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\section{Fake Rate}
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\label{sec:FR}
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\subsection{Intro}
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The Fake Rate (FR) method has been described in a
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separate analysis note~\cite{ref:FR} and applied
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to a number of Monte Carlo studies\cite{ref:topdil2009},
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\cite{ref:WW},\cite{ref:SSSusy}. Briefly, jet data
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is used to measure a lepton FR as a function of
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lepton $P_T$ and $|\eta|$ which is defined as the probability
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for a lepton candidate passing loose cuts to also pass the
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analysis cuts. Leptons passing loose cuts are called
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``Fakeable Objects'' (FO).
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In a given analysis the FR is then used to estimate the
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background due to ``fake'' leptons\footnote{Here ``fake'' leptons
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refer to truly fake leptons as well as leptons from heavy flavor decays.}
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as follows:
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\begin{itemize}
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\item Events are selected using all analysis cuts, except
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for the lepton selection. Dilepton backgrounds
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with one real lepton and one fake lepton are estimated by
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selecting one lepton passing the full lepton selection
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and one failing it but passing the FO selection.
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Backgrounds with two fake leptons (QCD) are estimated by requiring
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both lepton candidates to pass the FO selection and fail
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the full selection.
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\item For the QCD backgrounds considered in this note,
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each event is weighted by the product of the two factors
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of FR/(1-FR), where FR is the Fake
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Rate for each of the two FO.
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\item The sum of the weights over the selected events is the
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background prediction.
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\end{itemize}
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\subsection{Fakeable Object Definitions}
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\label{sec:FODefinition}
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Fakeable Objects are defined starting from the full
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lepton selection by relaxing some combination of the identification
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and isolation requirements. Since most muons in QCD events are
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from heavy flavor decays, relaxing the identification requirements
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is not very useful. Thus our muon FO definition consists mainly
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of relaxing the isolation requirement. On the other hand, for
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electrons we have more freedom and we use three separate definitions
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of the FR. Broadly speaking these correspond to relaxing either
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isolation, or ID, or both. Our FO definitions are given below
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Muon FO definition: relax the following muon requirements from
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Section~\ref{sec:muID}:
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\begin{itemize}
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\item $\chi^2$/ndof of global fit $<$ 50 (was $<$ 10).
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\item Transverse impact parameter with respect to the beamspot
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$<$ 2 mm (was $<$ 200 $\mu$m).
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\item $Iso < 0.4$ (was $<$ 0.15).
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\end{itemize}
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Electron V1 FO definition: relax the following electron requirements from
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Section~\ref{sec:eleID}:
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\begin{itemize}
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\item Remove the VBTF90 requirement.
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\item $Iso < 0.4$ (was $<$ 0.15).
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\item The impact parameter cut was removed (used to be $<$ 400 $\mu$m).
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\end{itemize}
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Electron V2 FO definition: relax the following electron requirements from
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Section~\ref{sec:eleID}:
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\begin{itemize}
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\item Remove the VBTF90 requirement.
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\item The impact parameter cut was removed (used to be $<$ 400 $\mu$m).
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\end{itemize}
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Electron V3 FO definition: relax the following electron requirements from
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Section~\ref{sec:eleID}:
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\begin{itemize}
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\item $Iso < 0.4$ (was $<$ 0.15).
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\item The impact parameter cut was removed (used to be $<$ 400 $\mu$m).
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\end{itemize}
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\subsection{Measuring the FR on jet data}
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\label{sec:FRjet}
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The FR is measured by studying lepton candidates in jet data.
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The jet data come from the JetMetTau and JetTauMonitor
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Primary/Secondary datasets, see Section~\ref{sec:datasets}.
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We measure the FR on different jet triggers. The stability
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of the FR in these different jet samples is a measure of the
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robustness of our procedure. We select the following triggers:
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\begin{itemize}
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\item HLT\_L1\_Jet6U from JetMetTauMonitor
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\item HLT\_L1\_Jet10U from JetMetTauMonitor
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\item HLT\_Jet15U from JetMetTau
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\item HLT\_Jet30U from JetMetTau
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\end{itemize}
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In order to eliminate a possible trigger bias, we scan the list
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of HLT objects associated with the given jet trigger. If there
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is only one such object above threshold, we only consider
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lepton candidates well separated ($\Delta R > 1$) from it.
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Furthermore, even on jet triggers it is possible to collect
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some $W$ and $Z$ leptonic decays. These events would cause
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a significant bias to the FR at high $P_T$. To minimize
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this problem, we reject events with tcMet $>$ 20 GeV and
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events with two opposite sign FO that make a mass within
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20 GeV of the $Z$ mass
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We believe that the bias from leftover $W$ and
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Drell Yan events in the jet sample is small, but it remains to
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be estimated.
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The FR are measured as a function of $P_T$ and $|\eta|$.
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The FR projections on the $P_T$ and $|\eta|$ axes for the muon FR and for
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the three (V1, V2, V3) electron FR in the different trigger samples
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are displayed in
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{\color{red} Figures~\ref{fig:muFR} and~\ref{fig:eleFR} (old)} and
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{\color{blue}Figures~\ref{fig:muFR2} and~\ref{fig:eleFR2} (new)}.
