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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 explouts the fact that
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\met 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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In 30 pb$^{-1}$ we expect $\approx$ 1 SM event in
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the signal region. The expectations from the LMO
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and LM1 SUSY benchmark points are 15.1 and
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6.0 events respectively. {\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 \met and
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\met$/\sqrt{\rm SumJetPt}$ are nearly uncorrelated.
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This is 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}[tb]
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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}[bt]
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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 SumJetPt
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vs. MET$/\sqrt{\rm SumJetPt}$ for SM Monte Carlo. Here we also
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show our choice of ABCD regions. {\color{red} Derek, I
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do not know if this is SM or $t\bar{t}$ only.}}
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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 30 pb$^{-1}$. The ABCD method is accurate
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to about 10\%.
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\begin{table}[htb]
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\begin{center}
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\caption{\label{tab:abcdMC} Expected SM Monte Carlo yields for
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30 pb$^{-1}$ in the ABCD regions.}
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\begin{tabular}{|l|c|c|c|c||c|}
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\hline
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Sample & A & B & C & D & AC/D \\ \hline
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ttdil & 6.9 & 28.6 & 4.2 & 1.0 & 1.0 \\
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Zjets & 0.0 & 1.3 & 0.1 & 0.1 & 0.0 \\
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Other SM & 0.5 & 2.0 & 0.1 & 0.1 & 0.0 \\ \hline
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total MC & 7.4 & 31.9 & 4.4 & 1.2 & 1.0 \\ \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 data 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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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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forward in the $W$ rest frame, 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 result
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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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\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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\begin{center}
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\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|}
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\hline
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& True $t\bar{t}$ dilepton & $t\to W\to\tau$& other SM & GEN or & Lepton $P_T$ & \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 & \\
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2&Y & N & N & GEN & Y & N & \\
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3&Y & N & N & GEN & Y & Y & \\
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4&Y & N & N & RECOSIM & Y & Y & \\
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5&Y & Y & N & RECOSIM & Y & Y & \\
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6&Y & Y & Y & RECOSIM & Y & Y & \\
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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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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 and 3) do not have a significant additional effect.
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Going from GEN to RECOSIM there is a significant change in observed/predicted.
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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.}. Finally, contamination from non $t\bar{t}$
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events can have a significant impact on the BG prediction. The changes between
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lines 5 and 6 of Table~\ref{tab:victorybad} is driven by only {\color{red} 3}
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Drell Yan events that pass the \met selection.
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An additional source of concern is that the CMS Madgraph $t\bar{t}$ MC does
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not include effects of spin correlations between the two top quarks.
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We have studied this effect at the generator level using Alpgen. We find
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that the bias is a the few percent level.
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Based on the results of Table~\ref{tab:victorybad}, we conclude that the
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naive data driven background estimate based on $P_T{\ell\ell)}$ needs to
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be corrected by a factor of {\color{red} $1.4 \pm 0.3$ (We need to
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decide what this number should be)}. The quoted
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uncertainty is based on the stability of the Monte Carlo tests under
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variations of event selections, choices of \met algorithm, etc.
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\subsection{Signal Contamination}
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\label{sec:sigcont}
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All data-driven methods are principle subject to signal contaminations
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in the control regions, and the methods described in
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Sections~\ref{sec:abcd} and~\ref{sec:victory} are not exceptions.
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Signal contamination tends to dilute the significance of a signal
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present in the data by inflating the background prediction.
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It is hard to quantify how important these effects are because we
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do not know what signal may be hiding in the data. Having two
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independent methods (in addition to Monte Carlo ``dead-reckoning'')
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adds redundancy because signal contamination can have different effects
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in the different control regions for the two methods.
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For example, in the extreme case of a
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new physics signal
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with $P_T(\ell \ell) = \met$, an excess of ev ents would be seen
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in the ABCD method but not in the $P_T(\ell \ell)$ method.
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The LM points are benchmarks for SUSY analyses at CMS. The effects
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of signal contaminations for a couple such points are summarized
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in Table~\ref{tab:sigcontABCD} and~\ref{tab:sigcontPT}.
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Signal contamination is definitely an important
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effect for these two LM points, but it does not totally hide the
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presence of the signal.
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\begin{table}[htb]
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\begin{center}
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\caption{\label{tab:sigcontABCD} Effects of signal contamination
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for the background predictions of the ABCD method including LM0 or
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LM1. Results
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are normalized to 30 pb$^{-1}$.}
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\begin{tabular}{|c||c|c||c|c|}
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\hline
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SM & LM0 & BG Prediction & LM1 & BG Prediction \\
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Background & Contribution& Including LM0 & Contribution & Including LM1 \\ \hline
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x & x & x & x & x \\
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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}[htb]
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\begin{center}
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\caption{\label{tab:sigcontPT} Effects of signal contamination
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for the background predictions of the $P_T(\ell\ell)$ method including LM0 or
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LM1. Results
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are normalized to 30 pb$^{-1}$.}
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\begin{tabular}{|c||c|c||c|c|}
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\hline
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SM & LM0 & BG Prediction & LM1 & BG Prediction \\
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Background & Contribution& Including LM0 & Contribution & Including LM1 \\ \hline
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x & x & x & x & x \\
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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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|
