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\section{Data Driven Background Estimation Methods}
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\label{sec:datadriven}
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We have developed two data-driven methods to
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estimate the background in the signal region.
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The first one exploits the fact that
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SumJetPt and \met$/\sqrt{\rm SumJetPt}$ are nearly
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uncorrelated for the $t\bar{t}$ background
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(Section~\ref{sec:abcd}); the second one
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is based on the fact that in $t\bar{t}$ the
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$P_T$ of the dilepton pair is on average
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nearly the same as the $P_T$ of the pair of neutrinos
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from $W$-decays, which is reconstructed as \met in the
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detector.
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%{\color{red} I took these
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%numbers from the twiki, rescaling from 11.06 to 30/pb.
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%They seem too large...are they really right?}
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\subsection{ABCD method}
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\label{sec:abcd}
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We find that in $t\bar{t}$ events SumJetPt and
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\met$/\sqrt{\rm SumJetPt}$ are nearly uncorrelated,
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as demonstrated in Figure~\ref{fig:uncor}.
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Thus, we can use an ABCD method in the \met$/\sqrt{\rm SumJetPt}$ vs
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sumJetPt plane to estimate the background in a data driven way.
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%\begin{figure}[bht]
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%\begin{center}
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%\includegraphics[width=0.75\linewidth]{uncorrelated.pdf}
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%\caption{\label{fig:uncor}\protect Distributions of SumJetPt
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%in MC $t\bar{t}$ events for different intervals of
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%MET$/\sqrt{\rm SumJetPt}$.}
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%\end{center}
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%\end{figure}
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\begin{figure}[bht]
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\begin{center}
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\includegraphics[width=0.75\linewidth]{uncor.png}
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\caption{\label{fig:uncor}\protect Distributions of SumJetPt
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in MC $t\bar{t}$ events for different intervals of
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MET$/\sqrt{\rm SumJetPt}$. h1, h2, and h3 refer to the MET$/\sqrt{\rm SumJetPt}$
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intervals 4.5-6.5, 6.5-8.5 and $>$8.5, respectively.}
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\end{center}
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\end{figure}
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\begin{figure}[tb]
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\begin{center}
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\includegraphics[width=0.5\linewidth, angle=90]{abcdMC.pdf}
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\caption{\label{fig:abcdMC}\protect Distributions of MET$/\sqrt{\rm SumJetPt}$ vs.
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SumJetPt for SM Monte Carlo. Here we also show our choice of ABCD regions.}
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\end{center}
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\end{figure}
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Our choice of ABCD regions is shown in Figure~\ref{fig:abcdMC}.
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The signal region is region D. The expected number of events
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in the four regions for the SM Monte Carlo, as well as the BG
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prediction AC/B are given in Table~\ref{tab:abcdMC} for an integrated
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luminosity of 35 pb$^{-1}$. The ABCD method with chosen boundaries is accurate
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to about 20\%. As shown in Table~\ref{tab:abcdsyst}, we assess systematic uncertainties
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by varying the boundaries by an amount consistent with the hadronic energy scale uncertainty,
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which we take as $\pm$5\% for SumJetPt and $\pm$2.5\% for MET/$\sqrt{\rm SumJetPt}$, since the
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uncertainty on this quantity partially cancels due to the fact that it is a ratio of correlated
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quantities. Based on these studies we assess a correction factor $k_{ABCD} = 1.2 \pm 0.2$ to the
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predicted yield using the ABCD method.
