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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Backgrounds}\label{section:BG}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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This section reviews our estimation of background in the $4\ell$ analysis. We discuss expected yields and the predicted $m(4\ell)$ shapes for the various background channels, both of which are used in the limit calculation described in Section~\ref{sec:limit}. Irreducible EWK backgrounds are estimated with Monte Carlo. Our estimates of ``fake'' backgrounds associated with $Z+jet$ and heavy flavor backgrounds from $Zb\bar{b}/c\bar{c}$ and $t\bar{t}$ are data-driven.
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%_________________________________________________________________
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\subsection{Electroweak Backgrounds}\label{sec:EWK}
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%_________________________________________________________________
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We use the $ZZ \rightarrow 4\ell$, $WZ \rightarrow 3\ell$ and $Z\gamma$ MC samples listed in Table~\ref{tab:MC} to estimate the yields and shapes of these backgrounds. Each of the MC acceptances ($\alpha$) we determine from the MC are efficiency-corrected following the same procedure described for signal in Section~\ref{section:signalEff}. This provides a corrected $\alpha_{c}$ for each process that we use to predict yields :
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\begin{eqnarray}
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N^{exp}_{i} & = & \alpha^{c}_{i}\int\mathcal{L}(k_{i}\sigma_{i})
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\end{eqnarray}
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The cross sections and K-factors used in formula above are taken from Table~\ref{tab:xsec}. Table~\ref{tab;MCBG} lists the corrected acceptances and expected yields for each of the irreducible backgrounds. Figure~\ref{fig:MCshapes} shows a yield-normalized histogram stack of the corresponding $m(4\ell)$ distributions.
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|c|c|}
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\hline
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process & $\alpha^{c}$ & $N^{exp}$ \\
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\hline
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$ZZ*$ & ~ & ~ \\
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$WZ$ & ~ & ~ \\
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$Z\gamma$ & ~ & ~ \\
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\hline
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\end{tabular}
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\caption{{\bf MC Background Yields.}\small{blah.}\label{tab:MCBG}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{figs/HF1.png}
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\caption{MC Background Shapes.\label{fig:MCshapes} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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Our current estimate of $WZ$ background is entirely MC-based. In addition to the leptons from $W$ and $Z$ decay, an additional ``fake'' lepton is needed for this process to contribute in the $4\ell$ signal region. Given the difficulty in simulating processes that result in fake leptons, we perform an additional data-driven estimate of background using the ``fakeable object'' method described below. The method is described in Appendix~\ref{app:WZfake}. Results from the procedure given in Table~\ref{tab:WZfake}. The data-driven estimates are consistent with MC predictions within uncertainties.
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|c|c|}
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\hline
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$4e$ & $4\mu$ & $2e2\mu$ \\
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\hline
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$X\pm Y$ & $Z\pm Y$ & $Z\pm Y$ \\
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\hline
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\end{tabular}
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\caption{{\bf Data-driven Expected $WZ$ Yields.}\small{blah.}\label{tab:WZfake}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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%_________________________________________________________________
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\subsection{Instrumental/Fake Backgrounds}\label{sec:Zj}
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%_________________________________________________________________
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We estimate $Z + jets$, $t\bar{t}$ and $Zb\bar{b}/c\bar{c}$ backgrounds using the ``fakeable object'' technique~\cite{fakeable}. We define fakerates with respect to loosely identified lepton candidates, which are henceforth referred to as {\it denominator objects}. Our electron and muon denominator definitions are given in Table~\ref{tab:fo}.
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|c|c|c|}
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\hline
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\multicolumn{2}{|c|}{Electron} & \multicolumn{2}{|c|}{Muon} \\
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\hline
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variable & requirement & variable & requirement \\
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\hline
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~ & ~ & $p_{T}$ & $> 5\rm~GeV$ \\
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~ & ~ & type & $\rm Global~||~Tracker$ \\
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~ & ~ & $|d_{0}|$ & $< 2\rm~cm$ \\
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~ & ~ & $\sigma(p_{T})/p_{T}$ & $< 0.4$ \\
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~ & ~ & $Iso^{pf}_{0.3}$ & $< 0.7$ \\
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\hline
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\end{tabular}
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\caption{{\bf Denominator Object Definitions.}\small{blah.}\label{tab:fo}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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We calculate fakerates ($\epsilon_{FR}(p_{T},\eta)$) from samples of events that pass a combination of single and double lepton triggers (XXX for electrons, \verb|HLT_Mu8| or \verb|HLT_Mu13| for muons). In both cases we veto events with $MET > 20\rm~GeV$ or $m_{T} > 35\rm~GeV$ or that contain two or more denominator objects with $p_{T} > 10\rm~GeV$ to reduce contributions from W's and Z's. The samples are enriched in background by selecting only those denominator objects opposite ($\Delta R(\eta,\phi) > 1.0$) a reconstructed $p_{T} > 35\rm~GeV$ jet. Figure~\ref{fig:FR} shows the electron and muon fakerates determined by this procedure as a function of $p_{T}$.
