line}. \n$\n\\item Free loop relation: $T\\, \\cup \\, \\bigcirc = \\delta \\, T, $ where $T\\, \\cup \\, \\bigcirc$ means the union of a tangle diagram $T$ and a closed loop having no crossings with $T$.\n\\end{enumerate}\n\nLet $E_j$ and $G_j$ denote the following $(n,n)$--tangle diagrams:\n$$\nE_j = \\inlinegraphic[scale=.7]{ordinary_E_j}\\qquad\nG_j = \\inlinegraphic[scale= .7]{ordinary_G_j} \n$$\nMorton and Wassermann \\cite{Morton-Wassermann} showed that the assignments $e_j \\mapsto E_j$ and $g_j \\mapsto G_j$ determine an isomorphism from $\\bmw n(S; \\rho, q, \\delta)$ to\n$\\kt n(S; \\rho, q, \\delta)$. Given this, we will no longer distinguish between the BMW algebras and the Kauffman tangle algebras. (However, we remark that it is possible to use our techniques to recover\nthe theorem of Morton and Wasserman, using only results in the original paper of Birman and Wenzl; we prove the analogous isomorphism theorem for the cyclotomic BMW algebras in Section \\ref{The cyclotomic Birman--Wenzl--Murakami (BMW) algebras}, and the result for the ordinary BMW algebras is a special case.)\n\n\\subsubsection{Brief history of the BMW algebras} The origin of the BMW algebras was in knot theory. Kauffman defined \\cite{Kauffman} an invariant of regular isotopy for links in $S^3$, determined by skein relations. Birman and Wenzl ~\\cite{Birman-Wenzl} and Murakami ~\\cite{Murakami-BMW} then defined the BMW algebras in order to give an algebraic setting for the Kauffman invariant. The BMW algebras were implicitly modeled on algebras of tangles. The definition of the Kauffman tangle algebra was made explicit by Morton and Traczyk ~\\cite{Morton-Traczyk}, who also showed that $\\kt n(S; \\rho, q, \\delta)$ is free as an $S$--module of rank\n$(2n-1)!!$. Morton and Wassermann ~\\cite{Morton-Wassermann} showed that the BMW algebras and Kauffman tangle algebras are isomorphic.\n\nXi showed ~\\cite{Xi-BMW} that the tangle basis of Morton and Traczyk is a cellular basis. Enyang has exhibited two cellular bases of BMW algebras; the first ~\\cite{Enyang1} is a tangle type basis, and the second ~\\cite{Enyang2} is a basis indexed by up--down tableaux, which demonstrates the coherence of the cellular structures on $(\\bmw n)_{n \\ge 0}$.\n\n\n\\subsubsection{Some properties of the BMW algebras} \nIn the following, we write $\\bmw n$ for \\break $\\bmw n(S; \\rho, q, \\delta)$.\n\n The BMW algebras have an algebra involution \n$i$ uniquely determined by $i(e_j) = e_j$ and $i(g_j) = g_j$ for all $j$. The action of $i$ on tangle diagrams is by the rotation through the axis $y = 1\/2$. (It is by rotation rather than reflection, since the reflection would take $g_j \\mapsto g_j^{-1}$.)\n\nFor $n \\ge 0$, there is a unique homomorphism $\\iota$ from $\\bmw n$ to $\\bmw {n+1}$ determined by $e_i \\mapsto e_i$ and $g_i \\mapsto g_i$ for $1 \\le i \\le n-1$. On the level of tangle diagrams, the map is given by adding\na new vertical strand connecting $\\p {n+1}$ and $\\pbar {n+1}$, as for the Brauer algebras.\n\n \n\n\nFor $n \\ge 1$, a map ${\\rm cl}$ from $(n, n)$--tangle diagrams to \n$(n-1, n-1)$--tangle diagrams can be defined as for Brauer diagrams. The linear extension of this map respects\nregular isotopy and the Kauffman skein relations, so determines a linear map from \n$\\bmw n$ to $\\bmw {n-1}$. We have $i \\circ {\\rm cl} = \n {\\rm cl} \\circ i$ and $ {\\rm cl} \\circ \\iota = \\delta \\, x$. Moreover, for $x \\in \\bmw{n}$, we have\n $x = {\\rm cl} ( \\iota(x) e_n)$, so it follows that $\\iota: \\bmw n \\to \\bmw {n+1}$ is injective.\n The involution $i$ and inclusion $\\iota$ satisfy $i\\circ \\iota = \\iota\\circ i$.\n Using $\\iota$, we identify $\\bmw n$ as a subalgebra of $\\bmw {n+1}$. \n\n \n If $\\delta$ is invertible in $S$, we can define $\\varepsilon_n = (1\/\\delta) {\\rm cl}$, which is a conditional expectation, that is, an unital $\\bmw {n-1}$--$\\bmw {n-1}$ bimodule map. We have $\\varepsilon_{n+1} \\circ \\iota(x) = x$ for $x \\in \\bmw n$.\n \n The ideal $J$ in $\\bmw n$ generated by $e_{n-1}$ contains $e_j$ for all $j$ because of the relations\n $e_j e_{j+1} e_j = e_j$. It follows from the BMW relations that $\\bmw n\/J $ is isomorphic to the Hecke algebra $H_n(S; q^2)$ with the quadratic relation $g_j - g_j^{-1} = q - q^{-1}$, or \n $(g_j - q) (g_j + q^{-1}) = 0$. \n \n \\begin{lemma} \\label{lemma: axiom 6 for BMW} \\mbox{}\n \\begin{enumerate}\n\\item \nFor $n \\ge 2$, $e_{n} \\bmw {n} e_{n} = \\bmw {n-1} e_{n}$.\n\\item $e_1 \\bmw 1 e_1 = \\delta\\, \\bmw 0 \\,e_1$\n\\item For $n \\ge 1$, \n$e_{n}$ commutes with $ \\bmw {n-1} $. \n\\end{enumerate}\n \\end{lemma}\n \n \\begin{proof} The proof is the same as that of Lemma \\ref{lemma: Brauer axiom 6} for the Brauer algebras, using the tangle realization of the BMW algebras.\n \\end{proof}\n \n \\begin{lemma} \\label{Bn e(n-1) = B(n-1) e(n-1) for BMW algebras}. For $n \\ge 1$, \n$\\bmw {n+1} \\,e_{n} = \\bmw {n} \\,e_{n}$. Moreover, \n $x \\mapsto x e_{n}$ is injective from $\\bmw {n}$ to $\\bmw {n} e_{n}$.\n\\end{lemma}\n\n\\begin{proof} According to ~\\cite{Birman-Wenzl}, Lemma 3.1, any $(n+1, n+1)$--tangle is already in $\\bmw {n}$, or it can be written as a linear combination of elements\n$a \\chi_{n} b$, with $a, b \\in \\bmw {n}$ and $\\chi_{n} \\in \\{e_{n}, g_{n}\\}$. Given this, the proof of \nthe lemma is the same as the proof of Lemma \\ref{B(n+1) e(n) = B(n) e(n) for Brauer algebras} \n for the Brauer algebras, using the tangle relations and untwisting relations of Definition \\ref{definition: BMW algebra} in place of similar identities for the Brauer algebras.\n\\end{proof}\n \n \\subsubsection{Verification of the framework axioms for the BMW algebras}\n The generic or universal ground ring for the BMW algebras is\n $$\n R = {\\mathbb Z}[{\\bm \\rho}^{\\pm1}, {\\bm q}^{\\pm1}, {\\mathbold \\delta}]\/\\langle {\\bm \\rho}^{-1} - {\\bm \\rho} = ({\\bm q}^{-1} - {\\bm q})({\\mathbold \\delta} -1) \\rangle,\n $$\n where ${\\bm \\rho}$, ${\\bm q}$, and ${\\mathbold \\delta}$ are indeterminants over ${\\mathbb Z}$. Suppose that $S$ is an appropriate ground ring for the BMW algebras; that is, \n $S$ is a commutative unital ring with invertible elements $\\rho$ and $q$ and an element $\\delta$ satisfying $\\rho^{-1} - \\rho = (q^{-1} -q)(\\delta -1)$. Then $\\bmw n(S; \\rho, q, \\delta) \\cong \\bmw n(R; {\\bm \\rho}, {\\bm q}, {\\mathbold \\delta}) \\otimes_R S$. \n\n$R$ is an integral domain whose field of fractions is $F \\cong {\\mathbb Q}({\\bm \\rho}, {\\bm q})$ (with ${\\mathbold \\delta} = \\break\n ({\\bm \\rho}^{-1} - {\\bm \\rho})\/({\\bm q}^{-1} - {\\bm q}) + 1$ in $F$.) We write $\\bmw n$ for \n $\\bmw n(R; {\\bm \\rho}, {\\bm q}, {\\mathbold \\delta})$ and $H_n$ for $H_n(R; {\\bm q}^2)$ in this section.\n\n\\begin{proposition}\\label{proposition: framework axioms for BMW}\n The two sequences of algebras $(\\bmw n)_{n \\ge 0}$ and $(H_n)_{n \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}.\n\\end{proposition}\n\n\\begin{proof} According to example \\ref{example: Hn coherent tower} , $(H_n)_{n \\ge 0}$ is a coherent tower of cellular algebras, so axiom (\\ref{axiom Hn coherent}) holds. Axioms (\\ref{axiom: involution on An}) and (\\ref{axiom: A0 and A1}) are evident.\n$\\bmw n^F$ is semisimple by ~\\cite{Birman-Wenzl}, Theorem 3.7, or ~\\cite{Wenzl-BCD}, Theorem 3.5. Thus axiom (\\ref{axiom: semisimplicity}) holds.\n\nWe observed above that $\\bmw n\/\\bmw n e_{n-1} \\bmw n \\cong H_n$; it is easy to check that the isomorphism respects the involutions. Thus axiom (\\ref{axiom: idempotent and Hn as quotient of An}) holds. Axiom \n(\\ref{axiom: en An en}) follows from Lemma \\ref{lemma: axiom 6 for BMW} and axiom (\\ref{axiom: An en}) from Lemma \\ref{Bn e(n-1) = B(n-1) e(n-1) for BMW algebras}.\nFinally, axiom (\\ref{axiom: e(n-1) in An en An}) holds again because of the relation $e_{n-1} e_n e_{n-1} = e_{n-1}$.\n\\end{proof}\n\n\\begin{corollary} Let $S$ be any ground ring for the BMW algebras, with parameters $\\rho$, $q$, and \n$\\delta$. \nThe sequence of BMW algebras $(\\bmw n(S; \\rho, q, \\delta))_{n \\ge 0}$ is a coherent tower of cellular algebras.\n$\\bmw n(S; \\rho, q, \\delta)$ has cell modules indexed by all Young diagrams of size $n$, $n-2$, $n-4, \\dots$. The cell module labeled by\na Young diagram $\\lambda$ has a basis labeled by up--down tableaux of length $n$ and shape $\\lambda$.\n\\end{corollary}\n\n\n\\subsection{The cyclotomic Birman--Wenzl--Murakami (BMW) algebras}\n\\label{The cyclotomic Birman--Wenzl--Murakami (BMW) algebras}\n\n\\subsubsection{Definition of the cyclotomic BMW algebras}\n\nIn general, our notation will follow ~\\cite{GH3}. In order to simplify statements, we establish the following convention.\n\n\\begin{definition} Fix an integer $r \\ge 1$. A {\\em ground ring} \n$S$ is a commutative unital ring with parameters $\\rho$, $q$, $\\delta_j$ ($j \\ge 0$), and\n$u_1, \\dots, u_r$, with $\\rho$, $q$, and $u_1, \\dots, u_r$ invertible, and with $\\rho^{-1} - \\rho= (q^{-1} -q) (\\delta_0 - 1)$.\n\\end{definition}\n\n\n\n\\begin{definition} \\label{definition: cyclotomic BMW}\nLet $S$ be a ground ring with\nparameters $\\rho$, $q$, $\\delta_j$ ($j \\ge 0$), and\n$u_1, \\dots, u_r$.\nThe {\\em cyclotomic BMW algebra} $\\bmw{n, S, r}(u_1, \\dots, u_r)$ is the unital $S$--algebra\nwith generators $y_1^{\\pm 1}$, $g_i^{\\pm 1}$ and\n$e_i$ ($1 \\le i \\le n-1$) and relations:\n\\begin{enumerate}\n\\item (Inverses)\\unskip\\kern.55em\\ignorespaces $g_i g_i^{-1} = g_i^{-1} g_i = 1$ and \n$y_1 y_1^{-1} = y_1^{-1} y_1= 1$.\n\\item (Idempotent relation)\\unskip\\kern.55em\\ignorespaces $e_i^2 = \\delta_0 e_i$.\n\\item (Affine braid relations) \n\\begin{enumerate}\n\\item[\\rm(a)] $g_i g_{i+1} g_i = g_{i+1} g_ig_{i+1}$ and \n$g_i g_j = g_j g_i$ if $|i-j| \\ge 2$.\n\\item[\\rm(b)] $y_1 g_1 y_1 g_1 = g_1 y_1 g_1 y_1$ and $y_1 g_j =\ng_j y_1 $ if $j \\ge 2$.\n\\end{enumerate}\n\\item[\\rm(4)] (Commutation relations) \n\\begin{enumerate}\n\\item[\\rm(a)] $g_i e_j = e_j g_i$ and\n$e_i e_j = e_j e_i$ if $|i-\nj|\n\\ge 2$. \n\\item[\\rm(b)] $y_1 e_j = e_j y_1$ if $j \\ge 2$.\n\\end{enumerate}\n\\item[\\rm(5)] (Affine tangle relations)\\vadjust{\\vskip-2pt\\vskip0pt}\n\\begin{enumerate}\n\\item[\\rm(a)] $e_i e_{i\\pm 1} e_i = e_i$,\n\\item[\\rm(b)] $g_i g_{i\\pm 1} e_i = e_{i\\pm 1} e_i$ and\n$ e_i g_{i\\pm 1} g_i= e_ie_{i\\pm 1}$.\n\\item[\\rm(c)\\hskip1.2pt] For $j \\ge 1$, $e_1 y_1^{ j} e_1 = \\delta_j e_1$. \n\\vadjust{\\vskip-\n2pt\\vskip0pt}\n\\end{enumerate}\n\\item[\\rm(6)] (Kauffman skein relation)\\unskip\\kern.55em\\ignorespaces $g_i - g_i^{-1} = (q - q^{-1})(1- e_i)$.\n\\item[\\rm(7)] (Untwisting relations)\\unskip\\kern.55em\\ignorespaces $g_i e_i = e_i g_i = \\rho ^{-1} e_i$\n and $e_i g_{i \\pm 1} e_i = \\rho e_i$.\n\\item[\\rm(8)] (Unwrapping relation)\\unskip\\kern.55em\\ignorespaces $e_1 y_1 g_1 y_1 = \\rho e_1 = y_1 \ng_1 y_1 e_1$.\n\\item[\\rm(9)](Cyclotomic relation) \\unskip\\kern.55em\\ignorespaces $(y_1 - u_1)(y_1 - u_2) \\cdots (y_1 - u_r) = 0$.\n\\end{enumerate}\n\\end{definition}\n\nThus, a cyclotomic BMW algebra is the quotient of the affine BMW algebra ~\\cite{GH1}, by the cyclotomic relation $(y_1 - u_1)(y_1 - u_2) \\cdots (y_1 - u_r) = 0$. \n\n\n \\subsubsection{Geometric realization} \\label{subsubsection: cyclotomic BMW geometric realization}\n We recall from ~\\cite{GH1} that the affine BMW algebra is isomorphic to the affine Kauffman tangle algebra, which is an algebra of ``affine tangle diagrams,\" modulo Kauffman skein relations. An affine $(n,n)$--tangle diagram is just an ordinary $(n+1, n+1)$--tangle diagram with a fixed\n vertical strand connecting $\\p 1$ and $\\pbar 1$, as in the following figure.\n $$\n\\inlinegraphic[scale=1.5]{affine-4-tangle}\n$$\n The affine Kauffman tangle algebra is generated by the following affine tangle diagrams:\n$$\nX_1 = \\inlinegraphic[scale= .7]{X1}\n\\qquad\nG_i = \\inlinegraphic[scale=.6]{G_i}\\qquad\nE_i = \\inlinegraphic[scale= .7]{E_i} .\n$$\n\n One can also define a cyclotomic Kauffman tangle algebra \n $\\kt{n, S, r}(u_1, \\dots, u_r)$ as the quotient of the affine Kauffman tangle algebra by a cyclotomic skein relation, which is a ``local\" version of the cyclotomic relation of \n Definition \\ref{definition: cyclotomic BMW} (9). See\n ~\\cite{GH2} for the precise definition. We denote the images of $X_1$, $E_i$ and $G_i$ in the cyclotomic Kauffman tangle algebra by the same letters. \nThe assignments\n $e_i \\mapsto E_i$, $g_i \\mapsto G_i$ and $y_1 \\mapsto \\rho X_1$ defines a surjective homorphism\n from $\\varphi : \\bmw {n, S, r}(u_1, \\dots, u_r) \\to \\kt{n, S, r}(u_1, \\dots, u_r)$, see ~\\cite{GH2}, page 1114. \n \n It is shown in ~\\cite{GH2, GH3} and in ~\\cite{Wilcox-Yu3} that\n the the map $\\varphi$ is an isomorphism, assuming admissibility conditions on the ground ring (see Section \\ref{subsubsection: admissibility}). However, we are {\\em not} going to assume this result here, but will give a new proof of the isomorphism. \n \n\n\n\\subsubsection{Brief history of cyclotomic BMW algebras} Affine and cyclotomic BMW algebras were introduced by H\\\"aring--Oldenberg ~\\cite{H-O2} and have recently been studied by three groups of mathematicians:\nGoodman and Hauschild Mosley ~\\cite{GH1, GH2, GH3, goodman-2008}, Rui, Xu, and Si ~\\cite{rui-2008, rui-2008b}, and Wilcox and Yu ~\\cite{Wilcox-Yu, Wilcox-Yu2, Wilcox-Yu3, Yu-thesis}. Under (slightly different) admissibility assumptions on the ground ring (see Section \\ref{subsubsection: admissibility}) all three groups have shown that the algebra\n$\\bmw{n, S, r}$ is free over $S$ of rank $r^n (2n-1) ! !$ and in fact is cellular. (Wilcox and Yu produced cellular basis satisfying the strict equality $i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda$, while the other groups only established cellularity in the weaker sense of Definition \\ref{gl cell}.) The cellular bases produced by all three groups are essentially tangle bases, i.e., cyclotomic analogues of the basis of Morton, Traczyk, and Wassermann for the ordinary BMW algebras. Goodman \\& Hauschild Mosley and Wilcox \\& Yu have shown that the algebras can be realized as algebras of tangles, when the ground ring is admissible. Rui et.\\ al.\\ have achieved additional representation theoretic results. Further background on cyclotomic BMW algebras, motivation for the study of these algebras, relations to other mathematical topics (quantum groups, knot theory), and further literature citations can be found in ~\\cite{GH2} and in the other papers cited above.\n\n\n\\subsubsection{Advantages of our approach to cellularity}\nOne of our motivations in undertaking the current work was to produce a Murphy type cellular basis for the cyclotomic BMW algebras, indexed by up--down tableaux. As mentioned in the introduction, this has not been done previously, and it would be involved to extend Enyang's method for ordinary BMW algebras ~\\cite{Enyang2} to the cellular case. \n\n It turns out that our proof of cellularity is actually more direct than the previous proofs cited above, in that it bypasses the lengthy proof (in ~\\cite{GH2}, Proposition 3.7, or ~\\cite{Wilcox-Yu3}, Theorem 3.2) that these algebras have a finite spanning set of the appropriate cardinality. Our method does not depend the isomorphism of the cyclotomic BMW algebras and cyclotomic Kauffman tangle algebras ~\\cite{GH2, GH3} or ~\\cite{Wilcox-Yu3}; in fact, we can give a new proof of this isomorphism. \n \n One might say that the difficulty in our proof has been displaced, because instead of the finite spanning set result cited above, we require Mathas' recent theorem on coherence of cellular structures for cyclotomic Hecke algebras ~\\cite{mathas-2009}.\n\n\n\\subsubsection{Admissibility conditions on the ground ring.} \\label{subsubsection: admissibility}\n\nThe cyclotomic BMW algebras can be defined over arbitrary ground rings. However, it is necessary to impose conditions on the parameters in order to get a satisfactory theory.\n\nOne can see by a simple computation why one has to expect conditions on the parameters. First, one can show that there are elements $\\delta_{-j}$ in the ground ring $S$ for $j \\ge 1$ such that\n$e_1 y_1^{-j} e_1 = \\delta_{-j} \\,e_1$; moreover, $\\delta_{-j}$ is a polynomial in\n$\\rho^{-1}$, $q - q^{-1}$, and $\\delta_0, \\delta_1, \\dots, \\delta_j$; see \\cite{GH3}, Lemma 2.5.\nIf one now multiplies the cyclotomic relation, Definition \\ref{definition: cyclotomic BMW} (9), by $y_1^a$ and\npre-- and post--multiplies by $e_1$, one gets \n$\n(\\sum_{k = 0}^r a_k \\delta_{k + a} ) e_1 = 0,\n$\nfor $a \\in {\\mathbb Z}$, where the $a_k$ are signed elementary symmetric polynomials in $u_1, \\dots, u_r$. Therefore, either $e_1$ is a torsion element over $S$, or the following {\\em weak admissibility conditions} hold:\n$$\n\\sum_{k = 0}^r a_k \\delta_{k + a} = 0, \\quad \\text{for $a \\in {\\mathbb Z}$}.\n$$\nIf $S$ is a field and the weak admissibility conditions do not hold, then $e_1 = 0$; it follows that all the $e_i$ are zero, and the algebra reduces to the cyclotomic Hecke algebra over $S$ with parameters $q^2$ and\n$u_1, \\dots, u_r$.\n\nThe weak admissibility conditions are complicated and not strong enough to give satisfactory results on the representation theory of the algebras. Therefore, one wishes to find conditions that are both simpler and stronger. \nTwo apparently different conditions have been proposed, one by Wilcox and Yu ~\\cite{Wilcox-Yu}, and another by Rui and Xu ~\\cite{rui-2008}. It has been shown in ~\\cite{goodman-admissibility} that the two conditions are equivalent in the case of greatest interest, when $S$ is an integral domain with $q - q^{-1} \\ne 0$. \nWe consider only this case from now on.\n\n\\begin{definition} Let $S$ be an integral ground ring with\nparameters $\\rho$, $q$, $\\delta_j$ ($j \\ge 0$) and $u_1, \\dots, u_r$, with $q - q^{-1} \\ne 0$.\nOne says that $S$ is {\\em admissible} (or that the parameters are {\\em admissible}) if $\\{e_1, y_1 e_1, \\dots, y_1^{r-1} e_1\\} \\subseteq \\bmw{2, S, r}$ is linearly independent over $S$.\n\\end{definition}\n\n\nIt is shown in ~\\cite{Wilcox-Yu} that admissibility is equivalent to finitely many (explicit) polynomial relations on the parameters. Moreover, these relations give $\\rho$ and $(q - q^{-1}) \\delta_j$ as Laurent polynomials\nin the remaining parameters $q, u_1, \\dots, u_r$; see ~\\cite{Wilcox-Yu} and ~\\cite{GH3} for details.\n\n\\subsubsection{Morphisms of ground rings and a universal admissible ground ring} \n\\label{subsubsection: morphisms and generic ring}\nWe consider what are the appropriate morphisms between ground rings for cyclotomic BMW algebras. The obvious notion would be that of a ring homomorphism taking parameters to parameters; that is, if $S$ is a ground ring with parameters $\\rho$, $q$, \netc., and $S'$ another ground ring with parameters $\\rho'$, $q'$, etc., then a morphism $\\varphi : S \\to S'$ would be required to map $\\rho \\mapsto \\rho'$, \n$q \\mapsto q'$, etc. \n\nHowever, it is better to require less, for the following reason: The parameter $q$ enters into the cyclotomic BMW\n relations only in the expression $q^{-1} -q$, and the transformation $q \\mapsto -q^{-1}$ leaves this expression invariant. Moreover, the transformation $g_i \\mapsto -g_i$, $\\rho \\mapsto -\\rho$, $q \\mapsto -q$ (with all other generators and parameters unchanged) leaves the cyclotomic BMW relations unchanged. \n\n\nTaking this into account, we arrive at the following notion:\n\n\n\\begin{definition} \\label{definition: parameter preserving}\nLet $S$ be a ground ring with\nparameters $\\rho$, $q$, $\\delta_j$ ($j \\ge 0$), and\n$u_1, \\dots, u_r$.\nLet $S'$ be another ground ring with parameters $\\rho'$, $q'$, etc. \n\nA unital ring homomorphism $\\varphi : S \\rightarrow S'$ is a {\\em morphism of ground rings} if it maps\n$$\n\\begin{cases}\n&\\rho \\mapsto \\rho', \\text{ and}\\\\\n& q \\mapsto q' \\text{ or } q \\mapsto -{q'}^{-1},\n\\end{cases}\n$$\nor\n$$\n\\begin{cases}\n&\\rho \\mapsto -\\rho', \\text{ and}\\\\\n& q \\mapsto -q' \\text{ or } q \\mapsto {q'}^{-1},\n\\end{cases}\n$$\nand strictly preserves all other parameters.\n\\end{definition}\n\n\n Suppose there is a morphism of ground rings $\\psi : S \\rightarrow S'$.\n Then $\\psi$ extends to a \nhomomorphism from \n$\\bmw{n, S, r}$ to $\\bmw{n, S', r}$. Moreover, $\\bmw{n, S, r} \\otimes_S S' \\cong \\bmw{n,S', r}$ as $S'$--algebras. These statements are discussed in ~\\cite{GH3}, Section 2.4.\n\nLet $S$ be a ground ring with admissible parameters $\\rho$, $q$, $\\delta_j$ ($j \\ge 0$), and\n$u_1, \\dots, u_r$. Then \n$$\n\\rho, -q^{-1}, \\delta_j \\ (j \\ge 0), \\text{ and } u_1, \\dots, u_r\n$$\nand\n$$\n-\\rho, -q, \\delta_j \\ (j \\ge 0), \\text{ and } u_1, \\dots, u_r\n$$\nare also sets of admissible parameters. \n Suppose that $S$ is an integral ground ring with admissible parameters, with $q - q^{-1} \\ne 0$, and that $S'$ is another integral ground ring; \n if $\\varphi : S \\rightarrow S'$ is \na morphism of ground rings such that\n $\\varphi(q - q^{-1}) \\ne 0$, then $S'$ is also admissible.\n \n It is easy to show (see \\cite{GH3}, Theorem 3.19) that there is a universal integral admissible ground ring $R$, with parameters ${\\bm \\rho}$, ${\\bm q}$, ${\\mathbold \\delta}_j$ ($j \\ge 0$), and ${\\bm u}_1, \\dots, {\\bm u}_r$, with the following properties:\n \\begin{enumerate}\n \\item The parameters ${\\bm q}$, ${\\bm u}_1$, \\dots, ${\\bm u}_r$ of \n$R$ are algebraically independent over ${\\mathbb Z}$.\n\\item $R$ is generated as a ring by \n ${\\bm q}^{\\pm 1}$, ${\\bm \\rho} \\powerpm$, ${\\mathbold \\delta}_0$, ${\\mathbold \\delta}_1$, \\dots ${\\mathbold \\delta}_{r-1} $, and ${\\bm u}_1^{\\pm 1}, \\dots, {\\bm u}_r^{\\pm 1}$.\n\\item Whenever $S$ is an integral ground ring with admissible\nparameters, with $q - q^{-1} \\ne 0$, there exists a morphism of ground rings from $R$ to $S$; thus\n$\\bmw{n, S, r} \\cong \\bmw{n, R, r} \\otimes_{R} S$. \n\\item The field of fractions of $R$ is ${\\mathbb Q}({\\bm q}, {\\bm u}_1, \\dots, {\\bm u}_r)$.\n\\item Let $\\bm p = \\prod_{j = 1}^r {\\bm u}_j$. Then one has ${\\bm \\rho} = \\bm p$ if $r$ is even and\n${\\bm \\rho} = {\\bm q}^{-1} \\bm p$ if $r$ is odd. Since ${\\bm \\rho}^{-1} - {\\bm \\rho}= ({\\bm q}^{-1} -{\\bm q}) ({\\mathbold \\delta}_0 - 1)$, and ${\\bm q}$, ${\\bm u}_1$, \\dots, ${\\bm u}_r$ are algebraically independent,\none has ${\\mathbold \\delta}_0 \\ne 0$. \n \\end{enumerate}\n \n \\subsubsection{Some properties of cyclotomic BMW and Kauffman tangle algebras.} We restrict attention to the case of an integral admissible ground ring $S$ with $q - q^{-1} \\ne 0$. We write $\\bmw n$ for $\\bmw {n, S, r}(u_1, \\dots, u_r)$ and $\\kt n$ for $\\kt {n, S, r}(u_1, \\dots, u_r)$.\n \n The cyclotomic BMW algebras have an algebra involution \n$i$ uniquely determined by $i(e_j) = e_j$ and $i(g_j) = g_j$ for all $j$, and $i(y_1) = y_1$. Likewise, the\n cyclotomic Kauffman tangle algebras have an algebra involution $i$, whose action on affine tangle diagrams is by the rotation through the axis $y = 1\/2$. The surjective homomorphism $\\varphi : \\bmw n \\to \\kt n$ respects the involutions. \n \n For $n \\ge 0$, there is a homomorphism (of involutive algebras) $\\iota$ from $\\bmw n$ to $\\bmw {n+1}$ determined by $e_i \\mapsto e_i$ and $g_i \\mapsto g_i$ for $1 \\le i \\le n-1$, and $y_1 \\mapsto y_1$; it is not clear\n {\\em a priori} that $\\iota$ is injective. \n \n Likewise, there is a homomorphism (of involutive algebras) $\\iota$ from $\\kt n$ to $\\kt {n+1}$. \n On the level of affine tangle diagrams, the map is given by adding\na new vertical strand connecting $\\p {n+1}$ and $\\pbar {n+1}$, as for the Brauer algebras. This map\nis injective, as we will now explain. \n\n For $n \\ge 1$, a map ${\\rm cl}$ from affine $(n, n)$--tangle diagrams to affine \n $(n-1, n-1)$--tangle diagrams can be defined as for Brauer diagrams and ordinary tangle diagrams. The linear extension of this map respects\nregular isotopy and all the skein relations defining the cyclotomic Kauffman tangle algebras, so determines a linear map from \n$\\kt n$ to $\\kt {n-1}$. (See ~\\cite{GH1}, Section 2.7, and ~\\cite{GH2}, Section 3.3 for details.) \n The map ${\\rm cl}$ respects the involutions, $i \\circ {\\rm cl} = \n {\\rm cl} \\circ i$. Moreover, for $x \\in \\kt{n}$, we have\n $x = {\\rm cl} ( \\iota(x) e_n)$, so it follows that $\\iota: \\kt n \\to \\kt {n+1}$ is injective.\n Using $\\iota$, we identify $\\kt n$ as a subalgebra of $\\kt {n+1}$. \n\n If $\\delta_0$ is invertible in $S$, we can define $\\varepsilon_n = (1\/\\delta_0) {\\rm cl}$, which is a conditional expectation, that is, an unital $\\kt {n-1}$--$\\kt {n-1}$ bimodule map. We have $\\varepsilon_{n+1} \\circ \\iota(x) = x$ for $x \\in \\bmw n$.\n\n\n\\subsubsection{The cyclotomic Hecke algebra} \nWe recall the definition of the affine and cyclotomic Hecke algebras, see ~\\cite{ariki-book}.\n\n\\begin{definition}\nLet $S$ be a commutative unital ring with an invertible element $q$. The {\\em affine Hecke algebra} \n$\\ahec{n,S}(q^2)$ \nover $S$\nis the $S$--algebra with generators $t_1, g_1, \\dots, g_{n-1}$, with relations:\n\\begin{enumerate}\n\\item The generators $g_i$ are invertible, satisfy the braid relations, and $g_i - g_i^{-1} = (q - q^{-1})$.\n\\item The generator $t_1$ is invertible, $t_1 g_1 t_1 g_1 = g_1 t_1 g_1 t_1$ and $t_1$ commutes with $g_j$ for $j \\ge 2$.\n\\end{enumerate}\nLet $u_1, \\dots, u_r$ be additional elements in $S$. The {\\em cyclotomic Hecke algebra}\n\\break $\\hec{n, S, r}(q^2; u_1, \\dots, u_r)$ is the quotient of the affine Hecke algebra $\\ahec{n, S}(q^2)$ by the polynomial relation $(t_1 - u_1) \\cdots (t_1 - u_r) = 0$.\n\\end{definition}\n\nWe remark that since the generator $t_1$ can be rescaled by an arbitrary invertible element of $S$, only the ratios of the parameters $u_i$ have invariant significance in the definition of the cyclotomic Hecke algebra. The affine and cyclotomic Hecke algebras have unique algebra involutions determined by\n$g_i \\to g_i$ and $t_1 \\to t_1$. \n\nNow let $S$ be a ground ring with parameters $\\rho$, $q$, $\\delta_j$, and $u_1, \\dots, u_r$. For each $n$, let $I_n$ be the two sided ideal in $\\bmw{n, S, r}$ generated by $e_{n-1}$. \nBecause of the relations $e_j e_{j\\pm1} e_j = e_j$, the ideal $I_n$ is generated by any $e_i$ ($1 \\le i \\le n-1$) or by all of them. It is easy to check that the quotient of $\\bmw{n, S, r}$ by $I_n$ is isomorphic (as involutive algebras) to the cyclotomic Hecke algebra $\\hec{n, S, r}(q^2; u_1, \\dots, u_r)$.\n\n\nLet $\\lambdabold = (\\lambda^{(1)}, \\dots, \\lambda^{(r)})$ be an $r$--tuple of Young diagrams. The total size of $\\lambdabold$ is\n$|\\lambdabold| = \\sum_i |\\lambda^{(i)}|$. If ${\\bm \\mu}$ and $\\lambdabold$ are $r$--tuples of Young diagrams of total size $f-1$ and $f$ respectively, we write ${\\bm \\mu} \\subset \\lambdabold$ if ${\\bm \\mu}$ is obtained from $\\lambdabold$ by removing one box from one component of $\\lambdabold$.\n\n\\begin{theorem}[\\cite{ariki-book}] \\label{theorem: cyclotomic hecke split semisimple}\nLet $F$ be a field. The cyclotomic Hecke algebra $\\hec{n, F, r}(q^2; u_1, \\dots, u_r)$ is split semisimple for all $n$ as long as $q$ is not a proper root of unity and, for all\n$i \\ne j$,\n$u_i\/u_j$ is not an integer power of $q$ . In this case, the simple components of \\break $\\hec{n, F, r}(q; u_1, \\dots, u_r)$ are labeled by $r$--tuples of Young diagrams of total size $n$, and a simple $\\hec{n, F, r}$ module $V_\\lambdabold$ decomposes as a\n$\\hec{n-1, F, r}$ module as the direct sum of all $V_{\\bm \\mu}$ with ${\\bm \\mu} \\subset \\lambdabold$.\n\\end{theorem}\n\nLet us call the branching diagram for the cyclotomic Hecke algebras, as described in the theorem, the\n{\\em $r$--Young lattice}. Note that, as for the usual Young's lattice, the $r$--Young lattice has no multiple edges. \n\n\n\n\n \\begin{theorem}[Ariki, Koike, Dipper, James, Mathas] \\label{propositon: cyclotomic Hecke coherent tower}\n The sequence of cyclotomic Hecke algebras $(\\hec{n, S, r}(q^2; u_1, \\dots, u_r))_{n \\ge 0}$ is a coherent tower of cellular algebras.\n \\end{theorem}\n \n \\begin{proof} Write $\\hec n$ for $ \\hec{n, S, r}(q^2; u_1, \\dots, u_r)$. Ariki and Koike showed that the cyclotomic Hecke algebras are free as $S$ modules ~\\cite{ariki-koike}, which implies that $\\hec n$ imbeds naturally in \n $\\hec {n+1}$. Moreover, the algebras $\\hec n$ have involutions that are consistent with the inclusions. Dipper, James and Mathas ~\\cite{dipper-james-mathas} constructed a cellular basis of the cyclotomic Hecke algebras, generalizing the Murphy basis of ordinary Hecke algebras. Ariki and Mathas\n showed ~\\cite{ariki-mathas}, Proposition 1.9, that restrictions of cell modules from $\\hec {n+1}$ to $\\hec n$ have cell filtrations. Finally, Mathas has shown ~\\cite{mathas-2009} that the module obtained from inducing a cell module from\n $\\hec n$ to $\\hec {n+1}$ has a cell filtration.\n \\end{proof}\n\n \\subsubsection{Verification of the framework axioms for the cyclotomic BMW algebras}\nLet $R$ be the generic admissible integral ground ring, with parameters \n${\\bm \\rho}$, ${\\bm q}$, ${\\mathbold \\delta}_j$ ($j \\ge 0$), and ${\\bm u}_1, \\dots, {\\bm u}_r$, as introduced at the end of Section \\ref{subsubsection: morphisms and generic ring}. In this section, \nwe write $\\bmw n$ for $\\bmw{n, R, r}({\\bm u}_1, \\dots, {\\bm u}_r)$,\n $\\kt n$ for $\\kt{n, R, r}({\\bm u}_1, \\dots, {\\bm u}_r)$,\nand $\\hec n$ for\n$\\hec{n, R, r}({\\bm q}^2; {\\bm u}_1, \\dots, {\\bm u}_r)$. Recall that the field of fractions of $R$ is\n$F = {\\mathbb Q}({\\bm q}, {\\bm u}_1, \\dots, {\\bm u}_r)$. Let $\\bmw n^F = \\bmw n \\otimes_R F$, and similarly for the other algebras. \n\nIf we would assume the isomorphism of $\\bmw n$ and $\\kt n$, then we could verify the framework axioms for the pair of sequences $(\\bmw n)_{n \\ge 0}$ and $(\\hec n)_{n \\ge 0}$ without difficulty,\nusing elementary observations and some deeper results from the literature, and consequently apply Theorem \\ref{main theorem} to the cyclotomic BMW algebras. However, we wish to give\nan independent proof of the isomorphism. Consequently, we have to verify the framework axioms and prove the isomorphism $\\bmw n \\cong \\kt n$ inductively, in tandem with the inductive step in the proof of Theorem \\ref{main theorem}. \n\n\\begin{lemma} \\label{lemma: Axiom on A0 for cyclotomic BMW}\n $\\bmw 0 \\cong \\kt 0 \\cong R$.\n\\end{lemma}\n\n\\begin{proof} Wilcox and Yu ~\\cite{Wilcox-Yu3}, Proposition 6.2, show that $\\kt 0$ is a free $R$ module with basis $\\{\\emptyset\\}$, where\n$\\emptyset$ denotes the empty affine tangle diagram, \nwhich is also the identity element of $\\kt 0$.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma: isomorphism varphi implies injectivity of iota}\n If for some $n$ and for some admissible ground ring $S$, we have\n$\\varphi : \\bmw n^S \\to \\kt n^S$ is an isomorphism, then $\\iota: \\bmw n^S \\to \\bmw {n+1}^S$ is injective.\n\\end{lemma}\n\n\\begin{proof} $\\varphi \\circ \\iota = \\iota \\circ \\varphi : \\bmw n^S \\to\n\\kt {n+1}^S$ is injective, because $\\varphi: \\bmw n^S \\to \\kt n^S$ and $\\iota: \\kt n^S \\to \\kt {n+1}^S$ are injective. Thus\n $\\iota : \\bmw n^S \\to \\bmw {n+1}^S$ is injective.\n\\end{proof} \n\n\n\\begin{lemma} \\label{lemma: generic semisimplicty of cycotomic BMW} For all $n \\ge 0$, $\\bmw n^F \\cong \\kt n^F$, $\\bmw n^F$ is split semisimple of dimension \\break $r^n (2n - 1)!!$, and $\\iota : \\bmw n^F \\to \\bmw {n+1}^F$ is injective. \n\\end{lemma}\n\n\\begin{proof} This is proved in ~\\cite{GH3}, Theorem 4.8. We stress that the result is independent of the finite spanning set theorem, ~\\cite{GH2}, Proposition 3.7. One thing that is not made clear in the proof of \n~\\cite{GH3}, Theorem 4.8 is why $\\iota : \\bmw n^F \\to \\bmw {n+1}^F$ is injective. But if one assumes\ninductively that the conclusions of the theorem hold for $\\bmw f^F, f \\le n$, for some fixed $n$, and in particular that\n$\\varphi : \\bmw n^F \\to \\kt n^F$ is an isomorphism, then $\\iota : \\bmw n^F \\to \\bmw {n+1}^F$ is injective by Lemma \\ref{lemma: isomorphism varphi implies injectivity of iota}.\nOne can then continue with the proof of the inductive step of ~\\cite{GH3}, Theorem 4.8. \n\\end{proof} \n\n\\begin{lemma} \\label{lemma: cardinality of basis of bmw n}\nIf for some $n$, $\\bmw n$ is a free $R$--module, then its rank is $r^n (2n-1)!!$.\n\\end{lemma}\n\n\\begin{proof} $x \\mapsto x \\otimes 1$ takes an $R$--basis of $\\bmw n$ to an $F$--basis of\n$\\bmw n \\otimes_R F = \\bmw n^F$. \n\\end{proof}\n\n\\begin{lemma} \\label{lemma: finite spanning set implies isomorphism and freeness}\n If for some $n$, $\\bmw n$ has a spanning set $A$ of cardinality $r^n (2n -1)!!$, \nthen $\\varphi : \\bmw n \\to \\kt n$ is an isomorphism, and $A$ is an $R$--basis of $\\bmw n$. \n\\end{lemma}\n\n\\begin{proof} Say $\\bmw n$ has a spanning set $A$ of cardinality $r^n (2n -1)!!$. To prove both conclusions, it suffices to show that $\\varphi(A)$ is linearly independent in $\\kt n$. But $$ \\{\\varphi(a) \\otimes 1: a \\in A\\} \\subseteq \\kt n \\otimes_R F = \\kt n^F$$ is a spanning set of cardinality $r^n (2n -1)!!$, which is the dimension of $\\kt n^F$, according to Lemma \\ref{lemma: generic semisimplicty of cycotomic BMW}. Therefore $\\{\\varphi(a) \\otimes 1: a \\in A\\} $ is linearly independent in $\\kt n^F$, and hence $\\varphi(A)$ is linearly independent in $\\kt n$. \n\\end{proof} \n\n\n\n\n\\begin{lemma} \\label{lemma: W1 isomorphic to KT1 and to H1}\n$\\bmw 1 \\cong \\kt 1 \\cong \\hec 1$, $\\bmw 1$ is a free $R$--module of rank $r$, and both\n$\\iota : \\bmw 0 \\to \\bmw 1$ and $\\iota : \\bmw 1 \\to \\bmw 2$ are injective. \n\\end{lemma}\n\n\\begin{proof} By definition, $\\bmw 1 \\cong \\hec 1 \\cong R[X]\/((X-u_1)\\cdots (X-u_r))$, and these algebras are free $R$--modules of rank $r$. Hence $\\varphi : \\bmw 1 \\to \\kt 1$ is an isomorphism by\nLemma \\ref{lemma: finite spanning set implies isomorphism and freeness}. The injectivity statements follow from Lemma \\ref{lemma: isomorphism varphi implies injectivity of iota}.\n\n \\end{proof}\n\n\\begin{lemma} \\label{lemma: axioms 2 6 7 for cyclotomic BMW} \nSuppose that for some $n \\ge 1$ one has $\\bmw k \\cong \\kt k$ for $0 \\le k \\le n$. Then\nthe maps $\\iota: \\bmw {k} \\to \\bmw {k+1}$ are injective for $0 \\le k \\le n$. Using the maps $\\iota$, regard $\\bmw k$ as a subalgebra of $\\bmw {k+1}$ for $0 \\le k \\le n$. One has:\n \\begin{enumerate}\n \\item $ {\\mathbold \\delta}_0 R \\,e_1 \\subseteq\n e_1 \\bmw 1 e_1 \\subseteq R \\,e_1$.\n \n \n \\item \nFor $2 \\le k \\le n$, $e_{k} \\bmw {k} e_{k} = \\bmw {k-1} e_{k}$.\n\n\\item For $1 \\le k \\le n$, \n$e_{k}$ commutes with $ \\bmw {k-1} $. \n\\item For $1 \\le k \\le n$, \n$\\bmw {k+1} \\,e_{k} = \\bmw {k} \\,e_{k}$. Moreover, \n $x \\mapsto x e_{k}$ is injective from $\\bmw {k}$ to $\\bmw {k} e_{k}$.\n\\end{enumerate}\n \\end{lemma}\n \n \\begin{proof} The statement about injectivity of the maps $\\iota$ follows from Lemma \\ref{lemma: isomorphism varphi implies injectivity of iota}. \n \n Point (1) follows from the relations $e_1 y_1^j e_1 = {\\mathbold \\delta}_j e_1$ for $j \\ge 0$. \n Point (2) and the first part of point (4) follows from the corresponding facts for the affine BMW algebras, ~\\cite{GH1}, Proposition 3.17, and Proposition 3.20. \nPoint (3) follows from the defining relations for the cyclotomic BMW algebras. For the injectivity statement\nin point (4), note that for $x \\in \\bmw k$, \n$$\n{\\rm cl}(\\varphi(x e_k)) = {\\rm cl}(\\varphi(x) E_k) = \\varphi(x).\n$$ \nSince $\\varphi : \\bmw k \\to \\kt k$ is injective, so is $x \\mapsto x e_k$. \n\n\\end{proof}\n\n\n\\vbox{\n\\begin{theorem}\\label{proposition: framework axioms for CBMW} \\mbox{}\n\\begin{enumerate}\n\\item\n The two sequences of algebras $(\\bmw k)_{k \\ge 0}$ and $(H_k)_{k \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}.\n \\item For all $k \\ge 0$, $\\varphi : \\bmw k \\to \\kt k$ is an isomorphism, and $\\iota: \\bmw k \\to \\bmw {k+1}$ is injective.\n \\item The conclusions of Theorem \\ref{main theorem} are valid for the sequence $(\\bmw k)_{k \\ge 0}$. \n \\end{enumerate}\n\\end{theorem}\n}\n\n\\begin{proof} According to Proposition \\ref{propositon: cyclotomic Hecke coherent tower}, $(H_k)_{k \\ge 0}$ is a coherent tower of cellular algebras, so axiom (\\ref{axiom Hn coherent}) of the framework axioms holds. Axiom (\\ref{axiom: A0 and A1}) holds by Lemmas \\ref{lemma: Axiom on A0 for cyclotomic BMW} and \\ref{lemma: W1 isomorphic to KT1 and to H1}. \nAxiom (\\ref{axiom: semisimplicity}) holds by Lemma \\ref{lemma: generic semisimplicty of cycotomic BMW}. We observed above that $\\bmw k\/\\bmw k e_{k-1} \\bmw k \\cong H_k$ as involutive algebras; thus axiom (\\ref{axiom: idempotent and Hn as quotient of An}) holds. \nAxiom (\\ref{axiom: e(n-1) in An en An}) holds because of the relation $e_{k-1} e_k e_{k-1} = e_{k-1}$.\n\nSuppose that for some $n \\ge 0$, it is known that the maps $\\varphi : \\bmw k \\to \\kt k$ are isomorphisms for $0 \\le k \\le n$. Then, from Lemma \\ref{lemma: axioms 2 6 7 for cyclotomic BMW} , we have the following versions of framework axioms (\\ref{axiom: involution on An}), (\\ref{axiom: en An en}) and (\\ref{axiom: An en}):\n\\par\\noindent (2$'$) \\ \\ $\\bmw k$ is an $i$--invariant subalgebra of $\\bmw {k+1}$ for $0 \\le k \\le n$. \n\\par\\noindent (6$'$) \\ \\ For $1 \\le k \\le n$, $e_{k}$ commutes with $\\bmw {k-1}$ and $e_{k} \\bmw {k} e_{k} \\subseteq \\bmw {k-1} e_{k}$.\n\\par\\noindent (7$'$) \\ \\ For $ 1 \\le k \\le n$, $\\bmw {k+1} \te_{k} = \\bmw {k} e_{k}$, and the map $x \\mapsto x e_{k}$ is injective from\n$\\bmw {k}$ to $\\bmw {k} e_{k}$.\n\nNow we consider the following:\n\\par\\smallskip\n\\noindent {\\bf Claim:}\\quad\n For all $n \\ge 0$, \\par\n\\noindent (a) \\ \\\nfor $0\\le k \\le n$, the maps $\\varphi : \\bmw k \\to \\kt k$ are isomorphisms,\nand therefore $\\bmw k$ may be regarded as an $i$--invariant subalgebra of $\\bmw {k+1}$, and\n\\par\\noindent (b) \\ \\ the statements (1) --(4) of Theorem \\ref{main theorem} hold for the finite tower $(\\bmw k)_{0 \\le k \\le n}$.\n\\smallskip\n\nFor $n = 0$ and $n = 1$, the claim follows from Lemmas \\ref{lemma: Axiom on A0 for cyclotomic BMW} and \\ref{lemma: W1 isomorphic to KT1 and to H1}. We assume the claim holds for some $n \\ge 1$ and show that it also holds for $n + 1$. \nThen, by the discussion above, the framework axioms hold for the finite tower\n$(\\bmw k)_{0\\le k \\le n}$ with axioms (\\ref{axiom: involution on An}), (\\ref{axiom: en An en}) and (\\ref{axiom: An en}) replaced by the finite versions (2$'$), (6$'$), and (7$'$).\nNow the inductive step in the proof of Theorem \\ref{main theorem} goes through without change and\nyields part (b) of the claim for the tower $(\\bmw k)_{0 \\le k \\le n+1}$. In particular, \n$\\bmw {n+1}$ is a cellular algebra; the cardinality of its cellular basis is $r^{n+1} (2n + 1)!!$,\nby Lemma \\ref{lemma: cardinality of basis of bmw n}. But then\nLemma \\ref{lemma: finite spanning set implies isomorphism and freeness} gives that\n$\\varphi: \\bmw {n+1} \\to \\kt {n+1}$ is an isomorphism, so part (a) of the claim also holds for $n + 1$. \n\\end{proof}\n\n\\begin{corollary} Let $S$ be any admissible integral ground ring with $q - q^{-1} \\ne 0$. \n\\begin{enumerate}\n\\item\nThe sequence of cyclotomic BMW algebras $(\\bmw {n, S, r})_{n \\ge 0}$ is a coherent tower of cellular algebras.\n$\\bmw {n, S, r}$ has cell modules indexed by all $r$--tuples of Young diagrams of total size $n$, $n-2$, $n-4, \\dots$. The cell module labeled by\nan $r$--tuple of Young diagrams $\\lambdabold$ has a basis labeled by up--down tableaux of length $n$ and shape $\\lambdabold$.\n\\item $\\bmw {n, S, r} \\cong \\kt {n, S, r}$ for all $n \\ge 0$. \n\\end{enumerate}\n\\end{corollary}\n\n\n\\begin{remark} It is possible to combine our results with the results of Wilcox and Yu\n~\\cite{Wilcox-Yu2} to obtain\nMurphy type bases of the cyclotomic BMW algebras that are strictly cellular, i.e. $i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda $ for all $\\lambda, s, t$. To do this, all we need, according to Remark \\ref{remark: conditions for cell net to give strict cellular basis}, is an $i$--invariant $R$--module complement to the ideal\n$\\breve {\\bmw n}^{(\\lambdabold, n)}$ in $ {\\bmw n}^{(\\lambdabold, n)}$. However, one can check that the\nideals $\\breve {\\bmw n}^{(\\lambdabold, n)}$ and $ {\\bmw n}^{(\\lambdabold, n)}$ for our cellular structure are the same as for the cellular structure of Wilcox and Yu, and therefore, since their cellular basis satisfies the strict equality $i(c_{s, t}^\\lambda) = c_{t, s}^\\lambda $ for all $\\lambda, s, t$, the desired\n$i$--invariant $R$--module complement exists. \n\\end{remark}\n\n \\begin{remark} Our framework also applies to \n the degenerate\ncyclotomic BMW algebras (cyclotomic Nazarov Wenzl algebras) studied in ~\\cite{ariki-mathas-rui}. For the details, see ~\\cite{GG2}.\n \\end{remark} \n\n \n\\subsection{The walled Brauer algebras} \\label{subsection: walled Brauer algebras}\n\\subsubsection{Definition of the walled Brauer algebras} \nLet $S$ be a commutative ring with identity, with a distinguished element $\\delta$.\nThe walled (or rational) Brauer algebra $B_{r, s}(S, \\delta)$\n is a unital subalgebra of the Brauer algebra $B_{r+s}(S, \\delta)$ spanned by certain Brauer diagrams.\n Divide the $r+s$ top vertices into a left cluster consisting of the leftmost $r$ vertices and a right cluster consisting of the remaining $s$ vertices, and similarly for the bottom vertices. The walled Brauer diagrams are those in which no vertical strand connects a left vertex and a right vertex, and every horizontal strand connects a left vertex and a right vertex. (If we draw a vertical line--the wall--separating left and right vertices, then vertical strands are forbidden to cross the wall, and horizontal strands are required to cross the wall.) One can easily check that the span of walled Brauer diagrams is a unital subalgebra of $B_{r+s}(S, \\delta)$.\n \n \\subsubsection{Brief history of the walled Brauer algebras} The walled Brauer algebras were introduced by Turaev ~\\cite{turaev-operator-invariants} and by Koike ~\\cite{koike-tensor-products}, and studied by Benkart et.\\ al.\\ ~\\cite{benkart-walled-brauer} and by Nikitin ~\\cite{nikitin-walled-brauer}. \n The walled Brauer algebras arise in connection with the invariant theory of the general linear group acting on mixed tensors. \n Cellularity of walled Brauer algebras was proved by Green and Martin ~\\cite{green-martin-tabular} and by Cox et.\\ al.\\ ~\\cite{cox-walled-brauer}; the latter authors show that\n walled Brauer algebras can be arranged into coherent cellular towers.\n \n \\subsubsection{Some properties of the walled Brauer algebras} The walled Brauer algebra $B_{r, s}$ is invariant under the involution $i$ of the Brauer algebra $B_{r+s}$. Moreover, the inclusion map\n $\\iota : B_{r+s} \\to B_{r+s+1}$ maps $B_{r, s}$ to $B_{r, s+1}$, and the closure map\n ${\\rm cl} : B_{r+s} \\to B_{r + s -1}$ maps $B_{r, s}$ to $B_{r, s-1}$, when $s \\ge 1$. If $\\delta$ is invertible, $\\varepsilon_{r, s} = (1\/\\delta)\\, {\\rm cl} : B_{r, s} \\to B_{r, s-1}$ is a conditional expectation, and, of course, the trace $\\varepsilon$ on $B_{r+s}$ restricts to a trace on $B_{r, s}$. \n \n The Brauer algebras have an involutive inner automorphism $\\rho$ which maps each Brauer diagram to its reflection in the vertical line $x = 1\/2$. (We might as well take the vertical line to coincide with our wall.)\n It is clear that $\\rho$ restricts to an isomorphism from $B_{r, s}$ to $B_{s, r}$. Given this, we can define ``left versions\" of $\\iota$, ${\\rm cl}$ and $\\varepsilon_{r, s}$ by\n $\\iota' = \\rho \\circ \\iota \\circ \\rho : B_{r, s} \\to B_{r+1, s}$, ${\\rm cl}' = \\rho \\circ {\\rm cl} \\circ \\rho : B_{r, s} \\to B_{r-1, s}$, and $\\varepsilon' = \\rho \\circ \\varepsilon \\circ \\rho : B_{r, s} \\to B_{r-1, s}$. Note that\n $\\iota'$ adds a vertical strand on the left, and ${\\rm cl}'$ partially closes diagrams on the left.\n \n Let $e_{a, b}$ be the Brauer diagram with horizontal strands connecting $\\p a$ to $\\p {b}$ and\n $\\pbar a$ to $\\pbar {b}$ and vertical strands connecting $\\p j$ to $\\pbar j$ for all $j \\ne a, b$.\n One can easily check the following properties: \n \\begin{lemma} \\mbox{} \\label{lemma: properties of e(r, s) in walled brauer algebra}\n \\begin{enumerate}\n \\item $e_{a, b}^2 = \\delta e_{a, b}$.\n \\item $e_{a, b} \\,e_{a, b\\pm 1}\\, e_{a, b} = e_{a, b}$\n and $e_{a, b} \\,e_{a\\pm 1, b}\\, e_{a, b} = e_{a, b}$. \n \\item For $e_{a, b} \\in B_{r, s}$, $\\iota(e_{a, b}) = e_{a, b}$ and $\\iota'(e_{a, b}) = e_{a+1, b+1}$.\n \\item For $x \\in B_{r, s+1}$, we have $e_{1, r+ s+2} \\,\\iota'(x)\\, e_{1, r+ s+2} = \\iota'\\circ \\iota\\circ{\\rm cl}(x)\\, e_{1, r+ s+2}$.\n \\item For $x \\in B_{r+1, s}$, we have $e_{1, r+ s+2} \\,\\iota(x)\\,e_{1, r+ s+2}= \\iota'\\circ \\iota\\circ {\\rm cl}'(x) \\,e_{1, r+ s+2} $.\n \\item $e_{1, r+s + 2}$ commutes with $\\iota'\\circ \\iota(x)$ for all $x \\in B_{r, s}$.\n \\end{enumerate}\n \\end{lemma}\n \n The following statement is also easy to check:\n \n \\begin{lemma} \\label{lemma: quotient of A_n by J for walled brauer}\nThe ideal $J$ in $B_{r, s}(S, \\delta)$ generated by $e_{1, r+s}$ is the ideal spanned by diagrams with fewer than $r + s$ through strands, and $B_{r, s}(S, \\delta)\/J \\cong S (\\mathfrak S_r \\times \\mathfrak S_s)$.\n \\end{lemma}\n \n\n \n \\begin{lemma} \\mbox{} \\label{lemma: An e(n-1) equals A(n-1) e(n-1) for walled Brauer}\n \\begin{enumerate}\n \\item $B_{r, s+1}\\, e_{1, r+s+1} = \\iota(B_{r, s}) \\,e_{1, r+s+1}$.\n \\item $B_{r+1, s} \\,e_{1, r+s+1} = \\iota'(B_{r, s})\\, e_{1, r+s+1}$.\n \\end{enumerate}\n \\end{lemma}\n \n \\begin{proof} To prove part (1), we\n have to show that if $d$ is a diagram in $B_{r, s+1}$, then there is a diagram\n $d' \\in \\iota(B_{r, s})$ such that $d \\,e_{1, r+s+1} = d' \\,e_{1, r+s+1}$. We can suppose that $d$ is not already in $ \\iota(B_{r, s})$; therefore, the vertex $ \\pbar {r+s + 1}$ in $d$ is connected to some vertex $v$ other than $\\p 1$ and $\\p {r+s + 1}$. There are two cases to consider. \n \n\n \n The first is that the vertices $\\p 1$ and $\\p {r+s + 1}$ are not connected to each other in $d$; let $a$ and $b$ be the vertices connected to\n $\\p 1$ and $\\p {r+s + 1}$. Now let $d'$ be the diagram in which $a$ and $b$ are connected to each other; $\\p {r + s + 1}$ is connected to $ \\pbar {r+s + 1}$; $\\p 1$ is connected to $v$; and all other strands are as in $d$. Then we have $d \\,e_{1, r+s+1} = d' \\,e_{1, r+s+1}$.\n The case that the vertices $\\p 1$ and $\\p {r+s + 1}$ are connected to each other is similar and will be omitted.\n \n Part (2) is proved by applying the map $\\rho$ to both sides of the equality in part (1) and then\n interchanging the roles of $r$ and $s$.\n \\end{proof}\n \n \n A unital trace $\\varepsilon$ on an $S$--algebra $A$ is {\\em non-degenerate} if for every non--zero $x \\in A$ there exists a $y \\in A$ such that $\\varepsilon(xy) \\ne 0$.\n \n \\begin{lemma} \\mbox{} \\label{lemma: non-degeneracy of trace on Brauer and walled Brauer}\n \\begin{enumerate}\n \\item\n The trace $\\varepsilon$ on $B_n({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is non-degenerate, for any $n$.\n \\item The trace $\\varepsilon$ on $B_{r, s}({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is non-degenerate, for any $r, s$.\n \\end{enumerate}\n \\end{lemma}\n \n \\noindent {\\em Sketch of proof.} The argument for part (1) is from ~\\cite{Morton-Traczyk}. It suffices to show that\n the determinant of the Gram matrix $\\varepsilon(d d')_{d, d'}$, where $d, d'$ run over the list of all Brauer diagrams (in some order), is non-zero. Recall that $\\varepsilon(d d')$ is ${\\bm q}^{c(d d')- n}$, where\n $c(d d')$ is the number of components in the tangle obtained by closing all the strands of $d d'$. \n One can check that $c(d \\,i(d) ) = n$ and $c(d d') < n$ for all diagrams other than $i(d)$. Therefore, each\n row and column of the Gram matrix has exactly one entry equal to $1$ and all other entries have the form \n ${\\bm q}^{-k}$ for some $k >0$. \n \n The argument for part (2) is identical.\n \\qed\n \n \\medskip\n\n \n \\subsubsection{Verification of the framework axioms for the walled Brauer algebras}\n \nTo fit the walled Brauer algebras to our framework, we have to reduce the double sequence of algebras to a single sequence. We adopt the following scheme, as in ~\\cite{nikitin-walled-brauer}, or\n~\\cite{cox-walled-brauer}: \n Fix some integer $t \\ge 0$. For any $S$ and $\\delta \\in S$, we consider the sequence of walled Brauer algebras\n $A_n = A_n(S, \\delta)$, where\n $A_{2 k}(S, \\delta) = B_{k, k+ t}(S, \\delta)$, and $A_{2 k + 1}(S, \\delta) = B_{k, k+t + 1}(S, \\delta)$, with the inclusions\n $$\n A_{2 k} \\stackrel{\\iota}{\\longrightarrow} A_{2k + 1} \\stackrel{\\iota'}{\\longrightarrow} A_{2k + 2}.\n $$\nWe put $f_{2k-1} = e_{1, 2k+t} \\in A_{2k}$ and $f_{2k} = e_{1, 2k+ t + 1} \\in A_{2k+1}$. \n We identify $A_n$ as a subalgebra of $A_{n+1}$ via these embeddings.\n With these conventions, Lemma \\ref{lemma: properties of e(r, s) in walled brauer algebra}, points (2) and (3) give $f_n f_{n\\pm 1} f_n = f_n$.\n Moreover, if we write ${\\rm cl}_n = {\\rm cl}$ when $n$ is even and ${\\rm cl}_n = {\\rm cl}'$ when $n$ is odd, then\n we have $f_{n-1} x f_{n-1} = {\\rm cl}_{n-1}(x) f_{n-1} $ for $x \\in A_{n-1}$, by Lemma \\ref{lemma: properties of e(r, s) in walled brauer algebra}, points (4) and (5). Point (6) of the Lemma says that $f_{n-1}$ commutes with\n $A_{n-2}$.\n \n \n If $J$ is the ideal in $A_n$ generated by\n $f_{n-1}$, then we have $A_{2k}\/J \\cong S(\\mathfrak S_k \\times \\mathfrak S_{k + t})$, and $A_{2k + 1}\/J \\cong S(\\mathfrak S_k \\times \\mathfrak S_{k + t + 1})$. So we set $Q_{2k}(S) = S(\\mathfrak S_k \\times \\mathfrak S_{k + t})$ and $Q_{2k + 1}(S) = S(\\mathfrak S_k \\times \\mathfrak S_{k + t + 1})$, with the natural embeddings.\n \n Since $A_0 = B_{0, t} \\cong S \\mathfrak S_t$, and $A_1 = B_{0, t+1} \\cong S \\mathfrak S_{t + 1}$, we cannot hope to satisfy our framework axiom (\\ref{axiom: A0 and A1}). However, we can replace axiom (\\ref{axiom: A0 and A1}) with the weaker\n \n \\medskip\n \\noindent(3$'$) \\quad $A_0 \\cong Q_0$, and $A_1 \\cong Q_1$.\n \\medskip\n \n\\noindent We also have to drop our usual convention (see Definition \\ref{def of branching}) regarding branching diagrams that the \n$0$--th row of the branching diagram has a single vertex. Our conclusions will have to be modified, but not severely.\n \n \\medskip\n We now take $R = {\\mathbb Z}[{\\mathbold \\delta}]$ and $\\delta = {\\mathbold \\delta}$. $R$ is the generic ground ring for\n walled Brauer algebras; if $S$ is any commutative unital ring with parameter $\\delta$, then\n $B_{r, s}(S, q) = B_{r, s}(R, {\\bm q}) \\otimes_R S$. Let $F = {\\mathbb Q}({\\mathbold \\delta})$.\n In the remainder of this section, we write\n $A_n = A_n(R, {\\mathbold \\delta})$ and $Q_n = Q_n(R)$. (Recall that $Q_n(R) = R(\\mathfrak S_k \\times \\mathfrak S_{k + t})$ if $n = 2k$, and\n$ Q_n(R) = R(\\mathfrak S_k \\times \\mathfrak S_{k + t + 1})$ if $n = 2 k + 1$.)\n \n \n \\begin{lemma} \\label{lemma: generic semisimplicity for walled Brauer}\n The walled Brauer algebra $B_{r, s}({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is split semisimple.\n \\end{lemma}\n \n \\noindent{\\em Sketch of proof.} It suffices to show that (for any $t$) the algebras in the sequence $A_n$ are split semisimple. This was proved by Nikitin in ~\\cite{nikitin-walled-brauer}, following Wenzl's method for the Brauer algebra in ~\\cite{Wenzl-Brauer}. \n Nikitin's proof involves obtaining the weights of the trace $\\varepsilon$, but little detail is given. For our purposes, we can bypass this issue, and use Lemma \\ref{lemma: non-degeneracy of trace on Brauer and walled Brauer} instead. Then the method of proof of Theorem 3.2 from ~\\cite{Wenzl-Brauer} applies.\n \\qed\n \\medskip \n \n \\begin{proposition} \\label{proposition: framework axioms for walledBrauer} The two sequence of $R$--algebras $(A_n)_{n \\ge 0}$ and $(Q_n)_{n \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}, with axiom (3) replaced by (3\\,$'$), specified above, and with the elements\n $f_n$ taking the role of the elements $e_n$ in the list of framework axioms.\n\\end{proposition}\n\n\\begin{proof} The sequence $(Q_n)_{n \\ge 0}$ is clearly a coherent tower of cellular algebras, so axiom (\\ref{axiom Hn coherent}) holds. Axiom (\\ref{axiom: involution on An}) is evident, and we have remarked about substituting axiom (\\ref{axiom: A0 and A1}$'$) for axiom (\\ref{axiom: A0 and A1}). \n$A_n^F$ is split semisimple by Lemma \n\\ref{lemma: generic semisimplicity for walled Brauer}. Thus axiom (\\ref{axiom: semisimplicity}) holds.\n\nWe have $f_{n-1}$ is an essential idempotent with $i(f_{n-1}) = f_{n-1}$. We have \\break $A_n\/ (A_n f_{n-1} A_n) \\cong Q_n$ by Lemma \\ref{lemma: quotient of A_n by J for walled brauer}, which gives axiom \n(\\ref{axiom: idempotent and Hn as quotient of An}).\n\nWe have seen that $f_{n-1}$ commutes with $A_{n-2}$ and $f_{n-1} A_{n-1} f_{n-1} \\subseteq A_{n-2} f_{n-1}$. Moreover, if $x \\in A_{n-2}$, then $f_{n-1} x f_{n-1} = {\\mathbold \\delta} x f_{n-1}$, so\n$f_{n-1} A_{n-1} f_{n-1} \\supseteq {\\mathbold \\delta} A_{n-2} f_{n-1}$. Therefore, \n$f_{n-1} A_{n-1}^F f_{n-1} = A_{n-2}^F f_{n-1}$, so axiom (\\ref{axiom: en An en}) holds.\n\nAxiom (\\ref{axiom: An en}) results from Lemma \\ref{lemma: An e(n-1) equals A(n-1) e(n-1) for walled Brauer}, and\naxiom (\\ref{axiom: e(n-1) in An en An}) from $f_{n-1} f_n f_{n-1} = f_{n-1}$.\n\\end{proof}\n\n\\begin{remark} The branching diagram for the sequence\n$(Q_n^F)$ is the following: Each row has vertices labeled by pairs of Young diagrams; on an even row $2k$, the the first Young diagram in a pair has $k$ boxes and the second $k+t$ boxes; on an odd row $2k+1$, the first Young diagram has $k$ boxes and the second $k+t+1$ boxes; finally, there is an edge between pairs of Young diagrams in successive rows that differ by exactly one box.\n\\end{remark}\n\n\\begin{corollary} \\mbox{} Let $S$ be any commutative unital ring with parameter $\\delta$.\n\\begin{enumerate}\n\\item The walled Brauer algebras $B_{r, s}(S, \\delta)$ are cellular algebras. \n\\item The family is coherent in the sense that the restriction of a cell module from $B_{r, s}(S, \\delta)$ to $B_{r-1, s}(S, \\delta)$ or to $B_{r, s-1}(S, \\delta)$ and induction of a cell module from $B_{r, s}(S, \\delta)$ to\n$B_{r+1, s}(S, \\delta)$ or to $B_{r, s+1}(S, \\delta)$ have filtrations by cell modules. \n\\item The cell modules\nof $B_{r, s}(S, \\delta)$ are labeled by pairs of Young diagrams $(\\lambda^{(1)}, \\lambda^{(2)})$, where\n$|\\lambda^{(2)}| - |\\lambda^{(1)}| = s - r$ and $|\\lambda^{(2)}| + |\\lambda^{(1)}| \\le s + r$.\n\\end{enumerate}\n\\end{corollary}\n\n\nA basis for any cell module for $B_{r, s}$ can be labeled by paths on a certain branching diagram.\nSuppose without loss of generality that $t = s - r \\ge 0$. Let $(A_n)_{n \\ge 0}$ and $(Q_n)_{n \\ge 0}$ be the two sequences of algebras defined above, depending on $t$, so in particular, $B_{r, s} = A_{2 r}$.\nLet $\\mathfrak{B}_0$ be the branching diagram for $(Q_n^F)_{n \\ge 0}$, which was described above, and let \n$\\mathfrak{B}$ be that obtained by reflections from $\\mathfrak{B}_0$. On the $0$--th row, $\\mathfrak{B}$ has vertices labeled\nby all pairs $(\\emptyset, \\lambda)$, where $\\lambda$ is a Young diagram of size $t$. Finally, augment $\\mathfrak{B}$ with a copy of Young's lattice up to the $(t-1)$--st level, with vertices labeled by pairs $(0, \\mu)$ with $0 \\le |\\mu| \\le t-1$. The pairs of Young diagrams labeling the cell modules of $B_{r, s}$\nare located on the $r + s$--th row of the augmented branching diagram, and a basis of any cell module can be labeled by paths on the augmented branching diagram from $(\\emptyset, \\emptyset)$ to the pair in question.\n\nWe note that several of the results of Section 3 of ~\\cite{cox-walled-brauer} follow from the application of our method to the walled Brauer algebras.\n\n\\subsection{Partition algebras} \n\n\\subsubsection{Definition of the partition algebras} Let $n$ be an integer, $n \\ge 1$. Let \n$[\\p n] = \\break\\{\\p 1, \\dots, \\p n\\}$ and $[\\pbar n] = \\{ \\pbar 1, \\dots, \\pbar n\\}$ be disjoint sets of size $n$, and \nlet $X_n$ be the family of all set partitions of $[\\p n] \\cup [\\pbar n]$. \n\nWe can represent an element $x$ of $X_n$ by any graph with vertex set equal to $[\\p n] \\cup [\\pbar n]$ whose connected components are the blocks or classes of the partition $x$. We picture such a graph as a diagram\nin the rectangle $\\mathcal R$, with the vertices in $[\\p n]$ arranged on the top edge and those in\n$[\\pbar n]$ arranged on the bottom edge of $\\mathcal R$, as in the tangle diagrams discussed in Section \\ref{subsection: preliminaries on tangle diagrams}.