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1   \section{Signal extraction}
2 < %\label{sec:gen}
3 < The probability to misidentify a jet as a muon is very low while for
4 < the case of the electron, $\pi^0$ in jets can be misidentified as
5 < electrons. When considering the subtraction of the background in this
6 < analysis, we will mainly concentrate on the final state where the $W$
7 < is decaying to an electron. For the muon case, the subtraction can be
8 < done using Monte Carlo sample and assigning a large error on this
9 < estimation.
10 <
11 < \subsection{Z+jets background fraction}
12 < The main background remaining even after having applied all our selection
13 < in the case of the $W$ decaying to electron is the $Z+jets$
14 < production. As signal and background have a \Z boson in the final
15 < state, we will concentrate on the third lepton which is an electron in
16 < this study.
17 <
18 < In order to select the \Z candidate in the events, we apply loose
19 < criteria. The loose sample contain a given number of signal events
20 < which contains a third isolated electron and a given number of
21 < background events which do not contains a third isolated
22 < electron. When we apply the tight criteria the fraction of signal and
23 < background events is changing according to the efficiency of the
24 < criteria. This can be expressed by this formula:
25 < \begin{eqnarray}
26 < N_{loose} & = & \hspace*{0.9cm}               N_e +   \hspace*{0.9cm}   N_{j} \\
27 < N_{tight} & = & \epsilon_{tight} N_e  + p_{fake}  N_{j}
28 < \end{eqnarray}
29 < Where $N_{loose}$ and $N_{tight}$ are the numbers of events in
30 < the loose and tight samples, respectively, $N_e$ is the number of events with a third
31 < isolated electron, $N_j$ is the number of events without a third
32 < isolated electron, $\epsilon_{tight}$ is the efficiency of the tight
33 < criteria on electron, $p_{fake}$ is the probability for a jet identified
34 < as a loose electron to be also identified as a tight electron.  By
35 < solving this set of equations we obtain:
36 < $$
37 < N_e     = \frac{N_{tight}-p_{fake} N_{loose}} { \epsilon_{tight} -p_{fake}} \ \ \ \mbox{and} \ \ \
38 < N_{j} = \frac{ \epsilon_{track} N_{loose} - N_e}{  \epsilon_{tight} -p_{fake}}
39 < $$
40 <
41 < The estimation of $\epsilon_{tight}$ and $p_{fake}$ can be done using control
42 < samples, containing electrons and jet respectively with high purity. The Tag and probe
43 < method is used to determine signal efficiency $\epsilon_{tight}$. The rate
44 < will be derived from $Z \to e^+e^-$ samples. In the following section we
45 < describe the method used to determine $p_{fake}$.
46 <
47 < \subsection{Determination of $p_{fake}$}
2 > \label{sec:SignalExt}
3 > We separate backgrounds into two categories: one with a genuine \Z boson
4 > from $\Z+jets$ processes, and the other without a genuine \Z boson from
5 > $t\bar{t}$ and $\W+jet$ production. The latter source can be estimated from
6 > the invariant mass of the \Z boson candidate, where the background events
7 > with no genuine \Z boson should not produce a \Z mass peak and should
8 > be relatively smooth.
9 >
10 > \subsection{Study of the background without a genuine \Z boson}
11 > We estimate the background due to events without a genuine \Z boson from
12 > fitting a \Z candidate invariant mass to a Gaussian function convoluted
13 > with a Breit-Wigner function. The background is parameterized as a straight
14 > line. An example of a fit for $3e$ category is given in Fig.~\ref{fig:ZFit}.
15 > Both number of signal and background events are calculated for the
16 > invariant mass range between 81 and 101 GeV. In Table~\ref{tab:FitVsMC}
17 > we summarize the number of background events obtained from the fit
18 > and from the Monte Carlo truth information.