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.45\linewidth]{figs/frMu.png}
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\includegraphics[width=0.45\linewidth]{figs/HF1.png}
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\caption{ {\bf Muon and Electron Fake Rates.}\label{fig:FR} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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We estimate the contribution of the $\ell\elljj$ backgrounds in our signal region by applying the fakerates in events that contain a good Z1. We select denominator objects that fail our full lepton identification/isolation selection to prevent bias from real leptons. We then loop over pairs of denominator objects, weight each leg with $\epsilon_{FR}(p_{T},\eta)/(1-\epsilon_{FR}(p_{T},\eta))$ and apply the Z2 kinematic requirements ($12\rm~GeV < m(Z2) < 120$) on the pair. The denominator in the weight term accounts for the fact the we only consider denominator objects that fail full lepton selection. The weighted pairs that pass the Z2 kinematic selection are summed to provide an estimate of the $\ell\ell\jj$ background. Table~\ref{tab:fakes} lists the number of Z1 + $2\times~fakes$ we determine for the $1.6~\rm fb{-1}$ dataset.
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%\begin{eqnarray}
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% \Sigma_{i=0}^{Nd}~\Sigma_{j=i+1}^{Nd}~\frac{\epsilon(\eta^{i},p_{T}^{i})}{1-\epsilon(\eta^{i},p_{T}^{i})}~\frac{\epsilon(\eta^{j},p_{T}^{j})}{1-\epsilon(\eta^{j},p_{T}^{j})} &=& N(Z+j)
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%\end{eqnarray}
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|c|}
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\hline
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\multicolumn{2}{|c|}{Z1-Inclusive $Z+jets$ Yields} \\
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\hline
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$Z1 + \mu\mu$ & $0.057 \pm X$ \\
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$Z1 + ee$ & $X \pm Y$ \\
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\hline
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\multicolumn{2}{|c|}{Final-State $Z+jets$ Yields} \\
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\hline
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$4\mu$ & $0.044 \pm X$ \\
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$4e$ & $X \pm Y$ \\
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$2e2\mu$ & $(0.013 + Z) \pm Y$ \\
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\hline
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\end{tabular}
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\caption{{\bf Expected Fakes.}\label{tab:fakes}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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%In the second calculation, we require three fully selected leptons (two from the Z1 plus one additional) and then perform a single loop that associates the denominator objects with additional lepton. As before, we weight the demoninators with $\epsilon_{FR}(p_{T},\eta)$, apply kinematic selection and sum. The additional lepton with which the denominators are paired is either a fake (Z+jets) or real (WZ+jets) lepton. In the case where the additional lepton is fake over-represented because each jet leg has a probability of $\epsilon_{FR}$ to passing the tight selection.to jets The sum includes both contributions, counting picks a This sum double counts the contributions from are summed in our signal region to
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%\begin{eqnarray}
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% \ell~\Sigma_{i=0}^{Nd}~\epsilon(\eta^{i},p_{T}^{i}) &=& 2\times N(Z+j) + N(WZ)
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%\end{eqnarray}
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It is difficult to obtain a reliable estimate of the $Z + jets$ $m(4\ell)$ distribution with the limited number of events containing a good Z1 and two denominator objects. We loosen the denominator and Z2 selections to better understand the shape. For the muon-channel, we relax the isolation requirement in the denominator definition and remove the opposite-sign requirement in Z2 selection. For electrons we {\bf XXX}. With these modifications we obtain the $m(4\ell)$ distributions shown Figure~\ref{fig:fakeshapes}. We fit the shapes with Landau distributions and obtain acceptable values for goodness-of-fit. Consequently, we also choose Landau distributions for modeling the $m(4l)$ distribution of our $Z+jets$ predictions, as is shown in Figure~\ref{fig:fakeshapes}. We take differences with respect to the high-statistics distributions as a systematic uncertainties on our predicted shapes.
