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\documentclass[prd,amsmath,amssymb]{revtex4}
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%\documentclass[prd,twocolumn,showpcs,amsmath,amssymb]{revtex4}

\usepackage{subfigure}
\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math

\newcommand{\invpb}{pb$^{-1}$}
\newcommand{\dzero}{D\O}
\newcommand{\metz}{\mbox{$\not\!\!E_{{\rm T}}^{z}$}}
\newcommand{\met}{\mbox{$\not\!\!E_{{\rm T}}$}}
\newcommand{\alpgen}{{\sc alpgen}}
\newcommand{\pythia}{{\sc Pythia}}
\newcommand{\geant}{{\sc geant}}
\newcommand{\ttbar}{\mbox{$t\bar{t}$}}

\begin{document}
%Version: 1.8  \hfill Send comments to d0-run2eb-010@fnal.gov \\
%Author(s): Mikhail Arov, Dhiman Chakraborty  \hfill   by November 29, 2006
% remove the following for publication 
\begin{figure}
\leftline{\includegraphics[scale=0.5]{d0logo.eps}\hfill D\O note 5234-CONF}
\end{figure}

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\vspace*{1.5cm}

\title{Measurement of $\sigma(p\bar{p}\rightarrow t\bar{t})$ in $\tau+jets$
channel \\ \vspace*{2.0cm}}

% throughout the internal review, the note will be authored by individuals
% who contributed to the paper
%\author{Mikhail Arov (NIU/NICADD), Dhiman Chakraborty (NIU/NICADD) \vspace*{0.5cm}}
% once the note is approved for conferences, the following author will be used
 \author{The D\O\ Collaboration}
 \affiliation{URL http://www-d0.fnal.gov}


% use the official authorlist for publication
%\input list_of_authors_r2.tex

\date{\today}
           
\begin{abstract}
% remove the space for publication
\vspace*{3.0cm}
This note presents a new measurement of $p\bar p \to t \bar tX$ production 
at $\sqrt{s}=1.96$ TeV using 350 pb$^{-1}$ of data collected with the
D\O\ detector between 2002 and 2005. We focus on the final state
where the $W$ boson from one of the top quarks decays into a 
$\tau$ lepton and its associated neutrino, while the other decays into 
a quark-antiquark pair. We aim to select those events in which the
$\tau$ lepton subsequently decays to one or three charged hadrons, zero
or more neutral hadrons and a tau neutrino (the charge conjugate 
processes are implied in all of the above). The observable signature
thus consists of a narrow calorimeter shower with associated track(s)
characteristic of a hadronic tau decay, four or more jets, of which
two are initiated by $b$ quarks accompanying the $W$'s in the top quark
decays, and a large net missing momentum in the transverse plane due to 
the energetic neutrino-antineutrino pair that leave no trace in the 
detector media. The preliminary result for the measured cross section is:

\begin{center}\[\sigma (t\overline{t}) = 
5.1\;\;_{-3.5}^{+4.3}\;\;({\textrm{stat}})\;\;_{-0.7}^{+0.7}\;\;({\textrm{syst}})\;\;
\pm0.3\;\;({\textrm{lumi}.})\;\; {\textrm{pb}}\]
\par\end{center}


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\vspace*{4.0cm}
\centerline{\em Preliminary Results for Fall 2006 Conferences}
\end{abstract}

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%\pacs{Valid PACS appear here} 

\maketitle

\newpage
\section{\label{sec:level1}Introduction}

The top quark is the heaviest fundamental particle in nature. Precise measurements of its production rate and other properties allow us to perform precision tests of QCD predictions. In addition, any deviations from the Standard Model (SM) may signal the presence of new physics. In this respect, the decays of the top quark into the $\tau$ lepton are especially interesting, since the $\tau$ is the heaviest lepton. Any non-standard flavor- and mass-dependent couplings could produce a very significant effect in this channel. An interesting example is the charged Higgs boson, which appears in
extensions of the SM Higgs sector to 2HDMs (Two-Higgs Doublet Models),
and is required in MSSM (Minimal Supersymmetric Standard Model) \cite{Charged Higgs Theory}. Since the Higgs-fermion
coupling is proportional to the latter's mass,
decays to the heavy $\tau$ lepton would be much more frequent than those to the lighter
$e$ and $\mu$. 
This prompts us to search for $H^{+}$ decaying to $\tau$ leptons.
D\O\ and CDF performed such searches in Run I \cite{CDF Charged Higgs,D0 Charged Higgs}.
The measurement of the SM process presented here is an important ingredient
to extending the reach of such searches in addition to performing a crucial test of the Standard Model.

Theoretical computations of $\sigma(p\bar{p}\rightarrow t\bar{t})$
are constantly improving. The latest published NNLO cross section is
6.8$\pm$0.4 pb \cite{NNLO}. The decay mode to $\tau+jets$ has a branching
fraction of 0.15, the same as the $e+jets$ and $\mu+jets$
channels. However, secondary $e$'s and $\mu$'s from  leptonic decays of a $\tau$ lepton are 
difficult to distinguish from prompt (primary) ones at a hadron collider. 
Hence, in this analysis we only try to identify the events in which the
$\tau$ subsequently decays to one or three charged hadrons, zero
or more neutral hadrons and a $\nu_\tau$ (the charge conjugate 
processes are implied in all of the above). 
These account for only 0.65 of the $\tau$ lepton branching ratio, so we expect the number of events 
to be lower than in the $e+jets$ and $\mu+jets$ channels.

 This note presents the first measurement of the $t\bar{t}$ cross section in this channel, using the data collected during Run II by the D\O\ detector. The integrated luminosity of our data sample was $349\pm23$ pb$^{-1}$ \cite{alljet}. After careful optimization of triggers, object 
identification and using an artificial neural network for the final
full-event pattern recognition, we are able to extract the faint
signal from the very large background, which is dominated by
mismeasured QCD events.

