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\section{Background Estimates from Data}
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\label{sec:datadriven}
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To look for possible BSM contributions, we define 2 signal regions that preserve about
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0.1\% of the dilepton $t\bar{t}$ events, by adding requirements of large \MET\ and \Ht:
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\begin{itemize}
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\item high \MET\ signal region: \MET $>$ 275~GeV, \Ht $>$ 300~GeV,
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\item high \Ht\ signal region: \MET $>$ 200~GeV, \Ht $>$ 600~GeV.
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\end{itemize}
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For the high \MET\ (high \Ht) signal region, the MC predicts 2.6 (2.5) SM events,
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dominated by dilepton $t\bar{t}$; the expected LM1 yield is 17 (14) and the
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expected LM3 yield is 4.3 (4.3). The signal regions are indicated in Fig.~\ref{fig:met_ht}.
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We use three independent methods to estimate from data the background in the signal region.
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The first method is a novel technique based on the ABCD method, which we used in our 2010 analysis~\cite{ref:ospaper},
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and exploits the fact that \HT\ and $y$ are nearly uncorrelated for the $t\bar{t}$ background;
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this method is referred to as the ABCD' technique. First, we extract the $y$ and \Ht\ distributions
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$f(y)$ and $g(H_T)$ from data, using events from control regions which are dominated by background.
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Because $y$ and \Ht\ are weakly-correlated, the distribution of events in the $y$ vs. \Ht\ plane is described by:
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\begin{equation}
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\frac{\partial^2 N}{\partial y \partial H_T} = f(y)g(H_T),
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\end{equation}
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allowing us to deduce the number of events falling in any region of this plane. In particular,
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we can deduce the number of events falling in our signal regions defined by requirements on \MET\ and \Ht.
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We measure the $f(y)$ and $g(H_T)$ distributions using events in the regions indicated in Fig.~\ref{fig:abcdprimedata}
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Next, we randomly sample values of $y$ and \Ht\ from these distributions; each pair of $y$ and \Ht\ values is a pseudo-event.
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We generate a large ensemble of pseudo-events, and find the ratio $R_{S/C}$, the ratio of the
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number of pseudo-events falling in the signal region to the number of pseudo-events
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falling in a control region defined by the same requirements used to select events
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to measure $f(y)$ and $g(H_T)$. We then
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multiply this ratio by the number events which fall in the control region in data
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to get the predicted yield, ie. $N_{pred} = R_{S/C} \times N({\rm control})$.
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To estimate the statistical uncertainty in the predicted background, we smear the bin contents
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of $f(y)$ and $g(H_T)$ according to their uncertainties. We repeat the prediction 20 times
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with these smeared distributions, and take the RMS of the deviation from the nominal prediction
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as the statistical uncertainty. We have studied this technique using toy MC studies based on
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event samples of similar size to the expected yield in data for 1 fb$^{-1}$.
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Based on these studies we correct the predicted backgrounds yields by factors of 1.2 $\pm$ 0.5
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(1.0 $\pm$ 0.5) for the high \MET\ (high \Ht) signal region.
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The second background estimate, henceforth referred to as the dilepton transverse momentum ($\pt(\ell\ell)$) method,
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is based on the idea~\cite{ref:victory} that in dilepton $t\bar{t}$ events the
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\pt\ distributions of the charged leptons and neutrinos from $W$
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decays are related, because of the common boosts from the top and $W$
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decays. This relation is governed by the polarization of the $W$'s,
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which is well understood in top
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decays in the SM~\cite{Wpolarization,Wpolarization2} and can therefore be
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reliably accounted for. We then use the observed
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$\pt(\ell\ell)$ distribution to model the $\pt(\nu\nu)$ distribution,
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which is identified with \MET. Thus, we use the number of observed
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events with $\HT > 300\GeV$ and $\pt(\ell\ell) > 275\GeV$
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($\HT > 600\GeV$ and $\pt(\ell\ell) > 200\GeV^{1/2}$ )
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to predict the number of background events with
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$\HT > 300\GeV$ and $\MET > 275\GeV$ ($\HT > 600\GeV$ and $\MET > 200\GeV$).
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In practice, two corrections must be applied to this prediction, as described below.
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%
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% Now describe the corrections
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%
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The first correction accounts for the $\MET > 50\GeV$ requirement in the
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preselection, which is needed to reduce the DY background. We
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rescale the prediction by a factor equal to the inverse of the
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fraction of events passing the preselection which also satisfy the
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requirement $\pt(\ell\ell) > 50\GeVc$.
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For the \Ht\ $>$ 300 GeV requirement corresponding to the high \MET\ signal region,
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we determine this correction from data and find $K_{50}=1.5 \pm 0.3$.
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For the \Ht\ $>$ 600 GeV requirement corresponding to the high \Ht\ signal region,
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we do not have enough events in data to determine this correction with statistical
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precision, so we instead extract it from MC and find $K_{50}=1.3 \pm 0.2$.
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The second correction ($K_C$) is associated with the known polarization of the $W$, which
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introduces a difference between the $\pt(\ell\ell)$ and $\pt(\nu\nu)$
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distributions. The correction $K_C$ also takes into account detector effects such as the hadronic energy
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scale and resolution which affect the \MET\ but not $\pt(\ell\ell)$.
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The total correction factor is $K_{50} \times K_C = 2.2 \pm 0.9$ ($1.7 \pm 0.6$) for the
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high \MET (high \Ht) signal regions, where the uncertainty includes the MC statistical uncertainty
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in the extraction of $K_C$ and the 5\% uncertainty in the hadronic energy scale~\cite{ref:jes}.
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Our third background estimation method is based on the fact that many models of new physics
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produce an excess of SF with respect to OF lepton pairs. In SUSY, such an excess may be produced
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in the decay $\chi_2^0 \to \chi_1^0 \ell^+\ell^-$ or in the decay of $Z$ bosons produced in
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the cascade decays of heavy, colored objects. In contrast, for the \ttbar\ background the
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rates of SF and OF lepton pairs are the same, as is also the case for other SM backgrounds
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such as $W^+W^-$ or DY$\to\tau^+\tau^-$. We quantify the excess of SF vs. OF pairs using the
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quantity
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\begin{equation}
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\label{eq:ofhighpt}
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\Delta = R_{\mu e}N(ee) + \frac{1}{R_{\mu e}}N(\mu\mu) - N(e\mu),
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\end{equation}
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where $R_{\mu e} = 1.13 \pm 0.05$ is the ratio of muon to electron selection efficiencies,
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evaluated by taking the square root of the ratio of the number of
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$Z \to \mu^+\mu^-$ to $Z \to e^+e^-$ events in data, in the mass range 76-106 GeV with no jets or
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\met\ requirements. The quantity $\Delta$ is predicted to be 0 for processes with
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uncorrelated lepton flavors. In order for this technique to work, the kinematic selection
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applied to events in all dilepton flavor channels must be the same, which is not the case
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for our default selection because the $Z$ mass veto is applied only to same-flavor channels.
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Therefore when applying the OF subtraction technique we also apply the $Z$ mass veto also
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to the $e\mu$ channel.
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All background estimation methods based on data are in principle subject to signal contamination
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in the control regions, which tends to decrease the significance of a signal
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which may be present in the data by increasing the background prediction.
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In general, it is difficult to quantify these effects because we
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do not know what signal may be present in the data. Having two
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independent methods (in addition to expectations from MC)
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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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BSM signal with identical distributions of $\pt(\ell \ell)$ and \MET, an excess of events might be seen
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in the ABCD' method but not in the $\pt(\ell \ell)$ method.
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