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\section{Signal extraction} |
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%\label{sec:gen} |
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The probability to misidentify a jet as a muon is very low while for |
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the case of the electron, $\pi^0$ in jets can be misidentified as |
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electrons. When considering the subtraction of the background in this |
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analysis, we will mainly concentrate on the final state where the $W$ |
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is decaying to an electron. For the muon case, the subtraction can be |
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done using Monte Carlo sample and assigning a large error on this |
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estimation. |
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|
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\subsection{Z+jets background fraction} |
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The main background remaining even after having applied all our selection |
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in the case of the $W$ decaying to electron is the $Z+jets$ |
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production. As signal and background have a \Z boson in the final |
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state, we will concentrate on the third lepton which is an electron in |
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this study. |
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|
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In order to select the \Z candidate in the events, we apply loose |
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criteria. The loose sample contain a given number of signal events |
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which contains a third isolated electron and a given number of |
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background events which do not contains a third isolated |
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electron. When we apply the tight criteria the fraction of signal and |
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background events is changing according to the efficiency of the |
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criteria. This can be expressed by this formula: |
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\begin{eqnarray} |
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N_{loose} & = & \hspace*{0.9cm} N_e + \hspace*{0.9cm} N_{j} \\ |
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N_{tight} & = & \epsilon_{tight} N_e + p_{fake} N_{j} |
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\end{eqnarray} |
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Where $N_{loose}$ and $N_{tight}$ are the numbers of events in |
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the loose and tight samples, respectively, $N_e$ is the number of events with a third |
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isolated electron, $N_j$ is the number of events without a third |
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isolated electron, $\epsilon_{tight}$ is the efficiency of the tight |
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criteria on electron, $p_{fake}$ is the probability for a jet identified |
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as a loose electron to be also identified as a tight electron. By |
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solving this set of equations we obtain: |
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$$ |
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N_e = \frac{N_{tight}-p_{fake} N_{loose}} { \epsilon_{tight} -p_{fake}} \ \ \ \mbox{and} \ \ \ |
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N_{j} = \frac{ \epsilon_{track} N_{loose} - N_e}{ \epsilon_{tight} -p_{fake}} |
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$$ |
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|
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The estimation of $\epsilon_{tight}$ and $p_{fake}$ can be done using control |
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samples, containing electrons and jet respectively with high purity. The Tag and probe |
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method is used to determine signal efficiency $\epsilon_{tight}$. The rate |
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will be derived from $Z \to e^+e^-$ samples. In the following section we |
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describe the method used to determine $p_{fake}$. |
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|
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\subsection{Determination of $p_{fake}$} |
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\label{sec:SignalExt} |
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We separate backgrounds into two categories: one with a genuine \Z boson |
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from $\Z+jets$ processes, and the other without a genuine \Z boson from |
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$t\bar{t}$ and $\W+jet$ production. The latter source can be estimated from |
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the invariant mass of the \Z boson candidate, where the background events |
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with no genuine \Z boson should not produce a \Z mass peak and should |
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be relatively smooth. |
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|
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\subsection{Study of the background without a genuine \Z boson} |
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We estimate the background due to events without a genuine \Z boson from |
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fitting a \Z candidate invariant mass to a Gaussian function convoluted |
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with a Breit-Wigner function. The background is parameterized as a straight |
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line. An example of a fit for $3e$ category is given in Fig.~\ref{fig:ZFit}. |
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Both number of signal and background events are calculated for the |
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invariant mass range between 81 and 101 GeV. In Table~\ref{tab:FitVsMC} |
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we summarize the number of background events obtained from the fit |
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and from the Monte Carlo truth information. |
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|
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\begin{figure}[!bp] |
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\begin{center} |
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\scalebox{0.4}{\includegraphics{figs/FitBkg3eTight.eps}} |
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\caption{The invariant mass distribution of the $Z$ boson candidate that is fit to a signal |
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parameterized as a Gaussian function convoluted with a Breit-Wigner function and |
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a background, parameterized as a straight line.} |
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\label{fig:ZFit} |
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\end{center} |
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\end{figure} |
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|
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\begin{table}[!tb] |
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\begin{center} |
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\begin{tabular}{|l|c|c|c|c|c|c|c|} \hline |
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& \multicolumn{2}{c|}{Background with genuine \Z} & \multicolumn{4}{c|}{Background without |
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genuine \Z boson} \\ |
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Channel & $\Z+jets$ & $\Z b\bar{b}$ & $t\bar{t}$ & $\W+jets$ & $t\bar{t}$ + $\W+jets$ & Fit result \\ \hline |
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$3e$ Loose & 148.6 & 41.8 & 5.4 & 1.2 & 6.6& 4.4 $\pm$ 1.2 \\ \hline |
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$3e$ Tight & 51.9 & 17.3 & 3.3 & 1.3 & 4.6 & 3.7 $\pm$ 1.1 \\ \hline |
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$2\mu 1e$ Loose & 184.3 & 51.2 & 6.3 & 0 & 6.3 & 2.8 $\pm$ 1.0\\ \hline |
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$2\mu 1e$ Tight & 50.2 & 18.9 & 2.8 & 0 & 2.8 & 4.6 $\pm$ 1.3\\ \hline |
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$2e1\mu$ & 7.3 & 8.0 & 3.8 & 1.3 & 4.1 & 0.7 $\pm$ 0.5\\ \hline |
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$3\mu$ & 1.8 & 7.8 & 1.2 & 0 & 1.2 & 0.5 $\pm$ 0.4\\ \hline |