\section{\label{sec:detector}The D\O\ Detector}

The D\O\ Run II detector~\cite{d0det} is comprised of the following main components: the central tracking system, 
the liquid-argon/uranium calorimeter and the muon spectrometer. 

The central tracking system includes a silicon microstrip tracker (SMT) and a central fiber tracker (CFT), 
both located inside a 2~T superconducting solenoid magnet.
The SMT is designed to provide efficient tracking and vertexing capability at pseudorapidities of $|\eta| < 3$. 
The system has a six-barrel longitudinal structure, each with a set of four layers arranged axially around the 
beampipe, and interspersed with 16 radial disks. A typical pitch of 50-80 $\mu$m of the SMT strips allows a 
precision determination of the three-dimensional track impact parameter with respect to the primary vertex, which is 
the key component of the lifetime based $b$-jet tagging algorithms. 
The CFT has eight coaxial barrels, each supporting two 
doublets of overlapping scintillating fibers of 0.835 mm diameter, one doublet being parallel to the collision axis, 
and the other alternating by $\pm 3^{\circ}$ relative to the axis. 

The calorimeter is divided into a central section (CC) providing coverage out to $|\eta| \approx 1$ and two end 
calorimeters (EC) extending coverage to $|\eta| \approx 4$, each housed in separate cryostats. 
Scintillators placed between the CC and EC provide sampling of showers at $1.1 < |\eta| < 1.4$.

The muon system, covering $|\eta| < 2$, resides beyond the calorimetry and consists of 
three layers of tracking detectors and scintillating trigger counters. 
Moving radially outwards, the first layer is placed inside the 1.8 T toroid magnets, and the two following 
layers are located outside the magnets.

 The \dzero\ trigger is a three-level trigger system. Level 1 is a hardware trigger, while levels 2 and 3 are software filters.
 

\section{Object Identification \label{sec:objects}}
The most important objects for this analysis are jets and hadronic $\tau$ lepton candidates.
Jets are reconstructed using an iterative algorithm, which integrates the energy observed in the calorimeter in a cone with radius $\Delta R = \sqrt{{\delta y}^{2} + {\delta \phi}^{2}} = 0.5$, where $y$ is detector rapidity and $\phi$ is the angle in the plane transverse to the beam axis. 
These jets are required to be in the fiducial region $|\eta| < 2.4$, with $p_{T} > 20$ GeV/$c$.
The energy of the jets after reconstruction is corrected to represent the true jet energy \cite{alljet}.
An equivalent correction is derived for Monte Carlo events. 

\subsection{\label{sub:tau--ID}$\tau$ ID}

\subsubsection{Tau decay modes}

The $\tau$ lepton has several decay channels, classified by the
number of charged particles (tracks) and EM clusters associated with it \cite{PDG}
:

\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 ($\tau\rightarrow\pi^{-}\nu_{\tau}$), BR =12\% .
\item single charged hadron + $\geq1$ neutral particle (i.e.,  $\tau\rightarrow\rho^{-}\nu_{\tau}\rightarrow (n\pi^0+\pi^{-})\nu_{\tau}$)
, BR = 38\% .
\item 3 charged hadrons + $\geq0$ neutral hadrons, BR = 15\% (so-called
{}``3-prong'' decays). 
\end{itemize}

\subsubsection{Tau ID variables}

At D{\O}, $\tau$ leptons are identified in their hadronic modes as narrow ($\Delta R = 0.3$ cone) jets, isolated and matched
to charged tracks. The (most important) discriminating variables
are \cite{tau ID}:

\begin{itemize}
\item Profile - $\frac{E_{T}^{1}+E_{T}^{0}}{\sum_{i}E_{T}^{i}}$, 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 $y - \phi$ of radius $R$ around the calorimeter cluster centroid.
\item Track isolation, defined as scalar $\sum p_{T}$ of non-$\tau$ tracks in a $y - \phi$
cone of 0.5 around the calorimeter cluster centroid.
\end{itemize}
Using these and other variables, three Neural Networks (NNs) are trained to
identify three types of $\tau$ lepton ($\pi$-type, $\rho$-type and 3-prong)

The output of these NNs provides a set of three variables ({\tt{nnout}} = 1,2,3)
to be used to select the $\tau$ lepton in the event. The types roughly correspond to the $\tau$ lepton decay modes. Type 1 contains only one charged track, type 2 has one charged track and  one or more electromagnetic clusters and type 3 has more than one track (multi-prong decays). High values of NN correspond to the physical $\tau$ leptons, while low ones
should indicate jets misidentified as $\tau$'s (fakes). For more details, see Ref.~\cite{tau ID}.

\subsection{The Secondary Vertex Tagging Algorithm \label{sec:svt}}

The $\ttbar$ final state contains two $b$-jets, while jets in QCD and W+jets events originate most often from light quarks or gluons. Requiring at least one jet in the
event to be $b$-tagged is therefore a very powerful method of background rejection. The $b$-tagging algorithm used in this measurement is a
secondary vertex tagging algorithm (SVT), which explicitly reconstructs vertices that are displaced from the primary vertex. In the fitting procedure, 
the SVT uses only good quality tracks that have an impact parameter significance $>$ 3. Also, only secondary vertices which are displaced
from the primary vertex in the plane transverse to the beam line by more than 7 standard deviations are considered in this measurement. 

\paragraph{$b$-tagging efficiency}

It is known \cite{bID} that the $b$-tagging, when applied directly to Monte Carlo, gives an efficiency that is higher than data. In order to account for this factor, SVT has been parameterized
on $t\overline{t}\rightarrow\mu+jets$ MC and $\mu+jets$ data to
compute the correction factor, which scales the MC-derived efficiency. As a result,
we obtain the MC tagging probability and the data-corrected one 
\cite{bID}. It can be noted that the data-corrected efficiency is
indeed noticeably (30\%) lower than what we would expect by applying the SVT directly to MC.


