cmsnotes/StopSearch/exampleBG.tex

\section{Example BG prediction calculation}
\label{BGexample}

The calculation of the background prediction is a bit complicated.
Here we walk the reader through a concrete example. 

 {\bf NB: the numbers
in this section corresponded to the numbers in V3 of the analysis note. 
They will not be updated, because this is meant as an illustration only.}.

The main background is $t\bar{t}$.
The main idea is to normalize to the $M_T$ peak region ($50 < M_T < 80$ GeV).
This eliminates dependence on \ttbar\ cross-section, luminosity,
trigger efficiency, JES, lepton ID, etc.  This gets a bit complicated because
the $M_T$ peak region, while dominantly \ttbar\ lepton $+$ jets,
also includes some \wjets, \ttdl, rare processes, etc.  Also, we want
to minimize our need to understand the effect of the isolated 
track veto on \ttsl.  As a result we actually define two $M_T$ peak
regions: one before and one after applying the isolated track veto.
Then the \ttdl\ background is normalized the the ``before veto'' region,
and the \ttsl\ and \wjets\ background are normalized to the ``post veto''
region.

This complex procedure is important for the high statistics signal regions 
with relatively low \met\ requirements, eg, SRA.  For these SRs we want to keep the 
systematics low in order to be sensitive to low mass stop; for the signal regions
with hard cuts on \met\, this is less important.  However, we apply the same 
procedure to all SRs.

For concreteness, we show the calculation for SRA, electron channel.  The MC and data
event counts used in the background calculation are collected in Table~\ref{tab:bgexample}.
Note that the background uncertainties have already been described 
in Section~\ref{sec:systematics}.  The one tricky point to keep in mind is that 
when the \wjets\ and rare cross-sections are changed by their assumed uncertainties
(50\% each), the whole calculation describe below is repeated in order to take care
of all the correlations properly.


\begin{table}[!h]
\begin{center}
\begin{tabular}{l||c|c|c|}
\hline
Sample              &  $M_T$ peak, before trk veto  &  $M_T$ peak, after trk veto  & Signal Region A \\
\hline
\ttdl\                & $290 \pm 6$                          & $116 \pm 4$                       & $261 \pm 6$  \\
\ttsl\  (1\Lep)   & $2899 \pm 19$                      & $2595 \pm 18$                   &   not used    \\
\wjets\             & $252 \pm 28$                        & $236 \pm 28$                     &  not used   \\
Rare                  & $89 \pm 5$                            & $62 \pm 4$                         & $26 \pm 2$ \\
\hline 
Total                & $3530 \pm 35$                      & $3009 \pm 34$                    & not used \\
\hline
Data                & 3358                                       & 2787                                     & not used  \\
\hline
\end{tabular}
\caption{ Data and MC event counts used to predict the background in SRA, for electron events.
Uncertainties are statistical only.  The trigger efficiency has been applied to the MC samples.  In the
case of \ttdl\, the $K_3$ and $K_4$ factors of Section~\ref{sec:jetmultiplicity} have also been applied.
\label{tab:bgexample}}
\end{center}
\end{table}


\subsection{Central value of dilepton background}
A ``before veto'' scale factor is defined from the second column in Table~\ref{tab:bgexample} as the factor by which 
all MC except the ``rare'' need to be scaled up in order to have data/MC agreement.  This is 

\noindent $SF_{pre} = (3358 - 89)/(2899 + 252 + 290) = 0.950$.

Then the \ttdl\ background prediction is the number of events predicted by the MC  in SRA (261 from the
last column of SRA), rescaled by $SF_{pre}$.  The result for the central value is 248 events.

\subsection{Central value of the \ttsl\ background}
\label{sec:cenvttlj}
A ``post veto'' scale factor is defined from the third column in Table~\ref{tab:bgexample} as the factor by which
the \ttsl\ and the \wjets\ backgrounds need to be scaled to have data/MC agreement.  

\noindent $SF_{post} = (2787 - 62 - SF_{pre} \cdot 116) / (2595 + 236) = 0.924$.

Then the \ttsl\ background is obtained as the product of the following three factors

\begin{itemize}
\item $SF_{post} = 0.924$ as obtained above
\item 2595, from the third column of Table~\ref{tab:bgexample}
\item The tail-to-peak ratio $R_{top} = 0.045$ from Table~\ref{tab:ttpcorr}
%\item The tail-to-peak ratio scale factor $SFR_{top} = 1.89 \pm 0.56$ from Table~\ref{tab:cr2yields}.
\end{itemize}

The result for the central value is 108 events.

\subsection{Central value for the \wjets\ background}

It is calculated as the product of
\begin{itemize}
\item $SF_{post} = 0.924$ from Section~\ref{sec:cenvttlj}
\item 236, from the third column of Table~\ref{tab:bgexample}
\item The tail-to-peak ratio $R_{wjet} = 0.066$ from Table~\ref{tab:ttpcorr}
%\item The tail-to-peak ratio scale factor $SFR_{wjet} = 1.64 \pm 0.38$ from Table~\ref{tab:cr1yields}.
\end{itemize}

The result for the central value is 14.3 events.

