Notes/ElectronID/Variables.tex

Electron is identified by combining information from the CMS
tracker and ECAL. Our initial (preselection) requirement is that
the electron candidate in addition to the kinematics criteria
described in the Section above, is also identified as a GSF electron,
{\it i.e.}, it has a GSF track matched with the ECAL Super Cluster (SC).
The efficiency of the selection criteria is measured with respect
to these initial requirements.

The focus of this note is to optimize official loose and tight identification
criteria to identify electrons from $Z$ and $W$ decays, respectively. 
This can be achieved by optimizing the thresholds, optimizing the 
discriminants that have more background rejection power, and selecting 
the variables that are not highly correlated with each other. 
The latter allows keeping the number of variables in the criteria to a 
minimum, which results in a simpler set of requirements with smaller 
systematic uncertainties, and thus more robust in the startup conditions.

Variables that allow discriminating electrons from em-jets can be roughly
divided into two classes: matching/shower-shape and isolation
discriminants. In the following we treat these two classes separately to
follow the existing egamma POG identification scheme. The variables are
described in the next two subsections.

\subsection{Identification variables}
\label{ss:matching}
\subsubsection{Track-ECAL matching}
An electron GSF track should be well-matched to the ECAL Super Cluster (SC), while
a $\pi^0$ energy deposit in ECAL is not necessarily matched well with a GSF track, which
is usually belong to a charged pion. Thus, a spatial match between a track and a SC can be a good
discriminant. This match is described in azimuthal and pseudorapidity planes and denoted as
$\Delta_\phi$ and $\Delta_\eta$, respectively.

\subsubsection{ECAL energy width}
An electron brems extensively in CMS tracker, which results in a rather wide shower in azimuthal 
plane due to a strong CMS magnetic field. However, the width  of a shower in pseudorapidity 
plane remains very narrow and can discriminate against jets, which tend to have rather large
$\eta$ and $\phi$ energy widths. We consider two parameterization of the shower width:
the official one $\sigma_{\eta\eta} = \sqrt{CovEtaEta}$ which describes the width of the highest-energy
basic cluster of the SC, further referred to as a seed cluster, and an energy-weighted $\eta$-width of 
the SC, defined as
\begin{equation}
\label{eq:etawidth}
\sigma_\eta = \frac{1}{E_{SC}} \sqrt{\sum_{ECAL~SC~RecHits} E_i(\eta_i - \eta_{SC})^2}.
\end{equation}

\subsubsection{E/p-based variables}
Electrons should deposit all of their energy in the ECAL detector, thus the track momentum
at the outer edge of tracker $p_{out}$ should be of the same order as an energy of the seed EM 
cluster $E_{seed}$ of the electron's SC.  The em-content of a jet is carried mostly by neutral 
pions which do not have much correlation to the momentum of charged particles in the vicinity. 
Thus, a ratio $E_{seed}/p_{out}$ can be a good discriminant. We also considered an official
version of this discriminant $E_{SC}/p_{in}$, where $E_{SC}$ is a SC energy, and $p_{in}$
is the initial momentum of a charged particle. 

\subsubsection{H/E variables}
One can form a powerful discriminant by using the HCAL and ECAL energies associated 
with an electron candidate, as electrons tend to deposit very little or no energy in HCAL,
while jets produce a wide energy deposition in the HCAL. An official variable used in
``Robust" criteria is the ratio of HCAL and ECAL energies: $H/E$. It peaks around 0 for
electrons and can be quite large for em-jets. We also form a variable that also takes into
account the width of the HCAL energy deposition by making a ratio of the energy deposited
in HCAL and ECAL in the cone of $\Delta R < 0.3$ which is not included in the SC, normalized
to the SC energy as follows

\begin{equation}
\label{eq:emhad}
EMHAD =\frac{1}{E_{SC}}\left(\sum_{\Delta R=0.3}E^{ecal}_{RecHit} + 
                                                      \sum_{\Delta R=0.3}E^{hcal}_{RecHit} - E_{SC}\right).
\end{equation}

\subsubsection{Track isolation requirements}
As electrons from $W$ and $Z$ boson decays are isolated, requiring electron isolated from tracking
activity can significantly suppress em-jets that usually have a large number of soft tracks around the
leading $\pi^0$. It also can suppress real electrons from semi-leptonic decays of $b$ quarks which
tend to be non-isolated as well. We consider several versions of the track isolation requirements:

\begin{equation}
\label{eq:trkIsoN}
IsoN_{trk} = \frac{1}{p_T(e)}\left(\sum_{\Delta R=0.3} p_T(trk) - \sum_{\Delta R=0.05} p_T(trk)\right),
\end{equation}

and non-normalized version of the above:

\begin{equation}
\label{eq:trkIso}
Iso_{trk} = \sum_{\Delta R=0.3} p_T(trk) - \sum_{\Delta R=0.05} p_T(trk).
\end{equation}

