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\section{Systematic uncertainties}
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\label{sec:systematic}
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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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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~\cite{Zmumu}~\cite{Zee}, the number of \Z events is estimated of the
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order of 50k per 100 pb$^{-1}$ of data analyzed. To determine the
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trigger efficiency ``tag-and-probe'' method~\cite{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 mis-measurement 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 mis-measured 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 leptons 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|} \hline
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Source & Systematic uncertainty,\% \\ \hline
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Luminosity & 10.0 \\
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Trigger & 1.0 \\
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Lepton reconstruction & 2.0 \\
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Electron charge determination & 2.0 \\
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Muon charge determination & 1.0 \\
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Lepton energy scale & 1.0 \\
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Electron identification & 4.0 \\
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Muon identification & 2.0 \\
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PDF uncertainties & 4.0 \\
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$M_{T}(W)$ requirement & 10.0 \\ \hline
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\end{tabular}
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\end{center}
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\caption{Systematic uncertainties for $pp\rightarrow \WZ$ process
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estimated for a scenario of 300~\invpb of integrated luminosity data sample.}
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\label{tab:sys}
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\end{table}
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We assign 100\% systematic uncertainty on the instrumental backgrounds without
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genuine \Z boson. This correspond to 7\% effective systematic uncertainty on the final result.
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The systematic uncertainty on the number of the genuine \Z boson background
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events $\Delta N_j^t$ estimated using the matrix method described in Section~\ref{sec:D0Matrix}
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is calculated as
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\begin{equation}
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\left(\Delta N_j ^{t}\right)^2 = \left(\frac{p\left(N_t - pN_l\right)}{\left(\epsilon -p\right)^2}\right)^2 \Delta \epsilon^2
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+\left(\frac{\epsilon\left(\epsilon N_{l}-N_{t}\right)}{\left(\epsilon -p\right)^2}\right)^2 \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 $N_t$ and $N_l$ are the numbers of observed events in tight and loose samples
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after the \ZZ and \Z$\gamma$ backgrounds have been subtracted. $\Delta N_t$ and $\Delta N_l$
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are the systematic uncertainties associated with this subtraction. We take those as
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100\% of the estimated physics background from the Monte Carlo simulation. Finally,
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$\epsilon$ and $p$ are genuine and misidentified ``loose'' lepton efficiency to
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satisfy ``tight'' requirements.
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We summarize full systematic uncertainties in Table~\ref{tab:FullSys} for each
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individual signature. The systematic uncertainty is comparable to the statistical
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uncertainty which is roughly 30\% for each channel. Improvement in understanding
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of the physics and instrumental backgrounds without genuine \Z bosons, that are
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currently subtracted with overly conservative 100\%, as well as
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understanding of the MET, better measurement of the $p_{fake}$ will
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allow to decrease the overall systematic uncertainty with real data.
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\begin{table}[!tb]
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\begin{center}
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\begin{tabular}{|l|c|c|c|} \hline
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Channels & Modeling, \% & Background estimation, \% & Total, \% \\ \hline
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$3e$ & 21 & 27 & 34 \\
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$2e1\mu$ & 19 & 16 & 25 \\
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$2\mu1e$ & 17 & 31 & 35 \\
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$3\mu$ & 17 & 12 & 21 \\ \hline
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\end{tabular}
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\end{center}
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\caption{Total systematic uncertainty for identification of $pp\rightarrow WZ$ production.}
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\label{tab:FullSys}
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\end{table}
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