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\section{Tail-to-Peak ratio for lepton $+$ jets top and W events}
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\label{sec:ttp}
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An important component
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of the background calculation is the ratio of the number of events with $M_T$ in the signal region
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to the number of events with $50 < M_T < 80$~GeV.
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As discussed in Section~\ref{sec:ljbg-general}, these ratios are different for \wjets\ and
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top events.
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\begin{table}[!h]
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\begin{center}
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{\footnotesize
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\begin{tabular}{l||c|c|c|c|c|c|c}
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\hline
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Sample & SRA & SRB & SRC & SRD & SRE & SRF & SRG\\
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\hline
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\hline
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\multicolumn{8}{c}{Muons} \\
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\hline
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$R^{MC}_{top}$ & $0.015 \pm 0.001$ & $0.035 \pm 0.002$ & $0.021 \pm 0.002$ & $0.021 \pm 0.004$ & $0.025 \pm 0.007$ & $0.015 \pm 0.009$ & $0.021 \pm 0.015$ \\
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$R^{MC}_{wjet}$ & $0.040 \pm 0.001$ & $0.071 \pm 0.003$ & $0.062 \pm 0.004$ & $0.064 \pm 0.006$ & $0.065 \pm 0.009$ & $0.067 \pm 0.012$ & $0.065 \pm 0.016$ \\
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\hline
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\hline
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\hline
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\multicolumn{8}{c}{Electrons} \\
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\hline
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$R^{MC}_{top}$ & $0.015 \pm 0.001$ & $0.031 \pm 0.002$ & $0.026 \pm 0.003$ & $0.025 \pm 0.005$ & $0.009 \pm 0.005$ & $0.021 \pm 0.012$ & $0.034 \pm 0.024$ \\
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$R^{MC}_{wjet}$ & $0.040 \pm 0.002$ & $0.075 \pm 0.004$ & $0.067 \pm 0.005$ & $0.063 \pm 0.007$ & $0.061 \pm 0.010$ & $0.067 \pm 0.015$ & $0.070 \pm 0.021$ \\
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\hline
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\end{tabular}}
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\caption{ Ratio of MC events in the \mt-tail over events in the \mt-peak for
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\ttsl\ (also used for 1-lepton single top) and \wjets. These are
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derived before applying the b-tagging requirement.
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\label{tab:ttp}}
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\end{center}
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\end{table}
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The MC values of these ratios are shown in Table~\ref{tab:ttp}. The e and $\mu$ channel results are averaged before corrections are made.
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The MC value of $R^{MC}_{wjet}$ is corrected based on the studies of CR1 (Section~\ref{sec:cr1}), which
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lead to the data/MC scale factor $SFR_{wjet}$ (Table~\ref{tab:cr1yields}). The corrected $R_{wjet}$ is thus given by $R^{MC}_{wjet} \times SFR_{wjet}$.
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%$SFR_{top}$ (Table~\ref{tab:cr2yields})
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There is no similar scale factor to correct the MC value of $R^{MC}_{top}$ due to the lack of events in CR2 (Section~\ref{sec:cr2}).
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We must therefore use a different procedure to derive a corrected value of $R_{top}$.
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We start by defining optimistic (too small) and pessimistic (too large) predictions for $R_{top}$.
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For the pessimistic prediction, we use the \wjets\ MC tail-to-peak ratio and data/MC scale factor, $R^{MC}_{wjet}$ and $SFR_{wjet}$ (i.e. the pessimistic prediction is the same as $R_{wjet}$).
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This prediction is too large because in \wjets\ events the $M_T$ tail comes from
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off-shell Ws and resolution effects, while in top events to first order
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only resolution effects matter.
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For the optimistic prediction, we use the \ttsl\ MC tail-to-peak ratio $R^{MC}_{top}$, but take the \wjets\ data/MC scale factor $SFR_{wjet}$.
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This prediction is too small because
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the true top scale factor is to first order the same as for on-shell Ws,
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while $SFR_{wjet}$ is a weighted average of the
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scale factor for on-shell Ws (which is $>1$) and the
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scale factor for off-shell Ws (which is close to 1 as it is well modeled by MC).
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The final prediction for $R_{top}$ is given by the average of the optimistic and pessimistic predictions, and
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the systematic uncertainty is given by half the difference between the two.
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The corrected values of $R_{wjet}$ and $R_{top}$ and their uncertainties are given in Table~\ref{tab:ttpcorr}.
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\begin{table}[!h]
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\begin{center}
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{\footnotesize
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\begin{tabular}{l||c|c|c|c|c|c|c}
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\hline
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Sample & SRA & SRB & SRC & SRD & SRE & SRF & SRG\\
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\hline
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\hline
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$R_{top}$ & $0.045 \pm 0.023$ & $0.074 \pm 0.031$ & $0.055 \pm 0.031$ & $0.042 \pm 0.028$ & $0.041 \pm 0.036$ & $0.052 \pm 0.049$ & $0.053 \pm 0.066$ \\
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$R_{wjet}$ & $0.066 \pm 0.015$ & $0.101 \pm 0.022$ & $0.081 \pm 0.025$ & $0.061 \pm 0.029$ & $0.064 \pm 0.042$ & $0.082 \pm 0.062$ & $0.075 \pm 0.088$ \\
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\hline
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\end{tabular}}
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\caption{ Corrected values of $R_{wjet}$ and $R_{top}$. Both statistical and systematic uncertainties are included.
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\label{tab:ttpcorr}}
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\end{center}
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\end{table}
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\clearpage |