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\end{tabular} |
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\end{center} |
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\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. The ``Loose'' and ``Tight'' selection criteria applied for $W\rightarrow e\nu$ final state only. |
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%I AM NOT SURE I UNDERSTAND WHAT IS WRITTEN HERE |
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% 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. |
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} |
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\label{tab:FitVsMC} |
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\end{table} |
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|
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|
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\subsection{Estimation of the background with genuine \Z boson} |
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\label{sec:D0Matrix} |
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All of the instrumental background events with real \Z boson come from |
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$\Z + jets$ processes where one of the jets is misidentified as a |
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lepton. The same principle will be used for electron and muon while |
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the background is coming from differente sources. |
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The probability to misidentify a jet as a muon is very low in CMS, while that |
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for the case of electron can be quite high as jets with large electromagnetic |
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energy fraction can be misidentified as electrons. $\Z+jet$ background |
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is especially high for \WZ\ signal with $\W\to e\nu$. Thus, it is imperative |
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to have a reliable estimation of this background from data to avoid |
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unnecessary systematic uncertainties due to Monte Carlo description of data |
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in startup conditions. Therefore, in the following we describe the data-driven |
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estimation of the $\Z+jets$ background for the $\ell^+\ell^- e$ |
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categories. A similar study for the remaining $\ell^+\ell^- \mu$ categories |
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are in progress. However, as the $\Z+jets$ background is sufficiently small, it is |
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possible to use Monte Carlo simulation to estimate $\Z+jet$ background with |
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early data, without incurring significant systematic uncertainty due to data modeling. |
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|
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\subsubsection{$\Z+jets$ background fraction} |
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To estimate the fraction of the $\Z+jets$ events in data |
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we apply a method, commonly referred to as ``matrix'' method. |
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The idea of a method is to apply ``Loose'' identification criteria |
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on the third lepton after \Z boson candidate is identified |
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and count the number of the observed events, $N_{loose}$. |
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These events contain events with real electrons $N_{e}$ |
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and events with misidentified jets $N_j$: |
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\begin{equation} |
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\label{eq:matrixEq1} |
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N_{loose} = N_e + N_j. |
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\label{eq:matrix_1} |
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\end{equation} |
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|
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If we are to apply ``Tight'' selection on the third lepton, the number |
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of the observed events $N_{tight}$ would change as following |
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\begin{equation} |
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\label{eq:matrixEq2} |
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N_{tight} = \epsilon_{tight} N_e + p_{fake} N_j, |
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\label{eq:matrix_2} |
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\end{equation} |
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where $\epsilon_{tight}$ and $p_{fake}$ are efficiency of ``Tight'' |
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criteria with respect to ``Loose'' requirements for electrons and |
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misidentified jets, respectively. As $N_{loose}$ and $N_{tight}$ |
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are directly observable, to extract the number of $Z+jet$ events |
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in the final sample, one needs to measure $\epsilon_{tight}$ |
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and $p_{fake}$ in control data samples. Two possible ways |
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to estimate these values are given below. |
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|
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\subsubsection{Determination of $\epsilon_{tight}$} |
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|
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% UNDEERSTAND THIS FIT: THIS CANNOT BE SHOWN AS IT IS!!! |
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%\begin{figure}[bt] |
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% \begin{center} |
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% \scalebox{0.8}{\includegraphics{figs/tag_probe_fit.eps}} |
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% \caption{Invariant mass of the \Z boson candidate for ``Tight-Tight'' (a) |
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% and ``Tight-Loose'' (b) electron selections fitted to a Gaussian with |
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% bifurcated Breit-Wigner functions.} |
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% \label{fig:tagprobe} |
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% \end{center} |
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%\end{figure} |
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|
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To estimate the $\epsilon_{tight}$ we apply ``tag-and-probe'' method |
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using $\Z \to e^+e^-$ from \Z+jets Chowder sample, including \W+jets |
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and $t\bar{t}$ as background. \Z mass distribution is separated for two cases where |
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electrons from \Z decay either both pass ``Tigh'' selection (``Tight-Tight'' case), or only |
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one passes the ``Tight'' selection, while the other electron passes ``Loose'' but not ``Tight'' |
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selection (``Tight-Loose'' case). |
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%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 |
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%and straight line for a background model. \Z mass distribution and fit are shown in \ref{fig:tagprobe}. |
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|
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Equation for determination of signal efficiency is given as |
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\begin{equation} |
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\epsilon_{tight}=\frac{ 2*(N_{TT}-B_{TT}) }{ (N_{TL}-B_{TL})+2*(N_{TT}-B_{TT}) } |
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\end{equation} |
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|
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where $N_{TT}$,$B_{TT}$,$N_{TL}$ and $B_{TL}$ are, respectively, number of signal+background |
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and background events for ``Tight-Tight'' and ``Loose-Tight'' electron combinations. |
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We estimated an efficiency $\epsilon_{tight}=0.98 \pm 0.01$. |
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|
