| 1 |
\section{Data Driven Background Estimation Methods}
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| 2 |
\label{sec:datadriven}
|
| 3 |
We have developed two data-driven methods to
|
| 4 |
estimate the background in the signal region.
|
| 5 |
The first one exploits the fact that
|
| 6 |
SumJetPt and \met$/\sqrt{\rm SumJetPt}$ are nearly
|
| 7 |
uncorrelated for the $t\bar{t}$ background
|
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(Section~\ref{sec:abcd}); the second one
|
| 9 |
is based on the fact that in $t\bar{t}$ the
|
| 10 |
$P_T$ of the dilepton pair is on average
|
| 11 |
nearly the same as the $P_T$ of the pair of neutrinos
|
| 12 |
from $W$-decays, which is reconstructed as \met in the
|
| 13 |
detector.
|
| 14 |
|
| 15 |
|
| 16 |
%{\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 Fig.~\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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| 35 |
%MET$/\sqrt{\rm SumJetPt}$.}
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%\end{center}
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%\end{figure}
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|
| 39 |
\begin{figure}[bht]
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| 40 |
\begin{center}
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| 41 |
\includegraphics[width=0.75\linewidth]{uncor.png}
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| 42 |
\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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| 45 |
intervals 4.5-6.5, 6.5-8.5 and $>$8.5, respectively. }
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| 46 |
\end{center}
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| 47 |
\end{figure}
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| 48 |
|
| 49 |
\begin{figure}[tb]
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| 50 |
\begin{center}
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| 51 |
\includegraphics[width=0.75\linewidth]{ttdil_uncor_38X.png}
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\caption{\label{fig:abcdMC}\protect Distributions of MET$/\sqrt{\rm SumJetPt}$ vs.
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| 53 |
SumJetPt for SM Monte Carlo. Here we also show our choice of ABCD regions. The correlation coefficient
|
| 54 |
${\rm corr_{XY}}$ is computed for events falling in the ABCD regions.}
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| 55 |
\end{center}
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| 56 |
\end{figure}
|
| 57 |
|
| 58 |
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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 background
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prediction A $\times$ C / B are given in Table~\ref{tab:abcdMC} for an integrated
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luminosity of 34.0 pb$^{-1}$. In Table~\ref{tab:abcdsyst}, we test the stability of
|
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observed/predicted with respect to variations in the ABCD boundaries.
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Based on the results in Tables~\ref{tab:abcdMC} and~\ref{tab:abcdsyst}, we assess
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a systematic uncertainty of 20\% on the prediction of the ABCD method.
|
| 67 |
|
| 68 |
%As shown in Table~\ref{tab:abcdsyst}, we assess systematic uncertainties
|
| 69 |
%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
|
| 71 |
%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
|
| 73 |
%predicted yield using the ABCD method.