\n\nLet $S$ be any commutative ring with identity, with a distinguished element $\\delta$. \nWe define a product on $X_n$ as follows: \nLet $x$ and $y$ be elements of $X_n$. Realize $y$ as a set partition of $[\\p n] \\cup [\\p n']$ (with $[\\p n']$ the set of bottom vertices). Realize $x$ as a set partition of $[\\p n'] \\cup [\\pbar n]$ (with $[\\p n']$ the set of top vertices).\nLet $E_x$ and $E_y$ be the corresponding equivalence relations, regarded as equivalence relations on\n$[\\p n] \\cup [\\p n'] \\cup [\\pbar n]$. Let $E$ be the smallest equivalence relation on $[\\p n] \\cup [\\p n'] \\cup [\\pbar n]$ containing $E_x \\cup E_y$. Let $r$ be the number of equivalence classes of $E$ contained in $[\\p n']$.\nLet $E_{x y}$ be the equivalence relation obtained by restricting $E$ to $[\\p n] \\cup [\\pbar n]$, and let\n$z$ be the corresponding set partition of $[\\p n] \\cup [\\pbar n]$. Then $x y$ is defined to be $\\delta^r z$. \n\n\n Here is an example of two set partitions represented by graphs and their product.\n\n$$\ny = \\inlinegraphic[scale= .3]{partition1}\n\\qquad\nx = \\inlinegraphic[scale=.3]{partition2}\\qquad\nxy = \\delta \\ \\inlinegraphic[scale= .3]{partition3} \n$$ \n\nWe let $A_{2n}(S, \\delta)$ be the free $S$ module with basis $X_n$. We give $A_{2n}(S, \\delta)$ the bilinear product extending the product defined on $X_n$. One can check the multiplication is associative. Note that $A_0(S, \\delta) \\cong S$. \nFor $n \\ge 1$, the multiplicative identity of $A_{2n}(S, \\delta)$ is the partition with blocks $\\{ \\p i, \\pbar i\\}$ for $1 \\le i \\le n$. \n\n\nFor $n \\ge 1$, Let $X'_n \\subset X_n$ be the family of set partitions with $\\p n$ and $\\pbar n$ in the same block. The\n$S$--span of $X'_n$ is a unital subalgebra of $A_{2n}(S, \\delta)$, which we denote by \\break$A_{2n -1}(S, \\delta)$. \n\nThe algebras $A_k(S, \\delta)$ for $k \\ge 0$ are called the {\\em partition algebras}. \n\nNote that the set partitions $x \\in X_n$ each of whose blocks has size $2$ can be identified with Brauer diagrams on $2n$ vertices, and the product of such diagrams in the Brauer algebra $B_n(S, \\delta)$ agrees with the product in $A_{2n}$. Thus $B_n(S, \\delta)$ can be identified with a unital subalgebra of $A_{2n}(S, \\delta)$.\n\n\\subsubsection{Brief history of the partition algebras} The partition algebras $A_{2n}$ were introduced independently by Martin ~\\cite{martin-partition-JKTR, martin-structure-j.algebra} and Jones ~\\cite{jones-partition}. Partition algebras arise as centralizer algebras for the symmetric group $\\mathfrak S_k$ acting\nas a subgroup of ${\\rm GL}(k, {\\mathbb C})$ on tensor powers of ${\\mathbb C}^k$ ~\\cite{jones-partition, martin-partition-potts}.\nThe algebras $A_{2n+1}$ have been used as an auxiliary device for studying the partition algebras, by Martin and others. Halverson and Ram ~\\cite{halverson-ram-partition} emphasized putting the even and odd algebras on an equal footing, which reveals the role played by the basic construction.\nCellularity of the partition algebras was proved in ~\\cite{Xi-Partition, doran-partition, wilcox-cellular}. For further literature citations, see the review article ~\\cite{halverson-ram-partition}.\n\n\\subsubsection{Some properties of the partition algebras} Fix a ground ring $S$ and $\\delta \\in S$.\nIn this section write $A_k$ for $A_k(S, \\delta)$. \n\n\nFor $n \\ge 1$, $A_{2n-1}$ is defined as a subalgebra of \n$A_{2n}$. The map $\\iota : X_{n} \\to X'_{n+1}$ which adds the additional block\n$\\{\\p {n+1}, \\pbar {n+1}\\}$ to $x \\in X_n$ is an imbedding; the linear extension of $\\iota$ to $A_{2n}$\nis a unital algebra monomorphism into $A_{2n + 1}$. Using $\\iota$, we identify $A_{2n}$ with its image in $A_{2n + 1}$. \n\n\nFor $n \\ge 1$, let $p_{2n -1} \\in A_{2n}$ \nbe the set partition of $[\\p n] \\cup [\\pbar n]$ with blocks $\\{\\p n\\}$, $\\{\\pbar n\\}$, and $\\{\\p i, \\pbar i\\}$ for $1 \\le i \\le n-1$, . The element\n$p_{2n -1}$ satisfies $p_{2n-1}^2 = \\delta\\, p_{2n-1}$. Let $p_{2n} \\in A_{2n + 1}$ be the set partition of $[\\p {n+1}] \\cup [\\pbar {n+1}]$ with blocks $\\{ \\p n, \\p {n+1}, \\pbar n, \\pbar {n+1}\\}$ and $\\{\\p i, \\pbar i\\}$ for $1 \\le i \\le n-1$. Then $p_{2n} $ is an idempotent. \n\nHere are graphs representing the $p_k$ for $k$ even and odd:\n$$\np_8 = \\inlinegraphic[scale= .3]{e8}\n\\qquad\np_9 = \\inlinegraphic[scale=.3]{e9}\\\n$$ \nOne can check that \n\\begin{equation}\np_k p_{k \\pm 1} p_k = p_k \\quad \\text{for all $k$}.\n\\end{equation}\n\nDefine an involution $i$ on $X_n$ by interchanging $\\p j$ with $\\pbar j$ for each $j$. The map $i$ reflects a graph $d(x)$ representing $x \\in X_n$ in the line $y = 1\/2$. The linear extension of $i$ to \n$A_n$ is an algebra involution. Note that $X'_n$ and $A_{2n-1}$ are invariant under $i$. \nThe embeddings of $A_k$ in $A_{k+1}$ commute with the involutions. The elements $p_k$ are \ninvariant under $i$. \n\nDefine a map ${\\rm cl} : X_n \\to X'_n$ by merging the blocks containing $\\p n$ and $\\pbar n$, and define\n${\\rm cl} : A_{2n} \\to A_{2n-1}$ as the linear extension of the map ${\\rm cl} : X_n \\to X'_n$. \n\n\nDefine a map \n${\\rm cl} : X'_n \\to A_{2n-2}$ as follows: For $x \\in X'_n$, if $\\{\\p n, \\pbar n\\}$ is a block of $x$, then\n${\\rm cl}(x) = \\delta \\,x'$, where $x' \\in X_{n-1}$ is obtained by removing the block $\\{\\p n, \\pbar n\\}$.\nOtherwise, ${\\rm cl}(x) \\in X_{n-1}$ is obtained by intersecting each block of $x$ with $[\\p {n-1}] \\cup [\\pbar {n-1}]$. Define \n${\\rm cl} : A_{2n-1} \\to A_{2n-2}$ as the linear extension of the map ${\\rm cl} : X'_n \\to A_{2n-2}$.\n\nOne can check that for all $k$, ${\\rm cl} : A_k \\to A_{k-1}$ is a non--unital $A_{k-1}$--$A_{k-1}$ bimodule map. Moreover, ${\\rm tr} = {\\rm cl} \\circ {\\rm cl} \\circ \\cdots \\circ {\\rm cl} : A_k \\to A_0 \\cong S$ is a non--unital trace. The trace ${\\rm tr}$ can be computed as follows: given $x \\in X_n$, let $d(x)$ be any graph representing $x$ and let $d'(x)$ be the graph augmented by drawing edges between each pair\nof vertices $\\{\\p j, \\pbar j\\}$; then ${\\rm tr}(x) = \\delta^r$, where $r$ is the number of components of $d'(x)$.\n\nThe maps ${\\rm cl}$ commute with the algebra involutions $i$, and ${\\rm tr} (a) = {\\rm tr}(i(a))$. \nMoreover,\n\\begin{equation}\np_k x p_k = {\\rm cl}(x) p_k \\quad \\text{for all $x \\in A_k$, $k \\ge 1$}.\n\\end{equation}\n\nIf $\\delta$ is invertible, define $\\varepsilon_{2n} : A_{2n} \\to A_{2n -1}$ by $\\varepsilon_{2n} = {\\rm cl}$, and\n$\\varepsilon_{2n-1} : A_{2n-1} \\to A_{2n -2}$ by $\\varepsilon_{2n-1} = \\delta^{-1} \\, {\\rm cl}$. Then the maps \n$\\varepsilon_k$ are unital conditional expectations, and the map $\\varepsilon = \\varepsilon_1 \\circ \\cdots \\varepsilon_k : A_k \\to A_0 \\cong S$ is a unital trace.\n\n\n\\def{\\rm pn}{{\\rm pn}}\nLet $x \\in X_n$. Call a block of $x$ a {\\em through block} if the block has non--empty intersection with both\n$[\\p n ]$ and $[\\pbar n]$. The number of through blocks of $x$ is called the propagating number\nof $x$, denoted ${\\rm pn}(x)$. Clearly, ${\\rm pn}(x) \\le n$ for all $x \\in X_n$. The only $x \\in X_n$ with propagating number equal to $n$ are Brauer diagrams with only vertical strands, i.e.\\ permutation diagrams.\n\n If $x, y \\in X_n$ and $x y = \\delta^r z$, then\n${\\rm pn}(z) \\le \\min\\{ {\\rm pn}(x), {\\rm pn}(y)\\}$. Hence the span of the set of $x \\in X_n$ with\n${\\rm pn}(x) < n$ is an ideal $J_{2n} \\subset A_{2n}$. Moreover, $J_{2n -1} := J_{2n} \\cap A_{2n -1}$ is the span\nof $x \\in X'_n$ with ${\\rm pn}(x) < n$. \n\n\\begin{lemma} \\label{lemma: axiom 5 for partition algebras}\n\n For $n \\ge 1$, $A_{2n}\/J_{2n} \\cong S \\mathfrak S_n$, and $A_{2n-1}\/J_{2n-1} \\cong S \\mathfrak S_{n-1}$, as algebras with involution.\n\\end{lemma}\n\n\\begin{proof} The span of permutation diagrams is a linear complement to $J_{2n}$, and is an $i$--invariant subalgebra of $A_{2n}$ isomorphic to $S \\mathfrak S_n$; hence, $A_{2n}\/J_{2n} \\cong S \\mathfrak S_n$. The span of permutation diagrams $\\pi$ with $\\pi(n) = n$ is a linear complement to $J_{2n-1}$ in $A_{2n-1}$; hence\n$A_{2n-1}\/J_{2n-1} \\cong S \\mathfrak S_{n-1}$.\n\\end{proof}\n\n\n\n\n\\begin{lemma} For $k \\ge 2$, $J_{k} = A_{k-1} p_{k-1} A_{k-1}$. \n\\end{lemma}\n\n\\begin{proof} It is straightforward to check that if $x \\in X_n$ has propagating number strictly less than $n$, then \n$x$ can be factored as $x = x' p_{2n-1} x''$, with $x', x'' \\in X'_n$. Likewise, if $n \\ge 2$ and $x \\in X'_n$\nhas propagating number strictly less than $n$, then $x$ can be factored as\n$x = x' p_{2n-2} x''$ with $x', x'' \\in X'_{n-1}$. \n\\end{proof}\n\n\\begin{lemma}\\label{lemma: axiom 6 for partition algebras}\n \\mbox{}\n \\begin{enumerate}\n\n\\item \nFor $k \\ge 3$, $p_{k-1} A_{k-1} p_{k-1} = A_{k-2} p_{k-1}$.\n \\item $p_1 A_1 p_1 = \\delta\\, A_0 \\,p_1$.\n\\item For $k \\ge 2$, \n$p_{k-1}$ commutes with $ A_{k-2} $. \n\\end{enumerate}\n \\end{lemma}\n\n\\begin{proof} Let $x \\in A_{2n}$ with $n \\ge 1$. Then $p_{2n-1} x p_{2n-1}$ is contained in the span of $y \\in X_n$ such\nthat $\\{\\p n\\}$ and $\\{ \\pbar n\\}$ are blocks of $y$, and any such $y$ can be written as $y = z p_{2n-1}$, where $z \\in A_{2n-2}$. \n\nNow consider $x \\in A_{2n + 1}$ with $n \\ge 1$. Then\n$p_{2n} x p_{2n}$ is contained in the span of $y \\in X'_{n+1}$ such that $\\{\\p n, \\p {n+1}, \\pbar n, \\pbar {n+1}\\}$ is contained in one block of $y$. Any such $y$ can be written as $y = z p_{2n}$ where\n$z \\in A_{2n-1}$. \n\nThis shows that $p_{k-1} A_k p_{k-1} \\subseteq A_{k-2} p_{k-1}$ for all $k \\ge 3$.\nOn the other hand, if $x \\in A_{k-2}$ then $x p_{k-1} = x p_{k-1} p_{k-2} p_{k-1} =\np_{k-1} x p_{k-2} p_{k-1} \\in p_{k-1} A_{k} p_{k-1}$, so $p_{k-1} A_k p_{k-1} \\supseteq A_{k-2} p_{k-1}$.\nThis proves (1).\n\nPoints (2) and (3) are easy to check.\n\\end{proof}\n\n\n\n\\begin{lemma} \\label{lemma: axiom 7 for partition algebras}\nFor $k \\ge 2$, $A_k p_{k-1} = A_{k-1} p_{k-1}$. Moreover, \n $x \\mapsto x e_{k-1}$ is injective from $ A_{k-1}$ to $A_{k}$.\n\\end{lemma}\n\n\\begin{proof} For $k =2$, we have $A_2 p_1 = S p_1 = A_1 p_1$. For $k \\ge 3$, we have\n$$\n\\begin{aligned}\nA_k p_{k-1} &= A_k p_{k-1} p_{k-2} p_{k-1} \\\\& \\subseteq J_k p_{k-1} = A_{k-1} p_{k-1} A_{k-1} p_{k-1} \n\\\\& \\subseteq\n A_{k-1} A_{k-2} p_{k-1} = A_{k-1} p_{k-1}.\n \\end{aligned}\n $$\n Checking $k$ odd and even separately, one can check that $x = {\\rm cl}(x p_{k-1})$ for $k \\ge 2$ and\n $x \\in A_{k-1}$.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma: non-degeneracy of trace partition algebra}\n The trace $\\varepsilon$ on $A_k({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is non--degenerate.\n\\end{lemma}\n\n\\begin{proof} For any set partition $x \\in X_n$, let $r(x)$ be the number of blocks of $x$. Let $E_x$\nbe the equivalence relation on\n$[\\p n] \\cup [\\pbar n]$ whose equivalence classes are the blocks of $x$. \n\n For any $x, y \\in X_n$, define an integer $r(x, y)$ as follows: Let $E(x, y)$ be the smallest equivalence relation on $[\\p n] \\cup [\\pbar n]$ containing $E_x \\cup E_{i(y)}$ and let $r(x, y)$ be the number of equivalence classes of $E(x, y)$. Clearly, $r(x, y) \\le \\min\\{r(x), r(y)\\}$. Moreover, if $r(x) = r(y)$, then\n $r(x, y) < r(x)$ unless $y = i(x)$, and $r(x, i(x)) = r(x)$. \n \n It is not hard to see that ${\\rm tr}(x y) = {\\mathbold \\delta}^{r(x, y)}$, so $\\varepsilon(x, y) = {\\mathbold \\delta}^{r(x, y)-n}$.\n It follows that the Gram determinant $\\det(\\varepsilon(x y))_{x, y}$ is a Laurent polynomial that has a unique\n term of highest degree namely $\\pm \\prod_x \\varepsilon(x\\, i(x))$. In particular the Gram determinant is non--zero.\n This shows that the trace on $A_{2n}({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is non--degenerate, and the same method shows that the restriction of the trace to $A_{2n-1}({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is non--degenerate.\n \\end{proof}\n\n\\begin{lemma} \\label{lemma: generic semisimplicity for partition algebras} $A_k({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta})$ is split semisimple. The branching diagram for \\break\n$(A_k({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta}))_{k \\ge 0}$ has vertices on levels $2n$ and $2n+1$ labeled by all Young diagrams of size $j$, $0 \\le j \\le n$. There is an edge connecting $\\lambda$ on level $2n$ and $\\mu$ on level\n$2n \\pm 1$ if, and only if, $\\lambda = \\mu$ or $\\mu$ is obtained by removing one box from $\\lambda$. \\end{lemma}\n\n\\begin{proof} This is proved by Martin ~\\cite {martin-partition-JKTR}. It can also be proved using the method of Wenzl from ~\\cite{Wenzl-Brauer}, using\nLemma \\ref{lemma: non-degeneracy of trace partition algebra}.\n\\end{proof}\n\n\\subsubsection{Verification of framework axioms for the partition algebras}\n\nWe take $R = {\\mathbb Z}[{\\mathbold \\delta}]$, where ${\\mathbold \\delta}$ is an indeterminant. Then $R$ is the universal ground ring for the partition algebras; for any commutative ring $S$ with distinguished element $\\delta$, we have\n$A_k(S, \\delta) \\cong A_k(R, {\\mathbold \\delta})\\otimes_R S$. Let $F = {\\mathbb Q}({\\mathbold \\delta})$ denote the field of fractions of $R$. Write $A_k = A_k(R, {\\mathbold \\delta})$. Define $Q_{2n} = Q_{2n + 1} = R \\mathfrak S_n$.\n\n\\begin{proposition} \\label{proposition: framework axioms for partition algebras} The two sequence of $R$--algebras $(A_k)_{k \\ge 0}$ and $(Q_k)_{k \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}.\n\\end{proposition}\n\n\\begin{proof}\nAccording to Example \\ref{example: Hn coherent tower}, $(Q_k)_{k \\ge 0}$ is a coherent tower of cellular algebras, so axiom (\\ref{axiom Hn coherent}) holds.\nFramework axioms (\\ref{axiom: involution on An}) and (\\ref{axiom: A0 and A1}) are evident.\n$A_k^F$ is split semisimple by Lemma \\ref{lemma: generic semisimplicity for partition algebras}. This verifies axiom (\\ref{axiom: semisimplicity}).\n\nWe take $p_{k-1} \\in A_k$ to be the element defined in the previous section. Then \n$p_{k-1}$ is an $i$--invariant essential idempotent.\nWith $J_k = A_k p_{k-1} A_k$, we have $A_k\/J_k \\cong Q_k$ as algebras with involution by Lemma \\ref{lemma: axiom 5 for partition algebras}. \n This verifies axiom (\\ref{axiom: idempotent and Hn as quotient of An}).\n \nAxiom (\\ref{axiom: en An en}) follows from Lemma \\ref{lemma: axiom 6 for partition algebras}, and axiom (\\ref{axiom: An en}) from \n Lemma \\ref{lemma: axiom 7 for partition algebras}. Axiom (\\ref{axiom: e(n-1) in An en An}) holds because $p_{n-1} p_n p_{n-1} = p_{n-1}$.\n \\end{proof}\n\n\\begin{corollary} For any commutative ring $S$ and for any $\\delta \\in S$, \nthe sequence of partition algebras $(A_n(S, \\delta))_{n \\ge 0}$ is a coherent tower of cellular algebras.\n$A_n(S, \\delta)$ has cell modules indexed by all Young diagrams of size $j$, $0 \\le j \\le n$.\n The cell module labeled by\na Young diagram $\\lambda$ has a basis labeled by paths on the branching diagram for $(A_k({\\mathbb Q}({\\mathbold \\delta}), {\\mathbold \\delta}))_{k \\ge 0}$, described in Lemma \\ref{lemma: generic semisimplicity for partition algebras}.\n\\end{corollary}\n\n\\subsection{Contour algebras}\nWe define generalizations of the {\\em contour algebras} of Cox et.\\ al.\\ ~\\cite{cox-towers}, which in turn include several sorts of diagram algebras. The algebras are obtained as a sort of wreath product of the Jones--Temperley--Lieb algebras with some other algebra $A$ with involution; varying $A$ gives a wide variety of examples.\n\n\\subsubsection{Definition of contour algebras} Let $S$ be a commutative ring with distinguished element $\\delta$. Let $A$ be an $S$--algebra with involution $i$ and with a unital $S$--valued trace $\\varepsilon$. We first define the $A$--Temperley--Lieb algebras $T_n(A)$ and then the contour algebras $\\cont n d A$ as subalgebras of $T_n(A)$. In case we need to emphasize the ground ring $S$ and parameter $\\delta$, we\nwrite $\\cont n d {A, S, \\delta}$. \n\nAn $A$--Temperley--Lieb diagram is a Temperley--Lieb (TL) diagram with strands labeled by elements of $A$. For convenience, we adopt the convention that an unlabeled strand is the same as a strand labeled with the identity of $A$.\n\nWe will define the product of two $A$--Temperley--Lieb diagrams. \nFirst we note that ordinary TL diagrams have an inherent orientation. Label the top vertices of a TL diagram by\n$\\p 1, \\dots, \\p n$ and the bottom vertices by $\\pbar 1, \\dots, \\pbar n$. Place a small arrow pointing down at each\nodd numbered vertex (top or bottom) and a small arrow pointing up at each even numbered vertex. Then because of the planarity of TL diagrams, each strand of a TL diagram must connect one arrow pointing into the rectangle $\\mathcal R$ of the diagram with one arrow pointing out of $\\mathcal R$; the strand can be thought of as oriented from the inward pointing arrow to the outward pointing arrow. When two TL diagrams are multiplied by stacking, the orientation of composed strands agrees. \n\nNow consider two $A$--Temperley--Lieb diagrams $X$ and $Y$. To form the product $X Y$, stack $Y$ over $X$ as for tangles, forming a composite diagram $X\\circ Y$. Label each non--closed composite strand with the product of the labels of its component strands from $X$ and $Y$, taken in the order of their occurrence as the strand is traversed according to its orientation. For each closed strand $s$ in $X\\circ Y$, let $\\varepsilon(s)$ be the trace of the product of\nthe labels of its component strands; the product is unique up to cyclic permutation of the factors, so the trace \nis uniquely determined. Let $r$ be the number of closed strands and let $Z$ be the labeled diagram obtained by removing all the closed strands. Then $XY = \\delta^r ( \\prod_s \\varepsilon(s))\\, Z$. \n\nAs an $S$--module, $T_n(A)$ is $A^{\\otimes n} \\otimes T_n(S, \\delta) = \\bigoplus_x (A^{\\otimes n} \\otimes x)$, where the sum is over ordinary Temperley--Lieb diagrams $x$. We identify a simple tensor $a_1 \\otimes \\cdots \\otimes a_n \\otimes x$\nwith a labeling of $x$ with the labels $a_1, \\dots, a_n$. We have to specify how to place the labels. We fix an ordering of the vertices, for example $\\p 1 < \\cdots < \\p n < \\pbar 1 < \\cdots \\pbar n$, and then order the strands of $x$ according to the order of the initial vertex of each (oriented) strand. The simple tensor\n$a_1 \\otimes \\cdots \\otimes a_n \\otimes x$\nis identified with the diagram with underlying TL diagram $x$, with the $j$--th strand of $x$ labeled by $a_j$ for each $j$. \n\n\nFix TL diagrams $x$ and $y$. The product of $A$--Temperley--Lieb diagrams with underlying TL diagrams $x$ and $y$, defined above, determines a multilinear map $A^{2n} \\to A^{\\otimes n} \\otimes xy$, and hence a bilinear\nmap $( A^{\\otimes n} \\otimes x) \\times ( A^{\\otimes n} \\otimes y) \\to A^{\\otimes n} \\otimes xy$. This product extends to a bilinear product on $T_n(A)$, which one can check to be associative.\n\nNext we define an involution on $T_n(A)$. Define $i$ on an $A$--labeled TL diagram by flipping the diagram over the line $y = 1\/2$ and applying the involution in $A$ to the label of each strand. For a fixed TL diagram $x$, this gives a multilinear map from $A^n$ to $A^{\\otimes n} \\otimes i(x)$, and hence a linear map from\n$A^{\\otimes n} \\otimes x$ to $A^{\\otimes n} \\otimes i(x)$. Now $i$ extends to a linear map on \n$T_n(A)$. One can check that $i$ is an algebra involution.\n\nThis completes the definition of the $A$--Temperley--Lieb algebra, as an algebra with involution.\n\nNext we define the $A$--contour algebras. We assign a depth to each strand in an ordinary TL diagram $x$, as follows: Draw a curve from a point on a given strand $s$ to the western boundary of $\\mathcal R$, having only transverse intersections with\nany strands of $x$. The depth of $s$ is the minimum, over all such curves $\\gamma$, of the number of points of intersection of $\\gamma$ with the strands of $x$ (including $s$). The depth of an $A$--labeled TL diagram is the maximum depth of the strands with non--identity labels.\n\nFix $d \\le n$. As an $S$--module $\\cont n d A$ is the span of those $A$--labeled TL diagrams of depth no greater than $d$. It is easy to check as in\n~\\cite{cox-towers} Lemma 2.1 that $\\cont n d A$ is an $i$--invariant subalgebra of $T_n(A)$. \n\nFor $a \\in A$ and $1 \\le j \\le n$ let $a^{(j)}$ be the identity TL diagram in $T_n(A)$ with the $j$--th strand labeled with $a$ (and the other strands unlabeled). We have $a\\power j$ and $b \\power k$ commute if $j \\ne k$. \nAlso $a \\power j$ commutes with $e_k$ unless $j \\in \\{k, k+1\\}$ and $e_k a \\power k = e_k a \\power {k+1}$, and, likewise, $ a \\power k e_k= a \\power {k+1} e_k$. \nNote that $a \\mapsto a \\power k$ is an algebra homomorphism if $k$ is odd, but an algebra anti-homomorphism if $k$ is even.\n\n\\begin{lemma} \\label{lemma; generating contour algebras}\n$\\cont n d A$ is generated as an algebra by $e_1, \\dots, e_{n-1}$ and by \n $\\{ a \\power k : 1 \\le k \\le d\\}$. \n\\end{lemma}\n\n\\noindent{\\em Sketch:} It is enough to show that if $x$ is a Temperley--Lieb diagram and $X = x a \\power k$ has depth\n$r$, then $X$ can be rewritten as a product of $a \\power r$ and TL diagrams. First one can check that\n$X$ can be written as $x_1 \\, x_2 a \\power {k'} \\, x_3$ where the $x_i$ are TL diagrams, $x_2$ is a product of commuting $e_i$'s, and the depth of $ x_2 \\,a \\power {k'} $ is $r$. Finally, it suffices to show that\n$ x_2 \\,a \\power {k'} $ can be written as a product of TL diagrams with $a \\power r$. We give an example that\ncaptures the idea: $e_1 e_3 a \\power 6$ has depth $2$. We have\n$$\n\\begin{aligned}\ne_1 e_3 a \\power 6 &= (e_1 e_3) (e_2 e_4) (e_1 e_3) a \\power 6 \\\\\n&= (e_1 e_3) (e_2 e_4) (e_3 e_5) (e_2 e_4) (e_1 e_3) a \\power 6 \\\\\n&= (e_1 e_3) (e_2 e_4) (e_3 e_5)a \\power 2 (e_2 e_4) (e_1 e_3),\n\\end{aligned}\n$$\nby repeated use of the relations listed before the statement of the lemma.\n\n\\subsubsection{Brief history of contour algebras} The contour algebras introduced by Cox et.\\ al.\\ ~\\cite{cox-towers} are the special case with $A$ the group algebra of the cyclic group ${\\mathbb Z}_m$. On the other hand, the\n$A$--Temperley--Lieb algebras $T_n(A)$ have been considered in ~\\cite{jones-planar}, Example 2.2. The contour subalgebras of $T_n(A)$ were discussed in ~\\cite{green-martin}. \n\n\n\\subsubsection{Some properties of $A$--Temperley--Lieb and contour algebras} We deal with the contour algebras and the\n$A$--Temperley--Lieb algebras together; regard $T_n(A)$ as $\\cont n {\\infty} A$. \n\n We define maps $\\iota: \\cont n d A \\to \\cont {n+1} d A$ as for other classes of \ndiagram or tangle algebras, and likewise maps ${\\rm cl} : \\cont n d A \\to \\cont {n-1} d A$; if closing the rightmost strand of an $A$--Temperley--Lieb diagram produces a closed loop, remove the loop and multiply the resulting diagram\nby $\\delta$ times the trace of the product of labels along the loop. The map $\\iota$ is injective, since\n$x = {\\rm cl}( \\iota(x) e_n)$ for $x \\in \\cont n d A$. The maps $\\iota$ and ${\\rm cl}$ commute with the involutions.\n\nIf $\\delta$ is invertible in $S$, we can define $\\varepsilon_n = (1\/\\delta) {\\rm cl} : \\cont n d A \\to \\cont {n-1} d A$, which is a unital\nconditional expectation. We have\n${\\varepsilon_{n+1}}\\circ\\iota(x) = x$ for $x \\in \n\\cont n d A$. The map $\\varepsilon = \\varepsilon_1\\circ \\cdots \\circ \\varepsilon_n : \\cont n d A \\to \\cont 0 d A \\cong S$ is a normalized trace. The value of $\\varepsilon$ on an $A$-Temperley--Lieb diagram $X$ with $n$ strands is obtained as follows: first close all the strands of $X$ by introducing new curves joining $\\p j$ to $\\pbar j$ for all $j$; let $r$ be the number of closed loops in the resulting\ndiagram; then $\\varepsilon(X) = \\delta^{r - n} \\prod_s \\varepsilon(s)$, where the product is over the collection of closed loops $s$, and $\\varepsilon(s)$ denotes the trace in $A$ of the product of labels along the loop $s$.\n\nThe span $J$ of $A$--Temperley--Lieb diagrams of depth $\\le d$ and with at least one horizontal strand is an ideal in $\\cont n d A$. \nBy Lemma \\ref{lemma; generating contour algebras}, any $A$--Temperley--Lieb diagram with depth $ \\le d$ can be written as a word in the $e_i$'s and in elements $a \\power k$ with $k \\le d$; the diagram is in $J$ if, and only if,\nsome $e_i$ appears in the word. Thus $J$ is the ideal generated by the $e_i$'s. Because of the relations\n$e_i e_{i \\pm 1} e_i = e_i$, $J$ is generated by $e_{n-1}$. The quotient $\\cont n d A\/ J$ is isomorphic\n(as algebras with involution) to the subalgebra generated by the $a \\power k$ with $k \\le d$, and thus to $A^{\\otimes d}$ if $n \\ge d$ and\n$A^{\\otimes n}$ if $n < d$. \n\n \\begin{lemma} \\label{lemma: axiom 6 for contour} \\mbox{}\n \\begin{enumerate}\n\\item \nFor $n \\ge 3$, $e_{n-1} \\, \\cont {n-1} d A \\, e_{n-1} = \\cont {n-2} d A \\, e_{n-1}$.\n\\item $e_1 \\, \\cont 1 d A \\, e_1 = \\delta\\, S \\,e_1$\n\\item For $n \\ge 2$, \n$e_{n-1}$ commutes with $ \\cont {n-2} d A$. \n\\end{enumerate}\n \\end{lemma}\n \n \\begin{proof} The proof is the same as that of Lemma \\ref{lemma: Brauer axiom 6} for the Brauer algebras.\n \\end{proof}\n \n \\begin{lemma} \\label{Bn e(n-1) = B(n-1) e(n-1) for contour algebras} For $n \\ge 2$, \n$\\cont n d A \\,e_{n-1} = \\cont {n-1} d A \\,e_{n-1}$. Moreover, \n $x \\mapsto x e_{n-1}$ is injective from $\\cont {n-1} d A$ to $\\cont {n-1} d A \\,e_{n-1}$.\n\\end{lemma}\n\n\\begin{proof} Any $A$--TL diagram in $\\cont n d A$ is either already in $\\cont {n-1} d A$, or it can be written\nas $ \\alpha \\chi \\beta$, with $\\alpha, \\beta \\in \\cont {n-1} d A$, and \n$\\chi \\in \\{ e_{n-1} , a \\power n \\}$ if $n \\le d$, or $\\chi = e_{n-1}$ if $n > d$. \n\nThe remainder of the proof is the same as the proof of Lemma \\ref{B(n+1) e(n) = B(n) e(n) for Brauer algebras} for the Brauer algebras, using the identities: $a \\power {n} x e_{n-1} = x a \\power {n-1} e_{n-1}$, and $e_{n-1} x e_{n-1} = {\\rm cl}(x) e_{n-1}$ for $x \\in \\cont {n-1} d A$. \n\\end{proof}\n\n\n\\subsubsection{Hypotheses on the algebra $A$} \\label{subsubsection: hypotheses on A for generic contour algebras}\n\nWe will suppose that the algebra $A$ has a generic version defined over an integral domain $R_0$. \nLet $F_0$ be the field of fractions of $R_0$. We suppose that $A = A(R_0)$ satisfies the following hypotheses:\n\\begin{enumerate}\n\\item $A = A(R_0)$ is cellular.\n\\item $A(F_0) = A(R_0) \\otimes_{R_0} F_0$ is split semisimple.\n\\item The trace $\\varepsilon$ on $A({R_0})$ is non--degenerate.\n\\end{enumerate}\n\nWe take $R = R_0[{\\mathbold \\delta}]$, where ${\\mathbold \\delta}$ is an indeterminant, and \nlet $F = F_0({\\mathbold \\delta})$ denote the field of fractions of $R$. \nWe will show that $(\\cont n d {A, R, {\\mathbold \\delta}})_{n \\ge 0} $ is a coherent tower of cellular algebras. \n\n\n\\subsubsection{Special instances} The cellular algebra $A$ in Section \\ref{subsubsection: hypotheses on A for generic contour algebras} can be taken to be the generic version of any of the diagram or tangle algebras treated in this paper. $A$ could be taken to be a generic Hecke algebra or cyclotomic Hecke algebra, or the group ring of a symmetric group over $R_0 = {\\mathbb Z}$. \n\nThe contour algebras of Cox et.\\ al.\\ ~\\cite{cox-towers} are recovered by taking\n$R_0 = {\\mathbb Z}[ {\\mathbold \\delta}_1, \\dots, {\\mathbold \\delta}_{m-1}]$ and $A$ the group algebra of ${\\mathbb Z}_m$ over $R_0$. The trace on $A$ is determined by $\\varepsilon([k]) = {\\mathbold \\delta}_k$ for $[k] \\ne [0]$ and $\\varepsilon([0] )= 1$. The parameter $\\delta_0$ in ~\\cite{cox-towers} becomes identified with our $\\delta$. \n\n\n\\subsubsection{Verification of the framework axioms for contour algebras} Adopt the hypotheses and notation of Section \\ref{subsubsection: hypotheses on A for generic contour algebras}. \n\n\\begin{lemma} \\label{lemma: non degenerate trace on contour algebras}\n The trace $\\varepsilon$ on $\\cont n d {A, F, {\\mathbold \\delta}}$ is non--degenerate.\n\\end{lemma}\n\n\\begin{proof} We take any basis $\\mathbb A$ of $A$ over $F_0$ with $\\bm 1 \\in \\mathbb A$. As a basis $\\mathbb B$ of $\\cont n d A$ over $F$ we take all $n$--strand TL diagrams decorated up to depth $d$ with elements of $\\mathbb A$. We consider the modified Gram determinant $\\det [\\varepsilon( X i(Y))]_{X, Y \\in \\mathbb B}$. If $X$ and $Y$ have different underlying TL diagrams, then $\\varepsilon( X i(Y)) \\in \\delta^{-1} F_0$. \n\nNext consider matrix entries $\\varepsilon( X i(Y))$ where $X$ and $Y$ have the same underlying TL diagram, say $x$. Suppose $x$ has $\\ell$ strands at depth $d$ or less and these strands are decorated by basis elements\n$a_{1}, \\dots, a_{\\ell}$ in $X$, respectively $b_{1}, \\dots, b_{\\ell}$ in $Y$. Then $\\varepsilon(X i(Y)) = \\prod_{j=1}^{\\ell} \\varepsilon(a_{j} i(b_{j}))$. The determinant of the square submatrix of $ [\\varepsilon( X i(Y))]$ consisting of those entries for which $X$ and $Y$ both have underlying TL diagram $x$ is therefore $D^{\\ell}$, where\n$D$ is the determinant of $[\\varepsilon(a i(b))]_{a, b \\in \\mathbb A}$. \nIt follows that $\\det [\\varepsilon( X i(Y))]_{X, Y \\in \\mathbb B}$ is equal to a power of $D$ modulo\n$\\delta^{-1} R_{0}$, and is therefore non--zero. \n\\end{proof}\n\nConsider $$Q_{n} = \\cont n d A\/ J \\cong \\begin{cases} A^{\\otimes n} &\\text{if $n < d$} \\\\\nA^{\\otimes d} &\\text{if $n \\ge d$.} \\\\ \\end{cases}\n$$\nBy the assumptions in Section \\ref{subsubsection: hypotheses on A for generic contour algebras},\n$Q_{n}(R)$ is cellular and $Q_{n}(F)$ is split semisimple. Moreover, it is easy to see that\n$(Q_{n})_{n \\ge 0}$ is a coherent tower of cellular algebras.\n\n\\begin{lemma} \\label{lemma: generic semisimplicity for contour algebras} \n$\\cont n d {A, F, {\\mathbold \\delta}}$ is split semisimple for all $n$. \n\\end{lemma}\n\n\\begin{proof} The method of Wenzl from ~\\cite{Wenzl-Brauer} applies, using the non--degeneracy of the trace and the split semisimplicity of $Q_{n}(F)$ for all $n$. \n\\end{proof}\n\n\\begin{proposition} The pair of sequences $(\\cont n d {A, R, {\\mathbold \\delta}})_{n \\ge 0}$ and\n$(Q_{n}(R))_{n \\ge 0}$ satisfy the framework axioms of Section \\ref{subsection: framework axioms}.\nHence, $(\\cont n d {A, R, {\\mathbold \\delta}})_{n \\ge 0}$ is a coherent tower of cellular algebras. \n\\end{proposition}\n\n\\begin{proof}\nWe observed above that $(Q_k)_{k \\ge 0}$ is a coherent tower of cellular algebras, so axiom (\\ref{axiom Hn coherent}) holds.\nFramework axioms (\\ref{axiom: involution on An}) and (\\ref{axiom: A0 and A1}) are evident.\nFramework axiom (\\ref{axiom: semisimplicity}) follows from Lemma \\ref{lemma: generic semisimplicity for contour algebras}. \n\nThe elements $e_{k}$ are $i$--invariant essential idempotents. \nWith $J = \\cont k d A e_{k-1} \\cont k d A$, we have $\\cont k d A\/J \\cong Q_k$ as algebras with involution. \n This verifies axiom (\\ref{axiom: idempotent and Hn as quotient of An}).\n Axiom (\\ref{axiom: en An en}) follows from Lemma \\ref{lemma: axiom 6 for contour}, and axiom (\\ref{axiom: An en}) from \n Lemma \\ref{Bn e(n-1) = B(n-1) e(n-1) for contour algebras}. Axiom (\\ref{axiom: e(n-1) in An en An}) holds because $e_{n-1} e_n e_{n-1} = e_{n-1}$.\n \\end{proof}\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Spin-1\/2 fermions in quantum field theory}\n \\label{sec:spinhalf}\n\nWe begin these lectures with a treatment of spin-$\\ifmath{\\tfrac12}$ fermions in\nquantum field theory. In most introductory courses in relativistic quantum field\ntheory, the student first encounters fermion fields in the treatment of a\nrelativistic theory of electrons and photons. The electron is\nrepresented by a four-component Dirac fermion field, and the free field electron\nLagrangian yields the Dirac equation. The four components represent\ntwo degrees of freedom corresponding to the electron \nand two degrees\nof freedom corresponding to the positron. Feynman rules for quantum\nelectrodynamics are developed and the vector-like nature of the $e^+\ne^-$ coupling to photons leads to some important simplifications.\n\nThe theory of electroweak interactions involves chiral\ninteractions of fermions with gauge bosons. Left-handed and\nright-handed fermions transform differently under the electroweak\ngauge group, which may appear strange to students trained to think in\nterms of four-component Dirac fermions. Nevertheless, after electroweak\nsymmetry breaking, the mass-eigenstate fermion fields can be\nidentified. All massive fermion states are charged under U(1)$_{\\rm\n EM}$ and are thus represented by Dirac fermion fields. The\nneutrinos are massless, but only the left-handed neutrinos and\nright-handed antineutrinos are present in the theory. Thus, one can\nstill use four-component fermion fields (by applying the\nappropriate chiral projection operators on the neutrino fields). Hence, \nthe four-component techniques of quantum\nelectrodynamics are easily accommodated and Feynman rules for the\nfermion fields are obtained in a straightforward manner. \n \nHowever, the observation of\nneutrino mixing phenomena implies that neutrinos are massive, which\nrequires new physics beyond the Standard Model of the electroweak\ninteractions. Models of neutrino mass often include neutral\nself-conjugate fermion states with two degrees of freedom, called\nMajorana fermions. Such\nstates can be described using four-component fermion fields that are\nconstrained by an appropriate conjugation condition. However,\nthe resulting field theory description of systems of Majorana and\nDirac fermions is somewhat awkward. Moreover, the \nFeynman rules for interacting Majorana fermions require some care.\n\nReturning to first principles, one can ask how spin-$\\ifmath{\\tfrac12}$ fermions arise\nin quantum field theory. In Section~\\ref{sec:spin_half_rep}, we\nshall demonstrate that the fundamental building blocks employed in \nconstructing spin-$\\ifmath{\\tfrac12}$ quantum fields are\ntwo-component spinors corresponding to the two-dimensional representations of the\nLorentz group. \nA neutral Majorana fermion is then represented\nby a two-component fermion field.\nDirac fermions arise when one considers\ntheories of two mass-degenerate two-component fermions, which can\nbe combined to make a charged four-component Dirac fermion. This is\ncompletely analogous to the case of spin-0 bosons, in which a neutral\nboson is represented by a real scalar field and a charged boson is\nrepresented by a complex scalar field (whose real and imaginary parts\nconstitute two mass-degenerate real scalars).\n\nThe development of two-component spinor technology has a number of\nbenefits. First, it provides an elegant unified description of\nMajorana and Dirac fermions. Second, it is very convenient to employ\nthe two-component spinor formalism in theories of chiral interactions.\nFinally, it will prove especially useful in developing the formalism\nof supersymmetry, which is the main focus of these lectures.\n\nBecause most students see the four-component spinor formalism first\nand are therefore more familiar with it, we shall devote Section~\\ref{sec:24} to the translation between the two- and four-component formalisms. \nFinally, in Section~\\ref{sec:Feynman} we demonstrate how Feynman rules involving four-component\nfermion fields can be extended to incorporate Majorana fermions.\n\nThis section is based on a comprehensive review of Dreiner, Haber and\nMartin\\cite{Dreiner:2008tw}, where many references to the original\nliterature can be found.\n\n\n \n\n\n\n\n\n\n\\subsection{Two-component spinor technology}\n\\label{sec:spin_half_rep}\n\n\n\\subsubsection{Orthochronous Lorentz transformations}\n\n\nQuantum spin-$\\ifmath{\\tfrac12}$ fields transform\nunder a two-dimensional irreducible representation of the Lorentz\ngroup. Thus, we first examine the properties that define a Lorentz transformation\\cite{sexl}.\nUnder an active\nLorentz transformation, $\\Lambda^\\mu{}_\\nu$, a four-vector $p^\\mu$\ntransforms as\n\\begin{equation}\np^{\\prime\\,\\mu}=\\Lambda^\\mu{}_\\nu p^\\nu\\,.\n\\end{equation} \nThe condition that $g_{\\mu\\nu}p^\\mu p^\\nu$ is invariant under\nLorentz transformations implies that\\footnote{In our conventions, the Minkowski metric tensor is\n$g_{\\mu\\nu}={\\rm diag}(1\\,,\\,-1\\,,\\,-1\\,,\\,-1)$.} \n\\begin{equation} \\label{lambdarelation}\n\\Lambda^\\mu{}_\\nu g_{\\mu\\rho}\\Lambda^\\rho{}_\\lambda=g_{\\nu\\lambda}.\n\\end{equation}\nThat is, $\\Lambda\\in$\\,O(3,1). \\Eq{lambdarelation} implies that $\\Lambda$ possesses the following two\nproperties: (i)~$\\rm{det}~\\Lambda=\\pm 1$ and (ii)~$|\\Lambda^0{}_0|\\geq\n1$. Thus, Lorentz transformations fall into four disconnected\nclasses denoted by a pair of signs, $\\left(\\rm{sgn}[\\rm{det}~\\Lambda]\\,,\\,\n\\rm{sgn}[\\Lambda^0{}_0]\\right)$. The proper\northochronous Lorentz transformations correspond to $(+,+)$ and are\ncontinuously connected to the identity.\n \n The most general proper orthochronous\nLorentz transformation, characterized by a\nrotation angle $\\theta$ about an axis $\\mathbold{\\widehat n}$\n($\\mathbold{\\vec\\theta}\\equiv\\theta \\mathbold{\\widehat n}$) and a\nboost vector $\\mathbold{\\vec \\zeta}\\equiv\n\\mathbold{\\hat v}\\tanh^{-1}\\beta$ (where $\\mathbold{\\hat{v}}\\equiv\n\\mathbold{\\vec{v}}\/|\\mathbold{\\vec{v}}|$ is the unit velocity vector and\n$\\beta\\equiv |\\mathbold{\\vec v}|\/c$),\\footnote{Henceforth, we shall\n work in particle physics\nunits where $\\hbar=c=1$.}\nis a $4\\times 4$\nmatrix given by:\n\\begin{equation} \\label{lambda44}\n\\Lambda=\\exp\\left(-\\ifmath{\\tfrac12} i\\theta^{\\alpha\\beta}s_{\\alpha\\beta}\\right)\n=\\exp\\left(\n-i\\mathbold{{\\vec\\theta}\\cdot}\\boldsymbol{\\vec s}\n-i\\mathbold{{\\vec\\zeta}\\cdot}\\boldsymbol{\\vec k}\\right)\\,,\n\\end{equation}\nwhere $\\theta^{\\alpha\\beta}$ is antisymmetric, with\n$\\theta^i \\equiv \\ifmath{\\tfrac12}\\epsilon^{ijk} \\theta_{jk}$,\n$\\zeta^i\\equiv\\theta^{i0}=-\\theta^{0i}$, and\n\\begin{equation} \\label{explicitsmunu}\n(s_{\\alpha\\beta})^\\mu{}_\\nu=i(g_\\alpha{}^\\mu\\,g_{\\beta\\nu}-g_\\beta{}^\\mu\n\\,g_{\\alpha\\nu})\\,,\n\\end{equation}\nwith $s^i\\equiv\\ifmath{\\tfrac12}\\epsilon^{ijk}s_{jk}$ and $k^i\\equiv s^{0i}=-s^{i0}$.\nWe have employed a notation where the lower case Latin indices $i,j,k=1,2,3$ and $\\epsilon^{123}=+1$.\n\nNote that the $s^{\\mu\\nu}$ are antisymmetric $4\\times 4$ matrices, {\\it i.e.},\n$s^{\\mu\\nu}=-s^{\\nu\\mu}$, and satisfy the\ncommutation relations,\n\\begin{equation} \\label{eq:comm-rels}\n[s^{\\alpha\\beta},s^{\\rho\\sigma}] = i(g^{\\beta\\rho}\\,s^{\\alpha\\sigma} -\ng^{\\alpha\\rho}\\,s^{\\beta\\sigma} - g^{\\beta\\sigma}\\,s^{\\alpha\\rho} +\ng^{\\alpha\\sigma}\\,s^{\\beta\\rho} ).\n\\end{equation}\nIt follows from \\eqs{lambda44}{explicitsmunu} that an\ninfinitesimal orthochronous Lorentz transformation is\ngiven by\n\\begin{equation} \\label{inflambda4}\n\\Lambda^\\mu{}_\\nu\\simeq\\delta^\\mu{}_\\nu+\\theta^\\mu{}_\\nu\n\\simeq (\\mathds{1}_{4\\times 4}-i\\mathbold{{\\vec\\theta}\\cdot}\\boldsymbol{\\vec s}\n-i\\mathbold{{\\vec\\zeta}\\cdot}\\boldsymbol{\\vec k})^\\mu{}_\\nu\\,,\n\\end{equation}\nwhere $\\mathds{1}_{4\\times 4}$ is the $4\\times 4$ identity matrix, \nand we have used $\\theta^\\mu{}_\\nu=-\\theta_\\nu{}^\\mu$.\n\n\n\\subsubsection{Finite-dimensional Representations of the Lorentz Group}\n\n\nA generic spin-$s$ field $\\Phi$ transforms as\n\\begin{equation}\n\\Phi(x) \\rightarrow \\Phi'(x^{\\prime}) = M_R(\\Lambda)\\Phi(x)\\,,\n\\end{equation}\nwhere $M_R\\equiv\\exp\\bigl(-\\ifmath{\\tfrac12} i \\theta_{\\mu\\nu}S^{\\mu\\nu}\\bigr)$ and\nthe $S_{\\mu\\nu}$ constitute finite-dimensional irreducible matrix\nrepresentations of the Lie algebra of the Lorentz group. The $S^{\\mu\\nu}$ satisfy \nthe same commutation relations as the $s^{\\mu\\nu}$ given in \\eq{eq:comm-rels}.\nIt is convenient to denote the six independent generators defined by the\n$S^{\\mu\\nu}$ as\n\\begin{equation} \\label{jkdef}\nS^i \\equiv\\ifmath{\\tfrac12} \\epsilon^{ijk} S_{jk}\\,,\\qquad\\qquad K^i \\equiv S^{0i}\\,,\n\\end{equation}\nwhere $i,j,k=1,2,3$. The $S^i$ generate\nthree-dimensional rotations in space and the $K^i$ generate the\nLorentz boosts. It then follows that\n\\begin{equation}\nM_R\\equiv\\exp\\left(-i\\mathbold{{\\vec\\theta}\\!\\cdot\\!}\\boldsymbol{\\vec\nS} -i\\mathbold{{\\vec\\zeta}\\!\\cdot\\!}\\boldsymbol{\\vec K}\\right)\\,.\n\\end{equation}\nThe $S^i$ and $K^i$ satisfy the\ncommutation relations,\n\\begin{align}\n[S^i\\, , \\,S^j] &= \\epsilon^{ijk} S^k\\,,\\\\\n [S^i\\, , \\,K^j] &= \\epsilon^{ijk} K^k\\,,\\\\\n [K^i\\, , \\,K^j] &= - \\epsilon^{ijk} S^k\\,.\n\\end{align}\nWe define the following linear combinations of the generators,\n\\begin{equation}\n\\mathbold{\\vec S_+} \\equiv\\ifmath{\\tfrac12} (\\mathbold{\\vec S}+\ni\\mathbold{\\vec K})\\,,\\qquad\\quad\n\\mathbold{\\vec S_-} \\equiv\\ifmath{\\tfrac12}\n(\\mathbold{\\vec S}- i\\mathbold{\\vec K}),\n\\end{equation}\nwhich satisfy the commutation relations, \n\\begin{align}\n[S_+^i\\,,\\,S_+^j] &= i\\epsilon^{ijk}S_+^k\\,, \\\\\n[S_-^i \\, , \\, S_-^j ] & = i \\epsilon^{ijk}S_-^k\\,, \\\\\n[S_{\\pm}^i\\,,\\,S_{\\mp}^j ] &= 0\\,,\n\\end{align}\ncorresponding to two independent (complexified) SU(2) Lie algebras.\nThus, the representations of the Lorentz algebra are characterized by $(s_1,s_2)$, where\nthe $s_i$ are half-integers.\nFor example, $(0,0)$ corresponds to a scalar field and\n$(\\ifmath{\\tfrac12},\\ifmath{\\tfrac12})$ corresponds to a four-vector field.\n\n\\subsubsection{Two-component spinors}\n\nSpin-1\/2 fermion fields transform under the spinor representations, $(\\ifmath{\\tfrac12},0)$ corresponding to $\\boldsymbol{\\vec{S}}_+=\\ifmath{\\tfrac12}\\boldsymbol{\\vec\\sigma}$ and\n$\\boldsymbol{\\vec{S}}_-=0$,\n and $(0,\\ifmath{\\tfrac12})$ corresponding to $\\boldsymbol{\\vec{S}}_+=0$ and\n$\\boldsymbol{\\vec{S}}_-=\\ifmath{\\tfrac12}\\boldsymbol{\\vec\\sigma}$. That is, the \nLorentz transformation matrices acting on spinor fields may be written in terms of the Pauli spin matrices $\\sigma^1$, $\\sigma^2$, and $\\sigma^3$ as follows,\n\\begin{equation} \\label{halfzero}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n(\\ifmath{\\tfrac12}, 0): \\hspace{1.2cm} M=\\exp\\left(-\\nicefrac{i}{2} \\mathbold{\\vec\\theta\\!\\cdot\\!\\vec\\sigma}-\\ifmath{\\tfrac12}\\mathbold{\\vec\\zeta\n\\!\\cdot\\!\\vec\\sigma}\\right),\n\\end{equation}\nwhich via a similarity transformation is equivalent to the matrix representation, $(M^{-1})^{{\\mathsf T}} =i\\sigma^2 M (i\\sigma^2)^{-1}$, and\n\\begin{equation} \\label{zerohalf}\n(0,\\ifmath{\\tfrac12}): \\hspace{1cm} [M^{-1}]^\\dagger=\\exp\\left(-\\nicefrac{i}{2}\n\\mathbold{\\vec\\theta\\!\\cdot\\!\\vec\\sigma}\n+\\ifmath{\\tfrac12}\\mathbold{\\vec\\zeta\\!\\cdot\\!\\vec\\sigma}\\right), \n\\end{equation}\nwhich via a similarity transformation is equivalent to the matrix representation, $M^*=i\\sigma^2 [M^{-1}]^\\dagger (i\\sigma^2)^{-1}$.\n\nThus, the Lorentz transformation law for two-component $(\\ifmath{\\tfrac12},0)$ fields can be written in two equivalent ways, \n\\begin{equation}\n \\xi'_\\alpha=M_\\alpha{}^\\beta\\,\\xi_\\beta\\,,\\qquad\\quad\n\\xi^{\\prime\\,\\alpha}=[(M^{-1})^{{\\mathsf T}}]^\\alpha{}_\\beta\\,\\xi^\\beta\\,,\n\\end{equation}\nwhere $\\alpha,\\beta=1,2$.\nLikewise, the Lorentz transformation law for two-component $(0,\\ifmath{\\tfrac12})$ fields can be written in two equivalent ways, \n\\begin{equation}\n\\xi^{\\prime\\,\\dagger\\,\\dot\\alpha}=\n[(M^{-1})^\\dagger]^{\\dot\\alpha}{}_{\\dot\\beta}\\,\\xi^{\\dagger\\,\\dot\\beta}\n\\,,\\qquad\\quad \n\\xi^{\\prime\\,\\dagger}_{\\dot\\alpha}=\n[M^*]_{\\dot{\\alpha}}{}^{\\dot\\beta}\\xi^\\dagger_{\\dot\\beta}\\,.\n\\end{equation}\nThe $(0,\\ifmath{\\tfrac12})$ fields are related to the $(\\ifmath{\\tfrac12},0)$ fields by hermitian conjugation,\n\\begin{equation}\n\\xi^\\dagger_{\\dot\\alpha}\\equiv(\\xi_\\alpha)^\\dagger\\,,\\qquad\\quad \\xi^{\\dagger\\,\\dot\\alpha}\\equiv(\\xi^\\alpha)^\\dagger\\,.\n\\end{equation}\nIt is conventional to employ undotted indices for the spinor\ncomponents of $(\\ifmath{\\tfrac12},0)$ fields and dotted indices for the spinor components of $(0,\\ifmath{\\tfrac12})$ fields.\n\nAs noted below \\eqs{halfzero}{zerohalf}, respectively,\neach of the two equivalent representation matrices, $M$ and\n$(M^{-1})^{{\\mathsf T}}$ in the case of $(\\ifmath{\\tfrac12},0)$, and $(M^{-1})^\\dagger$\nand $M^*$ in the case of $(0,\\ifmath{\\tfrac12})$,\nare related by a similarity transformation involving the antisymmetric matrices,\n\\begin{equation}\ni\\sigma^2=\\left(\\begin{matrix} \\phantom{-} 0&\\quad 1\\\\\n-1&\\quad\n0\\end{matrix}\\right)=\\epsilon^{\\alpha\\beta}=\\epsilon^{\\dot\\alpha\\dot\\beta}\\,,\n\\end{equation}\nand\n\\begin{equation}\n(i\\sigma^2)^{-1}=-i\\sigma^2=\\epsilon_{\\alpha\\beta}=\\epsilon_{\\dot\\alpha\\dot\\beta}\\,,\n\\end{equation}\nwhich define the epsilon symbols with undotted and dotted indices. Note that the epsilon symbols with raised and lowered\nindices differ by an overall sign. \nMoreover, they can be used to\nraise and lower the \nspinor indices,\n\n\\begin{equation} \\label{raiseindex}\n\\xi^\\alpha =\\epsilon^{\\alpha\\beta}\\,\\xi_\\beta\\,,\\qquad\n\\xi_\\alpha =\\epsilon_{\\alpha\\beta}\\,\\xi^\\beta,\\qquad\n\\xi^{\\dagger\\,\\dot\\alpha}\n=\\epsilon^{\\dot\\alpha\\dot\\beta}\\,\\xi^\\dagger_{\\dot\\beta}\\,,\\qquad\n\\xi^\\dagger_{\\dot\\alpha}\n=\\epsilon_{\\dot\\alpha\\dot\\beta}\\,\\xi^{\\dagger\\,\\dot\\beta}.\n\\end{equation}\n\n\nThe products of two epsilon symbols with undotted and with dotted indices,\nrespectively, satisfy,\n\\begin{align}\n&\\epsilon_{\\alpha\\beta} \\epsilon^{\\gamma\\delta} =\n-\\delta_\\alpha^\\gamma \\delta_\\beta^\\delta\n+\\delta_\\alpha^\\delta \\delta_\\beta^\\gamma\n,\\\\\n&\\epsilon_{\\dot{\\alpha}\\dot{\\beta}} \\epsilon^{\\dot{\\gamma}\\dot{\\delta}} =\n-\\delta_{\\dot{\\alpha}}^{\\dot{\\gamma}}\\delta_{\\dot{\\beta}}^{\\dot{\\delta}}\n+\\delta_{\\dot{\\alpha}}^{\\dot{\\delta}}\\delta_{\\dot{\\beta}}^{\\dot{\\gamma}}\\,,\n\\end{align}\nwhere $\\delta_{\\dot\\alpha}^{\\dot\\beta}=\\delta_\\alpha^\\beta$\nand the two-index symmetric Kronecker delta symbol \nwith undotted indices is defined by\n$\\delta^1_1=\\delta^2_2=1$ and $\\delta_1^2=\\delta_2^1=0$.\nIn particular,\n\\begin{equation}\n\\epsilon_{\\alpha\\gamma}\\,\\epsilon^{\\gamma\\beta}=\\delta_\\alpha^\\beta\\,,\\qquad\\quad\n\\epsilon_{\\dot\\alpha\\dot\\gamma}\\,\\epsilon^{\\dot\\gamma\\dot\\beta}=\\delta_{\\dot\\alpha}^{\\dot\\beta}\\,.\n\\end{equation}\n\n\nFinally, we introduce the $\\sigma$-matrices:\n\\begin{align}\n\\sigma^\\mu_{\\alpha\\dot\\beta}=(\\mathds{1}_{2\\times 2}\\,;\\,\n\\mathbold{\\vec\\sigma}) \\,,\\qquad\n\\overline{\\sigma}^{\\mu\\,\\dot\\alpha\\beta}=(\\mathds{1}_{2\\times 2}\\,;\\, \n-\\mathbold{\\vec\\sigma})\\,,\n\\end{align}\nwhere $\\mathds{1}_{2\\times 2}$ is the $2\\times 2$ identity matrix. The spinor index\nstructure derives from the relations,\n\\begin{equation} \\label{MM}\n(M^\\dagger)^{\\dot\\alpha}{}_{\\dot\\beta}\\overline{\\sigma}^{\\mu\\dot\\beta\\gamma}\nM_\\gamma{}^\\delta=\\Lambda^\\mu{}_\\nu\\overline{\\sigma}^{\\nu\\,\\dot\\alpha\\delta}\\,,\\qquad\n(M^{-1})_\\alpha{}^\\beta\\sigma^\\mu_{\\beta\\dot\\gamma}[(M^{-1})^\\dagger]^{\\dot\\gamma}{}_{\\dot\\delta}=\\Lambda^\\mu{}_\\nu\\sigma^\\nu_{\\alpha\\dot\\delta}\\,.\n\\end{equation}\nNote that the matrix $M$ and its inverse have the same spinor index\nstructure (and likewise for the matrix $M^\\dagger$ and its inverse).\n\nWe will sometimes find it useful to relate the $\\sigma^\\mu$ and $\\overline{\\sigma}^\\mu$ matrices using the identities\n\\begin{equation}\n\\sigma^\\mu_{\\alpha{\\dot{\\alpha}}} = \\epsilon_{\\alpha\\beta}\n\\epsilon_{\\dot{\\alpha}\\dot{\\beta}} \\overline{\\sigma}^{\\mu\\,\\dot{\\beta}\\beta}\n\\,, \\qquad\\quad \\overline{\\sigma}^{\\mu\\,\\dot{\\alpha}\\alpha} =\n\\epsilon^{\\alpha\\beta} \\epsilon^{\\dot{\\alpha}\\dot{\\beta}}\n\\sigma^{\\mu}_{\\beta\\dot{\\beta}}\\,.\n\\end{equation}\nThe significance of $\\sigma^\\mu$ is that Lorentz 4-vectors can be built\nfrom spinor bilinears. For example, $\\chi^\\alpha \\of{x} \\sigma^\\mu_{\\alpha \\dot{\\beta}} \\xi^{\\dot{\\beta}}\\of{x} $ transforms as a Lorentz 4-vector,\n\\begin{Eqnarray}\n\\chi^{\\,\\prime\\,\\alpha}(x')\\sigma^\\mu_{\\alpha\\dot\\beta}\n\\xi^{\\prime\\,\\dagger\\,\\dot\\beta}(x')\n&=&\\chi^\\alpha(x)\n[M^{-1}\\sigma^\\mu(M^{-1})^\\dagger]_{\\alpha\\dot\\beta}\\xi^{\\dagger\\,\\dot\\beta}(x)\n \\\\\n&\n=&\\Lambda^\\mu{}_\\nu\\,\\chi(x)^\\alpha\\sigma^\\nu_{\\alpha\\dot\\beta}\n\\xi^{\\dagger\\,\\dot\\beta}(x)\n\\,,\n\\end{Eqnarray}\nafter making use of \\eq{MM}. Spinor\nindices can be suppressed by adopting a summation\nconvention where we contract indices as follows:\n\\begin{equation} \\label{contract}\n{}^\\alpha{}_\\alpha\\qquad {\\rm and} \\qquad\n{}_{\\dot{\\alpha}}{}^{\\dot{\\alpha}}\\,.\n\\end{equation}\nFor example,\n\\begin{Eqnarray} \\xi\\eta\n&\\equiv & \\xi^\\alpha\\eta_\\alpha ,\n\\\\\n\\xi^\\dagger \\eta^\\dagger &\\equiv & \\xi^\\dagger_{\\dot\\alpha} \\eta^{\\dagger\\,\\dot\n\\alpha} ,\n\\\\\n\\xi^\\dagger\\overline{\\sigma}^\\mu\\eta &\\equiv & \\xi^\\dagger_{\\dot{\\alpha}}\n\\overline{\\sigma}^{\\mu\\dot{\\alpha}\\beta}\\eta_\\beta , \n\\\\\n\\xi\\sigma^\\mu \\eta^\\dagger &\\equiv & \\xi^{{\\alpha}} \\sigma^{\\mu}_{\\alpha\n\\dot \\beta} \\eta^{\\dagger\\,\\dot \\beta} .\n \\end{Eqnarray} \nIn particular, for\nanticommuting spinors,\n\\begin{Eqnarray}\n\\eta\\xi\\equiv\\eta^\\alpha\\xi_\\alpha&=&-\\xi_\\alpha\\eta^\\alpha=+\\xi^\\alpha\\eta_\\alpha=\\xi\\eta\\,.\\\\\n\\eta^\\dagger\\xi^\\dagger\\equiv \\eta^\\dagger_{\\dot\\alpha}{\\xi^\\dagger}^{\\dot\\alpha}&=&-{\\xi^\\dagger}^{\\dot\\alpha} \\eta^\\dagger_{\\dot\\alpha}=\\xi^\\dagger_{\\dot\\alpha} {\\eta^\\dagger}^{\\dot\\alpha}=\\xi^\\dagger\\eta^\\dagger\\,.\n\\end{Eqnarray}\n\nThe behavior of spinor products\nunder hermitian conjugation is noteworthy,\n\\begin{equation}\n(\\xi \\Sigma \\eta)^\\dagger = \\eta^\\dagger \\reversed{\\Sigma} \\xi^\\dagger\\,,\n\\quad (\\xi \\Sigma \\eta^\\dagger)^\\dagger = \\eta \\reversed{\\Sigma}\n\\xi^\\dagger\\,,\n\\quad (\\xi^\\dagger \\Sigma \\eta)^\\dagger = \\eta^\\dagger \\reversed{\\Sigma}\n\\xi\\,,\n\\end{equation}\nwhere in each case $\\Sigma$ stands for any sequence of alternating\n$\\sigma$ and $\\overline{\\sigma}$ matrices, and $\\reversed{\\Sigma}$ is\nobtained by reversing the order of the $\\sigma$\nand $\\overline{\\sigma}$ matrices that appear in $\\Sigma$.\n\nFrom the sigma matrices, one can construct the\nantisymmetrized products,\n\\begin{align}\n(\\sigma^{\\mu\\nu})_\\alpha{}^\\beta\n&\\equiv \\tfrac14 i\\left(\\sigma^\\mu{}_{\\!\\!\\!\\!\\alpha\\dot{\\gamma}}\n\\overline{\\sigma}^{\\nu\\dot{\\gamma}\\beta}-\\sigma^\\nu{}_{\\!\\!\\!\\!\\alpha\\dot{\\gamma}}\n\\overline{\\sigma}^{\\mu\\dot{\\gamma}\\beta}\\right)\\,,\n\\\\\n(\\overline{\\sigma}^{\\mu\\nu})^{\\dot{\\alpha}}{}_{\\dot{\\beta}} &\\equiv\n\\tfrac14 i\\left(\\overline{\\sigma}^\\mu{}^{\\dot{\\alpha}\\gamma}\n\\sigma^\\nu{}_{\\!\\!\\!\\!\\gamma\\dot{\\beta}}-\\overline{\\sigma}^\\nu{}^{\\dot{\\alpha}\\gamma}\n\\sigma^\\mu{}_{\\!\\!\\!\\!\\gamma\\dot{\\beta}}\\right)\\,. \n\\end{align}\n\nWith this notation, we may write the $(\\ifmath{\\tfrac12},0)$ and $(0,\\ifmath{\\tfrac12})$ transformation\nmatrices, respectively, as\n\\begin{Eqnarray}\nM&=&\\exp\\left(-\\ifmath{\\tfrac12} i\\theta^{\\mu\\nu}\\sigma_{\\mu\\nu}\\right)\\,,\n \\\\\n(M^{-1})^\\dagger &=&\n\\exp\\left(-\\ifmath{\\tfrac12} i\\theta^{\\mu\\nu}\\overline{\\sigma}_{\\mu\\nu}\\right)\\,,\n\\end{Eqnarray} \nwhere the $\\theta^{\\mu\\nu}$ are defined below \\eq{lambda44}.\n\n\nConsider a pure boost of an on-shell two-component spinor from its rest frame to\nthe frame where $p^\\mu=(E_{\\boldsymbol{p}}\\,,\\,\\boldsymbol{\\vec\np})$, with $E_{\\boldsymbol{p}}=(|{\\mathbold{\\vec p}}|^2+m^2)^{1\/2}$.\nIn this case, setting $\\theta^{ij}=0$ (corresponding to no rotation),\nwe obtain,\n \\begin{Eqnarray}\nM&=&\\exp\\left(-\\ifmath{\\tfrac12}\\mathbold{\\vec\\zeta\\!\\cdot\\!\\vec\\sigma}\\right)\n=\\sqrt{\\frac{p\\!\\cdot\\!\\sigma}{m}}=\\frac{(E_{\\boldsymbol{p}}+m)\\mathds{1}_{2\\times 2}\n-\\mathbold{\\vec\\sigma\\!\\cdot\\!\\vec\np}}{\\sqrt{2m(E_{\\boldsymbol{p}}+m)}}\n\\,,\\label{mboost}\n\\\\\n(M^{-1})^\\dagger&=&\\exp\\left(+\\ifmath{\\tfrac12}\\mathbold{\\vec\\zeta\\!\\cdot\\!\\vec\\sigma}\\right)\n=\\sqrt{\\frac{p\\!\\cdot\\!\\overline{\\sigma}}{m}}=\\frac{(E_{\\boldsymbol{p}}+m) \\mathds{1}_{2\\times 2}\n+\\mathbold{\\vec\\sigma\\!\\cdot\\!\\vec\np}}{\\sqrt{2m(E_{\\boldsymbol{p}}+m)}} \\,.\\label{mstarboost}\n \\end{Eqnarray} \nThe matrix square roots, $\\sqrt{p\\!\\cdot\\!\\sigma}$ and\n$\\sqrt{p\\!\\cdot\\!\\overline{\\sigma}}$, appearing in \\eqs{mboost}{mstarboost} are defined to be the unique non-negative\ndefinite hermitian matrices\nwhose squares are equal\nto the non-negative definite hermitian matrices\n${\\color{Red}\\ominus} p\\!\\cdot\\!\\sigma$ and ${\\color{Red}\\ominus} p\\!\\cdot\\!\\overline{\\sigma}$,\nrespectively.\\footnote{Note that ${\\color{Red}\\ominus} p\\!\\cdot\\!\\sigma$\nand ${\\color{Red}\\ominus} p\\!\\cdot\\!\\overline{\\sigma}$ are non-negative\nmatrices due to the implicit mass-shell condition\nsatisfied by $p^\\mu$.}\n\n\n\\subsubsection{Useful identities}\n\n\nThe following identities can be used to systematically simplify\nexpressions involving products of $\\sigma$ and $\\overline{\\sigma}$\nmatrices,\n\\begin{align}\n& \\sigma^\\mu_{\\alpha\\dot{\\alpha}}\n\\overline{\\sigma}_\\mu^{\\dot{\\beta}\\beta} = {\\color{Red}\\ominus} 2 \\delta_{\\alpha}^{\\beta}\n\\delta^{\\dot{\\beta}}_{\\dot{\\alpha}}, \\label{sigid1}\n\\\\\n& \\sigma^\\mu_{\\alpha\\dot{\\alpha}} \\sigma_{\\mu\\beta\\dot{\\beta}} =\n{\\color{Red}\\ominus} 2 \\epsilon_{\\alpha\\beta}\n\\epsilon_{\\dot{\\alpha}\\dot{\\beta}}\\,, \\label{sigid2} \\\\\n&\n\\overline{\\sigma}^{\\mu\\dot{\\alpha}\\alpha}\n\\overline{\\sigma}_\\mu^{\\dot{\\beta}\\beta} = {\\color{Red}\\ominus} 2 \\epsilon^{\\alpha\\beta}\n\\epsilon^{\\dot{\\alpha}\\dot{\\beta}}\\,, {}\\label{sigid3}\n\\\\\n& {[\\sigma^\\mu\\overline{\\sigma}^\\nu + \\sigma^\\nu \\overline{\\sigma}^\\mu\n]_\\alpha}^\\beta = {\\color{Red}\\ominus} 2g^{\\mu\\nu}\n\\delta_{\\alpha}^{\\beta}\\,, {}\\label{sigid4}\n\\\\\n&[\\overline{\\sigma}^\\mu\\sigma^\\nu + \\overline{\\sigma}^\\nu \\sigma^\\mu\n]^{\\dot{\\alpha}}{}_{\\dot{\\beta}} = {\\color{Red}\\ominus} 2g^{\\mu\\nu}\n\\delta^{\\dot{\\alpha}}_{\\dot{\\beta}}\\,, {}\\label{sigid5}\n\\\\\n& \\sigma^\\mu \\overline{\\sigma}^\\nu \\sigma^\\rho = {\\color{Red}\\ominus} g^{\\mu\\nu}\n\\sigma^\\rho \\oplus g^{\\mu\\rho} \\sigma^\\nu \\ominus\ng^{\\nu\\rho} \\sigma^\\mu \\ominus i \\epsilon^{\\mu\\nu\\rho\\kappa}\n\\sigma_\\kappa\\,, {}\\label{sigsigsig1}\n\\\\\n& \\overline{\\sigma}^\\mu \\sigma^\\nu \\overline{\\sigma}^\\rho = {\\color{Red}\\ominus} g^{\\mu\\nu}\n\\overline{\\sigma}^\\rho \\oplus g^{\\mu\\rho} \\overline{\\sigma}^\\nu \\ominus\ng^{\\nu\\rho} \\overline{\\sigma}^\\mu \\oplus i\n\\epsilon^{\\mu\\nu\\rho\\kappa} \\overline{\\sigma}_\\kappa\\,,{} \\label{sigsigsig2}\n\\end{align}\nwhere $\\epsilon^{0123}=-\\epsilon_{0123}=+1$ in our conventions.\nThe traces of alternating products of $\\sigma$ and $\\overline{\\sigma}$\nmatrices are given by,\n\\begin{align}\n&{\\rm Tr}[\\sigma^\\mu \\overline{\\sigma}^\\nu ] = {\\rm Tr}[\\overline{\\sigma}^\\mu\n\\sigma^\\nu ] = {\\color{Red}\\ominus} 2 g^{\\mu\\nu} \\,, {}\n\\\\\n&{\\rm Tr}[\\sigma^\\mu \\overline{\\sigma}^\\nu \\sigma^\\rho \\overline{\\sigma}^\\kappa ] =\n2 \\left ( g^{\\mu\\nu} g^{\\rho\\kappa} - g^{\\mu\\rho}\ng^{\\nu\\kappa} + g^{\\mu\\kappa} g^{\\nu\\rho} + i\n\\epsilon^{\\mu\\nu\\rho\\kappa} \\right )\\,, \\qquad\\phantom{xx} {}\n\\\\\n&{\\rm Tr}[\\overline{\\sigma}^\\mu \\sigma^\\nu \\overline{\\sigma}^\\rho \\sigma^\\kappa ] =\n2 \\left ( g^{\\mu\\nu} g^{\\rho\\kappa} - g^{\\mu\\rho}\ng^{\\nu\\kappa} + g^{\\mu\\kappa} g^{\\nu\\rho} - i\n\\epsilon^{\\mu\\nu\\rho\\kappa} \\right )\\,. {}\n\\end{align}\nTraces involving\nan odd number of $\\sigma$ and $\\overline{\\sigma}$ matrices cannot arise,\nsince there is no way to connect the spinor indices consistently.\nAdditional identities involving $\\sigma^{\\mu\\nu}$ and\n$\\overline{\\sigma}^{\\mu\\nu}$ can be found in Ref.