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In this section, we will assign systematics errors to this |
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analysis. The assignement of systematics is expected to be |
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conservatives. |
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|
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\subsection{Experimental Systematics} |
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|
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The experimental systematics errors expected that will affect the |
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signal and standard model background are: |
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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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|
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\subsection{Modeling systematics} |
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|
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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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|
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\begin{itemize} |
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\item For trigger selection, a systematics of 1\% is assigned. Even |
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though the efficiency of the signal is greater than 99\%, the trigger |
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path used for both muons and electron expect the leptons to be |
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isolated. As the isolation depends on the occupancy of the events, |
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the alignment of the tracker (when considering tracker isolation |
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variables) and noise in the calorimeters (when considering a |
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calorimetric isolation), this value is expected to be conservative. |
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|
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\item 3\% error is assigned on electron/muons reconstruction. Both of |
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them are link to alignment of the track in order to reconstruct the |
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leptons. A systematics of 2\% is assigned for the determination of |
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the charge of the electron candidate while 1\% for the muon as the |
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electron problem is coming from the high probability of emission of |
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photons. |
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|
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\item A systematics of 1\% will be assigned for the measurement of |
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the lepton energy. |
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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 analysed. To determine the |
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trigger efficiency ``tag-and-probe'' method~\cite{TP} will be used. |
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|
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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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|
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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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|
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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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|
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\item 4\% of systematics are considered for the electron |
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identification, 2\% for the muon case. |
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\item {\it Luminosity}: we estimate luminosity uncertainty of 10\%. |
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\end{itemize} |
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|
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The PDF uncertainties on the signal has been determined in~\cite{OldNote}. |
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The uncertainty was found to be: |
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\begin{equation} |
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\Delta \sigma_+ ^{tot} = 3.9\% \hspace{0.9cm} \Delta \sigma_- ^{tot} = 3.5\% |
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\end{equation} |
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The systematic uncertainties are summarized in Table~\ref{tab:sys}. |
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|
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The luminosity error is expected to be 10\%. |
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|
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The table~\ref{tab:sys} resume all systematics considered. |
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|
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\begin{table}[!] |
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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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Systematics Source (in \%) & Cross Section & Signficance \\ \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 & 3.0 & 3.0\\ |
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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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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 & + 3.9\\ |
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&- 3.5 & - 3.5 \\ |
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$M_{T}(W)$ requirement & 10 & 10 \\ \hline |
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\end{tabular} |
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|
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\end{center} |
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\caption{Systematics in percent for $pp\rightarrow WZ$ cross section measurement and significance estimation for 1 fb$^-1$ of integrated luminosity.} |
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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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|
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\subsection{Background Substraction Systematics} |
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\subsection{Systematic uncertainties due to background estimation method} |
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|
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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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|
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|
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Two methods will be used to substract the different background. The |
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main background is the production $Z+jets$. Such background can be |
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estimated using data as presented in section~\ref{sec:SignalExt}. For |
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the $t\bar{t}$ background, we can use safely the side band around the |
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$Z$ mass in order to evaluate it. |
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|
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If we consider an error of xx\% on the fake rate and an error of xx\% |
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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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|
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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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|
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%From the fit, we will consider a systematics error of 10\%. |
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|
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If we consider an error $\Delta p$ |
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%of 4\% |
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on the fake rate and an error $\Delta \epsilon$ |
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%of 1\% |
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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} = \frac{\sqrt{(p[N_{t} - p(N_{l}+N_{t})])^2 \times \Delta \epsilon^2 |
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+(\epsilon[\epsilon(N_{l}+N_{t})-N_{t}]^2 \times \Delta p^2 |
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+ (p\epsilon)^2 \times N_{l} + [p(\epsilon -1 )]^2 \times N_{t}}}{\epsilon_{t} - p} |
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%\Delta N_j ^{tight} = \frac{\sqrt{(p_{fake}[N_{tight} - p_{fake}(N_{loose}+N_{tight})])^2 \dot \Delta \epsilon^2 |
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%+(\epsilon[\epsilon(N_{loose}+N_{tight})-N_{tight}]^2 \dot \Delta p_{fake}^2 |
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%+ (p_{fake}\epsilon)^2 \dot N_{loose} + [p_{fake}(\epsilon -1 )]^2 \dot N_{tight}}}{\epsilon_{tight} - p_{fake}} |
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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 $N_{t}$ and $N_{l}$ represents respectivement the number of |
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events in the tight sample and in the loose sample and if they are |
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greater than 25.$\epsilon$ represent efficiency for a loose electron |
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to pass the tight criteria, $\Delta \epsilon$ the error on this |
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value.$p$ gives the probability for a fake loose electron to pass also |
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the tight criteria and $\Delta p$ its error. |
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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. |
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%, $\Delta \epsilon$ the error on this value. |
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$p$ gives the probability for a fake loose electron to |
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pass also the tight criteria. |
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%and $\Delta p$ its error. |
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|
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%The overall error from the background substraction is XXX %18\%. |
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|
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\subsection{Summary of Systematics} |
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|
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An example of the method is given on figure~\ref{fig:Fitbkg}. The |
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number of estimated background compare to the true value is shown on |
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table~\ref{tab:FitbkgSub}. |
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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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|
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We assign a systematics error of 20\%. |
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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 & +9.3\% / - 9.2\% +10\% = +13.7\% / -13.6\% & +9.3\% / - 9.2\% \\ |
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2e1$\mu$ & +8.7\% / - 8.5\% +10\% = +13.3\% / -13.1\% & +8.7\% / - 8.5\% \\ |
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1e2$\mu$ & +7.6\% / - 7.4\% +10\% = +12.7\% / -12.4\% & +7.6\% / - 7.4\% \\ |
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3$\mu$ & +6.7\% / - 6.5\% +10\% = +12.0\% / -11.9\% & +6.7\% / - 6.5\% \\\hline |
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\end{tabular} |
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|
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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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|