|
| 74 |
|
| 75 |
|
| 76 |
%{\color{red} Avi wants some statement about stability
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| 77 |
%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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| 80 |
\begin{table}[ht]
|
| 81 |
\begin{center}
|
| 82 |
\caption{\label{tab:abcdMC} Expected SM Monte Carlo yields for
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| 83 |
34.0~pb$^{-1}$ in the ABCD regions, as well as the predicted yield in
|
| 84 |
the signal region given by A $\times$ C / B. Here `SM other' is the sum
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| 85 |
of non-dileptonic $t\bar{t}$ decays, $W^{\pm}$+jets, $W^+W^-$,
|
| 86 |
$W^{\pm}Z^0$, $Z^0Z^0$ and single top.}
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| 87 |
\begin{tabular}{lccccc}
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%%%official json v3, 38X MC (D6T ttbar and DY)
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\hline
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| 90 |
sample & A & B & C & D & PRED \\
|
| 91 |
\hline
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$t\bar{t}\rightarrow \ell^{+}\ell^{-}$ & 8.44 $\pm$ 0.18 & 32.83 $\pm$ 0.35 & 4.78 $\pm$ 0.14 & 1.07 $\pm$ 0.06 & 1.23 $\pm$ 0.05 \\
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$Z^0 \rightarrow \ell^{+}\ell^{-}$ & 0.17 $\pm$ 0.08 & 1.18 $\pm$ 0.22 & 0.04 $\pm$ 0.04 & 0.12 $\pm$ 0.07 & 0.01 $\pm$ 0.01 \\
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SM other & 0.53 $\pm$ 0.03 & 2.26 $\pm$ 0.11 & 0.23 $\pm$ 0.03 & 0.07 $\pm$ 0.01 & 0.05 $\pm$ 0.01 \\
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\hline
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total SM MC & 9.14 $\pm$ 0.20 & 36.26 $\pm$ 0.43 & 5.05 $\pm$ 0.14 & 1.27 $\pm$ 0.10 & 1.27 $\pm$ 0.05 \\
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| 97 |
\hline
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| 98 |
\end{tabular}
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| 99 |
\end{center}
|
| 100 |
\end{table}
|
| 101 |
|
| 102 |
|
| 103 |
|
| 104 |
\begin{table}[ht]
|
| 105 |
\begin{center}
|
| 106 |
\caption{\label{tab:abcdsyst}
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| 107 |
Results of the systematic study of the ABCD method by varying the boundaries
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| 108 |
between the ABCD regions shown in Fig.~\ref{fig:abcdMC}. Here $x_1$ is the lower SumJetPt boundary and
|
| 109 |
$x_2$ is the boundary separating regions A and B from C and D, their nominal values are 125 and 300~GeV,
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| 110 |
respectively. $y_1$ is the lower MET/$\sqrt{\rm SumJetPt}$ boundary and
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| 111 |
$y_2$ is the boundary separating regions B and C from A and D, their nominal values are 4.5 and 8.5~GeV$^{1/2}$,
|
| 112 |
respectively.}
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| 113 |
\begin{tabular}{cccc|c}
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| 114 |
\hline
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| 115 |
$x_1$ & $x_2$ & $y_1$ & $y_2$ & Observed/Predicted \\
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\hline
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| 117 |
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| 118 |
nominal & nominal & nominal & nominal & $1.00 \pm 0.08$ \\
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| 119 |
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| 120 |
+5\% & +5\% & +2.5\% & +2.5\% & $1.08 \pm 0.11$ \\
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| 121 |
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+5\% & +5\% & nominal & nominal & $1.04 \pm 0.10$ \\
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| 123 |
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| 124 |
nominal & nominal & +2.5\% & +2.5\% & $1.03 \pm 0.09$ \\
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| 125 |
|
| 126 |
nominal & +5\% & nominal & +2.5\% & $1.05 \pm 0.10$ \\
|
| 127 |
|
| 128 |
nominal & -5\% & nominal & -2.5\% & $0.95 \pm 0.07$ \\
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| 129 |
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| 130 |
-5\% & -5\% & +2.5\% & +2.5\% & $1.00 \pm 0.08$ \\
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| 131 |
|
| 132 |
+5\% & +5\% & -2.5\% & -2.5\% & $0.98 \pm 0.09$ \\
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\hline
|
| 134 |
\end{tabular}
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| 135 |
\end{center}
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| 136 |
\end{table}
|
| 137 |
|
| 138 |
\subsection{Dilepton $P_T$ method}
|
| 139 |
\label{sec:victory}
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| 140 |
This method is based on a suggestion by V. Pavlunin\cite{ref:victory},
|
| 141 |
and was investigated by our group in 2009\cite{ref:ourvictory}.