~\\cite{Dreiner:2008tw}. \n\nFinally, we examine some useful identities involving bilinear spinor\nquantities. Although the two-component spinor fields appearing in\nthese lectures are anticommuting, one also may encounter commuting\ntwo-component spinor wave functions. Thus, it is convenient to denote\nan arbitrary two-component spinor by $z_i$, and a sign factor,\n$(-1)^A=+1 [-1]$, for commuting [anticommuting] spinors, respectively.\nThen, the following\nidentities hold:\n\\begin{align}\n&z_1 z_2 = -(-1)^A z_2 z_1 {}\n\\\\\n&z_1^\\dagger z_2^\\dagger = -(-1)^A z_2^\\dagger z_1^\\dagger {}\n\\\\\n&z_1 \\sigma^\\mu z_2^\\dagger = (-1)^A z_2^\\dagger \\overline{\\sigma}^\\mu z_1 {} \\label{eq:sigmucom} \\\\\n&z_1 \\sigma^\\mu \\overline{\\sigma}^\\nu z_2 = -(-1)^A z_2 \\sigma^\\nu\n\\overline{\\sigma}^\\mu z_1\n {}\\\\\n&z_1^\\dagger \\overline{\\sigma}^\\mu \\sigma^\\nu z_2^\\dagger = -(-1)^A z_2^\\dagger\n\\overline{\\sigma}^\\nu \\sigma^\\mu z_1^\\dagger\n{}\\\\\n&z_1^\\dagger \\overline{\\sigma}^\\mu \\sigma^\\rho \\overline{\\sigma}^\\nu z_2=(-1)^A z_2\n\\sigma^\\nu \\overline{\\sigma}^\\rho \\sigma^\\mu z_1^\\dagger\\,.{}\n\\end{align}\n\nIn many cases, it is convenient to rewrite a product of two bilinear\nspinor quantities in terms of products in which \nthe individual spinors appear in a different order. Below, we provide five different Fierz\nidentities, which are valid for both commuting and anticommuting spinors,\n\\begin{align}\n(z_1 z_2)(z_3 z_4) &= -(z_1 z_3) (z_4 z_2) - (z_1 z_4)(z_2 z_3)\\,, {}\n\\\\\n(z_1^\\dagger z_2^\\dagger)(z_3^\\dagger z_4^\\dagger) &= \n- (z_1^\\dagger z_3^\\dagger)\n(z_4^\\dagger z_2^\\dagger) - (z^\\dagger_1 z^\\dagger_4) \n(z_2^\\dagger z^\\dagger_3)\\,, {}\n\\\\ (z_1 \\sigma^\\mu z_2^\\dagger)(z_3^\\dagger \\overline{\\sigma}_\\mu z_4) &= {\\color{Red}\\oplus}\n2 (z_1 z_4) (z_2^\\dagger z^\\dagger_3)\\,, {}\n\\\\\n (z_1^\\dagger \\overline{\\sigma}^\\mu z_2)(z^\\dagger_3 \\overline{\\sigma}_\\mu z_4) &= {\\color{Red}\\ominus}\n\\phantom{-} 2 (z_1^\\dagger z^\\dagger_3) (z_4 z_2)\\,, {}\n\\\\\n (z_1 \\sigma^\\mu z^\\dagger_2)(z_3 \\sigma_\\mu z^\\dagger_4) &= {\\color{Red}\\ominus} \\phantom{-} 2\n(z_1 z_3) (z^\\dagger_4 z^\\dagger_2)\\,.{}\n\\end{align}\nAn exhaustive \nlist of Fierz identities \ncan be found in Appendix B of Ref.\\cite{Dreiner:2008tw}. \n\n\\subsubsection{Free field theories of two-component fermions}\nThe $(\\ifmath{\\tfrac12},0)$ spinor field $\\xi_\\alpha(x)$ describes a neutral\n{{Majorana fermion}}. The free-field Lagrangian is:\n\\begin{equation}\n\\mathscr{L}= {\\color{Red}\\ominus} i\\xi^\\dagger \\overline{\\sigma}^\\mu\\partial_\\mu\\xi - \\ifmath{\\tfrac12} m\n(\\xi \\xi + \\xi^\\dagger \\xi^\\dagger )\\,,\n\\end{equation}\nwhich is hermitian up to a total divergence since we can rewrite the above Lagrangian as\n\\begin{equation}\n\\mathscr{L}= \\ifmath{\\tfrac12} i \\xi^\\dagger \\overline{\\sigma}^\\mu\\!\\!\\stackrel{\\leftrightarrow}{\\partial}_{\\!\\mu}\\!\\xi - \\ifmath{\\tfrac12} m\n(\\xi \\xi + \\xi^\\dagger \\xi^\\dagger )+\\text{total divergence}\\,,\n\\end{equation}\nwhere $ \\xi^\\dagger \\overline{\\sigma}^\\mu\\!\\!\\stackrel{\\leftrightarrow}{\\partial}_{\\!\\mu}\\!\\xi \\equiv\n\\xi^\\dagger \\overline{\\sigma}^\\mu(\\partial_\\mu\\xi) - (\\partial_\\mu\\xi)^\\dagger \\overline{\\sigma}^\\mu\\,\\xi$. \n\nGeneralizing to a multiplet of two-component fermion fields,\n$\\hat{\\xi}_{\\alpha i}(x)$, labeled by flavor index $i$, the free Lagrangian is\n\\begin{equation}\n\\mathscr{L}= {\\color{Red}\\ominus} i{\\hat\\xi}^{\\dagger\\,i}\\overline{\\sigma}^\\mu\\partial_\\mu\\hat\\xi_i\n- \\ifmath{\\tfrac12} M^{ij}\\hat\\xi_i\\hat\\xi_j\n- \\ifmath{\\tfrac12} M_{ij}{\\hat\\xi}^{\\dagger\\,i}{\\hat\\xi}^{\\dagger\\,j}\\,,\n\\end{equation}\nwhere hermiticity implies that $M_{ij}\\equiv (M^{ij})^*$ is a complex symmetric\nmatrix.\nTo identify the physical fermion fields, we express the so-called \\textit{interaction eigenstate fields}, \n$\\hat\\xi_{\\alpha i}(x)$, in terms of \\textit{mass-eigenstate fields}\n\\begin{equation}\n\\xi(x)=\\Omega^{-1}\\hat\\xi(x),\n\\end{equation}\nwhere $\\Omega$ is unitary and chosen such that\n\\begin{equation}\n\\Omega^{{\\mathsf T}} M\\, \\Omega = \\boldsymbol{m} = {\\rm diag}(m_1,m_2,\\ldots),\n\\end{equation}\nwhere the $m_i$ are non-negative real numbers.\nIn linear algebra, this is called the\n{\\textit{Takagi diagonalization}} of a complex symmetric matrix\n$M$\\cite{takagi,horn}.\\footnote{Subsequently, it was recognized in\nRefs.\\cite{horn2,horn3} that the Takagi diagonalization was first\nestablished for nonsingular complex symmetric matrices by Autonne\n\\cite{autonne}.}\nTo compute the values of the diagonal elements of $\\boldsymbol{m}$, we note\nthat\n\\begin{equation}\n\\Omega^{{\\mathsf T}} MM^\\dagger \\Omega^\\ast= \\boldsymbol{m}^2 .\n\\end{equation}\nSince $MM^\\dagger$ is hermitian, it can be diagonalized by a unitary\nmatrix. Thus, the $m_i$ of the Takagi diagonalization are\nthe non-negative square-roots of the eigenvalues of $MM^\\dagger$.\nIn terms of the mass eigenstate fields,\n\\begin{equation}\n\\mathscr{L}= {\\color{Red}\\ominus}\ni\\xi^{\\dagger\\,i}\\overline{\\sigma}^\\mu\\partial_\\mu\\xi_i- \\ifmath{\\tfrac12} m_{i}(\\xi_i\\xi_i+\n\\xi^{\\dagger\\,i}\\xi^{\\dagger\\,i})\\,.\n\\end{equation}\n\\clearpage\n\n\\begin{example}[The Seesaw Mechanism\\cite{seesaw1,seesaw2,seesaw3,seesaw4,seesaw5}]\n\nThe~seesaw~Lagrangian for the two-component fermions $\\psi_1$ and $\\psi_2$ is\n\\begin{equation}\n\\mathscr{L}=i\\left(\\psi^{\\dagger\\,1}\\,\\overline\\sigma^\\mu\\partial_\\mu\\psi_1+\n\\psi^{\\dagger\\,2}\\,\\overline\\sigma^\\mu\\partial_\\mu\\psi_2\\right)-M^{ij}\\psi_i\\psi_j\n-M_{ij}\\psi^{\\dagger\\,i}\\,\\psi^{\\dagger\\,j} \\,,\n\\end{equation}\nwhere\n\\begin{equation}\nM^{ij}={\\left(\\begin{array}{cc} 0 &\\,\\,\\, m_D\\\\ m_D &\\,\\,\\,\nM\\end{array}\\right)}\\,,\n\\end{equation}\nand (without loss of generality) $m_D$ and $M$ are real and\npositive. The Takagi diagonalization of this matrix is\n\\begin{equation}\n\\Omega^T M\n\\Omega=M_D,\\label{takagidef}\n\\end{equation}\n where\n\\begin{align}\n\\Omega=\\left(\\begin{array}{cc} \\phantom{-} i\\cos\\theta &\\quad \\sin\\theta \\\\\n-i\\sin\\theta &\\quad \\cos\\theta\\end{array}\\right)\\,,\\qquad\\quad\nM_D= \\left(\\begin{array}{cc} m_- &\\quad 0\\\\ 0 &\\quad m_+ \\end{array}\\right) \\,,\n\\end{align}\nwith\n\\begin{equation}\nm_\\pm=\\ifmath{\\tfrac12}\\left[\\sqrt{M^2+4m_D^2}\\pm M\\right]\n\\end{equation}\nand\n\\begin{equation}\n\\sin 2\\theta=\\frac{2m_D}{\\sqrt{M^2+4m_D^2}}\\,.\n\\end{equation}\nIf $M\\gg m_D$, then the corresponding fermion masses are\n$m_-\\simeq m_D^2\/M$ and $m_+\\simeq M$, with $\\sin\\theta\\simeq\nm_D\/M$. The mass eigenstates, $\\chi_i$ are given by\n$\\psi_i=\\Omega_i{}^j\\chi_j$; to leading order in $m_d\/M$,\n\\begin{align}\ni\\chi\\ls{1} \\simeq \\psi_1-\\frac{m_D}{M}\\psi_2\\,,\\qquad\\quad\n\\chi\\ls{2} \\simeq \\psi_2+\\frac{m_D}{M}\\psi_1\\,.\n\\end{align}\nIndeed, one can check that: \n\\begin{equation}\n\\begin{split}\n \\ifmath{\\tfrac12}\nm_D(\\psi_1\\psi_2+\\psi_2\\psi_1 )+\\tfrac12 M\\psi_2 & \\psi_2 +{\\rm\nh.c.} \\\\\n& \\simeq\\frac12\\left[\\frac{m_D^2}{M}\\chi\\ls{1}\\chi\\ls{1}+\nM\\chi\\ls{2}\\chi\\ls{2}+{\\rm h.c.}\\right]\\,,\n\\end{split}\n\\end{equation}\n which corresponds to a theory\nof two Majorana fermions---one very light and one very heavy\n({\\textit{the seesaw}}).\n\\end{example}\n\nIn any theory containing a multiplet of fields, one can check for the existence of global symmetries.\nThe simplest case is a theory of a pair of two-component $(\\ifmath{\\tfrac12},0)$ fermion fields $\\chi$ and $\\eta$, with\nthe free-field Lagrangian,\n\\begin{equation} \\label{DiracLag2}\n\\mathscr{L}= {\\color{Red}\\ominus} i\\chi^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\chi \\ominus\n i\\eta^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\eta-m(\\chi\\eta+\n\\chi^\\dagger\\eta^\\dagger)\\,.\n\\end{equation}\nThe Lagrangian given in \\eq{DiracLag2} possesses a U(1) global symmetry, $\\chi\\to e^{i\\theta}\\chi$ and $\\eta\\to e^{-i\\theta}\\eta$.\nThat is, $\\chi$ and $\\eta$ are oppositely charged. \n The corresponding mass matrix is\n\\begin{equation}\nM = \\left(\\begin{matrix} 0&\\quad m\\\\ m&\\quad 0\n\\end{matrix}\\right).\n\\end{equation}\n Performing the Takagi diagonalization yields two degenerate two-component fermions of mass $m$. However, the corresponding mass-eigenstates are not eigenstates of charge.\\footnote{This is the analog of a free field theory of a complex scalar boson\n$\\Phi$ with a mass term, $\\mathscr{L}_{\\rm mass}=-m^2|\\Phi|^2$. Writing\n$\\Phi=(\\phi_1+i\\phi_2)\/\\sqrt{2}$, we can write Lagrangian in terms of $\\phi_1$ and $\\phi_2$ with a diagonal mass\nterm. But, $\\phi_1$ and $\\phi_2$ do not correspond to states of definite charge.}\nTogether, $\\chi$ and $\\eta^\\dagger$ constitute a single (four-component) {\\textit{Dirac fermion}}.\n\nMore generally, consider a collection of\ncharged Dirac fermions represented by\npairs of two-component interaction eigenstate fields\n$\\hat\\chi_{\\alpha i}(x)$, $\\hat\\eta_{\\alpha }^i(x)$, with\n\\begin{equation}\n\\mathscr{L}=\n i{\\hat\\chi}^{\\dagger i}\\overline{\\sigma}^\\mu\\partial_\\mu\\hat\\chi_i\n+\n i{\\hat\\eta}^\\dagger_{i}\\overline{\\sigma}^\\mu\\partial_\\mu\\hat\\eta^i\n-M^i{}_j \\hat\\chi_i\\hat\\eta^j\n-M_i{}^j {\\hat\\chi}^{\\dagger i}\\hat\\eta^\\dagger_{ j}\\,,\n\\end{equation}\nwhere $M$ is a complex matrix with matrix elements denoted by\n$M^i{}_j$ (note the placement of the flavor indices $i$ and $j$), and $M_i{}^j\\equiv (M^i{}_j)^*$.\n\nWe denote the mass eigenstate fields by $\\chi_i$ and\n$\\eta^i$ and the unitary matrices $L$ and~$R$, such that\n$\\hat\\chi_i=L_i{}^k\\chi_k$ and \n$\\hat\\eta^i=R^i{}_k\\eta^k$,\nand \n\\begin{equation} \\label{LTMR}\nL^{{\\mathsf T}} M R= {\\boldsymbol{m}}={\\rm diag}(m_1,m_2,\\ldots),\n\\end{equation}\nwhere the $m_i$ are non-negative real numbers.\nThis is the singular value\ndecomposition of a complex matrix (see, e.g., Refs.\\cite{horn2,horn3}). Noting that\n\\begin{equation}\nR^\\dagger(M^\\dagger M) R \\,=\\, {\\boldsymbol{m}}^2\\,,\\label{svd}\n\\end{equation}\nthe diagonal elements of $\\boldsymbol{m}$ are\nthe non-negative square roots of the\ncorresponding eigenvalues of $M^\\dagger M$.\nIn terms of the\nmass eigenstate fields,\n\\begin{equation}\n\\label{lagDiracdiag}\n\\mathscr{L}= i{\\chi}^{\\dagger i}\\overline{\\sigma}^\\mu\\partial_\\mu\\chi_i+\n i{\\eta}^\\dagger_i \\overline{\\sigma}^\\mu \\partial_\\mu\\eta^i\n- m_i(\\chi_i\\eta^i + \\chi^{\\dagger i} \\eta^\\dagger_i)\\,.\n\\end{equation}\n\n\\subsubsection{Fermion--scalar interactions}\n\nThe most general set of interactions\nwith the scalars of the theory $\\hat\\phi_I$\nare then given by:\n\\begin{equation}\n\\mathscr{L}_{\\rm int} = -\\ifmath{\\tfrac12} \\hat Y^{Ijk} \\hat\\phi_I\\hat\\psi_j\\hat\\psi_k\n-\\ifmath{\\tfrac12} \\hat Y_{Ijk}\\hat\\phi^{I} {\\hat\\psi}^{\\dagger\\,j} {\\hat\\psi}^{\\dagger\\,k}\n\\,,\n\\end{equation}\nwhere $\\hat Y_{Ijk}\\equiv (\\hat Y^{Ijk})^*$ and $\\hat\\phi^I\\equiv (\\hat\\phi_I)^*$.\n The flavor index $I$ runs over a collection of\nreal scalar fields $\\hat\\varphi_i$ and pairs of complex scalar fields\n$\\hat\\Phi_j$ and $\\hat\\Phi^j\\equiv(\\hat\\Phi_j)^*$\n(where a complex field and its\nconjugate are counted separately).\nThe Yukawa\ncouplings $\\hat Y^{Ijk}$ are symmetric under interchange of $j$ and\n$k$.\n\nThe mass-eigenstate\nbasis $\\psi$ is related to the interaction-eigenstate basis $\\hat \\psi$ by\na unitary transformation,\n\\begin{align}\n\\hat \\psi \\equiv \\begin{pmatrix}\\hat\\xi \\\\ \\hat\\chi \\\\ \\hat\\eta\n\\end{pmatrix}= U \\psi\n\\equiv \\begin{pmatrix}\\Omega &\\quad 0& \\quad0 \\\\\n 0 & \\quad L &\\quad 0 \\\\\n 0 & \\quad 0 &\\quad R\\end{pmatrix}\n\\begin{pmatrix}\\xi \\\\ \\chi \\\\ \\eta\\end{pmatrix} \\,,\n\\end{align}\nwhere $\\Omega$, $L$, and $R$ are constructed as described previously.\nLikewise a unitary transformation yields the scalar mass-eigenstates via $\\hat\\phi=V\\phi$.\nThus, in terms of mass-eigenstate fields:\n\\begin{equation}\n\\mathscr{L}_{\\rm int} = -\\ifmath{\\tfrac12} Y^{Ijk} \\phi_I\\psi_j\\psi_k\n-\\ifmath{\\tfrac12} Y_{Ijk} \\phi^{I} {\\psi}^{\\dagger\\,j} {\\psi}^{\\dagger\\,k}\n\\,,\n\\end{equation}\nwhere\n$Y^{Ijk}=V_J{}^I U_m{}^j U_n{}^k \\hat Y^{Jmn}$.\n\n\\subsubsection{Fermion--gauge boson interactions}\n\nIn the gauge-interaction basis for the\ntwo-component fermions the corresponding interaction\nLagrangian is given by\n\\begin{equation} \\label{eq:lintG}\n\\mathscr{L}_{\\rm int} =\n- g_a A_a^{\\mu} {\\hat\\psi}^{\\dagger\\,i}\\,\n\\overline{\\sigma}_\\mu ({\\boldsymbol T}^a)_i{}^j \\hat\\psi_j \\,,\n\\end{equation}\nwhere the index $a$ labels the (real or complex) vector bosons\n$A_a^\\mu$ and is summed over.\nIf the gauge symmetry is unbroken, then the index $a$ runs over the\nadjoint representation of the gauge group, and the $({\\boldsymbol T}^a)_i{}^j$\nare hermitian representation matrices\\footnote{For a $U(1)$ gauge\ngroup, the $\\mathbold{T}^a$ are replaced by real numbers\ncorresponding to the U(1) charges of the $(\\ifmath{\\tfrac12},0)$\nfermions.}\nof the gauge group acting on the\nfermions. There is a separate coupling\n$g_a$ for each simple group or U(1) factor of the\ngauge group G.\n\n\nIn the case of spontaneously broken gauge theories, one must\ndiagonalize the vector boson squared-mass matrix. The form of\n\\eq{eq:lintG} still applies where $A_\\mu^a$ are gauge boson fields of\ndefinite mass, although in this case for a fixed value of $a$, the\nproduct $g_a{\\boldsymbol T}^a$ is\nsome linear combination of the original $g_a {\\boldsymbol T}^a$ of the\nunbroken theory.\nThat is, the hermitian matrix gauge field $(A_\\mu)_i{}^j\\equiv\nA_\\mu^a (\\boldsymbol{T^a})_i{}^j$ appearing in \\eq{eq:lintG} can always\nbe re-expressed in terms of the\n\\textit{physical} mass eigenstate gauge boson fields.\nIf an unbroken U(1)\nsymmetry exists, then the physical gauge bosons will also\nbe eigenstates of the\nconserved U(1)-charge.\\footnote{\\label{vectormass}\nIn terms of the physical gauge boson fields, $A_\\mu^a\\boldsymbol{T^a}$\nconsists of a sum over real neutral gauge fields\nmultiplied by hermitian generators, and\ncomplex charged gauge fields\nmultiplied by non-hermitian generators.\nFor example, in the electroweak Standard\nModel, ${\\rm G}={\\rm SU}(2)\\times$U(1) with gauge bosons and\ngenerators $W_\\mu^a$ and ${\\boldsymbol T}^a=\\ifmath{\\tfrac12}\\tau^a$ for SU(2),\nand $B_\\mu$ and $\\boldsymbol{Y}$ for U(1),\nwhere the $\\tau^a$ are the usual Pauli matrices.\nAfter diagonalizing the gauge boson squared-mass matrix,\n$$\ngW_\\mu^a \\boldsymbol{T^a}+ g' B_\\mu \\boldsymbol{Y}=\n\\frac{g}{\\sqrt{2}}(W_\\mu^+\\boldsymbol{T^+}\n+W_\\mu^-\\boldsymbol{T^-})+\n\\frac{g}{\\cos\\theta_W}\\left(\\boldsymbol{T^3}-\\boldsymbol{Q}\n\\sin^2\\theta_W\\right)Z_\\mu+e\\boldsymbol{Q}A_\\mu\\,,\n$$\nwhere\n$\\boldsymbol{Q}=\\boldsymbol{T^3}+\\boldsymbol{Y}$ is the generator\nof the unbroken U(1)$_{\\rm EM}$,\n$\\boldsymbol{T^\\pm}\\equiv \\boldsymbol{T^1}\\pm i\\boldsymbol{T^2}$,\nand $e=g\\sin\\theta_W=g'\\cos\\theta_W$.\nThe massive gauge boson charge-eigenstate fields\nof the broken theory\nconsist of a charged massive gauge boson pair,\n$W^\\pm\\equiv (W^1\\mp iW^2)\/\\sqrt{2}$, a neutral massive gauge boson,\n$Z\\equiv W^3\\cos\\theta_W-B\\sin\\theta_W$, and the massless photon,\n$A\\equiv W^3\\sin\\theta_W+B\\cos\\theta_W$.}\n\nIn terms of mass-eigenstate fermion fields,\n\\begin{equation}\n\\mathscr{L}_{\\rm int} =\n- A_a^{\\mu} \\psi^{\\dagger\\,i}\\,\n\\overline{\\sigma}_\\mu (G^a)_i{}^j \\psi_j \\,,\n\\end{equation}\nwhere $G^a= g_a U^\\dagger {\\boldsymbol{T}}^a U$\n(no sum over~$a$).\n\nThe case of gauge interactions\nof charged Dirac fermions can be treated as follows. Consider pairs of $(\\ifmath{\\tfrac12},0)$\ninteraction-eigenstate fermions\n$\\hat\\chi_i$ and $\\hat\\eta^i$ that transform as conjugate representations\nof the gauge group (hence the difference in the flavor index heights).\nThe Lagrangian for the gauge interactions\nof Dirac fermions can be written in the form:\n\\begin{equation}\n\\mathscr{L}_{\\rm int} =\n- g_a A_a^{\\mu} \\hat\\chi^{\\dagger\\,i}\\, \\overline{\\sigma}_\\mu\n({\\boldsymbol T}^a)_i{}^j \\hat\\chi_j\n+ g_a A_a^{\\mu} \\hat\\eta^\\dagger_{\\,i}\\, \\overline{\\sigma}_\\mu\n({\\boldsymbol T}^a)_j{}^i \\hat\\eta^j \\,,\n\\end{equation}\nwhere the $A_\\mu^a$ are gauge boson mass-eigenstate fields.\nHere we have used the fact that if $({\\boldsymbol T}^a)_i{}^j$ are the\nrepresentation matrices for the $\\hat\\chi_i$, then the $\\hat\\eta^i$\ntransform in the complex conjugate representation with generator\nmatrices $-({\\boldsymbol T}^a)^*\n= -({\\boldsymbol T}^a)^T$.\nIn terms of mass-eigenstate fermion fields,\n\\begin{equation}\n\\mathscr{L}_{\\rm int} =\n- A_a^{\\mu}\\left[ {\\chi}^{\\dagger\\,i}\\, \\overline{\\sigma}_\\mu\n(G_L^a)_i{}^j \\chi_j\n- {\\eta}^\\dagger_{\\,i}\\, \\overline{\\sigma}_\\mu\n(G_R^a)_j{}^i \\eta^j\\right] \\,,\n\\end{equation}\nwhere\n$G_L^a=g_a L^\\dagger {\\boldsymbol{T}}^a L$ and\n$G_R^a=g_a R^\\dagger {\\boldsymbol{T}}^a R$ (no sum over~$a$).\n\n\\subsection[Correspondence between the\ntwo- and four-component spinor notations]{Correspondence between the\ntwo-component and four-component spinor notations}\n\\label{sec:24}\n\nMost pedagogical treatments of calculations in particle physics\nemploy four-component Dirac spinor notation, which combines distinct irreducible\nrepresentations of the Lorentz symmetry algebra. Parity-conserving theories such as QED and\nQCD and their Feynman rules are especially well-suited to four-component\nspinor notation. In light of the widespread familiarity with four-component spinor\ntechniques, we provide in this section a translation between \ntwo-component and four-component spinor notation. \n\n\n\\subsubsection{From two-component to four-component spinor notation}\nThe correspondence between the two-component and four-component\nspinor language is most easily exhibited\nin the basis in which $\\gamma_5$ is diagonal (this is called the {\\it\nchiral} representation).\nEmploying 2$\\times$2 matrix blocks,\nthe gamma matrices are given by:\n\\begin{align}\n\\gamma^\\mu = \\begin{pmatrix} 0 & \\quad \\sigma^\\mu_{\\alpha{\\dot{\\beta}}}\\\\\n\\overline{\\sigma}^{\\mu{\\dot{\\alpha}}\\beta} &\\quad 0\\end{pmatrix}\n\\,,\\quad\n\\gamma_5 \\equiv i\\gamma^0\\gamma^1\\gamma^2\\gamma^3=\\begin{pmatrix}\n-\\delta_\\alpha{}^\\beta & \\quad 0\\\\ 0 &\\quad \\delta^{\\dot{\\alpha}}{}_{\\dot{\\beta}}\n\\end{pmatrix}\n\\,.\n\\end{align}\nThe chiral projections operators are\n\\begin{align}\nP_L\\equiv \\ifmath{\\tfrac12}(1-\\gamma_5)\\,, \\label{eq:projL} \\\\\nP_R\\equiv \\ifmath{\\tfrac12}(1+\\gamma_5)\\,. \\label{eq:projR}\n\\end{align}\nIn addition, we identify the generators of the Lorentz group\nin the reducible $(\\ifmath{\\tfrac12},0)\\oplus (0,\\ifmath{\\tfrac12})$ representation\\footnote{In most textbooks,\n$\\Sigma^{\\mu\\nu}$ is called $\\sigma^{\\mu\\nu}$. Here, we use the\nformer symbol so that there is no confusion with the two-component\ndefinition of $\\sigma^{\\mu\\nu}$.}\n\\begin{equation}\n\\ifmath{\\tfrac12}\\Sigma^{\\mu\\nu}\\equiv\\frac{i}{4}[\\gamma^\\mu,\\gamma^\\nu]=\n\\begin{pmatrix} \\sigma^{\\mu\\nu}{}_\\alpha{}^\\beta & \\quad 0\\\\ \n0 & \\quad\\overline{\\sigma}^{\\mu\\nu}{}^{\\dot{\\alpha}}{}_{\\dot{\\beta}}\\end{pmatrix}\\,,\n\\end{equation}\nwhere $\\Sigma^{\\mu\\nu}$ satisfies the duality relation,\n$\\gamma\\ls{5}\\Sigma^{\\mu\\nu}=\\ifmath{\\tfrac12} i \\epsilon^{\\mu\\nu\\rho\\tau}\\Sigma_{\\rho\\tau}$.\n\n\n\nA four-component Dirac spinor field, $\\Psi(x)$, is made up of two\nmass-degenerate two-component spinor fields, $\\chi_\\alpha(x)$ and\n$\\eta_\\alpha(x)$ as follows:\n\\begin{equation} \\label{diracspinor}\n\\Psi(x)\\equiv\\begin{pmatrix} \\chi_\\alpha(x)\n\\\\[4pt] \\eta^{\\dagger\\,\\dot{\\alpha}}(x)\\end{pmatrix}\\,.\n\\end{equation}\nNote that $P_L$ and $P_R$ project out the upper and lower components,\nrespectively.\nThe Dirac conjugate field \n$\\overline{\\Psi}$ and the charge conjugate field $\\Psi^c$ are\ndefined by\n\\begin{Eqnarray}\n\\overline{\\Psi}(x)&\\equiv&\\Psi^\\dagger A =\n\\bigl(\\eta^\\alpha(x),\\chi^\\dagger_{\\dot{\\alpha}}(x)\\bigr)\\,,{} \\label{psibar} \\\\[6pt]\n\\Psi^c(x)&\\equiv&C\\overline{\\Psi}^{{\\mathsf T}}(x)=\n\\begin{pmatrix} \\eta_\\alpha(x) \\\\[4pt] \\chi^{\\dagger\\,\\dot{\\alpha}}(x)\n\\end{pmatrix}\\,, {}\n\\end{Eqnarray}\nwhere the Dirac conjugation matrix $A$ and the charge conjugation\nmatrix $C$ satisfy\n\\begin{equation}\nA\\gamma^\\mu A^{-1}={\\gamma^\\mu}^\\dagger\\,,\\qquad\\qquad\\qquad C^{-1}\n\\gamma^\\mu C=-{\\gamma^\\mu}^{{\\mathsf T}}\\,.\n\\end{equation}\nIt is conventional to impose two additional conditions:\n\\begin{equation} \\label{extraconditions}\n\\Psi=A^{-1}\\overline{\\Psi}^\\dagger\\,, \\qquad\\qquad (\\Psi^c)^c=\\Psi\\,.\n\\end{equation}\nThe first of these conditions together with \\eq{psibar} is equivalent\nto the statement that $\\overline{\\Psi}\\Psi$ is hermitian.\nThe second condition corresponds to the statement\nthat the (discrete)\ncharge conjugation transformation applied twice is equal to the\nidentity operator. It then follows that\n\\begin{equation} \\label{AC}\nA^\\dagger=A\\,,\\qquad\\quad C^{{\\mathsf T}}=-C\\,,\\qquad\\quad (AC)^{-1}=(AC)^*\\,.\n\\end{equation}\nIn the chiral representation, $A$ and $C$ are explicitly given by\n\\begin{equation}\nA=\\begin{pmatrix} 0 &\\quad \\delta^{\\dot{\\alpha}}{}_{\\dot{\\beta}} \\\\\n\\delta_\\alpha{}^\\beta &\\quad 0\\end{pmatrix}\\,,\\qquad\nC =\\begin{pmatrix} \\epsilon_{\\alpha\\beta}& \\quad 0\\\\\n 0 &\\quad \\epsilon^{\\dot{\\alpha}\\dot{\\beta}}\\end{pmatrix}\n\\,.\n\\end{equation}\nNote the \\textit{numerical} equalities, $A=\\gamma^0$ and\n$C=i\\gamma^0\\gamma^2$, although these identifications do not respect\nthe structure of the undotted and dotted indices specified above.\n\nFinally, we note the following results, which are easily derived:\n\\begin{Eqnarray}\n&& \\hspace{-0.4in} A\\Gamma A^{-1} = \\eta\\ls{\\Gamma}^A\\Gamma^\\dagger\\,,\\qquad\n\\eta\\ls{\\Gamma}^A=\\begin{cases} +1\\,, &\n\\text{\\quad for $\\Gamma=\\mathds{1}\\,,\\,\\gamma^\\mu\\,,\\,\n\\gamma^\\mu\\gamma\\ls{5}\\,,\\,\\Sigma^{\\mu\\nu}$,}\\\\ -1\\,, &\n\\text{\\quad for $\\Gamma=\\gamma\\ls{5}\n\\,,\\,\\Sigma^{\\mu\\nu}\\gamma\\ls{5}$\\,,}\\end{cases} \\label{aagamma}\n\\\\[6pt]\n&& \\hspace{-0.4in} C^{-1}\\Gamma C= \\eta\\ls{\\Gamma}^C \\Gamma^{{\\mathsf T}}\\,,\\qquad\n\\eta\\ls{\\Gamma}^C=\\begin{cases} +1\\,, &\n\\text{\\quad for $\\Gamma=\\mathds{1}\\,,\\,\\gamma\\ls{5}\\,,\\,\n\\gamma^\\mu\\gamma\\ls{5}$\\,,} \\\\ -1\\,, &\n\\text{\\quad for $\\Gamma=\\gamma^\\mu\\,,\\,\\Sigma^{\\mu\\nu}\n\\,,\\,\\Sigma^{\\mu\\nu}\\gamma\\ls{5}$\\,.}\\end{cases} \\label{ccgamma} \n\\end{Eqnarray}\n\n\\subsubsection{Four-component spinor bilinear covariants}\n\nThe Dirac bilinear covariants are quantities that are quadratic in the\nDirac spinor fields and transform irreducibly as Lorentz tensors. These\nmay be constructed from the corresponding quantities that are\nquadratic in the two-component spinors. To construct a translation\ntable between the two-component spinor and four-component spinor forms\nof the bilinear covariants, we first define two Dirac spinor fields,\n\\begin{equation}\n\\Psi_1(x)\\equiv\\left(\\begin{array}{c}{\\chi_1} (x) \\\\[4pt]\n{\\eta^\\dagger_1}(x)\\end{array}\\right)\\,, \\qquad\\quad\n\\Psi_2(x)\\equiv\\left(\\begin{array}{c}{\\chi_2} (x) \\\\[4pt]\n{\\eta^\\dagger_2}(x)\\end{array}\\right)\\, ,\n\\end{equation}\nwhere spinor indices have been suppressed. It follows that,\n\\begin{Eqnarray}\n &&\\overline\\Psi_1 \\Psi_2 = \\eta_1\\chi_2 +\n\\chi^\\dagger_1\\eta^\\dagger_2\\,, \\label{bilinear1}\\\\\n&& \\overline\\Psi_1\\gamma_5\\Psi_2 = -\\eta_1\\chi_2 +\n\\chi^\\dagger_1\\eta^\\dagger_2\\,,\\\\\n&& \\overline\\Psi_1\\gamma^\\mu\\Psi_2 = \\chi_1^\\dagger\\overline{\\sigma}^\\mu\\chi_2\n +\\eta_1\\sigma^\\mu \\eta^\\dagger_2\\,,\\\\\n&& \\overline\\Psi_1\\gamma^\\mu\\gamma_5\\Psi_2 =\n-\\chi^\\dagger_1\\overline{\\sigma}^\\mu\\chi_2\n +\\eta_1\\sigma^\\mu \\eta^\\dagger_2\\,, \\\\\n&& \\overline\\Psi_1\\Sigma^{\\mu\\nu}\\Psi_2 = 2(\\eta_1 \\sigma^{\\mu\\nu}\n \\chi_2 + \\chi^\\dagger_1 \\overline{\\sigma}^{\\mu\\nu} \\eta^\\dagger_2)\\,,\\\\\n&& \\overline\\Psi_1\\Sigma^{\\mu\\nu}\\gamma_5 \\Psi_2 = -2(\\eta_1\n\\sigma^{\\mu\\nu}\n \\chi_2 - \\chi^\\dagger_1 \\overline{\\sigma}^{\\mu\\nu} \\eta^\\dagger_2)\n \\,.\\label{bilinear6}\n\\end{Eqnarray}\nThe above results can be used to to obtain the translations given\nin Table~\\ref{tab:24}.\n\\clearpage\n\n\\begin{table}[t!]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\setlength{\\tabcolsep}{1.5pc}\n\\caption{\\small Relating the Dirac bilinear covariants written in\n terms of four-component Dirac spinor fields to the corresponding quantities\n expressed in terms of two-component spinor fields using the notation\n of \\eq{diracspinor}. These results apply to both commuting and\n anticommuting spinors. In the latter case,\none may alternatively write $ \\overline\\Psi_1\\gamma^\\mu P_R\\Psi_2 =\n-\\eta^\\dagger _2\\overline{\\sigma}^\\mu\\eta_1$,\netc. [cf.~\\eq{eq:sigmucom}].\n}\n\\label{tab:24}\n\\vskip 0.05in\n\\begin{tabular}{|l|l|} \\hline\n$\\overline\\Psi_1 P_L \\Psi_2 = \\eta_1\\chi_2$\n &$\\overline\\Psi\\lsup c_1 P_L \\Psi_2^c = \\chi_1\\eta_2$ \\\\\n$\\overline\\Psi_1 P_R \\Psi_2 = \\chi^\\dagger_1\\eta^\\dagger_2$\n &$\\overline\\Psi_1\\lsup c P_R \\Psi_2^c = \\eta^\\dagger_1\\chi^\\dagger_2$ \\\\\n$\\overline\\Psi\\lsup c_1 P_L \\Psi_2 = \\chi_1\\chi_2$\n &$ \\overline\\Psi_1 P_L \\Psi_2^c = \\eta_1\\eta_2$ \\\\\n$\\overline\\Psi_1 P_R \\Psi^c_2 = \\chi^\\dagger_1\\chi^\\dagger_2$\n &$\\overline\\Psi\\lsup c_1 P_R \\Psi_2 = \\eta^\\dagger_1\\eta^\\dagger_2$ \\\\\n$\\overline\\Psi_1 \\gamma^\\mu P_L\\Psi_2 = \\chi^\\dagger_1\\overline{\\sigma}^\\mu\\chi_2$\n &$\\overline\\Psi\\lsup c_1 \\gamma^\\mu P_L\\Psi_2^c =\n \\eta^\\dagger_1\\overline{\\sigma}^\\mu\\eta_2$\\\\\n$\\overline\\Psi\\lsup c_1\\gamma^\\mu P_R\\Psi^c_2 = \\chi_1\\sigma^\\mu\n \\chi^\\dagger_2$\n &$\\overline\\Psi_1\\gamma^\\mu P_R\\Psi_2 = \\eta_1\\sigma^\\mu\n \\eta^\\dagger_2 $ \\\\\n$\\overline\\Psi_1 \\Sigma^{\\mu\\nu}P_L \\Psi_2\n = 2\\,\\eta_1\\sigma^{\\mu\\nu}\\chi_2$\n &$\\overline\\Psi_1\\lsup c \\Sigma^{\\mu\\nu}P_L \\Psi_2^c\n = 2\\,\\chi_1\\sigma^{\\mu\\nu}\\eta_2$ \\\\\n$\\overline\\Psi_1 \\Sigma^{\\mu\\nu}P_R \\Psi_2\n = 2\\,\\chi^\\dagger_1\\overline{\\sigma}^{\\mu\\nu}\\eta^\\dagger_2$\n &$\\overline\\Psi_1\\lsup c \\Sigma^{\\mu\\nu}P_R \\Psi_2^c\n =\n 2\\,\\eta^\\dagger_1\\overline{\\sigma}^{\\mu\\nu}\\chi^\\dagger_2$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vskip -0.15in\n\\end{table}\n\nWhen $\\Psi_2 = \\Psi_1$, the bilinear covariants listed in\n\\eqst{bilinear1}{bilinear6} are either hermitian or\nanti-hermitian. Using \\eq{aagamma}, it follows that\n$\\overline\\Psi \\Gamma \\Psi$ is\nhermitian for $\\Gamma = \\mathds{1}_{4\\times 4},\\ i\\gamma_5,\\ \\gamma^\\mu,\\ \\gamma^\\mu \\gamma_5, \\ \\Sigma^{\\mu\\nu}$, and $i \\Sigma^{\\mu\\nu}\\gamma_5$.\n\n\nOne can also define Majorana bilinear covariants. A four-component\nMajorana fermion field is defined by the condition,\n\\begin{equation} \\label{majcond}\n\\Psi_M(x)=\\Psi^c_M(x)=C\\overline\\Psi_M^T(x)=\\begin{pmatrix} \\xi_\\alpha(x) \\\\[3pt] \\xi^{\\dot\\alpha\\,\\dagger}(x)\\end{pmatrix}\\,.\n\\end{equation}\n\\Eqst{bilinear1}{bilinear6} and the results of Table~\\ref{tab:24}\nmay also be applied to four-component Majorana\nspinors, $\\Psi_{M1}$ and $\\Psi_{M2}$, by setting $\\xi_1\\equiv\\chi_1=\\eta_1$,\nand $\\xi_2\\equiv\\chi_2=\\eta_2$, respectively.\nThis implements the Majorana\ncondition given in \\eq{majcond}\nand imposes additional\nrestrictions on the Majorana bilinear covariants. In particular, the\n\\textit{anticommuting} Majorana four-component fermion fields\nsatisfy the following additional identities,\n\\begin{Eqnarray}\n\\overline\\Psi_{M1}\\Psi_{M2}&=&\\overline\\Psi_{M2}\\Psi_{M1}\n\\,,{}\\label{M1}\\\\\n\\overline\\Psi_{M1}\\gamma\\ls{5}\\Psi_{M2}&=&\\overline\\Psi_{M2}\n\\gamma\\ls{5}\\Psi_{M1}\\,,{}\\label{M2}\\\\\n\\overline\\Psi_{M1}\\gamma^\\mu \\Psi_{M2}&=&\n-\\overline\\Psi_{M2}\\gamma^\\mu\\Psi_{M1}\\,,{}\\label{M3}\\\\\n\\overline\\Psi_{M1}\\gamma^\\mu\\gamma\\ls{5} \\Psi_{M2}&=&\n\\overline\\Psi_{M2}\\gamma^\\mu\\gamma\\ls{5}\\Psi_{M1}\\,,{}\\label{M4}\\\\\n\\overline\\Psi_{M1}\\Sigma^{\\mu\\nu} \\Psi_{M2}&=&\n-\\overline\\Psi_{M2}\\Sigma^{\\mu\\nu}\\Psi_{M1}\\,,{}\\label{M5}\\\\\n\\overline\\Psi_{M1}\\Sigma^{\\mu\\nu}\\gamma\\ls{5} \\Psi_{M2}&=&\n-\\overline\\Psi_{M2}\\Sigma^{\\mu\\nu}\\gamma\\ls{5}\\Psi_{M1}\\,. {}\\label{M6}\n\\end{Eqnarray}\n\nIf\n$\\Psi_{M1}=\\Psi_{M2}\\equiv\\Psi_M$, then \\eqst{M1}{M6} yield\n\\begin{equation}\n\\overline\\Psi_{M}\\gamma^\\mu\n\\Psi_{M}=\\overline\\Psi_{M}\\Sigma^{\\mu\\nu}\n\\Psi_{M}=\\overline\\Psi_{M}\\Sigma^{\\mu\\nu}\\gamma\\ls{5} \\Psi_{M}=0\\,.\\\\\n\\end{equation}\nOne additional useful result for Majorana fermion fields is:\n\\begin{equation}\n\\overline\\Psi_{M1}\\gamma^\\mu P_L\\Psi_{M2}=\n-\\overline\\Psi_{M2}\\gamma^\\mu P_R\\Psi_{M1}\\,.\n\\end{equation}\n\n\\subsection{Feynman Rules for Dirac and Majorana fermions}\n\\label{sec:Feynman}\n\n\nThe application of four-component fermion\ntechniques in parity-violating theories is straightforward for\nprocesses involving Dirac fermions. However, the inclusion of\nMajorana fermions involves some subtleties that require elucidation.\nIn light of the widespread familiarity with four-component spinor\ntechniques, we shall develop four-component fermion Feynman rules\n that treat Dirac and Majorana fermions on equal \nfooting\\cite{Dreiner:2008tw,Gates:1987ay,Denner:1992me,Kleiss:2009hu}.\\footnote{\nFor a comprehensive set of \ntwo-component fermion Feynman rules, see Ref.~\\cite{Dreiner:2008tw}.}\n\n\nConsider first the Feynman rule for the four-component fermion\npropagator.\nVirtual Dirac fermion lines can either correspond to $\\Psi$ or\n$\\Psi^c$. Here, there is no ambiguity in the propagator Feynman rule,\nsince for free Dirac fermion fields,\n\\begin{equation}\n\\left\\langle 0\\right|T[\\Psi(x)\\overline{\\Psi}(y)]\n\\left|0\\right\\rangle=\n\\left\\langle 0\\right |T[\\Psi^c(x)\\overline{\\Psi^c}(y)]\n\\left|0\\right\\rangle\\,,\n\\end{equation}\nso that the Feynman rules for the propagator of a $\\Psi$ and $\\Psi^c$\nline, exhibited below, are identical.\nThe same rule also applies to a four-component Majorana fermion $\\Psi_M$.\n\\begin{center}\n\\begin{picture}(200,50)(-135,-16)\n\\thicklines\n\\LongArrow(-110,25)(-70,25)\n\\ArrowLine(-130,15)(-50,15)\n\\put(-90,30){$p$}\n\\put(20,10){$\\displaystyle\n {\\frac{i(\\slashchar{p}\\ominus m)}\n {p^2 \\oplus m^2 \\ominus i\\epsilon}}$}\n\\end{picture}\n\\end{center}\n\n\\vspace{-0.2in}\nConsider next a set of neutral Majorana fermions $\\Psi_{Mi}$ and\ncharged Dirac fermions $\\Psi_i$,\n\\begin{equation}\n\\Psi_{Mi} =\n\\begin{pmatrix}\n\\xi_i\n\\\\[4pt]\n\\xi^\\dagger_i\n\\end{pmatrix},\n\\qquad\n\\Psi_i =\n\\begin{pmatrix}\n\\chi_i \\\\[4pt]\n\\eta^\\dagger_i\n\\end{pmatrix},\n\\end{equation}\n interacting with a neutral scalar $\\phi$ or\nvector boson $A_\\mu$. The interaction Lagrangian in terms of two-component\nfermions is\n\\begin{Eqnarray} \\!\\!\\!\\!\\!\\!\\!\\! \\!\\!\\!\\!\\!\\!\\!\\!\n\\mathscr{L}_{\\rm int} &=& -\\ifmath{\\tfrac12}(\\lambda^{ij}\\xi_i\\xi_j+\\lambda_{ij}\n\\xi^{\\dagger\\,i}\\xi^{\\dagger\\,j})\\phi-(\\kappa^i{}_j\\chi_i\\eta^j+\\kappa_i{}^j\n\\chi^{\\dagger\\,i}\\eta^\\dagger_j)\\phi\\ {} \\nonumber \\\\\n&&\\, -G_i{}^j\\,\\xi^{\\dagger\\,i}\\overline{\\sigma}^\\mu\\xi_j A_\\mu\n-[(G_L)_i{}^j\\chi^{\\dagger\\,i}\\overline{\\sigma}^\\mu\\chi_j\n+(G_R)_i{}^j\\eta^{\\dagger\\,i}\\overline{\\sigma}^\\mu\\eta_j]A_\\mu\\,,{} \\label{lint1}\n\\end{Eqnarray}\nwhere $\\lambda$ is a complex symmetric matrix with \n$\\lambda^{ij}\\equiv\\lambda^*_{ij}$,\n$\\kappa$ is an arbitrary complex matrix with $\\kappa_i{}^j\\equiv (\\kappa^i{}_j)^*$,\nand $G$, $G_L$ and $G_R$\nare hermitian matrices.\nConverting to four-component spinor notation (see Problem 1), the resulting Feynman rules \nare shown below.\n\\clearpage\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{picture}(200,68)(40,0)\n\\DashLine(10,40)(60,40)5\n\\ArrowLine(60,40)(100,70)\n\\ArrowLine(100,10)(60,40)\n\\Text(30,30)[]{$\\scriptstyle\\phi$}\n\\Text(70,20)[]{$\\scriptstyle\\Psi_{Mj}$}\n\\Text(70,67)[]{$\\scriptstyle\\Psi_{Mi}$}\n\\Text(140,40)[l]{$-i(\\lambda^{ij}P_L+\\lambda_{ij} P_R)$}\n\\end{picture}\n\\end{center}\n\\vspace{0.2in}\n\n\\begin{center}\n\\begin{picture}(200,68)(40,0)\n\\Photon(60,40)(10,40){3}{5}\n\\ArrowLine(60,40)(100,70)\n\\ArrowLine(100,10)(60,40)\n\\Text(30,25)[]{$\\scriptstyle A_\\mu$}\n\\Text(70,20)[]{$\\scriptstyle\\Psi_{Mj}$}\n\\Text(70,67)[]{$\\scriptstyle\\Psi_{Mi}$}\n\\Text(140,40)[l]{$-i\\gamma_\\mu[G_i{}^j P_L-G_j{}^i P_R]$}\n\\end{picture}\n\\end{center}\n\\vspace{0.2in}\n\\begin{center}\n\\begin{picture}(200,68)(40,0)\n\\DashLine(10,40)(-40,40)5\n\\ArrowLine(10,40)(50,70)\n\\ArrowLine(50,10)(10,40)\n\\Text(-20,30)[]{$\\scriptstyle\\phi$}\n\\Text(20,20)[]{$\\scriptstyle\\Psi_j$}\n\\Text(20,67)[]{$\\scriptstyle\\Psi_i$}\n\\DashLine(160,40)(110,40)5\n\\ArrowLine(160,40)(200,70)\n\\ArrowLine(200,10)(160,40)\n\\Text(75,40)[]{or}\n\\Text(130,30)[]{$\\scriptstyle\\phi$}\n\\Text(170,15)[]{$\\scriptstyle\\Psi^{cj}$}\n\\Text(170,65)[]{$\\scriptstyle\\Psi^{ci}$}\n\\Text(240,40)[l]{$-i(\\kappa^i{}_j P_L+\\kappa_j{}^i P_R)$}\n\\end{picture}\n\\end{center}\n\\vspace{0.2in}\n\\begin{center}\n\\begin{picture}(200,68)(40,0)\n\\Photon(60,40)(10,40){3}{5}\n\\ArrowLine(60,40)(100,70)\n\\ArrowLine(100,10)(60,40)\n\\Text(30,25)[]{$\\scriptstyle A_\\mu$}\n\\Text(70,20)[]{$\\scriptstyle\\Psi_j$}\n\\Text(70,67)[]{$\\scriptstyle\\Psi_i$}\n\\Text(140,40)[l]{$-i\\gamma_\\mu[(G_L)_i{}^j P_L+(G_R)_i{}^j P_R]$}\n\\end{picture}\n\\end{center}\n\\vspace{0.2in}\n\\begin{center}\n\\begin{picture}(200,68)(40,0)\n\\Photon(60,40)(10,40){3}{5}\n\\ArrowLine(60,40)(100,70)\n\\ArrowLine(100,10)(60,40)\n\\Text(50,90)[]{or}\n\\Text(200,90)[]{or}\n\\Text(30,25)[]{$\\scriptstyle A_\\mu$}\n\\Text(70,15)[]{$\\scriptstyle\\Psi^{cj}$}\n\\Text(70,65)[]{$\\scriptstyle\\Psi^{ci}$}\n\\Text(140,40)[l]{$\\phantom{-} i\\gamma_\\mu[(G_L)_i{}^j P_L+(G_R)_i{}^j P_R]$}\n\\end{picture}\n\\end{center}\n\\end{figure}\n\nThe arrows on the Dirac fermion lines depict the flow of the\nconserved charge. A Majorana fermion is self-conjugate, so\nits arrow simply reflects the structure of $\\mathscr{L}_{\\rm int}$;\n{\\it i.e.}, $\\overline\\Psi_M$ [$\\Psi_M$] is represented by\nan arrow pointing out of [into] the vertex. The arrow directions\ndetermine the placement of the $u$ and $v$ spinors in an\ninvariant amplitude.\n\nFor vertices involving Dirac fermions, one has a choice of either\nusing the Dirac field or its charge conjugated field. The Feynman\nrules corresponding to these two choices are related, due to the\nfollowing identity, \n\\begin{equation} \\label{CC}\n\\overline\\Psi^c_i\\Gamma\\Psi^c_j=-\\Psi_i^T C^{-1}\\Gamma C\\overline\\Psi_j^T=\n\\overline\\Psi_j C\\Gamma^T\nC^{-1}\\Psi_i=\\eta^C\\ls{\\Gamma}\\overline\\Psi_j\\Gamma \\Psi_i\\,,\n\\end{equation}\nwhere we have used \\eq{ccgamma}. Note that the extra minus sign that\narises in the penultimate step above is due to the anticommutativity\nof the fermion fields.\n\n\nNext, consider the interaction of fermions with charged bosons $\\Phi$ and $W$ (assumed\nto have charge equal to that of $\\chi$ and $\\eta^\\dagger$). The corresponding interaction Lagrangian is given by:\n\\begin{Eqnarray}\n\\mathscr{L}_{\\rm int} &=&\n-\\Phi[(\\kappa_1)^i{}_j\\xi_i\\eta^j\n+(\\kappa_2)_{ij}\\xi^{\\dagger i} \\chi^{\\dagger j}] \n -\\Phi^\\dagger[(\\kappa_2)^{ij}\\xi_i\\chi_j\n+(\\kappa_1)_i{}^j\\xi^{\\dagger i}_i \\eta^{\\dagger}_j] \\nonumber\n\\\\\n&&\n\\oplus W_\\mu[(G_1)_j{}^i\\chi^{\\dagger j}\\overline{\\sigma}^\\mu\\xi_i\n-(G_2)_{ij}\\xi^{\\dagger i}\\overline{\\sigma}^\\mu \\eta^j] \\nonumber \\\\\n&& \\oplus W_\\mu^\\dagger[(G_1)^j{}_i\\xi^{\\dagger i}\\overline{\\sigma}^\\mu\\chi_j\n-(G_2)^{ij}\\eta^{\\dagger}_j\\overline{\\sigma}^\\mu\\xi_i]\\,,\\label{lint2}\n\\end{Eqnarray}\nwhere $\\kappa_1$, $\\kappa_2$, $G_1$ and $G_2$ \nare complex matrices. Converting to four-component spinor notation,\nthe corresponding Feynman rules are:\n\\begin{center}\n\\begin{picture}(200,78)(20,0)\n\\DashArrowLine(-60,40)(-10,40)5\n\\ArrowLine(-10,40)(30,70)\n\\ArrowLine(30,10)(-10,40)\n\\Text(-50,30)[]{$\\scriptstyle\\Phi$}\n\\Text(-10,18)[]{$\\scriptstyle\\Psi_{Mi}$}\n\\Text(-10,67)[]{$\\scriptstyle\\Psi_{j}$}\n\\Text(45,40)[l]{or}\n\\DashArrowLine(80,40)(130,40)5\n\\ArrowLine(170,70)(130,40)\n\\ArrowLine(130,40)(170,10)\n\\Text(100,30)[]{$\\scriptstyle\\Phi$}\n\\Text(140,18)[]{$\\scriptstyle\\Psi_{Mi}$}\n\\Text(140,67)[]{$\\scriptstyle\\Psi^{cj}$}\n\\Text(200,40)[l]{$-i(\\kappa_1{}^i{}_j P_L+\\kappa_{2ij} P_R)$}\n\\end{picture}\n\\end{center}\n\\vspace{0.04in}\n\\begin{center}\n\\begin{picture}(200,68)(20,0)\n\\DashArrowLine(-10,40)(-60,40)5\n\\ArrowLine(30,70)(-10,40)\n\\ArrowLine(-10,40)(30,10)\n\\Text(-40,30)[]{$\\scriptstyle\\Phi$}\n\\Text(0,18)[]{$\\scriptstyle\\Psi_{Mi}$}\n\\Text(0,67)[]{$\\scriptstyle\\Psi_{j}$}\n\\Text(45,40)[l]{or}\n\\DashArrowLine(130,40)(80,40)5\n\\ArrowLine(130,40)(170,70)\n\\ArrowLine(170,10)(130,40)\n\\Text(100,30)[]{$\\scriptstyle\\Phi$}\n\\Text(140,18)[]{$\\scriptstyle\\Psi_{Mi}$}\n\\Text(140,67)[]{$\\scriptstyle\\Psi^{cj}$}\n\\Text(200,40)[l]{$-i(\\kappa_2{}^{ij}P_L+\\kappa_{1i}{}^j P_R)$}\n\\end{picture}\n\\end{center}\n\\vspace{0.04in}\n\\begin{center}\n\\begin{picture}(200,78)(20,0)\n\\Photon(-20,40)(30,40){3}{5}\n\\ArrowLine(30,40)(70,70)\n\\ArrowLine(70,10)(30,40)\n\\ArrowLine(5.05,40)(4.95,40)\n\\Text(0,30)[]{$\\scriptstyle W$}\n\\Text(40,18)[]{$\\scriptstyle\\Psi_{Mj}$}\n\\Text(40,67)[]{$\\scriptstyle\\Psi_{i}$}\n\\Text(160,40)[l]{$-i\\gamma^\\mu(G_{1i}{}^jP_L-G_{2ji} P_R)$}\n\\end{picture}\n\\end{center}\n\\vspace{0.3in}\n\\begin{center}\n\\begin{picture}(200,68)(20,0)\n\\Photon(-20,40)(30,40){3}{5}\n\\ArrowLine(70,70)(30,40)\n\\ArrowLine(30,40)(70,10)\n\\ArrowLine(5.05,40)(4.95,40)\n\\Text(0,30)[]{$\\scriptstyle W$}\n\\Text(40,18)[]{$\\scriptstyle\\Psi_{Mj}$}\n\\Text(40,67)[]{$\\scriptstyle\\Psi^{ci}$}\n\\Text(170,40)[l]{$i\\gamma^\\mu(G_{2ji} P_R -G_{1i}{}^j P_L)$}\n\\Text(20,95)[]{or}\n\\Text(210,95)[]{or}\n\\end{picture}\n\\end{center}\n\\vspace{0.3in}\n\\begin{center}\n\\begin{picture}(200,68)(20,0)\n\\Photon(30,40)(-20,40){3}{5}\n\\ArrowLine(70,70)(30,40)\n\\ArrowLine(30,40)(70,10)\n\\ArrowLine(4.95,40)(5.05,40)\n\\Text(0,30)[]{$\\scriptstyle W$}\n\\Text(40,18)[]{$\\scriptstyle\\Psi_{Mj}$}\n\\Text(40,67)[]{$\\scriptstyle\\Psi_{i}$}\n\\Text(160,40)[l]{$-i\\gamma^\\mu(G_1{}^i{}_j P_L-G_{2}{}^{ji} P_R)$}\n\\end{picture}\n\\end{center}\n\\vspace{0.3in}\n\\begin{center}\n\\begin{picture}(200,68)(20,0)\n\\Photon(30,40)(-20,40){3}{5}\n\\ArrowLine(30,40)(70,70)\n\\ArrowLine(70,10)(30,40)\n\\ArrowLine(4.95,40)(5.05,40)\n\\Text(20,95)[]{or}\n\\Text(210,95)[]{or}\n\\Text(0,30)[]{$\\scriptstyle W$}\n\\Text(40,18)[]{$\\scriptstyle\\Psi_{Mj}$}\n\\Text(40,67)[]{$\\scriptstyle\\Psi^{ci}$}\n\\Text(170,40)[l]{$i\\gamma^\\mu(G_{2}{}^{ji} P_R -G_{1}{}^i{}_j P_L)$}\n\\end{picture}\n\\end{center}\n\nWhen the interaction Lagrangians given in \\eqs{lint1}{lint2} are\nconverted to four-component spinor notation (see Problems 1 and 2 at\nthe end of this section), there is an equivalent form in which\n$\\mathscr{L}_{\\rm int}$ is written in terms of charge-conjugated Dirac\nfour-component fields [after using \\eq{CC}]. Thus, the Feynman rules involving\nDirac fermions can take two possible forms, as\nshown above.\nAs previously noted, the direction of an\narrow on a Dirac fermion line indicates the\ndirection of the fermion charge flow (whereas the arrow on\nthe Majorana fermion line is unconnected to charge flow).\nHowever, we are free to choose either a\n$\\Psi$ or $\\Psi^c$ line to represent a Dirac fermion at any place in a\ngiven Feynman graph.\\footnote{Since the charge of $\\Psi^c$ is opposite\nin sign to the charge\nof $\\Psi$, the corresponding arrow directions of the $\\Psi$ and $\\Psi^c$\nlines must point in opposite directions.}\nFor any decay or scattering process,\na suitable choice of either the $\\Psi$-rule or the $\\Psi^c$-rule\nat each vertex (the choice can be different at different vertices)\nwill guarantee that\nthe arrow directions on fermion lines flow continuously through\nthe Feynman diagram. Then, to evaluate an invariant amplitude,\none should traverse \\textit{any} continuous fermion\nline (either $\\Psi$ or $\\Psi^c$)\nby moving antiparallel to the direction of the fermion arrows.\n\n\n\nFor a given process, there may be a number of distinct\nchoices for the arrow directions on the Majorana fermion lines,\nwhich may depend on whether one represents a given Dirac fermion by\n$\\Psi$ or $\\Psi^c$.\nHowever, different choices do {\\it not} lead to independent Feynman\ndiagrams.\nWhen computing an invariant amplitude, one\nfirst writes down the relevant\nFeynman diagrams with no arrows on any Majorana\nfermion line. The number of distinct graphs contributing to the\nprocess is then determined. Finally, one makes some choice for\nhow to distribute the arrows on the Majorana fermion lines\nand how to label Dirac fermion lines (either as the field $\\Psi$ or its\ncharge conjugate $\\Psi^c$) in a manner consistent\nwith the Feynman rules for the vertices previously given.\nThe end result for the invariant\namplitude (apart from an overall unobservable phase)\ndoes not depend on the choices\nmade for the direction of the fermion arrows.\n\nUsing the above procedure, the Feynman rules for the\nexternal fermion wave functions are the same for Dirac and Majorana fermions:\n\\begin{itemlist}\n\\item\n$u(\\boldsymbol{\\vec p},s)$: incoming $\\Psi$ [or $\\Psi^c$]\nwith momentum $\\boldsymbol{\\vec p}$ parallel to the arrow direction,\n\\item\n$\\bar u(\\boldsymbol{\\vec p},s)$: outgoing $\\Psi$ [or $\\Psi^c$] with\nmomentum $\\boldsymbol{\\vec p}$ parallel to the arrow direction,\n\\item\n$v(\\boldsymbol{\\vec p},s)$: outgoing $\\Psi$ [or $\\Psi^c$] with\nmomentum $\\boldsymbol{\\vec p}$ anti-parallel to the arrow direction,\n\\item\n$\\bar v(\\boldsymbol{\\vec p},s)$: incoming $\\Psi$ [or $\\Psi^c$] with\nmomentum $\\boldsymbol{\\vec p}$ anti-parallel to the arrow direction.\n\\end{itemlist}\n\n\n\nWe now consider the application of the Feynman rules presented above\nto some $2\\to 2$ scattering processes involving a Majorana fermion\neither as an external state or as an internal line.\n\n\\begin{example}[$\\boldsymbol{\\Psi(p_1)\\Psi(p_2)\\to\\Phi(k_1)\\Phi(k_2)}$ via $\\boldsymbol{\\Psi_M}$-exchange]\n\nHere, $\\Phi$ is a charged scalar.\nThe contributing Feynman graphs are:\n\n\\vspace{12pt}\n\n\\begin{picture}(450,85)(125,-25)\n\\thicklines\n\\ArrowLine(185,-15)(125,-15)\n\\DashArrowLine(185,-15)(245,-15){5}\n\\ArrowLine(125,45)(185,45)\n\\DashArrowLine(185,45)(245,45){5}\n\\ArrowLine(185,45)(185,-15)\n\\put(165,12){$\\Psi_M$}\n\\put(130,50){$\\Psi$}\n\\put(130,-25){$\\Psi^c$}\n\\ArrowLine(360,-15)(300,-15)\n\\DashLine(360,-15)(390,15){5}\n\\DashArrowLine(390,15)(420,45){5}\n\\ArrowLine(300,45)(360,45)\n\\DashLine(360,45)(390,15){5}\n\\DashArrowLine(390,15)(420,-15){5}\n\\ArrowLine(360,45)(360,-15)\n\\put(340,12){$\\Psi_M$}\n\\put(305,50){$\\Psi$}\n\\put(305,-25){$\\Psi^c$}\n\\end{picture}\n\n\\vspace{12pt}\n\n\\noindent\nFollowing the arrows on the fermion lines in reverse,\nthe invariant amplitude is given by,\n\\begin{Eqnarray} \\label{mex2}\ni\\mathcal{M}&=&\n(-i)^2\\bar v(\\boldsymbol{\\vec p}_2,s_2)(\\kappa_1 P_L+\\kappa_2^* P_R)\n\\left[\\frac{i(\\slashchar{p_1}-\\slashchar{k_1}+m)}{t-m^2} \n+\\frac{i(\\slashchar{k_1}-\\slashchar{p_2}+m)}{u-m^2}\\right]\\nonumber \\\\\n&&\\qquad \\times (\\kappa_1 P_L+\\kappa_2^* P_R) u(\\boldsymbol{\\vec p}_1,s_1)\\,,{}\n\\end{Eqnarray}\nwhere $t\\equiv (p_1-k_1)^2$, $u\\equiv (p_2-k_1)^2$ and\n$m$ is the Majorana fermion mass. The sign of each diagram is\ndetermined by the relative permutation of spinor wave functions\nappearing in the amplitude (the overall sign of the amplitude is\nunphysical).\nIn the present example, in\nboth terms appearing in \\eq{mex2}, the spinor wave functions appear in \nthe same order (first $\\boldsymbol{\\vec p}_2$ and then\n$\\boldsymbol{\\vec p}_1$),\nimplying a relative plus sign between the two terms. \n\n\nOne can check that \\textrm{$i\\mathcal{M}$} is antisymmetric under\ninterchange of the two initial electrons. \nThis is most easily verified by\ntaking the transpose of the invariant amplitude (the latter is a\ncomplex number whose value is not changed by transposition).\nIt is convenient to adopt the convention in which the (commuting) $u$\nand $v$ spinor wave functions are related via,\n\\begin{Eqnarray}\nv(\\boldsymbol{\\vec p},s) &=& C\\bar u(\\boldsymbol{\\vec p},s)^{{\\mathsf T}}\n\\,,\\qquad\\qquad\\quad\nu(\\boldsymbol{\\vec p},s) = C\\bar v(\\boldsymbol{\\vec p},s)^{{\\mathsf T}}\\,,\n\\label{uvspinrelation1} \\\\\n\\bar v(\\boldsymbol{\\vec p},s) &=& -u(\\boldsymbol{\\vec p},s)^{{\\mathsf T}}C^{-1}\n\\,,\\qquad\\quad\\,\n\\bar u(\\boldsymbol{\\vec p},s) = -v(\\boldsymbol{\\vec p},s)^{{\\mathsf T}}C^{-1}\\,.\n\\label{uvspinrelation2}\n\\end{Eqnarray}\nwhere $C$ is the charge conjugation matrix.\nUsing \\eqs{uvspinrelation1} {uvspinrelation2}, \nthe transposed amplitude can be simplified by employing the relation,\n\\begin{align}\n\\bar v(\\boldsymbol{\\vec p}_2,s_2)\\Gamma u(\\boldsymbol{\\vec p}_1,s_1)=\n-\\eta^C\\ls{\\Gamma}\n\\bar v(\\boldsymbol{\\vec p}_1,s_1)\\Gamma u(\\boldsymbol{\\vec\n p}_2,s_2)\\,, \\label{vgamu}\n\\end{align}\nwhich is a consequence of \\eq{ccgamma}.\n\n\n\\end{example}\n\n\\begin{example}[$\\boldsymbol{\\Psi(p_1)\\Psi^c(p_2)\\!\\to\\!\\Psi_M(p_3)\\Psi_M(p_4)}$~\\!via\\! charged\\! $\\boldsymbol{\\Phi}$-exchange]\n\nIn addition to a possible $s$-channel annihilation graph,\nthe contributing Feynman graphs can be represented by \neither diagram set (i) or diagram set (ii) shown below, where each set\ncontains a $t$-channel and $u$-channel graph, respectively.\n\\clearpage\n\n\\noindent Diagram set (i):\n\n\\begin{picture}(450,85)(130,-10)\n\\thicklines\n\\ArrowLine(125,-15)(185,-15)\n\\ArrowLine(185,-15)(245,-15)\n\\ArrowLine(125,45)(185,45)\n\\ArrowLine(185,45)(245,45)\n\\DashArrowLine(185,45)(185,-15){5}\n\\put(220,50){$\\Psi_M$}\n\\put(220,-25){$\\Psi_M$}\n\\put(130,50){$\\Psi$}\n\\put(130,-25){$\\Psi^c$}\n\\ArrowLine(290,-15)(350,-15)\n\\Line(380,15)(350,-15)\n\\ArrowLine(380,15)(410,45)\n\\ArrowLine(290,45)(350,45)\n\\Line(350,45)(380,15)\n\\ArrowLine(380,15)(410,-15)\n\\DashArrowLine(350,45)(350,-15){5}\n\\put(410,50){$\\Psi_M$}\n\\put(410,-25){$\\Psi_M$}\n\\put(295,50){$\\Psi$}\n\\put(295,-25){$\\Psi^c$}\n\\end{picture}\n\n\\vskip 0.5in\n\\noindent\nDiagram set (ii):\n\n\\begin{picture}(450,85)(130,-10)\n\\thicklines\n\\ArrowLine(185,-15)(125,-15)\n\\ArrowLine(245,-15)(185,-15)\n\\ArrowLine(125,45)(185,45)\n\\ArrowLine(185,45)(245,45)\n\\DashArrowLine(185,45)(185,-15){5}\n\\put(220,50){$\\Psi_M$}\n\\put(220,-25){$\\Psi_M$}\n\\put(130,50){$\\Psi$}\n\\put(130,-25){$\\Psi$}\n\\ArrowLine(350,-15)(290,-15)\n\\Line(380,15)(350,-15)\n\\ArrowLine(410,45)(380,15)\n\\ArrowLine(290,45)(350,45)\n\\Line(350,45)(380,15)\n\\ArrowLine(380,15)(410,-15)\n\\DashArrowLine(350,45)(350,-15){5}\n\\put(410,50){$\\Psi_M$}\n\\put(410,-25){$\\Psi_M$}\n\\put(295,50){$\\Psi$}\n\\put(295,-25){$\\Psi$}\n\\end{picture}\n\\vskip 0.5in\n\nThe amplitude is evaluated by following the arrows on the fermion\nlines in reverse. Either diagram set (i) or set (ii) may be chosen to evaluate the invariant amplitude.\nWe again employ \\eq{ccgamma} to derive the relation,\n\\begin{equation} \\label{vgamv}\n\\bar v(\\boldsymbol{\\vec p}_2,s_2)\\Gamma v(\\boldsymbol{\\vec p}_4,s_4)=\n-\\eta^C\\ls{\\Gamma}\n\\bar u(\\boldsymbol{\\vec p}_4,s_4)\\Gamma u(\\boldsymbol{\\vec p}_2,s_2)\\,,\n\\end{equation}\nwhich can be used in comparing the invariant amplitude obtained by\nusing diagram sets (i) and (ii). One can check that \nthe invariant amplitudes resulting from diagram sets\n(i) and (ii) differ by an overall minus sign, which is unphysical.\nThe overall minus sign arises due to the fact that the corresponding\norder of the spinor wave functions\ndiffers by an odd permutation [e.g.,\nfor the $t$-channel graphs, compare 3142 and\n3124 for (i) and (ii) respectively]. For the same\nreason, there is a relative minus sign between the $t$-channel and\n$u$-channel graphs for either diagram set [e.g., compare 3142\nand 4132 in diagram set(i)].\n\n\nIf $s$-channel annihilation contributes, its contribution to the\ninvariant amplitude is easily obtained.\nRelative to the $t$-channel graph of diagram set\n(ii) above, the $s$-channel graph shown below\ncomes with an extra minus sign (since 2134 is odd with respect to 3124).\n\\end{example}\n\n\\begin{picture}(450,95)(28,-40)\n\\SetScale{0.8}\n\\thicklines\n\\ArrowLine(185,15)(125,-25)\n\\ArrowLine(305,-25)(245,15)\n\\ArrowLine(125,55)(185,15)\n\\ArrowLine(245,15)(305,55)\n\\DashLine(185,15)(245,15){5}\n\\put(220,45){$\\Psi_M$}\n\\put(220,-25){$\\Psi_M$}\n\\put(110,45){$\\Psi$}\n\\put(110,-25){$\\Psi$}\n\\end{picture}\n\n\n\\vskip -0.05in\n\nIn the computation of the unpolarized cross-section, non-standard spin\nprojection operators can arise in the evaluation of the interference\nterms (see Appendix D of Reference \\cite{Haber:1984rc}), such as\n\\begin{Eqnarray}\n \\sum_s u({\\boldsymbol{\\vec p}},s) v^T({\\boldsymbol{\\vec p}},s)\n = (\\slashchar{p} + m)C^T\\,, \\qquad\n\\sum_s \\bar{u}^T({\\boldsymbol{\\vec p}},s)\n\\bar{v}({\\boldsymbol{\\vec p}},s)\n= C^{-1}(\\slashchar{p} - m)\\,,\\nonumber \n\\end{Eqnarray}\nwhich requires additional manipulation of the charge conjugation\nmatrix~$C$. However, these non-standard spin projection operators can be\navoided by judicious use of spinor wave function product relations of the kind\nobtained in \\eqs{vgamu}{vgamv}.\n\n\n\n\n\n\\subsection{Problems}\n\\begin{problem}\nConvert the interaction Lagrangian given by \\eq{lint1} to\nfour-component spinor notation.\nShow that the end result is \n\\begin{Eqnarray}\n\\mathscr{L}_{\\rm int}&=& -\\ifmath{\\tfrac12}(\\lambda^{ij}\\overline{\\Psi}_{Mi} P_L\\Psi_{Mj}\n+\\lambda_{ij}\\overline{\\Psi}_{M}\\llsup{i}P_R\\Psi_{M}\\llsup{j})\\phi\n-\\overline{\\Psi}\\llsup{\\,j}(\\kappa^i{}_j P_L +\\kappa_i{}^j P_R)\\Psi_{i}\\phi \\nonumber \\\\\n&& \\oplus\\ifmath{\\tfrac12}\\overline{\\Psi}_{Mi}\\gamma^\\mu\\left[(G^a)_i{}^j P_L-(G^a)_j{}^i P_R\\right]\n\\Psi_{Mj} \\nonumber \\\\\n&& \n\\oplus \\left[(G_L^a)_i{}^j\\overline{\\Psi}\\llsup{\\,i}\\gamma^\\mu P_L\\Psi_{j}\n+(G_R^a)_i{}^j\\overline{\\Psi}\\llsup{\\,i}\\gamma^\\mu\nP_R\\Psi_{j}\\right]A^a_\\mu\\,, \\label{lint14}\n\\end{Eqnarray}\nwhere the $\\Psi_{Mj}$ are a set of (neutral) Majorana\nfour-component fermions and the $\\Psi_{j}$ are a set of Dirac four-component fermions.\n\\end{problem}\n\n\\begin{problem}\nConvert the interaction Lagrangian given by \\eq{lint2} to\nfour-component spinor notation.\nShow that the end result is \n\\begin{Eqnarray} \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\mathscr{L}_{\\rm int} &=&\n-\\left[(\\kappa_1)^i{}_j\\overline{\\Psi}\\llsup{\\,j}P_L\\Psi_{Mi}\n+(\\kappa_2)_{ij}\\overline{\\Psi}\\llsup{j}P_R\\Psi_{M}^i\\right]\\Phi\n\\nonumber \\\\\n&&\n\\oplus \\left[(G_1)_j{}^i\\overline{\\Psi}\\llsup{\\,j}\\gamma^\\mu P_L\\Psi_{Mi}\n+(G_2)_{ij}\\overline{\\Psi}\\llsup{\\,j}\\gamma^\\mu P_R\\Psi_{M}^i\\right]W_\\mu\n+{\\rm h.c.} \\label{lintc4}\n\\end{Eqnarray}\n\\end{problem}\n\\begin{problem}\nDerive \\eq{vgamu}. Then,\nverify that the invariant amplitude given by \\eq{mex2} is\nantisymmetric under the interchange of the two initial electrons.\n\\end{problem}\n\n\\begin{problem}\nDerive \\eq{vgamv}. Then, verify that the invariant amplitude \nfor the scattering process considered in Example 3 obtained\nfrom diagram sets (i) and (ii), respectively, differ by an overall\nminus sign.\n\\end{problem}\n\n\\section{Supersymmetric gauge theories}\n\\label{sec:gaugetheories}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\setcounter{equation}{0}\n\nIn this section, we discuss the supersymmetric extension of \ngauge theories. We begin with the vector superfield $V$, which contains\nthe gauge fields as well as their supersymmetric partners, the\ngauginos. We discuss the behavior of $V$ under a gauge\ntransformation, and the gauge-invariant interaction terms that couple\nthe vector superfield with one or more \nchiral superfields. Both abelian and non-abelian gauge groups are\ntreated. Finally, we construct the SUSY Lagrangians corresponding to QED and a\nnon-Abelian SUSY Yang-Mill theory coupled to supersymmetric matter.\n\n\\subsection{Vector superfields}\n\nImposing a reality condition on a complex superfield (which is a\ncovariant constraint with respect to SUSY transformations), we obtain the so-called real vector superfield,\n\\begin{align}\nV(x,\\theta,\\theta^\\dagger)=V^\\dagger(x,\\theta,\\theta^\\dagger)\\,,\n\\end{align}\nwhich will be employed in constructing supersymmetric gauge theories.\nExpanding in $\\theta$ and $\\theta^\\dagger$,\n\\begin{Eqnarray}\nV&=& \\ C + i\\theta\\chi-i\\theta^\\dagger\\chi^\\dagger +\\ifmath{\\tfrac12} i\n\\theta\\theta(M+iN)-\\ifmath{\\tfrac12} i \\theta^\\dagger \\theta^\\dagger (M-iN)\n+\\theta\\sigma^\\mu\\theta^\\dagger V_\\mu \\nonumber \\\\\n&&+\ni(\\theta\\theta)\\theta^\\dagger\\bigl(\\lambda^\\dagger-\\ifmath{\\tfrac12}\ni\\,\\overline{\\sigma}^\\mu\\partial_\\mu\\chi\\bigr)-i(\\theta^\\dagger\n\\theta^\\dagger)\\theta(\\lambda-\\ifmath{\\tfrac12}\ni\\sigma^\\mu\\partial_\\mu\\chi^\\dagger\\bigr) \\nonumber \\\\\n&& +\\ifmath{\\tfrac12}(\\theta\\theta)(\\theta^\\dagger \n\\theta^\\dagger)\\bigl(D-\\ifmath{\\tfrac12}\\Box C\\bigr)\\,,\n\\label{VectorSF}\n\\end{Eqnarray}\nwhere $C$, $M$, $N$, $D$ and $V_\\mu$ are real bosonic fields,\nand $\\chi$ and $\\lambda$ are two-component fermion fields. The\nvarious factors of $i$ and $\\ifmath{\\tfrac12}$ are conventional, and the\nparticular linear combination of fields chosen as coefficients of\n$(\\theta\\theta)\\theta^\\dagger$, $(\\theta^\\dagger\\thetabar)\\theta$ and\n$(\\theta\\theta)(\\theta^\\dagger\\thetabar)$ are convenient for later\npurposes [cf.~footnote~\\ref{fn38}].\n \nNote that the superfield $V$ is dimensionless, in which case it\nfollows that the dimensions of the component fields are $[V_\\mu]=1$ and\n$[\\lambda]=\\tfrac32$, as expected, whereas $[C]=[D]=0$, and $[\\chi]=\\ifmath{\\tfrac12}$\nafter making use of the dimensions of the Grassmann coordinates,\n$[\\theta]=[\\theta^\\dagger]=-\\ifmath{\\tfrac12}$.\n\nThe real vector field $V_\\mu$ is a candidate for a gauge\nboson of an abelian U(1) gauge theory. The corresponding field strength tensor is given by\n\\begin{align}\nF_{\\mu\\nu}=\\partial_\\mu V_\\nu-\\partial_\\nu V_\\mu\\,.\n\\end{align}\nIndeed, this can be shown to be one of the components of the \\textit{field strength superfield}, which is defined by\n\\begin{align}\n\\mathcal{W}_\\alpha=-\\tfrac{1}{4} {\\overline{D}}^{2} D_\\alpha V\\,.\\label{Walpha}\n\\end{align}\nNote that $\\overline{D}_{\\dot\\beta}\\mathcal{W}_\\alpha=0$, so that $\\mathcal{W}_\\alpha$ is a spinor chiral superfield. Evaluating the above expression, and expressing it in the chiral representation,\n\\begin{align}\n\\mathcal{W}_\\alpha(y,\\theta,\\theta^\\dagger)=\n-i\\lambda_\\alpha + \\theta_\\alpha D \n-\\ifmath{\\tfrac12} i (\\sigma^\\mu\\overline{\\sigma}^\\nu\\theta)_\\alpha F_{\\mu\\nu}\n-\\theta\\theta (\\sigma^\\mu\\partial_\\mu \\lambda^\\dagger)_\\alpha ,\\label{Wdef}\n\\end{align}\nwhere $y\\equiv x-i\\theta \\sigma^\\mu\\theta^\\dagger$. The fermionic partner\nof the gauge boson, called the gaugino, is represented by the\ntwo-component spinor field $\\lambda$. Remarkably, the fields $C$,\n$M$, $N$ and $\\chi$ that are coefficients in the Taylor expansion of the vector\nsuperfield $V$ do not appear in \\eq{Wdef}. The reason for this will\nbecome apparent in Section~\\ref{sec:gauge}.\n\n\nOne can work out the SUSY transformation laws of the fields, $\\lambda$,\n$F_{\\mu\\nu}$ and~$D$,\nby matching component fields on both sides of the following equation,\n \\begin{align}\n \\delta_{\\xi} \\mathcal{W}_\\alpha=-i(\\xi \\widehat{Q}+\\xi^\\dagger \\widehat{Q}^\\dagger)\\mathcal{W}_\\alpha.\n \\end{align}\n The end result is\n\\begin{align}\n \\delta_{\\xi}\\lambda_\\alpha&= i\\xi_\\alpha D+\\ifmath{\\tfrac12}(\\sigma^\\mu\\overline{\\sigma}^\\nu)_\\alpha{}^\\beta\\xi_\\beta F_{\\mu\\nu}\\,, \\\\\n \\delta_{\\xi} F_{\\mu\\nu}&=i\\partial_\\mu(\\xi\\sigma_\\nu\\lambda^\\dagger-\\lambda\\sigma_\\nu\\xi^\\dagger)\n -i\\partial_\\nu(\\xi\\sigma_\\mu\\lambda^\\dagger-\\lambda\\sigma_\\mu\\xi^\\dagger)\\,, \\\\\n \\delta_{\\xi} D&=\\partial_\\mu(\\xi\\sigma^\\mu\\lambda^\\dagger+\\lambda\\sigma^\\mu\\xi^\\dagger)\\,.