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 defined in Appendix~\ref{a:OfficialCriteria} 
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 produced by 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}
\label{ss:etawidth}
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 create rather large
clusters in both $\eta$ and $\phi$ directions. We consider two parameterization of the shower width:
the one, used in official electron identification developed by egamma POG, $\sigma_{\eta\eta} = \sqrt{CovEtaEta}$ which describes the width of the highest-energy basic cluster of the SC, further referred 
to as a seed cluster, and an energy-weighted width of SC in $\eta$ direction, 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}
\label{ss:ep}
Electron should deposit all of its 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 $E_{seed}$.  Jet is reconstructed combining information from CMS tracker and hadronic calomitere (HCAL). It interact with ECAL much less intensively and most of its energy, carried after tracker, is deposited in HCAL. Hence, $p_{out}$ and $E_{seed}$ differ significantly for the jets and can be used to discriminate them from electrons.  
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. A variable used in official
``Robust" as well as ``Loose" and ``Tight" 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 form a variable that also 
takes into account the width of the ECAL 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}
\label{ss:trackIso}
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 
using CTF tracks around the electron candidates.

\begin{equation}
\label{eq:trkIsoN}
Iso^{trk}_{Norm.} = \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 with the larger outer cone, similar to that defined by electroweak group:~\footnote{Our analysis trees do not store information that would allow us to test
the isolation variable with an annulus of $0.02 < \Delta R < 0.6$. However, the performance
of a track isolation in an annulus of $0.05 < \Delta R < 0.5$ is similar.}
\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}


\subsection{Optimization method and strategy}
The focus of this work is to define two simple selection criteria: ``Simple Loose" and ``Simple Tight"
to select electrons from $Z \to ee$ and $W \to e\nu_e$ processes, respectively. We start building 
the ``Simple Loose" electron criteria based on the discriminants utilized in the official ``Robust'' 
and adding more variables from the ``Loose" criteria (or their more powerful equivalents described 
above), and tuning the thresholds to keep the efficiency of each criterion high. We change the order
of how the requirements are applied to assess the level of correlation between different criteria.
We try to retain only the variables that are as less correlated as possible to keep the number
of criteria to a minimum.

We design the ``Simple Tight'' requirements by applying additional criteria on top of the 
unchanged ``Simple Loose''. We chose not to modify thresholds of ``Simple Loose'' criteria
to reduce possible systematic uncertainties and to make the process of extracting the 
efficiency from data easier. 

Both ``Simple Tight'' and ``Simple Loose'' criteria are designed separately for ECAL 
barrel and endcap. We consider electron candidates with $p_T > 10$ GeV. The signal
electron sample was obtained from the $WZ$ Monte Carlo simulation, and the misidentified
jets are obtained from the "Gumbo" soup where we exclude $\gamma+X$ processes.

\subsection{Tuning  the ``Simple Loose" criteria}
The first two criteria of ``Robust'' requirements are $\Delta \eta$ and $\Delta \phi$.
The distributions of these variables for signal and background together with the
em-jet separation power are given in Figs.~\ref{fig:DeltaEta}~and~\ref{fig:DeltaPhi}.
We apply both requirements with the thresholds of 0.009(0.007) and 0.05(0.05) for
$\Delta \eta$ and $\Delta \phi$ for barrel(endcap), respectively. 

\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/deltaEta.eps}
        \vspace{-10mm}
    \caption{Distributions of $\Delta \eta$ for electrons (blue solid line) and misidentified jets 
    (red dashed line) in barrel (top left) and endcap (top right). 
    The performance of these discriminants in terms of efficiencies to select
     true electrons and misidentified jets are given below.
    \label{fig:DeltaEta}}
  \end{center}
\end{figure}


\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/deltaPhi.eps}
        \vspace{-10mm}
    \caption{The same as Fig.~\ref{fig:DeltaEta}, but for $\Delta \phi$.
    \label{fig:DeltaPhi}}
  \end{center}
\end{figure}

The third discriminating variable included in the ``Robust'' criteria describes the width
of the EM shower. We study two different parameterizations of this variable
described in~\ref{ss:etawidth}. The performance of their discriminating power is given in
Fig.~\ref{fig:SigmaEE_vs_EtaWidth}. 

\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/Eff_EtaWidth_SigmaEtaEta.eps}
        \vspace{-10mm}
    \caption{The background efficiency $v.s.$ signal efficiency in barrel (left) and endcap (right) 
    for $\eta$ width of SC (red) and $\sigma_{\eta\eta}$ (blue).
    \label{fig:SigmaEE_vs_EtaWidth}}
  \end{center}
\end{figure}