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| 142 |
The idea is that in dilepton $t\bar{t}$ events the lepton and neutrinos
|
| 143 |
from $W$ decays have the same $P_T$ spectrum (modulo $W$ polarization
|
| 144 |
effects). One can then use the observed
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| 145 |
$P_T(\ell\ell)$ distribution to model the sum of neutrino $P_T$'s which
|
| 146 |
is identified with the \met.
|
| 147 |
|
| 148 |
Then, in order to predict the $t\bar{t} \to$ dilepton contribution to a
|
| 149 |
selection with \met$+$X, one applies a cut on $P_T(\ell\ell)+$X instead.
|
| 150 |
In practice one has to rescale the result of the $P_T(\ell\ell)+$X selection
|
| 151 |
to account for the fact that any dilepton selection must include a
|
| 152 |
moderate \met cut in order to reduce Drell Yan backgrounds. This
|
| 153 |
is discussed in Section 5.3 of Reference~\cite{ref:ourvictory}; for a \met
|
| 154 |
cut of 50 GeV, the rescaling factor is obtained from the MC as
|
| 155 |
|
| 156 |
\newcommand{\ptll} {\ensuremath{P_T(\ell\ell)}}
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| 157 |
\begin{center}
|
| 158 |
$ K = \frac{\int_0^{\infty} {\cal N}(\ptll)~~d\ptll~}{\int_{50}^{\infty} {\cal N}(\ptll)~~d\ptll~} = 1.5$
|
| 159 |
\end{center}
|
| 160 |
|
| 161 |
|
| 162 |
%%%TO BE REPLACED
|
| 163 |
%Given the integrated luminosity of the
|
| 164 |
%present dataset, the determination of $K$ in data is severely statistics
|
| 165 |
%limited. Thus, we take $K$ from $t\bar{t}$ Monte Carlo as
|
| 166 |
|
| 167 |
%\begin{center}
|
| 168 |
%$ K_{MC} = \frac{\int_0^{\infty} {\cal N}(\met)~~d\met~}{\int_{50}^{\infty} {\cal N}(\met)~~d\met~}$
|
| 169 |
%\end{center}
|
| 170 |
|
| 171 |
%\noindent {\color{red} For the 11 pb result we have used $K$ from data.}
|
| 172 |
|
| 173 |
There are several effects that spoil the correspondance between \met and
|
| 174 |
$P_T(\ell\ell)$:
|
| 175 |
\begin{itemize}
|
| 176 |
\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
|
| 178 |
anti-parallel. Thus the $P_T(\nu\nu)$ distribution is harder
|
| 179 |
than the $P_T(\ell\ell)$ distribution for top dilepton events.
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| 180 |
\item The lepton selections results in $P_T$ and $\eta$ cuts on the individual
|
| 181 |
leptons that have no simple correspondance to the neutrino requirements.
|
| 182 |
\item Similarly, the \met$>$50 GeV cut introduces an asymmetry between leptons and
|
| 183 |
neutrinos which is only partially compensated by the $K$ factor above.
|
| 184 |
\item The \met resolution is much worse than the dilepton $P_T$ resolution.
|
| 185 |
When convoluted with a falling spectrum in the tails of \met, this results
|
| 186 |
in a harder spectrum for \met than the original $P_T(\nu\nu)$.
|
| 187 |
\item The \met response in CMS is not exactly 1. This causes a distortion
|
| 188 |
in the \met distribution that is not present in the $P_T(\ell\ell)$ distribution.
|
| 189 |
\item The $t\bar{t} \to$ dilepton signal includes contributions from
|
| 190 |
$W \to \tau \to \ell$. For these events the arguments about the equivalence
|
| 191 |
of $P_T(\ell\ell)$ and $P_T(\nu\nu)$ do not apply.
|
| 192 |
\item A dilepton selection will include SM events from non $t\bar{t}$
|
| 193 |
sources. These events can affect the background prediction. Particularly
|
| 194 |
dangerous are high $P_T$ Drell Yan events that barely pass the \met$>$ 50
|
| 195 |
GeV selection. They will tend to push the data-driven background prediction up.
|
| 196 |
Therefore we estimate the number of DY events entering the background prediction
|
| 197 |
using the $R_{out/in}$ method as described in Sec.~\ref{sec:othBG}.