\\label{delD}\n\\end{align}\nNote that the mass dimension of the $D$-term is given by $[D]=2$. Hence, \ndimensional analysis implies that $\\delta_\\xi D$ must be a total\nderivative, which is confirmed in \\eq{delD}.\nFrom the above transformation laws, we conclude that $(\\lambda\\,,\\,\\lambda^\\dagger\\,,\\,F_{\\mu\\nu}\\,,\\,D)$ forms an irreducible supermultiplet (corresponding to superhelicity $1$).\n\nTo obtain the Lagrangian for the SUSY U(1) gauge theory, note that\n\\begin{align}\n\\tfrac14[\\mathcal{W}^\\alpha \\mathcal{W}_\\alpha]_F+{\\rm h.c.}&=\\ifmath{\\tfrac12} i(\\lambda\\sigma^\\mu\\partial_\\mu\\lambda^\\dagger+\\lambda^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\lambda)+\\ifmath{\\tfrac12} D^2-\\tfrac14 F_{\\mu\\nu}F^{\\mu\\nu} \\nonumber \\\\[5pt]\n&=i\\lambda^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\lambda+\\ifmath{\\tfrac12} D^2-\\tfrac14\n F_{\\mu\\nu}F^{\\mu\\nu}+\\text{total derivative}. \\label{WF}\n\\end{align}\nThis is the kinetic energy term for a U(1) gauge field $V_\\mu$ and its\ngaugino superpartner $\\lambda$. Both the gauge boson and gaugino are\nmassless. The real scalar field $D$ is not dynamical; it is an\nauxiliary field.\n\nThe action corresponding to the Lagrangian of\n\\eq{WF} can be written as an integral over half of superspace. In\nparticular, \\eq{d2theta} yields,\n\\begin{align}\n\\mathscr{L}=\\tfrac14\\int d^2\\theta\\, \\mathcal{W}^\\alpha \\mathcal{W}_\\alpha+{\\rm h.c.}\n\\end{align}\nOne can show that $[\\mathcal{W}^\\alpha \\mathcal{W}_\\alpha]_F$ and its hermitian\nconjugate term differ only by a total derivative. Hence, both terms\ncontribute equally to the action, which is given by\n\\begin{align}\nS=\\tfrac12\\int d^4 x\\,d^2\\theta\\, \\mathcal{W}^\\alpha \\mathcal{W}_\\alpha\\,.\n\\end{align}\nIt is sometimes convenient to turn this integral into an integration\nover the full superspace. Using a trick analogous to the one employed\nin \\eq{2to4}, we end up with,\n\\begin{align}\nS=\\tfrac12\\int d^4 x\\,d^2\\theta\\, \\left(-\\tfrac14 \\overline{D}\\lsup{2}\\right)(D^\\alpha V) \\mathcal{W}_\\alpha\n=\\tfrac12\\int d^4 x\\,d^2\\theta\\,d^2\\theta^\\dagger(D^\\alpha V) \\mathcal{W}_\\alpha\n\\,,\n\\end{align}\nafter using \\eq{Walpha} to rewrite one factor of $\\mathcal{W}^\\alpha$ in\nterms of $V$.\n\nIt is instructive to count the degrees of freedom in the irreducible\nsupermultiplet, $(\\lambda\\,,\\,\\lambda^\\dagger\\,,\\,F_{\\mu\\nu}\\,,\\,D)$. \nOn-shell, there are two real fermionic degrees\nof freedom \nassociated with the massless gaugino, after imposing the\nLagrange field equations,\\footnote{Starting with two complex (or\n equivalently four real) degrees\n of freedom for the two-component gaugino field $\\lambda$,\n \\eq{gauginoDirac} relates the spinor components $\\lambda_1$ and $\\lambda_2$, thereby\n reducing the number of real degrees of freedom from four to two.}\n\\begin{equation}\ni\\overline{\\sigma}\\lsup{\\mu\\alpha\\dot\\beta}\\partial_\\mu\\lambda_\\beta=0\\,.\\label{gauginoDirac}\n\\end{equation}\nThis matches the two real bosonic degrees of freedom corresponding\nto the two transverse polarizations of the massless gauge boson.\n\nTo count\n the off-shell bosonic degrees of freedom,\none must take into account the Bianchi identity,\\footnote{Although it\n appears that the Bianchi identity yields four constraints, since the\n spacetime index $\\mu$ is a free index, in fact only\n three constraints are independent. This is because one of the four\n constraints is redundant due to the identity,\n$\\epsilon^{\\mu\\nu\\rho\\sigma}\\partial_\\mu\\partial_\\nu F_{\\rho\\sigma}=0$,\nwhich is automatically satisfied as a result of the antisymmetry of the Levi-Civita tensor.\nPhysically, the Bianchi identity implies that the three components of the\n electric field vector determine the three components of the magnetic field vector.}\n\\begin{equation}\n\\epsilon^{\\mu\\nu\\rho\\sigma}\\partial_\\nu F_{\\rho\\sigma}=0\\,,\\label{bianchi}\n\\end{equation}\nwhich is satisfied independently of the field equations. This\nidentity reduces the number of real degrees of freedom in the real\nantisymmetric tensor $F_{\\mu\\nu}$ from\nsix to three. Adding in the one real degree of freedom associated\nwith $D$, we end up with a total of four real bosonic degrees of freedom,\nwhich matches the four real off-shell fermionic degrees of freedom corresponding to $\\lambda$ and~$\\lambda^\\dagger$. \n\n\n\n\n\\subsection{Gauge invariance}\n\\label{sec:gauge}\nThe vector superfield $V$ contains the familiar gauge field\n$V_\\mu$. But it also includes other component fields $C$, $\\chi$, $M$\nand $N$, whose meaning is less obvious. As we will see, these latter fields\nturn out to be gauge artifacts. Thus, we must examine how gauge\ntransformations of the gauge field theory get promoted to gauge transformations of the\nvector superfield $V$.\n\nLet $\\Lambda(x,\\theta,\\theta^\\dagger)$ be a chiral superfield (\\textit{i.e.}, $\\overline{D}_{\\dot\\alpha}\\Lambda=0$) and let \n$\\Lambda^\\dagger(x,\\theta,\\theta^\\dagger)$ be the corresponding antichiral superfield. Consider the transformation,\n\\begin{align}\nV\\to V+i(\\Lambda-\\Lambda^\\dagger)\\,.\n\\label{eq:Vgaugetransform}\n\\end{align}\nWe assert that \\eq{eq:Vgaugetransform} is a supersymmetric\ngeneralization of the gauge transformation of an abelian gauge theory,\nhenceforth called a super gauge transformation.\n\nWith the help of \\eq{eq:Dcomms}, it is straightforward to show that\nthe field strength superfield, $\\mathcal{W}_\\alpha$, is invariant under a super\ngauge transformation.\nMoreover, if the Taylor series of $\\Lambda(x,\\theta,\\theta^\\dagger)$ is\nwritten as\\footnote{In contrast to the chiral superfield $\\Phi$ in\n \\eq{eq:chiralSF} whose mass dimension is 1, the chiral superfield $\\Lambda$ is dimensionless,\n as required for consistency in light of \\eq{eq:Vgaugetransform}.}\n\\begin{align}\n\\begin{split}\n\\Lambda(x,\\theta,\\theta^\\dagger)=&\n\\widetilde{A}(x) + \\sqrt{2}\\,\\theta \\widetilde{\\psi}(x) + \\theta\\theta \\widetilde{F}(x)-i \\theta\\sigma^\\mu\\theta^\\dagger \\partial_\\mu \\widetilde{A}(x) \\\\\n& - \\frac{i}{\\sqrt{2}} (\\theta\\theta) \n\\theta^\\dagger \\overline{\\sigma}^\\mu\\, \\partial_\\mu \\widetilde{\\psi}(x)-\\tfrac{1}{4}(\\theta\\theta)(\\theta^\\dagger\\thetabar) \\square \\widetilde{A}(x)\\,, \n\\end{split}\n\\end{align}\nthen the impact of the super gauge transformation given by\n\\eq{eq:Vgaugetransform} on the component fields of $V$\nis,\\footnote{\\label{fn38} The invariance\nof $\\lambda$ and $D$ under super gauge transformations is a\nconsequence of the particular choices made for the coefficients of\n$(\\theta\\theta)\\theta^\\dagger$, $(\\theta^\\dagger\\thetabar)\\theta$ and\n$(\\theta\\theta)(\\theta^\\dagger\\thetabar)$ in \\eq{VectorSF}.}\n\\begin{align}\n C \n & \n \\to C+i(\\widetilde{A}-\\widetilde{A}^*)\\,,\\\\\n\\chi\n&\n\\to \\chi+ \\sqrt{2}\\, \\widetilde{\\psi} \\, , \\\\\nM+iN \n&\n\\to M+iN+2\\widetilde{F}\\,,\\\\\nV_\\mu\n&\n \\to V_\\mu+\\partial_\\mu(\\widetilde{A}+\\widetilde{A}^*)\\,, \\\\\n \\lambda\n &\n \\to \\lambda\\,,\\label{lambdaGT} \\\\\nD \n&\n\\to D\\,.\\label{DGT}\n\\end{align}\nIndeed, under a super gauge transformation, the gauge field $V_\\mu$\ntransforms by an ordinary gauge transformation. Moreover the \nfield strength tensor $F_{\\mu\\nu}=\\partial_\\mu V_\\nu-\\partial_\\nu\nV_\\mu$, the gaugino\nfield $\\lambda$, and the auxiliary field $D$ are gauge invariant as\none would anticipate (consistent with the fact that the field strength\nsuperfield $\\mathcal{W}$ is gauge invariant).\n\n\nOne particularly useful gauge choice is to choose $\\widetilde A$,\n$\\widetilde\\psi$ and $\\widetilde F$ such that \n\\begin{equation}\nC=\\chi=M=N=0\\,.\\label{WZ}\n\\end{equation}\nThis is called the {{Wess-Zumino (WZ) gauge}}\\cite{Wess:1974jb}. \nThe existence of such a gauge implies that the fields $C$,\n$\\chi$, $M$, and $N$ are gauge artifacts, as previously stated.\nThe main drawback of the WZ gauge is that it is not a\nsupersymmetric gauge choice. That is, starting from the WZ gauge and\nperforming a SUSY transformation on the component fields of the vector\nsuperfield~$V$ will yield new component fields that do not satisfy the WZ gauge\ncondition. \n\nThe main benefit of the WZ gauge is that it provides enormous\nsimplification in many practical computations. In particular, applying\nthe WZ gauge condition [\\eq{WZ}] to the vector superfield given in \\eq{VectorSF},\n\\begin{align}\nV_{\\rm WZ}=\\theta\\sigma^\\mu\\theta^\\dagger V_\\mu+\ni(\\theta\\theta)(\\theta^\\dagger\\bar{\\lambda})-i(\\theta^\\dagger\\thetabar)(\\theta\\lambda)+\\ifmath{\\tfrac12}(\\theta\\theta)(\\theta^\\dagger\\thetabar)D\\,.\n\\end{align}\nComputing the square of $V_{\\rm WZ}$ with the help of \\eq {eq:r3} yields,\n\\begin{align}\nV^2_{\\rm WZ}(x,\\theta,\\theta^\\dagger)=\\ifmath{\\tfrac12} (\\theta\\theta)(\\theta^\\dagger\\thetabar)V_\\mu V^\\mu\\,.\n\\end{align}\nand $V^n_{\\rm WZ}(x,\\theta,\\theta^\\dagger)=0$ for $n=3,4,5,\\dots$. This\nimplies that the Taylor series for the exponential of $V_{\\rm WZ}$ is\na finite series and\ncontains only three terms,\n\\begin{align}\n\\exp(2gV_{\\rm WZ})=1+2gV_{\\rm WZ}+2g^2V^2_{\\rm WZ}\\,. \\label{e2gV}\n\\end{align}\nThis result will be especially important when we consider\ngauge-invariant interactions in Section~\\ref{GI}.\n\nFinally, we consider the implications of $R$-invariance. \nSince $V$ is a real superfield, it follows from \\eq{Rtrans} that,\n\\begin{align}\n\\widehat{R}V(x,\\theta,\\theta^\\dagger)=V(x,e^{-ia}\\theta,e^{ia}\\theta^\\dagger)\\,.\n\\end{align}\nIn the Wess-Zumino gauge, the $R$ transformations of the component fields are given by\n\\begin{align}\nV_\\mu&\\to V_\\mu\\,, \\\\\n\\lambda&\\to e^{ia}\\lambda\\,, \\\\\nD&\\to D\\,.\n\\end{align}\nThe Lagrangian of \\eq{WF} for the SUSY gauge theory is invariant under\n$R$ transformations.\nIn the present context, the presence of $R$-invariance is associated with the chiral symmetry of the massless gaugino.\n\n\n\n\\subsection{Gauge-invariant interactions}\n\\label{GI}\nSuppose that $\\Phi$ is a chiral superfield that is charged under the U(1) gauge group.\nThen the gauge transformations of the chiral superfield and the corresponding antichiral superfield are given by,\n\\begin{align}\n\\Phi\\to e^{-2ig\\Lambda}\\Phi\\,,\\qquad\\quad \\Phi^\\dagger\\to e^{2ig\\Lambda^\\dagger}\\Phi^\\dagger\\,,\\label{eq:Phigauge}\n\\end{align}\nwhere $\\Lambda$ is the chiral superfield gauge\ntransformation parameter introduced in\n\\eq{eq:Vgaugetransform}.\nIn the presence of gauge interactions, the kinetic energy term for the\nchiral superfield given by \\eq{phiphiD},\n\\begin{align}\n\\mathscr{L}_{\\rm KE}=[\\Phi^\\dagger \\Phi]_D=\\int d^4 \\theta\\,\\Phi^\\dagger\\Phi\\,,\n\\end{align}\nis not gauge invariant. But this deficiency is easily repaired. A\ngauge-invariant kinetic energy term \nwith respect to the gauge transformations given in\n\\eqs{eq:Vgaugetransform}{eq:Phigauge} is given by,\n\\begin{align}\n\\mathscr{L}_{\\rm KE}=[\\Phi^\\dagger e^{2gV}\\Phi]_D=\\int d^4 \\theta\\,\\Phi^\\dagger e^{2gV}\\Phi\\,.\n\\label{eq:LKE}\n\\end{align}\nThe proof is left as an exercise (see Problem \\ref{pr:invKE}).\n\n\nNormally, the exponential, $\\exp(2gV)$, would yield an infinite series\nof terms. But, the series terminates in the Wess-Zumino gauge, as\nindicated in \\eq{e2gV}, and we get \n\\begin{align}\n\\begin{split}\n\\mathscr{L}_{\\rm KE}=&(\\mathcal{D}_\\mu A)(\\mathcal{D}^\\mu A)^\\dagger+\ni \\psi^\\dagger \\overline{\\sigma}^\\mu \\mathcal{D}_\\mu \\psi+F^\\dagger F\\\\\n&\n+ig\\sqrt{2}(A^\\dagger\\lambda\\psi-A\\lambda^\\dagger\\psi^\\dagger)\n+gAA^\\dagger D+\\text{total derivative}\\,,\\label{KEint}\n\\end{split}\n\\end{align}\nwhere $\\mathcal{D}_\\mu\\equiv\\partial_\\mu+igV_{\\mu}$ is the usual gauge-covariant derivative.\nThe presence of the Yukawa interaction of the scalar-fermion-gaugino\nis especially noteworthy, with a coupling proportional to the gauge\ncoupling $g$. This is a consequence of supersymmetry, which relates\nthe gauge and Yukawa couplings that otherwise would be independent.\n\n\nAnother manifestation of SUSY is revealed when we consider the terms\nof the Lagrangian involving the auxiliary fields $F$ and $D$. \nConsider the Lagrangian of the interacting gauge theory that consists\nof contributions from \\eqs{WF}{KEint}. We can isolate those terms\nthat involve $F$ and $D$ explicitly, \n\\begin{align}\n\\mathscr{L} = \\biggl\\{\\tfrac14[\\mathcal{W}^\\alpha \\mathcal{W}_\\alpha]_F+{\\rm h.c.}\\biggr\\}+[\\Phi^\\dagger e^{2gV}\\Phi]_D \n =\\ldots+\nF^\\dagger F+\\ifmath{\\tfrac12} D^2+gAA^\\dagger D\\,.\\label{FD}\n\\end{align}\nSolving the Lagrange field equations for $F$ and $D$,\n\\begin{align}\n& \\frac{\\partial\\mathscr{L}}{\\partial F}=0\\qquad\\Longleftrightarrow\\qquad F=0\\,, \\\\\n& \\frac{\\partial\\mathscr{L}}{\\partial\n D}=0\\qquad\\Longleftrightarrow\\qquad D=-gA^\\dagger A\\,.\\label{DAA}\n\\end{align}\nInserting these results back into \\eq{FD} [where the terms not\nexplicitly given can be found in \\eqs{WF}{KEint}] yields the Lagrangian in terms of its physical fields,\n\\begin{align}\n\\begin{split}\n\\mathscr{L} =&-\\tfrac14 F_{\\mu\\nu}F^{\\mu\\nu}\n+i\\lambda^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\lambda+(\\mathcal{D}_\\mu A)(\\mathcal{D}^\\mu A)^\\dagger \n+ i \\psi^\\dagger \\overline{\\sigma}^\\mu \\mathcal{D}_\\mu \\psi \\\\\n& +i\\sqrt{2}\\,g(A^\\dagger\\lambda\\psi-A\\lambda^\\dagger\\psi^\\dagger)\n-\\ifmath{\\tfrac12} g^2 (A^\\dagger A)^2\\,.\n\\end{split}\n\\end{align}\nThus, a potential for the scalar field $A$ has been generated,\n\\begin{align}\nV_{\\rm scalar}=\\ifmath{\\tfrac12} g^2 (A^\\dagger A)^2\\,.\n\\end{align}\n\nThere is one more possible term, called the Fayet-Iliopoulos term\\cite{Fayet:1974jb},\nthat can appear in a renormalizable SUSY U(1) gauge theory Lagrangian,\n\\begin{align}\n\\mathcal{L}_{\\rm FI}=2\\xi[V]_D= \\xi D+\\text{total divergence}\\,.\n\\end{align}\nThis modifies the form of $D$ obtained in \\eq{DAA},\n\\begin{align}\nD=-gA^\\dagger A-\\xi\\,,\n\\end{align}\nwhich in turn modifies the scalar potential,\n\\begin{align}\nV_{\\rm scalar}=\\ifmath{\\tfrac12}\\bigl[gA^\\dagger A+\\xi\\bigr]^2\\,.\n\\end{align}\nThe existence of a quartic scalar coupling proportional to the square of the gauge coupling (in the presence or absence of a Fayet-Iliopoulos term) is another manifestation of SUSY.\n\n\\subsection{Generalizing to more than one chiral superfield}\n\nWith only one chiral superfield, it was not possible to include a superpotential $W(\\Phi)$ in our gauge theory, since \n$W$ is a holomorphic function of a charged field and hence not gauge-invariant.\nBut, a theory with more than one charged chiral superfield can admit a gauge invariant superpotential.\n\nFor example, consider a set of charged chiral superfields $\\Phi_i$ with U(1) charges $q_i$, which transform under U(1) as\n\\begin{align}\n\\Phi_i\\to e^{-2igq_i\\Lambda}\\Phi_i\\,.\n\\end{align}\nSuppose that a gauge-invariant superpotential can be constructed, $W(\\Phi_i)$. When we solve for the auxiliary field $F_i$, we will obtain\n\\begin{align}\nF_i=-\\left(\\frac{dW}{dA_i}\\right)^\\dagger\\,, \\label{fsubi}\n\\end{align}\nas before [cf.~\\eq{f}], which provides the $F$-term contributions to the scalar potential,\n\\begin{align}\nV_{\\rm scalar}\\ni \\sum_i \\left|\\frac{dW}{dA_i}\\right|^2\\,.\\label{niF}\n\\end{align}\n\nWhen we solve for the auxiliary field $D$, we obtain a contribution from each scalar $A_i$,\n\\begin{align}\nD=-\\xi-\\sum_i q_i g A_i^\\dagger A_i\\,. \\label{DFI}\n\\end{align}\nThe corresponding $D$-term contributions to the scalar potential are\n\\begin{align}\nV_{\\rm scalar}\\ni\\ifmath{\\tfrac12}\\left[\\xi+\\sum_i gq_iA^\\dagger A\\right]^2\\,.\\label{niD}\n\\end{align}\nIncluding both the $F$-term and $D$-term contributions yields the\nfollowing scalar potential,\n\\begin{align}\nV_{\\rm scalar}=\\sum_i \\left|\\frac{dW}{dA_i}\\right|^2+\\ifmath{\\tfrac12}\\left[\\xi+\\sum_i gq_iA^\\dagger A\\right]^2\\,,\\label{vscalar1}\n\\end{align}\nwhich can also be conveniently written as\n\\begin{align}\nV_{\\rm scalar}=\\sum_i F_i^\\dagger F_i+\\ifmath{\\tfrac12} D^2\\,, \\label{vscalar2}\n\\end{align}\nwhere $F$ and $D$ are given by \\eqs{fsubi}{niD}, respectively.\nNote that the form of the scalar potential [either \\eq{vscalar1} or\n(\\ref{vscalar2})] makes clear that $V_{\\rm scalar}\\geq 0$. This\nobservation will play an important role in the theory of supersymmetry\nbreaking, which is treated in Section~\\ref{SSB}.\n\nThe above results can now be used to construct the supersymmetric\nextension of QED. The superfield content of SUSY-QED consists of a real vector superfield $V$, a chiral superfield $\\Phi_+$ with charge $q=1$, and a chiral superfield $\\Phi_-$ with charge $q=-1$. The unique renormalizable, gauge-invariant superpotential is\n\\begin{equation} \\label{Wsqed}\nW(\\Phi_+,\\Phi_-)=m\\Phi_+\\Phi_-\\,.\n\\end{equation}\nThe $R$-charges of both $\\Phi_+$ and $\\Phi_-$ can be chosen to be $+1$, in which case the theory is also $R$-invariant.\nThe construction of the SUSY-QED Lagrangian is left as an exercise\n(see Problem \\ref{pr:SUSYQED}).\n\n\n\n\\subsection{\\mbox{SUSY \\!Yang-Mills theory coupled to supermatter}}\n\nThe construction of the supersymmetric generalization of Yang-Mills\ntheory, \\textit{i.e.},~a non-abelian gauge theory coupled to matter, is more\ncomplicated than the case of an abelian gauge theory treated in\nprevious sections. In this subsection, we will summarize the main\nmodifications. The reader can fill in the details with the help of Refs.\\cite{Bailin,Sohnius:1985qm}.\n\n\nConsider a non-abelian compact simple Lie group G, with generators $T^a$ that satisfy commutation relations,\n\\begin{align}\n\\bigl[T^a\\,,\\,T^b\\bigr]=if_{abc}T^c\\,.\n\\end{align}\nIt is convenient to normalize the generators of the defining (fundamental)\nrepresentation of G such that,\n\\begin{align}\n\\Tr(T^a T^b)=\\ifmath{\\tfrac12}\\delta_{ab}.\n\\end{align}\n\nThe vector superfield, $V^a$, possesses an adjoint index $a$, which\nruns over the generators of G.\nThus, we can define the matrix gauge superfield,\n\\begin{align}\nV\\equiv V^a T^a\\,.\n\\end{align}\nThe gauge transformation law for $V$ given in \\eq{eq:Vgaugetransform}\nis significantly more complicated in the case of a non-abelian gauge theory, \n\\begin{align}\ne^{2gV} \\longrightarrow e^{-2ig\\Lambda^\\dagger}e^{2gV} e^{2ig\\Lambda^\\dagger},\\label{eq:Vnonabelian}\n\\end{align}\nwhere $\\Lambda\\equiv (\\Lambda^a T^a)_{ij}$ is the matrix chiral superfield gauge\ntransformation parameter.\n\nThe chiral superfields are now multiplets corresponding to representation $R$\nof the gauge group G, transforming as\\footnote{When acting on the $\\Phi_i$, one employs the generators $T^a$ in the representation $R$.}\n\\begin{align}\n\\Phi_i\\to \\left(e^{-2ig\\Lambda}\\right)_{ij}\\Phi_j\\,,\n\\end{align}\nwhich provides the\ngeneralization of \\eq{eq:Phigauge} to a nonabelian gauge group.\nNote that $\\Phi^\\dagger_i\\left(e^{2gV}\\right)_{ij}\\Phi_j$ is\ngauge-invariant, if the gauge transformation law for $V$ is given by \\eq{eq:Vnonabelian}.\n\nLikewise, we define a matrix version of \nthe nonabelian field-strength superfield, $\\mathcal{W}_\\alpha\\equiv \\mathcal{W}_\\alpha^a T^a$, where\n\\begin{align}\n\\mathcal{W}_\\alpha=-\\frac{1}{8g}\\overline{D}\\lsup{2}e^{-2gV}D_\\alpha e^{2gV}\\,.\\label{eq:Wnonabelian}\n\\end{align}\nUnlike the abelian case, $\\mathcal{W}_\\alpha$ is not gauge-invariant. However it transforms as an adjoint field,\n\\begin{align}\n\\mathcal{W}_\\alpha\\to e^{-2ig\\Lambda}\\mathcal{W}_\\alpha e^{2ig\\Lambda}\\,,\n\\end{align}\nso that $\\Tr(\\mathcal{W}^\\alpha \\mathcal{W}_\\alpha)$ is gauge-invariant.\nIn the WZ gauge,\\footnote{In contrast to the abelian case, the\n expansion of $\\mathcal{W}_\\alpha^a$ in terms of its component fields in the\n nonabelian case will necessarily contain gauge artifacts. After imposing\n the WZ gauge condition, the expansion of $\\mathcal{W}_\\alpha^a$ in terms of\n its component fields resembles the corresponding expression of \n SUSY abelian gauge theory [cf.~\\eq{Wdef}].}\nwhen expanded in component fields, $\\mathcal{W}_\\alpha^a$ depends only on the\nphysical fields, $\\lambda^a$, $F^{\\mu\\nu a}$ and the auxiliary field $D^a$,\n\\begin{equation}\n\\mathcal{W}_\\alpha^a=-i\\lambda_\\alpha^a+\\theta_\\alpha D^a-\\ifmath{\\tfrac12} i(\\sigma^\\mu\\overline{\\sigma}^\\nu\\theta)_\\alpha F_{\\mu\\nu}^a-\\sigma^\\mu(\\mathscr{D}_{\\mu ab}\\lambda^{\\dagger b})_\\alpha \\,\\theta\\theta\\,,\\label{Wna}\n\\end{equation}\nwhere \n\\begin{equation}\n\\mathscr{D}_{\\mu ab}\\equiv\\delta_{ab}\\partial_\\mu+g f_{abc} V_\\mu^c\\,,\n\\end{equation}\n is the gauge-covariant derivative in the adjoint representation, and \n\\begin{equation}\nF_{\\mu\\nu}^a=\\partial_\\mu V_\\nu^a-\\partial_\\nu V_\\mu^a-gf_{abc}V_\\mu^b V_\\nu^c\n\\end{equation}\nis the nonabelian field strength tensor.\n\n\\subsection{The SUSY Lagrangian}\nThe Lagrangian for SUSY Yang-Mills theory coupled to supermatter is given by\n\\begin{align}\n\\begin{split}\n\\mathscr{L}= \\left[\\ifmath{\\tfrac12}\\int d^2\\theta\\,\\Tr(\\mathcal{W}^\\alpha \\mathcal{W}_\\alpha)+{\\rm h.c.}\\right]+\\int d^4\\theta \\,\n\\Phi^\\dagger e^{2gV}\\Phi \n+\\left[\\int d^2\\theta\\,W(\\Phi_k)+{\\rm h.c.}\\right]. \\label{SUSYYMLag}\n\\end{split}\n\\end{align}\nIn contrast to the abelian gauge theory, no Fayet-Iliopoloulos term\nis allowed since $[D^a]_D$ carries an adjoint index and thus is not\ngauge invariant. The superpotential $W(\\Phi_k)$ is assumed to be a\ngauge-invariant holomorphic function of the chiral superfields. The\nchiral superfields $\\Phi_k$ taken together transform under a reducible\n$d$-dimensional representation $R=\\oplus_k R_k$ of the gauge group G,\nwhere $d=\\sum_k {\\rm dim}~R_k$.\nIn terms of component fields, \\eq{SUSYYMLag} yields\n\\begin{align}\n\\begin{split}\n\\mathscr{L}=&-\\tfrac14 F_{\\mu\\nu}^a F^{\\mu\\nu a}+i\\lambda^{\\dagger a}\\overline{\\sigma}^\\mu(\\mathscr{D}_\\mu\\lambda)^a+\\ifmath{\\tfrac12} D^a D^a+F_i^\\dagger F_i \n+(\\mathcal{D}_\\mu A)_i(\\mathcal{D}^\\mu A)_i^\\dagger \\\\\n& +i \\psi_i^\\dagger\n\\overline{\\sigma}^\\mu (\\mathcal{D}_\\mu \\psi)_i +gA_i^\\dagger T_{ij}^a A_j D^a \n+ig\\sqrt{2}(A_i^\\dagger T^a_{ij}\\psi_j\\lambda^a-\\lambda^{\\dagger a}\\psi^\\dagger_i T^a_{ij} A_j)\n \\\\\n&\n+F_i\\frac{dW}{dA_i}+F^\\dagger_i\\left(\\frac{dW}{dA_i}\\right)^\\dagger\n-\\ifmath{\\tfrac12} \\frac{d^2 W}{dA_i dA_j}\\psi_i\\psi_j-\n\\ifmath{\\tfrac12} \\left(\\frac{d^2 W}{dA_i dA_j}\\right)^\\dagger\\psi^\\dagger_i\\psi^\\dagger_j \\,,\n\\end{split}\n\\label{eq:LSUSYcomponents}\n\\end{align}\nwhere there is an implicit sum over repeated indices, and \nthe labels $i$ and $j$ run over $1,2,\\ldots, d$.\nThe corresponding covariant derivative, when acting\non the component fields $A_i$ and $\\psi_i$, is $\\mathcal{D}_\\mu=\\mathds{1}\\partial_\\mu+igT^a V_\\mu^a$,\nwhere $\\mathds{1}$ is the $d\\times d$ identity matrix and the generators $T^a$ are in the reducible \nrepresentation $R$ of the group G.\n\nNote that the interactions of the matter fermions and the gauginos\nwith the gauge fields are dictated by gauge invariance (via the\ngauge covariant derivative) and do not depend on supersymmetry.\nIn contrast, the Yukawa interaction of the gaugino with the matter\nfermion and its scalar partner (with a coupling proportional to the gauge\ncoupling $g$) is a consequence of supersymmetry, and relates\nthe gauge and Yukawa couplings that otherwise would be independent.\n\n\nWe can now eliminate the auxiliary fields $F_i$ and $D^a$ by employing the\nLagrange field equations. We end up with\n\\begin{align}\nF_i=-\\left(\\frac{dW}{dA_i}\\right)^\\dagger\\,,\\qquad\\quad D^a=-gA_i^\\dagger T^a_{ij} A_j\\,.\\label{FandD}\n\\end{align}\nSubstituting back into \\eq{eq:LSUSYcomponents} yields the following scalar potential,\n\\begin{align}\nV_{\\rm scalar}=\\sum_i \\left|\\frac{dW}{dA_i}\\right|^2+\\ifmath{\\tfrac12} g^2(A_i^\\dagger T^a_{ij} A_j)^2\\,.\\label{vscalar3}\n\\end{align}\nEquivalently, we can write:\n\\begin{align}\nV_{\\rm scalar}=\\ifmath{\\tfrac12} D^a D^a+\\sum_i F_i^\\dagger F_i\\,.\\label{vscalar4}\n\\end{align}\n\\Eqs{vscalar3}{vscalar4} provide the nonabelian generalization of\n\\eqs{vscalar1}{vscalar2}. As in the abelian case, $V_{\\rm scalar}\\geq 0$.\n\n\n\nIf we drop the requirement of renormalizability, then we can generalize the action of a SUSY-Yang Mills theory coupled to supermatter,\n\\begin{Eqnarray}\n\\mathscr{L} &=&\\ifmath{\\tfrac12}\\int d^4\\theta \\bigl[K(e^{2gV}\\Phi\\,,\\, \\Phi^\\dagger) +\nK(\\Phi\\,,\\, \\Phi^\\dagger e^{2gV})\\bigr] \n+\\left[\\int d^2\\theta\\,W(\\Phi_i)+{\\rm h.c.}\\right]\\nonumber \\\\\n&& +\\left[\\tfrac14\\int d^2\\theta\\,f_{ab}(\\Phi) \\mathcal{W}^{\\alpha\n a}\\mathcal{W}^b_\\alpha+{\\rm h.c.}\\right] \\label{susykahler}\n\\,,\n\\end{Eqnarray}\nwhere $K$ is the K\\\"ahler potential and $f_{ab}(\\Phi)$ is a holomorphic function of the chiral superfields called the \\textit{gauge kinetic function}.\nIn renormalizable global supersymmetry, the minimal versions of the K\\\"ahler potential and gauge kinetic function are used:\n\\begin{align}\nK(e^{2gV}\\Phi\\,,\\, \\Phi^\\dagger) &=\nK(\\Phi\\,,\\, \\Phi^\\dagger e^{2gV})=\\Phi^\\dagger e^{2gV}\\Phi\\,,\\\\\nf_{ab}(\\Phi) &=\\delta_{ab}\\,.\n\\end{align}\n\nThe generalization of the SUSY Lagrangian to a theory based on a\ngauge group that is a direct product of compact simple Lie groups and\nU(1) factors is straightforward. There is a gauge field strength\ntensor and a separate gauge coupling constant\ncorresponding to each group in the direct product. Details are left\nfor the reader.\n\n\n\n\n\\subsection{Problems}\n\\begin{problem}\nShow that $\\mathcal{W}_\\alpha$ is invariant under the gauge transformation of \\eq{eq:Vgaugetransform}.\n\\end{problem}\n\\begin{problem}\n\\label{pr:invKE}\nShow that the kinetic energy term given by \\eq{eq:LKE} is invariant under the gauge transformations for $\\Phi$ and $\\Phi^\\dagger$ given in \\eq{eq:Phigauge} and $V\\to V+i(\\Lambda-\\Lambda^\\dagger)$.\n\\end{problem}\n\n\n\\begin{problem}\n\\label{pr:SUSYQED}\nConstruct the full SUSY QED Lagrangian in the Wess-Zumino gauge. Show that the physical states of the theory consist of a Dirac fermion (the ``electron''), two complex scalar ``selectrons,'' usually denoted by $\\widetilde e_L$ and $\\widetilde e_R$, a massless photon, and a massless photino. \nCheck that the number of bosonic and fermionic degrees of freedom are equal, both off-shell and on-shell.\n\\end{problem}\n\n\\begin{problem}\nConsider the SUSY QED theory examined in Problem~\\ref{pr:SUSYQED}.\nHowever, this time\ndo \\textit{not} impose the Wess-Zumino gauge condition. Instead, explore the consequences of adding the following supersymmetric gauge fixing term\\cite{Ovrut:1981wa,Miller:1983pg,Dine:2016rxc}, \n\\begin{align}\n\\mathscr{L}_{\\rm GF}=-\\frac{1}{8\\alpha}\\bigl[(D^2 V)(\\overline{D}\\lsup2 V)\\bigr]_D\\,,\n\\end{align}\nwhere $\\alpha$ is the gauge fixing parameter.\n\\end{problem}\n\n\\begin{problem}\nStarting from the case where the gauge group G is nonabelian, show\nthat the gauge transformation law for the gauge superfield~$V$, as\ndeduced from \\eq{eq:Vnonabelian}, reduces to\n$V\\rightarrow V+i(\\Lambda-\\Lambda^\\dagger)$ in the abelian limit.\nLikewise, show that $\\mathcal{W}_\\alpha$ as given in \\eq{eq:Wnonabelian} reduces to\n$\\mathcal{W}_\\alpha=-\\tfrac{1}{4} {\\overline{D}}^{2} D_\\alpha V $ in the abelian limit.\n\\end{problem}\n\n\\begin{problem}\nEvaluate the contribution of the K\\\"ahler potential terms to the\nLagrangian given in \\eq{susykahler} in terms of the component fields.\nShow that your result reduces to \\eq{kahler} in the limit of $g\\to 0$. \n\\end{problem}\n\n\\begin{problem}\nEvaluate the contribution of the gauge kinetic function terms\nto the\nLagrangian given in \\eq{susykahler} in terms of the component fields.\nHow does your result simplify in the abelian limit?\n\\end{problem}\n\n\\begin{problem}\nStarting from \\eq{susykahler}, solve for the auxiliary fields $F_i$\nand~$D^a$ using the Lagrange field equations. Using these results, determine the form of the scalar\npotential that generalizes the results of \\eqs{vscalar3}{vscalar4}.\n\\end{problem}\n\n\n\\section{\\hbox{Supersymmetric extension of the Standard Model (MSSM)}}\n\\label{sec:MSSM}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\setcounter{equation}{0}\n\nWith the necessary SUSY technology now in hand, we are ready to study its realization in extensions to the SM. \nIn this section, we describe the minimal supersymmetric extension of\nthe Standard Model (MSSM). Much of the presentation of this section\nfollows Ref.\\cite{susy}, where many of the relevant references to the\noriginal literature can be found.\n\nIn Section~\\ref{sec:MSSMfields}, we begin by presenting the MSSM\nfield content. We then specify the\nSU(3)$\\times$SU(2)$\\times$U(1) gauge-invariant superpotential for the\nchiral superfields in Section~\\ref{sec:MSSMW}. Given the superfield formalism developed in\nSections~\\ref{sec:superspace} and \\ref{sec:gaugetheories}, all the\nsupersymmetric interactions of the theory are now determined.\nAt this stage, the supersymmetry is still an exact symmetry.\n\nWe introduce SUSY breaking in the MSSM in Section~\\ref{sec:MSSMSSB}.\nSince the fundamental origin of SUSY-breaking is \nunknown, we parametrize the SUSY-breaking by adding all possible\nsoft-SUSY-breaking terms consistent with the SU(3)$\\times$SU(2)$\\times$U(1)\ngauge symmetry and a discrete $B-L$ symmetry. In\nSection~\\ref{sec:count}, we count the number of\nparameters that govern the MSSM.\nThe resulting MSSM particle spectrum and Higgs boson spectrum are\nexhibited in Sections~\\ref{sec:MSSMspectrum} and \\ref{higgssector}, respectively.\nFinally, in Section~\\ref{sec:MSSMGU}, we demonstrate the unification of\ngauge couplings in the MSSM. \n\nAs in the SM, the neutrinos of the MSSM are massless.\nTo incorporate massive neutrinos, one can introduce \nright-handed neutrinos and employ the seesaw mechanism. It is then\na simple matter to extend the MSSM by adding a SM singlet superfield\nthat contains a right-handed neutrino and the corresponding sneutrino\nsuperpartner. We shall not present this construction in these\nlectures; for further details, see e.g.~Ref.\\cite{Dedes:2007ef}.\n\n\n\n\\subsection{Field content of the MSSM}\n\\label{sec:MSSMfields}\n\n\\subsubsection{MSSM superfields and their component fields}\nThe minimal supersymmetric extension of the Standard Model (MSSM)\ncontains the fields of the\ntwo-Higgs-doublet extension of the SM\nand their corresponding superpartners.\nThe gauge fields and their superpartners are contained in real vector supermultiplets.\nThese gauge supermultiplets consist of the \nSU(3)$\\times$SU(2)$\\times$U(1) gauge bosons and their\ngaugino fermionic superpartners.\nThe matter fields and their superpartners reside in chiral supermultiplets.\nThe three generations of quark and lepton supermultiplets\nconsist of left-handed\nquarks and leptons and\ntheir scalar superpartners (squarks and sleptons),\nand the corresponding antiparticles. \nThe Higgs supermultiplets\nconsist of two complex Higgs doublets, their\nhiggsino fermionic superpartners, and the\ncorresponding antiparticles.\nThe MSSM fields and their gauge quantum\nnumbers are shown in Table~\\ref{tab:MSSMcontent}. \n\\vskip -0.05in\n\\begin{table}[h!]\n\\caption{\\small\nThe fields of the MSSM and their\nSU(3)$\\times$SU(2)$\\times$U(1) quantum numbers are listed.\nThe electric charge is given in terms of the third component of\nthe weak isospin $T_3$ and U(1) hypercharge $Y$ by\n$Q=T_3+\\ifmath{\\tfrac12} Y$.\nFor simplicity, only one generation of quarks and leptons is exhibited.\nThe left-handed charge-conjugated quark and lepton fields are denoted\nby a superscript $c$. In particular, $f^c_L\\equiv\nP_Lf^c=P_LC\\bar{f}\\lsup{\\,{\\mathsf T}}=C\\bar{f}_R\\lsup{\\,{\\mathsf T}}$, following\nthe notation of Ref.\\cite{Langacker:1980js}, where $f$ is a\nfour-component fermion field.\nThe $L$ and $R$ subscripts\nof the squark and slepton fields indicate the chirality of the\ncorresponding fermionic superpartners.\n \\label{tab:MSSMcontent} }\n\\vskip 0.1in\n\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\\multicolumn{7}{|c|}{Field content of the MSSM} \\\\ \\hline\nSuper- & Super- & Bosonic & Fermionic & & & \\\\\nmultiplets & field & fields & partners &\nSU(3) & SU(2) & U(1) \\\\ \\hline\ngluon\/gluino & $\\widehat V_8$ & $g$ & $\\widetilde g$ & 8 & 1& $\\phantom{-} 0$ \\\\\ngauge boson\/ & $\\widehat V$ & $W^\\pm\\,,\\,W^0$ & $\\widetilde W^\\pm\\,,\\widetilde W^0$ & 1 & 3 & $\\phantom{-} 0$ \\\\\ngaugino & $\\widehat V^\\prime$ & $B$ & $\\widetilde B$ & 1 &1 & $\\phantom{-} 0$ \\\\ \\hline\nslepton\/ & $\\widehat L$ &$(\\widetilde\\nu_L, \\widetilde e^-_L)$ & $(\\nu,e^-)_L$ & 1 & 2 & $-1$ \\\\\nlepton & $\\widehat E^c$ & $\\tilde e^+_R$ & $e_L^c$ & 1 & 1 & $\\phantom{-}\n 2$ \\\\ \\hline\nsquark\/ & $\\widehat Q$ & $(\\widetilde u_L,\\widetilde d_L)$ & $(u,d)_L$ & 3 &2 & $\\phantom{-} 1\/3$ \\\\\nquark & $\\widehat U^c$ & $\\widetilde u_R^*$ & $u_L^c$ & $\\bar{3}$ &1 & $-4\/3$ \\\\\n & $\\widehat D^c$ & $\\widetilde d_R^*$ & $d_L^c$ & $\\bar{3}$ & 1 & $\\phantom{-} 2\/3$ \\\\ \\hline\nHiggs boson\/ & $\\widehat H_d$ & $(H^0_d\\,,\\,H_d^-)$ & $(\\widetilde H^0_d,\\widetilde H^-_d)$ & 1 & 2 & $-1$ \\\\ \nhiggsino & $\\widehat H_u$ & $(H^+_u\\,,\\,H^0_u)$ & $(\\widetilde H^+_u,\\widetilde H^0_u)$ & 1 & 2 & $\\phantom{-} 1$ \\\\ \\hline\n \\end{tabular}\n \\end{table}\n %\n\nTable~\\ref{tab:MSSMcontent} shows that one Higgs doublet superfield has hypercharge $-1$, and the other has hypercharge $+1$.\nThe distinction between hypercharge $\\pm 1$ is irrelevant in a\n non-supersymmetric quantum field theory, where complex scalar fields are\n always accompanied by their hermitian conjugates. However, in\nsupersymmetric models the distinction is important, because \nthe corresponding Higgs superfields are used to construct the\nsuperpotential. Since the superpotential \nmust be holomorphic, \\textit{i.e.}~depend only on chiral superfields and not\ntheir hermitian conjugates, it is important to keep track of the\nquantum numbers of the chiral superfields of the model.\n\n\n\n\n\\subsubsection{Anomaly cancellation and the second Higgs doublet}\n\\label{sec:ac}\n\nThe enlarged Higgs sector of the MSSM constitutes the minimal structure\nneeded to guarantee the cancellation of gauge\nanomalies generated by the \nhiggsino superpartners that can appear as internal lines in one-loop triangle diagrams with\nthree external electroweak gauge bosons.\n\nPotentially problematic anomalies arise from \none-loop $VVA$ and $AAA$ triangle diagrams with three external gauge bosons, and fermions running around the loop [where $V$ refers to a $\\gamma_\\mu$ (vector) vertex and $A$ refers to a $\\gamma_\\mu\\gamma\\ls{5}$ (axial vector) vertex].\nAn anomalous theory violates unitarity and fails as a consistent quantum field theory.\nThus, we need to make sure all gauge anomalies cancel when summed over\nall triangle diagrams with fixed external gauge fields\\cite{anomalies}. \n\nThe anomalies will cancel if \ncertain group theoretical constraints are satisfied. \nIn particular, the trace of the product of the relevant generators appearing at\nthe external vertices must vanish,\n\\begin{align}\n&\nW^i W^j B~\\text{triangle} \\qquad\\Longleftrightarrow \\qquad \\Tr(T_3^2 Y)=0\\,,\\nonumber \\\\\n&\nBBB~\\text{triangle} \\,\\,\\quad\\qquad\\Longleftrightarrow \\qquad\\quad\\!\\! \\Tr(Y^3)=0\\,.\\nonumber\n\\end{align}\nIn the Standard Model, the fermion contributions to \n $\\Tr(Y^3)$ sum to zero:\n\\begin{align}\n\\Tr(Y^3)_{\\rm SM}=3\\left(\\tfrac{1}{27}+\\tfrac{1}{27}-\\tfrac{64}{27}+\\tfrac{8}{27}\\right)-1-1+8=0\\,.\n\\end{align}\nIn contrast, in the MSSM, \nif we only add the higgsinos $(\\widetilde{H}_u^+ \\,,\\,\\widetilde{H}_u^0)$, the resulting\nanomaly factor is\n$\n\\Tr (Y^3)=\\Tr(Y^3)_{\\rm SM}+2,\n$\nleading to a gauge anomaly. To cancel this, we must add a second higgsino doublet with opposite hypercharge, $(\\widetilde{H}_d^0 \\,,\\,\\widetilde{H}_d^-)$.\n\nThere is an independent argument for requiring the second Higgs\ndoublet in the MSSM.\nWith only one Higgs doublet, one cannot\ngenerate mass for both ``up''-type and ``down''-type\nquarks (and charged leptons)\nin a way that is consistent with a holomorphic superpotential.\n\n\\subsubsection{Suppressed baryon and lepton number violation}\n\\label{sec:bml}\n\nIt is an experimental fact that baryon number $B$ and lepton number\n$L$ are, to a very good approximation, global symmetries of nature.\nIf neutrinos are Majorana fermions, then $L$-violation is present but strongly\nsuppressed, with neutrino masses of order $v^2\/M$, where $v$ is the\nscale of electroweak symmetry breaking and $M\\gg v$. No $B$-violation\nhas yet been experimentally observed. Moreover, the\ncurrent bounds on the\nproton lifetime suggest that the mass scale associated with baryon\nnumber violation cannot be below about $10^{16}$~GeV, which is a\ncharacteristic scale of grand unification.\n \nOne of the remarkable features of the SM is that the suppression of $B$ and $L$-violating\nprocesses is a natural feature of the model.\nThat is, the SM Lagrangian possesses an accidental\nglobal \\hbox{$B\\!\\!-\\!\\!L$} symmetry due to the fact that \nall renormalizable terms of the Lagrangian (with dimension four or less) \nthat can be composed of SM fields preserve the $B$ and $L$\nglobal symmetries. Indeed, $B$ and\n$L$-violating operators composed of SM fields must have\ndimension $d=5$ or\nlarger\\cite{Weinberg:1979sa,Wilczek:1979hc,Weldon:1980gi}.\n\nFor example, consider the dimension-five $L$-violating operator,\n\\begin{equation} \\label{L5}\n\\mathscr{L}_5=-\\frac{f_{mn}}{M}(\\epsilon^{ij}L_i^mH_j)(\\epsilon^{k\\ell}L_k^n\nH_\\ell)+{\\rm h.c.}\\,,\n\\end{equation}\nwhere $f$ is a coefficient that depends on the lepton generation (labeled by\n$m$ and $n$), $H_j$ is the complex Higgs doublet field and $L_i^a\\equiv\n(\\nu_L^a\\,,\\,\\ell_L^a)$ is the doublet \nof two-component lepton fields. \nAfter electroweak symmetry breaking, the neutral component\nof the doublet Higgs\nfield acquires a vacuum expectation value, and a Majorana mass\nmatrix for the neutrinos is generated. The dimension-five term given\nby \\eq{L5}\nis generated by new physics beyond\nthe SM at the scale $M$. Likewise, one can construct dimension-six \n$B$-violating operators composed of SM fields that allow, e.g.,\nfor proton decay, which is suppressed by $v^2\/M_{\\rm G}^2$. Such \nterms can be generated, e.g., in grand unified theories with a\ncharacteristic mass scale $M_{\\rm G}$.\nIn general, $B$ and\n$L$-violating effects are suppressed by $(v\/M)^{d-4}$, where\n$M$ is the characteristic mass scale of the physics that generates the\ncorresponding higher dimensional operator (of dimension $d$). \n\nUnfortunately, the suppression of $B$ and $L$-violation is not guaranteed in a generic\nsupersymmetric extension of the Standard Model. For example, it is\npossible to construct gauge invariant supersymmetric dimension-four\n$B$ and $L$-violating operators made up of fields of SM\nparticles and their superpartners. Such operators, if present in the\ntheory, would yield a proton decay rate many orders of magnitude\nlarger than the current experimental bound. \nTo avoid this catastrophic prediction, one can\nintroduce an additional symmetry\nin the supersymmetric theory that will eliminate the $B$ and\n$L$-violating operators of dimension\n$d\\leq 4$. Further details are provided in the next subsection.\nNevertheless, one must admit that the SM provides a more satisfying\nexplanation for approximate $B$ and $L$ conservation than does its\nsupersymmetric extension.\n\n\\subsection{The superpotential of the MSSM}\n\\label{sec:MSSMW}\nGiven the chiral and gauge superfield content of the MSSM, we must now specify the superpotential. The most general SU(3)$\\times$SU(2)$\\times$U(1) gauge-invariant superpotential (omitting the right-handed neutrino superfield) is\n\\begin{align}\n\\begin{split}\nW = &\\ (h_u)_{mn} \\widehat{Q}_m\\!\\cdot\\! \\widehat{H}_u\\, \\widehat{U}_n^c + (h_d)_{mn} \\widehat{H}_d\\!\\cdot\\!\\widehat{Q}_m\\, \\widehat{D}_n^c \\\\\n& + (h_e)_{mn} \\widehat{H}_d \\!\\cdot\\!\\widehat{L}_m\\,\\widehat{E}_n^c +\\mu \\widehat{H}_u\\!\\cdot\\! \\widehat{H}_d\\,+\\,W_{\\rm RPV},\\label{MSSMsuperpot}\n\\end{split}\n\\end{align}\nwhere $m$ and $n$ label the generations. That is, $h_u$, $h_d$ and $h_e$ are $3\\times 3$ matrix Yukawa couplings. Note that color indices have been suppressed, and we\n employ a dot product notation for the singlet combination of two SU(2) doublets. For example,\n\\begin{equation}\n \\widehat{H}_u\\!\\cdot\\! \\widehat{H}_d \\equiv \\epsilon^{ij}\\widehat{H}_{u\\,\\!i} \\widehat{H}_{d\\,\\!j}\n =\\widehat{H}_u^+ \\widehat{H}_d^--\\widehat{H}_u^0 \\widehat{H}_d^0\\,.\n \\end{equation}\nThe so-called $\\mu$-term above is the supersymmetric analog\nof the Higgs boson squared-mass term of the SM. \n\nIn addition to the supersymmetric generalization of the SM Yukawa\ncouplings and the $\\mu$-term, \nthe gauge symmetries of the superpotential also allow for a number of new terms that violate $B-L$ conservation.\nAs discussed in Section~\\ref{sec:bml},\nthis is in contrast to the SM where there are no $B$ or\n$L$-violating interactions at the renormalizable level.\nThe $B-L$ violating terms of the supersymmetric model arise due to the presence of $W_{\\rm RPV}$ in \\eq{MSSMsuperpot} and are\ngiven by,\n\\begin{align}\n\\begin{split}\nW_{\\rm RPV}=&\\ \n(\\lambda_L)_{pmn} \\widehat L_p \\widehat L_m \\widehat E^c_n\n+ (\\lambda_L^\\prime)_{pmn}\\widehat L_p \\widehat Q_m\\widehat D^c_n \\\\\n& +(\\lambda_B)_{pmn}\\widehat U^c_p \\widehat D^c_m \\widehat D^c_n\n+(\\mu_L)_p \\widehat H_u\\widehat L_p\\,.\n\\end{split}\n\\end{align}\nNote that the term \nproportional to $\\lambda_B$ violates $B$, while the other three terms\nviolate $L$. \nThe $L$-violating term proportional to $\\mu_L$ is the generalization of the\n$\\mu \\widehat H_u\\widehat H_d$ term,\nin which the $Y=-1$ Higgs supermultiplet $\\widehat H_d$ is replaced\nby the lepton supermultiplet $\\widehat L_p$. Indeed, if $L$ violation\nis present, then there is no distinction between $\\widehat{L}$ and $\\widehat{H}_d$, since the gauge quantum numbers of these two superfields are identical.\n\n\nIf all terms in $W_{\\rm RPV}$ were allowed, the resulting model would predict \na proton decay rate many orders of magnitude larger than the current\nexperimental bound.\nThis can be avoided by imposing an appropriate discrete symmetry that\nwould eliminate the undesirable terms in $W$.\n\nThe standard choice in constructing the MSSM is to set $W_{RPV}=0$.\nThere are a number of ways to accomplish this. First, one\none could directly impose a $B-L$ symmetry.\nAlternatively, one can set $W_{RPV}=0$ by introducing a matter parity, under which $\\widehat Q$, $\\widehat U^c$, $\\widehat D^c$, $\\widehat L$ and $\\widehat E^c$ are odd, and $\\widehat H_u$ and $\\widehat H_d$ are even. \nFinally, a third option is to impose an $R$-invariant superpotential. As discussed in Section~\\ref{Rinvariance},\n$W$ is $R$-invariant if the $R$ charges of the chiral superfields are\nchosen such that $R(W)=2$. Thus, if we choose $R$ charges of $+\\ifmath{\\tfrac12}$\nfor $\\widehat Q$, $\\widehat U^c$, $\\widehat D^c$, $\\widehat L$, $\\widehat E^c$ and $R$ charges\nof $+1$ for $\\widehat H_u$, $\\widehat H_d$, then the condition of $R$-invariance\nsets $W_{\\rm RPV}=0$.\n\nOne has to make sure that whichever symmetry one chooses to set\n$W_{\\rm RPV}=0$ is also consistent with the soft-SUSY-breaking terms\nthat are subsequently added to the model. In particular, in the case of the $R$-invariance, recall that $R(\\lambda)=1$, which forbids the gaugino mass term,\n\\begin{align}\nm_\\lambda(\\lambda\\lambda+\\lambda^\\dagger\\lambda^\\dagger).\\label{eq:gauginomass}\n\\end{align}\nBut phenomenology requires massive gauginos. This motivates the use of $R$-parity, described in the following subsection, rather than $R$-invariance.\n\n\n\\subsubsection{$R$-parity}\nThe gaugino mass term in \\eq{eq:gauginomass}\nis an allowed soft-SUSY-breaking term.\nIf this term is added \nto a theory with an $R$-invariant superpotential, then\nthe continuous U(1)$_R$ symmetry is broken down to a discrete $\\mathbb{Z}_2$ symmetry,\ncalled {$R$-parity}\\cite{Fayet:1976et,Farrar:1978xj}. One can check that the $R$-parity of a particle with baryon number $B$, lepton number $L$ and spin $S$ is given by\n\\begin{align}\nR=(-1)^{3(B-L)+2S}\\,.\\label{Rparity}\n\\end{align}\nIt is sufficient to impose $R$-parity invariance in order to set $W_{\\rm\n RPV}=0$,\\footnote{The effects of imposing matter parity and\n $R$-parity in the MSSM are identical for all\n renormalizable interactions.} \nwhich is equivalent to imposing the $B-L$ discrete symmetry.\nFor the remainder of these lectures, we shall assume that $R$-parity\nis conserved. \n\nOne can use \\eq{Rparity} to deduce the $R$-parity quantum numbers of\nall SM particles and their supersymmetric partners,\n\\begin{align}\nR=\\begin{cases} +1\\,, & \\quad \\text{for all SM particle particles}\\,,\\\\\n-1\\,,& \\quad \\text{for all superpartners}\\,.\\end{cases}\n\\end{align}\nThe conservation of $R$-parity in scattering\nand decay processes has a critical impact on supersymmetric\nphenomenology. \n For example, any initial state in a scattering\nexperiment will involve ordinary ($R$-even) particles.\nConsequently, it follows that supersymmetric particles must be\nproduced in pairs. In general, these particles are highly unstable\nand decay into lighter states. Moreover, $R$-parity invariance\nalso implies that\nthe lightest supersymmetric particle (LSP) is absolutely\nstable, and must eventually be produced\nat the end of a decay chain initiated by the decay of a heavy unstable\nsupersymmetric particle.\n\nIn order to be consistent with cosmological constraints, a stable LSP\nis almost certainly electrically and color neutral.\nConsequently, the LSP in an $R$-parity-conserving theory is weakly\ninteracting with ordinary matter, \\textit{i.e}\\!., it behaves like a stable heavy\nneutrino and will escape collider detectors without being directly\nobserved. Thus, the canonical signature for conventional\n$R$-parity-conserving supersymmetric theories is missing (transverse)\nenergy, due to the escape of the LSP. Moreover,\nthe stability of the LSP in $R$-parity-conserving supersymmetry\nmakes it a promising candidate for dark matter.\n\n\\subsubsection{MSSM parameters of the SUSY-conserving sector}\nThe parameters of the SUSY-conserving\nsector consist of: (i)~gauge couplings, $g_s$, $g$, and $g'$,\ncorresponding\nto the Standard Model gauge group SU(3)$\\times$SU(2)$\\times$U(1)\nrespectively; (ii)~a\nSUSY-conserving higgsino mass parameter\n$\\mu$; and (iii)~Higgs-fermion Yukawa coupling constants,\n$\\lambda_u$, $\\lambda_d$, and $\\lambda_e$, corresponding to\nthe couplings of one generation of left- and right-handed\nquarks and leptons and their\nsuperpartners to the Higgs bosons and higgsinos. Because there is no\nright-handed neutrino (or its superpartner) in the MSSM as defined\nhere, a Yukawa coupling $\\lambda_\\nu$ is not included.\nThe complex $\\mu$ parameter and Yukawa couplings\nenter via the most general renormalizable $R$-parity-conserving\nsuperpotential given by \\eq{MSSMsuperpot} with $W_{\\rm RPV}=0$.\n\n\n\nOne can now obtain the scalar potential from \\eq{vscalar4} as applied to\nthe MSSM,\n\\begin{align}\nV_{\\rm scalar}=\\ifmath{\\tfrac12}\\bigl[D^a D^a+(D')^2\\bigr]+F_i^* F_i\\,,\n\\end{align}\nwhere the index $a$ runs over the SU(3) and SU(2) gauge indices and\n$D'$ is the U(1)$_Y$ $D$-term.\nFocusing on the terms that depend on the Higgs boson fields, one\nobtains,\n\\clearpage\n\n\\begin{align}\nV_{\\rm Higgs}=|\\mu|^2\\bigl[|H_d|^2+|H_u|^2\\bigr]+\\tfrac18(g^2+g^{\\prime\\,2})\\bigl[|H_d|^2-|H_u|^2\\bigr]^2\n+\\ifmath{\\tfrac12} g^2|H_d^* H_u|^2\\,.\n\\end{align}\nClearly $\\vev{V_{\\rm Higgs}}\\equiv\\vev{0|V_{\\rm Higgs}|0}\\geq 0$, as expected. Moreover, $H_d=H_u=0$ minimizes the\nHiggs scalar potential, which yields $\\vev{V_{\\rm Higgs}}=0$, corresponding to a supersymmetric vacuum. Thus, there is no SU(2)$\\times$U(1) breaking at this stage.\nBut after introducing soft SUSY-breaking terms, some of which involve\nthe Higgs fields, it will then be possible to spontaneously break the\nSU(2)$\\times$U(1) symmetry. Consequently, SUSY breaking and electroweak symmetry breaking are intimately related in the MSSM.\n\n\n\\subsection{Supersymmetry breaking in the MSSM}\n\\label{sec:MSSMSSB}\n\nFollowing the rules of Girardello and Grisaru\\cite{Girardello:1981wz}\nthat were presented in Section~\\ref{GGrules}, we add the\nsoft-SUSY-breaking terms, consistent with the\nSU(3)$\\times$SU(2)$\\times$U(1) gauge symmetry and the assumed\n$R$-parity invariance (for a review, see Ref.\\cite{Chung:2003fi}). For simplicity, we consider in this section the case of one generation of quarks,\nleptons, and their scalar superpartners.\n\nThe supersymmetry-breaking\nsector contains the following sets of parameters:\n(i)~three complex\ngaugino Majorana mass parameters, $M_3$, $M_2$, and $M_1$, associated with\nthe SU(3), SU(2), and U(1) subgroups of the Standard Model;\n(ii)~five squark and slepton squared-mass parameters, $M^2_{\\widetilde Q}$,\n$M^2_{\\widetilde U}$, $M^2_{\\widetilde D}$, $M^2_{\\widetilde L}$, and $M^2_{\\widetilde E}$,\ncorresponding to the superpartners of the five electroweak multiplets of\nleft-handed fermion fields and their charge-conjugates, $(u, d)_L$, $u^c_L$,\n$d^c_L$, $(\\nu$, $e^-)_L$, and $e^c_L$\n[cf.~Table~\\ref{tab:MSSMcontent}]; and\n(iii)~three Higgs-squark-squark and Higgs-slepton-slepton trilinear\ninteraction terms, with complex coefficients $T_U\\equiv\\lambda_u A_U$,\n$T_D\\equiv\\lambda_d A_D$, and $T_E\\equiv\\lambda_e A_E$\n(which define the $A$-parameters). \nFollowing Ref.\\cite{Haber:1993wf}, it is conventional to separate out the\nfactors of the Yukawa couplings in defining the\n$A$-parameters, originally motivated by a simple class of\ngravity-mediated SUSY-breaking\nmodels\\cite{Hall:1983iz,Nilles:1983ge,Martin:1997ns}.\nWith this definition, if the $A$-parameters \nare parametrically of the same order (or smaller) relative\nto other supersymmetry-breaking mass parameters, then\nonly the third generation $A$-parameters will be\nphenomenologically relevant. \n\nFinally, we have\n(iv)~two real squared-mass parameters ($m_1^2$ and~$m_2^2$) and one \ncomplex squared-mass parameter, $m_{12}^2\\equiv \\mu B$\n(the latter defines the $B$-parameter), which appear in the \ntree-level scalar Higgs potential, \n\\begin{Eqnarray}\nV&=&(m_1^2+|\\mu|^2)H_d^\\dagger H_d+(m_2^2+|\\mu|^2)H_u^\\dagger\nH_u+(m_{12}^2H_u H_d+{\\rm\n h.c.}) \\nonumber \\\\\n&&\\qquad\\quad +\\ifmath{\\tfrac18}(g^2+g^{\\prime\\,2})(H_d^\\dagger H_d-H_u^\\dagger\nH_u)^2+\\ifmath{\\tfrac12}|H_d^\\dagger H_u|^2\\,.\\label{Hpot}\n\\end{Eqnarray}\nNote that the quartic Higgs couplings in \\eq{Hpot} are related to the gauge\ncouplings $g$ and $g'$ as a consequence of supersymmetry.\nThe breaking of the\nelectroweak symmetry SU(2)$\\times$U(1) to U(1)$_{\\rm EM}$ is\nonly possible after introducing the\nsupersymmetry-breaking Higgs squared-mass parameters $m_1^2$, $m_2^2$\n(which can be negative) and $m_{12}^2$.\nAfter minimizing the Higgs scalar potential,\nthese three squared-mass\nparameters can be re-expressed in terms of the two\nHiggs vacuum expectation values, $\\langle H_d^0\\rangle\\equiv v_d\/\\sqrt{2}$ \nand $\\langle H_u^0\\rangle\\equiv v_u\/\\sqrt{2}$,\nand the CP-odd Higgs mass $m_A$ [cf.~\\eqs{minbeta}{minconditions} below]. \nOne is always free to rephase the Higgs doublet fields such that $v_d$\nand $v_u$ are both real and positive.\n\nThe quantity, $v_d^2+v_u^2=\n4m_W^2\/g^2=(2G_F^2)^{-1\/2}\\simeq (246~{\\rm GeV})^2$, is fixed by the\nFermi constant, $G_F$, whereas the ratio\n\\begin{equation} \\label{eqtanbeta}\n\\tan \\beta = \\frac{v_u}{v_d}\n\\end{equation}\nis a free parameter such that $0\\leq\\beta\\leq\\pi\/2$.\nThe tree-level conditions for the scalar potential minimum\nrelate the diagonal and off-diagonal Higgs squared-mass parameters in terms\nof $m^2_Z=\\ifmath{\\tfrac14}(g^2+ g^{\\prime\\,2})(v_d^2+v_u^2)$, the angle~$\\beta$, and\nthe CP-odd Higgs mass $m_A$:\n\\begin{Eqnarray}\n\\sin 2\\beta &=& \\frac{2m_{12}^2}{m_1^2+m_2^2+2|\\mu|^2}=\\frac{2m_{12}^2}{m_A^2}\n\\,, \\label{minbeta} \\\\[6pt]\n\\ifmath{\\tfrac12} m_Z^2 &=& -|\\mu|^2+\\frac{m_1^2-m_2^2\\tan^2\\beta}{\\tan^2\\beta-1}\\,.\n\\label{minconditions}\n\\end{Eqnarray}\n\nAt this stage, one can already see the tension with naturalness, if\nthe SUSY parameters, $|m_1|$, $|m_2|$ and $|\\mu|$, are significantly larger than the scale of\nelectroweak symmetry breaking. In this case, $m_Z^2$ will be the\ndifference of two large numbers, requiring some fine-tuning of the\nSUSY parameters in order to produce the correct $Z$ boson mass. In\nthe literature, this tension is referred to as the little hierarchy\nproblem\\cite{little,little2,little3}, previous noted in Section~\\ref{quadratic}.\nOne must also guard against the existence of \ncharge and\/or color breaking global minima\ndue to non-zero vacuum expectation values for the squark and \ncharged slepton fields. This possibility can be avoided \nif the $A$-parameters are not unduly\nlarge\\cite{AlvarezGaume:1983gj,Frere:1983ag,Derendinger:1983bz,Gunion:1987qv,Chowdhury:2013dka,Hollik:2016dcm,Casas:1995pd}.\nAdditional constraints must also be respected to avoid directions in scalar field space in which\nthe full tree-level scalar potential can become unbounded from below\\cite{Casas:1995pd}.\n\n\\subsection{The MSSM parameter count}\n\\label{sec:count}\n\nThe total number of independent physical parameters\nthat define the MSSM (in its most general form) is\nquite large, primarily due to the\nsoft-supersymmetry-breaking sector. In particular, in the case of\nthree generations of quarks, leptons, and their superpartners,\n$M^2_{\\widetilde Q}$,\n$M^2_{\\widetilde U}$, $M^2_{\\widetilde D}$, $M^2_{\\widetilde L}$, and $M^2_{\\widetilde E}$\nare hermitian $3\\times 3$ matrices, and\n$A_U$, $A_D$, and $A_E$ are complex $3\\times 3$\nmatrices. In addition, $M_1$, $M_2$, $M_3$, $B$, and $\\mu$\nare in general complex parameters. Finally, as in the Standard Model, the\nHiggs-fermion Yukawa couplings, $\\lambda_f$ ($f\\!=\\!u$, $d$, and $e$),\nare complex $3\\times 3$ matrices that\nare related to the quark and lepton mass matrices via: $M_f=\\lambda_f\nv_f\/\\sqrt{2}$, where $v_e\\equiv v_d$ [with $v_u$ and $v_d$ as defined\nabove \\eq{eqtanbeta}].\n\nHowever, not all these parameters are physical.\nSome of the MSSM parameters can be eliminated by\nexpressing interaction eigenstates in terms of the mass eigenstates,\nwith an appropriate redefinition of the MSSM fields to remove unphysical\ndegrees of freedom. The analysis of Refs.\\cite{Dimopoulos:1995ju,Haber:2000jh} shows that the MSSM\npossesses 124 independent parameters. Of these, 18\ncorrespond to SM parameters\n(including the QCD vacuum angle, $\\theta_{\\rm QCD}$), one corresponds to\na Higgs sector parameter (the analogue of the SM\nHiggs mass), and 105 are genuinely new parameters of the model.\nThe latter include: five real parameters and three CP-violating phases in\nthe gaugino\/higgsino sector, 21 squark and slepton masses,\n36 real mixing angles to define the\nsquark and slepton mass eigenstates, and 40 CP-violating phases that\ncan appear in the squark and slepton interactions.\n\nUnfortunately, without additional restrictions on the 124 parameters,\nthe MSSM is not a\nphenomenologically viable theory. In particular, a generic point of\nthe MSSM parameter space typically exhibits:\n(i)~no conservation of the separate lepton numbers\n$L_e$, $L_\\mu$, and $L_\\tau$; (ii)~unsuppressed\nflavor-changing neutral currents (FCNCs)\\cite{Georgi:1986ku,Hall:1985dx};\nand (iii)~new sources of CP~violation\\cite{Khalil:2002qp} that are\ninconsistent with the experimental bounds.\nFor example, the strong suppression of FCNCs observed in nature implies\nthat the off-diagonal matrix elements of\nthe soft-SUSY-breaking squark and slepton squared-mass matrices\nare highly constrained\\cite{Chung:2003fi,RamseyMusolf:2006vr}.\n\nIn practice, various simplifying assumptions are imposed \non the SUSY-breaking sector to reduce the\nnumber of parameters to a more manageable form, such that\nthe constraints imposed by lepton and quark flavor changing and\nCP-violating processes are satisfied. For example,\nspecific models of gravity-mediated and gauge-mediated supersymmetry\nbreaking\\footnote{One of the benefits of GMSB models\n is that the SUSY-breaking is transmitted to the MSSM sector via\n gauge boson exchange, which is automatically flavor-conserving.