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% \title{Measuring the b-jet tagging efficiency using top anti-top events with ATLAS data}
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% \author[a]{Dominik Duda}
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% \affil[a]{Bergische Universit\"at Wuppertal}
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% \abstracttext{Many physics analyses with the ATLAS detector expect to have jets
%      originating from b-quarks in the final state. Algorithms that allow 
%      to identify such jets are thus of great importance and it is crucial 
%      to understand their performance with data-driven measurements of     
%      efficiencies and fake rates.
%      Since the top quark almost exclusively decays to a W boson and
%      a b-quark, a sample of top anti-top events is ideal for studying
%      the b-tagging performance.
%      Final states containing one or two leptons have been used to measure
%      the b-tagging efficiency, either by counting the number of b-tagged 
%      jets or by applying kinematical fit methods to extract a pure sample
%      of b-jets.
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\begin{document}

\Title{Measuring the b-jet tagging efficiency using top anti-top events with ATLAS data}
\bigskip\bigskip
\begin{raggedright}
{\it Dominik Duda on behalf of the ATLAS Collaboration.\\
Bergische Universit\"at Wuppertal\\
D-42119 Wuppertal, GERMANY}
\bigskip\bigskip
\end{raggedright}
%\linenumbers

\noindent
The identification of jets originating from $b$-quarks is relevant for many physics analyses with the 
ATLAS detector \cite{ATLAS}. Algorithms that allow to identify such jets are thus of great importance and it is crucial to understand their performance with data-driven measurements of efficiencies and fake rates. \\
Since the decay of a $t\bar{t}$ pair has a very clear signature and because the top quark almost exclusively decays to a $W$ boson and a $b$-quark, a sample of top anti-top events is ideal for calibrating the various $b$-tagging algorithms used in ATLAS analyses. The calibration results are presented in the form of data-to-simulation scale factors

\begin{align}
\kappa_{\varepsilon_b}^{\text{data}/\text{sim.}}=\frac{\varepsilon^{data}_b}{\varepsilon^{sim.}_b}
\end{align}
in which the measured $b$-tagging efficiency in data is divided by the $b$-tagging efficiency in simulated events. Currently three different $t\bar{t}$ calibration methods are in use, where both the lepton+jets and the dileptonic decay channels are considered \cite{btag}.

% \section{Measurement}
% \subsection{Kinematic selection method}
\noindent
Using the kinematic selection method the $b$-tagging efficiency can be estimated by
\begin{equation}\label{eq:method_kinsel_bigeq}
  \varepsilon_{b} = \frac{1}{f_{b-\mathrm{jets}}} \cdot  \Big(f_{b-\mathrm{tag}}-
  \varepsilon_{c} f_{c-\mathrm{jets}} - \varepsilon_{l} f_{l-\mathrm{jets}} - \varepsilon_{\mathrm{fake}}
f_{\mathrm{fake}}\Big)\;,
\end{equation}
where $f_{b-\mathrm{jets}}$, $f_{c-\mathrm{jets}}$ and $f_{l-\mathrm{jets}}$ are the fractions of $b$-, $c$- and light-flavour jets within the selected sample, while $f_{\mathrm{fake}}$ is the fraction of jets coming from the fake lepton background. The mistag efficiencies $\varepsilon_c$ for $c$- and $\varepsilon_l$ for light-flavour jets as well as the fractions of the various jet flavours are calculated using all the selected events in simulation, while the tagging efficiency of the jets coming from the fake lepton background $\varepsilon_{\mathrm{fake}}$ and the fraction of $b$-tagged jets $f_{b-\mathrm{tag}}$ are obtained from data.\\
 While the $b$-jet purity with the kinematic dileptonic selection reaches $40$-$80\;\%$, the lepton+jets selection includes a requirement that at least one jet in each event is $b$-tagged, to enhance the signal purity. If the leading jet (the second leading jet) in the lepton+jets analysis passes that $b$-tagging requirement, the $b$-tagging efficiency is measured on the three subleading jets (the leading jet). In the dilepton channel the $b$-tagging efficiency is instead extracted from the two leading jets.