|
| 198 |
\end{itemize}
|
| 199 |
|
| 200 |
We have studied these effects in SM Monte Carlo, using a mixture of generator and
|
| 201 |
reconstruction level studies, putting the various effects in one at a time.
|
| 202 |
For each configuration, we apply the data-driven method and report as figure
|
| 203 |
of merit the ratio of observed and predicted events in the signal region.
|
| 204 |
The results are summarized in Table~\ref{tab:victorybad}.
|
| 205 |
|
| 206 |
\begin{table}[htb]
|
| 207 |
\begin{center}
|
| 208 |
\caption{\label{tab:victorybad}
|
| 209 |
Test of the data driven method in Monte Carlo
|
| 210 |
under different assumptions, evaluated using Spring10 MC. See text for details.}
|
| 211 |
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
|
| 212 |
\hline
|
| 213 |
& True $t\bar{t}$ dilepton & $t\to W\to\tau$& other SM & GEN or & Lepton $P_T$ & Z veto & \met $>$ 50& obs/pred \\
|
| 214 |
& included & included & included & RECOSIM & and $\eta$ cuts & & & \\ \hline
|
| 215 |
1&Y & N & N & GEN & N & N & N & 1.90 \\
|
| 216 |
2&Y & N & N & GEN & Y & N & N & 1.64 \\
|
| 217 |
3&Y & N & N & GEN & Y & Y & N & 1.59 \\
|
| 218 |
4&Y & N & N & GEN & Y & Y & Y & 1.55 \\
|
| 219 |
5&Y & N & N & RECOSIM & Y & Y & Y & 1.51 \\
|
| 220 |
6&Y & Y & N & RECOSIM & Y & Y & Y & 1.58 \\
|
| 221 |
7&Y & Y & Y & RECOSIM & Y & Y & Y & 1.38 \\
|
| 222 |
\hline
|
| 223 |
\end{tabular}
|
| 224 |
\end{center}
|
| 225 |
\end{table}
|
| 226 |
|
| 227 |
|
| 228 |
\begin{table}[htb]
|
| 229 |
\begin{center}
|
| 230 |
\caption{\label{tab:victorysyst}
|
| 231 |
Summary of variations in $K_C$ due to the MET scale and resolution uncertainty, and to backgrounds other than $t\bar{t} \to$~dilepton.
|
| 232 |
In the first table, `up' and `down' refer to shifting the hadronic energy scale up and down by 5\%. In the second table, the quoted value
|
| 233 |
refers to the amount of additional smearing of the MET, as discussed in the text. In the third table, the normalization of all backgrounds
|
| 234 |
other than $t\bar{t} \to$~dilepton is varied.{\bf \color{red} UPDATE } }
|
| 235 |
\begin{tabular}{ lcccc }
|
| 236 |
\hline
|
| 237 |
MET scale & Predicted & Observed & Obs/pred \\
|
| 238 |
\hline
|
| 239 |
nominal & 0.92 $ \pm $ 0.09 & 1.27 $ \pm $ 0.10 & 1.39 $ \pm $ 0.18 \\
|
| 240 |
up & 0.90 $ \pm $ 0.09 & 1.58 $ \pm $ 0.10 & 1.75 $ \pm $ 0.21 \\
|
| 241 |
down & 0.70 $ \pm $ 0.06 & 0.96 $ \pm $ 0.09 & 1.37 $ \pm $ 0.18 \\
|
| 242 |
\hline
|
| 243 |
MET smearing & Predicted & Observed & Obs/pred \\
|
| 244 |
\hline
|
| 245 |
nominal & 0.92 $ \pm $ 0.09 & 1.27 $ \pm $ 0.10 & 1.39 $ \pm $ 0.18 \\
|
| 246 |
10\% & 0.88 $ \pm $ 0.09 & 1.28 $ \pm $ 0.10 & 1.47 $ \pm $ 0.19 \\
|
| 247 |
20\% & 0.87 $ \pm $ 0.09 & 1.26 $ \pm $ 0.10 & 1.44 $ \pm $ 0.19 \\
|
| 248 |