} \nintroduce a small number of fundamental parameters that provide the\nsource for SUSY-breaking for the MSSM,\nconsistent with the constraints due to flavor and CP violation.\nMore details can be found in Ref.\\cite{susy}.\n\nAn alternative approach, called the phenomenological MSSM (pMSSM) has\nbeen introduced\\cite{Djouadi:2002ze,Berger:2008cq}, which attempts to\nidentify the parameters most relevant for phenomenology, subject to\na number of simplifying assumptions.\nThe pMSSM is governed by 19 independent real supersymmetric\nparameters: the three gaugino\nmass parameters $M_1$, $M_2$ and $M_3$, the Higgs sector parameters $m_A$ and\n$\\tan\\beta$, the Higgsino mass parameter $\\mu$, five squark and slepton\nsquared-mass parameters for the degenerate first and second\ngenerations ($M^2_{\\widetilde Q}$, $M^2_{\\widetilde U}$, $M^2_{\\widetilde D}$,\n$M^2_{\\widetilde L}$ and $M^2_{\\widetilde E}$), the five\ncorresponding squark and slepton squared-mass parameters for\nthe third generation, and three third-generation $A$-parameters\n($A_t$, $A_b$ and $A_\\tau$).\\footnote{In Ref.\\cite{deVries:2015hva}, the number of pMSSM parameters\nis reduced to ten by assuming one common squark mass parameter for the\nfirst two generations, a second common squark mass parameter for the third\ngeneration, a common slepton mass parameter, and a common third generation\n$A$ parameter.} \nThe first and second generation $A$-parameters can be neglected as their\nphenomenological consequences are negligible. Such an approach \nassumes that new sources of flavor violation and\/or CP-violation\nare either absent or negligible.\\footnote{The pMSSM approach has been\n recently extended to include additional CP-violating\nSUSY-breaking parameters in Ref.\\cite{Berger:2015eba}.}\n\n\n\\subsection{The MSSM particle spectrum}\n\\label{sec:MSSMspectrum}\n\\subsubsection{ Spin-1\/2 superpartners}\n\nThe superpartners of the gauge and Higgs bosons are fermions,\nwhose names are obtained by appending ``ino'' to the end of the\ncorresponding SM particle name. The gluino is the\ncolor-octet Majorana fermion partner of the gluon\nwith mass $M_{\\widetilde g}=|M_3|$.\nThe superpartners of the electroweak gauge\nand Higgs bosons (the gauginos and higgsinos)\ncan mix due to SU(2)$\\times$U(1) breaking effects. As a result,\nthe physical states of definite mass are model-dependent linear combinations\nof the charged or neutral gauginos and higgsinos,\ncalled charginos and neutralinos, respectively\n(sometimes collectively called electroweakinos).\nThe charginos are Dirac fermions, and\nthe neutralinos are Majorana fermions.\n\n\nThe tree-level mixing of the charged gauginos ($\\widetilde W^\\pm$) and \nhiggsinos ($\\widetilde H_u^+$ and $\\widetilde H_d^-$) is governed \nby a $2\\times 2$ complex\nmass matrix,\n\\begin{align}\nM_C\\equiv \\begin{pmatrix}\n M_2\\quad\n & gv_u\/\\sqrt{2} \\\\\n gv_d\/\\sqrt{2} \\quad\n &\\mu \\end{pmatrix}\\,.\n\\end{align}\nThe physical chargino states and their\nmasses are obtained by\nperforming a singular value decomposition\nof the complex matrix $M_C$ [cf.~\\eq{LTMR}]:\n\\begin{align}\nU^* M_C V^{-1}={\\rm diag}(M_{\\widetilde\\chi^+_1}\\,,\\,M_{\\widetilde\\chi^+_2})\\,,\n\\end{align}\nwhere $U$ and $V$ are unitary matrices.\nThe physical chargino states are Dirac fermions and are denoted by\n$\\widetilde\\chi^\\pm_1$ and $\\widetilde\\chi^\\pm_2$. These are linear combinations of the\ncharged gaugino and higgsino states determined\nby the matrix elements of $U$ and $V$.\nThe chargino masses correspond to the singular values of\n$M_C$, \\textit{i.e.}, the positive square roots\nof the eigenvalues of $M_C^\\dagger M_C$,\n\\begin{align}\n\\begin{split}\n\\hspace{-0.1in}\nM^2_{\\widetilde\\chi^+_1,\\widetilde\\chi^+_2}=&\n\\ifmath{\\tfrac12} \\biggl\\{ |\\mu|^2+|M_2|^2+2m_W^2\\\\\n&\\quad\\left.\n\\mp\n\\sqrt{\\left(|\\mu|^2+|M_2|^2+2m_W^2\\right)^2 \n-4 |\\mu M_2 - m_W^2 \\sin2\\beta|^2}\\,\\,\n\\right\\rbrace\\,,\n\\end{split}\n\\end{align}\nwhere the states are ordered such that $M_{\\widetilde\\chi^+_1} \\leq M_{\\widetilde\\chi^+_2}$.\nThe relative phase of $\\mu$ and $M_2$ is physical and potentially observable.\n\nThe tree-level mixing of the neutral gauginos ($\\widetilde B$ and\n$\\widetilde W^0$) and \nhiggsinos ($\\widetilde H_d^0$ and $\\widetilde H_u^0$) is\ngoverned by a $4\\times 4$ complex symmetric mass\nmatrix,\n\\begin{align}\nM_N\\equiv \\begin{pmatrix}\n M_1\\quad & 0 \\quad & -\\ifmath{\\tfrac12} g' v_d \\quad & \\phantom{-}\\ifmath{\\tfrac12} g' v_u \\\\\n 0 \\quad & M_2 \\quad & \\phantom{-}\\ifmath{\\tfrac12} g v_d \\quad & -\\ifmath{\\tfrac12} g v_u \\\\\n-\\ifmath{\\tfrac12} g' v_d \\quad & \\phantom{-}\\ifmath{\\tfrac12} g v_d \\quad & 0 \\quad & -\\mu \\\\\n\\phantom{-}\\ifmath{\\tfrac12} g' v_u \\quad & -\\ifmath{\\tfrac12} g v_u \\quad & -\\mu \\quad & 0 \\end{pmatrix}\\,.\n\\end{align}\nTo determine the physical neutralino states and their masses,\none must perform a\nTakagi-diagonalization\nof the complex symmetric matrix $M_N$ [cf.~\\eq{takagidef}]:\n\\begin{align}\nW^T M_N W={\\rm diag}(M_{\\widetilde\\chi^0_1}\\,,\\,M_{\\widetilde\\chi^0_2}\\,,\\,M_{\\widetilde\\chi^0_3}\\,,\\,M_{\\widetilde\\chi^0_4})\\,,\n\\end{align}\nwhere $W$ is a unitary matrix.\nThe physical neutralino states are Majorana fermions, and are denoted by\n$\\widetilde\\chi^0_i$ ($i=1,\\ldots 4$), where the states are ordered such that\n$M_{\\widetilde\\chi^0_1}\\leqM_{\\widetilde\\chi^0_2}\\leqM_{\\widetilde\\chi^0_3}\\leqM_{\\widetilde\\chi^0_4}$.\nThe $\\widetilde\\chi^0_i$ are the linear combinations of the\nneutral gaugino and higgsino states determined\nby the matrix elements of $W$.\nThe neutralino masses correspond to the singular values of\n$M_N$, \\textit{i.e.}, the positive square roots\nof the eigenvalues of $M_N^\\dagger M_N$. \n\n\n\\subsubsection{Spin-0 superpartners}\n\nThe superpartners of the quarks and leptons are spin-zero\nbosons: the squarks, charged sleptons,\nand sneutrinos, respectively.\nFor a given Dirac fermion $f$, there are two superpartners, $\\widetilde\nf_L$ and $\\widetilde f_R$, where the $L$ and $R$ subscripts simply identify\nthe scalar partners that are related by supersymmetry to the left-handed and\nright-handed fermions, $f_{L,R}\\equiv\\ifmath{\\tfrac12}(1\\mp\\gamma_5)f$, respectively.\n(There is no $\\widetilde\\nu_R$ in the MSSM.)\nHowever, $\\widetilde f_L$--$\\widetilde f_R$ mixing is possible,\nin which case $\\widetilde f_L$ and $\\widetilde f_R$ are not mass\neigenstates. \n\nWe first consider the squarks and the sleptons.\nFor three generations of squarks, one\nmust diagonalize $6\\times 6$ matrices corresponding\nto the basis $(\\widetilde q_{iL}, \\widetilde q_{iR})$,\nwhere $i=1,2,3$ are the generation\nlabels.\nFor simplicity, only the one-generation case is illustrated\nin detail below.\n\nUsing the notation of the third family, the one-generation\ntree-level squark squared-mass matrix is given by\n\\begin{align}\n\\mathcal{M}^2 =& \\begin{pmatrix}\n M^2_{\\widetilde Q}+ m^2_q+ L_q\\quad\n & m_q X_q^* \\\\\n m_q X_q\\quad\n &M^2_{\\widetilde R}+ m^2_q+ R_q \\end{pmatrix}\\,,\\label{sqmassmat}\n \\end{align} \nwhere\n\\begin{align}\nX_q\\equiv A_q-\\mu^* (\\cot\\beta)^{2T_{3q}}\\,,\n\\end{align} \\label{Xtdef}\nand \n\\begin{align}\nT_{3q}=\\begin{cases} \\phantom{-}\\ifmath{\\tfrac12}\\,,\\quad \\text{for $q=t$}\\,,\\\\ -\\ifmath{\\tfrac12}\\,,\\quad \\text{for $q=b$}.\\end{cases}\n\\end{align}\n\n\nThe diagonal squared-masses are governed by soft-SUSY-breaking\nsquared-masses $M^2_{\\widetilde Q}$ and $M^2_{\\widetilde R}\\equiv\nM^2_{\\widetilde U}$ [$M^2_{\\widetilde D}$] for $q=t$~[$b$], the\ncorresponding quark masses $m_t$ [$m_b$], and electroweak correction terms:\n\\begin{align}\nL_q& \\equiv\n(T_{3q}-e_q\\sin^2\\theta_W)m_Z^2\\cos 2\\beta\\,,\\\\\nR_q& \\equiv\ne_q\\sin^2\\theta_W \\,m_Z^2\\cos 2\\beta\\,,\n\\end{align}\nwhere $e_q=\\tfrac23$ [$-\\tfrac13$] for $q=t$ [$b$].\n\nThe off-diagonal squark squared-masses are\nproportional to the corresponding quark masses and depend on\n$\\tan\\beta$, the\nsoft-SUSY-breaking $A$-parameters and the higgsino mass parameter\n$\\mu$.\nAssuming that the $A$-parameters\nare parametrically of the same order (or smaller) relative\nto other SUSY-breaking mass parameters, it then follows that\n$\\widetilde q_L$--$\\widetilde q_R$ mixing effects\nare small, with the possible exception of the third generation,\nwhere mixing can be enhanced by factors of $m_t$ and $m_b\\tan\\beta$.\n\n\n\nIn the case of third generation $\\widetilde q_L$--$\\widetilde q_R$\nmixing, the mass eigenstates (denoted by $\\widetilde q_1$ and\n$\\widetilde q_2$, with $m_{\\tilde q_1}m_Z$), the predicted upper bound for $m_h$\nis approximately given by\\cite{Haber:1996fp}\n\\begin{align}\nm_{h}^2{~\\raise.15em\\hbox{$<$}\\kern-.85em\\lower.35em\\hbox{$\\sim$}~} m_Z^2+\\frac{3g^2 m_t^4}{8\\pi^2m_W^2}\\left[\\ln\\left(\\frac{M_S^2}{m_t^2}\\right)+\\frac{X_t^2}{M_S^2}\n\\left(1-\\frac{X_t^2}{12M_S^2}\\right)\\right]\\,, \\label{hradcorr}\n\\end{align}\nwhere $X_t\\equiv A_t-\\mu\\cot\\beta$ governs stop mixing (taking $A_t$\nand $\\mu$ real for simplicity).\nThe Higgs mass upper limit is saturated when\n$\\tan\\beta$ is large [{\\it i.e.}, $\\cos^2 (2\\beta) \\sim 1$] and $X_t=\\sqrt{6}\\,\nM_S$, which defines the so-called maximal mixing scenario.\n\nA more complete treatment of the radiative corrections\\cite{Draper:2016pys}\nshows that\n\\eq{hradcorr} somewhat overestimates the true upper bound of $m_{h}$.\nThese more refined computations, which incorporate\nrenormalization group improvement, and the two-loop and\nleading three-loop contributions, yield an upper bound of $m_{h}{~\\raise.15em\\hbox{$<$}\\kern-.85em\\lower.35em\\hbox{$\\sim$}~} 135$~GeV in the\nregion of\nlarge $\\tan\\beta$ (with an accuracy of a few GeV)\nfor $m_t=175$~GeV and $M_S{~\\raise.15em\\hbox{$<$}\\kern-.85em\\lower.35em\\hbox{$\\sim$}~} 2$~TeV\\cite{Draper:2016pys},\nwhich is quite close to the observed value of the Higgs mass!\n\nIn certain cases, radiative corrections also can significantly modify the tree-level\nYukawa couplings. For a review of such effects, see e.g., Ref.\\cite{Carena:2002es}. \n\n\n\n\n\\subsection{Unification of gauge couplings}\n\\label{sec:MSSMGU}\n\n\n\nGrand unification theory (GUT) predicts the unification of gauge couplings at some very high energy scale\\cite{Raby,guts,Langacker:1980js,Ross}. \nThe running of the couplings is dictated by the particle content of the effective theory that resides below the GUT scale. \nHowever, attempts to embed the Standard Model in an SU(5) or SO(10)\nunified theory do not quite succeed.\nIn particular, the three running gauge couplings (the strong QCD\ncoupling $g_s$ and the electroweak gauge couplings $g$ and $g'$) do not meet at a point, as shown by the dashed lines in Fig.~\\ref{fig:GUT}.\nIn contrast, in the case of the MSSM with superpartner masses of order\n1 TeV, the renormalization group evolution is modified above the\nSUSY-breaking scale. In this case, unification of gauge couplings\ncan be (approximately) achieved as illustrated by the red and blue\nlines in Fig.~\\ref{fig:GUT}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{images\/unified_couplings.eps}\n\\caption{\\small\nRenormalization group evolution of the inverse gauge couplings $\\alpha_a^{-1}(Q)$ in the\nStandard Model (dashed lines) and the MSSM (solid lines). In the MSSM\ncase, $\\alpha_3(m_Z)$ is varied between 0.121 and 0.117, and the\nsupersymmetric particle mass thresholds are between 500 GeV and 1.5 TeV, for the\nlower and upper solid lines, respectively. Two-loop effects are\nincluded. Taken from Ref.\\cite{Martin:1997ns}.\n}\n\\label{fig:GUT}\n\\end{figure}\n\nA quantitative assessment of the success of gauge coupling unification\ncan be performed as follows. \nSince the electroweak gauge couplings $g$ and $g'$ are very well\nmeasured, first focus on these two couplings. For a given low-energy\neffective theory (below the GUT scale), we use the renormalization\ngroup equations (RGEs) to determine the couplings $g$ and $g'$ as a\nfunction of the energy scale. We then define $M_{\\rm GUT}$ to be the\nscale at which these two couplings meet. \n\n We now assume that the unification of the three\ngauge couplings, $g_s$, $g$ and $g'$ occurs at $M_{\\rm GUT}$. Using\nthe RGEs for the gauge couplings, we can now run $g_s$ down to the\nelectroweak scale and compare with the experimentally measured value.\n\n\n\\subsubsection{Normalization of the U(1)$_{\\rm Y}$ coupling}\nIn electroweak theory, the overall normalization of the U(1)$_{\\rm Y}$ coupling is a matter of convention. But, if the GUT group is simple and nonabelian, then the relative normalization of the U(1)$_{\\rm Y}$ coupling to the SU(2) gauge coupling is fixed. \nWe denote the SU(3)$\\times$SU(2)$\\times$U(1)$_{\\rm Y}$ gauge couplings using the proper GUT normalization by $g_3$, $g_2$ and $g_1$ respectively. Our task is to relate $g_1$ with $g'$.\nTo do so, let us begin by\nconsidering the covariant derivative,\n\\begin{align}\nD_\\mu=\\partial_\\mu+i\\sum_a g_a T^a A_\\mu^a\\,.\n\\end{align}\nIf the gauge group is a direct product group, then different sets of generators $T^a$ are associated with with the different group factors, and we must use the appropriate $g_a$ depending on which generator it multiplies. \nIn particular, for SU(2)$\\times$U(1)$_{\\rm Y}$ (below the GUT scale), \n\\begin{align}\ng_a T^a A_\\mu^q\\ni gT^3 W_\\mu^3+g'\\frac{Y}{2}B_\\mu\\,.\n\\end{align}\nAbove the GUT scale, the corresponding terms of the covariant derivative are\n\\begin{align}\ng_a T^a A_\\mu^q\\ni g_U( T^3 W_\\mu^3+T^0B_\\mu)\\,,\n\\end{align}\nwhere $g_U$ is the gauge coupling of the unifying GUT group and $T^0$ is the properly normalized hypercharge generator. \nIn particular, the generators of the GUT group satisfy\n\\begin{align}\n\\Tr(T^a T^b) =T(R) \\delta^{ab}\\,,\\label{tab}\n\\end{align}\nwhere $T(R)$ is a constant that depends on the representation\n$R$.\\footnote{Once $T(R)$ is fixed for one representation, it is then\n determined for all other representations. It is standard practice\n to fix $T(R)=\\ifmath{\\tfrac12}$ for the defining (fundamental) representation, although the\nargument presented below is independent of this choice.} \nWe now set the two covariant derivatives above equal at the GUT scale,\n\\begin{align}\n g_U( T^3 W_\\mu^3+T^0B_\\mu)=gT^3 W_\\mu^3+g'\\frac{Y}{2}B_\\mu\\,.\n \\end{align}\nNoting that $g_U=g_3=g_2=g_1$ at the GUT scale, it\nfollows that\n $g_2=g$ and $g_1 T^0=g'(Y\/2)$. Since $T(R)$ only depends on the\n representation $R$, \\eq{tab} yields $\\Tr (T^3)^2=\\Tr(T^0)^2$. Thus,\n\\begin{align}\n g_1^2=g^{\\prime\\,2}\\,\\frac{\\Tr Y^2}{4\\Tr(T^3)^2}\\,.\\label{gone}\n\\end{align}\nThe relevant quantum numbers are provided in\nTable~\\ref{tab:two-component_fields}. \n \nThe traces in \\eq{gone} are evaluated by summing over one generation of fermions, under the assumption that it is made up of\ncomplete irreducible representations of the GUT group.\\footnote{In an\n SU(5) GUT, one\n generation of fermions make up a 10-dimensional and the complex\n conjugate of a 5-dimensional \n representation of SU(5). In an SO(10) GUT, one generation of fermions (including\n the right-handed neutrino) comprise a 16 dimensional spinor\n representation of SO(10).}\nUsing the results of Table~\\ref{tab:two-component_fields}, we simply\nadd up the last two columns. Including the appropriate color factor\nof 3 when tracing over the suppressed color index, \nwe obtain\n$\\Tr (T^3)^2=2$ and $\\Tr Y^2=\\tfrac{40}{3}$. Thus, \n\\eq{gone} yields\n \\begin{align}\ng_1^2=\\tfrac53 g^{\\prime\\,2}\\,.\n\\end{align}\n\n\\begin{table}\n\\centering\n\\caption{\\small The $T_3$ and $Y$ quantum numbers of the two-component\n fermion fields that make up one generation of SM fermions. In\n computing the corresponding traces, one must not forget the color\n factor of 3 that arises when tracing over the (suppressed) color\n index. \\label{tab:two-component_fields}}\n\\vskip 0.06in\n\\begin{tabular}{|ccccc|} \\hline\nTwo-component fields & $T_3$ & $Y$ & $\\Tr (T^3)^2$ & $\\Tr Y^2$ \\\\ \\hline\n$\\psi_{Q_1}$ & $\\phantom{-}\\ifmath{\\tfrac12}$ & $\\phantom{-}\\tfrac13$ & $3(\\tfrac14)$ & $3(\\tfrac19)$ \\\\[5pt]\n$\\psi_{Q_2}$ & $-\\ifmath{\\tfrac12}$ & $\\phantom{-}\\tfrac13$ & $3(\\tfrac14)$ & $3(\\tfrac19)$ \\\\[5pt]\n$\\psi_{U}$ & $\\phantom{-} 0$ & $-\\tfrac{4}{3}$ & $3(0)$ & $3(\\tfrac{16}{9})$ \\\\[5pt]\n$\\psi_{D}$ & $\\phantom{-} 0$ & $\\phantom{-} \\tfrac23$ & $3(0)$ & $3(\\tfrac{4}{9})$ \\\\[5pt]\n$\\psi_{L_1}$ & $\\phantom{-}\\ifmath{\\tfrac12}$ & $-1$ & $\\tfrac14$ & $1$ \\\\[5pt]\n$\\psi_{L_2}$ & $-\\ifmath{\\tfrac12}$ & $-1$ & $\\tfrac14$ & $1$ \\\\[5pt]\n$\\psi_{E}$ & $\\phantom{-} 0$ & $\\phantom{-} 2$ & $0$ & $4$ \\\\ \\hline\n \\end{tabular}\n \\end{table}\n\n\n\\subsubsection{Gauge coupling running}\nWe now examine the running of the gauge couplings\nin the one-loop approximation, where the gauge couplings $g_i$ obey the differential equation,\n \\begin{align}\n \\frac{dg_i^2}{dt}=\\frac{b_i g_i^4}{16\\pi^2}\\,,\\qquad \\text{for $i=1,2,3$},\\label{gRGE}\n\\end{align}\n where $t=\\ln Q^2$ and $Q$ is the energy scale. The solution to\n \\eq{gRGE} is\n \\begin{align}\n \\frac{1}{g_i^2(m_Z)}=\\frac{1}{g_U^2}-\\frac{b_i}{16\\pi}\\ln\\left(\\frac{m_Z^2}{M_{\\rm GUT}^2}\\right)\\,,\\label{RGEsol}\n\\end{align}\n where $M_{\\rm GUT}$ is the GUT scale at which the three gauge\n couplings unify. Using \\eq{RGEsol}, the following two equations are obtained:\n\\begin{align}\n \\sin^2\\theta_W(m_Z)=&\\frac{g^{\\prime\\,2}(m_Z)}{g^2(m_Z)+g^{\\prime\\,2}(m_Z)}=\\frac{\\tfrac35 g_1^2(m_Z)}{g^2(m_Z)+\\frac35 g_1^2(m_Z)} \\nonumber \\\\\n=&\\frac{3}{8}-\\frac{5}{32\\pi}\\,\\alpha(m_Z)(b_1-b_2)\\ln\\left(\\frac{M_{\\rm GUT}^2}{m_Z^2}\\right)\\,,\\label{sinw} \\\\[5pt]\n \\ln\\left(\\frac{M_{\\rm GUT}^2}{m_Z^2}\\right)=&\\frac{32\\pi}{5b_1+3b_2-8b_3}\\left(\\frac{3}{8\\alpha(m_Z)}-\\frac{1}{\\alpha_s(m_Z)}\\right)\\,,\\label{log}\n \\end{align}\n where $e=g\\sin\\theta_W$, $\\alpha\\equiv e^2\/4\\pi$ and $\\alpha_s\\equiv g_s^2\/4\\pi$.\n\nIt is convenient to introduce the parameter,\n\\begin{align} x\\equiv \\frac{1}{5}\\left(\\frac{b_2-b_3}{b_1-b_2}\\right)\\,.\n \\end{align}\n Then, \\eqs{sinw}{log} yield,\n\\begin{align}\n \\sin^2\\theta_W(m_Z)=\\frac{1}{1+8x}\\left[3x+\\frac{\\alpha(m_Z)}{\\alpha_s(m_Z)}\n\\right]\\,.\n\\end{align}\nOnce we know the value of $x$,\nwe can use the above equation to determine $\\alpha_s(m_Z)$ given the\nvalues of $\\sin^2\\theta_W$ and $\\alpha$, evaluated at $m_Z$,\n\\begin{equation}\n\\alpha_s(m_Z)=\\frac{\\alpha(m_Z)}{(1+8x)\\sin^2\\theta_W(m_Z)-3x}\\,.\\label{alphastrong}\n\\end{equation}\n\nThe value of $x$ is determined from the values of the $b_i$, which are given by the following formula,\n\\begin{align}\nb_i=\\tfrac{2}{3}T_f(R_k)\\prod_{j\\neq k} d_f(R_j)+\\tfrac{1}{6}T_s(R_k)\\prod_{j\\neq k} d_s(R_j)-\\tfrac{11}{3} C_2(G_i)\\,,\\label{bi}\n\\end{align}\nwhere $f$, $s$ stand for fermions and scalars, respectively, $d(R)$ is the dimension of the representation $R$, and the generators in representation $R$ satisfy,\n\\begin{align}\n\\Tr(T^a T^b)= T(R)\\delta^{ab}\\,,\\qquad\\quad (T^a T^a)_{ij}=C_2(G)\\delta_{ij}\\,.\n\\end{align}\nNote that,\n\\begin{align}\nT(R_1)=\\left[\\sqrt{\\tfrac{3}{5}}\\,\\ifmath{\\tfrac12} Y\\right]^2=\\tfrac{3}{20}Y^2\\,,\n\\end{align}\nwhere we have employed the properly normalized hypercharge generator, $\\sqrt{3\/5}\\,(Y\/2)$.\nIn addition, $C_2({\\rm G})=N$ for G$=$SU($N$), and $C_2({\\rm G})=0$ for G$=$U(1).\n\nOne can now assess the success or failure of gauge coupling unification\nin the SM and in the MSSM. For details, see Problems~\\ref{pr:GUT1}\nand \\ref{pr:GUT2}. As advertised in Fig.~\\ref{fig:GUT}, the gauge\ncouplings do not unify when the SM is extrapolated to the GUT scale.\nIn contrast, in the MSSM, the modified running of the gauge couplings\ndue to the supersymmetric partners of the SM particles results in\napproximate unification.\\footnote{For a more precise analysis, we\n should extend the calculations of this subsection to include\n two-loop running of the gauge couplings\\cite{Castano:1993ri}. One must also properly\n treat threshold corrections at the TeV scale\\cite{Martens:2011uha,Allanach:2014nba} (due to mass splittings\n among superpartners) and at the GUT scale\\cite{Lucas:1995ic}. The latter are quite\n model-dependent and allows some wiggle room in achieving precise\n gauge coupling unification.}\nThis success has often been touted as one of\nthe motivations for TeV-scale supersymmetry.\n\n\n\n\n\\subsection{Problems}\n\n\\begin{problem}\n\\label{pr:spectra}\nStarting with the SUSY Lagrangian for SUSY Yang Mills theory coupled\nto matter given in \\eq{eq:LSUSYcomponents}, \neliminate the auxiliary fields and obtain the Lagrangian of the MSSM prior to\nSUSY-breaking. For simplicity, you may consider only one generation\nof quarks and leptons and their superpartners.\nThen add in the soft-SUSY-breaking terms to obtain the\ncomplete MSSM Lagrangian. Using this result, verify the mass spectrum\nof the supersymmetric particles obtained in\nSection~\\ref{sec:MSSMspectrum}. \n\\end{problem}\n\n\\begin{problem}\nUsing the results of Problem~\\ref{pr:spectra}, verify the results\nobtained in Section~\\ref{higgssector} for the MSSM Higgs sector.\nWrite out the Feynman rules for the interaction of the Higgs bosons \nwith the gauge bosons and with the quarks and leptons.\n\\end{problem} \n \n\\begin{problem}\nUsing the results of Problem~\\ref{pr:spectra}, one can obtain the complete set of Feynman rules for the MSSM with one\ngeneration of quarks and leptons and their superpartners.\nWork out as many of the rules as you can and check your results against\nRef.\\cite{Rosiek:1989rs}. \n\\end{problem}\n\n\n\\begin{problem}\n\\label{pr:GUT1}\nAssuming $N_g$ generations of the quarks and leptons and $N_h$ copies\nof the SM Higgs boson, use \\eq{bi} to obtain\n\\begin{align}\nb_3=&\\tfrac{4}{3}N_g-11\\,,\\nonumber \\\\\nb_2=&\\tfrac{1}{6}N_h+\\tfrac{4}{3}N_g-\\tfrac{22}{3}\\,,\\nonumber \\\\\nb_1=&\\tfrac{1}{10}N_h+\\tfrac{4}{3}N_g\\,.\\nonumber\n\\end{align}\nFor the SM, we have $N_g=3$ and $N_h=1$. Check that $b_3=-7$, $b_2=-\\tfrac{19}{6}$ and $b_1=\\tfrac{41}{10}$. Consequently,\n\\begin{align}\nx=\\frac{23}{218}=0.1055\n\\end{align}\nIn particular, note that $x$ is independent of $N_g$.\n\\label{pr:bs}\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:GUT2}\n Show that the SM results of Problem~\\ref{pr:bs} are modified in the\n MSSM as follows:\n \\begin{align}\nb_3=&2N_g-9\\,,\\nonumber \\\\\nb_2=&\\tfrac{1}{2}N_h+2N_g-6\\,,\\nonumber \\\\\nb_1=&\\tfrac{3}{10}N_h+2N_g\\,.\\nonumber\n\\end{align}\nFor the MSSM, we have $N_g=3$ and $N_h=2$. Verify that $b_3=-3$, $b_2=1$ and $b_1=\\tfrac{33}{5}$, and consequently,\n$x=\\tfrac17$. Using the values for $\\alpha(m_Z)$ and $\\sin^2\\theta_W(m_Z)$ given in Ref.\\cite{pdg},\nevaluate $\\alpha_s$ using \\eq{alphastrong}.\nShow that for $x=\\tfrac17$ (as predicted by the\nMSSM), one obtains a value for $\\alpha_s(m_Z)$ that is quite close to\nthe current world average\\cite{pdg}. Using $x=0.1055$, check that the\ncorresponding SM prediction for $\\alpha_s(m_Z)$ is significantly lower than the observed value.\n\\end{problem}\n\n\n\n\n\\section{Superspace and Superfields}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\label{sec:superspace}\n\n\nIn the section we introduce superspace coordinates $\\theta$ and $\\theta^\\dagger$.\nThe concept of a supersymmetry transformation is then realized as a translation in superspace.\nWe construct superfields\\cite{Ferrara:1974ac,Salam:1974jj,Salam:1976ib}, which can be expanded in powers of $\\theta$ and\n$\\theta^\\dagger$; the corresponding expansion coefficients are the fields\nof a super\\-multiplet. By introducing the spinor covariant derivative, one is\nable to define the derivative of a superfield that is covariant with\nrespect to SUSY transformations. This allows us to define an\nirreducible chiral\nsuperfield by imposing a derivative constraint.\n\nEmploying this formalism, we demonstrate how to construct a\nSUSY Lagrangian for chiral superfields, and \nand show that the\nsupersymmetric action can be expressed as an integral over superspace. \nFinally, we discuss the improved ultraviolet behavior of SUSY and introduce the\ncelebrated non-renormalization theorem of $N=1$ supersymmetry\\cite{GRS,SeibergNR}.\n \n\\subsection{Superspace coordinates and translations}\n\\label{sec:supercoords}\n\nIn Section~\\ref{sec:SUSYalgebra} we indicated that we expect a SUSY\ntranslation to be similar to a space-time translation, where the SUSY generators $Q$, $Q^\\dagger$ replace the $P^\\mu$ of ordinary space-time translations:\n\\begin{align}\n \\delta_{\\xi}\\Phi(x)=i\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\Phi(x)\\bigr]\\,,\n \\end{align}\n for $\\Phi=A$, $\\psi$ or $F$. \n But what exactly is being translated? \n\\clearpage\n\nIn this subsection,\n we extend spacetime by introducing Grassmann coordinates, $\\theta^\\alpha$ and $\\theta^\\dagger_{\\dot\\alpha}$. The result is an 8-dimensional \\textit{superspace} with coordinates\n$(x^\\mu\\,,\\,\\theta^\\alpha\\,,\\,\\theta^\\dagger_{\\dot\\alpha})$. The\nGrassmann coordinates are anticommuting coordinates; i.e., they satisfy anticommutation relations,\n \\begin{align}\n \\{\\theta^\\alpha\\,,\\,\\theta^\\beta\\}=\\{\\theta^\\dagger_{\\dot\\alpha}\\,,\\,\\theta^\\dagger_{\\dot\\beta}\\}=\\{\\theta^\\alpha\\,,\\,\\theta^\\dagger_{\\dot\\beta}\\}=0\\,.\n \\end{align}\n\nOne can also define derivatives with respect to $\\theta$ and\n$\\theta^\\dagger$. It is convenient to introduce the following notation,\n\\begin{equation} \\label{dth1}\n\\partial_{\\alpha}\\equiv \\frac{\\partial}{\\partial\\theta^\\alpha}\\,,\\qquad\\qquad\n\\partial^\\dagger_{\\dot\\alpha}\\equiv \\frac{\\partial}{\\partial{\\theta^\\dagger}^{\\dot\\alpha}}\\,.\n\\end{equation}\nThe derivatives with respect to $\\theta$ and $\\theta^\\dagger$ are\ndefined in the obvious way,\n\\begin{equation}\n \\partial_\\alpha\\theta^\\beta=\\delta_\\alpha^\\beta\\,,\\qquad\\qquad \\partial^\\dagger_{\\dot\\alpha}{\\theta^\\dagger}^{\\dot\\beta}=\\delta_{\\dot\\alpha}^{\\dot\\beta}\\,.\n\\end{equation}\nIt then follows that\n\\begin{align}\n\\partial_\\alpha\\theta_\\beta=\\partial_\\alpha(\\epsilon_{\\beta\\gamma}\\theta^\\gamma)=-\\epsilon_{\\alpha\\beta}\\,,\\qquad \\partial^\\dagger_{\\dot\\alpha}\\theta^\\dagger_{\\dot\\beta}=\\partial^\\dagger_{\\dot\\alpha}(\\epsilon_{\\dot\\beta\\dot\\gamma}\\theta^{\\dagger\\dot\\gamma})=-\\epsilon_{\\dot\\alpha\\dot\\beta}\\,.\\label{dteps}\n\\end{align}\n\nDerivatives with respect to $\\theta$ and $\\theta^\\dagger$ satisfy a\nmodified Leibniz rule, \n\\begin{Eqnarray}\n\\partial_\\alpha(fg)&=&(\\partial_\\alpha f)g+(-1)^{\\varepsilon(f)}f(\\partial_\\alpha g)\\,,\\\\\n\\partial^\\dagger_{\\dot\\alpha}(fg)&=&(\\partial^\\dagger_{\\dot\\alpha} f)g+(-1)^{\\varepsilon(f)}f(\\partial^\\dagger_{\\dot\\alpha} g)\\,,\n\\end{Eqnarray}\nwhere \n\\begin{equation}\n\\varepsilon(f)=\\begin{cases} 0\\,,&\\quad \\text{if $f$ is Grassmann even}\\,,\\\\ 1\\,,&\\quad \\text{if $f$\n is Grassmann odd}\\,, \\end{cases}\n\\end{equation}\nand $f$ is Grassmann even [odd] if it is a product of an even\n[odd] number of anticommuting quantities.\nFor example,\n\\begin{Eqnarray}\n\\partial_\\alpha(\\theta\\theta)&=&\\partial_\\alpha\\bigl(\\epsilon_{\\gamma\\beta}\\theta^\\gamma\\theta^\\beta\\bigr)=\\epsilon_{\\gamma\\beta}(\\delta^\\gamma_\\alpha\\theta^\\beta-\\delta_\\alpha^\\beta\\theta^\\gamma)=2\\theta_\\alpha\\,,\\label{partialtt}\\\\\n\\partial_{\\dot\\alpha}(\\theta^\\dagger\\thetabar)&=&\n\\partial_{\\dot\\alpha}\\bigl(\\epsilon_{\\dot\\beta\\dot\\gamma}\n\\theta^{\\dagger\\dot\\gamma}\\theta^{\\dagger\\dot\\beta}\\bigr)=\n\\epsilon_{\\dot\\beta\\dot\\gamma}(\\delta^{\\dot\\gamma}_{\\dot\\alpha}\\theta^{\\dagger\\dot\\beta}-\\delta_{\\dot\\alpha}^{\\dot\\beta}\\theta^{\\dagger\\dot\\gamma})\n=-2\\theta_{\\dot\\alpha}^\\dagger\\,.\\label{partialtdtd}\n\\end{Eqnarray}\n\nLikewise, one conventionally defines,\n\\begin{align}\n\\partial^{\\alpha}\\equiv \\frac{\\partial}{\\partial\\theta_\\alpha}\\,,\\qquad\n\\partial^{\\dagger\\dot\\alpha}\\equiv \\frac{\\partial}{\\partial\\theta^\\dagger_{\\dot\\alpha}}\\,.\\label{dth2}\n\\end{align}\nHowever, one needs to be careful since this notation leads to an unexpected minus sign when relating\nthe derivatives of \\eqs{dth1}{dth2},\n\\begin{equation}\n\\partial^\\alpha =-\\epsilon^{\\alpha\\beta}\\partial_\\beta\\,,\\qquad\n\\partial^{\\dagger\\dot\\alpha}=-\\epsilon^{\\dot\\alpha\\dot\\beta}\\partial^\\dagger_{\\dot\\beta}\\,. \\label{eq:partialsign}\n\\end{equation}\nThis is the one case where the rule for raising a spinor index given\nin \\eq{raiseindex} does \\textit{not} apply.\n\n\n\nIn order to define translations in superspace, we shall\ngeneralize the translation operator $\\exp(ix\\!\\cdot\\! P)$ to the super-translation operator,\n \\begin{align}\n G(x,\\theta,\\theta^\\dagger)=\\exp(ix\\!\\cdot\\! P+\\theta Q+\\theta^\\dagger Q^\\dagger)\\,.\n \\end{align}\n We can now extend the field operator, $\\Phi(x)=\\exp(ix\\!\\cdot\\!\n P)\\Phi(0)\\exp(-ix\\!\\cdot\\! P)$ to a \\textit{superfield} operator,\n \\begin{align}\n \\Phi(x,\\theta,\\theta^\\dagger)=G(x,\\theta,\\theta^\\dagger)\\Phi(0,0,0)G^{-1}(x,\\theta,\\theta^\\dagger)\\,.\n \\end{align}\n %\n In this way, we can realize a supersymmetry transformation as a translation in superspace.\n %\n\n Using the Baker-Campbell-Hausdorff formula\\cite{BrianHall},\n \\begin{equation} \\label{BCH}\n\\exp(A)\\exp(B)=\\exp\\bigl(A+B+\\ifmath{\\tfrac12}[A\\,,\\,B]+\\cdots\\bigr)\\,, \n\\end{equation}\none can prove (see Problem \\ref{pr:two_super_translations}),\n \\begin{align}\n G(y,\\xi,\\xi^\\dagger)G(x,\\theta,\\theta^\\dagger)=G\\bigl(x+y+i(\\xi\\sigma\\theta^\\dagger-\\theta\\sigma\\xi^\\dagger),\\xi+\\theta,\\xi^\\dagger+\\theta^\\dagger\\bigr)\\,. \\label{GG}\n \\end{align}\n Note the appearance in \\eq{GG} of an extra non-trivial spacetime translation, $ i (\\xi \\sigma \\theta^\\dagger - \\theta \\sigma \\xi^\\dagger)$.\n %\nHence, it follows that\n \\begin{align}\n \\begin{split}\n &G(y,\\xi,\\xi^\\dagger)\\Phi(x,\\theta,\\theta^\\dagger) G^{-1}(y,\\xi,\\xi^\\dagger) \\\\\n & \\qquad = \\Phi\\bigl(x+y+i(\\xi\\sigma\\theta^\\dagger-\\theta\\sigma\\xi^\\dagger),\\xi+\\theta,\\xi^\\dagger+\\theta^\\dagger\\bigr)\\,.\n\\end{split}\n\\label{eq:Phixy}\n\\end{align}\nFor infinitesimal $y$, $\\xi$ and $\\xi^\\dagger$, we can approximate\n\\begin{equation}\nG(y,\\xi,\\xi^\\dagger)\\simeq \\mathds{1}+i(y\\!\\cdot\\! P+\\xi Q+\\xi^\\dagger Q^\\dagger)\\,,\n\\end{equation}\nwhich allows us to rewrite the left-hand side of \\eq{eq:Phixy} as\n\\begin{align}\n\\begin{split}\n& G\\of{y,\\xi,\\xi^\\dagger} \\Phi\\of{ x,\\theta,\\theta^\\dagger} G^{-1}\\of{y,\\xi,\\xi^\\dagger} \\\\\n&\\quad \\simeq \\of{ \\mathds{1}+i\\of{y\\!\\cdot\\! P+\\xi Q+\\xi^\\dagger Q^\\dagger} } \\Phi\\of{x,\\theta,\\theta^\\dagger} \\of{ \\mathds{1} - i \\of{ y\\!\\cdot\\! P + \\xi Q + \\xi^\\dagger Q^\\dagger }}\n\\end{split} \\nonumber\n\\\\\n& \\quad \\simeq \\Phi\\of{x,\\theta,\\theta^\\dagger} \n+ i y_\\mu \\sqof{ P^\\mu, \\Phi } + i \\sqof{ \\xi Q, \\Phi } + i \\sqof{ \\xi^\\dagger Q^\\dagger, \\Phi }.\n\\label{eq:GPhiG}\n\\end{align}\nOne can also Taylor expand the right-hand side of \\eq{eq:Phixy}, which\nto first order yields\n\\begin{align}\n\\begin{split}\n& \\Phi\\bigl(x+y+i(\\xi\\sigma\\theta^\\dagger-\\theta\\sigma\\xi^\\dagger),\\xi+\\theta,\\xi^\\dagger+\\theta^\\dagger\\bigr)\n\\\\\n & \\qquad \\qquad = \\Phi(x,\\theta,\\theta^\\dagger) + \\bigl[y^\\mu+i(\\xi\\sigma^\\mu\\theta^\\dagger-\\theta\\sigma^\\mu\\xi^\\dagger)\\bigr]\\partial_\\mu \\Phi(x,\\theta,\\theta^\\dagger) \\\\\n& \\qquad \\qquad \\quad \\qquad \\qquad\\,\\,\\,\\, +\\bigl(\\xi^\\alpha\\partial_\\alpha+\\xi^\\dagger\\partial^{\\dagger\\dot\\alpha}\\bigr)\\Phi(x,\\theta,\\theta^\\dagger)\\,,\n \\end{split}\n \\label{eq:Phixy2}\n \\end{align}\nwhere we have employed the derivatives defined in \\eq{dth1}.\nComparing the first-order terms of eqns.~(\\ref{eq:GPhiG}) and (\\ref{eq:Phixy2}),\n we end up with expressions for the following commutators,\n \\begin{align}\n \\bigl[\\Phi\\,,\\,P_\\mu\\bigr]&= i\\,\\partial_\\mu\\Phi\\,, \\label{eq:PPhi} \\\\\n \\big[\\Phi\\,,\\,\\xi Q\\bigr]&= i\\,\\xi^\\alpha\\left(\\partial_{\\alpha}+i(\\sigma^\\mu\\theta^\\dagger)_\\alpha\\partial_\\mu\\right)\\Phi\\,, \\label{eq:QPhi} \\\\\n \\big[\\Phi\\,,\\,\\xi^\\dagger Q^\\dagger\\bigr]&= -i\\left(\\partial^\\dagger_{\\dot\\alpha}+i(\\theta\\sigma^\\mu)_{\\dot\\alpha}\\partial_\\mu\\right)\\xi^{\\dagger\\,\\dot\\alpha}\\Phi\\,,\\label{eq:QbarPhi}\n\\end{align}\n\nThe above results motivate the introduction of the following differential operators,\n\\begin{align}\n\\widehat{P}_\\mu&=i\\partial_\\mu\\,,\\label{Phat}\\\\\n\\widehat{Q}_\\alpha&=i\\partial_\\alpha-(\\sigma^\\mu\\theta^\\dagger)_\\alpha\\partial_\\mu\\,, \\label{Qhat}\\\\\n\\widehat{Q}^\\dagger_{\\dot\\alpha}&=-i\\partial^\\dagger_{\\dot\\alpha}+(\\theta\\sigma^\\mu)_{\\dot\\alpha}\\partial_\\mu\\,,\\label{QDhat}\n\\end{align}\nwhich allow us to succinctly rewrite \\eqst{eq:PPhi}{eq:QbarPhi} as follows:\n \\begin{align}\n \\bigl[\\Phi\\,,\\,P_\\mu\\bigr]&=\\widehat{P}_\\mu\\Phi\\,, \\\\\n \\big[\\Phi\\,,\\,\\xi Q\\bigr]&=(\\xi\\widehat{Q})\\Phi\\,, \\label{stranslate1}\\\\\n \\big[\\Phi\\,,\\,\\xi^\\dagger Q^\\dagger\\bigr]&=(\\xi^\\dagger\\widehat{Q}^\\dagger)\\Phi\\,.\\label{stranslate2}\n\\end{align}\n\n\nIn \\eq{susytranslate}, we noted that the action of an infinitesimal SUSY transformation on any field $\\Phi\\of{x}$ was given by\n$\\delta_{\\xi}\\Phi(x)=i\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\Phi(x)\\bigr]$.\nIn light of \\eqs{stranslate1}{stranslate2}, we conclude that the action of an\ninfinitesimal SUSY transformation on a\nsuperfield $\\Phi\\of{x,\\theta,\\theta^\\dagger}$ is given by\n \\begin{align}\n \\delta_{\\xi}\\Phi(x,\\theta,\\theta^\\dagger)=-i(\\xi \\widehat{Q}+\\xi^\\dagger \\widehat{Q}^\\dagger)\\Phi(x,\\theta,\\theta^\\dagger)\\,.\\label{supertrans}\n\\end{align}\n\n\n\n\\subsection{Expansion of the superfield in powers of $\\theta$ and $\\theta^\\dagger$}\n\n\n\nConsider the Taylor expansion of a superfield,\n$\\Phi(x,\\theta,\\theta^\\dagger)$, in powers of $\\theta$ and~$\\theta^\\dagger$. The coefficients of this expansion will be functions of\n$x$, which can be interpreted as ordinary fields. Since $\\theta$ and\n$\\theta^\\dagger$ are anticommuting coordinates, this Taylor series\nterminates after a finite number of terms. In particular, since $\\theta$ and\n$\\theta^\\dagger$ are \nanticommuting two-component spinor\nquantities, it follows that\n$(\\theta_1)^2=(\\theta_2)^2=(\\theta^\\dagger_{\\dot 1})^2=(\\theta^\\dagger_{\\dot\n 2})^2=0$, whereas products such as $\\theta_1\\theta_2$ and\n$\\theta_1^\\dagger \\theta_2^\\dagger$ do not vanish. Indeed, it is easy\nto check that\n\\begin{align}\n\\theta^\\alpha\\theta^\\beta&=-\\ifmath{\\tfrac12}\\epsilon^{\\alpha\\beta}\\theta\\theta\\,,\\qquad\\qquad\n{\\theta^\\dagger}^{\\dot\\alpha}{\\theta^\\dagger}^{\\dot\\beta}=\\ifmath{\\tfrac12}\\epsilon^{\\dot\\alpha\\dot\\beta}\\theta^\\dagger\\thetabar\\,,\\nonumber \\\\\n\\theta_\\alpha\\theta_\\beta&=\\ifmath{\\tfrac12}\\epsilon_{\\alpha\\beta}\\theta\\theta\\,,\\qquad\\qquad\\phantom{-}\n{\\theta}^\\dagger_{\\dot\\alpha}{\\theta}^\\dagger_{\\dot\\beta}=-\\ifmath{\\tfrac12}\\epsilon_{\\dot\\alpha\\dot\\beta}\\theta^\\dagger\\thetabar\\,,\\nonumber\n\\end{align}\nwhere $\\theta\\theta\\equiv \\theta^\\alpha\\theta_\\alpha$ and $\\theta^\\dagger\\thetabar\\equiv\\theta^\\dagger_{\\dot\\alpha}\n{\\theta^\\dagger}^{\\dot\\alpha}$ following the convention of \\eq{contract}. \nProducts such as $\\theta_\\alpha\\theta_\\beta\\theta_\\gamma=0$, since the\nspinor indices can assume at most two different values. Finally, the\nfollowing three results are noteworthy (see Problem 17),\n\\begin{align}\n(\\theta\\sigma^\\mu\\theta^\\dagger)\\theta_\\beta&=-\\ifmath{\\tfrac12} \\theta\\theta(\\sigma^\\mu\\theta^\\dagger)_\\beta \\label{eq:r1} \\\\\n(\\theta\\sigma^\\mu\\theta^\\dagger)\\theta^\\dagger_{\\dot\\beta}&=-\\ifmath{\\tfrac12} \\theta^\\dagger\\thetabar(\\theta\\sigma^\\mu)_{\\dot\\beta} \\label{eq:r2} \\\\\n(\\theta\\sigma^\\mu\\theta^\\dagger)(\\theta\\sigma^\\nu\\theta^\\dagger)&=\\ifmath{\\tfrac12} g^{\\mu\\nu}(\\theta\\theta)(\\theta^\\dagger\\thetabar). \\label{eq:r3}\n\\end{align}\nSometimes, we shall write\n$\\theta\\theta\\theta^\\dagger\\theta^\\dagger\\equiv\n(\\theta\\theta)(\\theta^\\dagger\\thetabar)$. In such products, there should\nbe no ambiguity in omitting the parentheses.\n\n\nThe Taylor series expansion of a complex superfield\n$\\Phi(x,\\theta,\\theta^\\dagger)$ is therefore given by,\n\\begin{Eqnarray}\n\\Phi(x,\\theta,\\theta^\\dagger)&=& f(x) +\\theta\\zeta(x)+\\theta^\\dagger\\chi^\\dagger(x)+\\theta\\theta m(x)+\\theta^\\dagger\\thetabar n(x) +\\theta\\sigma^\\mu\\theta^\\dagger V_\\mu(x) \\nonumber\\\\\n&& \n+(\\theta\\theta)\\theta^\\dagger\\lambda^\\dagger(x)+(\\theta^\\dagger\\thetabar)\\theta\\lambda(x)+\\theta\\theta\\theta^\\dagger\\thetabar d(x)\\,,\\label{phitaylor}\n\\end{Eqnarray}\nwhere $f$, $m$, $n$, $V_\\mu$, and $d$ are complex commuting bosonic\nfields and $\\zeta$, $\\chi$, $\\lambda$ and $\\psi$ are anticommuting\ntwo-component fermionic fields. \nThe SUSY transformation laws of the component fields can now be easily\nobtained (see Problem~\\ref{pr:fmnV}) by comparing both sides of \\eq{supertrans}.\n\n\nHence, there are 16 bosonic and 16 fermionic real degrees of freedom.\nIf we impose the constraint, $\\Phi^\\dagger=\\Phi$, then $f$, $d$ and $V_\\mu$\nare real bosonic fields, $n^\\dagger=m$, $\\zeta=\\chi$ and $\\lambda=\\psi$. In this\ncase, there are 8 bosonic and 8 fermionic real degrees of freedom. In both cases, there are too many degrees of freedom to describe the supermultiplet of the Wess-Zumino model.\nThis is because an unconstrained complex superfield, $\\Phi(x,\\theta,\\theta^\\dagger)$, describes a\nreducible representation of the SUSY algebra. One must impose\nsupersymmetric constraints to project out an irreducible\nsupermultiplet.\\footnote{A real superfield $\\Phi$ yields an off-shell\n irreducible representation with superspin $j=\\ifmath{\\tfrac12}$. More on this\n in Section~\\ref{sec:gaugetheories}.}\n\nThe superfield defined in \\eq{phitaylor} is an example of a\n\\textit{bosonic} superfield, where the Taylor series coefficients of terms even in\nthe number of Grassmann coordinates are commuting bosonic fields and the coefficients of terms odd in\nthe number of Grassmann coordinates are anticommuting fermionic fields.\nSimilarly, one can define a \\textit{fermionic} superfield, where the\nTaylor series coefficients of terms even in\nthe number of Grassmann coordinates are anticommuting fermionic fields and the coefficients of terms odd in\nthe number of Grassmann coordinates are commuting bosonic fields.\n \n\\subsection{Spinor covariant derivatives}\n\n\nFor a superfield $\\Phi$, it is easy to check that neither\n$\\partial_\\alpha\\Phi$ nor $\\partial_{\\dot\\alpha}\\Phi$ is a superfield, since\n\\begin{align}\n\\partial_\\alpha(\\delta_{\\xi}\\Phi)\\neq \\delta_{\\xi}(\\partial_\\alpha\\Phi)\\,,\\qquad\\quad\n\\partial^\\dagger_{\\dot\\alpha}(\\delta_{\\xi}\\Phi)\\neq \\delta_{\\xi}(\\partial^\\dagger_{\\dot\\alpha}\\Phi)\\,.\n\\end{align}\nNote that if $\\Phi$ is a bosonic superfield, then the hermitian conjugate of $\\partial_\\alpha\\Phi$ is given\nby,\n\\begin{equation} \\label{daggers}\n(\\partial_\\alpha\\Phi)^\\dagger=-\\partial_{\\dot\\alpha}^\\dagger\n\\Phi^\\dagger\\,,\n\\end{equation}\nwhere the minus sign above is related to the minus sign in\n\\eq{dteps}.\n\\clearpage\n\nWe therefore introduce spinor covariant derivatives $D_\\alpha$ and\n$\\overline{D}_{\\dot\\alpha}$ such that $D_\\alpha\\Phi$ and\n$\\overline{D}_{\\dot\\alpha}\\Phi$ are superfields,\\footnote{Note that if\n $\\Phi$ is a bosonic superfield, then $D_\\alpha\\Phi$ and\n $\\overline{D}_{\\dot\\alpha}\\Phi$ are fermionic superfields.}\n which implies the\nfollowing conditions must be satisfied,\n\\begin{align}\nD_\\alpha(\\delta_{\\xi}\\Phi)=\\delta_{\\xi}(D_\\alpha\\Phi)\\,,\\qquad\\quad\n\\overline{D}_{\\dot\\alpha}(\\delta_{\\xi}\\Phi)=\\delta_{\\xi}(\\overline{D}_{\\dot\\alpha}\\Phi)\\,.\\label{dsuper}\n\\end{align}\nUsing \\eq{supertrans} to express $\\delta_{\\xi}\\Phi$ in terms of the operators $\\widehat{Q}$ and\n$\\widehat{Q}^\\dagger$ defined in \\eqs{Qhat}{QDhat}, respectively, one easily derives\n\\begin{align}\n\\{D_\\alpha\\,,\\,\\widehat{Q}_\\beta\\}=\\{D_\\alpha\\,,\\,\\widehat{Q}^\\dagger_{\\dot\\beta}\\}=\\{\\overline{D}_{\\dot\\alpha}\\,,\\,\\widehat{Q}_{\\beta}\\}=\\{\\overline{D}_{\\dot\\alpha}\\,,\\,\\widehat{Q}^\\dagger_{\\dot\\beta}\\}=0\\,.\\label{antis}\n\\end{align}\n\nTo fix the explicit forms for the spinor covariant derivatives, we choose\nthe normalization of $D_\\alpha$ so that it has the form\n$D_\\alpha=\\partial_\\alpha+\\ldots$, where the ellipsis refers to\ncorrection terms needed to satisfy \\eqs{dsuper}{antis}. In the case\nof $\\overline{D}_\\alpha$, it is customary to impose the condition,\n\\begin{equation} \\label{Dcond}\n(D_\\alpha\\Phi)^\\dagger=\\overline{D}_{\\dot\\alpha}\\Phi^\\dagger\\,,\n\\end{equation}\nwhere $\\Phi$ is a bosonic superfield, in which case\n$\\overline{D}_{\\dot\\alpha}=-\\partial_{\\dot\\alpha}^\\dagger+\\ldots$\n [cf.~\\eq{daggers}]. \n\nThe explicit forms for the spinor covariant derivatives that satisfy\nthe above conditions are given by, \n\\begin{align}\nD_\\alpha&=\\partial_\\alpha-i(\\sigma^\\mu\\theta^\\dagger)_\\alpha\\,\\partial_\\mu\\,, \\label{eq:D} \\\\\n\\overline{D}_{\\dot\\alpha}&=-\\partial^\\dagger_{\\dot\\alpha}+i(\\theta\\sigma^\\mu)_{\\dot\\alpha}\\,\\partial_\\mu\\,. \\label{eq:Db}\n\\end{align}\nIn particular, $D$ and $\\overline{D}$\nsatisfy the same anticommutation relations as $\\widehat{Q}$ and $\\widehat{Q}^\\dagger$ (see Problem~\\ref{pr:D}),\n\\begin{align}\n\\{D_\\alpha\\,,\\,D_\\beta\\}=\\{\\overline{D}_{\\dot\\alpha}\\,,\\,\\overline{D}_{\\dot\\beta}\\}=0 \\ \\ \\mathrm{and\\ \\ }\n\\{D_\\alpha\\,,\\,\\overline{D}_{\\dot\\beta}\\}=2i\\sigma^\\mu_{\\alpha\\dot\\beta}\\partial_\\mu.\n\\label{eq:Dcomms}\n\\end{align}\n\nOne can also define spinor covariant derivatives with a raised spinor index.\nIn this case, it is conventional to define,\n\\begin{align}\nD^\\alpha\\equiv \\epsilon^{\\alpha\\beta}D_\\beta&=-\\partial^\\alpha+i(\\theta^\\dagger\\overline{\\sigma}^\\mu)^\\alpha\\,\\partial_\\mu\\,, \\\\\n\\overline{D}\\lsup{\\dot\\alpha}\\equiv \\epsilon^{\\dot\\alpha\\dot\\beta}D_\\beta\n&=\\partial^{\\dagger\\dot\\alpha}-i(\\overline{\\sigma}^\\mu\\theta)^{\\dot\\alpha}\\,\\partial_\\mu\\,,\n\\end{align}\nwhere we have employed \\eq{eq:partialsign}. That is,\nthe spinor indices of $D_\\alpha$ and $\\overline{D}_{\\dot\\alpha}$ are raised\nin the\nconventional way according to \\eq{raiseindex}.\\footnote{This is in contrast to the\nrule for raising the spinor indices of $\\partial_\\alpha$ and\n$\\partial^\\dagger_{\\dot\\alpha}$ specified in \\eq{eq:partialsign}, where\nan extra minus sign appears.} \nThe following differential operators\nwill be useful later in these lectures,\n\\begin{Eqnarray}\nD^2&=&D^\\alpha D_\\alpha=-\\partial^\\alpha\\partial_\\alpha+2i(\\partial^\\alpha\\sigma^\\mu_{\\alpha\\dot\\beta}{\\theta^\\dagger}\\lsup{\\dot\\beta})\\partial_\\mu+\\theta^\\dagger\\thetabar\\,\\square\\,,\\label{DD}\\\\\n\\overline{D}\\lsup{\\,2}&=&\\overline{D}_{\\dot\\alpha}\\overline{D}\\lsup{\\,\\dot\\alpha}=-\\partial^\\dagger_{\\dot\\alpha}\\partial^{\\dagger\\dot\\alpha}+2i(\\theta^\\alpha\\sigma^\\mu_{\\alpha\\dot\\beta}\\partial^{\\dagger\\dot\\beta})\\partial_\\mu+\\theta\\theta\\square\\,,\\label{DbDb}\n\\end{Eqnarray}\nwhere $\\square\\equiv\\partial_\\mu\\partial^\\mu$. One can then derive\nthe following identity (see Problem~\\ref{pr:DDid}),\n\\begin{equation} \\label{DDid}\n[D^2,\\overline{D}\\lsup{2}]= 4i\\sigma^\\mu_{\\alpha\n\\dot{\\beta}}\\partial_\\mu[D^\\alpha,\\overline D\\lsup{\\,\\dot{\\beta}}]\\,.\n\\end{equation}\n\nWe have employed different notation for the conjugation of the\nvarious differential operators that appear in this subsection.\nThe relation of $\\widehat{Q}^\\dagger$ to $\\widehat{Q}$ is\n\\textit{hermitian conjugation} in the same sense that $\\hat{P}_\\mu\n=i\\partial_\\mu$ [defined in \\eq{Phat}] is an hermitian operator in\nquantum field theory with respect to the inner product defined by the\nintegration of complex fields over spacetime.\nThat is, the dagger on the differential operator $\\widehat{Q}^\\dagger$\ndenotes Hermitian conjugation with respect to the inner product\ndefined by the integration of complex superfields over\nsuperspace.\\footnote{For further details, see\n Refs.~\\cite{Sohnius:1985qm,Martin:1997ns}. Integration over\n superspace will be treated in Section~\\ref{integration}.}\n\n\nIn contrast, the relation of $\\overline{D}$ to $D$ is \\textit{complex\n conjugation} in the same sense that $\\partial_\\mu^*$ is the complex\nconjugate of $\\partial_\\mu$. In the latter case, the differential\noperator $\\partial_\\mu$ is a real operator.\nThat is, if we define $\\partial^*_\\mu$ \nto be the derivative operator that acts on the field $\\phi$ such that\n\\begin{align}\n\\of{\\partial_\\mu \\phi}^\\dagger = \\partial_\\mu^* \\phi^\\dagger,\n\\end{align}\nthen since $\\of{\\partial_\\mu \\phi}^\\dagger = \\partial_\\mu \\phi^\\dagger$, it\nfollows that\n$\\partial^*_\\mu=\\partial_\\mu$. In light of \\eq{Dcond},\nwe can therefore regard $\\overline{D}$ as the complex conjugate of $D$.\n\n\n\n\\subsection{Chiral superfields}\n\nA chiral superfield is obtained by imposing\nthe constraint $\\overline{D}_{\\dot\\alpha}\\Phi=0$ on a general superfield $\\Phi$. Such a constraint is covariant with respect to\nSUSY transformations, and the end result is an irreducible superfield that\ncorresponds to the superspin $j=0$ irreducible\nrepresentation of the SUSY algebra. \nUsing \\eq{eq:Db}, the constraint yields a differential equation,\n\\begin{align}\n\\overline{D}_{\\dot\\alpha}\\Phi=\\bigl[-\\partial^\\dagger_{\\dot\\alpha}+i(\\theta\\sigma^\\mu)_{\\dot\\alpha}\\,\\partial_\\mu\\bigr]\\Phi(x,\\theta,\\theta^\\dagger)=0\\,,\n\\end{align}\nwhose solution is of the form\n\\begin{align}\n\\Phi(x,\\theta,\\theta^\\dagger)=\\exp(-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\Phi(x,\\theta)\\,.\\label{eq:Phixththb}\n\\end{align}\n\nWe can expand $\\Phi(x,\\theta)$ in a\n (truncated) \nTaylor series in $\\theta$,\n\\begin{align}\n\\Phi(x,\\theta)=A(x)+\\sqrt{2}\\,\\theta\\psi(x)+\\theta\\theta F(x)\\,,\n\\end{align}\nwhere the factor of $\\sqrt{2}$ is conventional. Plugging this into\n\\eq{eq:Phixththb} and using the identity (see Problem~\\ref{pr:expexp}),\n\\begin{align}\n\\exp(-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)=1-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu-\\tfrac{1}{4}(\\theta\\theta)(\\theta^\\dagger\\thetabar)\\square,\n\\end{align}\nwe find after some algebraic manipulation a chiral superfield with the form,\n\\begin{align}\n\\begin{split}\n\\Phi(x,\\theta,\\theta^\\dagger) &=\nA(x) + \\sqrt{2}\\,\\theta \\psi(x) + \\theta\\theta F(x)-i \\theta\\sigma^\\mu\\theta^\\dagger \\partial_\\mu A(x)\\\\\n&\\quad - \\frac{i}{\\sqrt{2}} (\\theta\\theta) \n\\theta^\\dagger \\overline{\\sigma}^\\mu\\, \\partial_\\mu \\psi(x)-\\tfrac{1}{4}(\\theta\\theta)(\\theta^\\dagger\\thetabar) \\square A(x).\n\\end{split}\n\\label{eq:chiralSF}\n\\end{align}\nNote that the chiral superfield $\\Phi$ has dimension $[\\Phi]=1$, in which case it\nfollows that the dimensions of the component fields are $[A]=1$ and\n$[\\psi]=\\tfrac32$, as expected, whereas $[F]=2$\nafter making use of the dimensions of the Grassmann coordinates,\n$[\\theta]=[\\theta^\\dagger]=-\\ifmath{\\tfrac12}$.\n\n\nGiven a chiral superfield $\\Phi$, its hermitian conjugate,\n$\\Phi^\\dagger$, is an antichiral superfield, which is\ndefined by the SUSY-covariant constraint,\n$D_\\alpha\\Phi^\\dagger=0$.\nUsing \\eq{eq:D}, the latter constraint yields a differential equation,\n\\begin{align}\nD_{\\alpha}\\Phi^\\dagger=\\bigl[\\partial_{\\alpha}-i(\\sigma^\\mu\\theta^\\dagger)_{\\alpha}\\,\\partial_\\mu\\bigr]\\Phi^\\dagger(x,\\theta,\\theta^\\dagger)=0\\,,\n\\end{align}\nwhose solution is of the form\n\\begin{align}\n\\Phi^\\dagger(x,\\theta,\\theta^\\dagger)=\\exp(i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\Phi^\\dagger(x,\\theta^\\dagger)\\,.\\label{eq:Phixththb2}\n\\end{align}\n\nWe can expand $\\Phi^\\dagger(x,\\theta^\\dagger)$ in a\n (truncated) \nTaylor series in $\\theta^\\dagger$,\n\\begin{align}\n\\Phi^\\dagger(x,\\theta^\\dagger)=A^\\dagger(x)+\\sqrt{2}\\,\\theta^\\dagger\\psi^\\dagger(x)+\\theta^\\dagger\\thetabar F^\\dagger(x)\\,.\n\\end{align}\nPlugging this result into \\eq{eq:Phixththb2} and\nfollowing the same procedure as before, we end up with,\n\\begin{align}\n\\begin{split}\n\\Phi^\\dagger(x,\\theta,\\theta^\\dagger) &=\nA^\\dagger(x) + \\sqrt{2}\\,\\theta^\\dagger \\psi^\\dagger(x) + \\theta^\\dagger\\thetabar F^\\dagger(x)+i \\theta\\sigma^\\mu\\theta^\\dagger \\partial_\\mu A^\\dagger(x) \\\\\n&\\quad - \\frac{i}{\\sqrt{2}} (\\theta^\\dagger\\thetabar) \n\\theta\\sigma^\\mu\\, \\partial_\\mu \\psi^\\dagger(x)-\\tfrac{1}{4}(\\theta\\theta)(\\theta^\\dagger\\thetabar) \\square A^\\dagger(x)\\,.\n\\end{split}\n\\end{align}\nSince $\\Phi^\\dagger$ is the hermitian conjugate of $\\Phi$, we can\nidentify $A^\\dagger$, $\\psi^\\dagger$ and $F^\\dagger$ as the hermitian\nconjugates of $A$, $\\psi$ and $F$.\n\nIn calculations, it is often simpler to employ the so-called \\textit{chiral representation}, in which all superfield operators $\\mathcal{O}$ are modified according to\n\\begin{align}\n\\mathcal{O}_{\\rm chiral}=\\exp(i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\mathcal{O}\\exp(-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\,.\n\\end{align}\nIn the chiral representation,\n\\begin{Eqnarray}\n&& \\widehat{Q}_\\alpha=i\\partial_\\alpha\\,,\\qquad\\qquad \\ \\\n\\widehat{Q}^\\dagger_{\\dot\\alpha}=-i\\partial^\\dagger_{\\dot\\alpha}+2(\\theta\\sigma^\\mu)_{\\dot\\alpha}\\,\\partial_\\mu\\,,\\label{Qchiral}\n\\\\\n&&\\overline{D}_{\\dot\\alpha}=-\\partial^\\dagger_{\\dot\\alpha}\\,,\\qquad\\qquad \nD_{\\alpha}=\\partial_{\\alpha}-2i(\\sigma^\\mu\\theta^\\dagger)_{\\alpha}\\,\\partial_\\mu\\,.\\label{Dchiral}\n\\end{Eqnarray}\nThus, in the chiral representation, the requirement $\\overline{D}_{\\dot\\alpha}\\Phi=-\\partial^\\dagger_{\\dot\\alpha}\\Phi=0$ is simply the requirement that $\\Phi$ is independent of $\\theta^\\dagger$.\nIn the chiral representation, the chiral superfield will be denoted by\n\\begin{align}\n\\Phi_1(x,\\theta)=A(x)+\\sqrt{2}\\,\\theta\\psi(x)+\\theta\\theta F(x)\\,.\\label{phione}\n\\end{align}\nIt then follows that the general expression for a chiral superfield is\n\\begin{align}\n\\Phi(x,\\theta,\\theta^\\dagger) =\\exp(-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\Phi_1(x,\\theta)=\\Phi_1(x-i\\theta\\sigma^\\mu\\theta^\\dagger\\,,\\,\\theta)\\,.\n\\end{align}\nIt is convenient to define the shifted spacetime coordinate,\n\\begin{align}\ny \\equiv x - i \\theta \\sigma^\\mu \\theta^\\dagger,\n\\end{align}\nso that the chiral superfield is given by,\n\\begin{align}\n\\Phi\\of{x,\\theta,\\theta^\\dagger} = \\Phi_1\\of{y,\\theta}.\n\\end{align}\n\nThe SUSY transformation laws for the fields that appear in the chiral\nsuperfield can now be determined simply by inserting the expression\nfor $\\Phi$ in the chiral representation given by \\eq{phione} into\n\\eq{supertrans}. In performing the computation, one employs the\nchiral representation expressions for $\\widehat{Q}$ and\n$\\widehat{Q}^\\dagger$ given in \\eq{Qchiral}. You may verify (see Problem~\\ref{pr:chiraltrans})\nthat the result of this calculation coincides with the SUSY\ntransformation laws\ngiven previously in \\eqst{offshell1}{offshell3}.\n\nLikewise, one can define an antichiral representation in which \n\\begin{align}\n\\mathcal{O}_{\\rm antichiral}=\\exp(-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\mathcal{O}\\exp(i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\,.\n\\end{align}\nIn the antichiral representation,\n\\begin{align}\n\\begin{split}\n& \\widehat{Q}^\\dagger_{\\dot\\alpha}=-i\\partial^\\dagger_{\\dot\\alpha}\\,,\\qquad\\qquad \\ \\ \n\\widehat{Q}_\\alpha=i\\partial_{\\alpha}-2(\\sigma^\\mu\\theta^\\dagger)_{\\alpha}\\,\\partial_\\mu\\,,\n \\\\\n&D_{\\alpha}=\\partial_{\\alpha}\\,,\\qquad\\qquad \\quad\n\\overline{D}_{\\dot\\alpha}=-\\partial^\\dagger_{\\dot\\alpha}+2i(\\theta\\sigma^\\mu)_{\\dot\\alpha}\\,\\partial_\\mu\\,.\n\\end{split}\n\\end{align}\nThus, in the antichiral representation, the requirement $D_{\\alpha}\\Phi^\\dagger\n=\\partial_{\\alpha}\\Phi^\\dagger=0$ is simply the requirement that $\\Phi^\\dagger$ is independent of $\\theta$.\nIn the antichiral representation, the antichiral superfield will be denoted by\n\\begin{align}\n\\Phi_2(x,\\theta^\\dagger)=A^\\dagger(x)+\\sqrt{2}\\,\\theta^\\dagger\\psi^\\dagger(x)+\\theta^\\dagger\\thetabar F^\\dagger(x)\\,.\n\\end{align}\nIt then follows that the general expression for an antichiral superfield is\n\\begin{align}\n\\Phi^\\dagger(x,\\theta,\\theta^\\dagger) =\\exp(i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)\\Phi_2(x,\\theta^\\dagger)=\\Phi_2(x+i\\theta\\sigma^\\mu\\theta^\\dagger\\,,\\,\\theta^\\dagger)\\,.\n\\end{align}\nIt is convenient to define the shifted spacetime coordinate,\n\\begin{align}\ny^\\dagger \\equiv x + i \\theta \\sigma^\\mu \\theta^\\dagger,\n\\end{align}\nso that the antichiral superfield is given by,\n\\begin{align}\n\\Phi^\\dagger\\of{x,\\theta,\\theta^\\dagger} = \\Phi_2\\of{y^\\dagger,\\theta^\\dagger}.\n\\end{align}\n\n\n\\subsection{Constructing the SUSY Lagrangian}\n\n\\subsubsection{$F$-terms}\nUltimately, our goal is to construct an action that is invariant under SUSY. It is therefore sufficient to construct a Lagrangian that transforms under SUSY as a total derivative.\nIn the literature, it is common to use the nomenclature\n\\textit{$F$-term} to denote the coefficient of the\n$\\theta\\theta$ term of a superfield. This is sometimes explicitly\nindicated as follows,\n\\begin{align}\n[\\Phi]_{\\theta\\theta}=[\\Phi]_F=F. \\label{Fterm}\n\\end{align}\nRecall that in \\eq{offshell3}, we demonstrated that the auxiliary\nfield $F(x)$ transforms as a total derivative under the SUSY\ntransformation laws. But, this field is simply the coefficient of the\n$\\theta \\theta$ term of a chiral superfield! Indeed, the \n$F$-term of any chiral superfield transforms under a SUSY\ntransformation as a total derivative. This means that such terms (and\ntheir hermitian conjugates) are candidates for terms in a Lagrangian,\nwhich then yields an action that is invariant under SUSY.\n\nTo discover the relevant $F$-terms for constructing a SUSY Lagrangian,\nwe first prove an important theorem.\n\n\\begin{theorem}\nFor any positive integers $n$ and $m$, \nif $\\Phi$ is a chiral superfield, then so is $\\Phi^n$, whereas $\\Phi^n (\\Phi^\\dagger)^m$ is not a chiral superfield.\n\\end{theorem}\n\\begin{proof}\nWe first note that\n\\begin{equation}\n\\overline{D}_{\\dot\\alpha}\\Phi^n=n\\Phi^{n-1}\\overline{D}_{\\dot\\alpha}\\Phi=0,\n\\end{equation}\nwhich shows $\\Phi^n$ satisfies the defining constraint of a chiral superfield.\nA similar computation shows that $\\Phi^n (\\Phi^\\dagger)^m$ does not\nsatisfy the required constraint.\n\\end{proof}\n\n\nAn important consequence of the above theorem is that\n\\begin{equation}\n\\sum_{n\\geq 1} [a_n\\Phi^n]_F+{\\rm h.c.}\n\\end{equation}\nis a Lorentz scalar that transforms as a total divergence, and thus is a candidate for terms in a Lagrangian whose action is invariant under SUSY.\n\n\\subsubsection{Kinetic terms}\n\\label{kineticterms}\nTo construct the kinetic terms of the SUSY Lagrangian,\nwe define the operator $T$,\n\\begin{align}\nT\\Phi=-\\tfrac{1}{4}\\overline{D}\\lsup2\\Phi^\\dagger\\,,\\label{Tdef}\n\\end{align}\nwhere $\\overline{D}\\lsup2\\equiv \\overline{D}_{\\dot\\alpha}{\\overline{D}}\\lsup{\\dot\\alpha}$. Note\nthat $\\overline{D}_{\\dot\\alpha}(T\\Phi)=0$ (due to the anticommutation relations\nsatisfied by $\\overline{D}$), so that $T\\Phi$ is a chiral superfield.\nIn the chiral representation, with $\\Phi=A+\\sqrt{2}\\,\\theta\\psi+\\theta\\theta F$,\n\\begin{align}\nT\\Phi=F^\\dagger-i\\sqrt{2}\\,\\theta\\sigma^\\mu\\partial_\\mu\\psi^\\dagger-\\theta\\theta\\,\\square A^\\dagger\\,.\n\\end{align}\nHence, the $F$-component of $\\Phi T \\Phi$ is given by,\n\\begin{align}\n[\\Phi T\\Phi]_F & =-A\\square A^\\dagger+F^\\dagger F+i\\psi\\sigma^\\mu\\partial_\\mu\\psi^\\dagger\\nonumber\\\\\n&= (\\partial_\\mu A)(\\partial^\\mu A^\\dagger)+F^\\dagger F+i\\psi^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\psi+\\text{total derivative}\\,,\\label{KE}\n\\end{align}\nwhich we recognize as the kinetic energy term of the\nWess-Zumino Lagrangian [cf.