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\section{Signal extraction}
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\label{sec:SignalExt}
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We separate backgrounds into three categories: physics background
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from \Z$\gamma$ and \ZZ processes, with a genuine \Z boson from
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$\Z+jets$ processes, and finally without a genuine \Z boson from
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$t\bar{t}$ and $\W+jet$ production.
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The first physics background from \ZZ production is estimated from
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Monte Carlo simulation. We assigning a very conservative 100\%
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uncertainty on the estimated \ZZ contribution due to modeling of
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kinematics of the decay products. The second physics
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background from \Z$\gamma$ processes is estimated from Monte Carlo
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simulation as well, although it can be determined from data once the FSR
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$\Z\gamma$ signal is measured at CMS. We also assign a conservative 100\%
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systematic uncertainty on the \Z$\gamma$ background contribution due
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to modeling of the photon conversion probability.
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Instrumental backgrounds from processes with misidentified leptons
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attributed to \Z candidate decay products can be estimated from the
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side-bands of the \Z candidate invariant mass distribution. However,
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the background is relatively small. We estimate this background to contribute
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at about 6\% level to the \WZ signal sample and about 20\% of the full
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background processes. That corresponds to less than 8 events combined
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for all four signatures for integrated luminosity of 1 \invfb.
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Thus, it is impractical to infer this background from a sideband
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fit. Thus, we estimate this background from Monte Carlo simulation
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and we assign a conservative 100\% systematic uncertainty on the contribution
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due to modeling of the $t\bar{t}+jets$ and $\W+jets$ background in
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Monte Carlo simulation.
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The remaining, largest background is from \Z+misidentified lepton processes
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that we describe in details below.
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\subsection{Matrix method}
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\label{sec:D0Matrix}
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In this Section the ``loose'' and ``tight'' lepton requirements
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are defined in Section~\ref{sec:looseTight}.
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The idea of a method is to apply ``loose'' identification criteria
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on the third lepton after \Z boson candidate is identified
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and count the number of the observed events, $N_{loose}$.
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These events contain events with real leptons $N_{l}$
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and events with misidentified jets $N_j$:
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\begin{equation}
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\label{eq:matrixEq1}
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N_{loose} = N_l + N_j.
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\end{equation}
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If we are to apply ``tight'' selection on the third lepton, the number
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of the observed events $N_{tight}$ would change as following
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\begin{equation}
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\label{eq:matrixEq2}
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N_{tight} = \epsilon_{tight} N_l + p_{fake} N_j,
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\end{equation}
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where $\epsilon_{tight}$ and $p_{fake}$ are efficiency of ``tight''
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criteria with respect to ``loose'' requirements for leptons and
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misidentified jets, respectively. As $N_{loose}$ and $N_{tight}$
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are directly observable, to extract the number of $Z+jet$ events
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in the final sample, one needs to measure $\epsilon_{tight}$
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and $p_{fake}$ in control data samples as described in the next
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two sections.
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If $\epsilon_{tight}$ and $p_{fake}$ are measured as functions of some
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variable, for example, lepton $p_T$, then it is possible to estimate
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the background as function of $p_T$ by applying the matrix method to
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binned distribution of $dN_{tight}/dp_T$ and $dN_{loose}/dp_T$.
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\subsubsection{Determination of $\epsilon_{tight}$}
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To estimate the $\epsilon_{tight}$ we apply ``tag-and-probe'' method
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using $\Z \to \ell\ell$ from \Z+jets Chowder sample, including \W+jets
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and $t\bar{t}$ as background. \Z mass distribution is separated for two cases where
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leptons from \Z boson decay either both pass ``tight'' selection (``tight-tight'' case), or only
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one passes the ``tight'' selection, while the other electron passes ``loose'' but not ``tight''
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selection (``tight-loose'' case).