\paragraph{$c$-tagging efficiency}

An assumption is made that the correction factor obtained by dividing
the semi-leptonic $b$-tagging efficiency in data to the one in MC also
is correct for $c$ jets. Hence the MC-obtained inclusive $c$-tagging efficiency
is multiplied by this factor in order
to estimate the $c$-tagging probability.


\paragraph{Light jet tagging efficiency}

The $b$-tag fake rate from light quarks is computed by measuring the
negative tag rate. It is defined by the rate of appearance of secondary
vertices with a negative decay length significance. It is assumed that
the light quarks have an equal chance to produce a secondary vertex with positive and
negative decay length significance (due to finite resolution effects)
while the heavy flavor jets can only produce a SV with positive decay
length significance. However, this is not quite true and a special
scaling factor ($SF_{hf}$) is introduced to correct for the fraction
of heavy flavors among the jets with a negative decay length significance.
Another correction is for the presence of long lived particles
in light jets ($SF_{ll}$). Both factors are derived from Monte Carlo.


\section{Backgrounds}
Two main distinctive features of the signal determine which backgrounds are important. In order to be relevant, the process must have a
high ($>3$) number of jets as well as a sizeable ($>15$ GeV) $\not\!\! E_{T}$.
All of the candidate processes are listed in Table \ref{backgrounds}.
The cross sections listed include the branching fractions into $\tau$ leptons.

We can conclude that the two dominant background sources are QCD ({}``fake
$\tau$'') and $W$+4jets. These two sources were taken into account
in this analysis. QCD is derived from the data, while Monte Carlo simulation is used for $W$+jets.

This simulation utilizes events generated at $\sqrt{s} = 1.96$ TeV with the ALPGEN~1.2 \cite{alpgen} matrix element generator, assuming a top mass 
of 175~GeV/$c^{2}$ and the parton distribution function set CTEQ~6.1M \cite{pdf}. These events are then  processed through \pythia~6.2 \cite{pythia}. This package takes into account the gluon radiation and fragmentation effects and also performs the short lived particle decays, except for $b$ hadrons and $\tau$ leptons. EvtGen \cite{evtgen} is used to model the decays of $b$ hadrons. $\tau$ leptons are decayed using Tauola \cite{tauola}. The generated events are then processed through a full \geant~\cite{geant} simulation of the \dzero\ detector providing tracking hits, calorimeter cell energy and muon hit information. Multiple interactions are added to all events according to a Poisson distribution with a mean that is determined from the average instantaneous luminosity. The same reconstruction and object ID is applied to data and Monte Carlo events. The signal ($t\bar{t}$) Monte Carlo is prepared in the same way.


\begin{table}
\label{backgrounds} 
\caption{Background sources, relevant for the $\tau+jets$ analysis. The branching fraction
into hadronic $\tau$ has been applied}
\begin{ruledtabular}
\begin{tabular}{lcr}
\hline 
Background&Description&Cross Section\\
\hline 
$W+jjjj\rightarrow\tau\nu jjjj$&
has identical signature to the signal&
$\sim$18 pb \\
$Z/\gamma+jjj\rightarrow\tau\tau jjj$ &
$\tau$ lepton is usually found as a jet&
$\sim$2.6 pb\\
$WZ\rightarrow\tau\nu jj$&
needs two extra jets (can be gluon emission) &
$\sim$0.2 pb\\ 
$WW\rightarrow\tau\nu jj$&
needs two extra jet (can be gluon emission) &
$\sim$0.5 pb\\
single top &
small cross section, but has $b$ jets &
$\sim$0.5 pb\\
QCD &
any 4-jet event that doesn't have a real $\tau$ lepton in it&
$>$100 nb \\
\end{tabular}
\end{ruledtabular}
\end{table}



\section{Event selection}
For this analysis, we used the data collected with a 4-jet trigger, which required 4 jet candidates with $|\eta | < 3.6$
and $E_{T}$$>$10 GeV using a simple cone algorithm. This trigger was designed for the all-hadronic top decay mode and works well for our purposes, since an energetic hadronic $\tau$ lepton is always found as a jet candidate. The efficiency of this trigger had been parametrized and cross-checked on data and is applied as a weighting factor to the Monte Carlo events.

The analysis procedure involved several stages:

\begin{itemize}
\item Preselection (section \ref{sub:Preselection}). At least 4 jets and
$\not\!\! E_{T}$ significance $>$ 3 are required. 653,727 events are selected in the data, with a prediction of 109.9 $\pm$ 7.3 $t\bar{t}$ events, for a S:B $\approx$ 1:6,000. 
\item ID cuts (section \ref{sub:Results-of-the}) . At least one good $\tau$ lepton 
candidate and at least one tight SVT tag are required. We also required
$\geq2$ jets with $|\eta|<2.4$ and $p_{T}>20$ GeV/$c$. 216 events are selected
in the data, 9.3 $\pm$ 0.6 $t\bar{t}$ among them are expected.
S:B $\approx$ 1:58. 
\item Topological NN (section \ref{sub:NN-variables}). A sequence of two
feed-forward NN's have been trained and applied. The optimal cut on the
second NN has been found to be 0.6. With this final cut, we obtained
13 events in data with 4.9 $\pm$ 0.3 $t\bar{t}$ among them expected.
S:B $\approx$ 1:2.5. 
\end{itemize}
The $W$ background has been modeled using ALPGEN Monte Carlo simulation,
while QCD background estimates were extracted from the data using the procedure described
in section \ref{sub:QCD-modeling}. In order to optimize the selection we also used the $t\bar{t}$  Monte Carlo and normalized it to known (either theoretical or computed by ALPGEN) values. In the end we applied this optimal selection to measure the actual $t\bar{t}$ cross section in the data.


