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In this section, we estimate systematics uncertainties of the methods
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used in this analysis. We follow the rule of making conservative estimates
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throughout this section.
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\subsection{Modeling systematics}
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The sources of systematic uncertainties due to modeling of trigger,
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reconstruction, PDF, and luminosity are described below
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\begin{itemize}
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\item {\it Trigger}: the trigger path used to select four categories
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require leptons to be isolated. Though, the isolation criteria
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depends on the occupancy of the sub-detectors, the alignment of the
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tracker (when considering tracker isolation variables), and noise in
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the calorimeters (when considering a calorimetric isolation), the
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trigger efficiency is expected to be around 99\%, and therefore, a
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systematic uncertainty is conservatively estimated as 1\%. From the
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current analysis of $Z\rightarrow l^+l^-$ in
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CMS~\ref{Zmumu}~\ref{Zee}, the number of \Z events is estimated of the
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order of 50k per 100pb$^{-1}$ of data analysed. To determine the
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trigger efficiency ``tag-and-probe'' method~\ref{TP} will be used.
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\item {\it Reconstruction}: we assign 2\% systematic uncertainty per
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lepton due to initial tracker alignment which is of paramount
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importance to reconstruct leptons, 2\% and 1\% is assigned for the
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determination of the charge of the electron and muon candidates,
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respectively. We assigned a larger electron charge identification
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uncertainty due to much stronger Bremsstrahlung energy loss which
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makes the charge identification more difficult. The mismeasurement of
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the charge is of the order of 2\% in CMSSW\_1\_6\_7 release for
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electron. The estimation of the fraction with data will be done by
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looking at the \Z peak without opposite charge requirement. Then
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number of events within the \Z mass windows asking for two leptons of
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same sign will give us a estimate of the fraction of mismeasure sign
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leptons.
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\item {\it Lepton identification}: we assign 4\% of systematic
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uncertainty due to efficiency measurement from early data using
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``tag-and-probe'' method and 2\% for that for a muon. Additionally we
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assign a systematic uncertainty on lepton energy scale of 2\% per
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lepton. The letpons scale will be established using the \Z mass peak.
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\item {\it PDF uncertainties}: we estimate PDF uncertainties following prescription
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described in~\cite{OldNote}. The uncertainty is found to be
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$$ \Delta \sigma_+ ^{tot} = 3.9\% \hspace{0.9cm} \Delta \sigma_- ^{tot} = 3.5\% $$.
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\item {\it Luminosity}: we estimate luminosity uncertainty of 10\%.
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\end{itemize}
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The systematic uncertainties are summarized in Table~\ref{tab:sys}.
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\begin{table}[!tb]
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\begin{center}
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\begin{tabular}{|l|c|c|} \hline
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& \multicolumn{2}{c|}{Systematic uncertainty} \\
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Source & on the cross section,\% & on the signficance,\% \\ \hline
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Luminosity & 10.0 & - \\
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Trigger & 1.0 & 1.0\\
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Lepton reconstruction & 2.0 & 2.0\\
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Electron charge determination &2.0& 2.0\\
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Muon charge determination &1.0& 1.0\\
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Lepton energy scale& 1.0& 1.0\\
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Electron identification& 4.0 &4.0\\
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Muon identification& 2.0 &2.0\\
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PDF uncertainties& - & + 3.9\\
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& & - 3.5 \\ \hline
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\end{tabular}
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\end{center}
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\caption{Systematic uncertainties for $pp\rightarrow \WZ$ cross section measurement
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and significance estimation for 300 \invpb of integrated luminosity.}
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\label{tab:sys}
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\end{table}
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\subsection{Systematic uncertainties due to background estimation method}
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In the following we estimate a systematic uncertainty due to estimation
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of background using the matrix method described in Section~\ref{sec:D0Matrix} above.
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We present here, the result for the case where the $W$ is decaying via
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an electron.
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Two steps will be used to substract the different background: first,
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the non peaking background should be substracted, then the background
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$Z+jets$ will be determine using the method described
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in~\ref{sec:D0Matrix}.
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From the fit, we will consider a systematics error of 10\%.
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If we consider an error of 10\%
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on the fake rate and an error of 2\%
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on the efficiency on signal to go from loose to tight criteria, we can
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calculate the error on the estimated background as follow:
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\begin{equation}
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\Delta N_j ^{t} = \sqrt{\left(\frac{p\left(N_t - pN_l\right)}{\left(\epsilon -p\right)^2}\right)^2 \times \Delta \epsilon^2
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+\left(\frac{\epsilon\left(\epsilon N_{l}-N_{t}\right)}{\left(\epsilon -p\right)^2}\right)^2 \times \Delta p^2
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+ \frac{p^2\left(\epsilon^2\Delta N_{l}^2 - \Delta N_{t}^2\left(2\epsilon -1\right)\right)}{\left(\epsilon -p\right)^2}}
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\end{equation}
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where $N_{t}$,$\Delta N_{t}$ and $N_{l}$,$\Delta N_{l}$ represents
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respectivement the number of events in the tight sample and in the
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loose sample and their errors.$\epsilon$ represent efficiency for a
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loose electron to pass the tight criteria, $\Delta \epsilon$ the error
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on this value.$p$ gives the probability for a fake loose electron to
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pass also the tight criteria and $\Delta p$ its error.
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The overall error from the background substraction is XXX %18\%.
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\subsection{Summary of Systematics}
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In table~\ref{tab:FullSys}, the systematics errors are expressed for
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each channels.
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\begin{table}[!tb]
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\begin{center}
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\begin{tabular}{|l|c|c|} \hline
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Channels & Cross Section & Signficance \\ \hline
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3e & 8.4\% +10\% = 13.1\% & +9.3\% / - 9.2\% \\
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2e1$\mu$ & 7.7\% +10\% = 12.6\% & +8.7\% / - 8.5\% \\
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1e2$\mu$ & 6.5\% +10\% = 11.9\% & +7.6\% / - 7.4\% \\
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3$\mu$ & 5.5\% +10\% = 11.4\% & +6.7\% / - 6.5\% \\\hline
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\end{tabular}
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\end{center}
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\caption{Systematics per channels in percent for $pp\rightarrow WZ$ cross section measurement and significance estimation for 300 \invpb of integrated luminosity. These systematics do not include the background substraction.}
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\label{tab:FullSys}
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\end{table}
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