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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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[FILL IN SOME XX] |
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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 $60 < M_T < 100$~GeV. |
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As discussed in Section~\ref{sec:ljbg-general}, these ratios are different for $W +$ jets and |
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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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\end{center} |
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\end{table} |
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The MC value of these ratios are shown in Table~\ref{tab:ttp}. |
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In addition the studies of CR1 and CR2 (Sections~\ref{sec:cr1} and~\ref{sec:cr2}) |
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lead to data/MC scale factors |
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$SFR^{e}_{wjets}$ and $SFR^{\mu}_{wjets}$ (Table~\ref{tab:cr1yields}) and |
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$SFR^{e}_{top}$ and $SFR^{\mu}_{top}$ (Table~\ref{tab:cr2yields}) |
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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 $R_{wjet}$ 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}). |
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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 $R_{top}$ 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. |
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For the pessimistic prediction, we use the \wjets\ $M_T$ tail-to-peak ratio and data/MC scale factor, $R_{wjet}$ and $SFR_{wjet}$. |
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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\ $M_T$ tail-to-peak ratio $R_{top}$, 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 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 $R_{top}$ 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 |