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\subsubsection{Determination of $p_{fake}$} |
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|
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As the events will be most of the time triggered by the leptons coming |
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from \Z boson, we assume that the third lepton is unbiased toward |
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trigger requirement. Ideally we need a sample of pure multi-jet events |
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from \Z boson, we assume that the third lepton is unbiased toward the |
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trigger requirement. Ideally, we need a sample of pure multi-jet events |
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|
in order to compute the probability for a jet identified as a loose |
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electron to be also identified as a tight electron. In selecting such |
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|
a sample in data, one has to avoid any bias from the trigger |
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requirements on the loose electron candidate. |
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requirements on the ``Loose'' electron candidate. |
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|
%Such sample will not |
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|
%exist in data as they will be bias by the trigger requirement. |
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+ |
\begin{figure}[bt] |
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\begin{center} |
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\scalebox{0.6}{\includegraphics{figs/tight_eff_gumbo.eps}} |
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\caption{Fraction of electron candidates passing the ``Tight'' criteria |
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in multijet event. No trigger requirement has been applied.{\em NEW PLOT |
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WITH TRIGGER REQUIREMENTS TO COME}} |
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\label{fig:qcd_efftight_noHLT} |
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\end{center} |
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\end{figure} |
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|
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\begin{figure}[bt] |
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\begin{center} |
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\scalebox{0.6}{\includegraphics{figs/p0_p_fake_mu_fit.eps}} |
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\caption{Determination of $p_{fake}$ for muons. Top plot: $p_t$ spectrum |
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of muons passing the ``Loose'' and ``Tight'' criteria in $b\bar{b}$ events |
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accepted by electron triggers; bottom plot: |
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fraction of muon candidates passing the ``Tight'' criteria. A constant fit |
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is overlayed.} |
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\label{fig:mu_pfake} |
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\end{center} |
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\end{figure} |
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|
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|
|
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From a sample of multi-jet events triggered by an ``OR'' of multi-jet |
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triggers, we will select loose electron candidates that are not |
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matched to any of the triggering object |
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%we will reject the object matched with the triggering |
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%objects. This will allow us to have a unbiased sample of multi-jet |
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%events. For the purpose of the study, we have used CSA07 Gumbo |
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%samples, with Pythia ID filtering in order to keep only events from |
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%photon+jets, QCD and minimum bias events. |
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The removal of the object matched with the triggering object is done |
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using a matching cone of $\Delta R =0.2$. "Simple Loose" selection is |
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applied to each reconstructed electron from this sample of jets from |
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QCD and photon+jet. Then the tight criteria is applied on such loose |
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electrons and the $p_{fake}$ is simply the ratio of this two |
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population. This ratio, given as a function of $Pt$ and $\eta$, is |
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showed in plots \ref{fig:qcd_zjet_est}. |
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|
|
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\subsection{Determination of $\epsilon_{tight}$} |
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From a sample of multijet events triggered by an ``OR'' of multi-jet |
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triggers, we select a ``Loose'' electron candidate that are not |
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matched to any of the trigger objects. We also require the |
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electron candidate to be separated from the jet that satisfies |
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the trigger requirement by requiring the candidate to be separated |
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%by at least $\Delta R = 0.2$ |
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from the trigger object. |
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This allows us to obtain an unbiased sample of multijet events |
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where an electron candidate is likely to be either a converted |
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photon or a misidentified jet. The $p_{fake}$ function of $p_T$ |
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and $\eta$ is simply obtained by dividing the $p_T$ and $\eta$ |
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distributions for the electron candidate that satisfied ``Simple Tight'' |
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electron identification requirements to that for electron candidates |
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that satisfied ``Simple Loose''. As an example of such a distributions, |
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the ratio of ``Tight'' over ``Loose'' electrons in QCD events is |
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shown in Fgiure~\ref{fig:qcd_efftight_noHLT}. |
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|
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For muons, a similar procedure has been applied. Since the bulk |
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of background muons is coming from heavy quark decays, we select |
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a $b\bar{b}$ sample as control sample. As an exercise, we selected |
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electron-triggered events on a $b\bar{b}$ Monte Carlo sample, |
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required one ``loose electron'' in the event and looked for |
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for muon candidates that are not close to the electron candidate, |
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and determined $p_{fake}$ on this sample of muons. The $p_t$ spectrum |
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for ``loose'' and ``tight'' muons and their ratio is shown in |
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Figure~\ref{fig:mu_pfake}. |
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|
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|
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\subsubsection{Background determination results} |
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|
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Using the values of $\epsilon_{tight}$ and $p_{fake}$ obtained |
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from the methods described in the previous sections, we estimated |
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the backgrounds from genuine Z decays by solving equations |
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(\ref{eq:matrix_1}) and (\ref{eq:matrix_2}) for $N_j$. The estimated |
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background through this method is shown in Figure XXX. |
