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\section{Object Identification \label{sec:objects}}

\noindent In this section we describe the main objects used in this study: \met, 
jets, $b$ jets and hadronic tau candidates.


\subsection{\label{sub:tau--ID}\boldmath Taus}

%\subsubsection{Tau decay modes}

\noindent Taus are reconstructed in the D0 detector from energy in the calorimeter and one or more 
tracks.The tau reconstruction algorithm uses a cone of 
$\Delta R = \sqrt{{(\Delta \eta})^{2} + {(\Delta \phi})^{2}} < 0.5$ and an inner cone of $\Delta R < 0.3$
is used to calculate tau isolation variables. 

For us, the most important discriminating variables for $\tau$-leptons are \cite{tau-id}:

\begin{itemize}
\item Profile - $\frac{E_{T}^{1}+E_{T}^{2}}{\sum_{i}E_{T}^{i}}$, where
$E_{T}^{i}$ is the $E_{T}$ of the $i^{\rm th}$ highest $E_{T}$ tower in
the cluster.
\item Isolation, defined as $\frac{E(0.5)-E(0.3)}{E(0.3)}$, where $E(R)$
is the energy contained in a $\eta,\phi$ of radius $R$ around the calorimeter cluster centroid.
\item Track isolation, defined as scalar sum of the $p_{T}$'s of non-$\tau$ tracks in a $\eta,\phi$
cone of 0.5 around the calorimeter cluster centroid divided by similar sum for tracks associated with $\tau$.
\end{itemize}

Such variables are chosen based on the possible tau decays:


\begin{itemize}
\item electron or muon ($\tau\rightarrow e\nu_{e}\nu_{\tau}$ or $\tau\rightarrow\mu\nu_{\mu}\nu_{\tau})$,
BR = 35\% .
\item single charged hadron and no neutral hadrons ($\tau\rightarrow\pi^{-}\nu_{\tau}$), BR =12\% .
\item single charged hadron + $\geq1$ neutral 
hadron (i.e.,  $\tau\rightarrow\rho^{-}\nu_{\tau}\rightarrow (\pi^0\pi^{-})\nu_{\tau}$)
, BR = 38\% .
\item 3 charged hadrons + $\geq0$ neutral hadrons, BR = 15\% (so-called
{}``3-prong'' decays). 
\end{itemize}


\noindent Reconstruction of hadronic decay of taus  results in classification of a tau candidate
in one of the following three types \cite{PDG}:

\begin{enumerate}
\item {\bf Type1}: calorimeter cluster, one matched track and no associated EM subcluster. 
Mainly $\tau\rightarrow\pi^{-}\nu_{\tau}$.
\item {\bf Type2}: calorimeter cluster, one matched  track and one or more associated EM subclusters. 
Mainly $\tau\rightarrow\rho^{-}\nu_{\tau}\rightarrow \pi^0\pi^{-}\nu_{\tau}$.
\item {\bf Type3}: calorimeter cluster, two or more matched tracks and with or without EM subcluster. 
Mainly $\tau\rightarrow\pi^{-}\pi^{-}\pi^{+}(\pi^{0})\nu_{\tau}$.
\end{enumerate}

A seperate NN is trained to identify each type of tau.

The output of these NNs provides a set of three variables ({\tt{nnout}} = 1,2,3)
to be used to select the tau in the event. The types roughly 
correspond to the $\tau$ lepton decay modes. High values 
of NN correspond to the physical taus, while low ones
should indicate jets misidentified as taus (fakes). 

\subsection{\label{sub:jet--ID}\boldmath Jets}

\noindent Jets are identified using the Run II cone algorithm \cite{jet-id} with cone size of
$\Delta R < 0.5$. The jet algorithm T42 \cite{t42} is run before jet reconstruction
to remove isolated small energy deposits due to noise. D0 standard jet quality cuts \cite{jet-qual}
include L1 Trigger information, calorimeter electromagnetic fraction and coarse hadronic fraction.