\subsection{\label{sub:Preselection}Preselection}

The total number of events recorded by our trigger in this 349 pb$^{-1}$ data set is 17 million.
The main goal of preselection was to reduce this dataset, while imposing the
most obvious and straightforward requirements that characterize the signal
signature. Such characteristic features include the following:

\begin{itemize}
\item Moderate $\not\!\! E_{T}$ arising from both the W vertex and $\tau$
decay. 
\item At least 4 jets must be present. 
\item At least 1 $\tau$ lepton and 2 $b$ jets are present. 
\end{itemize}

$\met$ significance \cite{metl} is defined as measure
of the likelihood of $\met$ arising from physical sources,
rather than fluctuations in detector measurements. These fluctuations are predicted from measuring and parametrizing the object resolutions in the data. As can be observed
in Fig. \ref{cap:met_significance}, a cut on this variable proves to be an effective way to reduce
the data set. A cut of 3 was used for preselection.

\begin{figure}
\label{cap:metl}
\includegraphics[scale=0.6]{plots/metl}% Here is how to import EPS art
\caption{$\not\!\! E_{T}$ significance for QCD-dominated data (black), $W+jets$ (blue) and $t\bar{t}\rightarrow\tau+jets$ (red).}
\label{cap:met_significance} 
\end{figure}

Now we need to scale the original 10K events of the MC sample to 349
pb$^{-1}$. The total $t\bar{t}$ cross section is 6.8 pb \cite{NNLO}.
Taking into account the branching fraction to the hadronic $\tau+jets$
mode, the effective cross section comes out to be:
\begin{center}
$BR(\tau\rightarrow {\textrm{hadrons}})\cdot BR(t\bar{t}\rightarrow\tau+jets)\cdot\sigma(t\bar{t})=0.65\cdot0.15\cdot6.8=0.66$
pb
\end{center}

The relative flavor fractions of the $W+4jets$ process were taken
from ALPGEN simulation as ratios of the simulated cross sections. It
was then normalized to the measured total value of 4.5 $\pm$ 2.2
pb \cite{W+4j}.
Table \ref{presel} shows the results of the preselection for both
data and the backgrounds.

%
\begin{table}
\caption{Preselection results. Shown are the total acceptances (including
preselection) and the number of events scaled to 349 $\pm$ 23 pb$^{-1}$
(no systematic uncertainties except for this luminosity error are
included). The ALPGEN samples generation cuts are described in \cite{l+jets}. An estimate of QCD background not included.}
\begin{ruledtabular}
\begin{tabular}{cccc}
\hline 
& (\# passed)/(total \#)&
$\sigma$, pb&
\# passed scaled\\
\hline 
data&
653,727/17M&
&
653,727\\
$t\overline{t}\rightarrow\tau+jets$&
6,141/10,878&
0.821 $\pm$ 0.004&
109.93 $\pm$ 7.26\\
$Wbbjj\rightarrow$ $\tau\nu+bbjj$&
2,321/11,576&
0.222 $\pm$ 0.044&
9.98 $\pm$ 2.08\\
$Wccjj\rightarrow$ $\tau\nu+ccjj$&
2,289/10,995&
0.527 $\pm$ 0.059&
24.77 $\pm$ 3.22\\
$Wcjjj\rightarrow$ $\tau\nu+cjjj$&
2,169/10,435&
0.920 $\pm$0.087 &
42.23 $\pm$ 4.87\\
$Wjjjj\rightarrow$ $\tau\nu+jjjj$&
2,683/11,920&
14.14 $\pm$ 1.3&
720.33 $\pm$ 81.48 \\
\end{tabular}
\end{ruledtabular}
\label{presel}
\end{table}

\subsection{\label{sub:Results-of-the}Results of the ID cuts}

The next step was to apply the requirement of $\tau$- and $b$-tagging.
Table \ref{cap:btaggingandtau} shows the selection criteria that
we apply to data and MC and Table \ref{b and tau} shows the resulting selection efficiencies. Table \ref{b and tau (types) after eta} shows the breakdown of these events between type 2 and 3 $\tau$ (type 1 is disregarded due to its small contribution to the signal and large background). It can be noted that the S:B at this stage is 1:58, which is too small. In section \ref{sub:NN-variables} we will describe the topological NN used to enhance the signal content.

At this point, we use these ID algorithms to define 3 subsamples out of the
original preselected data sample:

\begin{itemize}
\item The {}``signal'' sample - require at least 1 $\tau$ lepton with $NN>0.95$
and at least one SVT tag (as in table \ref{cap:btaggingandtau}).
This is the main sample used for the measurement. It contains 268 events. 
\item The {}``$\tau$ veto sample'' - Same selection, but instead of $NN_{\tau}>0.95$,
$0<NN_{\tau}<0.5$ was required for $\tau$ lepton candidates and no events
with {}``good'' ($NN>0.8$) $\tau$ leptons were allowed. This sample is used
for the topological NN training. It contains 21,022 events. 
\item The {}``$b$ veto'' sample - at least 1 $\tau$ lepton with $NN>0.95$,
but NO SVT tags. This sample is to be used for the QCD prediction. It contains 4,642 events.
\end{itemize}
%
\begin{table}
\caption{$b$-tagging and $\tau$ ID. In the MC, we use the $b$-tagging certified
parameterization rather than actual $b$-tagging, that is, we applied the
$b$-tagging weight. We also used the triggering weight
as computed by the trigger efficiency parameterization.}
\begin{ruledtabular}
\begin{tabular}{ccc}
\hline 
&
{\scriptsize data}&
{\scriptsize taggingMC}\\
&
{\scriptsize $\geq1$ $\tau$ with $|\eta|<2.4$ and $p_{T}>20$ GeV/$c$}&
{\scriptsize $\geq1$ $\tau$ with $|\eta|<2.4$ and $p_{T}>20$ GeV/$c$}\\
&
{\scriptsize $\geq1$ SVT}&
{\scriptsize $TrigWeight\cdot bTagProb$}\\
&
{\scriptsize $\geq2$ jets with $|\eta|<2.4$ and $p_{T}>20$ GeV/$c$}&
{\scriptsize $\geq2$ jets with $|\eta|<2.4$ and $p_{T}>20$ GeV/$c$}\\
\end{tabular}
\end{ruledtabular}
\label{cap:btaggingandtau} 
\end{table}