As it can be seen, the $\sigma_{\eta\eta}$ variable is more powerful in barrel that can reject $\sim 70\%$
of em-jets while loosing only a couple of percents of signal. The performance in endcap is comparable
between the two. As $\sigma_{\eta\eta}$ is shown to be described very well by Monte 
Carlo~\footnote{$\sigma_{\eta\eta}$ was studied using test beam data and was shown to be described
in simulation very well.} we chose to select $\sigma_{\eta\eta} < 0.012 (0.026)$ for barrel (endcap). 
The plot with distributions of $\sigma_{\eta\eta}$ and the performance in barrel and endcap is given
in Fig.~\ref{fig:SigmaEtaEta}.
\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/sigmaEtaEta.eps}
        \vspace{-10mm}
    \caption{
    Distributions of $\sigma_{\eta\eta}$ for electrons (blue solid line) and misidentified jets 
    (red dashed line) in barrel (top left) and endcap (top right). 
    The performance of these discriminants in terms of efficiencies to select
     true electrons and misidentified jets are given below.
    \label{fig:SigmaEtaEta}}
  \end{center}
\end{figure}

The discriminants that we chose so far are used in official ``Robust'' criteria.
The last requirement from the official criteria is $H/E$ which we chose to keep
unchanged from the predefined GSF electron requirement ($H/E < 0.2$). 
It will be shown that $H/E$ variable is highly correlated with isolation requirements
that we will apply later, and thus it does not need to be tightened.

As all the discriminants from the ``Robust'' selection are considered, we start including
the variables utilized in the official ``Loose'' requirement. At first, we consider
$E_{seed}/p_{out}$ and $E/p$, described in Section~\ref{ss:ep}. 
The latter is used in the official ``Loose'' and ``Tight'' selections to categorize
electrons in $E/p$ $v.s.$ $fbrem$ phase space, where $fbrem$ is defined as
\begin{equation}
\label{eq:fbrem}
        fbrem = \frac{p_{in} - p_{out}}{p_{in}},
\end{equation}
where $p_{in}$ and $p_{out}$ are momenta measured at the inner and outer edge of 
the tracker, respectively. The performance of the $E_{seed}/p_{out}$ and $E/p$
are given in Fig.~\ref{fig:EseedOPout_vs_EoP}.

\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/Eff_EoP_EseedoPout.eps}
        \vspace{-10mm}
    \caption{Background $v.s.$ signal efficiency for barrel (left) and endcap (right) for $E/p$ (red) 
    and $E_{seed}/p_{out}$ (blue).
    \label{fig:EseedOPout_vs_EoP}}
  \end{center}
\end{figure}

$E_{seed}/p_{out}$ has a better performance and we include it to the ``Simple Loose''
requirement to be more than 0.9 for both barrel and endcap. The distributions of this
variable and discriminating performance is given in Fig.~\ref{fig:EseedOPout}.

\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/EseedPout.eps}
        \vspace{-10mm}
    \caption{$E_{seed}/p_{out}$ for electrons(blue solid line) and mis-identified jets (red dashed line) 
    in barrel (left top) and endcap (right top). The performance of the criteria for different thresholds
    is given in the bottom left plot for barrel, and bottom right plot for endcap.
    \label{fig:EseedOPout}}
  \end{center}
\end{figure}

Up to this point we utilized all of the variables of the ``Robust'' and ``Loose'' criteria with an exception
of a much looser $H/E$ requirement and substitution of $E/p$ with $E_{seed}/p_{out}$ variable.
As it is shown later in the note, the performance of this criteria with respect to the official ``Loose'' is
very comparable without complexity due to the categorization employed by the ``Loose'' criteria.
The improvement is mostly achieved by utilizing a more discriminating variables and tuned
thresholds.

It is possible to further improve the performance of the ``Simple Loose'' identification by applying
track isolation requirements.  We study the performance of three various track isolation requirements
described in Section~\ref{ss:trackIso} where we varied the cone sizes from 0.5 to 0.05 and select 
the most optimal values. The results for the variables with optimized cone sizes are given
in Fig.~\ref{fig:TrkIso_no_vs_norm}.

 \begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/Eff_trkISoNonNorm_trkIsoNorm_trkIsoEWK.eps}
        \vspace{-10mm}
    \caption{Performance of three track isolation variables for barrel (left) and endcap (right).
    \label{fig:TrkIso_no_vs_norm}}
  \end{center}
\end{figure}

The normalized track isolation variable, $Iso^{trk}_{Norm}$, offers the best performance, 
with an expense of making the efficiency to depend on the electron candidate $p_T$. 
As it is described in the Section~\ref{Results}, the requirement results in a loss of efficiency
at low $p_T$; however, the loss is small in the characteristic range of $p_T$ of electrons 
from $W$ and $Z$ boson decays.