Jets used in this analysis are required to have at least two primary vertex tracks associated to them
(vertex confirmed jets). This choice was motivated by a better agreement between data and
MC, a better modeling in the ICD region of the calorimeter and the fact that all b-ID
studies were done using this kind of jets. It implies that although a calorimeter cluster
is still reconstructed as a jet, it will be discarded if it has less than 2 associated PV tracks.
In order to correct the energies of reconstructed jets in data and MC back to
parton-level energies, we apply certified jet energy scale correction (JES)\cite{jes}. Additionally,
jets containing a muon with $\Delta R(\mu , jet) = 0.5$ from heavy quark decays are corrected to take into
account the momentum carried away by the muon and the neutrino \cite{jesmu}.


\subsection{\label{sub:met-id}\boldmath \met}

\noindent Presence of neutrinos in an event is inferred from an imbalance of the component 
net momentum in the plane perpendicular  to the beam (transverse plane).
This quantity is calculated from the transverse energies of all calorimeter cells that pass the 
T42 algorithm, except those of the coarse hadronic layers due to high noise level. However, they are included
in the case that they are clustered within a reconstructed jet. This raw \met is corrected for the energies
of other objects like photons, electrons, taus and jets. As muons deposit only a small portion of their
energy in the calorimeter, their momenta is subtracted from the \met vector.

\subsection{b jets \label{sec:nntag}}

Since the main sources of background in this analysis are QCD and $W$ + jets, requiring
the presence of at least one jet coming from a $b$-quark is a very powerful method of background 
rejection. The $b$-tagging algorithm used in this measurement is a
Neural Network (NN) tagging algorithm developed by the b-ID group \cite{bID-p20},
which combines 7 characteristic variables of SVT, JLIP and CSIP tagging algorithms into the NN 
discriminant. As in the previous analysis we have chosen the operating
point TIGHT, which is equivalent to requiring the NN discriminant output to be greater than 0.775.
Both the average efficiency and fake rate are comparable between this p20 version of the
algorithm and the version used in p17 \cite{bID-p17}.

\paragraph{$b$-tagging efficiency}

In data we apply the b-tagging algorithm directly to jets selected in our sample. In MC such ``direct tag''
is not done. Instead we have to apply a certain efficiency to MC samples. This inclusive b-decay efficiency ($\epsilon_{b}$)
is measured in data and it is the product of the probability to tag a b-jet in an MC sample ($\epsilon_{b}^{MC}$)
containing inclusive decays of the b quark and a scale factor. This data/MC scale factor is given by the ratio
of data semileptonic efficiency ($\epsilon^{DATA}_{b\rightarrow \mu}$) and a MC semileptonic efficiency ($\epsilon^{MC}_{b\rightarrow \mu}$).
This scale factor, which measures the effect on the tagging rate
caused by the differences in tracking between data and MC, is then used to properly scale
the MC-derived efficiency. It is assumed that such factor could be applied to any MC tagging efficiency \cite{bID-p20}.

\paragraph{$c$-tagging efficiency}

It is assumed that the same procedure adopted in the b-jet case is also valid for c-jets,
namely, a c-jet scale factor ($\epsilon^{DATA}_{b\rightarrow \mu}$)/($\epsilon^{MC}_{b\rightarrow \mu}$)
multiplies the probability to tag a c-jet in an MC sample ($\epsilon_{b}^{MC}$) to get the
c-tagging efficiency.


\paragraph{Light jet tagging efficiency}

The $b$-tag fake rate from light quarks is computed by measuring the
negative tag rate as defined in \cite{b_fake}. The method uses fits of
b-, c- and light jets tagging rates to binned data combined with sample
composition estimated from data. Binned NN output ($NN_{out}$) distributions
for b, c and light jets is fitted to data distribution using b- and c-jet efficiencies
provided by standard bID TRF's.



\paragraph{Taggability}

$b$-tag algorithm can't be applied to any jet, but only the ones that contain tracks. 
Such jets are called ``taggable'' and are defined by matching within $\Delta R$ $<$ 0.5 to 
a track jet, composed of at least two tracks. 
Just like b-tagging, taggability is different in data and MC, due to imperfect 
simulation of tracking system. To account for this, taggability rate functions are applied
to MC events. Such functions are parametrized in terms of jet $p_{T}$, jet $\eta$ and
primary vertex $z$. They were derived on the single top loose data samples, where the isolation
quality of the lepton is loose.The validity of these functions is tested by comparing observed
and predicted taggability to tight samples. The results show a good agreement as described in \cite{single top}. 

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