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\section{Estimation of the background contributions}
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\label{sec:moreDetailsBackground}
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In this section we provide information on an alternative method to estimate
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the background contribution to the \WZ signal where we estimate the instrumental
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background with no genuine \Z boson from the \Z candidate invariant mass
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side-band subtraction. The fit of the invariant mass distribution to the
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Gaussian and linear functions for ``loose'' and ``tight'' \W leptons is given
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in Fig.~\ref{fig:AllFits} using the full statistics of the CSA07 samples.
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The fit is performed using an addition of a convolution of a Gaussian and Breit-Wigner function
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and a line in order to fit the background. It has to be noticed that due to a lack
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of statistics in ``Chowder soup'' sample, all bins with 0 events from the sample
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have been modified in order to avoid to have a null uncertainty. The
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corresponding uncertainty in the bin with no events correspond to the weight
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of each process in the ``Chowder soup''. One can see that the uncertainties are
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large, and the fit is not really constrained.
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\begin{figure}[hbt]
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\begin{center}
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\scalebox{0.3}{\includegraphics{figs/Fit3eLoose.eps}\includegraphics{figs/Fit3eTight.eps}}\\
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\scalebox{0.3}{\includegraphics{figs/Fit2e1muLoose.eps}\includegraphics{figs/Fit2e1muTight.eps}}\\
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\scalebox{0.3}{\includegraphics{figs/Fit2mu1eLoose.eps}\includegraphics{figs/Fit2mu1eTight.eps}}\\
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\scalebox{0.3}{\includegraphics{figs/Fit3muLoose.eps}\includegraphics{figs/Fit3muTight.eps}}\\
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\caption{Invariant mass of \Z boson candidate for $3e$, $2e1\mu$, $2\mu1e$, and $3\mu$
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signatures (from top to bottom) for the lepton passing loose (left) and tight (right) identification
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criteria.}
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\label{fig:AllFits}
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\end{center}
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\end{figure}
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The linear fit takes into account not only the background with non-genuine
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\Z boson but it also accounts for some part of the \Z+jets and
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$Zb\bar{b}$ background as the $\gamma^*$ processes populate the sidebands as well.
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However, the results are still consistent within errors, as it can be seen by comparison
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of the last two columns in Table~\ref{tab:CompFit}.
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\begin{table}[h]
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\begin{center}
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\begin{tabular}{|l|c|c|c|c|c|c|c|} \hline
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& \multicolumn{2}{c|}{Background with genuine \Z} & \multicolumn{4}{c|}{Background without
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genuine \Z boson} \\
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Channel & $\Z+jets$ & $\Z b\bar{b}$ & $t\bar{t}$ & $\W+jets$ & Combined & Fit result \\ \hline
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$3e$ Loose &7.1 & 2.9 & 1.1 & 0.4 & 1.5 & 1.5$ \pm $3.0 \\\hline
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$3e$ Tight &2.0 & 1.2 & 0.6 & 0.4 & 1.0 & 1.1$ \pm $2.8 \\\hline
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$2e1\mu$ Loose &4.0 & 4.7 & 6.2 & 0.0 & 6.2 & 6.0$ \pm $4.1 \\\hline
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$2e1\mu$ Tight &0.0 & 0.1 & 0.7 & 0.0 & 0.7 & 1.0$ \pm $2.8 \\\hline
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$2\mu 1e$ Loose &10.1 & 2.9 & 0.8 & 0.0 & 0.8 & 1.6$ \pm $3.1 \\\hline
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$2\mu 1e$ Tight &1.8 & 1.3 & 0.6 & 0.0 & 0.6 & 1.0$ \pm $2.7 \\\hline
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$3\mu$ Loose &4.5 & 4.2 & 5.9 & 0.0 & 5.9 & 3.1$ \pm $3.5 \\\hline
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$3\mu$ Tight &0.1 & 0.3 & 0.3 & 0.0 & 0.3 & 0.5$ \pm $2.5 \\\hline
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\end{tabular}
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\end{center}
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\caption{Comparison between Monte Carlo truth information and the results
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of the fit for the background without genuine \Z boson. Number of events are
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obtained in the invariant mass range between 81 and 101 GeV. The ``loose'' and ``tight''
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selection criteria applied on the \W lepton candidate.}
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\label{tab:CompFit}
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\end{table}
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The comparison between estimated background and the MC truth information is provided
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in Table~\ref{tab:FinalXCLoose} for ``loose'' and \ref{tab:FinalXC} for ``tight'' lepton candidates.
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Within uncertainties the results agree with each other for every signature and every category of
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\W lepton identification. The agreement between predicted and MC truth background
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as function of the dilepton mass is given in Figs.~\ref{fig:FinalMatrix3e}-\ref{fig:FinalMatrix3mu}
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for all four signal categories respectively.