\subsection{Central value for the rare backgrounds}

This is 26 events from Table~\ref{tab:bgexample}

Revision:	1.2
Committed:	Thu Oct 18 22:33:21 2012 UTC (12 years, 7 months ago) by linacre
Content type:	application/x-tex
Branch:	MAIN
CVS Tags:	HEAD
Changes since 1.1:	+9 -10 lines
Log Message:	updated with new top ttp
#	User	Rev	Content
1	claudioc	1.1	\section{Example BG prediction calculation}
2			\label{BGexample}
3
4			The calculation of the background prediction is a bit complicated.
5			Here we walk the reader through a concrete example.
6
7			{\bf NB: the numbers
8	linacre	1.2	in this section corresponded to the numbers in V3 of the analysis note.
9	claudioc	1.1	They will not be updated, because this is meant as an illustration only.}.
10
11	linacre	1.2	The main background is $t\bar{t}$.
12	claudioc	1.1	The main idea is to normalize to the $M_T$ peak region ($50 < M_T < 80$ GeV).
13			This eliminates dependence on \ttbar\ cross-section, luminosity,
14			trigger efficiency, JES, lepton ID, etc. This gets a bit complicated because
15			the $M_T$ peak region, while dominantly \ttbar\ lepton $+$ jets,
16			also includes some \wjets, \ttdl, rare processes, etc. Also, we want
17			to minimize our need to understand the effect of the isolated
18			track veto on \ttsl. As a result we actually define two $M_T$ peak
19			regions: one before and one after applying the isolated track veto.
20			Then the \ttdl\ background is normalized the the ``before veto'' region,
21			and the \ttsl\ and \wjets\ background are normalized to the ``post veto''
22			region.
23
24			This complex procedure is important for the high statistics signal regions
25			with relatively low \met\ requirements, eg, SRA. For these SRs we want to keep the
26	linacre	1.2	systematics low in order to be sensitive to low mass stop; for the signal regions
27	claudioc	1.1	with hard cuts on \met\, this is less important. However, we apply the same
28			procedure to all SRs.
29
30			For concreteness, we show the calculation for SRA, electron channel. The MC and data
31			event counts used in the background calculation are collected in Table~\ref{tab:bgexample}.
32			Note that the background uncertainties have already been described
33			in Section~\ref{sec:systematics}. The one tricky point to keep in mind is that
34			when the \wjets\ and rare cross-sections are changed by their assumed uncertainties
35			(50\% each), the whole calculation describe below is repeated in order to take care
36			of all the correlations properly.
37
38
39			\begin{table}[!h]
40			\begin{center}
41			\begin{tabular}{l\|\|c\|c\|c\|}
42			\hline
43			Sample & $M_T$ peak, before trk veto & $M_T$ peak, after trk veto & Signal Region A \\
44			\hline
45			\ttdl\ & $290 \pm 6$ & $116 \pm 4$ & $261 \pm 6$ \\
46			\ttsl\ (1\Lep) & $2899 \pm 19$ & $2595 \pm 18$ & not used \\
47			\wjets\ & $252 \pm 28$ & $236 \pm 28$ & not used \\
48			Rare & $89 \pm 5$ & $62 \pm 4$ & $26 \pm 2$ \\
49			\hline
50			Total & $3530 \pm 35$ & $3009 \pm 34$ & not used \\
51			\hline
52			Data & 3358 & 2787 & not used \\
53			\hline
54			\end{tabular}
55			\caption{ Data and MC event counts used to predict the background in SRA, for electron events.
56			Uncertainties are statistical only. The trigger efficiency has been applied to the MC samples. In the
57			case of \ttdl\, the $K_3$ and $K_4$ factors of Section~\ref{sec:jetmultiplicity} have also been applied.
58			\label{tab:bgexample}}
59			\end{center}
60			\end{table}
61
62
63			\subsection{Central value of dilepton background}
64			A ``before veto'' scale factor is defined from the second column in Table~\ref{tab:bgexample} as the factor by which
65			all MC except the ``rare'' need to be scaled up in order to have data/MC agreement. This is
66
67			\noindent $SF_{pre} = (3358 - 89)/(2899 + 252 + 290) = 0.950$.
68
69			Then the \ttdl\ background prediction is the number of events predicted by the MC in SRA (261 from the
70			last column of SRA), rescaled by $SF_{pre}$. The result for the central value is 248 events.
71
72			\subsection{Central value of the \ttsl\ background}
73			\label{sec:cenvttlj}
74			A ``post veto'' scale factor is defined from the third column in Table~\ref{tab:bgexample} as the factor by which
75			the \ttsl\ and the \wjets\ backgrounds need to be scaled to have data/MC agreement.
76
77			\noindent $SF_{post} = (2787 - 62 - SF_{pre} \cdot 116) / (2595 + 236) = 0.924$.
78
79	linacre	1.2	Then the \ttsl\ background is obtained as the product of the following three factors
80	claudioc	1.1
81			\begin{itemize}
82			\item $SF_{post} = 0.924$ as obtained above
83			\item 2595, from the third column of Table~\ref{tab:bgexample}
84	linacre	1.2	\item The tail-to-peak ratio $R_{top} = 0.045$ from Table~\ref{tab:ttpcorr}
85			%\item The tail-to-peak ratio scale factor $SFR_{top} = 1.89 \pm 0.56$ from Table~\ref{tab:cr2yields}.
86	claudioc	1.1	\end{itemize}
87
88	linacre	1.2	The result for the central value is 108 events.
89	claudioc	1.1
90			\subsection{Central value for the \wjets\ background}
91
92			It is calculated as the product of
93			\begin{itemize}
94			\item $SF_{post} = 0.924$ from Section~\ref{sec:cenvttlj}
95			\item 236, from the third column of Table~\ref{tab:bgexample}
96	linacre	1.2	\item The tail-to-peak ratio $R_{wjet} = 0.066$ from Table~\ref{tab:ttpcorr}
97			%\item The tail-to-peak ratio scale factor $SFR_{wjet} = 1.64 \pm 0.38$ from Table~\ref{tab:cr1yields}.
98	claudioc	1.1	\end{itemize}
99
100			The result for the central value is 14.3 events.
101
102			\subsection{Central value for the rare backgrounds}
103
104			This is 26 events from Table~\ref{tab:bgexample}
105