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.45\linewidth]{figs/muFakeShape-4m.png}
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\includegraphics[width=0.45\linewidth]{figs/muFakeShape-2m.png}
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\includegraphics[width=0.45\linewidth]{figs/muFakeShape-4m.png}
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\caption{$\gamma_d$ Branching Ratios.\label{fig:fakeshapes} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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\subsection{Cross Check : Light Flavor }
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We cross-check our fakerates and procedures by attempting to predict the number of fakes contributing in an independent control region. We require one lepton candidate that passes our nominal lepton selection and $1+$ opposite-sign, same-flavor denominator objects. In the muon-channel, this selection provides a sample of nearly pure background, of which the primary contribution is $W+jet$ where the jet fakes a lepton. For electrons, charge misidentification is significant enough to result in a noticeable Z-peak. However, the contribution of fakes in the electron sample is easily estimated from a fit with a same-sign MC Z template and an exponential background PDF. Events selected in data are shown in Figures~\ref{fig:ssMuon} and (\ref{fig:ssEle}) as points. Table~\ref{tab:ssfakes} lists the total number of observed events in the muon-channel and the electron-channel background estimated from the fit.
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.45\linewidth]{figs/ssMuon-m4l.png}
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\includegraphics[width=0.45\linewidth]{figs/ssMuon-pt.png}
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\caption{{\bf Fakerate validation with same-sign muon events.}\small{blah.}\label{fig:ssMuon} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.45\linewidth]{figs/ssMuon-m4l.png}
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\includegraphics[width=0.45\linewidth]{figs/ssMuon-pt.png}
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\caption{{\bf Fakerate validation with same-sign electron events.}\small{blah.}\label{fig:ssEle} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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Next, we predict the number of same-sign fake events in these samples by applying our fakerates to the denominator objects. We loop over all denominator objects in these events that fail full lepton selection, weight each with the appropriate factor of $\epsilon_{FR}(p_{T},\eta)/(1-\epsilon_{FR}(p_{T},\eta))$ and sum. The expected $m(\ell\ell)$ and $p_{T}$ distributions we obtain from this procedure are shown as solid histograms in Figures~\ref{fig:ssMuon} and ~\ref{fig:ssEle}. Predicted yields are compared with observation in Table~\ref{tab:ssfakes}. The shape of the observed $m(\ell\ell)$ distribution is well described by our prediction. The normalizations agree to within $6\%$ (muons) and $X\%$ (electrons). {\bf Shouldn't take every cross check as a systematic ... but should we take this one? Assign it to sample depend somehow?}.
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c||c|c|}
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\hline
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channel & observed & predicted \\
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\hline
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\hline
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$same sign~\mu\mu$ & $1741$ & $1642.6 \pm X$ \\
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$same sign~ee$ & $X$ & $Z \pm Y$ \\
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\hline
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\end{tabular}
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\caption{{\bf Same-sign Control Yields.}\label{tab:ssfakes}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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\subsection{Cross Check : Heavy Flavor }
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\subsection{Cross Check : W/Z }
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%_________________________________________________________________
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\subsection{$t\bar{t}$, $Zb\bar{b}$, $Zc\bar{c}$}\label{sec:HF}
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%_________________________________________________________________
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Background from $t\bar{t}$ and $Zb\bar{b}/c\bar{c}$ involves real leptons from heavy flavor decays. While lepton multiplicity, decay length and kinematics are correctly simulated ({\bf reference?}), we can not rely on MC to accurately model the identification and isolation efficiency of leptons embedded in heavy flavor jets. We address this by determining a set of data-driven efficiency scale factors, $f_{HF}$, that we use these to correct MC predictions for $t\bar{t}$ and $Zb\bar{b}$, $Zc\bar{c}$ 4-lepton backgrounds.
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We calculate the efficiency scale factors using simulated $Zb\bar{b}$ and $t\bar{t}$ samples and a large-IP control region in data. The control region consists of events that contain a pair of preselected leptons passing our Z1 selection and at least two additional reconstructed ({\it i.e.:} denominator) lepton candidates with $\sigma_{IP} > 5$. We do not require the denominators to be opposite-charge, same-flavor or isolated.
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The left plot of Figure~\ref{fig:ZHF} compares the predicted and observed $m(Z1)$ distributions from the control regions in data and MC for events passing the denominator selection described above. Both shape and normalization agree; we observe $98$ events and predict $91.1 \pm XXX$. We then require the high-IP lepton candidates to additionally pass the more stringent lepton ID and isolation criteria used in our nominal Z2 selection. We predict $0.80 \pm XXX$ events and observe 2. Table~\ref{tab:HFSF} lists the predicted and observed number of individual Z2 muon/electron candidates that pass the denominator and Z2 lepton selections\footnote{Note that values in the ``pass'' column are calculated for {\it single} leptons, not pairs.} We calculate per-lepton efficiencies from the fraction of denominators that pass the Z2 requirements. Per-lepton efficiency scale factors $f_{HF}$ are determined from the ratio of data and MC efficiencies.