We also test the track isolation discriminant defined within electroweak group:
\begin{equation}
\label{eq:trkIsoEWK}
Iso_{trk}(EWK) = \sum_{\Delta R = 0.6} p_T(trk) - \sum_{\Delta R = 0.02} p_T(trk).
\end{equation}

%\subsubsection{ECAL and HCAL isolation requirements}
%We also consider a few ECAL
%
%In order to discriminate electron from em-jet isolation variable defined using both ECAL and HCAL is used.
%Jet deposit high fraction of its energy in HCAL and much less in ECAL. While electron deposits all its energy in ECAL
%with very small hadronic fraction, unless it is very energetic when longitudinal energy leakage appears.
%Isolation variable, defined as Eq.~\ref{eq:3}, is very similar by the content to the variable $H/E$,
%ratio of energy deposited HCAL behind the SC over the SC energy, which is used in official egamma POG
%electron identification criteria.


\subsection{Optimization method and strategy}
We start building the ``Loose" electron criteria based on the discriminants utilized in the official
``Robust'' and adding more variables from the ``Loose" criteria (or their more powerful variants described
above), and tuning the threshold to keep the efficiency of each criterion to be 
above 99\%. A similar optimization is done for the ``Tight" requirements, although, the efficiency
was not required to exceed 99\% for the a given criterion. Instead, we vary the thresholds and plot
signal efficiency $v.s.$ background efficiency and find a region in the plot that is closest to the
``perfect" performance corner that has 100\% signal and 0\% background efficiencies.

We study the performance of the requirements by applying them in sequential order, starting 
with the most simple and robust, and continuing to more complex 
ones. We also study the correlation between variables by changing the order they are 
applied to see if some of the variables are completely correlated with the others and can 
be omitted.

