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\section{Example BG prediction calculation}
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\label{BGexample}
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The calculation of the background prediction is a bit complicated.
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Here we walk the reader through a concrete example.
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{\bf NB: the numbers
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in this section corresponded to the numbers in V2 of the analysis note.
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They will not be updated, because this is meant as an illustration only.}.
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The main background is $t\bar{t}$
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The main idea is to normalize to the $M_T$ peak region ($50 < M_T < 80$ GeV).
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This eliminates dependence on \ttbar\ cross-section, luminosity,
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trigger efficiency, JES, lepton ID, etc. This gets a bit complicated because
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the $M_T$ peak region, while dominantly \ttbar\ lepton $+$ jets,
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also includes some \wjets, \ttdl, rare processes, etc. Also, we want
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to minimize our need to understand the effect of the isolated
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track veto on \ttsl. As a result we actually define two $M_T$ peak
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regions: one before and one after applying the isolated track veto.
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Then the \ttdl\ background is normalized the the ``before veto'' region,
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and the \ttsl\ and \wjets\ background are normalized to the ``post veto''
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region.
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This complex procedure is important for the high statistics signal regions
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with relatively low \met\ requirements, eg, SRA. For these SRs we want to keep the
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systematics low in order to be sensistive to low mass stop; for the signal regions
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with hard cuts on \met\, this is less important. However, we apply the same
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procedure to all SRs.
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For concreteness, we show the calculation for SRA, electron channel. The MC and data
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event counts used in the background calculation are collected in Table~\ref{tab:bgexample}.
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Note that the background uncertainties have already been described
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in Section~\ref{sec:systematics}. The one tricky point to keep in mind is that
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when the \wjets\ and rare cross-sections are changed by their assumed uncertainties
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(50\% each), the whole calculation describe below is repeated in order to take care
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of all the correlations properly.
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\begin{table}[!h]
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\begin{center}
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\begin{tabular}{l||c|c|c|}
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\hline
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Sample & $M_T$ peak, before trk veto & $M_T$ peak, after trk veto & Signal Region A \\
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\hline
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\ttdl\ & $290 \pm 6$ & $116 \pm 4$ & $261 \pm 6$ \\
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\ttsl\ (1\Lep) & $2899 \pm 19$ & $2595 \pm 18$ & not used \\
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\wjets\ & $252 \pm 28$ & $236 \pm 28$ & not used \\
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Rare & $89 \pm 5$ & $62 \pm 4$ & $26 \pm 2$ \\
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\hline
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Total & $3530 \pm 35$ & $3009 \pm 34$ & not used \\
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\hline
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Data & 3358 & 2787 & not used \\
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\hline
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\end{tabular}
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\caption{ Data and MC event counts used to predict the background in SRA, for electron events.
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Uncertainties are statistical only. The trigger efficiency has been applied to the MC samples. In the
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case of \ttdl\, the $K_3$ and $K_4$ factors of Section~\ref{sec:jetmultiplicity} have also been applied.
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\label{tab:bgexample}}
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\end{center}
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\end{table}
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\subsection{Central value of dilepton background}
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A ``before veto'' scale factor is defined from the second column in Table~\ref{tab:bgexample} as the factor by which
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all MC except the ``rare'' need to be scaled up in order to have data/MC agreement. This is
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\noindent $SF_{pre} = (3358 - 89)/(2899 + 252 + 290) = 0.950$.
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Then the \ttdl\ background prediction is the number of events predicted by the MC in SRA (261 from the
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last column of SRA), rescaled by $SF_{pre}$. The result for the central value is 248 events.
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\subsection{Central value of the \ttsl\ background}
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\label{sec:cenvttlj}
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A ``post veto'' scale factor is defined from the third column in Table~\ref{tab:bgexample} as the factor by which
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the \ttsl\ and the \wjets\ backgrounds need to be scaled to have data/MC agreement.
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\noindent $SF_{post} = (2787 - 62 - SF_{pre} \cdot 116) / (2595 + 236) = 0.924$.
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Then the \ttsl\ background is obtained as the product of the following four factors
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\begin{itemize}
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\item $SF_{post} = 0.924$ as obtained above
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\item 2595, from the third column of Table~\ref{tab:bgexample}
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\item The tail-to-peak ratio $R_{top} = 0.015$ from Table~\ref{tab:ttp} (we use the average of electrons
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and muons)
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\item The tail-to-peak ratio scale factod $SFR_{top} = 1.89 \pm 0.56$ from Table~\ref{tab:cr2yields}.
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\end{itemize}
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The result for the central value is 68 events.
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\subsection{Central value for the \wjets\ background}
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It is calculated as the product of
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\begin{itemize}
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\item $SF_{post} = 0.924$ from Section~\ref{sec:cenvttlj}
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\item 236, from the third column of Table~\ref{tab:bgexample}
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\item The tail-to-peak ratio $R_{wjet} = 0.04$ from Table~\ref{tab:ttp} (we use the average of electrons and muons)
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\item The tail-to-peak ratio scale factod $SFR_{wjet} = 1.64 \pm 0.38$ from Table~\ref{tab:cr1yields}.
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\end{itemize}
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The result for the central value is 14.3 events.
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\subsection{Central value for the rare backgrounds}
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This is 26 events from Table~\ref{tab:bgexample}
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