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.45\linewidth]{figs/HF1.png}
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\includegraphics[width=0.45\linewidth]{figs/HF1.png}
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\caption{$\gamma_d$ Branching Ratios.\label{fig:ZHF} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|ccc|ccc|c|}
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\hline
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~ & \multicolumn{3}{|c|}{Monte Carlo} & \multicolumn{3}{|c|}{Data} & ~ \\
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\hline
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type & denom & pass & $\epsilon~(\%)$ & denom & pass & $\epsilon~(\%)$ & $f_{HF}$ \\
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\hline
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$e$ & $34.0$ & $1.9$ & $5.6 \pm 3.9$ & $43$ & $4$ & $9.3 \pm 4.4$ & $1.67 \pm X$ \\
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$\mu$ & $148.3$ & $8.0$ & $5.4 \pm 1.9$ & $153$ & $12$ & $7.8 \pm 2.2$ & $1.46 \rm X$ \\
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\hline
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\end{tabular}
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\caption{Muon Isolation.\label{tab:HFSF}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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To a large extent lepton isolation and identification are uncorrelated with impact parameter. Therefore, we can use $f_{HF}$ to correct MC predictions for the number of heavy flavor events with low-IP leptons that pass our full event selection. First, we select events in MC with a good Z1 and two $\sigma(IP) < 4$ lepton candidates satisfying the denominator requirements given above. Next, the candidates are required to be opposite-charge and same flavor with $m(Z2) > 12\rm~GeV$, consistent with a good Z2. Estimates of heavy flavor background can be simply obtained by applying full lepton identification and isolation requirements to the denominator leptons. To reduce statistical uncertainties, we categorize the predictions according to Z2 lepton flavor and ignore the flavor of Z1. These results are given in the first column of Table~\ref{tab:HFyield}.
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|c|c|}
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\hline
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channel & uncorrected & corrected \\
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\hline
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$ee$ & $0.130 \pm 0.X$ & $0.364 \pm 0.X$ \\
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$\mu\mu$ & $0.095 \pm 0.X$ & $0.203 \pm 0.X$ \\
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\hline
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\end{tabular}
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\caption{(Un)Corrected Z2 Yields.\label{tab:HFyield}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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Assuming an equal number of electron and muon-channel Z1 decays and accounting for data/MC efficiency differences appropriate for each Z1 flavor ({\bf do this!!!}), we obtain the uncorrected predictions for the individual $4e$, $\mu4$ and $2e2\mu$ channels listed in Table~\ref{tab:HFyield2}. To obtain corrected predictions, we weight the total Z2 yields with two powers of the relevant $f_{HF}$ and repeat the procedure. Corrected $4e$, $\mu4$ and $2e2\mu$ yields are listed in the second column of Table~\ref{tab:HFyield2}.
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%-------------------------------------------------
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\begin{table}[tbh]
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\begin{center}
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\begin{tabular}{|c|c|c|}
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\hline
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channel & uncorrected & corrected \\
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\hline
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$4e$ & $0.065 \pm 0.X$ & $0.182 \pm 0.X$ \\
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$4\mu$ & $0.047 \pm 0.X$ & $0.102 \pm 0.X$ \\
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$2e2\mu$ & $0.113 \pm 0.X$ & $0.295 \pm 0.X$ \\
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\hline
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total & $0.225 \pm 0.X$ & $0.579 \pm 0.X$ \\
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\hline
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\end{tabular}
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\caption{(Un)Corrected Heavy Flavor Yields.\label{tab:HFyield2}}
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\end{center}
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\end{table}
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%-------------------------------------------------
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We determine a shape for heavy flavor background in the signal region from the distribution of $m(4\ell)$ in the high-IP control region. Neither data nor MC contain a sufficient number events with Z2 leptons that pass ID and isolation for us to directly predict the shape, so we instead take the Z1 + $2\times$ denominator shape from data as a template. Figure~\ref{fig:HFshape} compares the $m(4\ell)$ distribution for this selection in data and MC. The figure MC distributions are cross-section normalized and correspond to the $91.1$ events mentioned above. The leftmost plot shows that the predicted $m(4\ell)$ shape is not a good representation of the shape in data. The observed difference is consistent with a simulated energy scale that is too small for electrons below $20~\rm GeV$ (see Figure~\ref{fig:HFshapeScaled}). We choose the shape from data as our signal region template and assign a systematic shape uncertainties based on the observed deviation with the MC shape.
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{figs/m4l-HF.png}
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\caption{Heavy Flavor $m(4\ell)$ Shape.\label{fig:HFshape} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
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%-------------------------------------------------
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{figs/m4l-HF.png}
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\caption{Heavy Flavor $m(4\ell)$ Shape.\label{fig:HFshapeScaled} }
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\end{center}
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\end{figure}
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%-------------------------------------------------
|
