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\section{Signal and Background Modeling}
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\label{sec:gen}
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\subsection{Monte Carlo generators}
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The signal and background samples for the full detector simulation
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are generated with the leading order (LO) event generators
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{\sl PYTHIA}~\cite{Sjostrand:2003wg}, {\sl ALPGEN} and {\sl COMPHEP}.
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To accommodate the next-to-leading (NLO) effects, constant $k$-factors are applied
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except for the signal where a $p_T$-dependence has been taken into account.
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The $p_T$-dependent $k$-factor for the signal is estimated using
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the NLO cross section calculator {\sl MCFM}~\cite{Campbell:2005}.
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We estimate the PDF uncertainty on the cross-section using
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{\sl MC@NLO 3.1}~\cite{Frixione:2002ik} NLO event generator
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together with CTEQ6M PDF set.
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\subsection{Signal definition}
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The goal of this analysis is to study the associative production of the on-shell
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$\W$ and $\Z$ bosons that decay into three leptons and a neutrino. In the
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following we refer to a lepton to as either a muon or an electron, unless
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specified otherwise.
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Using {\sl MCFM} we estimate the total NLO $\WZ$ cross-section to be
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\begin{equation}
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\sigma_{NLO} ( pp \rightarrow W^+\Z; \sqrt{s}=14~{\rm TeV}) = 30.5~{\rm pb},
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\end{equation}
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\begin{equation}
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\sigma_{NLO} ( pp \rightarrow W^-\Z; \sqrt{s}=14~{\rm TeV}) = 19.1~{\rm pb}.
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\end{equation}
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The LO and NLO distributions of the \Z boson transverse momentum are
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shown in Fig.~\ref{fig:LOvsNLO}. The NLO/LO ratio, $k$-factor, is also presented on the figure,
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and it is increasing with $p_T(\Z)$. We take into account the $p_T$ dependence
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by re-weighting the LO Monte Carlo simulation as a function of the $p_T(\Z)$.
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%
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%
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%The $p_T$ dependence of the $k$-factor
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%becomes important when a proper NLO description of the $\Z$ boson transverse
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%momentum must be obtained, $e.g$ to measure the strength of the $WWZ$ coupling.
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%As the focus of this analysis is to prepare for the cross-section measurement,
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%we take a $p_{T}$-averaged value of the $k$-factor, equal to 1.84.
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\begin{figure}[!bt]
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\begin{center}
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% \scalebox{0.8}{\includegraphics{figs/k_faktor_for_Note.eps}}
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\caption{Top plot: comparison of $p_T(Z)$ distributions for NLO and LO; bottom plot: k factor }
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\label{fig:LOvsNLO}
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\end{center}
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\end{figure}
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%# for bbll:
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%#CS NLO ((Z/gamma*->l+l-)bb) = 830pb = 345 pb * 2.4, where:
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%#- 345 pb is LO CS calculated with precision of ~0.15%
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%#- 2.4 is MCMF calculated k-factor with precision ~30% (!)
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%# 830x0.173 (== XS x eff.) = 143.59pb
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\subsection{Signal and background Monte Carlo samples}
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The signal Monte Carlo sample is produced using {\sl PYTHIA}
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generator. The decay for the \W lepton is forced to $e\nu_e$,
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$\mu\nu_{\mu}$ or $\tau\nu_{\tau}$ final state, while the \Z decays
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into electrons or muons only.
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The background to the \WZ final state can be divided in physics and
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instrumental. The only physics background is from $Z^0Z^0$ production
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where one of the leptons is either mis-reconstructed or lost.
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The instrumental backgrounds are all due to mis-identified electron candidates
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from either jets or photons. These backgrounds include production of $\W$ and $\Z$ bosons
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with jets and $t\bar{t}$ processes and $Z^0\gamma$ process. The background from $W\gamma$
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production, where the $\gamma$ converts and produces a dielectron system is neglected
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due to a requirement on the $\ell^+\ell^-$ invariant mass to be consistent with the nominal \Z boson mass.
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All non-negligible instrumental backgrounds are summarized below.
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\begin{itemize}
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\item $\Z + jets$: this background is one of the dominant to the \WZ final state. Although
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the misidentification rate for a jet to be misidentified as a lepton is quite small, the
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$\Z+jets$ cross-section is 35 times larger than the signal one. We use the {\sl ALPGEN}
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generated official samples of $\Z+jet$ production Monte Carlo samples for different
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values of the jet transverse momentum.