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Equation for determination of signal efficiency is given as
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\begin{equation}
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\epsilon_{tight}=\frac{ 2(N_{TT}-B_{TT}) }{ (N_{TL}-B_{TL})+2(N_{TT}-B_{TT}) }
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\label{eq:errmatrix}
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\end{equation}
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where $N_{TT}$, $B_{TT}$, $N_{TL}$, and $B_{TL}$ are numbers of signal plus background
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and background events for ``tight-tight'' and ``loose-tight'' electron combinations,
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respectively. We estimated an efficiency for both electrons and muons to be
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$\epsilon_{tight}=0.98 \pm 0.01$.
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\subsubsection{Determination of $p_{fake}$}
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The probability of a jet to be misidentified as a lepton depends on
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$p_T$, $\eta$, and composition of quarks and gluons. Although the
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light quark and gluons have a very similar misidentification rate
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(see Fig.~\ref{fig:pfake_gg_qq}), this rate can be different for jets
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enriched with heavy quark jets. Thus, we propose to measure the
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$p_{fake}$ in the \W+jets sample, as jets in this sample has a very
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similar composition as those in \Z+jets process, the major background
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to the signal.
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\begin{figure}[bt]
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\begin{center}
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\scalebox{0.4}{\includegraphics{figs/pfake_gg_qq.eps}}
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\caption{The misidentification probability $p_{fake}$ measured in light-quark jets (left),
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and gluon jets (right) using multi-jet {\sl ALPGEN} Monte Carlo simulation.}
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\label{fig:pfake_gg_qq}
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\end{center}
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\end{figure}
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We select the \W+jets sample for electron misidentification study as follows:
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\begin{itemize}
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\item event must be triggered by the HLTSingleMuonIso trigger,
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\item event must not have a \Z boson candidate with an invariant mass between 50 and 120 GeV,
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\item event must have a muon candidate within the acceptance with $p_T >$ 20 GeV satisfying
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isolation and impact parameter significance requirements; the transverse mass of the muon candidate
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and the MET in the event must exceed 50 GeV,
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\item event must have only one electron candidate that satisfies loose identification requirements with $p_T > 20$ GeV,
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isolated from the muon candidate by $\Delta R > 0.1$,
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\item muon and electron must have the same charge to suppress $t\bar{t}$ events with real muon and electrons from \W
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boson decays.
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\end{itemize}
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The ``tight'' criteria is just an application of the above-mentioned requirements with an additional criterion
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on the electron candidate to pass ``tight'' electron identification criteria.
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The probability $p_{fake}$ is then measured as the ratio of the $p_T$ distribution of the ``tight'' electron
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candidates to the $p_T$ distribution of the ``loose'' electron candidates in the original sample. This
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probability agrees well with the one obtained from the multi-jet sample and is given in Fig.~\ref{fig:pfake_w}.
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The errors correspond to statistical errors expected for a data sample with integrated luminosity of 300 \invpb.
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Thus, it is possible to extract $p_{fake}$ with $\sim$10\% accuracy with early CMS data.
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\begin{figure}[hbt]
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\begin{center}
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\scalebox{0.6}{\includegraphics{figs/pFake_300pb_WX_TTbar.eps}}
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\caption{Probability of a misidentified jet from \W+jet processes that passed ``loose'' electron identification
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requirements to also pass tight ones as function of the candidate's $p_T$. The errors correspond to
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statistical errors expected for a 300 \invpb integrated luminosity. The distribution is also fit to a
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constant to extract the flat $p_{fake}$ factor. }
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\label{fig:pfake_w}
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\end{center}
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\end{figure}
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A similar method can be used to extract the $p_{fake}$ for the muon misidentification as well.
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\subsubsection{Cross-check of the $p_{fake}$ using multi-jet sample}
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In the following we perform an additional cross-check of $p_{fake}$ measured
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in different control sample, enriched with multi-jet processes. In the sample,
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selected with jet triggers, we select a ``loose'' electron candidates that are
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separated from the jet that satisfied the trigger requirement. These candidates
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are dominated by the misidentified light quark and gluon jets. The admixture
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of converted photons from $\gamma + jets$ is small at the low-$p_T$ range and is
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neglected.