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%{\color{red} Avi wants some statement about stability
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%wrt changes in regions. I am not sure that we have done it and
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%I am not sure it is necessary (Claudio).}
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\begin{table}[ht]
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\begin{center}
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\caption{\label{tab:abcdMC} Expected SM Monte Carlo yields for
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35 pb$^{-1}$ in the ABCD regions, as well as the predicted yield in
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the signal region given by A $\times$ C / B. Here `SM other' is the sum
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of non-dileptonic $t\bar{t}$ decays, $W^{\pm}$+jets, $W^+W^-$,
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$W^{\pm}Z^0$, $Z^0Z^0$ and single top.}
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\begin{tabular}{lccccc}
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\hline
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sample & A & B & C & D & A $\times$ C / B \\
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\hline
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$t\bar{t}\rightarrow \ell^{+}\ell^{-}$ & 7.96 $\pm$ 0.17 & 33.07 $\pm$ 0.35 & 4.81 $\pm$ 0.13 & 1.20 $\pm$ 0.07 & 1.16 $\pm$ 0.04 \\
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$Z^0 \rightarrow \ell^{+}\ell^{-}$ & 0.03 $\pm$ 0.03 & 1.47 $\pm$ 0.38 & 0.10 $\pm$ 0.10 & 0.10 $\pm$ 0.10 & 0.00 $\pm$ 0.00 \\
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SM other & 0.65 $\pm$ 0.06 & 2.31 $\pm$ 0.13 & 0.17 $\pm$ 0.03 & 0.14 $\pm$ 0.03 & 0.05 $\pm$ 0.01 \\
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\hline
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total SM MC & 8.63 $\pm$ 0.18 & 36.85 $\pm$ 0.53 & 5.07 $\pm$ 0.17 & 1.43 $\pm$ 0.12 & 1.19 $\pm$ 0.05 \\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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\begin{table}[ht]
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\begin{center}
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\caption{\label{tab:abcdsyst} Results of the systematic study of the ABCD method by varying the boundaries
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between the ABCD regions shown in Fig.~\ref{fig:abcdMC}. Here $x_1$ is the lower SumJetPt boundary and
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$x_2$ is the boundary separating regions A and B from C and D, their nominal values are 125 and 300~GeV,
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respectively. $y_1$ is the lower MET/$\sqrt{\rm SumJetPt}$ boundary and
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$y_2$ is the boundary separating regions B and C from A and D, their nominal values are 4.5 and 8.5~GeV$^{1/2}$,
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respectively.}
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\begin{tabular}{cccc|c}
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\hline
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$x_1$ & $x_2$ & $y_1$ & $y_2$ & Observed/Predicted \\
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\hline
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nominal & nominal & nominal & nominal & $1.20 \pm 0.12$ \\
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+5\% & +5\% & +2.5\% & +2.5\% & $1.38 \pm 0.15$ \\
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+5\% & +5\% & nominal & nominal & $1.31 \pm 0.14$ \\
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nominal & nominal & +2.5\% & +2.5\% & $1.25 \pm 0.13$ \\
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nominal & +5\% & nominal & +2.5\% & $1.32 \pm 0.14$ \\
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nominal & -5\% & nominal & -2.5\% & $1.16 \pm 0.09$ \\
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-5\% & -5\% & +2.5\% & +2.5\% & $1.21 \pm 0.11$ \\
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+5\% & +5\% & -2.5\% & -2.5\% & $1.26 \pm 0.12$ \\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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\subsection{Dilepton $P_T$ method}
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\label{sec:victory}
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This method is based on a suggestion by V. Pavlunin\cite{ref:victory},
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and was investigated by our group in 2009\cite{ref:ourvictory}.
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The idea is that in dilepton $t\bar{t}$ events the lepton and neutrinos
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from $W$ decays have the same $P_T$ spectrum (modulo $W$ polarization
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effects). One can then use the observed
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$P_T(\ell\ell)$ distribution to model the sum of neutrino $P_T$'s which
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is identified with the \met.
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Then, in order to predict the $t\bar{t} \to$ dilepton contribution to a
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selection with \met$+$X, one applies a cut on $P_T(\ell\ell)+$X instead.
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In practice one has to rescale the result of the $P_T(\ell\ell)+$X selection
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to account for the fact that any dilepton selection must include a
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moderate \met cut in order to reduce Drell Yan backgrounds. This
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is discussed in Section 5.3 of Reference~\cite{ref:ourvictory}; for a \met
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cut of 50 GeV, the rescaling factor is obtained from the MC as
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\newcommand{\ptll} {\ensuremath{P_T(\ell\ell)}}
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\begin{center}
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$ K = \frac{\int_0^{\infty} {\cal N}(\ptll)~~d\ptll~}{\int_{50}^{\infty} {\cal N}(\ptll)~~d\ptll~}$
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\end{center}
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Monte Carlo studies give values of $K$ that are typically between 1.5 and 1.6,
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depending on selection details.