%
\begin{table}
\caption{$b$-tagging and $\tau$ ID results. Shown are the total acceptances
(including preselection) and the number of events scaled to 349 pb$^{-1}$. An estimate of QCD 
background is not included.}
\begin{ruledtabular}
\begin{tabular}{cccc}
\hline 
&
{\small (\# passed)/(total \#)}&
{\small Acceptance}&
{\small \# passed scaled}\\
\hline 
{\small data}&
{\small 216/653,727}&
&
{\small 216}\\
{\small $t\overline{t}\rightarrow\tau+jets$}&
{\small 524.0/6141}&
{\small 0.0480 $\pm$ 0.0020}&
{\small 9.320 $\pm$ 0.620}\\
{\small $Wbbjj\rightarrow$ $\tau\nu+bbjj$}&
{\small 54.5/2321}&
{\small 0.0150 $\pm$ 0.0024}&
{\small 0.012 $\pm$ 0.002}\\
{\small $Wccjj\rightarrow$ $\tau\nu+ccjj$}&
{\small 13.3/2289}&
{\small 0.0039 $\pm$ 0.0012}&
{\small 0.034 $\pm$ 0.005}\\
{\small $Wcjjj\rightarrow$ $\tau\nu+cjjj$}&
{\small 8.0/2169}&
{\small 0.0025 $\pm$ 0.0010}&
{\small 0.160 $\pm$ 0.020}\\
{\small $Wjjjj\rightarrow$ $\tau\nu+jjjj$}&
{\small 3.3/2683}&
{\small 0.0009 $\pm$ 0.0006}&
{\small 0.860 $\pm$ 0.100}\\
\end{tabular}
\end{ruledtabular}
\label{b and tau} 
\end{table}


\begin{table}
\caption{$b$-tagging and $\tau$ ID results per type after the $\eta$ cut (as explained in section \ref{sub:QCD-modeling}).
Shown are the number of events predicted in signal and observed in the
data. An estimate of QCD background is not included.}
\begin{ruledtabular}
\begin{tabular}{ccc}
\hline 
&
Type 2&
Type 3\\ 
data&
91&
71\\
$t\overline{t}\rightarrow\tau+jets$&
5.61 $\pm$ 0.37&
2.81 $\pm$ 0.18\\
$W\rightarrow\tau\nu+jets$&
0.93 $\pm$ 0.04&
0.32 $\pm$ 0.01\\
\end{tabular}
\end{ruledtabular}
\label{b and tau (types) after eta}
\end{table}

\subsection{\label{sub:QCD-modeling}QCD modeling}

The difference between the total number of $t\bar{t}$ and $W$ events
and data has to be accounted for by QCD-initiated events, where a $\tau$ candidate
is a jet, mistakenly identified as a $\tau$ lepton. In order to estimate
this background contribution, the following strategy was employed.

We started with the {}``$b$ veto'' sample. This sample is dominated by multijet events in which a jet is
misidentified as a hadronic $\tau$. Since the $\tau$ candidates here are really jets, we can simply divide $\eta$ vs $p_{T}$ distributions of the $\tau$ candidates by the same distributions for jets bin by bin to parameterize the $\tau$ fake rate. In order to reduce this effect and minimize the statistical uncertainty, we performed a 2D fit to this distribution. This fit was then used for the QCD prediction.
The distributions in $\eta$ and $p_{T}$ have been separately fitted with $A(\eta)$ and $B(p_{T})$. The validity of using the normalized product of these two efficiency curves has been checked using closure tests. The result of this procedure can be observed in Fig. \ref{cap:taufakerate_fit_types_noeta}. For type 3 $\tau$ lepton candidates the $0.85<|\eta|<1.1$ region was excluded from the fit due to the poor performance of the $\tau$ ID Neural Net in this region. The final 2D parameterization of the $\tau$ fake rate (normalized product of $\eta$ and $p_{T}$ fits) is shown in Fig. \ref{cap:taufakerate_fit2D}.

%
\begin{figure}
{\tiny \subfigure[Type 2 fit]{\includegraphics[scale=0.5]{plots/fit_type2}}}{\tiny \par}
{\tiny \subfigure[Type 3 fit]{\includegraphics[scale=0.5]{plots/fit_type3_cut}}}{\tiny \par}


\caption{Fit of the $\eta$ and $p_{T}$ distributions of the $\tau$ fake
rate.}

\label{cap:taufakerate_fit_types_noeta} 
\end{figure}

\begin{figure*}
\subfigure[Type 2 2D fit]{\includegraphics[scale=0.4]{plots/type2_surf}}\subfigure[Type 3 2D Fit]{\includegraphics[scale=0.4]{plots/type3_surf}}
\caption{The 2D combined fit (in $\eta$ and $p_{T}$) of the $\tau$ fake
rate.}
\label{cap:taufakerate_fit2D} 
\end{figure*}

\subsubsection{Computing the QCD fraction}

We assume that probability for a jet to fake a $\tau$ lepton is simply $F(\eta,p_{T})=A(\eta)B(p_{T})$.
Then the probability that at least one of the jets in the event will
fake a $\tau$ can be computed as the following:

\begin{center}$P_{event}=1-\prod_{j}(1-F(p_{T}^{j},\eta^{j})).$\par\end{center}
By summing up such probabilities over the tagged data, we obtain the QCD
background estimate.

Using the results described in the previous section, we get $N_{QCD}=71.13\pm1.56$
for the $\tau$ type 2 and $N_{QCD}=77.46\pm0.80$ for the $\tau$
type 3, which agrees with the observed data (in Table \ref{b and tau (types) after eta})
fairly well. One can also observe (see Appendix) that the predicted
distributions of the main topological variables (section \ref{sub:NN-variables})
are in fairly good agreement with what is observed in the data.

