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The samples used to train the TaNC neural networks are typical of the signals
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and backgrounds found in common physics analyses using taus. The signal--type
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training sample is composed of reconstucted tau--candidates that are matched to
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generator level hadronic tau decays coming from simulated $Z \rightarrow
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\tau^{+}\tau^{-}$ events. The background training sample consists of reconstructed
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tau--candidates in simulated QCD $2\rightarrow2$ hard scattering events. For
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both signal and background samples, 20\% of the events are reserved as a
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statistically independent sample to evaluate the performance of the neural nets
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after the training is completed. The TaNC uses the ``MLP'' neural network
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implementation provided by the TMVA software package, described in ~\cite{TMVA}.
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The signal and background samples are split into five subsamples corresponding
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to each reconstructed decay mode. An additional selection is applied to each
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subsample by requiring a ``leading pion'': either a charged hadron or gamma
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candidate with transverse momentum greater than 5 GeV$/c$. A large number of
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QCD training events is required as the leading pion selection and the
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requirement that the decay mode match one of the dominant modes given in table
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~\ref{tab:decay_modes} are both effective discriminants. For each subsample,
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10000 signal and background tau--candidates are reserved to be used internally
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by the TMVA software to test for over--training. The number of signal and
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background entries used for each decay mode subsample is given in table
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~\ref{tab:trainingEvents}.
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\begin{table}
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\centering
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\begin{tabular}{lc|c|}
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\multirow{2}{*}{Decay Mode} & \multicolumn{2}{c}{Training Events} \\
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& Signal & Background \\
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\hline
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$\pi^{-}$ & blah & blah \\
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$\pi^{-}\pi^0$ & blah & blah \\
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$\pi^{-}\pi^0\pi^0$ & blah & blah \\
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$\pi^{-}\pi^{-}\pi^{+}$ & blah & blah \\
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$\pi^{-}\pi^{-}\pi^{+}\pi^0$ & blah & blah \\
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\hline
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Total number of events & blah & blah \\
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\hline
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Training preselection efficiency & blah & blah \\
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\end{tabular}
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\label{tab:trainingEvents}
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\caption{Number of events used for neural network training for each
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selected decay mode.}
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\end{table}
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The neural nets use asi nput variables the transverse momentum and $\eta$ of the
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tau--candidates. These variables are included as their correlations with other
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observables can increase the separation power of the ensemble of observables.
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For example, the opening angle in $\Delta R$ for signal tau--candidates is
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inversely related to the transverse momentum, while for background events the
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correlation is very small (see~\cite{DavisTau}). In the training signal and
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background samples, there is significant discrimination power in the $P_T$
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spectrum. However, it is desirable to eliminate any systematic dependence of
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the neural network output on $P_T$ and $\eta$, as a physics analysis that makes use
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of the TaNC algorithm will in general have a different $P_T$ and $\eta$
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spectrum. \fixme(this sentance sucks) The dependence on $P_T$ and $\eta$ is
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removed by applying a $P_T$ and $\eta$ dependent weight to the tau--candidates
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when training the neural nets.
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The weights are defined such that weighted $P_T-\eta$ two--dimensional distributions
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are identical for the signal and background samples after the ``leading pion''
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and decay mode selections have been applied. The weights are defined
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\begin{align}
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W(P_T, \eta) &=& {\rm less}(p_{sig}(P_T, \eta), p_{bkg}(P_T, \eta))\\
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w_{sig}(P_T, \eta) &=& W(P_T, \eta)/p_{sig}(P_T, \eta) \\
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w_{bkg}(P_T, \eta) &=& W(P_T, \eta)/p_{bkg}(P_T, \eta)
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\end{align}
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\fixme(workingpoint)
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