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%%%TO BE REPLACED
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%Given the integrated luminosity of the
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%present dataset, the determination of $K$ in data is severely statistics
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%limited. Thus, we take $K$ from $t\bar{t}$ Monte Carlo as
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%\begin{center}
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%$ K_{MC} = \frac{\int_0^{\infty} {\cal N}(\met)~~d\met~}{\int_{50}^{\infty} {\cal N}(\met)~~d\met~}$
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%\end{center}
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%\noindent {\color{red} For the 11 pb result we have used $K$ from data.}
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There are several effects that spoil the correspondance between \met and
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$P_T(\ell\ell)$:
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\begin{itemize}
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\item $Ws$ in top events are polarized. Neutrinos are emitted preferentially
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parallel to the $W$ velocity while charged leptons are emitted prefertially
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anti-parallel. Thus the $P_T(\nu\nu)$ distribution is harder
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than the $P_T(\ell\ell)$ distribution for top dilepton events.
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\item The lepton selections results in $P_T$ and $\eta$ cuts on the individual
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leptons that have no simple correspondance to the neutrino requirements.
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\item Similarly, the \met$>$50 GeV cut introduces an asymmetry between leptons and
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neutrinos which is only partially compensated by the $K$ factor above.
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\item The \met resolution is much worse than the dilepton $P_T$ resolution.
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When convoluted with a falling spectrum in the tails of \met, this results
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in a harder spectrum for \met than the original $P_T(\nu\nu)$.
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\item The \met response in CMS is not exactly 1. This causes a distortion
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in the \met distribution that is not present in the $P_T(\ell\ell)$ distribution.
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\item The $t\bar{t} \to$ dilepton signal includes contributions from
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$W \to \tau \to \ell$. For these events the arguments about the equivalence
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of $P_T(\ell\ell)$ and $P_T(\nu\nu)$ do not apply.
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\item A dilepton selection will include SM events from non $t\bar{t}$
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sources. These events can affect the background prediction. Particularly
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dangerous are high $P_T$ Drell Yan events that barely pass the \met$>$ 50
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GeV selection. They will tend to push the data-driven background prediction up.
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Therefore we estimate the number of DY events entering the background prediction
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using the $R_{out/in}$ method as described in Sec.~\ref{sec:othBG}.
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\end{itemize}
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We have studied these effects in SM Monte Carlo, using a mixture of generator and
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reconstruction level studies, putting the various effects in one at a time.
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For each configuration, we apply the data-driven method and report as figure
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of merit the ratio of observed and predicted events in the signal region.
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The results are summarized in Table~\ref{tab:victorybad}.
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\begin{table}[htb]
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| 194 |
\begin{center}
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| 195 |
\caption{\label{tab:victorybad} Test of the data driven method in Monte Carlo
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under different assumptions. See text for details.}
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\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
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\hline
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& True $t\bar{t}$ dilepton & $t\to W\to\tau$& other SM & GEN or & Lepton $P_T$ & Z veto & \met $>$ 50& obs/pred \\
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& included & included & included & RECOSIM & and $\eta$ cuts & & & \\ \hline
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1&Y & N & N & GEN & N & N & N & 1.90 \\
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2&Y & N & N & GEN & Y & N & N & 1.64 \\
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3&Y & N & N & GEN & Y & Y & N & 1.59 \\
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4&Y & N & N & GEN & Y & Y & Y & 1.55 \\
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5&Y & N & N & RECOSIM & Y & Y & Y & 1.51 \\
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6&Y & Y & N & RECOSIM & Y & Y & Y & 1.58 \\
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7&Y & Y & Y & RECOSIM & Y & Y & Y & 1.38 \\
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%%%NOTE: updated value 1.18 -> 1.46 since 2/3 DY events have been removed by updated analysis selections,
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%%%dpt/pt cut and general lepton veto
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\hline
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| 211 |
\end{tabular}
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| 212 |
\end{center}
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| 213 |
\end{table}
|
| 214 |
|
| 215 |
|
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The largest discrepancy between prediction and observation occurs on the first
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line of Table~\ref{tab:victorybad}, {\em i.e.}, at the generator level with no
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cuts. We have verified that this effect is due to the polarization of
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the $W$ (we remove the polarization by reweighting the events and we get
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good agreement between prediction and observation). The kinematical
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requirements (lines 2,3,4) compensate somewhat for the effect of W polarization.