By introducing the isolation requirement, we also reject a significant background from
non-isolated electrons from $Zb\bar{b}$ production, where $b$ quarks decay semileptonically.
The distribution of $Iso^{trk}_{Norm}$ for signal electrons and misidentified
light- and $b$-quark jets are given in Fig.~\ref{fig:TrackIsoChowd}

\begin{figure}[!tb]
\begin{center}
\includegraphics[width=16cm]{Figs/TrackIsoChowd.eps}
\vspace{-10mm}
\caption{The distribution of the normalized track isolation defined in Eq.~\ref{eq:trkIsoN} for signal
electrons (blue solid line), misidentified light-quark jets (red squares), and $b$-quark jets (black inverted
triangles) in barrel (left) and endcap (right).
\label{fig:TrackIsoChowd}
}
\end{center}
\end{figure}

We illustrate the performance of track isolation variable chosen for ``Simple Loose'' criteria
in Fig.~\ref{fig:trkIso}, and we set the requirement of $Iso^{trk}_{Norm}$ to be less than 
0.1 (0.2) for barrel(endcap).
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=16cm]{Figs/TrkIso.eps}
\vspace{-10mm}
\caption{The $Iso^{trk}_{Norm}$ discriminant for electrons (blue soild line) and misidentified jets (red dashed line) for barrel (top left) and endcap (top right). The signal $v.s.$ background efficiencies 
obtained by varying the threshold are displayed in bottom left (barrel) and bottom right (endcap) plots.
\label{fig:trkIso}
}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Tuning the ``Simple Tight" criteria}
The ``Simple Loose'' criteria utilize variables that describe spatial- and energy-matching of a 
track to a SC, lateral energy deposition profile of a SC, and a track isolation requirement.
Although, these requirements are sufficient to select a relatively clean sample of $Z\to ee$ events,
the discriminating power is insufficient to suppress the background from misidentified jets
from the $Z+jets$ processes. Therefore, we introduce a ``Simple Tight'' criteria that is based
on the ``Simple Loose'' one with an addition of the variable that takes into account the energy
deposition in the HCAL.

The variable that is used by the official criteria is $H/E$. We also study $EmHad$ variable
defined in Eq.~\ref{eq:emhad} which is a mix of $H/E$ and an HCAL+ECAL isolation in a 
relatively narrow cone of $\Delta R < 0.3$.~\footnote{Similarly to the track isolation variable,
we varied the cone size to select the most optimal variable.} 
As expected, these two variables are found to be 
correlated, although the $EmHad$ has much stronger discriminating power 
(see Fig.~\ref{fig:Eff_hoE_IsoEmHad}, due to an additional requirement on the lateral energy 
profile in the 0.3 cone. 

\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/Eff_HoE_IsoEmHad.eps}
        \vspace{-10mm}
    \caption{$H/E$ and $EmHad$ variables for barrel (left) and endcap (right).
    \label{fig:Eff_hoE_IsoEmHad}}
  \end{center}
\end{figure}

We chose to apply the $EmHad < 0.18 (0.1)$ requirement for barrel (endcap) 
to the ``Simple Loose'' set (see Fig.~\ref{fig:EmHadIso}). The obtained selection 
criteria is thus referred to as to ``Simple Tight''.  Applying other ECAL or HCAL isolation
criteria on top of ``Simple Tight'' does not result in any substantial improvement
in performance. Thus, we believe that $EmHad$ and tracker isolation are sufficient
to identify the isolated electrons in the $p_T$ range of those from $W$ and $Z$ boson decays.
 
\begin{figure}[!tb]
  \begin{center}
    \includegraphics[width=16cm]{Figs/EmHadIso.eps}
        \vspace{-10mm}
    \caption{$EmHad$ for electrons (solid blue line) and mis-identified jets (red dashed line) 
    in barrel (top left) and endcap (top right).  The performance for the discriminator is given below
    for barrel (bottom left) and endcap (bottom right).
    \label{fig:EmHadIso}}
  \end{center}
\end{figure}


\subsection{Summary of definitions of the ``Simple Loose'' and ``Simple Tight''}

To summarize the results, the ``Simple Loose'' criteria is a simple cut-based criteria
that utilize three discriminants from the official ``Robust'' ($\Delta\eta$, $\Delta\phi$, and 
$\sigma_{\eta\eta}$), a modified variable from the ``Loose''  (a replacement of $E/p$ with 
$E_{seed}/p_{out}$) and an additional track isolation requirement. The thresholds
have been also optimized to retain high signal efficiency.

The ``Simple Tight'' requirement is ``Simple Loose'' with an addition of an effective
$H/E$ and ECAL+HCAL isolation requirement. 

We summarize the thresholds for barrel and endcap in Table~\ref{tab:OurSelection}
and proceed with detailed analysis of the performance in the next Section.

 \begin{table}[htb]
 \caption{Thresholds and efficinecy of the criteria.}
  \label{tab:OurSelection}
  \begin{center}
  \begin{tabular}{|c|c|c|} \hline
                        & Barrel threshold & Endcap  \\ \hline
 $\Delta\eta < $  & 0.009                    & 0.007      \\ 
 $\Delta\phi < $  & 0.005                   & 0.005       \\ 
 $\sigma_{\eta\eta} < $  & 0.012 & 0.026 \\ 
 $E_{seed}/p_{out} > $  & 0.9 & 0.9 \\ 
 $Iso^{trk}_{Norm} <  $  & 0.1 & 0.2 \\
 $EmHad < $                & 0.18 & 0.1 \\ \hline
     \end{tabular}
    \end{center}
  \end{table}


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