30\% & 1.03 $ \pm $ 0.17 & 1.33 $ \pm $ 0.10 & 1.29 $ \pm $ 0.23 \\
|
| 249 |
40\% & 0.88 $ \pm $ 0.09 & 1.36 $ \pm $ 0.10 & 1.55 $ \pm $ 0.20 \\
|
| 250 |
50\% & 0.80 $ \pm $ 0.07 & 1.39 $ \pm $ 0.10 & 1.73 $ \pm $ 0.19 \\
|
| 251 |
\hline
|
| 252 |
non-$t\bar{t} \to$~dilepton bkg & Predicted & Observed & Obs/pred \\
|
| 253 |
\hline
|
| 254 |
ttdil only & 0.79 $ \pm $ 0.07 & 1.07 $ \pm $ 0.06 & 1.36 $ \pm $ 0.14 \\
|
| 255 |
nominal & 0.92 $ \pm $ 0.09 & 1.27 $ \pm $ 0.10 & 1.39 $ \pm $ 0.18 \\
|
| 256 |
double non-ttdil yield & 1.04 $ \pm $ 0.15 & 1.47 $ \pm $ 0.16 & 1.40 $ \pm $ 0.25 \\
|
| 257 |
\hline
|
| 258 |
\end{tabular}
|
| 259 |
\end{center}
|
| 260 |
\end{table}
|
| 261 |
|
| 262 |
The largest discrepancy between prediction and observation occurs on the first
|
| 263 |
line of Table~\ref{tab:victorybad}, {\em i.e.}, at the generator level with no
|
| 264 |
cuts. We have verified that this effect is due to the polarization of
|
| 265 |
the $W$ (we remove the polarization by reweighting the events and we get
|
| 266 |
good agreement between prediction and observation). The kinematical
|
| 267 |
requirements (lines 2,3,4) compensate somewhat for the effect of W polarization.
|
| 268 |
Going from GEN to RECOSIM, the change in observed/predicted is small.
|
| 269 |
% We have tracked this down to the fact that tcMET underestimates the true \met
|
| 270 |
% by $\approx 4\%$\footnote{We find that observed/predicted changes by roughly 0.1
|
| 271 |
%for each 1.5\% change in \met response.}.
|
| 272 |
Finally, contamination from non $t\bar{t}$
|
| 273 |
events can have a significant impact on the BG prediction.
|
| 274 |
%The changes between
|
| 275 |
%lines 6 and 7 of Table~\ref{tab:victorybad} is driven by 3
|
| 276 |
%Drell Yan events that pass the \met selection in Monte Carlo (thus the effect
|
| 277 |
%is statistically not well quantified).
|
| 278 |
|
| 279 |
An additional source of concern is that the CMS Madgraph $t\bar{t}$ MC does
|
| 280 |
not include effects of spin correlations between the two top quarks.
|
| 281 |
We have studied this effect at the generator level using Alpgen. We find
|
| 282 |
that the bias is at the few percent level.
|
| 283 |
|
| 284 |
Based on the results of Table~\ref{tab:victorysyst}, we conclude that the
|
| 285 |
naive data-driven background estimate based on $P_T{(\ell\ell)}$ needs to
|
| 286 |
be corrected by a factor of $ K_C = 1.4 \pm 0.2({\rm stat})$.
|
| 287 |
|
| 288 |
The dominant sources of systematic uncertainty in $K_C$ are due to non-$t\bar{t} \to$~dilepton backgrounds,
|
| 289 |
and the MET scale and resolution uncertainties, as summarized in Table~\ref{tab:victorysyst}.
|
| 290 |
The impact of non-$t\bar{t}$-dilepton background is assessed
|
| 291 |
by varying the yield of all backgrounds except for $t\bar{t} \to$~dilepton.