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The $p_{fake}$ function of $p_T$ and $\eta$ is obtained by dividing the $p_T$ and $\eta$
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distributions for the electron candidate that satisfied ``tight''
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electron identification requirements to that for electron candidates
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that satisfied ``loose''. We estimate the $p_{fake}=0.32 \pm 0.04$ for
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electrons.
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For muons, a similar procedure has been applied. Since the bulk
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of background muons is coming from heavy quark decays, we select
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a $b\bar{b}$ sample as control sample. As an exercise, we selected
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electron-triggered events on a $b\bar{b}$ Monte Carlo sample,
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required one ``loose electron'' in the event and looked for
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for muon candidates that are not close to the electron candidate,
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and determined $p_{fake}$ on this sample of muons. The $p_T$ spectrum
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for ``loose'' and ``tight'' muons and their ratio is shown in
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Figure~\ref{fig:mu_pfake}. The factor $p_{fake}$ for muons estimated
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in this way amounts to $0.08 \pm 0.01$.
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\begin{figure}[bt]
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\begin{center}
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\scalebox{0.6}{\includegraphics{figs/tight_eff_gumbo.eps}}
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\caption{Fraction of electron candidates passing the ``tight'' criteria
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in multi-jet event. No trigger requirement has been applied.}
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\label{fig:qcd_efftight_noHLT}
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\end{center}
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\end{figure}
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\begin{figure}[!bt]
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\begin{center}
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\scalebox{0.6}{\includegraphics{figs/p0_p_fake_mu_fit.eps}}
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\caption{Determination of $p_{fake}$ for muons. Top plot: $p_T$ spectrum
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of muons passing the ``loose'' and ``tight'' criteria in $b\bar{b}$ events
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in the ``Stew'' soup that corresponds to 20 \invpb of integrated luminosity,
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accepted by electron triggers; bottom plot: fraction of muon candidates
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passing the ``tight'' criteria. A constant fit is overlayed.}
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\label{fig:mu_pfake}
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\end{center}
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\end{figure}
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\subsubsection{Background determination results}
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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$\gamma$ - W+jets - $t\bar{t}$ & 18.1$\pm$1.7 & 17.7$\pm$6.2 & 22.3$\pm$1.4 & 20.0$\pm$5.9\\ \hline
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$N^{genuine~Z}$ (matrix method) & 10.1 $\pm$3.2 & 9.4 $\pm$6.7 & 14.5 $\pm$2.9 & 9.4 $\pm$6.4\\ \hline
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$N^{WZ}$ & 8.0 $\pm$3.6 & 8.3 $\pm$9.1 & 7.8 $\pm$3.2 & 10.6 $\pm$8.7\\ \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$\gamma$ - W+jets - $t\bar{t}$ &11.1$\pm$1.3 &8.2$\pm$0.9 &12.1$\pm$1.2 &10.5$\pm$0.8\\ \hline
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$N^{genuine~Z}$ (matrix method) &3.2 $\pm$1.7 &0.6 $\pm$0.8 &4.6 $\pm$2.0 &0.6 $\pm$0.9\\ \hline
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$N^{\WZ}$ &7.9 $\pm$2.1 &7.6 $\pm$1.2 &7.5 $\pm$2.3 &10.0$\pm$1.2\\ \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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Using the values of $\epsilon_{tight}$ and $p_{fake}$ obtained
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from the methods described in the previous sections, we estimated
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the backgrounds from genuine \Z decays by solving Eqs.~\ref{eq:matrixEq1}
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and \ref{eq:matrixEq2} for $N_j$. The comparisons between predicted and true MC
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backgrounds are given in Tables~\ref{tab:FinalNoFitloose} and \ref{tab:FinalNoFit}
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for ``loose'' and ``tight'' \W lepton, respectively.
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The agreement between estimated and MC true backgrounds is excellent.
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\clearpage
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