~\\eq{eq:LWZF}].\n\\subsubsection{Mass terms}\n\nTo construct the mass terms of the SUSY Lagrangian, the following\ntheorem is useful.\n\\begin{theorem}\nFor any chiral superfield $\\Phi$,\n\\begin{align}\n[\\Phi]_F=-\\tfrac{1}{4} D^2\\Phi\\biggl|_{\\theta=\\theta^\\dagger=0}=\\tfrac{1}{4}\\partial^\\alpha\\partial_\\alpha\\Phi\\biggl|_{\\theta=\\theta^\\dagger=0}\\,.\\label{phiF}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\n\\Eq{phiF} follows immediately from \\eq{DD}. \n\\end{proof}\nWe can compute the $F$ term of any holomorphic function of a chiral\nsuperfield, $W(\\Phi)$, as follows. After making judicious use of the\nchain rule,\n\\begin{align}\n[W(\\Phi)]_F&=\\tfrac{1}{4}\\partial^\\alpha\\partial_\\alpha W\\biggl|_{\\theta=\\theta^\\dagger=0} \n\t=\\tfrac{1}{4}\\partial^\\alpha\\frac{dW}{d\\Phi}\\partial_\\alpha\\Phi\\biggl|_{\\theta=\\theta^\\dagger=0} \\nonumber \\\\[6pt]\n&=\\frac{1}{4}\\biggl\\{\\left(\\frac{d^2 W}{d\\Phi^2}\\partial^\\alpha\\Phi\\partial_\\alpha\\Phi\\right)\n+\\frac{dW}{d\\Phi}\\partial^\\alpha\\partial_\\alpha\\Phi\\biggr\\}\\biggl|_{\\theta=\\theta^\\dagger=0} \\,.\\label{WPhi}\n\\end{align}\nNoting that $(\\partial^\\alpha\\Phi\\partial_\\alpha\\Phi)_{\\theta=\\theta^\\dagger=0}=-2\\psi\\psi$,\n\\eq{WPhi} yields,\n\\begin{align}\n[W(\\Phi)]_F &=-\\frac12\\left(\\frac{d^2 W}{d\\Phi^2}\\right)_{\\Phi=A}\\psi\\psi+\\left(\\frac{dW}{d\\Phi}\\right)_{\\Phi=A}F\\,.\n\\end{align}\nIntroducing the notation, $dW\/dA\\equiv (dW\/d\\Phi)_{\\Phi=A}$,\nit follows that\n\\begin{align}\n[W(\\Phi)]_F=-\\frac12 \\frac{d^2 W}{dA^2}\\psi\\psi+\\frac{dW}{dA}F\\,.\\label{rest}\n\\end{align}\nIn the jargon of SUSY, $W(\\Phi)$ is called the \\textit{superpotential}. For renormalizable theories, $\\!W(\\Phi)\\!$ is at most cubic in~$\\Phi$. \n\n\\subsubsection{The Wess-Zumino SUSY Lagrangian using $F$-terms}\nCollecting the results of \\eqs{KE}{rest}, we end up with,\n\\begin{Eqnarray}\n\\mathscr{L}&=&[\\Phi T\\Phi]_F+\\bigl\\{[W(\\Phi)]_F+{\\rm h.c.}\\bigr\\}\n\\nonumber \\\\[2pt]\n&=&(\\partial_\\mu A)^\\dagger(\\partial^\\mu A)+ i \\psi^\\dagger \\overline{\\sigma}^\\mu \\partial_\\mu \\psi +F\\frac{dW}{dA}+F^\\dagger\\left(\\frac{dW}{dA}\\right)^{\\!\\!\\dagger}\n+F^\\dagger F \\nonumber \\\\\n&&\\quad \n-\\frac12\\left[\\frac{d^2 W}{dA^2}\\,\\psi\\psi+\\left(\\frac{d^2 W}{dA^2}\\right)^{\\!\\!\\dagger}\\!\\!\\psi^\\dagger\\psi^\\dagger\\right]\\,,\\label{WZlagF}\n\\end{Eqnarray}\nafter dropping total derivative terms. We have thus recovered the\nWess-Zumino Lagrangian that was previously written down in \\eq{eq:LWZoriginal}.\n\n\nThe proof that the Wess-Zumino action is supersymmetric, or\nequivalently, $\\delta_{\\xi}\\mathcal{L}=\\partial_\\mu K^{\\prime\\,\\mu}$, is\nnow trivial since \n$\\mathscr{L}$ was constructed from $F$-terms, which\ntransform as total derivatives under SUSY transformations.\n\n\\subsubsection{An alternate form for the kinetic terms: $D$-terms and the K\\\"ahler potential}\n\\label{Kahler}\nThe approach of subsection~\\ref{kineticterms} is not the only supersymmetric way to construct the kinetic energy terms.\nConsider an unconstrained superfield $V(x,\\theta,\\theta^\\dagger)$.\n Expanding $V$ as a Taylor series in $\\theta$ and $\\theta^\\dagger$, the\n highest order nonvanishing term is proportional to $(\\theta\\theta)(\\theta^\\dagger\\thetabar)$. If we write \n\\begin{align}\nV(x,\\theta,\\theta^\\dagger)=\\cdots+(\\theta\\theta)(\\theta^\\dagger\\thetabar)D(x)\\,,\n\\end{align}\nthen one can show that $\\delta_{\\xi} D(x)$ is a total derivative\nusing dimensional analysis as we did for $\\delta_{\\xi} F(x)$ at the end of\nSection~\\ref{offshell}. Hence, $D$-terms can also provide suitable terms for a SUSY Lagrangian.\n\n\nWe shall denote the $D$-term by,\n\\begin{equation}\n[V]_{\\theta\\theta\\theta^\\dagger\\thetabar}=[V]_D=D\\,,\n\\end{equation}\nusing a notation analogous to that of \\eq{Fterm}. The relevant\ntheorem analogous to \\eq{phiF} is given below.\n\\begin{theorem}\nFor any superfield $V$,\n\\begin{equation}\n[V]_D =\\tfrac{1}{16}\\overline{D}\\lsup{2}D^2\n V\\biggl|_{\\theta=\\theta^\\dagger=0}=\\tfrac{1}{16}(\\partial^\\dagger_{\\dot\\alpha}\\partial^{\\dagger\\dot\\alpha})(\\partial^\\alpha\\partial_\\alpha)V\\biggl|_{\\theta=\\theta^\\dagger=0}\\,.\\label{VD}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\\Eq{VD} follows immediately from \\eqs{DD}{DbDb}. \n\\end{proof}\n\n\\noindent\nFor example, if $\\Phi$ is a chiral superfield, one can show that (see Problem~\\ref{pr:phistphiD}),\n\\begin{align}\n[\\Phi^\\dagger\\Phi]_D=(\\partial_\\mu A)(\\partial^\\mu A^\\dagger)+F^\\dagger F+i\\psi^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\psi+\\text{total derivative}\\,,\\label{phiphiD}\n\\end{align}\nwhich again reproduces the kinetic energy terms of the Wess-Zumino Lagrangian.\n\nIndeed, one can obtain candidate terms for a SUSY Lagrangian by\nconsidering the $\\theta\\theta\\theta^\\dagger\\thetabar$ component of an\narbitrary function of a chiral superfield and its complex conjugate.\nThis function, denoted by $K(\\Phi,\\Phi^\\dagger)$, is called the K\\\"ahler\npotential. Applying the chain rule as in our computation of\n$[W(\\Phi)]_F$ [cf.~\\eqst{WPhi}{rest}], one can calculate (see Problem~\\ref{pr:K}),\n\\begin{align}\n\\begin{split}\n[K(\\Phi,\\Phi^\\dagger)]_D=&\\frac{\\partial^2 K}{\\partial A\\partial A^\\dagger}\\biggl[(\\partial_\\mu A)(\\partial^\\mu A^\\dagger)+F^\\dagger F+\\ifmath{\\tfrac12} i\\psi^\\dagger\\overline{\\sigma}^\\mu\\!\\!\\stackrel{\\leftrightarrow}{\\partial}_{\\!\\mu}\\!\\psi\\biggr] \\\\\n&-\\frac12\\,\\frac{\\partial^3 K}{\\partial A\\partial A^{\\dagger\\,2}}\\biggl[F\\psi^\\dagger\\psi^\\dagger+i\\psi^\\dagger\\overline{\\sigma}^\\mu\\psi\\partial_\\mu A^\\dagger\\biggl] \\\\\n&-\\frac12\\,\\frac{\\partial^3 K}{\\partial A^2\\partial A^{\\dagger}}\\biggl[F^\\dagger\\psi\\psi-i\\psi^\\dagger\\overline{\\sigma}^\\mu\\psi\\partial_\\mu A\\biggl] \\\\\n&+\\frac14\\,\\frac{\\partial^4 K}{\\partial A^2\\partial A^{\\dagger\\,2}}(\\psi\\psi)(\\psi^\\dagger\\psi^\\dagger)+\\text{total derivative}\\,.\\label{kahler}\n\\end{split}\n\\end{align}\n\nWe conclude that the most general SUSY Lagrangian involving a chiral\nsuperfield $\\Phi$ is given by\n\\begin{align}\n\\mathscr{L}=[K(\\Phi,\\Phi^\\dagger)]_D+\\bigl\\{[W(\\Phi)]_F+{\\rm h.c.}\\bigr\\}\\,.\\label{eq:Lgeneral}\n\\end{align}\nThe auxiliary field $F$ can be determined via its classical\nfield equation, which yields\n\\begin{align}\nF=\\left(\\frac{\\partial^2 K}{\\partial A\\partial A^\\dagger}\\right)^{-1}\\left[\\frac12\\,\\frac{\\partial^3 K}{\\partial A^2\\partial A^{\\dagger}}\\psi\\psi-\\left(\\frac{dW}{dA}\\right)^\\dagger\\right]\\,.\\label{aux}\n\\end{align}\n\nThe case of $K(\\Phi,\\Phi^\\dagger)=\\Phi^\\dagger\\Phi$ reduces to the result of\n\\eq{phiphiD} and corresponds to the kinetic energy term of the\nWess-Zumino model as noted above. In this case, \\eq{aux} yields,\n\\begin{equation}\nF=-\\left(\\frac{dW}{dA}\\right)^\\dagger\\,,\n\\end{equation}\nwhich reproduces the result previously obtained in \\eq{f}.\n\nMore complicated K\\\"ahler potentials yield non-renormalizable Lagrangians. These arise in\nlow-energy effective field theories (that include operators of dimension greater than four),\nin supersymmetric $\\sigma$-models, and in supergravity. Such\napplications lie beyond the scope of these lectures.\n\n\\subsection{$R$-invariance}\n\\label{Rinvariance}\nRecall that the SUSY algebra can be extended by added adding a bosonic\nU(1)$_R$ generator $R$ such that [cf.~\\eqst{R1}{susyalg7}],\n\\begin{align}\n\\left[R\\,,\\,Q_\\alpha\\right]=-Q_\\alpha\\,,\\qquad\\quad\n\\left[R\\,,\\,Q^\\dagger_{\\dot\\alpha}\\right]=Q^\\dagger_{\\dot\\alpha}\\,.\\label{Rcommute}\n\\end{align}\nThe action of ${\\rm U}(1)_R$ on a superfield $\\Phi$ can be represented by a differential operator $\\widehat{R}$ acting on superspace,\n\\begin{align}\n[\\Phi\\,,\\,R]=\\widehat{R}\\Phi\\,,\n\\end{align}\nwhere\n\\begin{align}\n\\widehat{R}\\equiv \\theta^\\alpha\\partial_\\alpha-\\theta^\\dagger_{\\dot\\alpha}\\partial^{\\dagger\\dot\\alpha}-n\\,,\\qquad \\text{with $n\\in\\mathbb{R}$}\\,.\n\\end{align}\nWe call $n$ the \\textit{weight} (or $R$-charge) of the superfield $\\Phi$. (For a \\textit{real} superfield, only $n=0$ is possible.)\nUnder a ${\\rm U}(1)_R$ transformation,\n\\begin{align}\n\\delta_a\\Phi=ia[R\\,,\\,\\Phi]=-ia\\widehat{R}\\Phi\\,.\n\\end{align}\nActing on a superfield $\\Phi(x,\\theta,\\theta^\\dagger)$,\n\\begin{equation}\n\\widehat{R}\\,\\Phi(x,\\theta,\\theta^\\dagger)=e^{2ina}\\,\\Phi(x,e^{-ia}\\theta,e^{ia}\\theta^\\dagger)\\,,\\label{Rtrans}\n\\end{equation}\nThe differential operator $\\widehat{R}$ satisfies the identities,\n\\begin{Eqnarray}\nD_\\alpha \\widehat{R}&=&(\\widehat{R}+1)D_\\alpha\\,,\\\\\n\\overline{D}_{\\dot\\alpha}\\widehat{R}&=&(\\widehat{R}-1)\\overline{D}_{\\dot\\alpha}\\,.\n\\end{Eqnarray}\nHence, it follows that if $\\Phi$ is a chiral [antichiral] superfield, then\n$\\widehat{R}\\Phi$ is a chiral [antichiral] superfield.\n\nGiven a chiral superfield, $\\Phi=A+\\sqrt{2}\\,\\theta\\psi+\\theta\\theta\nF$, in the chiral representation, the ${\\rm U}(1)_R$ transformations of the component fields are:\n\\begin{align}\nA&\\to e^{ina}A\\,,\\\\\n\\psi &\\to e^{i(n-1)a}\\psi\\,,\\\\\nF&\\to e^{i(n-2)a}F\\,,\\label{RF}\n\\end{align}\nafter employing \\eq{Rtrans}.\n\n\n\\begin{theorem}\n\\label{Rtheorem}\nThe kinetic energy term $[\\Phi^\\dagger\\Phi]_D$ is automatically\n$R$-invariant, whereas $[W(\\Phi)]_F$ is $R$-invariant if and only if $W$ has $R$-charge equal to 2.\n\\end{theorem}\n\\begin{proof}\nIf $n=2$, then $F$ is invariant under a ${\\rm U}(1)_R$ transformation,\n in light of \\eq{RF}. This result applies to any $F$-term.\n\\end{proof}\n\n\\begin{example}\n[Wess-Zumino model with $\\boldsymbol{W(\\Phi)=\\ifmath{\\tfrac12} m\\Phi^2+\\tfrac13 g\\Phi^3}$]\nIf $m=0$, then the Wess-Zumino model is $R$-invariant with $n=\\tfrac13$.\nIf $g=0$, then the Wess-Zumino model is $R$-invariant with $n=\\tfrac12$.\nIf both $m\\neq 0$ and $g\\neq 0$, then the Wess-Zumino model is not $R$-invariant. \n\\end{example}\n\n\n\\subsection{Grassmann integration and the SUSY action}\n\\label{integration}\nA supersymmetric action can be written as an integral over superspace.\nFirst, we introduce integration over anticommuting Grassmann\nvariables. The rules of integration are\\cite{Berezin},\n\\begin{align} \\label{grules}\n\\int d\\theta=\\int d\\theta^\\dagger=0\\,,\\qquad \\int \\theta\\,d\\theta=\\int\n \\theta^\\dagger\\,d\\theta^\\dagger=1\\,.\n\\end{align}\nThat is, integration over Grassmann variables is in some sense equivalent to differentiation.\n\nIt is conventional to define\n\\begin{align}\nd^2\\theta&\\equiv -\\tfrac14 d\\theta^\\alpha d\\theta_\\alpha\\,,\\\\\nd^2\\theta^\\dagger&\\equiv-\\tfrac14 d\\theta^\\dagger_{\\dot\\alpha} d\\theta^{\\dagger\\dot\\alpha}\\,,\\\\\nd^4\\theta &\\equiv d^2\\theta d^2\\theta^\\dagger\\,,\n\\end{align}\nwhich yields the following non-zero integrals,\n\\begin{align}\n\\int d^2 \\theta\\, (\\theta\\theta)=\\int d^2\\theta^\\dagger\\,(\\theta^\\dagger\\thetabar)=\\int d^4\\theta\\,(\\theta\\theta)(\\theta^\\dagger\\thetabar)=1\\,.\n\\end{align}\n\n\nIt follows that for a chiral superfield,\n\\begin{align}\n\\int d^2\\theta\\, \\Phi(x,\\theta,\\theta^\\dagger)=\\int d^2\\theta\\, \\Phi_1(x,\\theta)=[\\Phi]_F=-\\tfrac14 D^2\\Phi\\biggl|_{\\theta=\\theta^\\dagger=0}\\,.\n\\label{d2theta}\n\\end{align}\nLikewise, for an arbitrary superfield $V(x,\\theta,\\theta^\\dagger)$,\n\\begin{align}\n\\int d^4\\theta\\, V(x,\\theta,\\theta^\\dagger)=[V]_D=\\tfrac{1}{16}\\overline{D}\\lsup{2} D^2 V\\biggl|_{\\theta=\\theta^\\dagger=0}\\,.\\label{d4theta}\n\\end{align}\nThus, the most general SUSY action involving a chiral superfield $\\Phi$ is\n\\begin{align}\nS=\\int d^4 x\\,d^4\\, \\theta K(\\Phi,\\Phi^\\dagger)+\\int d^4 x\\,d^2\\theta\\, W(\\Phi)\n+\\int d^4 x\\,d^2\\theta^\\dagger\\, W(\\Phi^\\dagger)\\,.\\label{S}\n\\end{align}\n\nGeneralizations to theories with multiple chiral superfields are\nstraightforward. In the more general case, $W$ is a holomorphic\nmultivariable function of the chiral superfields, and $K$ is a\nmultivariable function of the chiral superfields and their hermitian conjugates. For a renormalizable theory, $W$ is at most a cubic multinomial, \n\\begin{align}\nW(\\Phi_i)=\\sum_i a_i\\Phi_i+\\sum_{i,j} b_{ij}\\Phi_i\\Phi_j+\\sum_{i,j,k}c_{ijk}\\Phi_i\\Phi_j\\Phi_k\\,,\n\\end{align}\nand\n\\begin{align}\nK(\\Phi_i,\\Phi_i^\\dagger)=\\sum_i \\Phi_i^\\dagger\\Phi_i\\,.\\label{Ksimple}\n\\end{align}\n\nIn special cases, one can convert an integral over ``half'' of superspace (e.g. integrals over\n$d^4 x\\, d^2\\theta$) into an integral over the full superspace. The key\nobservation is that for an arbitrary superfield $V$,\n\\begin{align}\n\\int d^4x\\,d^2\\theta\\,V(x,\\theta,\\theta^\\dagger)=\\int d^4 x\\left(-\\tfrac14 D^2 V\\right)\\,.\\label{intV}\n\\end{align}\nOn the left-hand side of \\eq{intV}, the integration over $d^2 \\theta$ projects out all terms proportional to $\\theta\\theta$. On the right-hand side, $D^2\\!=-\\!\\partial^\\alpha\\partial_\\alpha$\nup to total derivative terms that can be dropped because we are integrating over $d^4 x$. Hence, $\\tfrac14 \\partial^\\alpha\\partial_\\alpha$ has the effect of projecting out all terms proportional to $\\theta\\theta$.\nLikewise,\n\\begin{align}\n\\int d^4x\\,d^2\\theta^\\dagger\\,V(x,\\theta,\\theta^\\dagger)=\\int d^4 x\\left(-\\tfrac14 \\overline{D}\\lsup{2} V\\right)\\,.\n\\end{align}\nHence, it follows that\n\\begin{align}\n\\int d^4 x\\,d^2\\theta\\left(-\\tfrac14 \\overline{D}\\lsup{2} V\\right)=\\int d^4x\\,d^4\\theta\\,V(x,\\theta,\\theta^\\dagger)\\,.\\label{2to4}\n\\end{align}\n\n\\Eqs{d2theta}{d4theta} identify integrals over\nhalf of superspace as $F$-terms and integrals over the full superspace\nas $D$-terms. However,\n\\eq{2to4} appears to blur the distinction between $D$-terms and $F$-terms.\nFor example, in the Wess-Zumino Lagrangian, the kinetic energy\nterm may be written as an $F$-term, $[\\Phi T\\Phi]_F$ [cf. \\eq{WZlagF}], or\nas a $D$-term, $[\\Phi^\\dagger\\Phi]_D$, as in\n \\eqs{phiphiD}{eq:Lgeneral}. However,\nconsider the case of a half superspace integral of the superpotential given\nin \\eq{S}. If we attempt to convert this into a full superspace\nintegral using \\eq{2to4}, the end result is\n\\begin{equation}\n\\int d^4 x \\,d^2\\theta\\,W(\\Phi) =-4\\int d^4 x\\, d^4\\theta\\,\\label{nonlocal}\n\\overline{D}\\lsup{-2} W(\\Phi)\\,.\n\\end{equation}\nDue to the inverse differential operator, the integrand on the right-hand side of \\eq{nonlocal} is a non-local\nfunctional of chiral superfields. This provides the distinction\nbetween $F$-terms and $D$-terms. In particular, any half superspace integral\nthat can be converted into a full superspace integral over a \\textit{local}\nfunctional of superfields will be called a $D$-term.\n\nHaving written the action in \\eq{S} as an integral over superspace (for\n$D$-terms) and half of superspace (for $F$-terms), one can obtain\nexpressions for the Green functions of quantum chiral\n(and antichiral) superfields. The corresponding two-point functions\nprovide expressions for the superspace propagators. One can then\nformulate a set of superspace Feynman rules and develop a\ndiagrammatic representation of the perturbative expansion of the\nGreen functions. This was first carried out by Grisaru, Ro\\u{c}ek, and\nSiegel\\cite{GRS}, and was applied to the perturbative computation of\nthe effective action. Indeed, such techniques are quite useful since a\nsingle supergraph (in which individual lines correspond to\nsuperfields) is equivalent to a large number of Feynman diagrams involving\nthe corresponding component fields.\nA comprehensive treatment of these methods are beyond the scope of these lectures.\nFor a pedagogical development of supergraphs and superspace Feynman rules, see e.g.~Refs.\\cite{Gates,Srivastava,Buchbinder,Pokorski}.\n\n\\subsection{Improved ultraviolet behavior of supersymmetry}\n\\label{sec:non-renorm}\n\nAn attractive feature of supersymmetric quantum field theories is that\ntheir ultraviolet divergences are better behaved, as compared to\nordinary quantum field theories.\nRef.\\cite{GRS} demonstrated that the loop corrections to the effective\naction of a supersymmetric theory of chiral superfields\ncan be expressed as an integral over the full superspace,\n\\begin{equation} \\label{effact}\n\\sum_n \\int d^4 x_1\\cdots d^4 x_n\\int d^4\\theta\\, g_n(x_1,\\ldots,x_n)\nF_1(x_1,\\theta,\\theta^\\dagger)\\cdots F_n(x_n,\\theta,\\theta^\\dagger)\\,,\n\\end{equation}\nwhere the $F_i(x_i,\\theta,\\theta^\\dagger)$ are local functionals of chiral\nand antichiral superfields and their covariant derivatives, and the\n$g_n$ are translationally invariant functions on Minkowski space.\n\n\\Eq{effact} implies that $D$-terms are renormalized but $F$-terms\nare not renormalized. Moreover, if $F$-terms are absent at\ntree-level, then they are not generated at the loop level. Hence, the\ntree-level K\\\"ahler potential is\nrenormalized by radiative corrections, whereas there are no loop\ncorrections to the tree-level superpotential. This is the famous\nnon-renormalization theorem of $N=1$ supersymmetry.\\footnote{The \nproof of the non-renormalization theorem implicitly assumes that the function $g_n$\nin \\eq{effact} is local. However, the non-renormalization theorem can fail if \nthe super\\-symmetric theory contains massless fields as shown in\nRefs.\\cite{West:1990rm,Jack:1990pd,Dunbar:1991fc}, due to infrared\ndivergences. For example, the inverse Laplacian operator\n$\\square^{-1}$ (from a massless propagator) can appear, resulting in\na non-local function $g_n$ in \\eq{effact}. One can show that\nthe non-renormalization theorem holds for the\nWilsonian effective action\\cite{Shifman:1986zi,Shifman:1991dz}, where\nthe infrared effects are cut off\\cite{SeibergNR,Poppitz:1996na}. \\label{fnW}}\nThe proof of the non-renormalization theorem in Ref.\\cite{GRS} relies\non the analysis of supergraphs in perturbation theory, and is beyond\nthe scope of these lectures. Heuristically, this\ntheorem is a consequence of an exact cancellation between fermion and boson\nloop contributions to the effective action due to supersymmetry.\n\nNote that the non-renormalization of the tree-level superpotential\nis simply a consequence of the fact that the integral of a\nproduct of chiral superfields over \\textit{all} of superspace in \\eq{effact} is zero due to\n\\eq{grules} [see Problem~\\ref{pr:half}]. Moreover, the assumption that\nthe $F_i$ in \\eq{effact} are \\textit{local} functionals of chiral and\nantichiral superfields is essential. Otherwise, one could employ\n\\eq{nonlocal} and erroneously claim the existence of loop corrections\nto the tree-level superpotential.\n\n\n\nWe now briefly explore the consequence of the non-renormalization of\nthe superpotential. Consider the action of the Wess-Zumino model,\n\\begin{align}\nS_{\\mathrm{WZ}} = \\int d^4 x \\int d^4 \\theta\\, \\Phi^\\dagger \\Phi + \\sqof{ \\int d^4 x \\int d^2\\theta \\of{\\ifmath{\\tfrac12} m \\Phi^2 + \\tfrac{1}{3} \\lambda\\Phi^3} \n+ {\\rm h.c.}}.\n\\end{align}\nThe non-renormalization theorem implies that renormalized fields and\nparameters are related to bare fields and parameters as follows\\cite{Cui},\n\\begin{equation} \\label{bare}\n\\Phi_R=Z^{-1\/2}\\Phi\\,,\\qquad\\quad m_R=Zm\\,,\\qquad\\quad\n\\lambda_R=Z^{3\/2}\\lambda\\,.\n\\end{equation}\nwhere the subscript $R$ indicates renormalized quantities and the\nbare quantities have no subscript. \\Eq{bare} is equivalent to the\nstatement that the superpotential is unrenormalized,\n$W_R(\\Phi_R)=W(\\Phi)$. \nThat is,\n\\begin{equation}\n\\ifmath{\\tfrac12} m_R\\Phi_R^2 +\\tfrac13 \\lambda_R\\Phi_R^3=\\ifmath{\\tfrac12} m\\Phi^2 +\\tfrac13\n\\lambda\\Phi^3\\,.\n\\end{equation}\nWave function renormalization is a consequence of the\nrenormalization of the K\\\"ahler potential ($\\Phi^\\dagger\\Phi$ in\nthe case of the Wess-Zumino model). \n\nThe non-renormalization theorem does \\textit{not} assert that\nthe parameters of the\nsuperpotential are not renormalized. Indeed, \\eq{bare} states that\nthe renormalization of the parameters $m$ and $\\lambda$ are governed\nby the wave function renormalization constant $Z$. Moreover, the\nwave function renormalization constants of the component fields of the chiral\nsuperfield are~equal (i.e., $A_R\\!\\!=Z^{-1\/2}A$ and\n$\\psi_R\\!=\\!Z^{-1\/2}\\psi$), as a consequence of supersymmetry.\n\n\n\n\nIn Ref.\\cite{SeibergNR}, Seiberg offered a more intuitive understanding of the\nnon-renormalization theorem, which also forbids nonperturbative\ncorrections to the Wilsonian effective action [cf.~footnote~\\ref{fnW}].\nSeiberg's argument draws on the symmetry and\nholomorphy\\footnote{The fact that the superpotential is a\n holomorphic function of chiral superfields plays a critical role in\nSeiberg's argument.\nIn contrast, the renormalization of the K\\\"ahler potential is\npossible because the latter is a function of chiral and antichiral\nsuperfields and hence is not holomorphic.}\n\\!of the\nsuperpotential. Consider again the example of the Wess-Zumino\nsuperpotential, $W(\\Phi)=\\ifmath{\\tfrac12} m\\Phi^2+\\tfrac13\\lambda\\Phi^3$.\nFollowing Ref.\\cite{SeibergNR}, one can think of $m$ and $\\lambda$ as the vacuum\nexpectation values of chiral superfields, so that $W$ must be\nholomorphic in $m$ and $\\lambda$ as well as in $\\Phi$. \nIn light of Theorem~\\ref{Rtheorem} in Section~\\ref{Rinvariance}, the theory is\ninvariant under an enhanced ${\\rm U}(1) \\times {\\rm U}(1)_R$\nsymmetry, with the charge assignments shown in\nTable~\\ref{tab:charges}.\n\\begin{table}[h!]\n\\begin{center}\n\\caption{\\small Charge assignments under the ${\\rm U}(1) \\times {\\rm U}(1)_R$ symmetry.}\n\\label{tab:charges}\n\\vskip 0.1in\n\\begin{tabular}{| l | r r r r |}\n\\hline\n\t& $\\Phi$\t& $\\Phi^\\dagger$\t& $m$\t& $\\lambda$\t\\\\\n\t\\hline\n${\\rm U}(1)$ & 1\t\t& $-1$\t\t\t\t& $-2$\t\t& $-$3 \t\t\t\\\\\n${\\rm U}(1)_R$\t& 1\t\t&\t1\t\t\t&\t0\t&\t$-$1\t\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nTo maintain the ${\\rm U}(1) \\times{\\rm }U(1)_R$ symmetry and holomorphy,\ncorrections to the Wilsonian effective superpotential must therefore be of the form\n\\begin{align}\nm \\Phi^2 f\\of{ \\frac{ \\lambda \\Phi }{ m} },\\label{wilson}\n\\end{align}\nwhere $f$ is an arbitrary holomorphic function. \n\\Eq{wilson} is valid for arbitrary $\\lambda$. Thus, we can take\n$|\\lambda|\\ll 1$, in which case perturbation theory should be valid.\nExpanding in powers of the coupling constant $\\lambda$, the perturbative expansion should have the\nfollowing form,\n\\begin{equation}\nW_{\\rm eff}=\\sum_{n=0}^\\infty\na_n\\frac{\\lambda^n}{m^{n-1}}\\Phi^{n+2}\\,.\\label{Weffective}\n\\end{equation}\nThe terms in $W_{\\rm eff}$ are represented diagrammatically by one\nparticle irreducible (1PI) supergraphs constructed from propagators and\nthree-point vertices proportional to $\\lambda$. However, one cannot construct a\none-loop (or higher) supergraph that behaves like\n$\\lambda^n\\Phi^{n+2}$. It is easy to show that tree-level diagrams\nwith $n+2$ external legs, $n$ vertices and $n-1$ propagators would\nbehave like $\\lambda^n\\Phi^{n+2}$. But, the only 1PI tree-level\ngraphs are those with either two or three external legs! Hence, we\nconclude that $a_0=\\ifmath{\\tfrac12}$, $a_1=\\tfrac13$ and $a_n=0$ for\n$n\\geq 2$.\\footnote{One can also conclude that $a_n=0$ for $n\\geq 2$\n by noting that the Wilsonian effective action $W_{\\rm eff}$ must have a smooth\n limit as $m\\to 0$.} That is\n$W_{\\rm eff}(\\Phi)=W_{\\rm tree}(\\Phi)$, which is the statement that the\nsuperpotential is not renormalized.\n\n\n\n\n\\subsection{Problems}\n\n\\begin{problem}\n\\label{pr:two_super_translations}\nProve that\n\\begin{align}\n G(y,\\xi,\\xi^\\dagger)G(x,\\theta,\\theta^\\dagger)=G\\bigl(x+y+i(\\xi\\sigma\\theta^\\dagger-\\theta\\sigma\\xi^\\dagger),\\xi+\\theta,\\xi^\\dagger+\\theta^\\dagger\\bigr)\\,. \\nonumber\n \\end{align}\n {\\sl HINT}: use the Baker-Campbell-Hausdorff formula given in \\eq{BCH}. \n \\end{problem}\n \n \n\\begin{problem}\nVerify that when acting on a superfield $\\Phi(x,\\theta,\\theta^\\dagger)$,\n$$\n\\{\\widehat{Q}_\\alpha\\,,\\,\\widehat{Q}_\\beta\\}=\\{\\widehat{Q}^\\dagger_{\\dot\\alpha}\\,,\\,\\widehat{Q}^\\dagger_{\\dot\\beta}\\}=0\\,,\\qquad\\quad\n\\{\\widehat{Q}_\\alpha\\,,\\,\\widehat{Q}^\\dagger_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}\\widehat{P}_\\mu\\,.\n$$\n\\end{problem}\n \n \n\\begin{problem}\n\\label{pr:3results}\nProve \\eqst{eq:r1}{eq:r3}. The last result is an example of a\nFierz identity (see, e.g., Appendix B of Ref.\\cite{Dreiner:2008tw} or\nAppendix A of Ref.\\cite{Bailin}).\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:fmnV}\nUsing \\eq{supertrans}, obtain the SUSY transformation laws for the\nbosonic component fields,\n$f$, $m$, $n$, $V_\\mu$, and $d$, and the fermionic component fields,\n$\\zeta$, $\\chi$, $\\lambda$ and $\\psi$, which appear in the complex\nsuperfield defined in \\eq{phitaylor}. \n\\end{problem}\n\n\\begin{problem}\nSuppose that $\\Phi$ is a bosonic superfield. Verify that\n\\eq{daggers} holds. Then, show that \\eqs{eq:D}{eq:Db} satisfy\n\\eq{Dcond}.\n\\end{problem}\n\n\\begin{problem}\nSuppose that $\\Phi$ is a fermionic superfield. Show that\n\\eqs{daggers}{Dcond} are modified as follows:\n$(\\partial_\\alpha\\Phi)^\\dagger=\\partial_{\\dot\\alpha}^\\dagger\\Phi^\\dagger$\nand $(D_\\alpha\\Phi)^\\dagger=-\\overline{D}_{\\dot\\alpha}\\Phi^\\dagger$. \n \\end{problem}\n\n\\begin{problem} \n\\label{pr:D}\nShow that the spinor covariant derivatives, as defined in \\eq{eq:D}\nand \\eq{eq:Db}, satisfy the following anticommutation relations,\n$\\{D_\\alpha\\,,\\,D_\\beta\\}=\\{\\overline{D}_{\\dot\\alpha}\\,,\\,\\overline{D}_{\\dot\\beta}\\}=0$ and\n$\\{D_\\alpha\\,,\\,\\overline{D}_{\\dot\\beta}\\}=2i\\sigma^\\mu_{\\alpha\\dot\\beta}\\partial_\\mu$.\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:DDid}\nDerive \\eq{DDid}.\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:expexp}\nProve that\n\\begin{align*}\n\\exp(-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu)=1-i\\theta\\sigma^\\mu\\theta^\\dagger\\,\\partial_\\mu-\\tfrac{1}{4}(\\theta\\theta)(\\theta^\\dagger\\thetabar)\\square,\n\\end{align*}\nwhere $\\square\\equiv\\partial_\\mu\\partial^\\mu.$\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:chiraltrans}\nUsing \\eq{supertrans}, one can\nobtain the SUSY transformation laws for the component fields $A$,\n$\\psi$ and $F$ in \\eq{eq:chiralSF}. Perform the calculation by\nworking in the chiral representation and show\nthat the SUSY transformation laws for $A$, $\\psi$ and $F$ coincide with\nthe results obtained previously in \\eqst{offshell1}{offshell3} for the fields of a superspin $j=0$\nsupermultiplet. \n\\end{problem}\n\n\\begin{problem}\n\\label{pr:phistphiD}\nIf $\\Phi$ is a chiral superfield, show that\n\\begin{align*}\n[\\Phi^*\\Phi]_D=(\\partial_\\mu A)(\\partial^\\mu A^*)+F^* F+i\\psi^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\psi+\\text{total derivative}\\,.\n\\end{align*}\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:K}\nDerive \\eq{kahler}.\n\\end{problem}\n\n\\begin{problem}\nA linear superfield\\cite{Ferrara:1974ac,Salam:1974jj}, $L(x,\\theta,\\bar\\theta)$,\nis defined as a constrained real scalar superfield that satisfies, $D^2 L(x,\\theta,\\bar\\theta)=\\overline D\\lsup{2}\nL(x,\\theta,\\bar\\theta)=0$. Identify the component fields\nthat make up the linear superfield. Show that $\\partial_\\mu V^\\mu=0$,\nwhere $V^\\mu$ is the component vector field of $L$. \nCheck that the number of fermion and boson degrees of freedom of the linear\nsuperfield are equal.\n[HINT: the identity given by \\eq{DDid} should be helpful.]\n\\end{problem}\n\n\n\\begin{problem}\nEmploying the operator $T$ defined in \\eq{Tdef}, show that\n\\begin{equation}\n\\int d^4 x\\,d^2\\theta\\, \\Phi T\\Phi = \\int d^4 x\\, d^4\\theta\n\\,\\Phi^\\dagger\\Phi\\,,\n\\end{equation}\nby converting the integral over half of superspace into an integral\nover the full superspace. Use the above result to conclude that\n$[\\Phi T \\Phi]_F=[\\Phi^\\dagger\\Phi]_D$.\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:half}\nIf $\\Phi$ is a chiral superfield and $\\Phi^\\dagger$ is an antichiral\nsuperfield, show that\n\\begin{equation}\n\\int d^4 x\\,d^4\\theta\\, \\Phi(x,\\theta,\\theta^\\dagger)=\\int d^4 x\\,d^4\\theta\\,\n\\Phi^\\dagger(x,\\theta,\\theta^\\dagger)=0\\,.\n\\end{equation}\n\\end{problem}\n\n\n\n\\section{Motivation for TeV-scale supersymmetry}\n\\label{sec:motivation}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\n\nThe Standard Model (SM) of particle physics has been remarkably\nsuccessful for describing the observed behavior of the fundamental\nparticles and their interactions\\cite{Langacker}. \nIndeed, there are no definitive departures from the Standard Model observed in experiments conducted at high energy collider facilities. \nNevertheless, some fundamental microscopic phenomena must necessarily lie outside of the purview of the SM.\nThese include: neutrinos with non-zero mass\\cite{numass}; dark matter\\cite{darkmatter}; the suppression of CP-violation in the strong interactions (the so-called strong CP problem\\cite{Kim:2008hd}); gauge coupling unification\\cite{guts}; the baryon asymmetry of the universe\\cite{White:2016nbo}; inflation in the early universe\\cite{inflation}; dark energy\\cite{darkenergy}; and the gravitational interaction. None of these phenomena can be explained within the framework of the SM alone.\n\nConsequently, the SM should be regarded at best as a low-energy effective field theory~\\cite{eft}, which is valid below some high energy scale. \nThat is, new high energy scales must exist where more fundamental physics resides.\nIn this section, we explain why one might expect to find this new\nphysics at the TeV scale. We discuss the \\textit{principle of\n naturalness}, and how supersymmetry provides a natural mechanism for\navoiding the quadratic sensitivity of the squared-masses of\nelementary scalar particles to ultraviolet physics.\n\n\n\\subsection{Why the \\textrm{TeV} scale?}\nThe classical gravitational interaction lies outside the SM. Using\nthe fundamental constants, $\\hbar$, $c$ and Newton's gravitational\nconstant $G_N$, one can construct a quantity with the units of energy\ncalled the Planck scale,\n\\begin{equation}\nM_{\\rm PL}c^2\\equiv \\left(\\frac{\\hbar c^5}{G_N}\\right)^{1\/2}\\simeq\n1.2\\times 10^{19}~{\\rm GeV}\\,.\n\\end{equation}\nThe significance of the Planck scale can be seen as follows.\nAt the Planck energy scale, the quantum mechanical\naspects of gravity can no longer be neglected.\nThe gravitational energy of a particle of mass $m$,\nevaluated at its Compton wavelength, $r_c=\\hbar\/(mc)$,\n\\begin{equation}\n\\Phi\\sim\\frac{G_N m^2}{r_c}=\\frac{G_N m^3 c}{\\hbar}{~\\raise.15em\\hbox{$<$}\\kern-.85em\\lower.35em\\hbox{$\\sim$}~} 2mc^2\\,,\n\\end{equation}\nmust be below $2mc^2$ to avoid particle-antiparticle pair\ncreation by the gravitational field. Hence, up to \n$\\mathcal{O}(1)$ constants, we conclude that $m{~\\raise.15em\\hbox{$<$}\\kern-.85em\\lower.35em\\hbox{$\\sim$}~} M_{\\rm\nPL}$.\\footnote{Note that for $m=M_{\\rm PL}$, the Schwarzschild radius\n$r_s\\equiv 2G_N m\/c^2\\simeq r_c$, which provides additional evidence\nthat the quantum mechanical nature of gravity cannot be neglected at\nenergy scales above the Planck scale.}\nSince particle-antiparticle pair creation is an inherently quantum\nmechanical phenomenon, \nquantum gravitational effects can no longer be ignored at the Planck scale. \nThus, the SM cannot be a\nfundamental theory of particles and interactions at energy scales of\norder the Planck scale and above. \n\n\nThere must be an energy scale $\\Lambda$ at which the Standard Model\nbreaks down. Based on the arguments given above, it follows that the\nupper bound on $\\Lambda$ is the Planck scale. But, it is possible\nthat $\\Lambda$ lies significantly below the Planck scale.\nFor example, a credible theory of neutrino masses (e.g., the type-I seesaw model~\\cite{numass}) posits the existence of a right-handed electroweak singlet Majorana neutrino of mass of order $10^{14}~{\\rm GeV}$. \nHenceforth, we shall define $\\Lambda$ to be the lowest energy scale at\nwhich the SM breaks down. \n\nThe predictions made by the SM depend on a number of parameters that\nmust be taken as input to the theory. These parameters cannot be\npredicted, since their values are \nsensitive to unknown ultraviolet (UV) physics.\nIn the 1930s, it was already appreciated\nthat a critical difference exists between the behavior of boson and\nfermion masses~\\cite{Weisskopf:1939zz}. Fermion masses are\nlogarithmically sensitive to UV physics~\\cite{Weisskopf:1934} due to\nthe chiral symmetry of massless fermions, which implies that the radiative\ncorrection to the tree-level fermion mass is of the form,\n\\begin{equation}\n\\delta m_F\\sim m_F\\ln(\\Lambda^2\/m_F^2)\\,,\n\\end{equation}\nwhich vanishes in the limit of $m_F\\to 0$.\nIn contrast, no such symmetry exists for bosons (in the absence of supersymmetry), and consequently we expect quadratic sensitivity of\nthe boson squared-mass to UV physics,\n$\\delta m^2_B\\sim \\Lambda^2\\,.$\n\n\n\nThese observations have important consequences for the fundamental physics\nthat describes the Higgs boson. \nIn the SM, the Higgs boson squared-mass is given by $m_h^2=\\lambda v^2$ and the W\nboson squared-mass is $m_W^2=\\tfrac{1}{4}g^2 v^2$, where\n$\\vev{\\Phi^0}=v\/\\sqrt{2}=174$~GeV is the vacuum\nexpectation value of the neutral Higgs field, $\\lambda$~is\nthe Higgs self-coupling [cf.~\\eq{vofphi}], and $g$ is the SU(2) gauge coupling.\nTogether, these imply that\n\\begin{equation}\n\\frac{m_h^2}{m_W^2}=\\frac{4\\lambda}{g^2}\\,,\n\\end{equation}\nwhich one would expect to be roughly of $\\mathcal{O}(1)$. The Higgs\nboson with mass 125~GeV\nsatisfies this expectation. \n\nHowever, the existence of the Higgs boson is a consequence of a spontaneously broken scalar potential, \n\\begin{equation}\nV(\\Phi)=-\\mu^2(\\Phi^\\dagger\\Phi)+\\ifmath{\\tfrac12}\\lambda(\\Phi^\\dagger\\Phi)^2\\,,\\label{vofphi}\n\\end{equation}\nwhere $\\mu^2=\\ifmath{\\tfrac12}\\lambda v^2$ at\nthe minimum of the scalar potential.\nThe parameter $\\mu^2$ is quadratically sensitive to $\\Lambda$. Hence, to obtain $v=246$~GeV in a theory\nwhere $v\\ll \\Lambda$ requires a significant fine-tuning of the ultraviolet parameters of the fundamental theory.\nIndeed, the one-loop contributions to the squared mass parameter $\\mu^2$ are expected to be of\norder $(g^2\/16\\pi^2)\\Lambda^2$. Setting this quantity to be of order of $v^2$ (to avoid an \\textit{unnatural} cancellation\nbetween the tree-level parameter and the loop corrections) yields\n\\begin{equation}\n\\Lambda\\simeq 4\\pi v\/g\\sim {O}(1~{\\rm TeV})\\,.\n\\end{equation}\nThus, a \\textit{natural} theory of electroweak symmetry breaking\n(EWSB) appears to require new TeV scale physics beyond the SM associated with the EWSB dynamics. \n\n\n\n\n\\subsection{The modern principle of naturalness}\nThis principle of naturalness was \nfirst introduced by Weisskopf in a paper published in 1939\\cite{Weisskopf:1939zz}.\nIn the abstract of this 1939 paper, Weisskopf wrote,\n``the self-energy of charged particles obeying Bose statistics is found to be quadratically divergent...,'' and concluded that in theories of elementary bosons, new phenomena must enter at an energy scale of $m\/e$ (where $e$ is the relevant coupling). \nIn modern particle physics, naturalness is often associated with the question, ``how do we understand the magnitude of the EWSB scale?''\nIn the absence of\nnew physics beyond the SM, its natural value would be the\nPlanck scale (or perhaps the grand unification scale or the seesaw scale that\ncontrols neutrino masses).\n\nThere have been a number of theoretical proposals to explain the origin of the EWSB energy scale:\n(1) naturalness is restored by a symmetry principle--supersymmetry (SUSY)--which ties the bosons to\nthe more well-behaved fermions\\cite{Witten,Susskind}; (2) the Higgs boson is an approximate Goldstone boson, the only other\nknown mechanism for keeping an elementary scalar light\\cite{dewsb}; (3) the Higgs boson is a composite scalar, with an inverse length of\norder the TeV-scale\\cite{dewsb}; (4) extra spatial dimensions beyond three provide new mechanisms\nfor naturally large hierarchies of scales\\cite{RS,extradim};\n(5)~classical scale invariance and its minimal violation via quantum\nanomalies\\cite{Bardeen:1995kv,Meissner:2006zh,Iso:2009ss,Tavares:2013dga,Gorsky:2014una,Helmboldt:2016mpi} can generate a Higgs mass via dimensional transmutation\\cite{Coleman:1973jx}; and\n(6)~the EWSB scale arises due to some vacuum selection mechanism\n(either anthropic\\cite{Agrawal:1998xa} or cosmological\\cite{Graham:2015cka,Arkani-Hamed:2016rle}).\nFinally, maybe none of these explanations are relevant, and the EWSB\nenergy scale \nis simply the result of some initial condition whose origin will never be discernible.\n\nOf course, these are lectures on supersymmetry. Thus, we shall\nmotivate SUSY at the TeV scale as a potential solution of the\nso-called hierarchy problem:\nwhy is the scale of EWSB so much smaller than the Planck\nscale?\n\n\n \\subsection{\\mbox{Avoiding quadratic UV-sensitivity \nwith elementary scalars}}\n\\label{quadratic}\nFirst, consider a lesson from history.\nThe electron self-energy in classical electromagnetism goes\nlike $e^2\/a$, where $a$ is the classical radius of the electron. For a\npoint-like electron, $a\\rightarrow 0$; hence the electron self-energy diverges linearly. In the quantum\ntheory, fluctuations of the electromagnetic fields (in the\n``single electron theory'') generate a quadratic divergence.\n If\nthese divergences are not canceled, one would expect \nQED to break down at an energy of order $m_e\/e$,\n far below the Planck scale.\n\nThe linear and quadratic divergences will cancel exactly if\none makes a bold hypothesis: the existence of the positron\n(with a mass equal to that of the electron but of opposite\ncharge).\nWeisskopf was the first to demonstrate this cancellation in\n1934\\cite{Weisskopf:1934}.\\footnote{Actually the cancellation was not present\nin the initial publication, but thanks to\na letter from Wendell Furry, the correct result was published in an erratum.}\nThis is an historical example in which \n a symmetry implies the existence of a partner particle that cancels\n the dangerously large UV contribution to the particle mass. \n\nThe motivation for SUSY may be viewed analogously\\cite{Hitoshi,Hitoshi2}, with the electron playing the role of SM particles and the\npositron playing the role of superpartners. SUSY\nassociates a fermionic superpartner with every SM particle and vice versa, thus doubling the SM spectrum. SUSY relates the self-energy of the\nelementary scalar boson to the self-energy of its fermionic partner. Since the latter is only logarithmically sensitive to~$\\Lambda$, we conclude\nthat the quadratic sensitivity of the scalar squared-mass to\nUV physics must exactly cancel.\nNaturalness is restored!\n\nHowever,\nsince no superpartners degenerate in mass with the\ncorresponding SM particles exist in nature, SUSY must be a broken symmetry.\nAlthough the fundamental origin of SUSY-breaking is yet to be understood,\nthe effective scale of SUSY-breaking cannot be much larger than of\norder a few TeV, \nif SUSY is responsible for the origin of the EWSB scale.\n\n\n\\enlargethispage{\\baselineskip}\nThe absence of any evidence for SUSY at the LHC\\cite{nosusy} is a cause for some\nconcern\\cite{susy}. This has led to some discussion of the so-called little\nhierarchy problem\\cite{little,little2,little3} which reflects the observation that the\neffective SUSY-breaking mass scale is somewhat separated from the scale of EWSB.\nNevertheless, if evidence for supersymmetric\nphenomena in the TeV or multi-TeV regime were to be eventually established at \nthe LHC or at a future collider facility\n(with an energy reach beyond the LHC\\cite{vlhc}), it would be viewed as a spectacularly\nsuccessful explanation of the large hierarchy between the (multi-)TeV scale and\nPlanck scale. In this case, the remaining little hierarchy would\nperhaps be regarded as a less pressing issue.\n\n\n\n\\section{Supersymmetry Breaking}\n\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\label{SSB}\n\nIf supersymmetry were an exact symmetry of nature, then particles\nand their superpartners, which differ in spin by half a unit, would be\ndegenerate in mass. Since superpartners have not (yet) been observed,\nsupersymmetry must be a broken symmetry. In light of the\nnon-observation of supersymmetric particles at the LHC,\nthe energy scale of supersymmetry breaking must lie above \n1~TeV. \n\nThe fundamental mechanism responsible for supersymmetry breaking is\npresently unknown. In Section~\\ref{sec:originsofSUSYbreaking}, we\ndescribe some general considerations related to SUSY breaking,\nand we examine several possible frameworks for the spontaneous\nbreaking of SUSY. In Section~\\ref{sumrule}, we examine constraints\non mass splittings within supermultiplets in the presence of\nSUSY-breaking.\nThe possible origins of SUSY-breaking dynamics is surveyed in Section~\\ref{SUSYdynamics}. \nFinally, in Section~\\ref{sec:softSUSYbreaking}, we examine a\nmore agnostic approach, in which the supersymmetry of the effective low energy theory at\nthe TeV scale is softly broken. In such an\napproach, we identify the possible soft-supersymmetry breaking terms\nthat can appear in the Lagrangian, without making assumptions about\ntheir fundamental origin.\n\n\\subsection{Spontaneous SUSY breaking}\n\\label{sec:originsofSUSYbreaking}\nIn Section~\\ref{SUSYalg}, we derived \\eq{pzero}, which states that the energy operator $P^0$ for a supersymmetric theory is given by\n\\begin{align}\nP^0= \\frac{1}{2 t } \\of{ Q_1 Q_1^\\dagger+Q_1^\\dagger Q_1+Q_2 Q_2^\\dagger+Q_2^\\dagger Q_2 }\\,,\\label{peezero}\n\\end{align}\nwhere $t$ is real and positive (conventionally, $t=2$). Since the right-hand side of \\eq{peezero} is\npositive semi-definite, it\nfollows that the vacuum energy is zero if and only if the vacuum is supersymmetric:\n\\begin{align}\n\\vev{0\\,|P^0\\,|\\,0}=0\\quad\\Longleftrightarrow\\quad Q_\\alpha\\ket{0}=0\\,.\n\\end{align}\nMoreover, assuming the absence of fermion condensation,\\footnote{That is, we assume the absence of a\nfermion bilinear covariant, with the properties of a Lorentz scalar,\nthat acquires a nonzero vacuum expectation value.}\nthe vacuum energy can be identified as the vacuum expectation value of\nthe scalar potential. That is, in the case of a supersymmetric vacuum,\n\\begin{align}\n\\vev{0\\,|P^0\\,|\\,0}=0\\quad\\Longleftrightarrow\\quad \\vev{0\\,|V_{\\rm scalar}\\,|\\,0} = 0\\,.\n\\end{align}\nTo appreciate the significance of $ \\vev{0\\,|V_{\\rm scalar}\\,|\\,0} = 0$,\nrecall \\eq{vscalar4}, which we repeat below for the\nconvenience of the reader,\n\\begin{align}\nV_{\\rm scalar}=\\ifmath{\\tfrac12} D^a D^a+\\sum_i F_i^* F_i\\,.\\label{DDFF}\n\\end{align}\nIt follows that if the vacuum is supersymmetric, then the vacuum expectation \nvalues of the auxiliary fields must vanish,\n\\begin{align}\n\\vev{0\\,|F_i\\,|\\,0}=\\vev{0\\,|D^a\\,|\\,0}=0.\n\\end{align}\n\nOne can reach the same conclusion by considering the transformation\nlaws of the field components of a superfield.\nFor a chiral superfield, the component fermion field transforms according to,\n\\begin{align}\n\\delta_\\xi\\psi_{\\alpha i}=i\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\psi_{\\alpha i}\\bigr]=\n- i\\sqrt{2}\\, (\\sigma^\\mu \\xi^\\dagger)_\\alpha\\> \\partial_\\mu A_i+\\sqrt{2}\\,\\xi_\\alpha F_i\\,.\n\\end{align}\nBy Lorentz invariance, $\\vev{0\\,|\\partial_\\mu A_i\\,|\\,0}=0$. Hence,\n\\begin{align}\n\\vev{0\\,|\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\psi_{\\alpha i}\\bigr]\\,|\\,0}=\\sqrt{2}\\,\\xi_\\alpha\\vev{0\\,|F_i\\,|\\,0}\\,.\n\\end{align}\nThus, if $Q_\\alpha\\ket{0} = 0$ and $Q^\\dagger_{\\dot\\alpha}\\ket{0}=0$, then $\\vev{0\\,|F_i\\,|\\,0}=0$.\nLikewise, for a real vector superfield, the component gaugino field transforms according to,\n\\begin{align}\n\\delta_{\\xi}\\lambda^a_\\alpha=i\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\lambda^a_\\alpha\\bigr]=\n i\\xi_\\alpha D^a+\\ifmath{\\tfrac12}(\\sigma^\\mu\\overline{\\sigma}^\\nu)_\\alpha{}^\\beta\\xi_\\beta F^a_{\\mu\\nu}\\,.\n\\end{align}\n Since $\\vev{0\\,|F_{\\mu\\nu}^a|\\,0}=0$ (again, by Lorentz invariance), it follows that\n\\begin{align}\n \\vev{0\\,|\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\lambda^a_\\alpha\\bigr]\\,|\\,0}=i\\xi_\\alpha\\vev{0\\,|D^a|\\,0}\\,.\n\\end{align}\nThus, if $Q_\\alpha\\ket{0}=0$ and $Q^\\dagger_{\\dot\\alpha}\\ket{0}=0$, then $\\vev{0\\,|D^a\\,|\\,0}=0$.\n %\n \nIf at least one of the components of the auxiliary fields $F_i$ or\n$D_a$ has a nonzero vacuum expectation value, then SUSY is\nspontaneously broken. Mechanisms of spontaneous SUSY breaking fall into two\npossible categories:\n$F$-type breaking, if $\\vev{0\\,|F_i\\,|\\,0}\\neq 0$ for some $i$, and\n$D$-type breaking if $\\vev{0\\,|D^a|\\,0}\\neq 0$ for some $a$.\n\n\\subsubsection{The O'Raifeartaigh mechanism ($F$-type breaking)}\n \n \n One way to spontaneously break SUSY is to construct a model in which\n it is impossible to simultaneously solve the Lagrange field equations\nfor all the components of the auxiliary fields, $F_i$. \nThis is the O'Raifeartaigh mechanism\\cite{ORaifeartaigh:1975nky},\\footnote{A well-known\n supersymmetric joke: a graduate student returns to the University\n for the fall semester after spending a month at TASI earlier in the summer.\n The professor says to\n the student, ``Welcome back! I see that one of the lecture courses you attended\n at TASI was an introduction to supersymmetry. So, did you learn\n anything useful from\n these lectures?'' The student replies, ``I learned how to spell\n O'Raifeartaigh's name.''}\nwhere the SUSY breaking arises\nentirely from a nonzero $F$-term vacuum expectation\nvalue.\\footnote{Implicitly, we are assuming here that if the $D$-term\n is present, then $\\vev{D^a}=0$.}\n\n\nConsider the set of equations,\n\\begin{align}\nF_i^\\dagger=-\\frac{dW}{dA_i}=0\\,.\\label{fdag}\n\\end{align}\nA solution to these equations corresponds to the existence of \na choice of the scalar fields, $A_i$, such that all the\nequations, $F_i^\\dagger=0$, are fulfilled. Suppose that a solution,\n$A_i=v_i$, solves these equations. In light of \\eq{DDFF}, this\nsolution must correspond to a minimum of the scalar potential, which\nwe identify as the vacuum (ground) state of the theory. Since\n$F_i^\\dagger=0$ implies that $F_i=0$, we can conclude\nthat $\\vev{0\\,|F_i\\,|\\,0}=0$ (for all $i$).\nIf no solution to \\eq{fdag} exists, then it must be true that\n$\\vev{0\\,|F_i\\,|\\,0}\\neq 0$ for some $i$. \nIn this latter case, SUSY must be spontaneously broken.\n\nThe simplest O'Raifeartaigh model that exhibits $F$-term SUSY breaking\ncontains three chiral superfields and\nis treated in Problem~\\ref{pr:Oraif}.\n\n\n\\subsubsection{$D$-type breaking via the Fayet-Iliopoulos term}\n\nConsider SUSY-QED with a superpotential given by \\eq{Wsqed} and a Fayet-Iliopoulos term. Using\n\\eqs{fsubi}{DFI}, the resulting scalar potential [\\eq{vscalar2}] is given by\n\\begin{equation} \\label{FI1}\nV_{\\rm\n scalar}=|F_+|^2+|F_-|^2+\\ifmath{\\tfrac12} D^2\\,,\n\\end{equation}\nwhere\n\\begin{equation} \\label{FI2}\nF_\\pm=-mA_\\pm\\,,\\qquad\\quad D=-g\\bigl(|A_+|^2-|A_-|^2\\bigr)-\\xi.\n\\end{equation}\nSuppose that $m^2>g\\xi$. One can check that the minimum of the scalar\npotential occurs for $\\vev{A_+}=\\vev{A_-}=0$. Moreover, at the scalar\npotential minimum, $\\vev{F_+}=\\vev{F_-}=0$, whereas $\\vev{D}=-\\xi\\neq\n0$. Thus, in this model SUSY breaking arises\nentirely from a nonzero $D$-term vacuum expectation value.\nAdditional aspects of this model are treated in Problems~\\ref{pr:FI}\nand \\ref{pr:FI2}.\n\n\\subsubsection{The goldstino}\n\\label{goldstino}\n\nFrom Goldstone's theorem, we know that the spontaneous breaking of a\ncontinuous symmetry (with bosonic generators) gives rise to a massless\nboson called the Nambu-Goldstone boson. Analogously, the spontaneous breaking\nof supersymmetry, whose algebra contains fermionic generators, gives rise to a\nmassless fermion called the Goldstone fermion, which is more\ncommonly known as the goldstino\\cite{Salam:1974zb}.\n\n\\begin{theorem}\nIf SUSY is spontaneously broken, then there exists a massless spin-1\/2 fermion in the spectrum called the \\textit{goldstino}.\n\\end{theorem}\n\\begin{proof}[Proof]\nAlthough this theorem can be proven rigorously, independently of\nperturbation theory, it is instructive to exhibit a proof based on a\ntree-level analysis of a SUSY nonabelian gauge theory coupled to supermatter.\nThe scalar potential is given by \\eq{DDFF} where [cf.~\\eq{FandD}],\n\\begin{align}\nF_i=-\\left(\\frac{dW}{dA_i}\\right)^\\dagger\\,,\\qquad\\quad D^a=-gA_i^\\dagger T^a_{ij} A_j\\,.\\label{FiDa}\n\\end{align}\nAt the scalar potential minimum, where $\\partial V\/\\partial A_j=0$, the scalar fields are equal to their vacuum expectation values, $A_j=\\vev{A_j}$.\nThen,\n\\begin{align}\n0=\\left(\\frac{ \\partial V}{\\partial A_j}\\right)_{\\vev{A}}=-gA_i^\\dagger T^a_{ij} D^a\\biggl|_{\\vev{A}}-\\sum_i\n\\frac{ \\partial^2 W}{\\partial A_i\\partial A_j}F_i\\biggl|_{\\vev{A}}\\,.\n\\end{align}\nHence,\n\\begin{align}\n\\sum_i\n\\left\\langle\\frac{ \\partial^2 W}{\\partial A_i\\partial A_j}\\right\\rangle\\vev{F_i}=-g\\vev{A_i}^\\dagger T^a_{ij} \\vev{D^a}\\,.\n\\label{eq:FD}\n\\end{align}\n\nThe superpotential $W$ must be a gauge invariant function of the\nchiral superfields. That is,\n\\begin{equation}\nW(\\Phi)=W(e^{-2ig\\Lambda}\\Phi)\\,.\n\\end{equation}\nwhere $\\Lambda\\equiv \\Lambda^a T^a$ is the matrix chiral superfield gauge\ntransformation parameter.\nTaking $\\Lambda^a$ infinitesimal and expanding to first order yields\n\\begin{equation}\n\\frac{dW}{d\\Phi_i}T^a_{ij}\\Phi_j=0\\,.\n\\end{equation}\nEvaluating the hermitian conjugate of this expression, setting $\\theta=\\theta^\\dagger=0$, and\ntaking the vacuum expectation value of the resulting equation, we end up with\n\\begin{equation} \\label{FTa}\n\\vev{F_i}T^a_{ji}\\vev{A_j}^\\dagger=0\\,.\n\\end{equation}\n\nThe fermion masses can be determined from the SUSY Lagrangian given by\n\\eq{eq:LSUSYcomponents} after setting the scalar fields to their\nvacuum expectation values,\n\\begin{align}\n-\\mathscr{L}_{\\rm mass} =&\\ifmath{\\tfrac12} \\left\\langle\\frac{ \\partial^2 W}{\\partial A_i\\partial A_j}\\right\\rangle \\psi_i\\psi_j\n-i\\sqrt{2}\\,g\\vev{A_i}^\\dagger T^a_{ij} \\psi_j\\lambda^a+{\\rm h.c.}\n\\\\\n=&\\ifmath{\\tfrac12}\\bigl(\\psi_i\\quad -i\\lambda^b\\bigr)\n\\begin{pmatrix} \\left\\langle\\displaystyle\\frac{ \\partial^2 W}{\\partial A_i\\partial A_j}\\right\\rangle & \\quad\n\\sqrt{2}\\,g\\vev{A_j}^\\dagger T^a_{ji} \\\\[25pt] \\sqrt{2}\\,g\\vev{A_i}^\\dagger T^b_{ij} & \\quad 0\\end{pmatrix}\n\\begin{pmatrix} \\psi_j \\\\[25pt] -i\\lambda^b\\end{pmatrix}\\,.\\label{fmatrix}\n\\end{align}\n\nUsing \\eqs{eq:FD}{FTa}, one can verify that the fermion mass matrix\ngiven in \\eq{fmatrix} possesses a zero eigenvalue,\n\\begin{align}\n\\begin{pmatrix} \\left\\langle\\displaystyle\\frac{ \\partial^2 W}{\\partial A_i\\partial A_j}\\right\\rangle & \\quad\n\\sqrt{2}\\,g\\vev{A_j}^\\dagger T^a_{ji} \\\\[25pt] \\sqrt{2}\\,g\\vev{A_i}^\\dagger T^b_{ij} & \\quad 0\\end{pmatrix}\n\\begin{pmatrix} \\vev{F_j} \\\\[25pt] \\frac{1}{\\sqrt{2}}\\vev{D^a}\\end{pmatrix}=0\\,,\\label{eigen}\n\\end{align}\nunder the assumption that at least one of the auxiliary field vacuum\nexpectation values is nonzero.\nThe corresponding eigenvector, $\\of{ \\vev{F_j},\n \\tfrac{1}{\\sqrt{2}}\\vev{D^a} }$, can be identified with the massless\ngoldstino, $\\widetilde{G}$. That is,\n\\begin{align}\n\\widetilde G=\\vev{F_j}\\psi_j-\\frac{i}{\\sqrt{2}}\\vev{D^a}\\lambda^a\\,.\n\\end{align}\n\n\nThe existence of the goldstino in the fermion mass spectrum is a\nconsequence of the assumption that the vacuum is not invariant under\nSUSY transformations, in which case at least one of the auxiliary field vacuum\nexpectation values is nonzero, as assumed below \\eq{eigen}.\nIn contrast, if the vacuum is supersymmetric, then $\\vev{F_j}=\\vev{D^a}=0$, in\nwhich case \\eqs{eq:FD}{FTa} are trivially\nsatisfied. Hence in this case, one cannot conclude that a zero\neigenvalue of the fermion mass matrix exists. \n\\end{proof}\n\n \n\\subsection{Mass Sum rules}\n\\label{sumrule}\n \nIf SUSY is broken, then there is no expectation that particles in a\nwould-be supermultiplet are degenerate in mass. If the SUSY breaking\nis spontaneous, then there is still some memory of supersymmetry\nin the properties of the SUSY-broken theory. In particular, the mass spectrum\nof the spontaneously broken SUSY theory satisfies certain sum rules that\nreflect the fact the spontaneous breaking of the supersymmetry is inherently soft\\cite{Ferrara:1979wa}.\n\nTo exhibit such sum rules, we return to the Lagrangian of the SUSY\nnonabelian gauge theory coupled to supermatter\ngiven in \\eq{eq:LSUSYcomponents}. We set the scalar fields and the\nauxiliary fields to their vacuum expectation values and compute the\nresulting tree-level mass spectrum.\n\nThe spin-1 masses arise from\n\\begin{align}\n\\mathscr{L}_{\\rm mass}= (\\mathcal{D}_\\mu A)(\\mathcal{D}^\\mu A)^\\dagger,\n\\end{align}\n where \n$\\mathcal{D}_\\mu=\\partial_\\mu+igT^a V^a_\\mu$. It is convenient to write the gauge boson squared-mass matrix as follows,\n\\begin{align}\n(M^2_1)_{ab}=2g^2\\vev{A^\\dagger_i}T^a_{ij}T^b_{jk}\\vev{A_k}=\n2\\left\\langle \\frac{\\partial D^a}{\\partial A_k^\\dagger}\\frac{\\partial D^b}{\\partial A_k}\\right\\rangle\\,,\n\\end{align}\nwhere we have made use of $D^a=-gA_i^\\dagger T^a_{ij}A_j$ [cf.~\\eq{FandD}].\nLikewise, we can rewrite the spin-1\/2 mass matrix [previously obtained\nin \\eq{fmatrix}] as,\n\\begin{align}\n M_{\\scalebox{.8}{$\\tfrac12$}}=\\begin{pmatrix} \\left\\langle -\\displaystyle\\frac{ \\partial F_i^\\dagger}{\\partial A_j}\\right\\rangle & \\quad\n-\\sqrt{2}\\, \\displaystyle\\left\\langle\\frac{ \\partial D^a}{\\partial A_i}\\right\\rangle\\\\[25pt]\n-\\sqrt{2}\\, \\displaystyle\\left\\langle\\frac{ \\partial D^b}{\\partial A_j}\\right\\rangle& \\quad 0\\end{pmatrix}\\,.\n\\end{align}\n\nThe spin-0 masses arise from the scalar potential, $V\\equiv V_{\\rm scalar}$. Identifying the\nterms quadratic in the scalar field,\n\\begin{equation}\n-\\mathscr{L}_{\\rm mass}=\\frac12\\bigl(A_i\\quad A_j^\\dagger\\bigr)\\begin{pmatrix}\n \\displaystyle\\left\\langle{\\frac{\\partial^2 V}{\\partial A_i\\partial A_k^\\dagger}}\\right\\rangle\\qquad \n \\displaystyle\\left\\langle{\\frac{\\partial^2 V}{\\partial A_i\\partial A_\\ell}}\\right\\rangle \\\\[15pt]\n \\displaystyle\\left\\langle{\\frac{\\partial^2 V}{\\partial A_j^\\dagger\\partial A_k^\\dagger}}\\right\\rangle\\qquad \n \\displaystyle\\left\\langle{\\frac{\\partial^2 V}{\\partial A_j^\\dagger\\partial\n A_\\ell}}\\right\\rangle\\end{pmatrix}\\begin{pmatrix} A_k^\\dagger \\\\[25pt]\n A_\\ell \\end{pmatrix}\\,.\n\\end{equation}\nThe scalar squared-mass matrix given above will be denoted by $M_0^2$.\n \nThe elements of the scalar squared-mass matrix can be\nrewritten in terms of derivatives of the auxiliary fields $F_i$ and\n$D^a$. For example, noting that \n\\eq{FiDa} implies that $F$\nis a function of $A^\\dagger$ (and likewise, $F^\\dagger$ is a\nfunction of $A$), then it follows from \\eq{DDFF} that \n\\begin{Eqnarray}\n\\frac{\\partial^2 V}{\\partial A_i\\partial\n A_k^\\dagger}\n&=& \\frac{\\partial F^\\dagger_m}{\\partial A_i}\\frac{\\partial\n F_m}{\\partial A^\\dagger_k} +\\frac{\\partial D^a}{\\partial\n A_k^\\dagger}\\frac{\\partial D^a}{\\partial\n A_i}+D^a\\,\\frac{\\partial^2 D^a}{\\partial A_k^\\dagger\\partial A_i}\\,.\n\\end{Eqnarray}\n\nOne can now evaluate the trace of the various squared-mass\nmatrices,\n\\begin{Eqnarray}\n\\Tr M_1^2&=&2\\left\\langle \\frac{\\partial D^a}{\\partial\n A_k^\\dagger}\\frac{\\partial D^a}{\\partial A_k}\\right\\rangle\\,, \\\\\n\\Tr M_{\\scalebox{.8}{$\\tfrac12$}}^\\dagger M_{\\scalebox{.8}{$\\tfrac12$}}^{\\phantom{\\dagger}}&=&\\left\\langle \\frac{\\partial F_i}{\\partial A_k^\\dagger}\\frac{\\partial\n F_i^\\dagger}{\\partial A^\\dagger_k}\\right\\rangle +4\\left\\langle \\frac{\\partial D^a}{\\partial\n A_k^\\dagger}\\frac{\\partial D^a}{\\partial A_k}\\right\\rangle\\,, \\\\\n\\Tr M_0^2&=& 2 \\left\\langle \\frac{\\partial F^\\dagger_i}{\\partial A_k}\\frac{\\partial\n F_i}{\\partial A^\\dagger_k}\\right\\rangle +2\\left\\langle \\frac{\\partial D^a}{\\partial\n A_k^\\dagger}\\frac{\\partial D^a}{\\partial\n A_k}\\right\\rangle+2\\left\\langle D^a\\frac{\\partial^2 D^a}\n {\\partial A^\\dagger_k \\partial A_k}\\right\\rangle, \\nonumber \\\\\n\\phantom{line} \\label{trmzero}\n\\end{Eqnarray}\nwhere there are implicit sums over each pair of repeated indices.