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\section{Counting Experiments}
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\label{sec:datadriven}
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To look for possible BSM contributions, we define 2 signal regions that reject all but
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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 6.4 (6.7). The signal regions are indicated in Fig.~\ref{fig:met_ht}.
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These signal regions are tighter than the one used in our published 2010 analysis since
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with the larger data sample they allow us to explore phase space farther from the core
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of the SM distributions.
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We perform counting experiments in these signal regions, and 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 which is a variation of the ABCD method, which we used in our 2010 analysis~\cite{ref:ospaper},
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and exploits the fact that \HT\ and $y \equiv \MET/\sqrt{H_T}$ 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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\label{eq:abcdprime}
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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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and predict the background yields in the signal regions using Eq.~\ref{eq:abcdprime}.
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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, the bin contents
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of $f(y)$ and $g(H_T)$ are smeared according to their Poisson uncertainties.
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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 background 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 $\pt(\ell\ell) > 200\GeV$ )
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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, we apply two corrections to this prediction, following the same procedure as in Ref.~\cite{ref:ospaper}.
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The first correction accounts for the fact that we require \met\ $>$ 50\GeV in the preselection
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but there is no corresponding requirement on \ptll; this correction
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is $K_{50}=1.5 \pm 0.3$ ($1.3 \pm 0.2$) for the high \MET\ (high \Ht) signal region.
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The second correction factor accounts for the $W$ polarization in \ttbar\ events, as well
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as detector effects such as hadronic energy scale; this correction is $K_C = 1.5 \pm 0.5$ ($1.3 \pm 0.4$) for the
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high \MET (high \Ht) signal region.
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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, while for the \ttbar\ background the
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rates of SF and OF lepton pairs are the same, as discussed in Sec.~\ref{sec:fit}.
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Here we perform a counting experiment, by quantifying 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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This quantity 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
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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 three
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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 three 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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