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\begin{table}[hbt]
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\begin{center}
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\begin{tabular}{lcccc} \hline \hline
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& 3e &2e1$\mu$ &2$\mu$1e &3$\mu$\\ \hline
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$N$ - ZZ -Z$\gamma$ &19.6$\pm$1.2&23.9$\pm$0.7&23.1$\pm$1.1&25.9$\pm$0.8\\ \hline
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$N^{non genuine~Z}$ (Fit) & 1.5$\pm$3.0& 6.0$\pm$4.1& 1.6$\pm$3.1& 3.1$\pm$3.5\\ \hline
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$N^{genuine~Z}$ (matrix method)&11.6$\pm$6.6&10.5$\pm$5.4&14.5$\pm$6.6&12.6$\pm$4.7\\ \hline
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$N^{\WZ}$ & 6.6$\pm$7.4& 7.3$\pm$6.8& 7.0$\pm$7.4&10.2$\pm$6.0\\ \hline
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\WZ from MC & 8.1 & 9.0 & 9.2 &11.3\\
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\hline
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\end{tabular}
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\caption{Expected number of selected events for an integrated luminosity of 300
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pb$^{-1}$ for the signal and estimated background with 81 GeV $< M_Z < $ 101 GeV
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with the full selection criteria applied but the requirement on the \W lepton which is
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required to pass only ``loose'' criteria.}
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\label{tab:FinalXCLoose}
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\end{center}
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\end{table}
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\begin{table}[hbt]
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\begin{center}
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\begin{tabular}{lcccc} \hline \hline
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& 3e &2e1$\mu$ &2$\mu$1e &3$\mu$\\ \hline
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$N$ - ZZ -Zgamma &12.1$\pm$1.1 &8.9$\pm$0.6 &12.8$\pm$1.0 &10.8$\pm$0.7\\ \hline
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$N^{non genuine Z}$ (Fit) & 1.1$\pm$2.8 &1.0$\pm$2.8 & 1.0$\pm$2.7 & 0.5$\pm$2.5\\ \hline
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$N^{genuine Z}$ (matrix method)& 3.7$\pm$1.8 &0.6$\pm$0.8 & 4.6$\pm$2.0 & 0.8$\pm$1.0\\ \hline
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$N^{WZ}$ & 7.3$\pm$3.5 &7.3$\pm$3.0 & 7.1$\pm$3.5 & 9.6$\pm$2.7\\ \hline
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\WZ from MC & 7.9 &8.1 & 9.0 & 10.1\\ \hline
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\end{tabular}
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\caption{Expected number of selected events for an integrated luminosity of 300
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pb$^{-1}$ for the signal and estimated background with 81 GeV $< M_Z < $ 101 GeV for
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the full selection criteria applied.}
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\label{tab:FinalXC}
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\end{center}
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\end{table}
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\begin{figure}[hbt]
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\begin{center}
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\scalebox{0.62}{\includegraphics{figs/MatrixMethod3eLooseTightZmassMWtCut.eps}}
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\caption{Comparison between background predicted with matrix method and MC truth information for the
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$3e$ channel as a function of the \Z boson candidate invariant mass for loose (a, c) and tight (b, d) requirements
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on the \W lepton for background (a, b) and signal (c, d).}
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\label{fig:FinalMatrix3e}
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\end{center}
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\end{figure}
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\begin{figure}[hbt]
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\begin{center}
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\scalebox{0.62}{\includegraphics{figs/MatrixMethod2e1muLooseTightZmassMWtCut.eps}}
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\caption{
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Comparison between background predicted with matrix method and MC truth information for the
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$2e1\mu$ channel as a function of the \Z boson candidate invariant mass for loose (a, c) and tight (b, d) requirements
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on the \W lepton for background (a, b) and signal (c, d).}
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\label{fig:FinalMatrix2e1mu}
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\end{center}
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\end{figure}
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\begin{figure}[hbt]
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\begin{center}
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\scalebox{0.62}{\includegraphics{figs/MatrixMethod2mu1eLooseTightZmassMWtCut.eps}}
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\caption{Comparison between background predicted with matrix method and MC truth information for the
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$2\mu 1e$ channel as a function of the \Z boson candidate invariant mass for loose (a, c) and tight (b, d) requirements
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on the \W lepton for background (a, b) and signal (c, d).}
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\label{fig:FinalMatrix2mu1e}
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\end{center}
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\end{figure}
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\begin{figure}[hbt]
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\begin{center}
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\scalebox{0.62}{\includegraphics{figs/MatrixMethod3muLooseTightZmassMWtCut.eps}}
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\caption{Comparison between background predicted with matrix method and MC truth information for the
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$3\mu$ channel as a function of the \Z boson candidate invariant mass for loose (a, c) and tight (b, d) requirements
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on the \W lepton for background (a, b) and signal (c, d).}
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\label{fig:FinalMatrix3mu}
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\end{center}
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\end{figure}
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