19 >
20 > \begin{figure}[!bp]
21 >  \begin{center}
22 >  \scalebox{0.4}{\includegraphics{figs/FitBkg3eTight.eps}}
23 >  \caption{The invariant mass distribution of the $Z$ boson candidate that is fit to a signal
24 >  parameterized as a Gaussian function convoluted with a Breit-Wigner function and
25 >  a background, parameterized as a straight line.}
26 >  \label{fig:ZFit}
27 >  \end{center}
28 > \end{figure}
29 >
30 > \begin{table}[!tb]
31 > \begin{center}
32 > \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline
33 >                    & \multicolumn{2}{c|}{Background with genuine \Z} & \multicolumn{4}{c|}{Background without
34 >                    genuine \Z boson} \\
35 > Channel    & $\Z+jets$ & $\Z b\bar{b}$ &   $t\bar{t}$ & $\W+jets$ & $t\bar{t}$ + $\W+jets$ & Fit result \\ \hline
36 > $3e$ Loose & 196.5 & 67.4 & 35.7 & 0 & 35.7& 37.8 \\ \hline
37 > $3e$ Tight & 78.9 & 38.5 & 28.1 & 0 & 28.1 & 32.9 \\ \hline  
38 > $2\mu 1e$ Loose & 189.6 & 52.6 & 4.7 & 0 & 4.7 & 5.7 \\ \hline
39 > $2\mu 1e$ Tight & 63.1 & 18.2 & 1.3 & 0 & 1.3 & 1.5 \\ \hline
40 > $2e1\mu$   & 10.4 & 9.1 & 30.7 & 2.9 & 33.6 & 29.2 \\ \hline
41 > $3\mu$     & 1.9 & 7.7 & 0.7 & 0 & 0.7 & 0\\ \hline
42 > \end{tabular}
43 >
44 > \end{center}
45 > \caption{Comparison between Monte Carlo truth information and the results of the fit for the background without genuine \Z boson. Number of events are obtained in the invariant mass range between 81 and 101 GeV.
46 > %I AM NOT SURE I UNDERSTAND WHAT IS WRITTEN HERE
47 > %The ``Loose'' and ``Tight'' selection criteria applied for $W\rightarrow e\nu$ final state only. One has to consider that this study as been perform on a smaller sample than the other part of the analysis a 10\% statistics error as to be counted until the study is performed on the whole samples.
48 > }
49 > \label{tab:FitVsMC}
50 > \end{table}
51 >
52 >
53 > \subsection{Estimation of the background with genuine \Z boson}
54 > \label{sec:D0Matrix}
55 > All of the instrumental background events with real \Z boson come from
56 > $\Z + jets$ processes where one of the jets is misidentified as a lepton.
57 > The probability to misidentify a jet as a muon is very low in CMS, while that
58 > for the case of electron can be quite high as jets with large electromagnetic
59 > energy fraction can be misidentified as electrons. $\Z+jet$ background
60 > is especially high for \WZ\ signal with $\W\to e\nu$. Thus, it is imperative
61 > to have a reliable estimation of this background from data to avoid
62 > unnecessary systematic uncertainties due to Monte Carlo description of data
63 > in startup conditions. Therefore, in the following we describe the data-driven
64 > estimation of the $\Z+jets$ background for the $\ell^+\ell^- e$
65 > categories. A similar study for the remaining $\ell^+\ell^- \mu$ categories
66 > are in progress. However, as the $\Z+jets$ background is sufficiently small, it is
67 > possible to use Monte Carlo simulation to estimate $\Z+jet$ background with
68 > early data, without incurring significant systematic uncertainty due to data modeling.
69 >
70 > \subsubsection{$\Z+jets$ background fraction}
71 > To estimate the fraction of the $\Z+jets$ events in data
72 > we apply a method, commonly referred to as ``matrix'' method.
73 > The idea of a method is to apply ``Loose'' identification criteria
74 > on the third lepton after \Z boson candidate is identified
75 > and count the number of the observed events, $N_{loose}$.
76 > These events contain events with real electrons $N_{e}$
77 > and events with misidentified jets $N_j$:
78 > \begin{equation}
79 > \label{eq:matrixEq1}
80 > N_{loose} = N_e + N_j.
81 > \end{equation}
82 >
83 > If we are to apply ``Tight'' selection on the third lepton, the number
84 > of the observed events $N_{tight}$ would change as following
85 > \begin{equation}
86 > \label{eq:matrixEq2}
87 > N_{tight} = \epsilon_{tight} N_e + p_{fake} N_j,
88 > \end{equation}
89 > where $\epsilon_{tight}$ and $p_{fake}$ are efficiency of ``Tight''
90 > criteria with respect to ``Loose'' requirements for electrons and
91 > misidentified jets, respectively. As $N_{loose}$ and $N_{tight}$
92 > are directly observable, to extract the number of $Z+jet$ events
93 > in the final sample, one needs to measure $\epsilon_{tight}$
94 > and $p_{fake}$ in control data samples. Two possible ways
95 > to estimate these values are given below.
