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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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\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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%{\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.}
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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 is accurate
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to about 20\%.
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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}[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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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}{l||c|c|c|c||c}
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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 & 33.07 & 4.81 & 1.20 & 1.16 \\
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$Z^0$ + jets & 0.00 & 1.16 & 0.08 & 0.08 & 0.00 \\
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SM other & 0.65 & 2.31 & 0.17 & 0.14 & 0.05 \\
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\hline
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total SM MC & 8.61 & 36.54 & 5.05 & 1.41 & 1.19 \\
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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 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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%%%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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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|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 & 1.18 \\
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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 2,3,4) 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. The changes between
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lines 6 and 7 of Table~\ref{tab:victorybad} is driven by 3
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Drell Yan events that pass the \met selection in Monte Carlo (thus the effect
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is statistically not well quantified).
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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 at the few percent level.
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%%%TO BE REPLACED
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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} $ K_{\rm{fudge}} =1.2 \pm 0.3$
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%(We still need to settle on thie exact value of this.
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%For the 11 pb analysis it is taken as =1.)} . 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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%For example, we find that observed/predicted changes by roughly 0.1
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%for each 1.5\% change in the average \met response.
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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 $ K_C = X \pm Y$.
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The value of this correction factor as well as the systematic uncertainty
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will be assessed using 38X ttbar madgraph MC. In the following we use
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$K_C = 1$ for simplicity. Based on previous MC studies we foresee a correction
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factor of $K_C \approx 1.2 - 1.5$, and we will assess an uncertainty
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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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For example, we find that observed/predicted changes by roughly 0.1
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for each 1.5\% change in the average \met response.
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\subsection{Signal Contamination}
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\label{sec:sigcont}
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All data-driven methods are in 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 events 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:sigcont}. 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:sigcont} Effects of signal contamination
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for the two data-driven background estimates. The three columns give
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the expected yield in the signal region and the background estimates
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using the ABCD and $P_T(\ell \ell)$ methods. Results are normalized to 35~pb$^{-1}$.}
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\begin{tabular}{lccc}
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\hline
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& Yield & ABCD & $P_T(\ell \ell)$ \\
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\hline
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SM only & 1.41 & 1.19 & 0.96 \\
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SM + LM0 & 7.88 & 4.24 & 2.28 \\
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SM + LM1 & 3.98 & 1.53 & 1.44 \\
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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: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|c|}
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%\hline
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%SM & BG Prediction & SM$+$LM0 & BG Prediction & SM$+$LM1 & BG Prediction \\
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%Background & SM Only & Contribution & Including LM0 & Contribution & Including LM1 \\ \hline
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%1.2 & 1.0 & 6.8 & 3.7 & 3.4 & 1.3 \\
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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|c|}
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%\hline
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%SM & BG Prediction & SM$+$LM0 & BG Prediction & SM$+$LM1 & BG Prediction \\
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%Background & SM Only & Contribution & Including LM0 & Contribution & Including LM1 \\ \hline
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%1.2 & 1.0 & 6.8 & 2.2 & 3.4 & 1.5 \\
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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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|
