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\appendix
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\section{Additional Cross Check on Background Estimation Studies}
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In figures~\ref{fig:AllFits}, the fit approximation of the invariant
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mass of \Z boson candidate is shown for each channel and for loose and
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tight criteria. The fit is performed using an addition of a
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convolution of a Gaussian and Breit-Wigner function and a line in
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order to fit the background.
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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{Invariante mass of \Z boson candidate for the different samples studied on the left when the lepton pass the loose criteria, on the right when the lepton pass the tight 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 will take into account the background with non-genuine
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\Z candidate but it will also account for some part of \Z+jets and
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$Zb\bar{b}$ background as the gamma$^*$ will populate the side band.
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The comparison can be seen 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 & $\Z+jets$ & $\Z b\bar{b}$ & $t\bar{t}$ & $\W+jets$ & $t\bar{t}$ + $\W+jets$ & Fit result \\ \hline
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$3e$ Loose & 7.1 & 2.8 & 1.1 & 0.4 & 1.5& 4.3 $\pm$ 1.8 \\ \hline
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$3e$ Tight & 1.9 & 1.2 & 0.6 & 0.4 & 1.0 & 3.2 $\pm$ 1.5 \\ \hline
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$2e1\mu$ Loose & 4.0 & 4.7 & 5.3 & 0.0 & 5.3 & 17.3 $\pm$ 3.8\\ \hline
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$2e1\mu$ Tight & 0.0 & 0.1 & 0.6 & 0.0 & 0.6 & 2.8 $\pm$ 1.6\\ \hline
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$2\mu 1e$ Loose & 10.0 & 2.8 & 0.8 & 0.0 & 0.8 & 4.6 $\pm$ 1.9\\ \hline
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$2\mu 1e$ Tight & 1.8 & 1.2 & 0.6 & 0.0 & 0.6 & 3.1 $\pm$ 1.5\\ \hline
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$3\mu$ Loose & 4.7 & 4.1 & 5.4 & 0.0 & 5.4 & 8.8 $\pm$ 2.8\\ \hline
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$3\mu$ Tight & 0.1 & 0.3 & 0.3 & 0.0 & 0.3 & 1.4 $\pm$ 1.2\\ \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 of the fit for the background without genuine \Z boson. Number of events are obtained in the invariant mass range between 81 and 101 GeV. The ``Loose'' and ``Tight'' selection criteria applied for third lepton considered.
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%I AM NOT SURE I UNDERSTAND WHAT IS WRITTEN HERE
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% One has to consider that this study as been perform on a smaller sample than the other part of the analysis a 10\% statistics error as to be counted until the study is performed on the whole samples.
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}
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\label{tab:CompFit}
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\end{table}
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Nevertheless in association with the matrix method the background is
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well estimated as one can see in table~\ref{tab:FinalXC}.
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\begin{table}[h]
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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_{Loose}$ - ZZ -Zgamma &19.7$\pm$1.1 &22.9$\pm$0.7 &22.9$\pm$1.1 &25.6$\pm$0.8 \\
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%$N_{Loose} ^{non genuine Z}$ (Fit) &1.0$\pm$1.5 &11.2$\pm$5.5 &3.1$\pm$2.4 & 4.8$\pm$3.7\\
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$N$ - ZZ -Zgamma &12.2$\pm$1.1 &8.7$\pm$0.6 &12.8$\pm$1.0 &11.1$\pm$0.7\\
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$N^{non genuine Z}$ (Fit)&3.2$\pm$1.5 &2.8$\pm$1.6 &3.1$\pm$1.5 &1.4$\pm$1.2\\
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$N^{genuine Z}$ (matrix method)& 2.8$\pm$1.5 &0.4$\pm$1.1 &4.2$\pm$1.7 &3.7$\pm$1.7\\\hline
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$N^{WZ}$ & 6.2$\pm$2.4 &5.5$\pm$2.0 &5.5$\pm$2.5 &6.0$\pm$2.2\\\hline
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\WZ from MC &7.9&8.0& 8.9 &10.1\\
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\hline
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\end{tabular}
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\caption{Expected number of selected events for an integrated luminosity of 300
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pb$^{-1}$ for the signal and estimated background with 81 GeV $< M_Z < $ 101 GeV.}
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\label{tab:FinalXC}
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\end{center}
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\end{table}
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