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\subsection{Further cross-checks}
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The test described in the previous Section illustrates the robustness of the
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matrix method to estimate the background correctly for a different
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jet flavor composition in the $\Z+jet$ sample.
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In the following we further scrutinize the details of the background estimation.
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We test the performance of the matrix method on a sample selected with the
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full selection criteria but the requirement on the \W candidate transverse mass.
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We also test a possibility of extracting the background without categorization
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of the instrumental background contributions into genuine/fake \Z bosons.
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\subsubsection{Background estimation without the \W boson transverse mass requirement}
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One of the ways to validate the matrix method is a comparison of its background
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prediction with the MC truth information at different stage of application of the
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\WZ signal selection criteria. We show that the matrix method works well with a very
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loose selection criteria (see the Section above). In the following we perform
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the comparison after applying the full selection criteria but the requirement on the
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\W candidate transverse mass.
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We repeat the procedure described in Section~\ref{sec:moreDetailsBackground} for
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every signature channels and provide the results of background estimation from processes
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without real \Z boson in Table~\ref{tab:FitNoMWt} and final results in
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Tables~\ref{tab:FinalNoMWtCutLoose} and \ref{tab:FinalNoMWtCut} for ``Loose''
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and ``Tight'' requirements on the \W lepton. The results agree with each other
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within one sigma of uncertainty.
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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 &44.6 & 12.7 & 1.6 & 0.4 & 2.0 & 6.6$\pm$4.2 \\\hline
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$3e$ Tight &13.9 & 5.0 & 0.8 & 0.4 & 1.2 & 3.8$\pm$3.5 \\\hline
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$2e1mu$ Loose &41.5 & 78.9 & 12.6 & 0 & 12.6 & 16.9$\pm$5.5 \\\hline
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$2e1mu$ Tight &1.0 & 2.0 & 0.9 & 0 & 0.9 & 1.5$\pm$3.2 \\\hline
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$2mu1e$ Loose &56.3 & 15.4 & 1.9 & 0 & 1.9 & 6.9$\pm$4.4 \\\hline
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$2mu1e$ Tight &17.3 & 5.6 & 0.8 & 0 & 0.8 & 4.1$\pm$2.5 \\\hline
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$3mu$ Loose &43.7 & 84.9 & 12.0 & 0 & 12.0 & 11.0$\pm$5.0 \\\hline
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$3mu$ Tight &0.8 & 2.3 & 0.3 & 0 & 0.3 & 0.8$\pm$2.8 \\\hline
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\end{tabular}
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\end{center}
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\caption{Comparison between Monte Carlo truth information and the results of the fit for the background
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without genuine \Z boson. Number of events are obtained in the invariant mass range between 81 and 101 GeV. The
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``Loose'' and ``Tight'' selection criteria applied on the \W lepton. No requirement is applied on the transverse
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\W candidate mass.}
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\label{tab:FitNoMWt}
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\end{table}
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\begin{table}[h]
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\begin{center}
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\begin{tabular}{lcccc} \hline \hline
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& 3e &2e1$\mu$ &2$\mu$1e &3$\mu$\\ \hline
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$N$ - ZZ -Z$\gamma$ &71.2$\pm$5.7 &147.2$\pm$1.5 & 87.2$\pm$4.7 & 157.7$\pm$2.0\\ \hline
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$N^{non genuine~Z}$ (Fit) &6.6$\pm$4.2 &16.9$\pm$5.5 & 6.9$\pm$ 4.4 & 11.0$\pm$5.0\\ \hline
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$N^{genuine~Z}$ (matrix method) &52.6 $\pm$14.8 &123.2 $\pm$8.8 & 70.3 $\pm$13.1 & 137.0 $\pm$ 8.9\\ \hline
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$N^{\WZ}$ &12.0$\pm$16.4 &7.0 $\pm$10.5 &10.0 $\pm$14.6 & 9.7 $\pm$10.4\\\hline
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\WZ from MC &12.0&14.2& 13.6 &17.2\\
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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 \invpb
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for the signal and estimated background for 81 GeV $< M_Z < $ 101 GeV and for ``Loose''
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\W lepton. No requirement is applied on the transverse \W candidate mass.}
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\label{tab:FinalNoMWtCutLoose}
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\end{center}
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\end{table}
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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$ - ZZ -Z$\gamma$ & 31.8 $\pm$5.5 & 16.3$\pm$1.0 & 36.9$\pm$4.5 & 18.5$\pm$1.3\\ \hline