\subsection{\label{sub:NN-variables}Topological NN}
Similarly to the all-jets analysis \cite{alljet}, we define 2 neural networks (NN) (the Multi Layer Perceptron \cite{MLPfit} program was used). The two NNs contain

\begin{enumerate}
\item 3 topological (aplanarity, sphericity and centrality) and
2 energy-based ($H_{T}$ and $\sqrt{s}$) variables. 
\item the output of the first NN, $W$ and $t$ mass likelihoods, 
$p_{T}$ and decay length significance of the $b$ jets.
\end{enumerate}
The kinematic and topological variables used are:

\begin{itemize}
\item $H_{T}$ - the scalar sum of all jet's $p_{T}$ (and $\tau$ lepton candidates). 
\item Sphericity and Aplanarity \cite{alljet} - these variables are formed from the eigenvalues
of the normalized Momentum Tensor of the jets in the event. These
are expected to be higher in the top pair events than in a typical
QCD event. 
\item Centrality, defined as $\frac{H_{T}}{H_{E}}$ , where $H_{E}$ is sum
of energies of the jets. 
\item Top and W mass likelihood - a $\chi^{2}$-like variable. $L\equiv\left(\frac{M_{3j}-m_{t}}{\sigma_{t}}\right)^{2}+\left(\frac{M_{2j}-M_{W}}{\sigma_{W}}\right)^{2}$,
where $m_{t}, M_{W},\sigma_{t},\sigma_{W}$ are top and W masses (175
GeV and 80 GeV respectively) and resolution values (45 GeV and 10
GeV respectively) \cite{alljet}. $M_{3j}$ and $M_{2j}$ are invariant masses composed
of the jet combinations, so as to minimize $L$. 
\item $p_{T}$ and lifetime significance of the leading $b$-tagged jet. 
\end{itemize}
Many of these variables (for instance the mass likelihood and aplanarity)
are only defined for events with 2 or more jets. Thus we require 2 jets with $p_{T}>20$ GeV/$c$ and $|\eta|<2.5$.
Plots of some of these variables, which also serves as an additional check of the 
agreement between the data and prediction, are included in the appendix.

The result of applying these NN to data is shown in Fig. \ref{cap:Result-of-applying}. 
The maximum signal significance is used to determine the optimal NN cut. The signal significance is
defined as 
%$\frac{Number\, of\, signal\, events}{\sqrt{Number\, of\, Signal+Background\, events}}$
$\frac{N_S}{\sqrt{N_S+N_B}}$, where $N_S$ and $N_B$ are the expected numbers of signal and 
background events respectively and is shown in Fig. \ref{signal-signifficance}. 
This significance reaches its maximum at $NN>0.9$ for both type 2 and 3. 
Therefore, this selection is used for the cross section measurement. 
The results are summarized in Table \ref{cap:RESULTS}.

%
\begin{figure}
\subfigure[ Type 2 $\tau$ lepton]{\includegraphics[scale=0.3]{plots/NNresult_tau2}}\subfigure[ Type 2 $\tau$ lepton (zoomed)]{\includegraphics[scale=0.3]{plots/NNresult_zoomed_tau2}}

\subfigure[ Type 3 $\tau$ lepton]{\includegraphics[scale=0.3]{plots/NNresult_tau3}}\subfigure[ Type 3 $\tau$ lepton (zoomed)]{\includegraphics[scale=0.3]{plots/NNresult_zoomed_tau3}}


\caption{Result of applying the NN cut. $t\bar{t}$, $W$ and QCD are plotted
incrementally in order to compare with the number of events observed in data.
Error bars include only statistical uncertainties. $\sigma(t\bar{t})=5.54$
pb is assumed. The right plot only shows the entries at high values of NN cut.
}

\label{cap:Result-of-applying} 
\end{figure}


%
\begin{figure}
\subfigure[Type~2]{\includegraphics[scale=0.3]{plots/NNresult_signiff_tau2}}
\subfigure[Type~3]{\includegraphics[scale=0.3]{plots/NNresult_signiff_tau3}}


\caption{$t\bar{t}\rightarrow\tau+jets$ signal significance.}

\label{signal-signifficance} 
\end{figure}


%
\begin{table}
\caption{The final result summary after the $NN>0.9$ cut. $\epsilon(t\bar{t})$
is the total signal acceptance. Uncertainties are statistical only.}
\begin{ruledtabular}
\begin{centering}\begin{tabular}{ccccccccc}

\hline 
Channel &
$N^{obs}$ &
${\mathcal{B}}$ &
$\int{\mathcal{L}}dt, pb^{-1}$ &
\multicolumn{2}{c|}{Backgrounds}&
$\varepsilon(t\bar{t})$ (\%) &
$s$ (7 pb) &
$s+b$ \\ 
\hline \\
$\tau$+jets type 2 &
5 &
0.1 &
349.3 &
$W\rightarrow\tau\nu$ &
0.60 $\pm$ 0.03&
1.57 $\pm$ 0.01 &
3.83$_{-0.51}^{+0.46}$ &
6.84$_{-0.51}^{+0.46}$ \\
&
&
&
&
fakes &
2.41 $\pm$ 0.09 &
&
&
\\ 
$\tau$+jets type 3 &
5 &
0.1 &
349.3 &
$W\rightarrow\tau\nu$ &
0.27 $\pm$ 0.01&
0.73 $\pm$ 0.01 &
1.80$_{-0.23}^{+0.22}$ &
4.39$_{-0.23}^{+0.22}$ \\
&
&
&
&
fakes &
2.33 $\pm$ 0.09 &
&
&
\\
\end{tabular}\par\end{centering}
\end{ruledtabular}
\label{cap:RESULTS} 
\end{table}