\nWe can simplify the last term of \\eq{trmzero} using $D^a=-gA_i^\\dagger\nT^a_{ij}A_j$ to obtain.\n\\begin{align}\n\\Tr M_0^2=2\\left\\langle \\frac{\\partial F_i}{\\partial A_k^\\dagger}\\frac{\\partial F_i^\\dagger}{\\partial A_k}\\right\\rangle\\\n+2\\left\\langle \\frac{\\partial D^a}{\\partial A_k^\\dagger}\\frac{\\partial D^a}{\\partial A_k}\\right\\rangle-2g\\vev{D^a}\\Tr T^a\\,.\n\\end{align}\nIt then follows that\n\\begin{equation} \\label{masssumrule}\n\\Tr(M_0^2-2M_{\\scalebox{.8}{$\\tfrac12$}}+3M_1^2)=-2g\\vev{D^a}\\Tr\nT^a\\,.\n\\end{equation}\n\nWe recognize the left-hand side of \\eq{masssumrule} as a\nsupertrace, which is defined as the following weighted sum of traces,\n\\begin{Eqnarray}\n&&\\phantom{line}\\nonumber \\\\[-10pt]\n&&\\Str M^2\\equiv \\sum_J (-1)^{2J} (2J+1) \\Tr M_J^2\\,,\\label{stracedef}\n\\end{Eqnarray}\nwhere \n$M_J^2$ is the squared-mass matrix of \\textit{real} spin-$J$\nfields.\\footnote{Note that complex fields are equivalent to two\n mass-degenerate real fields.}\nNote the $(-1)^{2J}$\nfactor, so that bosons contribute positively and fermions negatively\nto the sum over $J$. \nAs applied to a SUSY nonabelian gauge theory coupled to supermatter, the sum is taken over $J=0$, $\\ifmath{\\tfrac12}$ and 1. \nHence, \\eq{masssumrule} assumes the following simple form,\n\\begin{equation} \\label{eq:supersumrule}\n{\\rm Str}~M^2=-2g\\vev{D^a}\\Tr T^a\\,.\n\\end{equation}\n\nThe mass sum rule can \nprovide a useful check on the phenomenological viability of theories\nwith tree-level spontaneous supersymmetry breaking. \nLet us now see how this applies in several cases.\n\n\\subsection{The origin of SUSY-breaking dynamics}\n\\label{SUSYdynamics}\n\\subsubsection{Models of tree-level spontaneous SUSY breaking}\nIn the case of $F$-type breaking (\\textit{i.e.}, the O'Raifeartaigh model), in which $\\vev{F_i}\\neq 0$ and $\\vev{D^a}=0$,\n\\eq{eq:supersumrule} yields \n\\begin{equation} \\label{strzero}\n{\\rm Str}~M^2=0\\,.\n\\end{equation}\nFor example, consider the matter sector of SUSY-QED, which contains two chiral\nsupermultiplets [cf.~\\eq{Wsqed}]. The corresponding spectrum contains a\nfour-component Dirac electron and its two complex scalar superpartners, the\nselectrons (denoted by $\\widetilde e_1$ and $\\widetilde e_2$). If SUSY\nis spontaneously broken by an $F$-term vacuum expectation value, then \\eq{strzero}\nyields\n\\begin{align}\nm_{\\tilde{e}_1}^2 + m_{\\tilde{e}_2}^2 = 2 m_e^2 ,\n\\end{align}\nso that one selectron would be heavier than the electron and the other\nselectron would be lighter than the electron. Clearly, this is very\nbad for phenomenology, since experiment demands that all superpartner\nmasses must be significantly heavier than their SM counterparts.\n\nConsider next $D$-type breaking with $\\vev{F_i}=0$ and $\\vev{D^a}\\neq 0$ in a nonabelian gauge theory.\nIn this case, $\\Tr T^a=0$ and we again conclude that $\\Str M^2=0$.\nHowever, it turns out that when the scalar potential is minimized, it\nis always possible to find a vacuum in which $\\vev{D^a}=0$. Hence,\n$D$-term SUSY-breaking is not possible in this case (see Problem~\\ref{pr:holo}).\n\nFinally, consider $D$-type breaking in a gauge theory with a U(1)\nfactor. The Standard Model provides an example of this case. But in the\nStandard Model, the hypercharge generator satisfies $\\Tr Y=0$ when\nsummed over one generation of matter. Hence we again find that $\\Str\nM^2=0$. It is possible to construct models of $D$-type SUSY breaking\nvia the Fayet-Iliopoulos term~$\\xi$. In such models, $\\vev{D}$ is\nproportional to $\\xi$, as shown below \\eq{FI2}. However, no realistic models of this type are known.\n\nBased on the above considerations, we conclude that the mass sum rule\nseverely constrains tree-level SUSY-breaking models. Indeed, no\nphenomenologically realistic tree-level spontaneously broken SUSY model has ever been\nsuccessfully constructed. \n\n\\subsubsection{Gauge-mediated SUSY breaking}\nOne way to avoid the tyranny of the mass\nsum rule is to consider models in which the radiative corrections to\nthe tree-level masses are significant. In general, there is no reason\nwhy the radiative corrections should respect the tree-level relations\nderived in Section~\\ref{sumrule}. For example, one can construct models with two distinct\nsectors of supermatter, which are coupled by the exchange of gauge\nbosons. The particles of the Standard Model (SM) reside in one of the\nsupermatter sectors, whereas the source of SUSY-breaking (SSB) is located in\nthe second supermatter sector, whose characteristic mass scale, $M_{\\rm\nSSB},$ is\nassumed to be significantly above 1~TeV. Indeed, in this second supermatter\nsector, the masses of particles and their superpartners are split due\nto SUSY-breaking, while respecting the tree-level mass sum rule\nobtained in \\eq{eq:supersumrule}. In this case, tree-level SUSY-breaking\nis phenomenologically viable in light of the large\ncharacteristic mass scale $M_{\\rm SSB}$ that governs the SSB sector.\n\nIn such a setup, SUSY is unbroken in\nthe SM sector at tree level, in which case $\\Str M^2=0$ is trivially\nsatisfied (see Problem~\\ref{pr:exact}). However, there exist\nradiative corrections to the sum rule induced by loops involving the\nsupermatter of the SSB sector. These corrections are\nresponsible for SUSY-breaking in the SM sector\nand the corresponding\nmass splitting between the SM particles and their superpartners.\nMoreover, these mass splittings are totally radiative in nature and not\nsubject to the tree-level sum rule of \\eq{eq:supersumrule}. Models\ncan easily be constructed in which the masses of the SM superpartners\nare all raised above 1 TeV, thereby avoiding conflict with the current LHC searches. \n The end result is SUSY-breaking in the SM that is\nphenomenologically viable.\n\nIn the scenario outlined above, SUSY-breaking is communicated to the\nSM-sector via a messenger mechanism, in which the messengers consists\nof gauge bosons that couple both to the SM sector and the\nSSB sector. Models of this type provide examples of\ngauge-mediated SUSY breaking (GMSB).\nDetails of GMSB model building lie beyond the scope of these lectures.\nFor further details, you may consult Refs.~\\cite{Giudice:1998bp,Luty:2005sn,Shirman:2009mt}. \n\n\n \n\\subsubsection{Local supersymmetry and the super-Higgs mechanism}\n\nAnother way of evading the tyranny of the mass sum rule is to\nconsider models with \\textit{local} supersymmetry.\n\nIn these lectures, we have focused on theories with global\nsupersymmetry, where the anticommuting SUSY translation parameter\n$\\xi$ is independent of the position $x$. Suppose we attempt to\ngeneralize this to local supersymmetry, where $\\xi=\\xi(x)$. Since\nthe spinorial SUSY generators satisfy\n$\\{Q_\\alpha\\,,\\,\\overline{Q}_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu$, \na theory of local supersymmetry must also be invariant under local\nspacetime translations, in which the translation depends on the\nposition. A theory that possesses a local spacetime translation\nsymmetry is a theory of gravity! Hence, a locally supersymmetric\ntheory is a theory of gravity plus supersymmetry, \\textit{i.e.}, supergravity\\cite{sugra1,sugra2}.\n \nWe have already encountered the massless supermultiplet that contains\nthe spin-3\/2 gravitino and the spin-2 graviton. \nSuppose we couple this supermultiplet to ordinary supermatter. In\naddition, suppose that the local supersymmetry is broken, which will\ngenerate a mass splitting within the graviton supermultiplet. \nWe require that the graviton remain massless, while the gravitino acquires\nmass. This can be accomplished via the super Higgs mechanism\\cite{Deser:1977uq,Cremmer:1978iv}.\n\nWe have seen in Section~\\ref{goldstino} that in models of\nspontaneously-broken global supersymmetry, the spectrum includes a\nmassless goldstino.\nIn models of spontaneously-broken supergravity, the goldstino is ``absorbed''\nby the gravitino via the super-Higgs mechanism. \nInitially, a massless gravitino possesses only two helicity states,\n$\\lambda=\\pm\\tfrac32$. In the super-Higgs mechanism, the goldstino\nprovides $\\lambda=\\pm\\ifmath{\\tfrac12}$ helicity states for a massive gravitino. \nThat is, the goldstino is removed from the\nphysical spectrum and the gravitino acquires a mass\n(denoted by $m_{3\/2}$). The gravitino now possesses the four\nhelicity states, $\\lambda=\\pm\\tfrac32$, $\\pm\\ifmath{\\tfrac12}$, as expected\nfor a massive spin-$\\tfrac32$ particle.\n\n In spontaneously broken supergravity, the tree-level mass sum rule\n obtained in \\eq{eq:supersumrule} is modified. For example, if $N$ chiral supermultiplets are minimally coupled to supergravity, then\\cite{Cremmer:1982en},\n \\begin{align}\n \\Str M^2= (N-1)(2m_{3\/2}^2-\\kappa \\vev{D^a D^a})-2g\\vev{D^a} T^a\\,,\n\\end{align}\n where\n $\\kappa=(8\\pi G_N)^{1\/2}=(8\\pi)^{1\/2}M_{\\rm PL}^{-1}$. Typical models of interest have $\\vev{D^a}=0$, in which case\\cite{Cremmer:1982wb} ,\n \\begin{align}\n \\Str M^2= 2(N-1)m_{3\/2}^2\\,.\n \\end{align}\n If $m_{3\/2}{~\\raise.15em\\hbox{$>$}\\kern-.85em\\lower.35em\\hbox{$\\sim$}~} \\mathcal{O}(1~{\\rm TeV})$, then one expects the superpartner masses of SM particles to lie in the TeV regime.\n \n\\subsubsection{Gravity-mediated SUSY-breaking}\n\nConsider again the framework of two distinct sectors of supermatter that\nare initially uncoupled. We identify one of the sectors as the\nSM sector where the SM particles and their superpartners reside.\nIn the second so-called ``hidden'' sector, SUSY is spontaneously\nbroken. \n \nSupergravity models provide a natural mechanism for\ntransmitting the SUSY breaking of the hidden sector to the\nparticle spectrum of the SM sector. In models of gravity-mediated\nSUSY breaking, gravity is the messenger of\nsupersymmetry breaking\\cite{Nilles:1983ge ,Hall:1983iz}.\nMore precisely, SUSY breaking in the SM sector is mediated by effects of\ngravitational strength (suppressed by inverse powers of the Planck mass).\nThe induced mass splittings between the SM particles and their superpartners\nare of $\\mathcal{O}(m_{3\/2})$, whereas the gravitino couplings are\nroughly gravitational in strength.\n\nUnder certain theoretical assumptions\non the structure of the K\\\"ahler potential (the so-called sequestered form\nintroduced in Ref.\\cite{Randall:1998uk}), SUSY breaking is due\nentirely to the super-conformal (super-Weyl) anomaly,\nwhich is common to all supergravity models.\nThis approach is called anomaly-mediated supersymmetry breaking (AMSB).\nIndeed, anomaly mediation is more generic than originally conceived,\nand provides a ubiquitous source of supersymmetry breaking\\cite{DEramo:2012vvz,Harigaya:2014sfa}.\n\n\n \\subsection{A phenomenological approach: soft SUSY-breaking}\n \\label{sec:softSUSYbreaking}\n \n If SUSY-breaking arises due to gauge-mediated SUSY-breaking or\n gravity-mediated SUSY-breaking, then we can formally integrate out\n the SSB sector physics at the mass scale $M_{\\rm SSB}$ that\n characterizes the fundamental SUSY-breaking dynamics. For example, in the case\n of gravity-mediated SUSY breaking, we identify $M_{\\rm SSB}=M_{\\rm PL}$. In\n GMSB models, $M_{\\rm SSB}$ can be much smaller than $M_{\\rm PL}$ but still\n significantly larger than the scale of electroweak symmetry breaking.\n\n The end result is an effective broken supersymmetric theory whose Lagrangian consists \n of supersymmetric terms and explicit SUSY-breaking terms.\nThe explicit SUSY-breaking terms that are present in the effective low-energy\ntheory (which is valid at energy scales below $M_{\\rm SSB}$) are ``soft.''\n The meaning of soft in this context will be explained shortly.\n \n The phenomenological approach to SUSY-breaking takes the point of\n view that the fundamental dynamics of SUSY-breaking is unknown.\n Therefore, we should simply parameterize SUSY breaking in the\n low-energy effective theory\n by including all possible soft-SUSY-breaking terms. The coefficients\n of these terms will be taken to be arbitrary (to be determined by\n experiment). Ultimately, these parameters will provide clues to the\n structure of the fundamental dynamics that is responsible for SUSY-breaking.\n \n \\subsubsection{A catalog of soft-SUSY-breaking terms}\n\\label{GGrules}\nThe most general set of soft-SUSY-breaking terms in a super-Yang Mills theory coupled to supermatter\nwas first elucidated by Girardello and Grisaru in Ref.\\cite{Girardello:1981wz},\n\\begin{align}\n-\\mathscr{L}_{\\rm soft}=m_{ij}^2 A_i^\\dagger A_j+\\ifmath{\\tfrac12}\\bigl[m_{ab}\\lambda^a\\lambda^b+{\\rm h.c.}\\bigr]+\\bigl[w(A)+{\\rm h.c.}\\bigr]\\,,\\label{GGsoft}\n\\end{align}\nwhere there is an implicit sum over repeated indices. The scalar squared-mass matrix $m_{ij}^2$ is hermitian and the gaugino mass matrix $m_{ab}$ is complex symmetric. The function\n$w(A)$ is a holomorphic cubic multinomial of the scalar fields,\n\\begin{align}\nw(A)=c_i A_i+b_{ij}A_i A_j+a_{ijk}A_i A_j A_k\\,.\n\\end{align}\nNote that $c_i=0$ in the absence of any gauge singlet fields. In the\nliterature, the $b_{ij}$ are called the $B$-terms and the $a_{ijk}$\nare called the $A$-terms. Note the corresponding mass dimensions, $[b_{ij}]=2$ and $[a_{ijk}]=1$.\n\nDimension-4 terms are not included in \\eq{GGsoft}, since non-supersymmetric\ndimension-4 terms would constitute a hard breaking of supersymmetry\\cite{Martin:1999hc}.\nOne interesting feature of \\eq{GGsoft} is the absence of\nnon-supersymmetric fermion mass terms, $m_{ij}\\psi_i\\psi_j+{\\rm\n h.c.}$, and non-holomorphic cubic terms in the scalar fields (e.g.,\n$A_i A_j A_k^\\dagger$, etc.). Although such terms are technically soft \nin models with no gauge\nsinglets\\cite{Hall:1990ac,Jack:1999ud,Un:2014afa,Chattopadhyay:2016ivr,Ross:2016pml},\ntheses terms rarely arise in\nactual models of fundamental SUSY-breaking, or if present are highly\nsuppressed\\cite{Martin:1999hc}. Henceforth, we shall\nneglect them.\n\nIn general, there is no relation between $w(A)$ and the\nsuperpotential, which under the assumption of renormalizability has\nthe following generic form, \n\\begin{align}\nW(\\Phi)=\\kappa_i\\Phi_i+\\mu_{ij}\\Phi_i\\Phi_j+\\lambda_{ijk}\\Phi_i\\Phi_j\\Phi_k\\,.\n\\end{align}\nBut, some models of fundamental SUSY breaking yield the relations,\n\\begin{align}\nc_i=C\\kappa_i\\,, \\qquad\\quad b_{ij}=B\\mu_{ij}\\,,\\qquad\\quad a_{ijk}=A\\lambda_{ijk}\\,,\n\\end{align}\nwhich relate the coefficients of $w(A)$ to the coefficients of $W(\\Phi)$.\n\n \\subsubsection{Soft vs.~hard SUSY breaking and the reappearance of quadratic divergences}\n \n Consider the one-loop effective potential for a gauge theory coupled to matter,\n\\begin{align}\n V_{\\rm eff}(A)=V_{\\rm scalar}(A)+V^{(1)}(A)\\,.\n \\end{align}\n If we regulate the divergence of the one-loop correction by a\n momentum cutoff $\\Lambda$, then\\cite{HaberTASI}\n\\begin{align}\n V^{(1)}(A)=\\frac{\\Lambda^2}{32\\pi^2}\\Str\n M_i^2(A)+\\frac{1}{64\\pi^2}\\Str\\left\\{M_i^4(A)\\left[\\ln\\left(\\frac{M_i^2(A)}{\\Lambda^2}\\right)-\\frac12\\right]\\right\\}\\,,\\label{effpot} \n \\end{align}\n where $M_i^2(A)$ are the relevant squared-mass matrices for spin 0,\n $\\ifmath{\\tfrac12}$ and 1, in which the scalar vacuum expectation values are\n replaced by the corresponding scalar fields, $A$.\n\n\\Eq{effpot} implies that both in supersymmetric theories and in the case\nof spontaneously broken SUSY (assuming\n in the latter that all U(1) generators are traceless), we have\n $\\Str M^2=0$, in which case the quadratic divergences [i.e., the\n terms proportional to $\\Lambda^2$ in \\eq{effpot}] cancel exactly!\n In Ref.\\cite{Girardello:1981wz}, Girardello and Grisaru showed that if\n explicit SUSY breaking terms are present, then there \nis a catalog of possible explicit SUSY-breaking terms for which\n$\\Str M_i^2(A)$ is a constant \\textit{independent} of the scalar\nfields, $A$. Such terms shift the vacuum energy, but in the context\nof quantum field theory they have no observable effect. Terms with\nsuch properties are deemed ``soft,'' and are given in\n\\eq{GGsoft}.\\footnote{Non-holomorphic cubic terms and mass terms of\n fermions that reside in a chiral supermultiplet can generate\n quadratically divergent terms in $V^{(1)}$ that are linear in the\n scalar fields, $A$. However, if no gauge singlet fields exist in the\n model, then terms that are linear in $A$ are absent due to gauge invariance.} \nIn contrast, hard SUSY-breaking terms will generate quadratically\ndivergent terms in $V^{(1)}$ that are scalar-field-dependent.\nThis is a signal that some of the parameters of the low-energy effective theory\nare quadratically sensitive to UV physics.\n\n\\subsubsection{Soft SUSY-breaking: an effective theory perspective}\n \nConsider a set of light chiral superfields $\\Phi$ and a set of heavy\nchiral superfields $\\Omega$ associated with a mass scale $M\\equiv\nM_{\\rm SSB}$. Furthermore, assume that SUSY-breaking is generated by\nan $F$-term that resides in the SSB sector,\n\\begin{align}\n \\vev{F_\\Omega}=f\\neq 0\\,.\n \\end{align}\nOne can integrate out the physics of the SSB sector, as shown in the\nfollowing examples\\cite{Girardello:1981wz,Pomarol:1995np,Rattazzi:1995tc}.\n\n\\begin{example}\nConsider a holomorphic cubic multinomial of chiral superfields $\\Phi$,\nwhich we denote by $\\widetilde{w}(\\Phi)$.\nA possible term in the effective Lagrangian is\n\\begin{align}\n \\frac{1}{M}\\int d^2\\theta\\, \\Omega\\, \\widetilde{w}(\\Phi)\\,,\\label{Ow}\n \\end{align}\n since $\\Omega\\, \\widetilde{w}(\\Phi)$ is a term in the\n superpotential. \nThe factor of $M^{-1}$ appears on the basis of dimensional analysis.\nIn particular, note the mass dimensions, $[\\widetilde{w}]=3$, $[\\Omega]$=1 and $[\\int d^2\\theta]=1$. \n \n Since the vacuum expectation value of ${F_\\Omega}$, denoted by\n$\\vev{F_\\Omega}=f$, is nonzero, it follows that $\\vev{\\Omega}\\ni \\theta\\theta f$. Inserting this into \\eq{Ow} yields,\n \\begin{align}\n \\frac{1}{M}\\int d^2\\theta\\, \\theta\\theta f\\, \\widetilde{w}(\\Phi)=\\frac{f}{M}\\,\\widetilde{w}(A)\\,,\n \\end{align}\n which produces the term, $w(A)=(f\/M)\\widetilde{w}(A)$, in our\n catalog of $\\delta\\mathscr{L}_{\\rm soft}$ given in \\eq{GGsoft}.\n \nIn order to achieve soft-SUSY-breaking masses in the low-energy\neffective theory of order 1~TeV, one must require that\n$f\/M\\sim\\mathcal{O}(1~{\\rm TeV})$. For example, in gravity-mediated SUSY breaking, $M\\sim M_{\\rm PL}$, in which case $f\\sim (10^{11}~{\\rm GeV})^2$. Note that $f^{1\/2}$ identifies the energy scale of the fundamental SUSY breaking.\n \\end{example}\n\\begin{example}\nAnother possible term in the effective Lagrangian is\n\\begin{align}\n \\frac{1}{M^2}\\int d^4\\theta\\,\\Phi_i^\\dagger \\left(e^{2gV}\\right)_{ij}\\Phi_j\\,\\Omega^\\dagger \\Omega\\,,\n \\end{align}\nwhich would contribute to the K\\\"ahler potential. Setting $\\vev{\\Omega}=\\theta\\theta f$ and evaluating the result in the Wess-Zumino gauge,\n \\begin{align}\n \\frac{f^2}{M^2}\\int d^4\\theta\\,(\\theta\\theta)(\\theta^\\dagger\\thetabar)\\Phi_i^\\dagger \\left(e^{2gV}\\right)_{ij}\\Phi_j=\\frac{f^2}{M^2}A^\\dagger A\\,.\n \\end{align}\nThus, the low-energy effective theory contains a scalar squared-mass\nterm of order $f\/M$, which we again recognize as one of the\nsoft-SUSY-breaking terms of \\eq{GGsoft}.\n\n \\end{example}\n %\n \\begin{example}\n Finally, one additional possible term in the effective Lagrangian is\n \\begin{align}\n \\frac{1}{M}\\int d^2\\theta\\,\\Omega\\Tr(W^\\alpha W_\\alpha)\\,,\n \\end{align}\n which would contribute to the gauge kinetic function. Setting $\\vev{\\Omega}=\\theta\\theta f$,\n \\begin{align}\n \\frac{f}{M}\\int d^2\\theta\\,\\theta\\theta \\Tr(W^\\alpha W_\\alpha)=-\\frac{f}{M}\\Tr(\\lambda^\\alpha\\lambda_\\alpha)\\,,\n \\end{align}\n which yields a gaugino mass term of order $f\/M$.\n \\end{example}\n\nWe have thus demonstrated how the possible soft-SUSY-breaking terms of\n\\eq{GGsoft} can arise in the low-energy effective theory after\nintegrating out the physics associated with the SSB sector.\n \n \n \n\n\n \n \\subsection{Problems}\n \n\\begin{problem}\n\\label{pr:Oraif}\nAn O'Raifeartaigh model that exhibits $F$-term SUSY breaking\nmust involve at least three chiral superfields\\cite{ORaifeartaigh:1975nky}. \nOne of the simplest\nmodels of this type has the following superpotential,\n\\begin{equation}\nW(\\Phi_1,\\Phi_2,\\Phi_3)=\\lambda\\Phi_1(\\Phi_3^2-m^2)+\\mu\\Phi_2\\Phi_3\\,,\n\\end{equation}\nwhere $\\lambda$ is dimensionless and $\\mu$ and $m$ are mass parameters.\nEvaluate the corresponding $F$-terms, $F_1$, $F_2$ and $F_3$ and write\nout the scalar potential, $V_{\\rm scalar}$. Show that no solution for\nthe scalar fields $A_1$ $A_2$ and $A_3$ exist such that\n$F_1=F_2=F_3=0$. Conclude that SUSY is spontaneously broken. \n\\end{problem}\n\n\\begin{problem}\nFind the minimum of $V_{\\rm scalar}$ obtained in Problem~\\ref{pr:Oraif}, and verify that $\\langle\n0|V_{\\rm scalar}|0\\rangle > 0$. Identify the goldstino of this model.\nFinally, compute the mass spectrum of the fermions and bosons and\nverify that the mass sum rule, \\eq{strzero}, is satisfied.\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:FI}\nShow that in the case of SUSY-QED with a Fayet-Iliopoulos term and\n$m^2>g\\xi$ [cf.~\\eqs{FI1}{FI2}], SUSY is broken and the goldstino can be identified as the\nphotino (the supersymmetric partner of the photon). \nIn the case of $m^2g\\xi$ and\n$m^20$. In this frame,\nit is easy to show that $w=w^0(1;0,0,1)$. That is, in any Lorentz frame,\n\\begin{equation} \\label{helicitydef}\nw^\\mu=h P^\\mu\\,,\n\\end{equation}\nwhere $h$ is called the helicity operator. In particular,\n\\begin{equation}\n[h\\,,\\,P^\\mu]=[h\\,,\\,J^{\\mu\\nu}]=0\\,,\n\\end{equation}\nwhich means that the eigenvalues of $h$ can be used to label states of\nthe irreducible massless representations of the Poincar\\'e algebra.\nFrom \\eq{helicitydef}, we\nderive\\footnote{We define the differential operator\n$L^i\\equiv\\ifmath{\\tfrac12}\\epsilon^{ijk}L_{jk}$. Then, noting that\n$\\mathbold{\\vec L}=\\mathbold{\\vec x\\times\\vec P}$, it follows that $\\mathbold{\\vec{L}\\cdot\\vec{P}}=0$.\nHence, $\\mathbold{\\vec{J}\\cdot\\vec{P}}=(\\mathbold{\\vec{L}}+\\mathbold{\\vec{S}})\\cdot\\mathbold{\\vec{P}}=\\mathbold{\\vec{S}\\cdot\\vec{P}}$.}\n\\begin{equation} \\label{hdefinition}\nh=\\frac{w^0}{P^0}=\\frac{\\mathbold{\\vec{J}\\cdot\\vec{P}}}{P^0}\n=\\frac{\\mathbold{\\vec{S}\\cdot\\vec{P}}}{|\\boldsymbol{\\vec{P}}|}=\\boldsymbol{\\vec{S}\\!\\cdot\\!\\hat{P}}\\,,\n\\end{equation}\nafter noting that $P^0=|\\boldsymbol{\\vec{P}}|$ for massless states.\nEigenvalues of $h$ are called the helicity (and are denoted by\n$\\lambda$);\nits spectrum consists of non-negative half-integers,\n$\\lambda=0,\\pm\\ifmath{\\tfrac12},\\pm 1,\\ldots$.\nUnder a CPT transformation, $\\lambda\\to -\\lambda$. \nThus, in any quantum field theory realization of massless particles, \nboth~~$\\pm|\\lambda|$ helicity states must appear in the theory.\nIt is common to refer to a massless (positive energy) state\nof helicity $\\lambda$ as having spin $|\\lambda|$.\n\n\\subsection{The supersymmetry (SUSY) algebra}\n\\label{SUSYalg}\nIn the 1960s, Coleman and Mandula proved\na very powerful no-go theorem\nthat showed that in quantum field theories in $3+1$ dimensional\nspacetime with a mass gap, the only possible symmetry incorporating Poincar\\'e\ntransformations and a global internal symmetry group of transformations \nmust be a trivial tensor product of the two groups\\cite{Coleman:1967ad}. \nSubsequently, Haag, {\\L}opusza{\\'{n}}ski and Sohnius proved that the only\npossible extension of the Poincar\\'e algebra involves the addition\nof new fermionic generators that transform either as a $(\\ifmath{\\tfrac12},0)$ or\n$(0,\\ifmath{\\tfrac12})$ under the Lorentz algebra, denoted by $Q^i_{\\alpha}$ and\nits hermitian conjugate\n$Q^{\\dagger}_{\\dot\\alpha i}\\equiv (Q^i_\\alpha)^\\dagger$, respectively,\nwhere $i=1,2,\\ldots N$\\cite{Lopuszanski,Haag:1974qh}.\nIn these lectures, we shall focus exclusively on the case of $N=1$, in which case the subscript $i$\ncan be dropped.\n\nWe therefore begin by examining the structure of\nthe $N=1$ SUSY algebra, which is obtained by adding one\n$(\\ifmath{\\tfrac12},0)$ and one $(0,\\ifmath{\\tfrac12})$ generator to the Poincar\\'e algebra,\ndenoted by $Q_\\alpha$ and $Q^\\dagger_{\\dot\\alpha}$, respectively.\nThese two-component spinor generators have no explicit dependence\non the spacetime coordinate and are thus\ninvariant under spacetime translations. That is,\n\\begin{Eqnarray}\n\\exp\\left(-ia_\\mu P^\\mu\\right)Q_\\alpha \\exp\\left(ia_\\mu P^\\mu\\right)&=&Q_\\alpha\\,,\\\\[6pt]\n\\exp\\left(-ia_\\mu P^\\mu\\right)Q^\\dagger_{\\dot\\alpha} \\exp\\left(ia_\\mu P^\\mu\\right)&=&Q^\\dagger_{\\dot\\alpha}\\,,\n\\end{Eqnarray}\nwhere the $a_\\mu$ are real parameters. Working to first order in $a_\\mu$, it follows that\nthe spinor generators\nmust commute with the translation generator~$P^\\mu$,\n\\begin{equation} \\label{QP}\n[Q_\\alpha\\,,\\,P^\\mu]=[Q^\\dagger_{\\dot\\alpha}\\,,\\,P^\\mu]=0\\,.\n\\end{equation}\n\nThe commutation relations given in \\eq{QP} can also be deduced by\nemploying the following algebraic argument. Using the known\ntransformation properties of $Q_\\alpha$, $Q^\\dagger_{\\dot\\alpha}$ and\n$P^\\mu$ under the\nPoincar\\'e algebra, it follows that $[Q_\\alpha\\,,\\,P^\\mu]$ must consist\nof generators whose transformation properties are consistent with the\ntensor product,\n\\begin{equation}\n(\\ifmath{\\tfrac12},0)\\otimes(\\ifmath{\\tfrac12},\\ifmath{\\tfrac12})=(1,\\ifmath{\\tfrac12})\\oplus(0,\\ifmath{\\tfrac12})\\,,\n\\end{equation}\nunder the Poincar\\'e algebra. But according to the\nHaag-{\\L}opuszanski-Sohnius theorem, there are no $(1,\\ifmath{\\tfrac12})$ generators.\nThis argument still leaves open the possibility that\n$[Q_\\alpha\\,,\\,P^\\mu]\\propto \\sigma^\\mu_{\\alpha\\dot\\beta}Q^{\\dagger\\,\\dot\\beta}$.\nHowever, it can be shown using the Jacobi identity\n that\nthe proportionality constant must be zero.\n\nThe transformation properties of $Q_\\alpha$ and\n$Q^\\dagger_{\\dot\\alpha}$\nunder the Poincar\\'e algebra yield their\ncommutation relations with the $J^{\\mu\\nu}$,\n\\begin{equation}\n[Q_\\alpha\\,,\\,J^{\\mu\\nu}]=(\\sigma^{\\mu\\nu})_\\alpha{}^\\beta Q_\\beta\\,,\n\\qquad\\qquad\n[Q^\\dagger_{\\dot\\alpha}\\,,\\,J^{\\mu\\nu}]\n=-Q^\\dagger_{\\dot\\beta}(\\overline{\\sigma}^{\\mu\\nu})^{\\dot\\beta}{}_{\\dot\\alpha}\\,.\n\\end{equation}\nThe Coleman-Mandula theorem implies that one cannot obtain a consistent algebraic\nstructure by postulating commutation relations for the $Q_\\alpha$ and $Q^\\dagger_{\\dot\\alpha}$.\nHowever, by declaring $Q_\\alpha$ and $Q^\\dagger_{\\dot\\alpha}$ to be \\textit{fermionic}\ngenerators, one can postulate \\textit{anticommutation} relations for $Q_\\alpha$ and $Q^\\dagger_{\\dot\\alpha}$\nsuch that the generators $\\{P^\\mu\\,,\\,J^{\\mu\\nu}\\,,\\,Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\alpha}\\}$ form\na closed algebraic system. We therefore consider the three possible anticommutation relations,\nalong with their transformation properties with respect to the Poincar\\'e algebra,\n\\begin{Eqnarray}\n\\{Q_\\alpha\\,,\\,Q_\\beta\\}\\qquad & \\qquad (\\ifmath{\\tfrac12},0)\\otimes(\\ifmath{\\tfrac12},0)=(1,0)\\oplus(0,0)\\,,\\label{qqcomm1}\\\\\n\\{Q^\\dagger_{\\dot\\alpha}\\,,\\,Q^\\dagger_{\\dot\\beta}\\}\\qquad & \\qquad (0,\\ifmath{\\tfrac12})\\otimes (0,\\ifmath{\\tfrac12})=(0,1)\\oplus(0,0)\\,,\\label{qqcomm2}\\\\\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}\\qquad & \\qquad (\\ifmath{\\tfrac12},0)\\otimes (0,\\ifmath{\\tfrac12})=(\\ifmath{\\tfrac12},\\ifmath{\\tfrac12})\\,.\\label{qqcomm3}\n\\end{Eqnarray}\n\\Eqs{qqcomm1}{qqcomm2} imply that\n\\begin{Eqnarray}\n\\{Q_\\alpha\\,,\\,Q^\\beta\\}&=&s(\\sigma^{\\mu\\nu})_\\alpha{}^\\beta J_{\\mu\\nu}+k\\delta_\\alpha{}^\\beta\\mathds{1}\\,,\\label{QQsk1}\\\\[6pt]\n\\{Q^{\\dagger\\dot\\alpha}\\,,\\,Q^\\dagger_{\\dot\\beta}\\}\\ &=& s^* (\\overline{\\sigma}^{\\mu\\nu})^{\\dot\\alpha}{}_{\\dot{\\beta}}J_{\\mu\\nu}\n+k^*\\delta^{\\dot\\alpha}{}_{\\dot\\beta}\\mathds{1}\\,,\\label{QQsk2}\n\\end{Eqnarray}\nwhere $s$ and $k$ are complex numbers\nand \\eq{QQsk2} is the hermitian conjugate of \\eq{QQsk1}. Note that\nwe have raised and\/or lowered some of the spinor indices for convenience.\nSince $[Q_\\alpha,P^\\lambda]=[Q^\\dagger_{\\dot\\alpha},P^\\lambda]=0$ and $[J_{\\mu\\nu},P^\\lambda]\\neq 0$,\nit follows that $s=0$. If we now lower all spinor indices, \\eqs{QQsk1}{QQsk2} with $s=0$ yield\n\\begin{equation}\n\\{Q_\\alpha\\,,\\,Q_\\beta\\}=k\\epsilon_{\\beta\\alpha}\\mathds{1}\\,,\\qquad\\quad\n\\{Q^\\dagger_{\\dot\\alpha}\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=k^*\\epsilon^{\\dot\\beta\\dot\\alpha}\\mathds{1}\\,,\n\\end{equation}\nand we conclude that $k=0$, since the left-hand sides of the above equations are symmetric under\nthe interchange of spinor indices, whereas the right hand sides are antisymmetric.\nHence,\n\\begin{equation}\n\\{Q_\\alpha\\,,\\,Q_\\beta\\}=\\{Q^\\dagger_{\\dot\\alpha}\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=0\\,.\n\\end{equation}\n\n\\Eq{qqcomm3} implies that the remaining anticommutation relation must be of the form\n\\begin{equation} \\label{QQt}\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=t\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu\\,,\n\\end{equation}\nwhere $t$ is a complex number. Multiplying \\eq{QQt} by $\\overline{\\sigma}^{\\nu\\dot\\beta\\alpha}$ and using\n$\\Tr(\\sigma^\\mu\\overline{\\sigma}^\\nu)=2g^{\\mu\\nu}$, it follows that\n\\begin{equation} \\label{sigbarqq}\n\\overline{\\sigma}_\\mu^{\\dot\\beta\\alpha}\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=2tP_\\mu\\,.\n\\end{equation}\nIn particular, for $\\mu=0$, \\eq{sigbarqq} relates the energy $P^0$ to the SUSY generators:\n\\begin{equation} \\label{pzero}\n2tP^0=Q_1 Q_1^\\dagger+Q_1^\\dagger Q_1+Q_2 Q_2^\\dagger+Q_2^\\dagger Q_2\\,.\n\\end{equation}\nSince $P^0\\geq m$ for physical states of mass $m$ and the right-hand side of \\eq{pzero} is positive semi-definite,\nit follows that $t$ must be real and positive.\\footnote{We reject the possibility of $t=0$, in\nwhich case $Q=Q^\\dagger=0$ and the SUSY algebra reduces to the Poincar\\'e algebra.}\nOne can rescale the definition of the fermionic generators $Q$ and $Q^\\dagger$ such that\n$t=2$. In this convention,\n\\begin{equation}\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu\\,.\n\\end{equation}\n\nTo summarize, the $N=1$ SUSY algebra\nis spanned by the generators $\\{P^\\mu\\,,\\,J^{\\mu\\nu}\\,,\\,Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\alpha}\\}$, which\nsatisfy \\eqst{spoincarealg1}{spoincarealg3} and\n\\begin{Eqnarray}\n[Q_\\alpha\\,,\\,P^\\mu]&=&[Q^\\dagger_{\\dot\\alpha}\\,,\\,P^\\mu]=0\\,,\\label{susyalg1}\\\\\n\\left[Q_\\alpha\\,,\\,J^{\\mu\\nu}\\right]&=&(\\sigma^{\\mu\\nu})_\\alpha{}^\\beta Q_\\beta\\,,\\label{susyalg2}\\\\\n\\left[Q^\\dagger_{\\dot\\alpha}\\,,\\,J^{\\mu\\nu}\\right]&=&-Q^\\dagger_{\\dot\\beta}\n(\\overline{\\sigma}^{\\mu\\nu})^{\\dot\\beta}{}_{\\dot\\alpha}\\,,\\label{susyalg3}\\\\\n\\{Q_\\alpha\\,,\\,Q_\\beta\\}&=&\\{Q^\\dagger_{\\dot\\alpha}\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=0\\,,\\label{susyalg4}\\\\\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}&=&2\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu\\,.\\label{susyalg5}\n\\end{Eqnarray}\n\nNote that \\eqst{susyalg1}{susyalg5} are unchanged under the U(1) phase transformation,\n\\begin{equation}\nQ_\\alpha\\to e^{-i\\chi}Q_\\alpha\\,,\\qquad\\quad Q^{\\dagger}_{\\dot\\alpha}\\to e^{i\\chi}Q^{\\dagger}_{\\dot\\alpha}\\,,\n\\end{equation}\nwhereas the generators $P^\\mu$ and $J^{\\mu\\nu}$ are not transformed.\nOne can therefore extend the $N=1$ SUSY algebra by adding a bosonic generator $R$ such that\n\\begin{Eqnarray}\ne^{i\\chi R}Q_\\alpha e^{-i\\chi R}&=&e^{-i\\chi}Q_\\alpha\\,,\\label{R1}\\\\\ne^{i\\chi R}Q^\\dagger_{\\dot\\alpha} e^{-i\\chi R}&=&e^{i\\chi}Q^\\dagger_{\\dot\\alpha}\\,.\\label{R2}\n\\end{Eqnarray}\nExpanding out to first order in $\\chi$, one easily derives the commutation relations,\n\\begin{Eqnarray}\n\\left[R\\,,\\,Q_\\alpha\\right]&=&-Q_\\alpha\\,,\\label{susyalg6} \\\\\n\\left[R\\,,\\,Q^\\dagger_{\\dot\\alpha}\\right]&=&Q^\\dagger_{\\dot\\alpha}\\,.\\label{susyalg7}\n\\end{Eqnarray}\nWe therefore say that the generator $Q_\\alpha$ has an $R$-charge of $-1$. Since $P^\\mu$ and $J^{\\mu\\nu}$\nare uncharged under the U(1)$_R$ transformation, it follows that\n\\begin{equation} \\label{susyalg8}\n[R\\,,\\,P^\\mu]=[R\\,,\\,J^{\\mu\\nu}]=0\\,.\n\\end{equation}\nThus, \\eqst{spoincarealg1}{spoincarealg3},\n(\\ref{susyalg1})--(\\ref{susyalg5}) and (\\ref{susyalg6})--(\\ref{susyalg8})\ndefine the maximally extended\n$N=1$ SUSY algebra, which includes an additional\ncontinuous U(1)$_R$ symmetry.\n\n\n\n\n\\subsection{Representations of the $N=1$ SUSY algebra}\nIn Section~\\ref{sec:Poincare}, we identified the two Casimir operators of the Poincar\\'e\nalgebra, $P^2$ and $w^2$, and noted that the\nrepresentations of the Poincar\\'e algebra can be labeled by the eigenvalues of\nthe Casimir operators acting on the physical states.\nWe saw that \nthe massive\nrepresentations can be labeled by their mass and spin, $(m,s)$. For a fixed value of $m$, the\ncorresponding spin-$s$ representations are $(2s+1)$-dimensional.\nFor massless states, we defined\n the helicity operator\n$h=\\boldsymbol{\\vec{S}\\!\\cdot\\!\\hat{P}}$ [cf.~\\eq{hdefinition}], with\n eigenvalues $\\lambda=0,\\pm\\ifmath{\\tfrac12},\\pm 1\\,\\ldots$.\nWe also noted that $\\lambda$ changes sign under a CPT\ntransformation. Hence, the massless positive energy\nrepresentations of the Poincar\\'e algebra are specified by $|\\lambda|$.\nFor the case of $\\lambda=0$, the corresponding representation is one-dimensional.\nFor any non-zero \nchoice for $\\lambda$,\nthe corresponding representation is two-dimensional and reducible,\nas both $\\pm|\\lambda|$ helicity states must appear.\n\nThe unitary representations of the $N=1$ SUSY algebra can be determined\nby using similar techniques\\cite{Salam:1974za,Sokatchev:1975gg}. First, we identify the Casimir operators, which\ncommute with all the SUSY algebra generators, $\\{P^\\mu\\,,\\,J^{\\mu\\nu}\\,,\\,Q_\\alpha\\,,\\,Q^{\\dagger\\dot\\alpha}\\}$.\nIt is clear that $P^2$ is a Casimir operator, since $Q_\\alpha$ and $Q^{\\dagger\\dot\\alpha}$ commute\nwith $P^\\mu$. However, $w^2$ is \\textit{not} a Casimir operator of the SUSY\nalgebra. To establish this result, it is straightforward to use the (anti-)commutation\nrelations of the SUSY algebra to prove that:\n\\begin{equation} \\label{wQQ}\n\\left[w^\\mu\\,,\\,Q_\\alpha\\right]=i(\\sigma^{\\mu\\nu})_{\\alpha}{}^\\beta Q_\\beta P_\\nu\\,,\\qquad\\quad\n\\left[w^\\mu\\,,\\,Q^\\dagger_{\\dot\\alpha}\\right]= i(\\overline{\\sigma}^{\\mu\\nu})^{\\dot\\beta}{}_{\\dot\\alpha}Q^\\dagger_{\\dot\\beta}P_\\nu\\,.\n\\end{equation}\nUsing these results, it is straightforward to derive:\n\\begin{Eqnarray}\n[w^2\\,,\\,Q_\\alpha]&=&2i\\sigma^{\\mu\\nu}{}_\\alpha{}^\\beta Q_\\beta w_\\mu P_\\nu-\\tfrac{3}{4}P^2 Q_\\alpha\\,,\\label{wtwo} \\\\\n\\left[w^2\\,,\\,Q^\\dagger_{\\dot\\alpha}\\right]&=&2i\\overline{\\sigma}^{\\mu\\nu\\dot\\beta}{}_{\\dot\\alpha} Q^\\dagger_{\\dot\\beta} w_\\mu P_\\nu\n-\\tfrac{3}{4}P^2 Q^\\dagger_{\\dot\\alpha}\\,.\\label{wtwodag}\n\\end{Eqnarray}\nThus, $w^2$ does not commute with the fermionic generators of the SUSY algebra. One consequence\nof this result is that the representations of the SUSY\nalgebra consist of supermultiplets that contain particles of equal\nmass but with different spins.\n\nIn order to deduce the possible spins that make up an irreducible supermultiplet, we shall identify a second Casimir\noperator of the $N=1$ SUSY algebra. We begin by defining the operator\n\\begin{equation}\nB^\\mu\\equiv w^\\mu+\\tfrac{1}{4}Q^\\dagger\\overline{\\sigma}^\\mu Q\\,.\n\\end{equation}\nUsing \\eqss{susyalg4}{susyalg5}{wQQ}, one can derive\n\\begin{equation} \\label{BQQ}\n[B^\\mu\\,,\\,Q_\\alpha]=-\\ifmath{\\tfrac12} P^\\mu Q_\\alpha\\,,\\qquad\\qquad [B^\\mu\\,,\\,Q^\\dagger_{\\dot\\alpha}]=\\ifmath{\\tfrac12} P^\\mu Q^\\dagger_{\\dot\\alpha}\\,.\n\\end{equation}\nThe four-vector operator $B^\\mu$ possesses some of the properties of the Pauli-Lubanski vector $w^\\mu$.\nIn particular,\n\\begin{align} \n[B^\\mu\\,,\\,B^\\nu] &=i\\epsilon^{\\mu\\nu\\rho\\lambda}B_\\rho P_\\lambda ; \\label{BmuBnu} \\\\\n[B^\\mu, P^\\nu] & = 0; \\label{BP}\\\\\n[B^\\mu, J^{\\nu\\lambda} ] & = i \\of{ g^{\\mu\\nu} B^\\lambda - g^{\\mu\\lambda} B^\\nu } . \\label{Bvector}\n\\end{align}\nOne may be tempted to conjecture that $B^2\\equiv B_\\mu B^\\mu$\nis a Casimir operator of the SUSY algebra. However, $[B^2\\,,\\,Q_\\alpha]\\neq 0$, so we must look further.\nThe structure of \\eq{BQQ} suggests that we define\n\\begin{equation} \\label{Cmunu}\nC^{\\mu\\nu}\\equiv B^\\mu P^\\nu-B^\\nu P^\\mu\\,.\n\\end{equation}\nIt then follows that\n\\begin{equation}\n[C^{\\mu\\nu}\\,,\\,Q_\\alpha]=[C^{\\mu\\nu}\\,,\\,Q^\\dagger_{\\dot\\alpha}]=[C^{\\mu\\nu}\\,,\\,P^\\lambda]=0\\,,\n\\end{equation}\nwhere the first two commutators vanish as a consequence of \\eq{BQQ} and the last commutator\nvanishes as a consequence of \\eq{BP}. Moreover, \\eqs{spoincarealg2}{Bvector} imply that $P^\\mu$ and $B^\\mu$\nare Lorentz four-vectors, in which case $C^{\\mu\\nu}$ is a second-rank Lorentz tensor. Hence\n\\begin{equation} \\label{C2def}\nC^2\\equiv C_{\\mu\\nu} C^{\\mu\\nu}=2[B^2 P^2-(B\\!\\cdot\\! P)^2]\\,,\n\\end{equation}\nsatisfies\n\\begin{equation}\n[C^2\\,,\\,P^\\mu]=[C^2\\,,\\,J^{\\mu\\nu}]=[C^2\\,,\\,Q_\\alpha]=[C^2\\,,\\,Q^\\dagger_{\\dot\\alpha}]=0\\,.\n\\end{equation}\n\nWe conclude that $P^2$ and $C^2$ are the two\nCasimir operators of the $N=1$ SUSY algebra.\nRepresentations of the $N=1$ SUSY algebra can therefore be\nlabeled by the eigenvalues of $P^2$ and $C^2$ when acting on the\nphysical states.\\footnote{As in the case of the Poincar\\'e algebra,\nwe restrict our considerations to\nstates of non-negative energy $P^0$.} The eigenvalue of $P^2$ is\n$m^2$, where $m$ is the mass. To understand the physical meaning of $C^2$, we will consider massive and massless supermultiplets separately.\n\n\\subsubsection{Massive $N=1$ supermultiplets}\nTo see the physical interpretation of $C^2$, we first consider the case of $m\\neq 0$, so that we are free to evaluate the Lorentz scalar $C^2$ in the particle rest frame.\nIn this frame,\n\\begin{equation} \\label{bmudef}\nB^\\mu=(\\tfrac{1}{4}Q^\\dagger\\overline{\\sigma}^0 Q\\,;\\,mS^i+\\tfrac{1}{4}Q^\\dagger\\overline{\\sigma}^i Q),\n\\end{equation}\nwhere $S^i$ is defined in \\eq{jkdef}.\nWe then compute,\n\\begin{align}\nC^2 & = 2\\left[B^2 P^2-(B\\!\\cdot\\! P)^2\\right] \n =2m^2\\left[B^2-B_0^2\\right] \n = -2m^2 B^i B^i,\n\\end{align}\nwhere $B^iB^i\\equiv |\\boldsymbol{\\vec B}|^2$. \nMoreover, if we define the rest-frame operator,\n\\begin{equation} \\label{caljdef}\n\\mathcal{J}^i\\equiv \\frac{1}{m}B^i=S^i+\\frac{1}{4m}Q^\\dagger\\overline{\\sigma}^i Q\\,,\n\\end{equation}\nthen it follows from \\eq{BmuBnu} that\n\\begin{equation}\n[\\mathcal{J}^i\\,,\\,\\mathcal{J}^j]=i\\epsilon^{ijk} \\mathcal{J}^k\\,.\n\\end{equation}\n\nThe eigenvalues of $\\mathcal{J}^i \\mathcal{J}^i$ are $j(j+1)$ for $j=0,\\ifmath{\\tfrac12},1,\\tfrac{3}{2}\\,\\ldots$. Hence, the\neigenvalues of\n\\begin{equation}\nC^2=-2m^4 \\mathcal{J}^i \\mathcal{J}^i\n\\end{equation}\nare $-2m^4 j(j+1)$. We conclude that for positive energy, timelike $P^\\mu$,\nthe unitary irreducible representations of the $N=1$ SUSY algebra are labeled by $(m,j)$, where $j$ is called\nthe \\textit{superspin} of the supermultiplet.\nThe states of an irreducible $N=1$ massive supermultiplet of superspin $j$ are exhibited\nin Table~\\ref{massivesuperplet}. \nThe explicit construction of these\nstates and a discussion of their properties is presented in Section~\\ref{App}.\n\n\\begin{table}[t!]\n \\caption{\\small States of an $N=1$ massive supermultiplet of superspin $j$. An interpretation\nis provided for $j=s$ and $j=s+\\ifmath{\\tfrac12}$ where $s$ is a non-negative integer.\nThe bosonic and fermionic degrees of freedom (D.o.f.) of the supermultiplet coincide\nand is equal to $2(2j+1)$.\\label{massivesuperplet}}\n\\vskip 0.1in\n{\n\\addtolength\\tabcolsep{2pt}\n\\begin{tabular}{cccc}\n\\hline\nSpin & D.o.f. & Interpretation ($j=s$) & Interpretation ($j=s+\\ifmath{\\tfrac12}$) \\\\ \\hline\n$j$ & $2(2j+1)$ & complex spin-$s$ boson & ``complex'' spin-($s+\\ifmath{\\tfrac12}$) fermion \\\\\n$j+\\ifmath{\\tfrac12}$ & $2j+2$ & spin-($s+\\ifmath{\\tfrac12})$ fermion & real spin-$(s+1)$ boson \\\\\n$j-\\ifmath{\\tfrac12}$ & $2j$ & spin-($s-\\ifmath{\\tfrac12})$ fermion & real spin-$s$ boson \\\\ \\hline\n\\end{tabular}}\n \\end{table}\n\n\n\\begin{example}[The massive chiral supermultiplet, $\\boldsymbol{j=0}$]\n\nFor $j=0$, only $j_3=0$ is possible, in which case the massive\nsupermultiplet is made up of two states of spin 0 and two states of\nspin $\\ifmath{\\tfrac12}$. The two spin-0 states can be combined into a single complex\nscalar state, and the two spin-$\\ifmath{\\tfrac12}$ states can be identified as the two\ncomponents of a two-component Majorana fermion. In this case the\n$j-\\ifmath{\\tfrac12}$ row of Table~\\ref{massivesuperplet} is not relevant. \n\\end{example}\n\nIt can be shown (see Problem \\ref{pr:jhalf}) that the massive\nsupermultiplet of superspin $\\ifmath{\\tfrac12}$ consists of a (real) spin-1 boson,\na (real) spin-0 boson and two mass-degenerate Majorana fermions, which\ncan be combined into a single Dirac fermion (called a \\textit{complex}\nfermion in Table~\\ref{massivesuperplet}).\nAs expected, in both the $j=0$ and $j=\\ifmath{\\tfrac12}$ cases exhibited above,\nthe number of bosonic degrees of freedom of the\nsupermultiplet equals the number of fermionic degrees of freedom. \n\n\n\n\\subsubsection{Massless $N=1$ supermultiplets}\n\nWe now examine the case of zero-mass positive energy states, where $P^2=0$ and $P^0>0$. If one multiplies\n\\eq{susyalg5} by $P^\\rho P^\\lambda\\overline{\\sigma}_\\rho^{\\dot\\gamma\\alpha}\\overline{\\sigma}_\\gamma^{\\dot\\beta\\tau}$,\none can easily derive the anticommutation relation,\n\\begin{equation}\n\\{P^\\rho\\overline{\\sigma}_\\rho^{\\dot\\gamma\\alpha}Q_\\alpha\\,,\\,P^\\lambda Q^\\dagger_{\\dot\\beta}\\overline{\\sigma}^{\\dot\\beta\\tau}\\}=2P^2 P^\\mu\\overline{\\sigma}_\\mu^{\\dot\\gamma\\tau}\\,.\n\\end{equation}\nThus, for $P^2=0$ we have,\n\\begin{equation} \\label{PoperatorP}\n\\bra{\\Psi}\\{P^\\rho\\overline{\\sigma}_\\rho^{\\dot\\gamma\\alpha}Q_\\alpha\\,,\\,P^\\lambda Q^\\dagger_{\\dot\\beta}\\overline{\\sigma}^{\\dot\\beta\\tau}\\}\\ket{\\Psi}=0\\,,\n\\end{equation}\nfor any state $\\ket{\\Psi}$.\nIn the space of one-particle states, only positively-normed states exist. Noting that\n$(P^\\mu\\overline{\\sigma}_\\mu^{\\dot\\alpha\\beta}Q_\\beta)^\\dagger=P^\\mu Q^\\dagger_{\\dot\\beta}\\overline{\\sigma}_\\mu^{\\dot\\beta\\alpha}$,\n\\eq{PoperatorP} implies that as operators on the space of one-particle states,\n\\begin{equation} \\label{zeroops}\nP^\\rho\\overline{\\sigma}_\\rho^{\\dot\\gamma\\alpha}Q_\\alpha=\nP^\\lambda Q^\\dagger_{\\dot\\beta}\\overline{\\sigma}_\\lambda^{\\dot\\beta\\tau}=0\\,,\\qquad\n\\text{for}~~P^2=0\\,.\n\\end{equation}\nUsing this result, one can evaluate the Casimir operator $C^2$, defined in \\eq{C2def}, in the case of $P^2=0$.\nIn particular, using $w_\\mu P^\\mu = 0$ and \\eq{zeroops},\n\\begin{equation}\nC^2=-2(B\\!\\cdot\\! P)^2=-\\tfrac{1}{8}(Q^\\dagger_{\\dot\\alpha}\\overline{\\sigma}^{\\dot\\alpha\\beta}_\\mu Q_\\beta P^\\mu)^2=0\\,.\n\\end{equation}\n\nThe same conclusion can be obtained by choosing\nthe standard reference frame, $P^\\mu=P^0(1\\,;\\,0\\,,\\,0\\,,\\,1)$,\nfor lightlike four-vectors. In this reference frame, the anticommutators given in \\eqs{susyalg4}{susyalg5}\nsimplify to\n\\begin{Eqnarray}\n\\{Q_1\\,,\\,Q^\\dagger_1\\}&=& 0\\,,\\qquad\\quad\\,\\,\n\\{Q_2\\,,\\,Q^\\dagger_2\\}=4P_0\\,,\\label{qqmassless1}\\\\\n\\{Q_1\\,,\\,Q_1\\}&=&\\{Q_2\\,,\\,Q_2\\}=\\{Q_1\\,,\\,Q_2\\}=0\\,,\\label{qqmassless2}\\\\\n \\{Q^\\dagger_1\\,,\\,Q^\\dagger_1\\}&=&\\{Q^\\dagger_2\\,,\\,Q^\\dagger_2\\}=\\{Q^\\dagger_1\\,,\\,Q^\\dagger_2\\}=0\\,.\\label{qqmassless3}\n\\end{Eqnarray}\nHence,\n\\begin{equation}\nC^2=-2(B\\!\\cdot\\! P)^2=-\\ifmath{\\tfrac12} P_0^2(Q_1^\\dagger Q_1)^2=\\ifmath{\\tfrac12} P_0^2 Q_1^\\dagger Q_1^\\dagger Q_1 Q_1=0\\,.\n\\end{equation}\n\n\\Eq{zeroops} implies a number of other operator identities when acting on the space of one-particle states.\nUsing \\eq{susyalg5}, one easily derives\n\\begin{equation}\n[Q^\\alpha Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}]=4P_\\mu\\sigma^\\mu_{\\alpha\\dot\\beta}Q^\\alpha\\,,\\qquad\\quad\n[Q^\\dagger_{\\dot\\alpha}Q^{\\dagger\\,\\dot\\alpha}\\,,\\,Q_\\beta]=-4P_\\mu\\sigma^\\mu_{\\alpha\\dot\\beta}Q^{\\dagger\\,\\dot\\beta}\\,.\n\\end{equation}\nApplying \\eq{zeroops} then yields\n\\begin{equation} \\label{QQQQ}\n[Q^\\alpha Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}]=[Q^\\dagger_{\\dot\\alpha}Q^{\\dagger\\,\\dot\\alpha}\\,,\\,Q_\\beta]=0\\,,\\qquad\n\\text{for}~~P^2=0\\,.\n\\end{equation}\nThen, for any one-particle state $\\ket{\\Psi}$, \\eqss{susyalg4}{susyalg5}{QQQQ} yield\n\\begin{Eqnarray}\nP_\\mu\\sigma^\\mu_{\\alpha\\dot\\alpha}Q^\\beta Q_\\beta\\ket{\\Psi}&=&\\ifmath{\\tfrac12}\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\alpha}\\}Q^\\beta Q_\\beta\\ket{\\Psi}\n=\\ifmath{\\tfrac12} Q_\\alpha Q^\\dagger_{\\dot\\alpha}Q^\\beta Q_\\beta\\ket{\\Psi}\\nonumber \\\\\n&=&\\ifmath{\\tfrac12} Q_\\alpha [Q^\\dagger_{\\dot\\alpha}\\,,\\,Q^\\beta Q_\\beta]\\ket{\\Psi}=0\\,.\n\\end{Eqnarray}\nA similar computation of $P_\\mu\\sigma^\\mu_{\\alpha\\dot\\alpha}Q^\\dagger_{\\dot\\beta}Q^{\\dagger\\,\\dot\\beta}$\nallows us to conclude that\n\\begin{equation}\nP_\\mu Q^\\beta Q_\\beta\\ket{\\Psi}=P_\\mu Q^\\dagger_{\\dot\\beta}Q^{\\dagger\\,\\dot\\beta}\\ket{\\Psi}=0\\,,\\qquad \\text{for}~~P^2=0\\,,\n\\end{equation}\nafter multiplying through by $\\overline{\\sigma}_\\nu^{\\dot\\alpha\\alpha}$ and evaluating the resulting trace.\nAs we are only interested in positive energy states, we conclude that as operators on the space of one-particle states,\n\\begin{equation} \\label{QQQQ0}\nQ^\\beta Q_\\beta=Q^\\dagger_{\\dot\\beta}Q^{\\dagger\\,\\dot\\beta}=0\\,,\\qquad \\text{for}~~P^2=0~~\\text{and}~~P^0>0\\,.\n\\end{equation}\n\nIn order to identify the massless supermultiplets of one-particle states, it is convenient to define\n\\begin{equation} \\label{Lmudef}\nL^\\mu\\equiv \\ifmath{\\tfrac12}(w^\\mu+B^\\mu)=w^\\mu+\\tfrac{1}{8}Q^\\dagger\\overline{\\sigma}^\\mu Q\\,.\n\\end{equation}\nNote \n$[Q_\\alpha ,P^\\mu ]=[Q^\\dagger_{\\dot{\\alpha}}, P^\\mu ]=0$ and\n$[w_\\mu, P_\\nu ]=0$\nimply that\n\\begin{equation} \\label{PL}\n[P^\\mu\\,,\\,L^\\nu]=0\\,.\n\\end{equation}\nUsing \\eqss{susyalg4}{susyalg5}{wQQ}, one can easily derive\n\\begin{equation} \\label{LQQ}\n[L^\\mu\\,,\\,Q_\\alpha]=-\\tfrac{1}{4}(\\sigma^\\mu\\overline{\\sigma}^\\nu)_\\alpha{}^\\beta Q_\\beta P_\\nu\\,,\\qquad\n[L^\\mu\\,,\\,Q^\\dagger_{\\dot\\alpha}]=\\tfrac{1}{4}(\\overline{\\sigma}^\\nu\\sigma^\\mu)^{\\dot\\beta}{}_{\\dot\\alpha} Q^\\dagger_{\\dot\\beta}P_\\nu\\,.\n\\end{equation}\nA straightforward computation then gives:\n\\begin{equation} \\label{LLcomm}\n[L^\\mu\\,,\\,L^\\nu]=i\\epsilon^{\\mu\\nu\\rho\\lambda}(L_\\rho+\\tfrac{1}{16}Q^\\dagger\\overline{\\sigma}_\\rho Q)P_\\lambda\\,.\n\\end{equation}\nWhen $P^2=0$, we impose the results of \\eq{zeroops} to obtain\n\\begin{equation} \\label{Lprops0}\nP^\\mu L_\\mu=[L^\\mu\\,,\\,Q_\\alpha]=[L^\\mu\\,,\\,Q^\\dagger_{\\dot\\alpha}]=0\\,,\\qquad \\text{for}~~P^2=0\\,.\n\\end{equation}\nMoreover, if we employ the identity\n\\begin{equation}\n\\epsilon^{\\mu\\nu\\rho\\lambda}\\overline{\\sigma}_\\rho=\\ifmath{\\tfrac12} i(\\overline{\\sigma}^\\nu\\sigma^\\mu\\overline{\\sigma}^\\lambda-\\overline{\\sigma}^\\lambda\\sigma^\\mu\\overline{\\sigma}^\\nu)\\,,\n\\end{equation}\n[which is a consequence of \\eq{sigsigsig1}],\nit then follows from \\eq{zeroops} that\n\\begin{equation} \\label{epsQP}\n\\epsilon^{\\mu\\nu\\rho\\lambda}Q^\\dagger\\overline{\\sigma}_\\rho Q P_\\lambda=0\\,,\\qquad \\text{for}~~P^2=0\\,.\n\\end{equation}\n\n\nHence, in the massless case, \\eq{LLcomm} simplifies to\n\\begin{equation} \\label{LLcomm0}\n[L^\\mu\\,,\\,L^\\nu]=i\\epsilon^{\\mu\\nu\\rho\\lambda}L_\\rho P_\\lambda\\,,\\qquad \\text{for}~~P^2=0\\,.\n\\end{equation}\nFinally, we evaluate $L^\\mu L_\\mu$ for the positive energy massless one-particle states.\nAs in the analysis of the Poincar\\'e algebra, we shall assume that $w^\\mu w_\\mu=\\lim_{m\\to\n 0} (-m^2\\boldsymbol{\\vec S}\\llsup{\\,2})=0$.\nUsing \\eq{epsQP}, it follows that\n\\begin{equation}\nw^\\mu Q^\\dagger\\overline{\\sigma}_\\mu Q= -\\ifmath{\\tfrac12}\\epsilon^{\\mu\\nu\\rho\\lambda}J_{\\nu\\rho}P_\\lambda Q^\\dagger\\overline{\\sigma}_\\mu Q=0\\,.\n\\end{equation}\nIn light of \\eq{sigid3},\nwe obtain\n\\begin{Eqnarray}\n(Q^\\dagger\\overline{\\sigma}^\\mu Q)(Q^\\dagger\\overline{\\sigma}_\\mu Q)&=&2\\epsilon^{\\dot\\alpha\\dot\\gamma}\\epsilon^{\\beta\\tau}Q^\\dagger_{\\dot\\alpha}\nQ_\\beta Q^\\dagger_{\\dot\\gamma}Q_\\tau=2\\epsilon^{\\dot\\alpha\\dot\\gamma}\\epsilon^{\\beta\\tau}\nQ^\\dagger_{\\dot\\alpha}[2P_\\mu\\sigma^\\mu_{\\beta\\dot\\gamma}-Q^\\dagger_{\\dot\\gamma}Q_\\beta]Q_\\tau\\nonumber \\\\\n&=& 2(Q^\\dagger_{\\dot\\alpha}Q^{\\dagger\\,\\dot\\alpha})(Q^\\beta Q_\\beta)-4P^\\mu Q^\\dagger\\overline{\\sigma}_\\mu Q=0\\,,\n\\end{Eqnarray}\nafter applying the operator identities given in \\eqs{zeroops}{QQQQ0}. Hence,\n\\begin{equation} \\label{lmulmu}\nL^\\mu L_\\mu=0\\,,\\qquad \\text{for}~~P^2=0~~\\text{and}~~P^0>0\\,.\n\\end{equation}\n\nWhen $P^2=0$ and $P^0>0$, the properties of $L^\\mu$ [cf.~eqs.~(\\ref{PL}), (\\ref{Lprops0}), (\\ref{LLcomm0})\nand (\\ref{lmulmu})] match precisely the\nproperties of the Pauli-Lubanski vector.\nThus, we must solve the equations $L^2=P^2=L_\\mu\nP^\\mu=0$. In\na reference frame in which $P^\\mu=P^0(1\\,;\\,0\\,,\\,0\\,,\\,1)$ and $P^0>0$, \nit follows that $L^\\mu=L^0(1\\,;\\,0\\,,\\,0\\,,\\,1)$. Consequently, in any Lorentz frame,\n\\begin{equation} \\label{shelicitydef}\nL^\\mu=\\mathcal{K} P^\\mu\\,,\n\\end{equation}\nwhere $\\mathcal{K}\\equiv L^0\/P^0$ is called the superhelicity operator.\nMore explicitly, in a frame where $P^\\mu=P^0(1\\,;\\,0\\,,\\,0\\,,\\,1)$,\n\\begin{equation} \\label{mathcalkdef}\n\\mathcal{K}=h+\\frac{1}{8P^0}\\left(Q_1^\\dagger Q_1+Q_2^\\dagger Q_2\\right)\\,,\n\\end{equation}\nwhere $h\\equiv w^0\/P^0=\\boldsymbol{\\vec S\\!\\cdot\\!\\hat P}$ is the usual helicity operator acting\non massless one-particle states. By virtue of \\eqs{susyalg1}{Lprops0}, it follows that\n\\begin{equation} \\label{KQQ}\n[\\mathcal{K}\\,,\\,P^\\mu]=[\\mathcal{K}\\,,\\,Q_\\alpha]=[\\mathcal{K}\\,,\\,Q^\\dagger_{\\dot\\alpha}]=0\\,.\n\\end{equation}\n\nHence, the states of the massless supermultiplet are eigenstates of\n$\\mathcal{K}$, with possible\neigenvalues $\\kappa=0,\\pm\\ifmath{\\tfrac12},\\pm 1,\\pm\\tfrac{3}{2},\\ldots$. In contrast, $h$ does not commute with\n$Q_\\alpha$ and $Q^\\dagger_{\\dot\\alpha}$. Thus, the different states of the massless\nsupermultiplet will have different helicities.\nWe conclude that for positive energy, timelike $P^\\mu$,\nthe irreducible representations of the $N=1$ SUSY algebra are labeled by\nthe eigenvalue $\\kappa$ of the superhelicity operator, which is called\nthe \\textit{superhelicity} of the massless supermultiplet. Moreover, an $N=1$ massless supermultiplet with superhelicity $\\kappa$ consists of two massless\nstates with helicity $\\kappa$ and $\\kappa-\\ifmath{\\tfrac12}$, respectively.\\footnote{In the literature, it is more\ncommon to define $L^\\mu=(\\mathcal{K}+\\ifmath{\\tfrac12})P^\\mu$, in which case the helicities of the massless $N=1$ supermultiplet\nare $\\kappa+\\ifmath{\\tfrac12}$ and $\\kappa$ (e.g., see refs.~\\cite{Srivastava,Buchbinder}).\nIn our opinion, the definition of the superhelicity operator given in \\eq{shelicitydef} is cleaner.}\n\n\nAny quantum field theory realization of supersymmetry respects CPT symmetry. Since the helicity\nchanges sign under a CPT transformation, it follows that any\nirreducible massless supermultiplet with superhelicity~$\\kappa$ must\nbe accompanied by the corresponding CPT-conjugate states that make up an\nirreducible massless supermultiplet with superhelicity\n$-\\kappa+\\ifmath{\\tfrac12}$. \nHence, without loss of generality, we can restrict the possible values\nof the superhelicity to $\\kappa=\\ifmath{\\tfrac12},1,\\tfrac32,\\ldots$. \nThese results are summarized in\nTable~\\ref{masslesssuperplet}. The explicit construction of the\nstates of an irreducible massless supermultiplet and a discussion of their properties is presented in Section~\\ref{App}.\n\n\n\\begin{table}[t!]\n\\caption{\\small States of an $N=1$ massless supermultiplet of superhelicity $\\kappa$\nand the corresponding\nCPT conjugates which comprise an $N=1$ massless super\\-multiplet of superhelicity $-\\kappa+\\ifmath{\\tfrac12}$.\nAn interpretation\nis provided for $\\kappa=s$ and $\\kappa=s-\\ifmath{\\tfrac12}$, where $s$ is a positive integer.\nIn the special case of $\\kappa=\\ifmath{\\tfrac12}$, the scalar boson of the supermultiplet is complex, whereas for $\\kappa=1,\\tfrac{3}{2},2,\\ldots$,\nthe bosonic member of the supermultiplet is real\nwith nonzero spin. In all cases, the number of bosonic and fermionic degrees of\nfreedom (D.o.f.) coincide and are equal to\n2.\\label{masslesssuperplet}}\n\\vskip 0.1in\n{\n\\begin{tabular}{cccc} \\hline\nHelicities & D.o.f. & Interpretation ($\\kappa=s$) & Interpretation ($\\kappa=s-\\ifmath{\\tfrac12}$) \\\\ \\hline\n$\\kappa$\\,,\\,$-\\kappa$ & $2$ & spin-$s$ boson & spin-$(s-\\ifmath{\\tfrac12})$ fermion \\\\\n$\\kappa-\\ifmath{\\tfrac12}$\\,,\\,$-\\kappa+\\ifmath{\\tfrac12}$ & $2$ & spin-$(s-\\ifmath{\\tfrac12})$ fermion & spin-$(s-1)$ boson \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\n\n\n\\begin{example}[A massless chiral supermultiplet, with $\\boldsymbol{\\kappa=\\ifmath{\\tfrac12}}$]\nIncluding the CPT-conjugates, this supermultiplet contains two states of helicity 0, and two states of helicity $\\pm\\ifmath{\\tfrac12}$,\nrespectively, which yields a massless complex scalar and a\nmassless Majorana fermion. We recognize this as the massless limit of\na massive $j=0$ chiral supermultiplet. \n\\end{example}\n\n\\begin{example}[a massless gauge supermultiplet, with $\\boldsymbol{\\kappa=1}$]\nIncluding the CPT-conjugates, this supermultiplet contains two states of helicity $\\pm\\ifmath{\\tfrac12}$\nand two states of helicity $\\pm 1$, which yields a massless\nMajorana fermion and a massless spin-1 particle. This is a gauge supermultiplet\n (e.g the photino and the photon of\nsupersymmetric QED). \n\\end{example}\n\n\nIn Problem \\ref{pr:spin2}, you will show that\na massless supermultiplet with $\\kappa=2$ and its CPT-conjugates\ncontains\na massless spin-$\\tfrac{3}{2}$ and a massless spin 2 particle, which\nis realized in supergravity by the\ngravitino and the graviton, respectively. \n\n\n\n\n\n\\subsection{Consequences of super-Poincar\\'e invariance}\n\n\nA Poincar\\'e invariant quantum field theory respects the Poincar\\'e algebra generated\nby $\\{P^\\mu\\,,\\,J^{\\mu\\nu}\\}$, which satisfy commutation relations given by\n\\eqst{spoincarealg1}{spoincarealg3}. One of the basic postulates of\nPoincar\\'e-invariant quantum field theory states that a translationally-invariant,\nLorentz-invariant vacuum $\\ket{0}$ exists such that\\cite{Roman},\n\\begin{equation} \\label{pvacuum}\nP^\\mu\\ket{0}=0\\,,\\qquad\\quad J^{\\mu\\nu}\\ket{0}=0\\,.\n\\end{equation}\nIn particular, $\\bra{0}P^\\mu\\ket{0}=0$. Indeed if $\\bra{0}P^\\mu\\ket{0}\\neq 0$, then\nthe vacuum would not be invariant under Lorentz transformations. This is easily proven\nby taking the vacuum expectation value of \n\\begin{equation}\n\\exp\\left(\\tfrac{1}{2}i\\theta_{\\rho\\tau}J^{\\rho\\tau}\\right) P^\\mu\n \\exp\\left(-\\tfrac{1}{2}i\\theta_{\\rho\\tau}J^{\\rho\\tau}\\right)=\\Lambda^{\\mu}{}_{\\nu}P^\\nu\\,,\n\\end{equation}\nwhere the $\\theta_{\\rho\\tau}=-\\theta_{\\rho\\tau}$ parameterize the $4\\times 4$ Lorentz transformation\nmatrix $\\Lambda^\\mu{}_\\nu$\n[cf.