% \subsection{Kinematic Fit}
% 
% The kinematic fit method takes advantage of the event structure of the semileptonic ttbar decay by reconstructing both the hadronic and the leptonic top quark decay. The fit assigns the leading jets in the event to originate either from the W-boson or the $b$-quarks, and provides a probability measure that this is the correct assignment. When taking only the four leading jets into account 12 various permutations are possible. Only the permutation with the lowest      is chosen to form the ttbar candidate. 
% The sample is futher purified by requiring that the jet assumed to be the b-jet on the hadronic side of the event is b-tagged, while the two jets assigned to the W-boson are required to not be b-tagged. The measurement of the b-tag efficiency is then performed on the jet assigned to be the b-jet on the leptonic side of the event.
% 
% plot
% 
% 
% This jet is however not always a b-jet due to the kinematic fit assigning the jets wrongly. The fraction of wrong combinations in the signal sample is estimated using an orthogonal background sample, where one of the jets associated to the W-boson is required to be b-tagged. The number of events in the background sample with large fit       are normalized to the corresponding number of events in the signal sample.The       shape of the background sample is then used to estimate the fraction of background in the low       region of the signal sample. The b-tag output weight distribution in this background-subtracted signal sample is then used to derive the b-tag efficiency in data.
% \subsection{Tag counting method}
\noindent
In case of the tag counting method a Likelihood fit is performed to estimate the $b$-tagging efficiency by assuming that the expected number of events containing $n$ $b$-tagged jets is given by
\begin{eqnarray}
 \begin{split}
  <N_n> & =  \sum_{i,j,k} \bigg\{ ( \sigma_{t\bar{t}}\cdot \mathrm{BR} \cdot \mathcal{A}_{t\bar{t}} \cdot \mathcal{L}
  \cdot F_{ijk}^{t\bar{t}} + N_{\rm bkg}\cdot F_{ijk}^{\rm bkg}) \times  \\
  & \sum_{i'+j'+k'=n}
  \binom{i}{ i'} \, {\varepsilon_b}^{i'} \, (1- \varepsilon_b)^{i-i'}\cdot
  \binom{j}{ j'} \, {\varepsilon_c}^{j'} \, (1- \varepsilon_c)^{j-j'}\cdot
  \binom{k}{ k'} \, {\varepsilon_l}^{k'} \, (1- \varepsilon_l)^{k-k'} \bigg\}
\label{eqn:method_tagcount_lepjet}
 \end{split}
\end{eqnarray}
\noindent
where $i$, $j$ and $k$ are the number of $b$-, $c$- and light-flavour jets per event before applying $b$-tagging, while $i'$, $j'$ and $k'$ represent the number of those jets after $b$-tagging. $F_{ijk}$ is the fraction of events (before tagging) containing $i$ $b$-jets, $j$ $c$-jets and $k$ light-flavour jets. $BR$ is the branching ratio, $\mathcal{A}_{t\bar{t}}$ is the selection acceptance and $\mathcal{L}$ is the integrated luminosity. The $b$-tagging efficiency can then be determined by fitting this expected $n$-tag distribution to that observed in data.\\
The $b$-tagging calibration results for the kinematic selection and tag counting methods are shown in Figure \ref{fig:results}.

\begin{figure}[!h]
      \includegraphics[width=0.3\textwidth]{fig_13a.eps}
      \includegraphics[width=0.3\textwidth]{fig_13b.eps}
      \includegraphics[width=0.3\textwidth]{fig_17b.eps}
% \includegraphics[width=0.3\textwidth]{fig_17a.eps}
 \caption{$b$-tagging efficiencies in data and simulation (left), the corresponding scaling factors $\kappa_{\varepsilon_b}^{\text{data}/\text{sim.}}$ (middle) both measured with the kinematic selection method and the 2-dimensional contour of the $b$-tagging efficiency and the $t\bar{t}$ cross section from the tag counting method (right). All plots show results for the SV0 tagging algorithm at an operating point corresponding to $50\;\%$ signal efficiency in simulated $t\bar{t}$ events \cite{btag}.}
 \label{fig:results}
 \end{figure}

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% A description of these observations, obviously based on second-hand reports,








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\bibitem{ATLAS}
ATLAS Collaboration, 2008 JINST 3 S08003.

\bibitem{btag}
ATLAS Collaboration, \textit{Calibrating the $b$-Tag Efficiency and Mistag Rate in $35\; \mathrm{pb}^{-1}$ of Data with the ATLAS Detector}, ATLAS-CONF-2011-089, https://cdsweb.cern.ch/record/1356198.



%%CITATION = PWASA,13,1564;%%

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% The ATLAS Collaboration, “Expected performance of the ATLAS experiment: detector, trigger and physics" ,2009.
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% \speaker{D. Giovanni (University of Seville)}  My analysis indicates that the
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% \speaker{Reggiano} Professor Giovanni has discussed this hypothesis in several
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% \speaker{D. Anna (University of Seville)}  In fact, my colleague Giovanni 
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