\section{Systematic uncertainties}

 The most important systematic effects (except for the $b$-tagging,
which is treated independently) are summarized in Table \ref{cap:Syst}.
Most of them are associated with uncertainties on the corresponding quantities (Jet Energy Scale, Primary Vertex, Branching Ratio and Trigger simulation). The QCD systematics comes mainly from the error of the 2D fit of the $\tau$ fake rate. The $W\rightarrow\tau\nu$ uncertainty comes from two sources: the uncertainty of the cross section value and the conservative estimate of the error associated with the method that we used for modeling this background. 
%
\begin{table}
\caption{Systematic uncertainties on $\sigma(t\bar{t})$ (in pb).}
\begin{ruledtabular}
{\footnotesize }\begin{tabular}{ccc}
\hline 
Channel&
{\footnotesize $\tau$+jets type 2 }&
{\footnotesize $\tau$+jets type 3 }\\
{\footnotesize Jet Energy Scale }&
{\footnotesize $_{-0.27, +0.30}$ }&
{\footnotesize $_{-0.69, +0.53}$ }\\
{\footnotesize Primary Vertex }&
{\footnotesize $_{+0.037, -0.036}$ }&
{\footnotesize $_{+0.095, -0.093}$ }\\
{\footnotesize MC Stat }&
{\footnotesize $_{+0.25, -0.22}$ }&
{\footnotesize $_{+0.65, -0.58}$ }\\
{\footnotesize Trigger }&
{\footnotesize $_{-0.020, +0.0025}$ }&
{\footnotesize $_{-0.069, +0.0056}$ }\\
{\footnotesize Branching Ratio }&
{\footnotesize $_{+0.074, -0.071}$ }&
{\footnotesize $_{+0.19, -0.18}$ }\\
{\footnotesize QCD fake rate Parameterization }&
{\footnotesize $_{+0.17, -0.17}$ }&
{\footnotesize $_{+0.34, -0.34}$ }\\
$W\rightarrow\tau\nu$&
{\footnotesize $_{+0.19, -0.19}$ }&
{\footnotesize $_{+0.19, -0.19}$ }\\
\end{tabular}{\footnotesize \par}
\end{ruledtabular}
\label{cap:Syst} 
\end{table}

\subsection{$b$-tagging}

The effects of uncertainties in tagging $b$ jets are taken into account by varying the
systematic and statistical uncertainties on the MC tagging weights.
The resulting effect of all of the error sources on the final number
is summarized in table \ref{cap:$b$ tagging-systematics-sources},
along with the total $b$ ID systematic uncertainty (quoted in table \ref{cap:Syst}).
%
\begin{table}
\caption{$b$-tagging systematics sources}
\begin{ruledtabular}
\begin{tabular}{ccc}
\hline 
Channel&
{\footnotesize $\tau$+jets type 2 }&
{\footnotesize $\tau$+jets type 3 }\\
$b$-tagging&
{\tiny $_{-0.13, +0.076}$ }&
{\tiny $_{-0.26, +0.41}$ }\\
c-tagging&
{\tiny $_{-0.20, +0.16}$ }&
{\tiny $_{-0.48, +0.60}$ }\\
l-tagging&
{\tiny $_{-0.0051, +0.0051}$ }&
{\tiny $_{-0.014, +0.014}$ }\\
$SF_{hf}$&
{\tiny $_{-0.00036, +0.00036}$ }&
{\tiny $_{-0.00094, +0.00094}$ }\\
$SF_{ll}$&
{\tiny $_{-0.00036, +0.00036}$ }&
{\tiny $_{-0.00094, +0.00094}$ }\\
$\mu$ $b$-tagging (data)&
{\tiny $_{-0.091, +0.094}$ }&
{\tiny $_{-0.24, +0.25}$ }\\
$\mu$ $b$-tagging (MC)&
{\tiny $_{+0.11, -0.10}$ }&
{\tiny $_{+0.28, -0.25}$ }\\
taggability&
{\tiny $_{-0.048, +0.049}$ }&
{\tiny $_{-0.13, +0.13}$ }\\
\end{tabular}
\end{ruledtabular}
\label{cap:$b$ tagging-systematics-sources} 
\end{table}

\section{Cross section}

The cross section is defined as $\sigma=\frac{Number\, of\, signal\, events}{\varepsilon(t\bar{t})\cdot BR(t\bar{t}\rightarrow \tau+jets)\cdot Luminosity}$.
The results are the following:

\begin{center}$\tau$+jets type 2 (single prong) cross section: \[\sigma (t\overline{t}) = 
3.63\;\;_{-3.50}^{+4.72}\;\;({\textrm{stat}})\;\;_{-0.48}^{+0.49}\;\;({\textrm{syst}})\;\;\pm0.24\;\;({\textrm{lumi}})\;\; \rm{pb,}\]
 \par\end{center}

\begin{center}$\tau$+jets type 3 (multi-prong) cross section: \[\sigma (t\overline{t}) = 
9.39\;\;_{-7.49}^{+10.10}\;\;({\textrm{stat}})\;\;_{-1.18}^{+1.25}\;\;({\textrm{syst}})\;\;\pm0.61\;\;({\textrm{lumi}})\;\; \rm{pb.}\]
\par\end{center}

The combined cross section was estimated by minimizing the sum of
the negative log-likelihood functions (constructed from a Poisson distribution using the number of observed events and ``hypothetically'' observed events as the Poisson mean) for each channel. The functional
form of the likelihood function was the same as has been used for
the $e\mu$ channel \cite{emu}. The combined cross section yields

\begin{center}\[\sigma (t\overline{t}) = 
5.05\;\;_{-3.46}^{+4.31}\;\;({\textrm{stat}})\;\;_{-0.67}^{+0.68}\;\;({\textrm{syst}})\;\;\pm0.33\;\;({\textrm{lumi}})\;\; \rm{pb.}\]
\par\end{center}

\section{Summary \label{sec:summary}}

This note presents the measurement of $\sigma(t\bar{t})$ by the D\O\ collaboration in the Run
2a of the Fermilab Tevatron. The decay channel studied involves one hadronically decaying $\tau$
lepton, 2 $b$ jets, 2 light jets and $\not\!\! E_{T}$. The trigger
and the corresponding $349\pm23$ pb$^{-1}$ dataset were shared with
the all-jets channel \cite{alljet}.