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Going from GEN to RECOSIM, the change in observed/predicted is small.
|
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% We have tracked this down to the fact that tcMET underestimates the true \met
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% by $\approx 4\%$\footnote{We find that observed/predicted changes by roughly 0.1
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%for each 1.5\% change in \met response.}.
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Finally, contamination from non $t\bar{t}$
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events can have a significant impact on the BG prediction.
|
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%The changes between
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| 229 |
%lines 6 and 7 of Table~\ref{tab:victorybad} is driven by 3
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| 230 |
%Drell Yan events that pass the \met selection in Monte Carlo (thus the effect
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%is statistically not well quantified).
|
| 232 |
|
| 233 |
An additional source of concern is that the CMS Madgraph $t\bar{t}$ MC does
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| 234 |
not include effects of spin correlations between the two top quarks.
|
| 235 |
We have studied this effect at the generator level using Alpgen. We find
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| 236 |
that the bias is at the few percent level.
|
| 237 |
|
| 238 |
%%%TO BE REPLACED
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| 239 |
%Based on the results of Table~\ref{tab:victorybad}, we conclude that the
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| 240 |
%naive data driven background estimate based on $P_T{(\ell\ell)}$ needs to
|
| 241 |
%be corrected by a factor of {\color{red} $ K_{\rm{fudge}} =1.2 \pm 0.3$
|
| 242 |
%(We still need to settle on thie exact value of this.
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| 243 |
%For the 11 pb analysis it is taken as =1.)} . The quoted
|
| 244 |
%uncertainty is based on the stability of the Monte Carlo tests under
|
| 245 |
%variations of event selections, choices of \met algorithm, etc.
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| 246 |
%For example, we find that observed/predicted changes by roughly 0.1
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| 247 |
%for each 1.5\% change in the average \met response.
|
| 248 |
|
| 249 |
Based on the results of Table~\ref{tab:victorybad}, we conclude that the
|
| 250 |
naive data driven background estimate based on $P_T{(\ell\ell)}$ needs to
|
| 251 |
be corrected by a factor of $ K_C = X \pm Y$.
|
| 252 |
The value of this correction factor as well as the systematic uncertainty
|
| 253 |
will be assessed using 38X ttbar madgraph MC. In the following we use
|
| 254 |
$K_C = 1$ for simplicity. Based on previous MC studies we foresee a correction
|
| 255 |
factor of $K_C \approx 1.2 - 1.5$, and we will assess an uncertainty
|
| 256 |
based on the stability of the Monte Carlo tests under
|
| 257 |
variations of event selections, choices of \met algorithm, etc.
|
| 258 |
For example, we find that observed/predicted changes by roughly 0.1
|
| 259 |
for each 1.5\% change in the average \met response.
|
| 260 |
|
| 261 |
|
| 262 |
|
| 263 |
\subsection{Signal Contamination}
|
| 264 |
\label{sec:sigcont}
|
| 265 |
|
| 266 |
All data-driven methods are in principle subject to signal contaminations
|
| 267 |
in the control regions, and the methods described in
|
| 268 |
Sections~\ref{sec:abcd} and~\ref{sec:victory} are not exceptions.
|
| 269 |
Signal contamination tends to dilute the significance of a signal
|
| 270 |
present in the data by inflating the background prediction.
|
| 271 |
|
| 272 |
It is hard to quantify how important these effects are because we
|
| 273 |
do not know what signal may be hiding in the data. Having two
|
| 274 |
independent methods (in addition to Monte Carlo ``dead-reckoning'')
|
| 275 |
adds redundancy because signal contamination can have different effects
|
| 276 |
in the different control regions for the two methods.