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\begin{figure}[htb]
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\begin{center}
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{\color{red}
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\includegraphics[width=0.48\linewidth]{MuFakeRatesJune1.pdf}
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\includegraphics[width=0.48\linewidth]{muFReta.pdf}
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\caption{\label{fig:muFR}The muon fake rate as a function of $P_T$
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and $|\eta|$ in the different jet samples. (Old plots updated below).}
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}
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\end{center}
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\end{figure}
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\begin{figure}[htb]
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\begin{center}
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{\color{blue}
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\includegraphics[width=0.31\linewidth, angle=90]{MuFakeRatesJuly7.pdf}
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\includegraphics[width=0.31\linewidth, angle=90]{muFReta7July.pdf}
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\caption{\label{fig:muFR2}The muon fake rate as a function of $P_T$
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and $|\eta|$ in the different jet samples. Note that these now start at
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$P_T=$ 10 GeV instead of $P_T=$5 GeV as they did in the earlier analysis
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because of a preselection aplied in our data handling. For this reason
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the $\eta$-projections cannot be directly compared since the this is
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dominated by muons of $P_T$ near threshold. Note also that the
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muon identification requirements have changed a little bit, as
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described in Section 3.1.5. (New updated plots).}
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}
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\end{center}
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\end{figure}
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\begin{figure}[htb]
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\begin{center}
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{\color{red}
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\includegraphics[width=\linewidth]{ElFakeRatesJune1.pdf}
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\includegraphics[width=\linewidth]{eFReta.pdf}
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\caption{\label{fig:eleFR}The electron fake rates
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(V1,V2,V3) as a function of $P_T$ and $|\eta|$
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in the different jet samples. (Old plots updated below).}
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}
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\end{center}
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\end{figure}
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\begin{figure}[htb]
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\begin{center}
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{\color{blue}
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\includegraphics[width=0.4\linewidth,angle=90 ]{ElFakeRatesJuly7.pdf}
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\includegraphics[width=0.4\linewidth,angle=90]{eFRetaJuly7.pdf}
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\caption{\label{fig:eleFR2}The electron fake rates
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(V1,V2,V3) as a function of $P_T$ and $|\eta|$
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in the different jet samples.
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Note that the spike removal has been added to the electron
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selection since the earlier analysis. This is a very
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small effect. (New updated plots).}
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}
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\end{center}
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\end{figure}
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The fake rates are reasonably stable with respect to
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jet trigger variations, within the $\approx 50\%$ which
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we believe to be a realistic goal for the FR systematics.
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There is some evidence for the V2 electron FR to be more stable than
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the V1 and V3 versions. This is expected from earlier MC studies: the FO
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V2 isolation requirement is the same as in the full electron selection,
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and thus the dependence on the amount of jet activity near the electron
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candidate is minimized.
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The FR binning
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is quite coarse, because of the lack of statistics. This will improve
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as we collect more data. In any case, the
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FR does not appear to change very fast as a function of $P_T$.
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\clearpage
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\subsection{Loosening the isolation requirement for the muon FO}
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Loosening the isolation requirement for the muon FO would reduce the
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muon FR. This is in general ``a good thing''\texttrademark.
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However, the price one then pays is that the jet dependence of the
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FR increases. This is illustrated in Figure~\ref{fig:FRlooseIso}.
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In the $P_T \approx 8$ GeV bin, the ratio of FR in the two extreme triggers
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(HLT\_L1Jet6U and HLT\_Jet30U) increases from 1.6 (FO-Iso $<$ 0.4) to 2.0
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(FO-Iso $<$ 0.6) to 2.4 (FO-Iso $<$ 0.8) to 2.9 (FO-Iso $<$ 1.0). Given this
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behavior, for now we keep the FO isolation requirement to 0.4.
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\begin{figure}[htb]
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\begin{center}
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\includegraphics[width=0.7\linewidth,angle=90]{muonFR_differentIso.pdf}
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\caption{\label{fig:FRlooseIso}Muon FR as a function of $P_T$ in different
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trigger samples for different choices of the maximum isolation
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requirement on the muon FO (0.4, 0.6, 0.8, 1.0).}
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\end{center}
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\end{figure}
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\subsection{Bias due to lepton triggers?}
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Another possible bias is related to the fact that events used to
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measure the FR are collected with jet triggers. On the other hand
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the dilepton analysis is performed using events collected with
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the photon and muon trigger, see Section~\ref{sec:trigger}.
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If the FR depends on the firing of the muon/photon trigger,
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the situation becomes more complicated and
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we may be forced to define different FR depending on whether or
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not the muon/photon trigger fired.
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To test this hypothesis, we select events collected with the
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HLT\_L1\_Jet15U trigger; we then compare the FR computed
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in the standard way (Figures~\ref{fig:muFR} and~\ref{fig:eleFR})
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with the FR on the subset of leptons where the HLT\_Mu5 or
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HLT\_Photon10\_L1R trigger fired\footnote{We made sure that the
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trigger fired, and that the HLT object that fired the trigger was
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matched in $\Delta R$ to the lepton candidate.}.
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The results are displayed in Figures~\ref{fig:muFRtrg}
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and~\ref{fig:eleFRtrg}.
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\begin{figure}[htb]
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\begin{center}
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\includegraphics[width=0.7\linewidth]{MuFakeRatesMu5June1.pdf}
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\caption{\label{fig:muFRtrg}The muon fake rate in HLT\_L1\_Jet15U
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as a function of $P_T$ with and without a requirement on the HLT\_Mu5
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trigger.}
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\end{center}
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\end{figure}
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\begin{figure}[htb]
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\begin{center}
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\includegraphics[width=\linewidth]{ElFakeRatesPhoton10June1.pdf}
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\caption{\label{fig:eleFRtrg}The electron fake rates
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(V1,V2,V3) in HLT\_L1\_Jet15U
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as a function of $P_T$ with and without a requirement on the
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HLT\_Photon10\_L1R trigger.}
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\end{center}
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\end{figure}
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The statistics are not very good, but the lepton trigger bias
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does not seem to be a significant effect. We tentatively neglect it.
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\clearpage
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