The main challenge was to reject the overwhelming QCD background, while
at the same time properly handling the physical $W+4j$ irreducible
background. In order to achieve this goal, a NN-based $\tau$ ID and SVT $b$-tagging
algorithm was employed. In addition, the relevant topological
variables were combined into a NN trained to differentiate the
signal from QCD. The single and triple pronged $\tau$ lepton channels were treated as independent
channels and then combined.

The cross section was measured to be
\begin{center}\[\sigma (t\overline{t}) = 
5.05\;\;_{-3.46}^{+4.31}\;\;({\textrm{stat}})\;\;_{-0.67}^{+0.68}\;\;({\textrm{syst}})\;\;\pm0.33\;\;({\textrm{lumi}})\;\; \rm{pb.}\]
\par\end{center}

This result is in agreement with the theoretically predicted value
of 6.8$\pm$0.4 pb \cite{NNLO} as well as other D\O\ and CDF measurements
\cite{alljet,l+jets}. 

\begin{acknowledgments}
We thank the staffs at Fermilab and collaborating institutions, 
and acknowledge support from the 
DOE and NSF (USA);
CEA and CNRS/IN2P3 (France);
FASI, Rosatom and RFBR (Russia);
CAPES, CNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil);
DAE and DST (India);
Colciencias (Colombia);
CONACyT (Mexico);
KRF and KOSEF (Korea);
CONICET and UBACyT (Argentina);
FOM (The Netherlands);
PPARC (United Kingdom);
MSMT (Czech Republic);
CRC Program, CFI, NSERC and WestGrid Project (Canada);
BMBF and DFG (Germany);
SFI (Ireland);
Research Corporation,
Alexander von Humboldt Foundation,
and the Marie Curie Program.
\end{acknowledgments}

\begin{thebibliography}{99}
  \bibitem{Charged Higgs Theory}J. A. Coarasa, J. Guasch, and J. Sola, Report No. UAB-FT-451, 1999, hep-ph/9903212 .
  \bibitem{CDF Charged Higgs}D. Acosta {\it et. al.}, Phys. Rev. D {\bf 62}, 012004 (2000).
  \bibitem{D0 Charged Higgs}Direct Search for Charged Higgs Bosons in $t\bar{t}$ events, D\O\ Note 4357. 
  \bibitem{NNLO}N. Kidonakis, R. Vogt, Phys. Rev. D {\bf 68}, 114014 (2003). 
  \bibitem{alljet}Measurement of the $t\bar{t}\rightarrow all-jets$ production cross section, using Secondary Vertex Tagging, D\O\ Note 4879-CONF.
  \bibitem{d0det} V.M.~Abazov, {\it et al.} (D0 Collaboration), ``The upgraded D0 detector," 
Nucl. Instrum. and Methods A {\bf 565}, 463 (2006).
  \bibitem{PDG}S. Eidelman \textit{et al.} (Particle Data Group), Phys. Lett. B {\bf 592}, 1 (2004) and 2005 update.
  \bibitem{tau ID}D\O\ $\tau$ ID certification, D\O\ Note 4773.
  \bibitem{bID} D\O\ Secondary Vertex $b$-tagger certification, D\O\ Note 4796. 
  \bibitem{alpgen}
M.L. Mangano {\it et al.}, JHEP {\bf 0307}, 001 (2003), hep-ph/0206293.
  \bibitem{pdf}
J. Pumplin, D.R.~Stump, J.~Huston, H.L.~Lai, P.~Nadolsky, W.K. Tung, hep-ph/0201195.
  \bibitem{pythia}
T. Sj\"{o}strand {\bf et al.}, LU TP 01-21, hep-ph/0108264.
  \bibitem{evtgen} 
{\texttt http://charm.physics.ucsb.edu/people/lange/EvtGen/}
  \bibitem{tauola} S. Jadach, Z. Was, R. Decker, J.H. Kuehn, Comp. Phys. Commun. {\bf}, 361 (1993) (CERN TH-6793 preprint).
  \bibitem{geant} R. Brun and F. Carminati, CERN Program Library Long Writeup W5013 1993.
  \bibitem{l+jets}Measurement of the ttbar cross section in the lepton+jets
channel at $\sqrt{s}=1.96\, $TeV (topological) , D\O\ Notes 4667 and 4662.
  \bibitem{metl} $\met$ Significance Algorithm in Run II data, D\O\ Note 4254 
  \bibitem{W+4j}Measurement of $W^{\pm}\to e^{\pm}\nu$ + Inclusive
n-jet Cross Sections with CDF Data at Tevatron Run II, http://www-cdf.fnal.gov/physics/new/qcd/wjets/Wjet\_ana/Wjet\_ana.ps   .
  \bibitem{MLPfit}http://schwind.home.cern.ch/schwind/MLPfit.html   .
  \bibitem{emu}C.Clement {\it et al.}, Top quark production cross section in the lepton+track channel using secondary vertex b-tagging, D\O\ Note 5011.
\end{thebibliography}


\appendix
\section{Kinematic Distributions}

%
\begin{figure}[htb]
\includegraphics[scale=0.4]{CONTROLPLOTS/aplan_0_type2}
\includegraphics[scale=0.4]{CONTROLPLOTS/ht_0_type2}


\caption{2 of the 5 input variables of the first topological NN before the
NN cut ($\tau$ type 2). The Kolmogorov-Smirnov (KS) probabilities
are shown, indicating the quality of the agreement.}

\label{cap:The-nn0-input-small} 
\end{figure}

%
\begin{figure}
\includegraphics[scale=0.4]{CONTROLPLOTS/nn1_type2}


\caption{The resulting output of the second (final) NN ($\tau$ type 2).}

\label{cap:The-resulting-output_type2} 
\end{figure}
%
\begin{figure}
\includegraphics[scale=0.4]{CONTROLPLOTS/nn1_type3}


\caption{The resulting output of the second (final) NN ($\tau$ type 3).}

\label{cap:The-resulting-output_type3} 
\end{figure}

\end{document}
%
% ****** End of file apssamp.tex ******
