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\subsection{Dilepton studies in CR4}
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\label{sec:cr4}
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[DO WE NEED TO BETTER SPECIFY THE SELECTION FOR THIS REGION???]
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\subsubsection{Modeling of Additional Hard Jets in Top Dilepton Events}
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\label{sec:jetmultiplicity}
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[THIS SUBSUBSECTION IS DONE...MODULO THE LATEST PLOTS AND THE LATEST
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NUMBERS IN THE TABLE]
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Dilepton \ttbar\ events have 2 jets from the top decays, so additional
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jets from radiation or higher order contributions are required to
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enter the signal sample. The modeling of addtional jets in \ttbar\
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events is checked in a \ttll\ control sample,
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selected by requiring
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\begin{itemize}
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\item exactly 2 selected electrons or muons with \pt $>$ 20 GeV
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\item \met\ $>$ 100 GeV
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\item $\geq1$ b-tagged jet
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\item Z-veto
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\end{itemize}
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Figure~\ref{fig:dileptonnjets} shows a comparison of the jet
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multiplicity distribution in data and MC for this two-lepton control
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sample. After requiring at least 1 b-tagged jet, most of the
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events have 2 jets, as expected from the dominant process \ttll. There is also a
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significant fraction of events with additional jets.
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The 3-jet sample is mainly comprised of \ttbar\ events with 1 additional
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emission and similarly the $\ge4$-jet sample contains primarily
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$\ttbar+\ge2$ jet events.
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%Even though the primary \ttbar\
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%Madgraph sample used includes up to 3 additional partons at the Matrix
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%Element level, which are intended to describe additional hard jets,
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%Figure~\ref{fig:dileptonnjets} shows a slight mis-modeling of the
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%additional jets.
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\begin{figure}[hbt]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{plots/njets_all_met100_mueg.pdf}
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\includegraphics[width=0.5\linewidth]{plots/njets_all_met100_diel.pdf}%
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\includegraphics[width=0.5\linewidth]{plots/njets_all_met100_dimu.pdf}
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\caption{
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\label{fig:dileptonnjets}%\protect
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Comparison of the jet multiplicity distribution in data and MC for dilepton events in the \E-\M\
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(top), \E-\E\ (bottom left) and \M-\M\ (bottom right) channels.}
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\end{center}
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\end{figure}
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It should be noted that in the case of \ttll\ events
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with a single reconstructed lepton, the other lepton may be
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mis-reconstructed as a jet. For example, a hadronic tau may be
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mis-identified as a jet (since no $\tau$ identification is used).
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In this case only 1 additional jet from radiation may suffice for
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a \ttll\ event to enter the signal sample. As a result, both the
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samples with $\ttbar+1$ jet and $\ttbar+\ge2$ jets are relevant for
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estimating the top dilepton bkg in the signal region.
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%In this section we discuss a correction to $ N_{2 lep}^{MC} $ in Equation XXX
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%due to differences in the modelling of the jet multiplicity in data versus MC.
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%The same correction also enters $ N_{peak}^{MC}$ in Equation XXX to the extend that the
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%dilepton contributions to $ N_{peak}^{MC}$ gets corrected.
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%The dilepton control sample is defined by the following requirements:
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%\begin{itemize}
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%\item Exactly 2 selected electrons or muons with \pt $>$ 20 GeV
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%\item \met\ $>$ 50 GeV
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%\item $\geq1$ b-tagged jet
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%\end{itemize}
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%
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%This sample is dominated by \ttll. The distribution of \njets\ for data and MC passing this selection is displayed in Fig.~\ref{fig:dilepton_njets}.
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%We use this distribution to derive scale factors which reweight the \ttll\ MC \njets\ distribution to match the data. We define the following
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%quantities
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%
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%\begin{itemize}
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%\item $N_{2}=$ data yield minus non-dilepton \ttbar\ MC yield for \njets\ $\leq$ 2
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%\item $N_{3}=$ data yield minus non-dilepton \ttbar\ MC yield for \njets\ = 3
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%\item $N_{4}=$ data yield minus non-dilepton \ttbar\ MC yield for \njets\ $\geq$ 4
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%\item $M_{2}=$ dilepton \ttbar\ MC yield for \njets\ $\leq$ 2
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%\item $M_{3}=$ dilepton \ttbar\ MC yield for \njets\ = 3
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%\item $M_{4}=$ dilepton \ttbar\ MC yield for \njets\ $\geq$ 4
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%\end{itemize}
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%
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%We use these yields to define 3 scale factors, which quantify the data/MC ratio in the 3 \njets\ bins:
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%
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%\begin{itemize}
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%\item $SF_2 = N_2 / M_2$
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%\item $SF_3 = N_3 / M_3$
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%\item $SF_4 = N_4 / M_4$
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%\end{itemize}
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%
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%And finally, we define the scale factors $K_3$ and $K_4$:
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%
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%\begin{itemize}
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%\item $K_3 = SF_3 / SF_2$
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%\item $K_4 = SF_4 / SF_2$
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%\end{itemize}
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%
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%The scale factor $K_3$ is extracted from dilepton \ttbar\ events with \njets = 3, which have exactly 1 ISR jet.