|
| 292 |
The uncertainty is assessed as the larger of the differences between the nominal $K_C$ value and the values
|
| 293 |
obtained using only $t\bar{t} \to$~dilepton MC and obtained by doubling the non $t\bar{t} \to$~dilepton component,
|
| 294 |
giving an uncertainty of $0.03$.
|
| 295 |
|
| 296 |
The uncertainty in $K_C$ due to the MET scale uncertainty is assessed by varying the hadronic energy scale using
|
| 297 |
the same method as in~\cite{ref:top}, giving an uncertainty of 0.36.
|
| 298 |
We also assess the impact of the MET resolution
|
| 299 |
uncertainty on $K_C$ by applying a random smearing to the MET. For each event, we determine the expected MET resolution
|
| 300 |
based on the sumJetPt, and smear the MET to simulate an increase in the resolution of 10\%, 20\%, 30\%, 40\% and 50\%.
|
| 301 |
The results show that $K_C$ does not depend strongly on the MET resolution and we therefore do not assess any uncertainty.
|
| 302 |
|
| 303 |
Incorporating all the statistical and systematic uncertainties we find $K_C = 1.4 \pm 0.4$.
|
| 304 |
|
| 305 |
\subsection{Signal Contamination}
|
| 306 |
\label{sec:sigcont}
|
| 307 |
|
| 308 |
All data-driven methods are in principle subject to signal contaminations
|
| 309 |
in the control regions, and the methods described in
|
| 310 |
Sections~\ref{sec:abcd} and~\ref{sec:victory} are not exceptions.
|
| 311 |
Signal contamination tends to dilute the significance of a signal
|
| 312 |
present in the data by inflating the background prediction.
|
| 313 |
|
| 314 |
It is hard to quantify how important these effects are because we
|
| 315 |
do not know what signal may be hiding in the data. Having two
|
| 316 |
independent methods (in addition to Monte Carlo ``dead-reckoning'')
|
| 317 |
adds redundancy because signal contamination can have different effects
|
| 318 |
in the different control regions for the two methods.
|
| 319 |
For example, in the extreme case of a
|
| 320 |
new physics signal
|
| 321 |
with $P_T(\ell \ell) = \met$, an excess of events would be seen
|
| 322 |
in the ABCD method but not in the $P_T(\ell \ell)$ method.
|
| 323 |
|
| 324 |
|
| 325 |
The LM points are benchmarks for SUSY analyses at CMS. The effects
|
| 326 |
of signal contaminations for a couple such points are summarized
|
| 327 |
in Table~\ref{tab:sigcont}. Signal contamination is definitely an important
|
| 328 |
effect for these two LM points, but it does not totally hide the
|
| 329 |
presence of the signal.
|
| 330 |
|
| 331 |
|
| 332 |
\begin{table}[htb]
|
| 333 |
\begin{center}
|
| 334 |
\caption{\label{tab:sigcont} Effects of signal contamination
|
| 335 |
for the two data-driven background estimates. The three columns give
|
| 336 |
the expected yield in the signal region and the background estimates
|
| 337 |
using the ABCD and $P_T(\ell \ell)$ methods. Results are normalized to 34.0~pb$^{-1}$.}
|
| 338 |
\begin{tabular}{lccc}
|
| 339 |
\hline
|
| 340 |
& Yield & ABCD & $P_T(\ell \ell)$ \\
|
| 341 |
\hline
|
| 342 |
SM only & 1.3 & 1.3 & 0.9 \\
|
| 343 |
SM + LM0 & 7.4 & 4.4 & 1.9 \\
|
| 344 |
SM + LM1 & 3.8 & 1.6 & 1.4 \\
|
| 345 |
%SM only & 1.27 & 1.27 & 0.92 \\
|
| 346 |
%SM + LM0 & 7.39 & 4.38 & 1.93 \\
|
| 347 |
%SM + LM1 & 3.77 & 1.62 & 1.41 \\
|
| 348 |
\hline
|
| 349 |
\end{tabular}
|
| 350 |
\end{center}
|
| 351 |
\end{table}
|
| 352 |
|