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\item $t\bar{t}$: each of the top quarks decay into a $\W b$ pair producing at least two
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leptons and two $b$-quark jets. Although this process does not have a genuine $\Z$
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candidate and can be suppressed be a $\Z$ candidate invariant mass requirement,
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the probability for a $b$-quark jet to decay semi-leptonically and be misidentified
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as a lepton is higher than that from a light-quark jets. The cross-section of the $t\bar{t}$
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production is also exceed by about 15 times the cross-section of the \WZ production.
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Thus, this background is also one of the most dominant. We use the official $t\bar{t}$
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samples produced with {\sl ALPGEN} generator to estimate this background.
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\item $\Z + b\bar{b}$: this process is produced by the {\sl COMPHEP}
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generator and have a genuine $\Z$ candidate in the final state. One of the $b$-quark
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jets are misidentified as the third lepton from the $\W$ boson.
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\item $\W+jets$: in this process, the \W boson produces a genuine lepton,
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while the other two leptons are misidentified jets. As the misidentification
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probability is low, this channel does not contribute significantly to the \WZ
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final state. The additional \Z candidate invariant mass requirement suppresses
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this background further. We use the officially produced sample of $\W+jets$ processes
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for different number of jets in the final state generated by the {\sl ALPGEN}
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generator.
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\item $Z^0\gamma$: this process is calculated with {\sl PYTHIA}.
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\end{itemize}
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The background sources that have \Z bosons described above are simulated with the
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contribution from the virtual photon.
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All the samples we use in this study are a part of the CSA07 production and
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are generated using $\mathrm{CMSSW}\_1\_4_\_6$ using the full {\sl GEANT}
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simulation of the CMS detector. The digitization and reconstruction are
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done using a newer $\mathrm{CMSSW}\_1\_6_\_7$ release with a
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misalignment/miscalibration of the detector scenario expected
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to be achieved after collection of $\sim$ 100~pb$^{-1}$ of data.
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All {\sl ALPGEN} samples are mixed together in further referred to as to a
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``Chowder soup''.
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The summary of all datasets used for signal and background is given in
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Table~\ref{tab:MC}. We use the RECO production level to access to
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low-level detector information, such as reconstructed hits. This lets
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us to use full granularity of the CMS sub-detectors to use isolation
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discriminants.
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Analysis of the samples is done using CMSSW$\_1\_6\_7$ CMS software
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release. The information is stored in ROOT trees using a code in
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CVS:/UserCode/Vuko/WZAnalysis, which is based on Physics Tools candidates.
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\begin{table}[!tb]
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%\begin{tabular}{llllll} \hline
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%Sample & Generator & Sample name & Events & $\sigma \cdot \epsilon
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%\cdot k$ & k-factor \\ \hline WZ & Pythia &
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%/WZ/CMSSW\_1\_6\_7-CSA07-1195663763/RECO & 58897 & 0.585 pb & 1.92 \\
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%$Zb\bar{b}$ & COMPHEP &
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%/comphep-bbll/CMSSW\_1\_6\_7-CSA07-1198677426/RECO & 143.59 pb & 2.4
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%\\ ``Chowder'' & ALPGEN &
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%/CSA07AllEvents/CMSSW\_1\_6\_7-CSA07-Chowder-A1-PDAllEvents-ReReco-100pb/RECO
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%& 25 M & event weights & - \\
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\begin{tabular}{|c|c|c|c|c|} \hline
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Sample & cross section, pb & Events & Dataset name \\ \hline
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$\WZ$ & 1.12 & 59K & /WZ/CMSSW$\_1\_6\_7$-CSA07-1195663763\\ \hline
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$\Z b\bar{b}$ & 830*0.173 (NLO) & 1.9M & /comphep-bbll/CMSSW$\_1\_6\_7$-CSA07-1198677426\\ \hline
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Chowder & Event Weight & $\sim$ 21M & /CSA07AllEvents/\\ & & & CMSSW$\_1\_6\_7$-CSA07-Chowder-A1-PDAllEvents-ReReco
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-100pb\\ \hline
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$\Z\Z$ inclusive & 16.1 (NLO) & $\sim$ 140k & /ZZ$\_$incl/CMSSW$\_1\_6\_7$-CSA07-1194964234/RECO\\ \hline
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$\Z\gamma \rightarrow e^+e^-\gamma$ & 1.08 (NLO) & $\sim$125k &/Zeegamma/CMSSW$\_1\_6\_7$-CSA07-1198935518/RECO \\ \hline
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$\Z\gamma \rightarrow \mu^+\mu^-\gamma$ & 1.08 (NLO) & $\sim$ 93k & /Zmumugamma/CMSSW$\_1\_6\_7$-CSA07-1194806860/RECO\\ \hline
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\end{tabular}
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\label{tab:MC}
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\caption{Monte Carlo samples used in this analysis using 100 pb$^{-1}$ scenario}
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\end{table}
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