|
| 277 |
For example, in the extreme case of a
|
| 278 |
new physics signal
|
| 279 |
with $P_T(\ell \ell) = \met$, an excess of events would be seen
|
| 280 |
in the ABCD method but not in the $P_T(\ell \ell)$ method.
|
| 281 |
|
| 282 |
|
| 283 |
The LM points are benchmarks for SUSY analyses at CMS. The effects
|
| 284 |
of signal contaminations for a couple such points are summarized
|
| 285 |
in Table~\ref{tab:sigcont}. Signal contamination is definitely an important
|
| 286 |
effect for these two LM points, but it does not totally hide the
|
| 287 |
presence of the signal.
|
| 288 |
|
| 289 |
|
| 290 |
\begin{table}[htb]
|
| 291 |
\begin{center}
|
| 292 |
\caption{\label{tab:sigcont} Effects of signal contamination
|
| 293 |
for the two data-driven background estimates. The three columns give
|
| 294 |
the expected yield in the signal region and the background estimates
|
| 295 |
using the ABCD and $P_T(\ell \ell)$ methods. Results are normalized to 35~pb$^{-1}$.}
|
| 296 |
\begin{tabular}{lccc}
|
| 297 |
\hline
|
| 298 |
& Yield & ABCD & $P_T(\ell \ell)$ \\
|
| 299 |
\hline
|
| 300 |
SM only & 1.43 & 1.19 & 1.03 \\
|
| 301 |
SM + LM0 & 7.90 & 4.23 & 2.35 \\
|
| 302 |
SM + LM1 & 4.00 & 1.53 & 1.51 \\
|
| 303 |
\hline
|
| 304 |
\end{tabular}
|
| 305 |
\end{center}
|
| 306 |
\end{table}
|
| 307 |
|
| 308 |
|
| 309 |
|
| 310 |
%\begin{table}[htb]
|
| 311 |
%\begin{center}
|
| 312 |
%\caption{\label{tab:sigcontABCD} Effects of signal contamination
|
| 313 |
%for the background predictions of the ABCD method including LM0 or
|
| 314 |
%LM1. Results
|
| 315 |
%are normalized to 30 pb$^{-1}$.}
|
| 316 |
%\begin{tabular}{|c|c||c|c||c|c|}
|
| 317 |
%\hline
|
| 318 |
%SM & BG Prediction & SM$+$LM0 & BG Prediction & SM$+$LM1 & BG Prediction \\
|
| 319 |
%Background & SM Only & Contribution & Including LM0 & Contribution & Including LM1 \\ \hline
|
| 320 |
%1.2 & 1.0 & 6.8 & 3.7 & 3.4 & 1.3 \\
|
| 321 |
%\hline
|
| 322 |
%\end{tabular}
|
| 323 |
%\end{center}
|
| 324 |
%\end{table}
|
| 325 |
|
| 326 |
%\begin{table}[htb]
|
| 327 |
%\begin{center}
|
| 328 |
%\caption{\label{tab:sigcontPT} Effects of signal contamination
|
| 329 |
%for the background predictions of the $P_T(\ell\ell)$ method including LM0 or
|
| 330 |
%LM1. Results
|
| 331 |
%are normalized to 30 pb$^{-1}$.}
|
| 332 |
%\begin{tabular}{|c|c||c|c||c|c|}
|
| 333 |
%\hline
|
| 334 |
%SM & BG Prediction & SM$+$LM0 & BG Prediction & SM$+$LM1 & BG Prediction \\
|
| 335 |
%Background & SM Only & Contribution & Including LM0 & Contribution & Including LM1 \\ \hline
|
| 336 |
%1.2 & 1.0 & 6.8 & 2.2 & 3.4 & 1.5 \\
|
| 337 |
%\hline
|
| 338 |
%\end{tabular}
|
| 339 |
%\end{center}
|
| 340 |
%\end{table}
|
| 341 |
|