~\\eqs{lambda44}{explicitsmunu}]. \n Using $J^{\\mu\\nu}\\ket{0}=0$,\nit follows that\n\\begin{equation}\n\\bra{0}P^\\mu\\ket{0}=\\Lambda^{\\mu}{}_{\\nu}\\bra{0}P^\\nu\\ket{0}\\,,\n\\end{equation}\nwhich holds for all Lorentz transformations $\\Lambda$. Thus, it follows that $\\bra{0}P^\\mu\\ket{0}=0$.\n\nA super-Poincar\\'e invariant quantum field theory respects the SUSY algebra\ngenerated by\n$\\{P^\\mu\\,,\\,J^{\\mu\\nu}\\,,\\,Q_\\alpha\\,,\\,Q^{\\dagger\\dot\\alpha}\\}$.\nThe SUSY algebra generators satisfy\nthe commutation relations of the Poincar\\'e algebra and the\n(anti)commutation relations given by\n\\eqst{susyalg1}{susyalg5}. Two important consequences can be established:\n\\vskip 0.1in\n\n1. \\textit{The vanishing of the vacuum energy is a necessary and sufficient condition\nfor the existence of a global supersymmetric vacuum.}\n\\vskip 0.1in\n\n2. \\textit{In a theory governed by a supersymmetric action, for a fixed non-zero $P_\\mu$ the number of bosonic\nand fermionic degrees of freedom coincide.}\n\\vskip 0.1in\n\n\\noindent\nWe address these two results in the next two subsections.\n\n\n\\subsubsection{The vacuum energy of a globally supersymmetric theory}\n\nIn order to prove that the vanishing of the vacuum energy is a necessary and sufficient condition\nfor the existence of a global supersymmetric vacuum, we consider\nthe anticommutation relations of the fermionic generators of the SUSY\nalgebra,\n\\begin{equation} \\label{QQanti}\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu\\,.\n\\end{equation}\nFollowing the derivation of \\eq{pzero},\n\\begin{equation} \\label{QQpzero}\nP^0=\\tfrac{1}{4}\\left[Q_1 Q_1^\\dagger+Q_1^\\dagger Q_1+Q_2 Q_2^\\dagger+Q_2^\\dagger Q_2\\right]\\,.\n\\end{equation}\nSince the right-hand side of \\eq{pzero} is\npositive semi-definite (and neither $Q$ nor $Q^\\dagger$ is the zero operator), it\nfollows that\n\\begin{equation}\n\\vev{0\\,|P^0\\,|\\,0}=0\\quad\\Longleftrightarrow\\quad Q_\\alpha\\ket{0}=0\\,.\n\\end{equation}\nIn particular, $Q_\\alpha\\ket{0}=0$ implies that the vacuum is supersymmetric, in the same way\nthat $P^\\mu\\ket{0}=J^{\\mu\\nu}\\ket{0}=0$ imply that the vacuum is translationally-invariant\nand Lorentz-invariant.\\footnote{Equivalently,\n$\\bra{0}\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}\\ket{0}=0$,\nby covariance with respect to the SUSY algebra,\nsince there are no spinor quantities with one undotted and one dotted index\nthat can appear on the right hand side of this equation.\nHence, $Q_\\alpha\\ket{0}=0$, which then yields $\\bra{0}P^0\\ket{0}=0$.}\n\nHowever, this proof is troubling for two separate reasons. First, suppose that the action\nof the theory is invariant under supersymmetric transformations, but the vacuum is not\npreserved by supersymmetry. In this case, $Q_\\alpha\\ket{0}\\neq 0$, and we say that\nsupersymmetry is spontaneously broken. Then, \\eq{QQpzero} implies that\n$\\bra{0}P^0\\ket{0}>0$, which contradicts \\eq{pvacuum}. Thus, it appears that the\nspontaneous breaking of supersymmetry is not possible without breaking Lorentz invariance.\nPerhaps a more fundamental objection is that the concept of the vacuum energy is\nusually considered to be unphysical\nin non-gravitational theories, as it is commonly asserted that only energy differences are physical.\nThus, it seems to be a matter of convention to choose the vacuum energy such that $\\bra{0}P^0\\ket{0}=0$.\n\nTo overcome the objections raised above, we re-examine the concept of the vacuum energy\nin relativistic (non-gravitational) quantum field theory. Using the\nNoether procedure, the conserved\ncanonical energy-momentum tensor, $T^{(c)}_{\\mu\\nu}$ can be obtained,\nwhich satisfies $\\partial^\\mu T^{(c)}_{\\mu\\nu}=0$.\\footnote{The arguments given here do not depend on\nwhether one employs the canonical energy momentum tensor or the\nimproved symmetrized energy-momentum tensor.}\nOne can then formally compute the vacuum energy\ndensity by summing over the vacuum Feynman diagrams of the theory. By Lorentz covariance~\\cite{Witten},\n\\begin{equation}\n\\bra{0}T^{(c)}_{\\mu\\nu}\\ket{0}=\\mathcal{E}g_{\\mu\\nu}\\,,\n\\end{equation}\nwhere $\\mathcal{E}$ is typically UV divergent. Since the Hamiltonian density is identified\nas $\\mathscr{H}=T_{00}$, it follows that $\\mathcal{E}$ is the vacuum energy density.\nHowever, one is always free to define a new subtracted energy-momentum tensor,\n\\begin{equation}\nT_{\\mu\\nu}\\equiv T^{(c)}_{\\mu\\nu}-\\mathcal{E} g_{\\mu\\nu}\\,,\n\\end{equation}\nwhich is a Lorentz-covariant expression.\\footnote{For example, in the quantum theory\nof free fields, the vacuum energy is set to zero by defining the Hamiltonian density to be\nnormal ordered.}\nBy construction, $\\partial^\\mu T_{\\mu\\nu}=0$ and\n\\begin{equation}\n\\bra{0}T_{\\mu\\nu}\\ket{0}=0\\,.\n\\end{equation}\nThe energy-momentum tensor $T_{\\mu\\nu}$ plays\na distinguished role in relativistic quantum field theory, since it can be used\nto construct the generators of spacetime translations,\n\\begin{equation} \\label{vacp}\nP_\\mu=\\int d^3 x\\ T_\\mu{}^0\\,,\n\\end{equation}\nthat satisfy $\\bra{0}P_\\mu\\ket{0}=0$. Indeed, $P_\\mu$ defined by \\eq{vacp} is a four-vector with respect\nto Lorentz transformations.\nLikewise, one can construct a distinguished angular momentum tensor $M_{\\mu\\nu\\lambda}$ that\ncan be used to construct the generators of Lorentz transformations\n\\begin{equation}\nJ_{\\mu\\nu}=\\int d^3 x M_{\\mu\\nu}{}^0\\,,\n\\end{equation}\nwhich satisfy $\\bra{0}J_{\\mu\\nu}\\ket{0}=0$.\n\nHowever, in a supersymmetric theory, another choice of the energy-momentum tensor is\nnatural. The fermionic generators $Q_\\alpha$ and $Q^{\\dagger\\dot\\alpha}$ of the SUSY algebra\nare time-independent (conserved) quantities that are obtained by integrating the zeroth component\nof the supercurrents,\n\\begin{equation} \\label{Qint}\nQ_\\alpha=\\int d^3 x J_\\alpha^0\\,,\\qquad\\qquad Q^{\\dagger\\dot\\alpha}=\\int d^3 x J^{\\dagger\\dot\\alpha\\,0}\\,.\n\\end{equation}\nIn a theory governed by a supersymmetric Lagrangian, the supercurrents $J_\\alpha^\\mu$ and $J^{\\dagger\\dot\\alpha\\,\\mu}$\nare related by supersymmetry to\nan energy-momentum tensor, denoted by $T^{(\\rm SUSY)}_{\\mu\\nu}$.\nThen, the proper interpretation of \\eq{QQanti} is~\\cite{deWit}\n\\begin{equation}\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}\\int d^3 x\\, T^{(\\rm SUSY)}{}_\\mu{}^0\\,.\n\\end{equation}\nOne can then rewrite the above anticommutation relation as:\n\\begin{equation} \\label{revisedQQ}\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu+2E_0\\sigma^0_{\\alpha\\dot\\beta}\\,,\n\\end{equation}\nwhere $P_\\mu$ is defined by \\eq{vacp} and\n\\begin{equation} \\label{Ezero}\nE_0\\equiv\\int d^3 x\\,\\bra{0}T^{(\\rm SUSY)}{}_0{}^0\\ket{0}\\,.\n\\end{equation}\nIf $E_0=0$ (which corresponds to $T^{(\\rm SUSY)}_{\\mu\\nu}=T_{\\mu\\nu}$),\nthen we recover the standard SUSY algebra, and the\nvacuum is supersymmetric. If $E_0\\neq 0$, then \\eq{revisedQQ} is consistent with $\\bra{0}P^\\mu\\ket{0}=0$\n(which is required by the Lorentz-invariant vacuum) and with $Q_\\alpha\\ket{0}\\neq 0$. In particular,\n$E_0$ serves as an order parameter for broken supersymmetry.\n\nNote that $E_0\\geq 0$ since \\eq{revisedQQ} implies that:\n\\begin{equation}\nE_0=\\tfrac{1}{4}\\bra{0}Q_1 Q_1^\\dagger+Q_1^\\dagger Q_1+Q_2 Q_2^\\dagger+Q_2^\\dagger Q_2\\ket{0}\\geq 0\\,.\n\\end{equation}\nIn supersymmetric theories, it is common to call $E_0$ the vacuum energy. Thus, if supersymmetry is\nspontaneously broken, then this definition of the vacuum energy is not compatible with usual\nconventions of quantum field theory in which the vacuum energy is defined to be zero.\n\n\nAlthough the conclusions obtained above are correct, the derivation of \\eq{revisedQQ} is still somewhat formal.\nIndeed\nif the vacuum breaks supersymmetry, then the integrals in \\eq{Qint} do not converge when integrated\nover an infinite volume (this is an infrared divergence), so strictly\nspeaking the fermionic generators $Q_\\alpha$ and $Q^{\\dagger\\dot\\alpha}$ are undefined.\\footnote{Moreover,\ngiven a non-zero value for $\\bra{0}T^{(\\rm SUSY)}{}_0{}^0\\ket{0}$, which is a constant by\ntranslational invariance, one sees that $E_0$ defined in \\eq{Ezero} also diverges in the infinite volume limit.}\nNevertheless,\nthe supercurrents are conserved, as expected in a supersymmetric theory with no \\textit{explicit}\nsupersymmetry breaking. In section~\\ref{goldstino}, we will demonstrate that given a supersymmetric\nLagrangian, if the vacuum breaks supersymmetry\nthen a massless Goldstone fermion exists in the spectrum. The long range forces mediated by\nthis massless particle are responsible for the non-convergence of the integrals in \\eq{Qint}.\nEquivalently, in a spontaneously-broken globally supersymmetric theory, applying $Q_\\alpha$ to the vacuum\ncreates a zero-momentum massless fermionic state, which is a state of infinite norm~\\cite{weinberg3}.\n\n\\subsubsection{Equality of bosonic and fermionic degrees of freedom in\n super\\-symmetric theories}\n\nIn a theory governed by a supersymmetric action, for a fixed non-zero $P_\\mu$ the number of bosonic\nand fermionic degrees of freedom coincide. To prove this result, we first observe that the application\nof $Q_\\alpha$ or $Q^\\dagger_{\\dot\\alpha}$ to a physical state changes that state by adding half a unit of spin.\nAn explicit example of this behavior can be seen in \\eqs{massiveplet2}{massiveplet3}. We can summarize\nthis behavior in the following schematic equations,\n\\begin{equation}\nQ_\\alpha\\ket{B}=\\ket{F}\\,,\\qquad\\quad Q_\\alpha\\ket{F}=\\ket{B}\\,,\n\\end{equation}\nand similarly for the application of $Q^\\dagger_{\\dot\\alpha}$, where $\\ket{B}$ is a bosonic state\nand $\\ket{F}$ is a fermionic state. It is convenient to introduce an operator, denoted by $(-1)^F$,\nwith the following properties:\n\\begin{equation}\n(-1)^F\\ket{B}=\\ket{B}\\,,\\qquad\\quad (-1)^F\\ket{F}=-\\ket{F}\\,.\n\\end{equation}\nNote that\n\\begin{Eqnarray}\nQ_\\alpha(-1)^F\\ket{F}&=&-Q_\\alpha\\ket{F}=-\\ket{B}\\,, \\\\\n(-1)^F Q_\\alpha\\ket{F}&=&(-1)^F\\ket{B}=\\ket{B}\\,,\n\\end{Eqnarray}\nand similarly for the application of $Q^\\dagger_{\\dot\\alpha}$. It follows that $Q_\\alpha$\n[and $Q^\\dagger_{\\dot\\alpha}$] anticommute with $(-1)^F$,\n\\begin{equation} \\label{minusF}\n\\{Q_\\alpha\\,,\\,(-1)^F\\}=\\{Q^\\dagger_{\\dot\\alpha}\\,,\\,(-1)^F\\}=0\\,.\n\\end{equation}\n\nUsing \\eq{minusF}, we can evaluate the following trace over physical states,\n\\begin{Eqnarray}\n\\Tr\\left[(-1)^F\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}\\right]&=&\n\\Tr\\left[(-1)^F(Q_\\alpha Q^\\dagger_{\\dot\\beta}+Q^\\dagger_{\\dot\\beta}Q_\\alpha)\\right] \\nonumber \\\\\n&=& \\Tr\\left[-Q_\\alpha (-1)^F Q^\\dagger_{\\dot\\beta}+(-1)^F Q^\\dagger_{\\dot\\beta}Q_\\alpha\\right]\\nonumber \\\\\n&=& \\Tr\\left[-Q^\\dagger_{\\dot\\beta}Q_\\alpha (-1)^F+Q^\\dagger_{\\dot\\beta}Q_\\alpha (-1)^F\\right]\\nonumber \\\\\n&=&0\\,,\n\\end{Eqnarray}\nafter a cyclic permutation within the trace at the penultimate step.\nEmploying \\eq{susyalg5},\nwe conclude that\n\\begin{equation} \\label{traceF}\n\\Tr (-1)^F=0\\,,\\qquad \\text{for any fixed non-zero $P^\\mu$}\\,.\n\\end{equation}\nFor a fixed non-zero eigenvalue $p^\\mu$ obtained by applying the\nmomentum operator $P^\\mu$ to a physical\nstate, \n\\begin{equation}\n\\Tr (-1)^F=\\sum_{\\{r\\}} \\bra{p^\\mu,\\{r\\}}(-1)^F\\ket{p^\\mu,\\{r\\}}=N_B(p^\\mu)-N_F(p^\\mu)=0\\,,\n\\end{equation}\nwhere $\\{r\\}$ indicates all other quantum numbers of the physical state. Thus, the number of\nbosonic ($N_B$) and fermionic ($N_F$) degrees of freedom coincide.\n\nWe have already observed that \\eq{minusF} is satisfied by all\npositive energy representations of the SUSY algebra. The\nproof above demonstrates that the equality of bosonic and fermionic\ndegrees of freedom in supersymmetric theories is far more general.\nIndeed, the only case where this equality can break down is when~$P^\\mu=0$,\ncorresponding to the vacuum state of the supersymmetric theory.\\footnote{For\nexample, Witten showed that in an SU($N$) supersymmetric Yang-Mills\ntheory, $\\Tr (-1)^F=N$ for the supersymmetric ground\nstate~\\cite{Witten2}.}\n\n\n\n\\subsection{Supersymmetric theories of spin-0 and spin-\\ifmath{\\tfrac12}\\ particles}\n\nThe simplest supermultiplet contains a complex scalar and a\ntwo-component (Majorana) fermion, of common mass $m$. The case of\n$m\\neq 0$ corresponds to superspin $j\\!=\\!0$ and the case of $m\\!=\\!0$\ncorresponds to superhelicity $\\ifmath{\\tfrac12}$ and its CPT-conjugate.\n\n\\subsubsection{The Wess-Zumino Lagrangian}\n\nA Lagrangian that respects the SUSY algebra is given by\n\\begin{align}\n\\mathscr{L}=(\\partial_\\mu A)^\\dagger(\\partial^\\mu A)+ i \\psi^\\dagger \\overline{\\sigma}^\\mu \\partial_\\mu \\psi-\\left|\\frac{dW}{dA}\\right|^2-\\frac12\\left[\\frac{d^2 W}{dA^2}\\,\\psi\\psi+\\left(\\frac{d^2 W}{dA^2}\\right)^{\\!\\!\\dagger}\\!\\!\\psi^\\dagger\\psi^\\dagger\\right]\\,,\n\\label{eq:LWZoriginal}\n\\end{align}\nwhere $A$ is a complex scalar,\\footnote{Employing $A$ for a complex\n scalar field rather than $\\phi$ follows the notation first introduced\n in Ref.\\cite{WessBagger}. It should not be confused with the\n notation for a vector field, which will henceforth be denoted\n by $V$.} \n$\\psi$ and $\\psi^\\dagger$ are two-component spinors, and $W=W(A)$\n[called the \\textit{superpotential}]\nis a holomorphic function of $A$ (\\textit{i.e.}, a function of $A$ and \\textit{not}~$A^\\dagger$).\nIf $W(A)$ is (at most) a cubic polynomial in $A$, then the above\nLagrangian yields a renormalizable \nquantum field theory called the \\textit{Wess-Zumino model}.\nFor example, a simple quadratic superpotential,\n$W=\\ifmath{\\tfrac12} mA^2$, describes\na free theory of a complex scalar and a Majorana fermion of common mass $|m|$.\nAn interacting theory is\nobtained by including a cubic term in the superpotential,\n\\begin{align}\nW=\\ifmath{\\tfrac12} mA^2+\\tfrac{1}{3}g A^3\\,.\\label{wcubic}\n\\end{align}\nWithout loss of generality, we can assume that $m$ and $g$ are\nnon-negative (by appropriate rephasing of $A$ and $\\psi$). Then,\ninserting \\eq{wcubic} into \\eq{eq:LWZoriginal} yields the Wess-Zumino\nLagrangian,\n\\begin{align}\n\\begin{split}\n\\mathscr{L}&=\n(\\partial_\\mu A)^\\dagger(\\partial^\\mu A)+ i \\psi^\\dagger \\overline{\\sigma}^\\mu \\partial_\\mu \\psi-\\ifmath{\\tfrac12} m(\\psi\\psi+\\psi^\\dagger\\psi^\\dagger)-m^2(A^\\dagger A)\n \\\\\n&\\quad -g(A\\psi\\psi+A^\\dagger\\psi^\\dagger\\psi^\\dagger)-mg(A^\\dagger A)(A+A^\\dagger)-g^2(A^\\dagger A)^2 \\,.\n\\end{split} \\label{wzlag}\n\\end{align}\nAs expected, the boson and fermion are mass-degenerate. Moreover, SUSY imposes relations among the couplings. In this model, we see that the quartic scalar coupling is the square of the Yukawa (scalar-fermion-fermion) coupling. \n\n\n\nIn order to employ four-component Feynman rules, it is convenient to\nconvert the Wess-Zumino Lagrangian into four-component fermion form.\nWriting $A=(S+iP)\/\\sqrt{2}$, where $S$ and $P$ are hermitian fields, we obtain\n\\begin{align}\n\\begin{split}\n\\mathscr{L}&=\\ifmath{\\tfrac12} (\\partial_\\mu S)^2+\\ifmath{\\tfrac12} (\\partial_\\mu P)^2-\\ifmath{\\tfrac12} m^2(S^2+P^2)+\\ifmath{\\tfrac12} \\overline\\Psi_M(i\\gamma^\\mu\\partial_\\mu-m)\\Psi_M\n \\\\\n&\\quad -\\frac{g}{\\sqrt{2}}\\left[S\\Psi_M\\psi_M-iP\\Psi_M\\gamma\\ls{5}\\Psi_M\\right]-\\frac{mg}{\\sqrt{2}}S(S^2+P^2)\n-\\tfrac{1}{4}g^2(S^2+P^2)^2\\,.\n\\end{split}\n\\end{align}\nNote that this Lagrangian separately conserves C, P and T. We identify $S$ as a scalar and $P$ as a pseudoscalar.\n\n\\subsubsection{Invariance of the Wess-Zumino Lagrangian with respect\n to SUSY transformations}\n\nThe Wess-Zumino Lagrangian given by \\eq{wzlag} is invariant with respect to global supersymmetry transformations. Explicitly, these transformations depend on an\ninfinitesimal Grassmann (anticommuting) two-component spinor parameter $\\xi$ that is independent of the spacetime position $x$,\n\\begin{align}\n\\delta_\\xi A &= \\sqrt{2}\\,\\xi \\psi\\,,\\label{susytr1}\\\\\n\\delta_\\xi\\psi_\\alpha\n&=\n- i\\sqrt{2} (\\sigma^\\mu \\xi^\\dagger)_\\alpha\\> \\partial_\\mu A-\\sqrt{2}\\,\\xi_\\alpha\\left(\\frac{dW}{dA}\\right)^{\\!\\!\\dagger}\\,.\\label{susytr2}\n\\end{align}\nBy hermitian conjugation, one also obtains\n\\begin{align}\n\\delta_\\xi A^\\dagger &= \\sqrt{2}\\,\\xi^\\dagger \\psi^\\dagger\\,,\\label{susytr3}\n \\\\\n\\delta_\\xi\\psi^\\dagger_{\\dot{\\alpha}}\n&=\n i \\sqrt{2}(\\xi\\sigma^\\mu)_{\\dot{\\alpha}}\\> \\partial_\\mu A^\\dagger-\\sqrt{2}\\,\\xi^\\dagger_{\\dot\\alpha}\\left(\\frac{dW}{dA}\\right)\\,.\\label{susytr4}\n \\end{align}\n Applying these transformation laws to \\eq{wzlag}, one obtains a result of the form\n \\begin{align}\n \\delta_{\\xi}\\mathscr{L}=\\partial_\\mu K^\\mu\\,.\n \\label{eq:Kmu}\n \\end{align}\n That is, the action of the Wess-Zumino Model, $S=\\int d^4 x\\,\\mathcal{L}$, is invariant under global SUSY transformations; \\textit{i.e.}, $\\delta_{\\xi} S=0$.\n \n\n \n But, how do we know that the transformation laws just introduced correspond to SUSY transformations? Recall that for ordinary spacetime translations,\n \\begin{equation}\n e^{ia\\!\\cdot\\! P}\\Phi(x)e^{-ia\\!\\cdot\\! P}=\\Phi(x+a)\\,,\n \\end{equation}\n which in infinitesimal form is given by\n \\begin{equation}\n i\\bigl[P^\\mu\\,,\\,\\Phi(x)\\bigr]=\\partial^\\mu\\Phi(x)\\,,\n \\end{equation}\n where $\\Phi=A$ or $\\psi$. Equivalently, for an infinitesimal translation, \n \\begin{equation}\n \\delta_a\\Phi(x)\\equiv\\Phi(x+a)-\\Phi(x)\\simeq a^\\mu\\partial_\\mu\\Phi(x)=ia^\\mu\\bigl[P^\\mu\\,,\\,\\Phi(x)\\bigr]\\,.\n \\end{equation}\n Likewise, since $Q$ and $Q^\\dagger$ are the generators of SUSY-translations, we expect\n \\begin{equation} \\label{susytranslate}\n \\delta_{\\xi}\\Phi(x)=i\\bigl[\\xi Q+\\xi^\\dagger Q^\\dagger\\,,\\,\\Phi(x)\\bigr]\\,.\n \\end{equation}\nConsider the commutator of two SUSY-translations:\n \\begin{Eqnarray}\n (\\delta_{\\eta}\\delta_{\\xi}-\\delta_{\\xi}\\delta_{\\eta})\\Phi(x)&=&\\biggl[i(\\eta Q+\\eta^\\dagger Q^\\dagger)\\,,\\,\n \\bigl[i(\\xi Q+\\xi^\\dagger Q^\\dagger)\\,,\\,\\Phi(x)\\bigr]\\biggr] \n -(\\xi\\longleftrightarrow\\eta)\\nonumber\n \\\\\n &=&\\biggl[\\bigl[i(\\eta Q+\\eta^\\dagger Q^\\dagger)\\,,\\,i(\\xi Q+\\xi^\\dagger Q^\\dagger)\\bigr]\\,,\\,\\Phi(x)\\biggr]\\,,\n \\end{Eqnarray}\n after employing the Jacobi identity for the double commutators. Using the SUSY algebra,\n $$\n \\bigl[\\eta Q\\,,\\,\\xi^\\dagger Q^\\dagger\\bigr]=2(\\eta\\sigma^\\mu\\xi^\\dagger) P_\\mu\\,.\n $$\n Note that the anticommutator has been converted into a commutator due to the fact that $\\eta$ and $\\xi$ are anticommuting two-component spinors. Likewise, \n $$\n \\bigl[\\eta Q\\,,\\,\\xi Q\\bigr]=\\bigl[\\eta^\\dagger Q^\\dagger\\,,\\,\\xi^\\dagger Q^\\dagger\\bigr]=0\\,.\n $$\n Hence, we end up with \n \\begin{Eqnarray}\n \\bigl[\\delta_{\\eta}\\,,\\,\\delta_{\\xi}\\bigr]\\Phi(x)&=&2(\\xi\\sigma^\\mu\\eta^\\dagger-\\eta^\\dagger\\sigma^\\mu\\xi^\\dagger)\\bigl[P_\\mu\\,,\\,\\Phi(x)\\bigr] \\nonumber\n \\\\\n&=& -2i(\\xi\\sigma^\\mu\\eta^\\dagger-\\eta^\\dagger\\sigma^\\mu\\xi^\\dagger)\\partial_\\mu\\Phi(x)\\,.\n \\end{Eqnarray}\nLikewise, a similar computation yields,\n \\begin{Eqnarray}\n \\bigl[\\delta_{\\eta}\\,,\\,\\delta_{\\xi}\\bigr]A(x)&=&-2i(\\xi\\sigma^\\mu\\eta^\\dagger-\\eta^\\dagger\\sigma^\\mu\\xi^\\dagger)\\partial_\\mu A(x)\\,, \n \\\\[6pt]\n \\bigl[\\delta_{\\eta}\\,,\\,\\delta_{\\xi}\\bigr]\\psi_\\alpha(x)&=&-2i(\\xi\\sigma^\\mu\\eta^\\dagger-\\eta^\\dagger\\sigma^\\mu\\xi^\\dagger)\\partial_\\mu \\psi_\\alpha +R\\,, \\label{eq:Remainder}\n \\end{Eqnarray}\n where the remainder $R$ vanishes after imposing the classical field\n equations for $\\psi_\\alpha(x)$, as you will verify in Problem \\ref{pr:R}.\nWe conclude that the SUSY algebra is realized \\textit{on-shell}, \\textit{i.e.}, after employing the classical field equations.\n \n It is instructive to employ\n Noether's theorem, which states that an invariance of the action under\n a continuous symmetry implies the existence of a conserved current.\n Since we have explicitly identified the SUSY\n transformations, we can \nuse Noether's theorem to determine the corresponding conserved supercurrent. Using $\\delta_{\\xi}\\mathscr{L}=\\partial_\\mu K^\\mu$, the resulting conserved Noether supercurrents are\n \\begin{align}\n \\xi^\\alpha J_\\alpha^\\mu+\\xi^\\dagger_{\\dot\\alpha} J^{\\dagger\\,\\mu\\dot\\alpha}=\\sum_\\Phi \\delta_{\\xi}\\Phi\\,\\frac{\\delta\\mathscr{L}}{\\delta(\\partial_\\mu \\Phi)}-K^\\mu\\,,\n \\end{align}\n where the sum is taken over $\\Phi=A$, $\\psi$. Note that the supercurrent has both a Lorentz index and a spinor index.\n %\n Noether's theorem states that the supercurrent is conserved \\textit{after imposing the classical field equations}. That is, \n\\begin{equation}\n \\partial_\\mu J^\\mu_\\alpha=\\partial_\\mu J^{\\dagger\\,\\mu\\dot\\alpha}=0\\,.\n\\end{equation}\n \nThe supercharges are defined in the usual way (as previously noted):\n \\begin{align} \nQ_\\alpha=\\int d^3 x J_\\alpha^0\\,,\\qquad\\qquad Q^{\\dagger\\dot\\alpha}=\\int d^3 x J^{\\dagger\\dot\\alpha\\,0}\\,.\\label{QJ}\n\\end{align}\nThese are expressions that depend on the fields $A$ and $\\psi$.\nOne can now employ the canonical commutation relations of the boson field $A$ and the canonical anticommutation relations of the fermion field $\\psi$ to verify that \n\\begin{equation} \\label{QandCCR}\n\\{Q_\\alpha\\,,\\,Q_\\beta\\}=\\{Q^\\dagger_{\\dot\\alpha}\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=0\\,,\\qquad\\quad\n\\{Q_\\alpha\\,,\\,Q^\\dagger_{\\dot\\beta}\\}=2\\sigma^\\mu_{\\alpha\\dot\\beta}P_\\mu\\,,\n\\end{equation}\nwhere $P_\\mu$ is the Noether charge of spacetime translations given in \\eq{vacp}.\n\n \n \n\\subsection{The SUSY algebra realized off-shell}\n\\label{offshell}\n\nThe SUSY transformation laws of the Wess-Zumino Lagrangian exhibited in \\eqs{susytr1}{susytr2} are not in an\noptimal form for two reasons. First, in the case of a cubic\nsuperpotential $W(A)$, the transformation law for $\\psi_\\alpha$ is non-linear in the fields.\nSecond, the SUSY algebra is only realized on-shell.\nWe can address both these issues by introducing an auxiliary complex scalar field $F(x)$.\nConsider the alternative Lagrangian,\n\\begin{Eqnarray}\n\\mathscr{L}&=&(\\partial_\\mu A)^\\dagger(\\partial^\\mu A)+ i \\psi^\\dagger \\overline{\\sigma}^\\mu \\partial_\\mu \\psi\n+F^\\dagger F \n+F\\,\\frac{dW}{dA}+F^\\dagger\\left(\\frac{dW}{dA}\\right)^{\\!\\!\\dagger} \\nonumber \\\\\n&& -\\frac12\\left[\\frac{d^2 W}{dA^2}\\,\\psi\\psi+\\left(\\frac{d^2 W}{dA^2}\\right)^{\\!\\!\\dagger}\\!\\!\\psi^\\dagger\\psi^\\dagger\\right]\\,.\n\\label{eq:LWZF}\n\\end{Eqnarray}\nThe field $F(x)$ is auxiliary since $\\mathscr{L}$ does not depend on $\\partial_\\mu F$ and\n$\\partial_\\mu F^\\dagger$. That is, $F$ and $F^\\dagger$ are non-dynamical fields.\n\nWe can trivially solve for $F$ and $F^\\dagger$ using the classical field equations,\n\\begin{align}\n\\frac{\\partial\\mathscr{L}}{\\partial F}&=0\\qquad\\Longrightarrow \\qquad F^\\dagger=-\\frac{dW}{dA}\\,,\\label{fs}\n\\\\\n\\frac{\\partial\\mathscr{L}}{\\partial\n F^\\dagger}&=0\\qquad\\Longrightarrow\\qquad F=\n-\\left(\\frac{dW}{dA}\\right)^{\\!\\!\\dagger}\\,.\\label{f}\n\\end{align}\nHence, \\eqs{fs}{f} yield,\n\\begin{equation}\nF^\\dagger F+F\\,\\frac{dW}{dA}+F^\\dagger\\left(\\frac{dW}{dA}\\right)^{\\!\\!\\dagger}=-\\left|\\frac{dW}{dA}\\right|^2\\,.\n\\end{equation}\nPlugging this result back into \\eq{eq:LWZF},\nwe recover the general form of the Wess-Zumino Lagrangian given by \\eq{eq:LWZoriginal}.\n\nThe Lagrangian including the auxiliary fields given by \\eq{eq:LWZF} is\nalso invariant under SUSY translations. The appropriately modified\nSUSY transformation laws are now given by\n\\begin{align}\n\\delta_\\xi A &= \\sqrt{2}\\,\\xi \\psi\\,,\\label{offshell1}\n\\\\\n\\delta_\\xi\\psi_\\alpha\n&=\n- i\\sqrt{2} (\\sigma^\\mu \\xi^\\dagger)_\\alpha\\> \\partial_\\mu A+\\sqrt{2}\\,\\xi_\\alpha F\\,,\\label{offshell2}\n\\\\\n\\delta_{\\xi} F&=-i\\sqrt{2}\\,\\xi^\\dagger\\overline{\\sigma}^\\mu\\partial_\\mu\\psi\\,.\\label{offshell3}\n\\end{align}\nBy hermitian conjugation, one also obtains\n\\begin{align}\n\\delta_\\xi A^\\dagger &= \\sqrt{2}\\,\\xi^\\dagger \\psi^\\dagger\\,,\n \\\\\n\\delta_\\xi\\psi^\\dagger_{\\dot{\\alpha}}\n&=\n i \\sqrt{2}(\\xi\\sigma^\\mu)_{\\dot{\\alpha}}\\> \\partial_\\mu A^\\dagger+\\sqrt{2}\\,\\xi^\\dagger_{\\dot\\alpha}F^\\dagger\\,,\n \\\\\n\\delta_{\\xi} F^\\dagger&=i\\sqrt{2}(\\partial_\\mu\\psi^\\dagger)\\overline{\\sigma}^\\mu\\xi\\,.\n \\end{align}\n Applying these transformation laws to \\eq{eq:LWZF}, one obtains a result of the form\n \\begin{align}\n \\delta_{\\xi}\\mathscr{L}=\\partial_\\mu K^{\\prime\\,\\mu}\\,,\n \\label{eq:Kpmu}\n \\end{align}\nwhere the explicit form for $K^{\\prime\\,\\mu}$ is to be determined in\nProblem~\\ref{pr:kprime}. Moreover, as you will verify in Problem \\ref{pr:xieta},\n \\begin{align}\n \\bigl[\\delta_{\\eta}\\,,\\,\\delta_{\\xi}\\bigr]\\Phi(x)=-2i(\\xi\\sigma^\\mu\\eta^\\dagger-\\eta^\\dagger\\sigma^\\mu\\xi^\\dagger)\\partial_\\mu \\Phi(x)\\,,\n \\end{align}\n for $\\Phi=A$, $\\psi$ and $F$ \\textit{without} the need to impose the classical field equations.\nThus, the Wess-Zumino Lagrangian with auxiliary fields included as in\n\\eq{eq:LWZF} is invariant under SUSY translations, and the SUSY algebra is realized \\textit{off-shell}, \\textit{i.e.}, without requiring that the fields satisfy their classical field equations.\n\nThe following two observations will be particularly useful as we move\nforward. First, note that the mass dimensions of the fields are given\nby $[A]=1$, $[\\psi]=\\tfrac{3}{2}$ and $[F]=2$, which is consistent with\nthe requirement that $[\\mathscr{L}]=4$ (since the action is\ndimensionless in units of $\\hbar=1$). Then,\n\\eqst{offshell1}{offshell3} are dimensionally consistent if $[\\xi]=\\ifmath{\\tfrac12}$.\n Second, note that $\\delta_{\\xi} F$ given in \\eq{offshell3} is a total\n derivative. Indeed, $\\delta_{\\xi} F$ is a total derivative as a consequence of dimensional analysis and the linearity of the SUSY transformation laws. This implies that $\\delta_{\\xi} F$ must involve $\\partial_\\mu$, since $[\\partial_\\mu]=1$.\nAn important consequence of this observation is that\n$\\int \\!d^4x\\, F$ is invariant under SUSY transformations. \n \n \n\\subsection{Counting bosonic and fermionic degrees of freedom}\nIt is instructive to count both the on-shell and off-shell bosonic and\nfermionic degrees of freedom in the Wess-Zumino model, which\nis a theory of a complex scalar and a\ntwo-component fermion. \n\nA complex scalar possesses two real\ndegrees of freedom. Note that applying the classical field equations\n(in this case the inhomogeneous Klein-Gordon equation) does not affect\nthe number of scalar\ndegrees of freedom, but only the spacetime dependence of the scalar\nfield. The two-component fermion $\\psi_\\alpha$ possesses two complex degrees of\nfreedom, which yields four real degrees of\nfreedom.\\footnote{Equivalently, we can count $\\psi$ and $\\psi^\\dagger$ as four independent degrees of freedom.} \nApplying the classical field equations,\n\\begin{equation} \\label{diraceq}\ni\\overline{\\sigma}^\\mu\\partial_\\mu\\psi=\\left(\\frac{d^2 W}{dA^2}\\right)^{\\!\\!\\dagger}\\psi^\\dagger\\,,\n\\end{equation}\nwhich relate $\\psi$ and $\\psi^\\dagger$, thereby eliminating two of the\nfour degrees of freedom.\\footnote{If $d^2 W\/dA^2=0$, then\n $i\\overline{\\sigma}^\\mu\\partial_\\mu\\psi=0$ yields a relation between \n $\\psi_1$ and $\\psi_2$.}\nBy taking the derivative of \\eq{diraceq}, one can eliminate\n$\\psi^\\dagger$ using the hermitian conjugate of \\eq{diraceq}.\nThe resulting equation for $\\psi$ is the inhomogeneous Klein-Gordon\nequation, which does not further affect the number of\nfermionic degrees of freedom.\nThus, the Wess-Zumino model possesses two on-shell bosonic and two fermionic\ndegrees of freedom. \n\nThe counting of the off-shell degrees of freedom can be performed by examining the\nLagrangian [\\eq{eq:LWZF}] expressed in terms of the propagating and\nauxiliary fields. In this case, we count two real degrees of freedom for\nthe complex scalar, four real degrees of freedom for the two-component\nfermion and two real degrees of freedom for the complex auxiliary\nfield~$F$. That is, the Wess-Zumino model possesses four bosonic and four fermionic\noff-shell degrees of freedom. \n\nThus, the number of bosonic and fermionic degrees of freedom match in both on-shell and off-shell counting.\n\n\\subsection{Lessons from the Wess-Zumino Model}\n\nIn our study of the Wess-Zumino model, we provided a Lagrangian that\nincorporated the fields of a known supermultiplet. However, it was\nrather mysterious how this Lagrangian was obtained. It was\neven more mysterious how we came up with the correct SUSY\ntransformation laws for the various fields. \nMoreover, it was quite laborious to verify that the proposed SUSY\ntransformation laws satisfy the SUSY algebra and the action is \ninvariant under super-Poincar\\'e transformations.\n\nWe also learned that in order for the SUSY transformation laws to\nrespect the SUSY algebra off-shell, one must introduce additional\nauxiliary fields. One additional benefit of doing so is that the \ncorresponding SUSY transformation laws are now linear in all the fields. \nFor this reason, we introduced the auxiliary field $F$, which can be\nused to write down the SUSY translation-invariant quantity $\\int\\! d^4\nx \\, F(x)$. This observation actually provides an important clue for how to\nconstruct a SUSY Lagrangian.\n\nAs we shall demonstrate in Section~\\ref{sec:superspace}, it is possible to develop a formalism in which, starting with\na known supermultiplet, one can trivially construct a Lagrangian that\nis invariant under super-Poincar\\'e transformations. Moreover, this\nformalism will provide explicit forms for the SUSY transformation\nlaws that automatically respect the SUSY algebra.\n\n\n\n\\subsection{\\mbox{Appendix: Constructing the states of a supermultiplet}}\n\\label{App}\n\nIn this subsection, we provide further details on the construction of\nthe states of the massive and massless supermultiplets, which yields\nthe results presented in Tables~\\ref{massivesuperplet} and\n\\ref{masslesssuperplet}.\n\n\\subsubsection{States of a massive supermultiplet of superspin $j$}\n \n\nTo construct the states of the massive supermultiplet,\nwe note that in the rest frame, the anticommutators given in \\eqs{susyalg4}{susyalg5}\nsimplify to\n\\begin{Eqnarray}\n\\{Q_1\\,,\\,Q^\\dagger_1\\}&=&\\{Q_2\\,,\\,Q^\\dagger_2\\}=2m\\,,\\label{restframeQQ}\\\\\n\\{Q_1\\,,\\,Q_1\\}&=&\\{Q_2\\,,\\,Q_2\\}=\\{Q_1\\,,\\,Q_2\\}=0\\,,\\label{restframeanti} \\\\\n \\{Q^\\dagger_1\\,,\\,Q^\\dagger_1\\}&=&\\{Q^\\dagger_2\\,,\\,Q^\\dagger_2\\}=\\{Q^\\dagger_1\\,,\\,Q^\\dagger_2\\}=0\\,.\n \\end{Eqnarray}\n All states in a supermultiplet with superspin $j$ are simultaneous eigenstates of $P^2$,\n $\\mathcal{J}^i \\mathcal{J}^i$ and $\\mathcal{J}^3$ with eigenvalues $m^2$, $j(j+1)$ and\n $j_3$, respectively, where the possible values of $j_3$\n are $-j,-j+1,\\ldots,j-1,j$. \n\nFor a fixed value of the superspin $j$,\n there exists a distinguished state of the supermultiplet that is a \n simultaneous eigenstate of $P^2$, $\\mathcal{J}^i\\mathcal{J}^i$ and $\\mathcal{J}^3$,\n denoted by $\\ket{\\Omega}$, which satisfies\\footnote{Recall that if $\\ket{s,m_s}$ are eigenstates\n of $\\boldsymbol{\\vec S}\\llsup{\\,2}$ and $S^3$ with corresponding eigenvalues $s(s+1)$ and $m_s$\n respectively, then\n $$\n S_{\\pm}\\ket{s,m_s}=\\sqrt{(s\\mp m_s)(s\\pm m_s+1)}\\ket{s,m_s\\pm 1}\\,.\n $$\n }\n \\begin{equation} \\label{Omegastate}\n Q_\\beta\\ket{\\Omega}=0\\,,\\qquad\\quad S_+\\ket{\\Omega}=0\\,,\n \\end{equation}\n where $S_{\\pm}\\equiv S^1\\pm iS^2$.\n To verify that a state $\\ket{\\Omega}$ exists that is annihilated by $Q_\\beta$,\nlet us assume the contrary. Suppose that\n a simultaneous eigenstate of $P^2$, $\\mathcal{J}^i\\mathcal{J}^i$ and $\\mathcal{J}^3$,\n denoted by $\\ket{\\Psi}$, is not annihilated by $Q_\\beta$. In the rest frame, \\eq{BQQ}\n yields\n \\begin{equation} \\label{JQQ}\n [\\mathcal{J}^i\\,,\\,Q_\\beta]=[\\mathcal{J}^i\\,,\\,Q^\\dagger_{\\dot\\beta}]=0\\,,\n \\end{equation}\n so it follows that $Q_\\beta\\ket{\\Psi}$ is also a simultaneous eigenstate of $P^2$, $\\mathcal{J}^i\\mathcal{J}^i$\n and $\\mathcal{J}^3$. By assumption, $Q_\\beta\\ket{\\Psi}$ is not annihilated by $Q_\\alpha$, so we\n conclude that $Q_\\alpha Q_\\beta\\ket{\\Psi}$ is also a simultaneous eigenstate of $P^2$, $\\mathcal{J}^i\\mathcal{J}^i$\n and $\\mathcal{J}^3$. But we now arrive at a contradiction, since \\eq{restframeanti} yields\n \\begin{equation}\n Q_\\gamma\\left(Q_\\alpha Q_\\beta\\ket{\\Psi}\\right)=0\\,.\n \\end{equation}\n Consequently, there must be at least one state of the supermultiplet that satisfies\n$Q_\\beta\\ket{\\Omega}=0$. Using \\eqs{caljdef}{Omegastate}, it follows that\n\\begin{equation}\n \\mathcal{J}^i\\ket{\\Omega}=S^i\\ket{\\Omega}\\,.\n \\end{equation}\n If $S_+\\ket{\\Omega}=0$, then it follows that $\\ket{\\Omega}$ is also\n a simultaneous eigenstate of $\\boldsymbol{\\vec{S}}\\llsup{\\,2}$ and $S^3$ with corresponding eigenvalues\n $j(j+1)$ and $j$. Moreover, this state must be unique under the assumption that the\n superspin $j$ supermultiplet is an \\textit{irreducible} representation of the $N=1$ supersymmetry\n algebra.\n\n Note that \\eq{wQQ} when evaluated in the rest frame yields:\n \\begin{equation} \\label{siQcomm}\n [S^i\\,,\\,Q_\\alpha]=i\\sigma^{i0}{}_\\alpha{}^\\beta Q_\\beta\\,,\\qquad\\quad\n [S^i\\,,\\,Q^\\dagger_{\\dot\\alpha}]=i\\overline{\\sigma}^{i0\\dot\\beta}{}_{\\dot\\alpha}Q^\\dagger_{\\dot\\beta}\\,.\n \\end{equation}\nHence, one can define additional states of the supermultiplet,\n\\begin{equation}\n\\ket{\\Omega(j_3)}\\equiv (S_-)^{j-j_3}\\ket{\\Omega}\\,,\\qquad\\quad\\text{for}~~j_3=-j,-j+1,\\ldots,j-1,j\\,,\n\\end{equation}\nall of which satisfy\n\\begin{equation} \\label{Omegastatej3}\nQ_\\alpha\\ket{\\Omega(j_3)}=0\\,,\n\\end{equation}\nas a result of \\eq{siQcomm}. As before, $\\mathcal{J}^i\\ket{\\Omega(j_3)}=S^i\\ket{\\Omega(j_3)}$ as\na consequence of \\eqs{caljdef}{Omegastatej3}. It follows that\n$\\ket{\\Omega(j_3)}$ is also a simultaneous eigenstate of $\\boldsymbol{\\vec{S}}\\llsup{\\,2}$ and $S^3$ with corresponding eigenvalues\n $j(j+1)$ and~$j_3$. That is,\n \\begin{equation} \\label{Omegajj3}\n \\ket{\\Omega(j_3)}=\\ket{j,j_3}\\,,\n \\end{equation}\n where the rest-frame spin and its projection along the $z$-axis are explicitly indicated.\n\nStarting from $\\ket{\\Omega(j_3)}=\\ket{j,j_3}$, one can now construct the remaining states of the massive supermultiplet\nby considering the series of states for each possible value of $j_3$,\n $\n \\ket{\\Omega(j_3)}\\,,\\, Q^\\dagger_{\\dot\\alpha}\\ket{\\Omega(j_3)}\\,,\\, Q^\\dagger_{\\dot\\alpha}Q^\\dagger_{\\dot\\beta}\\ket{\\Omega(j_3)}\\,,\\,\\ldots\\,.\n $\nThis series of states terminates due to \\eq{restframeanti} and only four independent states survive (for a given fixed value of $j_3$),\n\\begin{equation} \\label{mstatesj}\n\\ket{\\Omega(j_3)}\\,,\\quad Q^\\dagger_1\\ket{\\Omega(j_3)}\\,,\\quad Q^\\dagger_2\\ket{\\Omega(j_3)}\\,,\\quad\n Q^\\dagger_1Q^\\dagger_2\\ket{\\Omega(j_3)}\\,.\n \\end{equation}\n\nAll the states of \\eq{mstatesj} are mass-degenerate (with mass $m\\neq 0$). The spins of\nthese states can be determined by applying the operators $\\boldsymbol{\\vec S}\\llsup{\\,2}$ and $S^3$.\nBy virtue of \\eq{Omegajj3}, we already know that $\\ket{\\Omega(j_3)}$ is a spin-$j$ state\nwith $S^3$-eigenvalue $j_3$. Next, one can use \\eq{siQcomm} to derive:\n\\begin{Eqnarray}\n[S^i\\,,\\,Q^\\dagger_{\\dot\\alpha}Q^\\dagger_{\\dot\\beta}]&=&iQ^\\dagger_{\\dot\\gamma}\\left[\\overline{\\sigma}^{i0\\dot\\gamma}{}_{\\dot\\alpha}\nQ^\\dagger_{\\dot\\beta}-\\overline{\\sigma}^{i0\\dot\\gamma}{}_{\\dot\\beta}Q^\\dagger_{\\dot\\alpha}\\right]\\,,\\\\\n\\left[\\boldsymbol{\\vec S}\\llsup{\\,2}\\,,\\,Q^\\dagger_{\\dot\\alpha}Q^\\dagger_{\\dot\\beta}\\right]&=&2iQ^\\dagger_{\\dot\\gamma}\\left[\n\\overline{\\sigma}^{i0\\dot\\gamma}{}_{\\dot\\alpha} Q^\\dagger_{\\dot\\beta}\n-\\overline{\\sigma}^{i0\\dot\\gamma}{}_{\\dot\\beta} Q^\\dagger_{\\dot\\alpha}\\right]S^i\\,.\n\\end{Eqnarray}\nIt immediately follows that:\n\\begin{Eqnarray}\n[S^i\\,,\\,Q_1^\\dagger Q_2^\\dagger]&=&iQ_1^\\dagger Q_2^\\dagger\\Tr \\sigma^{i0}=0\\,,\\label{SiQ1Q2}\\\\\n\\left[\\boldsymbol{\\vec S}\\llsup{\\,2}\\,,\\,Q_1^\\dagger Q_2^\\dagger\\right]&=&2iQ_1^\\dagger Q_2^\\dagger S^i\\Tr \\sigma^{i0}=0\\,.\n\\label{S2Q1Q2}\n\\end{Eqnarray}\nApplying \\eqs{SiQ1Q2}{S2Q1Q2} to the state $\\ket{\\Omega(j_3)}$, it follows that $Q_1^\\dagger Q_2^\\dagger\n\\ket{\\Omega(j_3)}$ is also a spin-$j$ state with $S^3$-eigenvalue $j_3$.\nThis result is easily understood. Noting that we can write\n\\begin{equation}\nQ_1^\\dagger Q_2^\\dagger=\\ifmath{\\tfrac12}\\epsilon^{\\dot\\alpha\\dot\\beta}Q^\\dagger_{\\dot\\alpha}Q^\\dagger_{\\dot\\beta}\\,,\n\\end{equation}\nit follows that $Q_1^\\dagger Q_2^\\dagger$ is a \\textit{scalar} operator. This is consistent with the\nfact that the antisymmetric part of the tensor product of two SU(2) spinor representations is an SU(2) singlet.\nThus, $Q_1^\\dagger Q_2^\\dagger\\ket{\\Omega(j_3)}$ and $\\ket{\\Omega(j_3)}$ possess the same eigenvalues\nwith respect to $\\boldsymbol{\\vec S}\\llsup{\\,2}$ and $S^3$.\n\nTo determine the properties of $Q_1^\\dagger\\ket{\\Omega(j_3)}$ and $Q_2^\\dagger\\ket{\\Omega(j_3)}$,\nwe first note that $Q_\\alpha$ is a spinor operator \nthat imparts spin-$\\ifmath{\\tfrac12}$ to any state it acts on. \nMoreover, \\eq{siQcomm} yields:\n\\begin{equation} \\label{Qdaghalf}\n\\hspace{-0.2in}\nS^3 Q_1^\\dagger\\ket{\\Omega(j_3)}=(j_3+\\ifmath{\\tfrac12})Q_1^\\dagger\\ket{\\Omega(j_3)}\\,,\n\\quad\nS^3 Q_2^\\dagger\\ket{\\Omega(j_3)}=(j_3-\\ifmath{\\tfrac12})Q_2^\\dagger\\ket{\\Omega(j_3)}.\n\\end{equation}\nHence, one can employ the standard results from the theory of angular momentum addition in quantum mechanics,\nwhich relates the tensor product basis to the total angular momentum basis. In particular,\n\\begin{equation}\n\\ket{j\\,,\\,m}=\\sum_{m_1,m_2}\\ket{j_1\\,,\\,m_1}\\otimes\\ket{j_2\\,,\\,m_2}\\vev{j_1\\,\\, j_2\\,;\\, m_1\\,\\, m_2\\,|\\,j\\,\\, m}\\,,\n\\end{equation}\nwhere $\\vev{j_1\\,\\, j_2\\,;\\, m_1\\,\\, m_2\\,|\\,j\\,\\, m}$ are the\nClebsch-Gordon (C-G) coefficients. We employ the Condon-Shortly\nphase conventions in which the C-G coefficients are real and symmetric. In the present application,\nwe require the following two C-G coefficients (taking the upper and lower\nsigns, respectively),\n\\begin{align}\n\\ket{\\ifmath{\\tfrac12}\\,,\\, \\pm\\ifmath{\\tfrac12}}\\otimes\\ket{j\\,,\\,\n m\\mp\\ifmath{\\tfrac12}}=&\\left(\\frac{j+\\ifmath{\\tfrac12}\\pm\n m}{2j+1}\\right)^{1\/2}\\!\\ket{j+\\ifmath{\\tfrac12}\\,,\\, m}\\nonumber \\\\\n&\\mp\\left(\\frac{j+\\ifmath{\\tfrac12}\\mp m}{2j+1}\\right)^{1\/2}\\!\\ket{j-\\ifmath{\\tfrac12}\\,,\\, m}, \n\\end{align}\nEqs.~(\\ref{Omegajj3}), (\\ref{Qdaghalf}), (\\ref{SiQ1Q2}) and\n(\\ref{S2Q1Q2}) imply that\n\\begin{align}\n& \\ket{\\Omega(j_3)}=\\ket{j\\,,\\,j_3}\\,, \\label{massiveplet1}\\\\\n& Q_1^\\dagger \\ket{\\Omega(j_3)}=\\left(\\frac{j+j_3+1}{2j+1}\\right)^{1\/2}\\ket{j+\\ifmath{\\tfrac12}\\,,\\,j_3+\\ifmath{\\tfrac12}}\n-\\left(\\frac{j-j_3}{2j+1}\\right)^{1\/2}\\ket{j-\\ifmath{\\tfrac12}\\,,\\,j_3+\\ifmath{\\tfrac12}}\\,, \\label{massiveplet2} \\\\\n& Q_2^\\dagger \\ket{\\Omega(j_3)}=\\left(\\frac{j-j_3+1}{2j+1}\\right)^{1\/2}\\ket{j+\\ifmath{\\tfrac12}\\,,\\,j_3-\\ifmath{\\tfrac12}}\n+\\left(\\frac{j+j_3}{2j+1}\\right)^{1\/2}\\ket{j-\\ifmath{\\tfrac12}\\,,\\,j_3-\\ifmath{\\tfrac12}}\\, , \\label{massiveplet3}\\\\\n& Q_1^\\dagger Q_2^\\dagger \\ket{\\Omega(j_3)}=\\ket{j\\,,\\,j_3}\\,.\\label{massiveplet4}\n\\end{align}\nIn particular, if $j_3\\neq j$ then \\eqs{massiveplet2}{massiveplet3} imply that $Q_1^\\dagger \\ket{\\Omega(j_3)}$\nand $Q_2^\\dagger \\ket{\\Omega(j_3)}$ are orthogonal linear combinations of spin-($j\\pm\\ifmath{\\tfrac12}$) states\n(although these states are eigenstates of $S^3$ as shown in \\eq{Qdaghalf}). If\n$j_3=\\pm j$ then $Q_1^\\dagger \\ket{\\Omega(j)}$ and $Q_2^\\dagger \\ket{\\Omega(-j)}$\nare states of spin-($j+\\ifmath{\\tfrac12}$), since both these states\nare eigenstates of $\\boldsymbol{\\vec S}\\llsup{\\,2}$ and $S^3$ with eigenvalues $(j+\\ifmath{\\tfrac12})(j+\\tfrac{3}{2})$\nand $\\pm(j+\\ifmath{\\tfrac12})$, respectively.\n\n\nNote that since $[P^2,Q_\\alpha]=[P^2,Q^\\dagger_{\\dot\\alpha}]=0$, it follows that \nall the states of the supermultiplet,\n$\\ket{\\Omega(j_3)}\\,,\\,Q^{\\dagger\\,1}\\ket{\\Omega(j_3)}\\,,\\,Q^{\\dagger\\,2}\\ket{\\Omega(j_3)}\\,,\\,Q^{\\dagger\\,1}\nQ^{\\dagger\\,2}\\ket{\\Omega(j_3)}$, are mass-degenerate, with common\nmass $m$.\nThe states of an $N=1$ massive supermultiplet of superspin $j$ are exhibited\nin Table~\\ref{massivesuperplet}.\n\nIn summary, there are $4(2j+1)$ mass-degenerate states in a massive\nsupermultiplet of superspin $j$, which are explicitly given by\n\\eqst{massiveplet1}{massiveplet4}, for $j_3=-j,-j+1,\\ldots,j-1,j$. In\ngeneral, a massive supermultiplet of superspin $j$ is made up of\n$2(2j+1)$ states of spin $j$, $2j+2$ states of spin $(j+\\ifmath{\\tfrac12})$ and\n$2j$ states of spin ($j-\\ifmath{\\tfrac12}$). The extra two states for the case of\nspin-$(2j+1)$ arise when $j_3=\\pm j$, in which cases $Q_1^\\dagger\n\\ket{\\Omega(j)}$ and $Q_2^\\dagger \\ket{\\Omega(-j)}$ are pure states of\nspin $(j+\\ifmath{\\tfrac12})$ as previously noted. Note that the number of\nfermionic and bosonic degrees of freedom of the massive supermultiplet\ncoincide and is equal to $2(2j+1)$. These results are summarized in Table~\\ref{massivesuperplet}.\n\n\n\\subsubsection{States of a massless supermultiplet of superhelicity $\\kappa$}\nTo construct the states of an irreducible massless supermultiplet, we\nchoose the standard reference frame, $P^\\mu=P^0(1\\,;\\,0\\,,\\,0\\,,\\,1)$, \nfor lightlike four-vectors.\nIn this reference frame, the anticommutators given in \\eqs{susyalg4}{susyalg5}\nsimplify to those exhibited in \\eqst{qqmassless1}{qqmassless3}. All the\nstates in the massless supermultiplet are\nsimultaneous eigenstates of $P^2$ and the superhelicity operator $\\mathcal{K}$, with eigenvalues\n$m^2$ and $\\kappa$, respectively, where the possible values of\n$\\kappa=0,\\pm\\ifmath{\\tfrac12},\\pm 1,\\pm\\tfrac32,\\ldots$.\n\nFor a fixed value of the superhelicity $\\kappa$, there exists a distinct state of the supermultiplet,\n denoted by $\\ket{\\Omega}$, that satisfies:\n \\begin{equation} \\label{Omegastate0}\n Q_\\beta\\ket{\\Omega}=0\\,,\\qquad\\quad \\mathcal{K}\\ket{\\Omega}=\\kappa\\ket{\\Omega}\\,.\n \\end{equation}\nTo verify that a state $\\ket{\\Omega}$ exists that is annihilated by $Q_\\beta$,\nlet us assume the contrary. \nSuppose that a state of\n the massless supermultiplet, denoted by $\\ket{\\Psi}$ exists that is not annihilated by\n$Q_\\beta$. Due to \\eq{KQQ}, it follow that $Q_\\beta\\ket{\\Psi}$ must also be a state of\nthe massless supermultiplet. Arguing as we did below \\eq{Omegastate}, we again arrive at a contradiction.\nConsequently, there must be at least one state of the supermultiplet that satisfies\n$Q_\\beta\\ket{\\Omega}=0$. Moreover, a state that satisfies\n\\eq{Omegastate0} must be unique under the assumption that the\n massless supermultiplet with superhelicity $\\kappa$ is an \\textit{irreducible} representation of the $N=1$ SUSY\n algebra.\n\nThe states of the massless supermultiplet are obtained by considering the series,\n\\begin{equation}\n\\ket{\\Omega}\\,,\\, Q^\\dagger_{\\dot\\alpha}\\ket{\\Omega}\\,,\\,Q^\\dagger_{\\dot\\beta} Q^\\dagger_{\\dot\\alpha}\\ket{\\Omega}\\,.\n\\end{equation}\nHowever, $Q^\\dagger_{\\dot\\beta} Q^\\dagger_{\\dot\\alpha}\\ket{\\Omega}=0$ as a result of\n\\eq{QQQQ0}, and $P^\\lambda Q^\\dagger_{\\dot\\beta}\\overline{\\sigma}_\\lambda^{\\dot\\beta\\tau}\\ket{\\Omega}=0$\nas a consequence of \\eq{zeroops}. Thus, in contrast to the massive\nsupermultiplet, the massless supermultiplet contains only two states.\nThese two states are eigenvalues of the helicity operator $h$.\nTo determine the corresponding helicities, we shall employ\nthe standard reference frame where $P^\\mu=P^0(1\\,;\\,0\\,,\\,0\\,,\\,1)$.\nSince \\eq{zeroops} yields $Q_1=Q^\\dagger_1=0$, it follows that the massless $N=1$ supermultiplet consists of the\ntwo states, $\\ket{\\Omega}$ and $Q^\\dagger_2\\ket{\\Omega}$. Using \\eqs{mathcalkdef}{Omegastate0},\nthe helicities of these two states can be determined,\n\\begin{Eqnarray}\nh\\ket{\\Omega}&=&\\left[\\mathcal{K}-\\frac{1}{8P^0}\\left(Q_1^\\dagger Q_1+Q_2^\\dagger Q_2\\right)\\right]\\ket{\\Omega}\n=\\kappa\\ket{\\Omega}\\,,\\label{helicityOmega}\\\\[8pt]\nhQ^\\dagger_2\\ket{\\Omega}&=&\\left[\\mathcal{K}-\\frac{1}{8P^0}\\left(Q_1^\\dagger Q_1+Q_2^\\dagger Q_2\\right)\\right]Q^\\dagger_2\\ket{\\Omega} \\nonumber \\\\\n&=& \\left[\\kappa Q_2^\\dagger-\\frac{1}{8P^0} Q_2^\\dagger\\left(2P_\\mu\\sigma^\\mu_{22}-Q_2^\\dagger Q_2\\right) \\right. \\nonumber \\\\\n&&\\qquad\\quad\n-\\left.\\!\\!\\frac{1}{8P^0} Q_1^\\dagger\\left(2P_\\mu\\sigma^\\mu_{12}-Q_2^\\dagger Q_1\\right)\\right]\\ket{\\Omega} \\nonumber \\\\\n&=& \\left[\\kappa-\\tfrac{1}{4}(\\sigma^0_{22}-\\sigma^3_{22})\\right]Q^\\dagger_2\\ket{\\Omega}\n=(\\kappa-\\ifmath{\\tfrac12}) Q^\\dagger_2\\ket{\\Omega}\\,.\\label{helicityQomega}\n\\end{Eqnarray}\nIndeed, the superhelicity $\\kappa$ is the maximal helicity of the massless $N=1$ supermultiplet.\nThus, an irreducible $N=1$ massless supermultiplet with superhelicity $\\kappa$ consists of two massless\nstates with helicity $\\kappa$ and $\\kappa-\\ifmath{\\tfrac12}$, respectively.\nThese results are summarized in Table~\\ref{masslesssuperplet}.\n\n\n\\subsection{Problems}\n\n\n\n\\begin{problem}\n\\label{pr:jhalf}\nShow that the massive $j=\\ifmath{\\tfrac12}$ supermultiplet corresponds to a real vector field, a real scalar field and a Dirac fermion field.\n\\end{problem}\n\n\\begin{problem}\nDerive the following three commutation relations:\n\\begin{equation}\n[B^\\mu\\,,\\,Q_\\alpha]=-\\ifmath{\\tfrac12} P^\\mu Q_\\alpha\\,,\\qquad\\qquad\n[B^\\mu\\,,\\,Q^\\dagger_{\\dot\\alpha}]=\\ifmath{\\tfrac12} P^\\mu\nQ^\\dagger_{\\dot\\alpha}\\,,\n\\end{equation}\n\\begin{equation}\n[B^\\mu\\,,\\,B^\\nu]=i\\epsilon^{\\mu\\nu\\rho\\lambda}B_\\rho P_\\lambda\\,,\n\\end{equation}\nwhere $B^\\mu$ is defined in \\eq{bmudef}.\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:Lcomms}\nDerive the following twp commutation relations,\n\\begin{align} \n[L^\\mu\\,,\\,Q_\\alpha]=-\\tfrac{1}{4}(\\sigma^\\mu\\overline{\\sigma}^\\nu)_\\alpha{}^\\beta Q_\\beta P_\\nu\\,,\\qquad\\quad\n[L^\\mu\\,,\\,Q^\\dagger_{\\dot\\alpha}]=\\tfrac{1}{4}(\\overline{\\sigma}^\\nu\\sigma^\\mu)^{\\dot\\beta}{}_{\\dot\\alpha}\n Q^\\dagger_{\\dot\\beta}P_\\nu\\,,\n\\end{align}\nwhere $L^\\mu$ is defined in \\eq{Lmudef}.\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:spin2}\nShow that a massless supermultiplet with $\\kappa=2$ and \nits CPT-conjugates corresponds to\na massless spin-$\\tfrac{3}{2}$ and a massless spin 2 particle, which\nis realized in supergravity by the\ngravitino and the graviton. \n\\end{problem}\n\n \\begin{problem}\nObtain the explicit form for $K^\\mu$ in \\eq{eq:Kmu}.\n\\end{problem}\n\\clearpage\n\n \\begin{problem}\n \\label{pr:R}\nObtain an explicit expression for $R(x)$ in \\eq{eq:Remainder}, and\nshow that it vanishes after imposing the classical field equations for\n$\\psi_\\alpha(x)$. Note that this computation is non-trivial and\nrequires a judicious application of \nFierz identities for two-component fermions (which can be found, e.g.,\nin Appendix B of Ref.~\\cite{Dreiner:2008tw}).\n\\end{problem}\n\n %\n\\begin{problem}\nObtain an explicit expression for $J^\\mu_\\alpha$ in terms of the fields $A$ and $\\psi$ in the Wess-Zumino model.\n\\end{problem}\n\n\n\\begin{problem}\nVerify, for the Wess-Zumino model, \nthat the Noether supercharges defined by \\eq{QJ} satisfy the SUSY algebra [cf.~\\eq{QandCCR}].\n\\end{problem}\n\n\\begin{problem}\n\\label{pr:kprime}\nObtain the explicit form for $K^{\\prime\\,\\mu}$ in \\eq{eq:Kpmu}.\n\\end{problem}\n\n\n\n\\begin{problem}\n\\label{pr:xieta}\nStarting from \\eqst{offshell1}{offshell3}, verify that\n \\begin{align*}\n \\bigl[\\delta_{\\eta}\\,,\\,\\delta_{\\xi}\\bigr]\\Phi(x)=-2i(\\xi\\sigma^\\mu\\eta^\\dagger-\\eta^\\dagger\\sigma^\\mu\\xi^\\dagger)\\partial_\\mu \\Phi(x)\\,,\n \\end{align*}\n for $\\Phi=A$, $\\psi$ and $F$ without the need to impose the classical field equations.\n \\end{problem}\n\n\n\\chapter*{Supersymmetric Theory and Models\n \\vspace{-24pt}\n\\else\n \\chapter[Supersymmetric Theory and Models]{Supersymmetric Theory and Models}\n\\fi\n\n\n\n\\author[]{Howard E.~Haber\\textsuperscript{1} and Laurel Stephenson Haskins\\textsuperscript{1,2}\n}\n\n\\address{\n\\textsuperscript{1}Santa Cruz Institute for Particle Physics,\\\\\nUniversity of California, Santa Cruz, CA 95064, USA\\\\\n\\vspace{6pt}\n\\textsuperscript{2}Racah Institute of Physics,\\\\\nHebrew University, Jerusalem 91904, Israel \n}\n\n\\begin{abstract}\nIn these introductory lectures, we review the theoretical tools used in constructing supersymmetric field theories and their application to physical models. \nWe first introduce the technology of two-component spinors, which is convenient for describing spin-$\\ifmath{\\tfrac12}$ fermions. After motivating why a theory of nature may be supersymmetric at the TeV energy scale, we show how supersymmetry (SUSY) arises as an extension of the Poincar\\'e algebra of spacetime symmetries. We then obtain \nthe representations of the SUSY algebra and discuss its simplest realization in the Wess-Zumino model. \nIn order to have a systematic approach for obtaining supersymmetric Lagrangians, we \nintroduce the formalism of superspace and superfields and recover the Wess-Zumino Lagrangian. These methods are then extended to encompass supersymmetric abelian and non-abelian gauge theories coupled to supermatter.\nSince supersymmetry is not an exact symmetry of nature, it must ultimately be broken. We discuss several mechanisms of SUSY-breaking (both spontaneous and explicit) and briefly survey various proposals for realizing SUSY-breaking in nature.\nFinally, we construct the\nthe Minimal Supersymmetric extension of the Standard Model (MSSM), and \nconsider the implications for the future of SUSY in particle physics.\n\\end{abstract}\n\\body\n\n\\tableofcontents\n\n\\section{Introduction to the TASI-2016 Supersymmetry Lectures}\n\\label{Intro}\nThese lectures were first presented at the 2016 Theoretical Advanced Study Institute (TASI-2016) in Boulder, CO.\nFour ninety-minute lectures were given, with the aim of presenting the basic theoretical techniques of supersymmetry\nneeded for the construction of a supersymmetric extension of the Standard Model of particle physics. \nThe lectures were pitched at an elementary level, assuming that\nthe students were well versed in quantum field theory, gauge theory and the Standard Model, but with no assumed prior knowledge \nof supersymmetry. Nevertheless, some aspects of these lectures may also be useful to the reader with some prior \nknowledge of supersymmetry.\n\nIt is possible to introduce the technology of supersymmetry theory using four-component spinor notation that is familiar to all students of quantum field theory. However, it is our view that employing two-component spinor notation greatly simplifies the presentation of the theoretical structure of supersymmetry in 3$+$1 spacetime dimensions.\nThus, in Section~\\ref{sec:spinhalf}, we introduce the two-component spinor notation in some detail and discuss how it is related to the better known four-component spinor notation.\nThis material is based heavily on a comprehensive review of Dreiner, Haber and Martin that is presented in Ref.\\cite{Dreiner:2008tw}. In this review, it is shown that practical calculations in quantum field theory can be carried out entirely within the framework of the two-component spinor notation, which include the development of Feynman rules for two-component spinors. However, at the end of Section 1, we are slightly less ambitious and revert to four-component fermion notation for the purpose of computing scattering and decay processes. In particular, we provide a translation between two and four-component spinor notation, and develop four-component spinor Feynman rules that treat both Dirac and Majorana fermions on the same footing.\n\nIn Section~\\ref{sec:motivation}, we present the motivation for TeV-scale supersymmetry. Namely, why is it that we feel compelled to introduce a supersymmetric extension of the Standard Model, despite the great success of the Standard Model in describing collider data and the absence of significant evidence for new physics beyond the Standard Model.\nWith this motivation in mind, we are ready to explore the theoretical aspects of supersymmetry.\n\nSince this is not a review article, we do not feel compelled to present a comprehensive list of references. Nevertheless, it is instructive to assemble a list of books and lecture notes on supersymmetry, many of which we have found quite useful in preparing these lectures. Thus, we draw your attention to the following books listed in Refs.\\cite{WessBagger,Gates,Srivastava,Piguet,Freund,MullerKirsten,West,Lopuszanski,Bailin,Buchbinder,Soni,Galperin,Polonsky,Mohapatra,Drees,Baer,Aitchison,Binetruy,Terning,MullerKirsten2,Labelle,Shifman,sugra1,weinberg3,MDine,Manoukian,sugra2,Raby} and the following reviews and lecture notes listed in Ref.\\cite{Taylor:1983su,Nilles:1983ge,Haber:1984rc,Sohnius:1985qm,Lahanas:1986uc,Haber:1993wf,Derendinger,Lykken,Martin:1997ns,Giudice:1998bp,bilalsusy,Petrov:2001hz,FigueroaO'Farrill:2001tr,Chung:2003fi,Luty:2005sn,RamseyMusolf:2006vr,Shirman:2009mt,GKane,susy,Bertolini:2013via}. The reader is warned that conventions vary widely among these references. Apart from the two possible choices for the spacetime metric (either the mostly minus metric used in these lectures or the mostly plus metric), there are many different choices in the definition of a variety of quantities, often involving different choices of signs. Of these many conventions, we believe that the ones employed in these lecture notes are probably closest to those that appear in Ref.\\cite{Sohnius:1985qm}.\\footnote{We also note that although Ref.\\cite{Martin:1997ns} employs the mostly plus metric, one can obtain a version of Martin's Supersymmetry Primer in the mostly minus metric by changing one line in the LaTeX source code. This alternative version of the Primer closely matches the conventions employed in these lectures.}\n\nIn Section~\\ref{sec:SUSYalgebra}, we show how the algebra of the Poincar\\'e group can be extended to obtain the supersymmetry (SUSY) algebra. The representations of the $N=1$ SUSY\nalgebra are elucidated, and the Wess-Zumino model is presented as the simplest realization of a supersymmetric field theory. In Section~\\ref{sec:superspace}, we take some of the mystery out of constructing a SUSY Lagrangian by introducing the concepts of superspace and superfields. This formalism allows one to construct supersymmetric field theories without any guesswork. In Section~\\ref{sec:gaugetheories}, the formalism of supersymmetric gauge theories is developed. In Section~\\ref{SSB}, we examine supersymmetry breaking, which is necessary for accommodating the observation that the elementary particles observed today are not each accompanied by an equal-mass superpartner.\nFinally, in Section~\\ref{sec:MSSM}, we construct the Minimal Supersymmetric extension of the Standard Model (MSSM). \nWe end these lectures in Section~\\ref{sec:future} with a brief discussion of what lies ahead for supersymmetry.\n\n\n\n\\input{Sections\/spin_half_fermions}\n\\input{Sections\/SUSYMotivations}\n\\input{Sections\/SUSY_first_steps}\n\\input{Sections\/Superspace_superfields}\n\\input{Sections\/SUSY_gauge_theories}\n\\input{Sections\/SUSY_breaking}\n\\input{Sections\/SUSYic_extension}\n\\input{Sections\/LHC}\n\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank Zackaria Chacko, Andrew Cohen, Michael Dine, Herbi Dreiner, Stephen Martin, Raman Sundrum, and John Terning for many enlightening discussions.\nH.E.H. is grateful to Rouven Essig and Ian Low for their invitation to present these lectures at TASI 2016, and their patience in waiting for these lecture notes to be completed. This work is supported in part by the U.S. Department of Energy grant number DE-SC0010107.\nL.S.H. is also supported by the Israel\nScience Foundation under grant no.~1112\/17.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}