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$N^{non genuine~Z}$ (Fit) & 3.8 $\pm$3.5 & 1.5$\pm$3.2 & 4.1$\pm$2.5 & 0.8$\pm$2.8\\ \hline
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$N^{genuine~Z}$ (matrix method) & 16.8 $\pm$5.7 & 7.4 $\pm$5.9 & 22.5 $\pm$7.1 & 8.2 $\pm$6.6\\ \hline
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$N^{\WZ}$ & 11.2 $\pm$8.7 & 7.5 $\pm$6.8 & 10.3 $\pm$8.8 & 9.5 $\pm$7.3\\ \hline
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\WZ from MC &11.6&12.3& 13.3 &14.9\\
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\hline
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\end{tabular}
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\caption{Expected number of data events for an integrated luminosity of 300 \invpb for the signal and estimated background for 81 GeV $< M_Z < $ 101 GeV and for ``Tight'' \W lepton. No requirement is applied
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on the transverse \W candidate mass.}
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\label{tab:FinalNoMWtCut}
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\end{center}
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\end{table}
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\subsubsection{Performance of the matrix method without background categorization}
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The performance of the matrix method depends on the validity of the following three assumptions:
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\begin{itemize}
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\item the contribution from processes with two or more misidentified jets is negligible,
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\item $p_{fake}$ should describe the probability of misidentified jets passing loose criteria to also
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pass tight lepton requirements in the background to the signal,
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\item the misidentified lepton is associated with the \W candidate decay.
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\end{itemize}
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The first assumption is true for the \WZ analysis, and the second one is true if we assume
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that the jet composition in the control sample used to establish $p_{fake}$ is the same as that
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in the background in the \WZ data sample. This can be achieved by using $\W+X$
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processes as a control sample, as described in Section~\ref{sec:WPFake}. The latter assumption
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is generally not true for $t\bar{t}$ processes, and therefore, we subtract background
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without genuine \Z bosons using the fit results of the \Z candidate invariant mass.
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However, after applying the full selection criteria, the contribution from the backgrounds
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without real \Z boson is negligible, and the fit results in an unacceptable large uncertainty
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for the 300 \invpb scenario. Thus, it is possible to neglect the combinatorial bias from $t\bar{t}$
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processes with small integrated luminosity sample and forgo the fit altogether. In the following
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we provide the results of estimation of the background without subtracting the estimated
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non-genuine \Z boson background.
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The comparisons between predicted and true MC backgrounds are given in Tables~\ref{tab:FinalNoFitLoose}
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and \ref{tab:FinalNoFit} for ``Loose'' and ``Tight'' \W lepton, respectively.
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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$ - 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^{genuine~Z}$ (matrix method) & 10.0 $\pm$2.5 & 15.8 $\pm$1.2 & 16.0 $\pm$2.4 & 15.8 $\pm$1.4\\ \hline
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$N^{WZ}$ & 9.6 $\pm$2.8 & 8.1 $\pm$1.3 & 7.1 $\pm$2.7 & 10.1 $\pm$1.6\\ \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 events for an integrated luminosity of 300 \invpb for the signal
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and estimated background for 81 GeV $< M_Z < $ 101 GeV with ``Loose'' \W lepton criteria.}
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\label{tab:FinalNoFitLoose}
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\end{center}
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\end{table}
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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$ - ZZ -Z$\gamma$ &12.1$\pm$1.1 &8.9$\pm$0.7 &12.8$\pm$1.0 &10.6$\pm$0.7\\ \hline
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$N^{genuine~Z}$ (matrix method) &3.2 $\pm$1.7 &0.9 $\pm$1.0 &5.1 $\pm$2.1 &0.9 $\pm$1.1\\ \hline
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$N^{\WZ}$ &8.9 $\pm$2.1 &8.0 $\pm$1.2 &7.7 $\pm$2.3 &9.9$\pm$1.3\\ \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 events for an integrated luminosity of 300 \invpb for the signal
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and estimated background for 81 GeV $< M_Z < $ 101 GeV and ``Tight'' \W lepton requirement.}
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\label{tab:FinalNoFit}
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\end{center}
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\end{table}
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The agreement between estimated and MC true backgrounds is excellent. Smaller systematic uncertainty
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associated with the linear fit also results in a higher discovery potential, as described in the next Section.
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|