96 >
97 > \subsubsection{Determination of $\epsilon_{tight}$}
98 >
99 > \begin{figure}[bt]
100 >  \begin{center}
101 >  \scalebox{0.8}{\includegraphics{figs/tag_probe_fit.eps}}
102 >  \caption{Invariant mass of the \Z boson candidate for ``Tight-Tight'' (a)
103 >  and ``Tight-Loose'' (b) electron selections fitted to a Gaussian with
104 >  bifurcated Breit-Wigner functions.}
105 >  \label{fig:tagprobe}
106 >  \end{center}
107 > \end{figure}
108 >
109 > To estimate the $\epsilon_{tight}$ we apply ``tag-and-probe'' method
110 > using $\Z \to e^+e^-$ from \Z+jets Chowder sample, including  \W+jets
111 > and $t\bar{t}$ as background. \Z mass distribution is separated for two cases where
112 > electrons from \Z decay either both pass ``Tigh'' selection (``Tight-Tight'' case), or only
113 > one passes the ``Tight'' selection, while the other electron passes ``Loose'' but not ``Tight''
114 > selection (``Tight-Loose'' case). To estimate signal in the selected \Z candidate invariant mass distribution, we fit it to a Gaussisan with bifurcated Breit-Wigner function as a signal
115 > and straight line for a background model. \Z mass distribution and fit are shown in \ref{fig:tagprobe}.
116 >
117 > Equation for determination of signal efficiency is given as
118 > \begin{equation}
119 > \epsilon_{tight}=\frac{ 2*(N_{TT}-B_{TT}) }{ (N_{TL}-B_{TL})+2*(N_{TT}-B_{TT}) }
120 > \end{equation}
121 >
122 > where $N_{TT}$,$B_{TT}$,$N_{TL}$ and $B_{TL}$ are, respectively, number of signal+background
123 > and background events for ``Tight-Tight'' and ``Loose-Tight'' electron combinations.
124 > We estimated an efficiency $\epsilon_{tight}=0.99 \pm  0.01$.
125 >
126 > \subsubsection{Determination of $p_{fake}$}
127  
128   As the events will be most of the time triggered by the leptons coming
129 < from \Z boson, we assume that the third lepton is unbiased toward
130 < trigger requirement. Ideally we need a sample of pure multi-jet events
129 > from \Z boson, we assume that the third lepton is unbiased toward the
130 > trigger requirement. Ideally, we need a sample of pure multi-jet events
131   in order to compute the probability for a jet identified as a loose
132   electron to be also identified as a tight electron. In selecting such
133   a sample in data, one has to avoid any bias from the trigger
134 < requirements on the loose electron candidate.
134 > requirements on the ``Loose'' electron candidate.
135   %Such sample will not
136   %exist in data as they will be bias by the trigger requirement.
137 + \begin{figure}[bt]
138 +  \begin{center}
139 +  \scalebox{0.6}{\includegraphics{figs/tight_eff_gumbo.eps}}
140 +  \caption{Fraction of electron candidates passing the ``Tight'' criteria
141 +    in multijet event. No trigger requirement has been applied.}
142 +  \label{fig:qcd_efftight_noHLT}
143 +  \end{center}
144 + \end{figure}
145 +
146 + From a sample of multijet events triggered by an ``OR'' of multi-jet
147 + triggers, we select a ``Loose'' electron candidate that are not
148 + matched to any of the trigger objects. We also require the
149 + electron candidate to be separated from the jet that satisfies
150 + the trigger requirement by requiring the candidate to be separated
151 + by at least $\Delta R = 0.2$ from the trigger object.
152 + This allows us to obtain an unbiased sample of multijet events
153 + where an electron candidate is likely to be either a converted
154 + photon or a misidentified jet. The $p_{fake}$ function of $p_T$
155 + and $\eta$ is simply obtained by dividing the $p_T$ and $\eta$
156 + distributions for the electron candidate that satisfied ``Simple Tight''
157 + electron identification requirements to that for electron candidates
158 + that satisfied ``Simple Loose''. Such distributions are given
159 + in Fig.~\ref{fig:qcd_zjet_est}.
160  
59 From a sample of multi-jet events triggered by an ``OR'' of multi-jet
60 triggers, we will select loose electron candidates that are not
61 matched to any of the triggering object
62 %we will reject the object matched with the triggering
63 %objects. This will allow us to have a unbiased sample of multi-jet
64 %events. For the purpose of the study, we have used CSA07 Gumbo
65 %samples, with Pythia ID filtering in order to keep only events from
66 %photon+jets, QCD and minimum bias events.
67 The removal of the object matched with the triggering object is done
68 using a matching cone of $\Delta R =0.2$. "Simple Loose" selection is
69 applied to each reconstructed electron from this sample of jets from
70 QCD and photon+jet. Then the tight criteria is applied on such loose
71 electrons and the $p_{fake}$ is simply the ratio of this two
72 population. This ratio, given as a function of $Pt$ and $\eta$, is
73 showed in plots \ref{fig:qcd_zjet_est}.
161  
75 \subsection{Determination of $\epsilon_{tight}$}

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