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%The scale factor $K_4$ is extracted from dilepton \ttbar\ events with \njets $\geq$ 4, which have at least 2 ISR jets.
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%Both of these scale factors are needed since dilepton \ttbar\ events which fall in our signal region (including
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%the \njets $\geq$ 4 requirement) may require exactly 1 ISR jet, in the case that the second lepton is reconstructed
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%as a jet, or at least 2 ISR jets, in the case that the second lepton is not reconstructed as a jet. These scale
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%factors are applied to the dilepton \ttbar\ MC only. For a given MC event, we determine whether to use $K_3$ or $K_4$
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%by counting the number of reconstructed jets in the event ($N_{\rm{jets}}^R$) , and subtracting off any reconstructed
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%jet which is matched to the second lepton at generator level ($N_{\rm{jets}}^\ell$); $N_{\rm{jets}}^{\rm{cor}} = N_{\rm{jets}}^R - N_{\rm{jets}}^\ell$.
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%For events with $N_{\rm{jets}}^{\rm{cor}}=3$ the factor $K_3$ is applied, while for events with $N_{\rm{jets}}^{\rm{cor}}\geq4$ the factor $K_4$ is applied.
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%For all subsequent steps, the scale factors $K_3$ and $K_4$ have been
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%applied to the \ttll\ MC.
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Table~\ref{tab:njetskfactors} shows scale factors ($K_3$ and $K_4$)
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used to correct the
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fraction of events with additional jets in MC to the observed fraction
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in data. These scale factors are calculated from Fig.~\ref{fig:dileptonnjets}
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as follows:
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\begin{itemize}
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\item $N_{2}=$ data yield minus non-dilepton \ttbar\ MC yield for \njets\ $\leq$ 2
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\item $N_{3}=$ data yield minus non-dilepton \ttbar\ MC yield for \njets\ = 3
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\item $N_{4}=$ data yield minus non-dilepton \ttbar\ MC yield for \njets\ $\geq$ 4
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\item $M_{2}=$ dilepton \ttbar\ MC yield for \njets\ $\leq$ 2
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\item $M_{3}=$ dilepton \ttbar\ MC yield for \njets\ = 3
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\item $M_{4}=$ dilepton \ttbar\ MC yield for \njets\ $\geq$ 4
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\end{itemize}
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\noindent then
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\begin{itemize}
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\item $SF_2 = N_2 / M_2$
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\item $SF_3 = N_3 / M_3$
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\item $SF_4 = N_4 / M_4$
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\item $K_3 = SF_3 / SF_2$
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\item $K_4 = SF_4 / SF_2$
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\end{itemize}
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\noindent This insures that $K_3 M_3/(M_2 + K_3 M_3 + K_4 M_4) = N_3 /
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(N_2+N_3+N_4)$ and similarly for the $\geq 4$ jet bin.
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The factors $K_3$ and $K_4$ are applied to the \ttll\ MC throughout the
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entire analysis, i.e.
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whenever \ttll\ MC is used to estimate or subtract
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a yield or distribution.
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%
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In order to do so, it is first necessary to count the number of
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additional jets from radiation and exclude leptons mis-identified as
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jets. A jet is considered a mis-identified lepton if it is matched to a
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generator-level second lepton with sufficient energy to satisfy the jet
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\pt\ requirement ($\pt>30~\GeV$). Then \ttll\ events that need two
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radiation jets to enter our selection are scaled by $K_4$,
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while those that only need one radiation jet are scaled by $K_3$.
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\begin{table}[!ht]
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\begin{center}
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\begin{tabular}{l|c}
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\hline
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Jet Multiplicity Sample
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& Data/MC Scale Factor \\
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\hline
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\hline
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N jets $= 3$ (sensitive to $\ttbar+1$ extra jet from radiation) &
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$K_3 = 0.97 \pm 0.03$\\
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N jets $\ge4$ (sensitive to $\ttbar+\ge2$ extra jets from radiation)
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& $K_4 = 0.91 \pm 0.04$\\
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\hline
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\end{tabular}
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\caption{Data/MC scale factors used to account for differences in the
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fraction of events with additional hard jets from radiation in
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\ttll\ events. \label{tab:njetskfactors}}
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\end{center}
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\end{table}
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\clearpage
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\subsubsection{Validation of the ``Physics'' Modelling of the \ttdl\
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MC in CR4}
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\subsubsection{sec:CR4-valid}
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[THE TEXT IN THIS SUBSECTION IS ESSENTIALLY COMPLETE]
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As mentioned above, $t\bar{t} \to $ dileptons where one of the leptons
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is somehow lost constitutes the main background.