 \subsection{Tuning ``Loose" criteria}
 \subsection{Tuning ``Tight" criteria}
 

Revision:	1.1
Committed:	Thu Jun 19 15:23:08 2008 UTC (16 years, 10 months ago) by beaucero
Content type:	application/x-tex
Branch:	MAIN
Log Message:	First Version
#	User	Rev	Content
1	beaucero	1.1	Electron is identified by combining information from the CMS
2			tracker and ECAL. Our initial (preselection) requirement is that
3			the electron candidate in addition to the kinematics criteria
4			described in the Section above, is also identified as a GSF electron,
5			{\it i.e.}, it has a GSF track matched with the ECAL Super Cluster (SC).
6			The efficiency of the selection criteria is measured with respect
7			to these initial requirements.
8
9			The focus of this note is to optimize official loose and tight identification
10			criteria to identify electrons from $Z$ and $W$ decays, respectively.
11			This can be achieved by optimizing the thresholds, optimizing the
12			discriminants that have more background rejection power, and selecting
13			the variables that are not highly correlated with each other.
14			The latter allows keeping the number of variables in the criteria to a
15			minimum, which results in a simpler set of requirements with smaller
16			systematic uncertainties, and thus more robust in the startup conditions.
17
18			Variables that allow discriminating electrons from em-jets can be roughly
19			divided into two classes: matching/shower-shape and isolation
20			discriminants. In the following we treat these two classes separately to
21			follow the existing egamma POG identification scheme. The variables are
22			described in the next two subsections.
23
24			\subsection{Identification variables}
25			\label{ss:matching}
26			\subsubsection{Track-ECAL matching}
27			An electron GSF track should be well-matched to the ECAL Super Cluster (SC), while
28			a $\pi^0$ energy deposit in ECAL is not necessarily matched well with a GSF track, which
29			is usually belong to a charged pion. Thus, a spatial match between a track and a SC can be a good
30			discriminant. This match is described in azimuthal and pseudorapidity planes and denoted as
31			$\Delta_\phi$ and $\Delta_\eta$, respectively.
32
33			\subsubsection{ECAL energy width}
34			An electron brems extensively in CMS tracker, which results in a rather wide shower in azimuthal
35			plane due to a strong CMS magnetic field. However, the width of a shower in pseudorapidity
36			plane remains very narrow and can discriminate against jets, which tend to have rather large
37			$\eta$ and $\phi$ energy widths. We consider two parameterization of the shower width:
38			the official one $\sigma_{\eta\eta} = \sqrt{CovEtaEta}$ which describes the width of the highest-energy
39			basic cluster of the SC, further referred to as a seed cluster, and an energy-weighted $\eta$-width of
40			the SC, defined as
41			\begin{equation}
42			\label{eq:etawidth}
43			\sigma_\eta = \frac{1}{E_{SC}} \sqrt{\sum_{ECAL~SC~RecHits} E_i(\eta_i - \eta_{SC})^2}.
44			\end{equation}
45
46			\subsubsection{E/p-based variables}
47			Electrons should deposit all of their energy in the ECAL detector, thus the track momentum
48			at the outer edge of tracker $p_{out}$ should be of the same order as an energy of the seed EM
49			cluster $E_{seed}$ of the electron's SC. The em-content of a jet is carried mostly by neutral
50			pions which do not have much correlation to the momentum of charged particles in the vicinity.
51			Thus, a ratio $E_{seed}/p_{out}$ can be a good discriminant. We also considered an official
52			version of this discriminant $E_{SC}/p_{in}$, where $E_{SC}$ is a SC energy, and $p_{in}$
53			is the initial momentum of a charged particle.
54
55			\subsubsection{H/E variables}
56			One can form a powerful discriminant by using the HCAL and ECAL energies associated
57			with an electron candidate, as electrons tend to deposit very little or no energy in HCAL,
58			while jets produce a wide energy deposition in the HCAL. An official variable used in
59			``Robust" criteria is the ratio of HCAL and ECAL energies: $H/E$. It peaks around 0 for
60			electrons and can be quite large for em-jets. We also form a variable that also takes into
61			account the width of the HCAL energy deposition by making a ratio of the energy deposited
62			in HCAL and ECAL in the cone of $\Delta R < 0.3$ which is not included in the SC, normalized
63			to the SC energy as follows
64
65			\begin{equation}
66			\label{eq:emhad}
67			EMHAD =\frac{1}{E_{SC}}\left(\sum_{\Delta R=0.3}E^{ecal}_{RecHit} +
68			\sum_{\Delta R=0.3}E^{hcal}_{RecHit} - E_{SC}\right).
69			\end{equation}
70
71			\subsubsection{Track isolation requirements}
72			As electrons from $W$ and $Z$ boson decays are isolated, requiring electron isolated from tracking
73			activity can significantly suppress em-jets that usually have a large number of soft tracks around the
74			leading $\pi^0$. It also can suppress real electrons from semi-leptonic decays of $b$ quarks which
75			tend to be non-isolated as well. We consider several versions of the track isolation requirements:
76
77			\begin{equation}
78			\label{eq:trkIsoN}
79			IsoN_{trk} = \frac{1}{p_T(e)}\left(\sum_{\Delta R=0.3} p_T(trk) - \sum_{\Delta R=0.05} p_T(trk)\right),
80			\end{equation}
81
82			and non-normalized version of the above:
83
84			\begin{equation}
85			\label{eq:trkIso}
86			Iso_{trk} = \sum_{\Delta R=0.3} p_T(trk) - \sum_{\Delta R=0.05} p_T(trk).
87			\end{equation}
88
89			We also test the track isolation discriminant defined within electroweak group:
90			\begin{equation}
91			\label{eq:trkIsoEWK}
92			Iso_{trk}(EWK) = \sum_{\Delta R = 0.6} p_T(trk) - \sum_{\Delta R = 0.02} p_T(trk).
93			\end{equation}
94
95			%\subsubsection{ECAL and HCAL isolation requirements}
96			%We also consider a few ECAL
97			%
98			%In order to discriminate electron from em-jet isolation variable defined using both ECAL and HCAL is used.
99			%Jet deposit high fraction of its energy in HCAL and much less in ECAL. While electron deposits all its energy in ECAL
100			%with very small hadronic fraction, unless it is very energetic when longitudinal energy leakage appears.
101			%Isolation variable, defined as Eq.~\ref{eq:3}, is very similar by the content to the variable $H/E$,
102			%ratio of energy deposited HCAL behind the SC over the SC energy, which is used in official egamma POG
103			%electron identification criteria.
104
105
106			\subsection{Optimization method and strategy}
107			We start building the ``Loose" electron criteria based on the discriminants utilized in the official
108			``Robust'' and adding more variables from the ``Loose" criteria (or their more powerful variants described
109			above), and tuning the threshold to keep the efficiency of each criterion to be
110			above 99\%. A similar optimization is done for the ``Tight" requirements, although, the efficiency
111			was not required to exceed 99\% for the a given criterion. Instead, we vary the thresholds and plot
112			signal efficiency $v.s.$ background efficiency and find a region in the plot that is closest to the
113			``perfect" performance corner that has 100\% signal and 0\% background efficiencies.
114
115			We study the performance of the requirements by applying them in sequential order, starting
116			with the most simple and robust, and continuing to more complex
117			ones. We also study the correlation between variables by changing the order they are
118			applied to see if some of the variables are completely correlated with the others and can
119			be omitted.
120
121			\subsection{Tuning ``Loose" criteria}
122			\subsection{Tuning ``Tight" criteria}
123
124