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The object of this test is to validate the $M_T$ distribution of this
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background by looking at the $M_T$ distribution of well identified
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dilepton events.
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We construct a transverse mass variable from the leading lepton and
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the \met\. We distinguish between events with leading electrons and
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leading muons.
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The $t\bar{t}$ MC is corrected using the $K_3$ and $K_4$ factors
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from Section~\ref{sec:jetmultiplicity}. It is also normalized to the
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total data yield separately for the \met\ requirements of signal
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regions A, B, C, and D. These normalization factors are listed
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in Table~\ref{tab:cr4mtsf} and are close to unity.
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The underlying \met\ and $M_T$ distributions are shown in
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Figures~\ref{fig:cr4met} and~\ref{fig:cr4rest}. The data-MC agreement
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is quite good. Quantitatively, this is also shown in Table~\ref{tab:cr4yields}.
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\begin{table}[!h]
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\begin{center}
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\begin{tabular}{l||c|c|c|c}
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\hline
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Sample & CR4A & CR4B & CR4C & CR4D \\
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\hline
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\hline
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Muon Data/MC-SF & $0.91 \pm 0.04$ & $0.94 \pm 0.07$ & $1.06 \pm 0.13$ & $1.03 \pm 0.22$ \\
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\hline
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\hline
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Electron Data/MC-SF & $0.95 \pm 0.04$ & $1.00 \pm 0.08$ & $0.85 \pm 0.12$ & $0.83 \pm 0.19$ \\
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\hline
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\end{tabular}
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\caption{ Data/MC scale factors for total yields, applied to compare
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the shapes of the distributions.
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The uncertainties are statistical only.
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\label{tab:cr4mtsf}}
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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}{l||c|c|c|c}
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\hline
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Sample & CR4A & CR4B & CR4C & CR4D \\
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\hline
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\hline
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Muon MC & $199 \pm 7$ & $102 \pm 6$ & $29 \pm 3$ & $8 \pm 1$ \\
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Muon Data & $187$ & $108$ & $34$ & $9$ \\
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\hline
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Muon Data/MC SF & $0.94 \pm 0.08$ & $1.06 \pm 0.12$ & $1.17 \pm 0.23$ & $1.09 \pm 0.40$ \\
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\hline
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\hline
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Electron MC & $203 \pm 8$ & $97 \pm 5$ & $26 \pm 2$ & $8 \pm 1$ \\
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Electron Data & $201$ & $102$ & $25$ & $5$ \\
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\hline
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Electron Data/MC SF & $0.99 \pm 0.08$ & $1.06 \pm 0.12$ & $0.97 \pm 0.21$ & $0.60 \pm 0.29$ \\
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\hline
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\end{tabular}
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\caption{ Yields in \mt\ tail comparing the MC prediction (after
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applying SFs) to data. The uncertainties are statistical only.
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\label{tab:cr4yields}}
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\end{center}
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\end{table}
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\begin{figure}[hbt]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/met_met50_leadmuo_nj4.pdf}%
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/met_met50_leadele_nj4.pdf}
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met100_leadmuo_nj4.pdf}%
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met100_leadele_nj4.pdf}
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\caption{
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Comparison of the \met\ (top) and \mt\ for $\met>100$ (bottom) distributions in data vs. MC for events
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with a leading muon (left) and leading electron (right)
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satisfying the requirements of CR4.
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\label{fig:cr4met}
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}
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\end{center}
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\end{figure}
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\begin{figure}[hbt]
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\begin{center}
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met150_leadmuo_nj4.pdf}%
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met150_leadele_nj4.pdf}
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met200_leadmuo_nj4.pdf}%
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met200_leadele_nj4.pdf}
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met250_leadmuo_nj4.pdf}%
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\includegraphics[width=0.5\linewidth]{plots/CR4plots/mt_met250_leadele_nj4.pdf}
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\caption{
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Comparison of the \mt\ distribution in data vs. MC for events
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with a leading muon (left) and leading electron (right)
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satisfying the requirements of CR4. The \met\ requirements used are
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150 GeV (top), 200 GeV (middle) and 250 GeV (bottom).
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\label{fig:cr4mtrest}
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}
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\end{center}
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\end{figure}
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\clearpage
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