
\vspace{0.1in}
\small
\begin{center}
{\it M. Roco}
\end{center}
\normalsize \nopagebreak

To identify a jet associated with a $b$ quark, a technique, referred to as
{\em '$b$ tagging'}, is performed by reconstructing the decay vertex of a
long-lived $b$ hadron within the jet.  The algorithm uses
the precise track reconstruction of the silicon tracker to identify
secondary vertices which are significantly displaced from the primary
interaction vertex.  The separation of a secondary vertex from the primary
$p \overline p$ interaction vertex depends on the spatial resolution of the
individual tracks associated to the secondary vertex and on the primary vertex
resolution.

This report presents the results of a study performed to estimate the
$b$-tagging efficiency in $\ttbar$ and Higgs events by finding
secondary vertices from $b$-quark decays.  The study is based on the
MCFAST~\cite{mcfast} package, a useful tool designed to provide a fast
simulation framework for detector design studies.  It has been
interfaced with either PYTHIA or ISAJET event generators for the
parton fragmentation and hadronization.  It performs parameterized
tracking where for each generated track, a covariance matrix is
assembled, which represents all the material and detector planes
traversed by the ideal track.  A reconstructed track is produced by
smearing the generated track parameters according to this covariance
matrix.

\vspace{0.2in}
{\bf Detector Simulation} \\ \nopagebreak

The implementation of the Run 2 D\O \ tracking detector geometry includes a
solenoid which produces a 2 Tesla uniform magnetic field along the $z$-axis.
Within the magnetic field of the solenoid are the tracking detectors for
charged particle identification.
Drift chambers are used to simulate the scintillating central fiber tracker.
A six barrel 4-layer geometry is used for the silicon microstrip tracker (SMT).
For this study layers 1 and 3 of the four innermost barrels have 90$^o$ 
stereo angle.

The outermost barrels have single-sided detectors and therefore only give
$r-\phi$ information.  Layers 2 and 4 of all six barrels have 2$^o$ stereo.
The barrel segments are separated by gaps containing F disks.
The barrels are symmetric around $z=0$ and extend from $-38$ cm to $+38$ cm.

The beam vertex is centered at the origin with a smearing of 30 $\mu m$ in
both $x$ and $y$ and 25 cm in $z$.  In this study only one interaction
per beam crossing is considered.

\vspace{0.2in}
{\bf Tagging Algorithm} \\ \nopagebreak

Given the relatively long lifetime of the $b$ quark, one can select jets
arising from $b$ quark hadronization by 'tagging' long-lived hadrons within
jets.  The decay of a long-lived hadron produces several charged tracks
emanating from a point, or secondary vertex, separated from the primary
$p \overline p$ interaction point.

Track displacement with respect to the primary vertex is measured
using the perpendicular distance at the point of closest approach, or
impact parameter $d_0$ shown in Fig.~\ref{f:ip2d}.  The sign of $d_0$
is given by the location of the beamline in the transverse plane,
relative to the particles' trajectory in this plane.  For positively
charged tracks, the sign of $d_0$ is positive if the location of the
beamline is outside the circle.  The impact parameter resolution
includes contributions coming from the intrinsic detector resolution,
multiple scattering, and uncertainties from the primary vertex
reconstruction.  The impact parameter significance, $\left | d0\right
|/ \sigma (d0)$, is a gaussian of width 1 with non-gaussian tails from
heavy flavor quark content and and tracking errors.  Fig.~\ref{f:ip}
shows the impact parameter and significance distributions for tracks
originating from the primary vertex (top plots) and those tracks
associated with the $b$ jets (bottom plots).  Tracks from the $b$
hadron decays populate the tails of the distributions.  Most particles
in an event originate from the primary vertex.

\begin{figure}
  \begin{center}
    \parbox{5.0in}{\epsfxsize=\hsize\epsffile{./btag/perigee-sec2.eps}}
  \end{center}
  \caption{The track displacement relative to the primary vertex is measured
	using the perpendicular distance of closest approach, or impact
	parameter $d0$.  The transverse decay length $L_{xy}$ is the
	distance between the primary vertex and the secondary vertex.}
  \label{f:ip2d}
\end{figure}

\begin{figure}
  \begin{center}
    \parbox{5.0in}{\epsfxsize=\hsize\epsffile{./btag/impact-v2.eps}}
  \end{center}
  \caption{The impact parameter and significance distributions of tracks
        originating from the primary vertex are shown in the top plots.
        In comparison, tracks associated with the $b$ jets are shown in
        the bottom plots.  Tracks from the $b$ hadron decays, shown as the
	dots, populate the tails of the distributions.}
  \label{f:ip}
\end{figure}

The algorithm loops over tracks associated to a jet of cone radius $r
> 0.5$ and $p_T >$ 15 GeV/c.   A tight selection criterion is applied
in the first pass.  If there are less than three tracks associated to
the reconstructed decay vertex, a looser selection is applied.  The
selection is based on the track impact parameter significance.
Fig.~\ref{f:rate} shows the efficiency of the track selection using a
sample of ISAJET $\ttbar$ events.  It gives the fraction of jets,
associated with either a $W$ or a $b$ hadron, which have at least
three significantly displaced tracks (solid curves) for increasing
$\left |d0\right |/ \sigma (d0)$ cuts.  The dashed curves show the
corresponding efficiency if the requirement on the minimum number of
tracks is relaxed to two.  These plots show that a tight $+$ loose
tagging scheme requiring at least two significantly displaced tracks
keeps about 55\% of the $b$ jets in the $\ttbar$ sample, while less
than 5\% of the jets associated with $W$ boson production ($W \to
\qq$), mainly from charm hadron decays, are retained.

\begin{figure}
  \begin{center}
    \parbox{5.0in}{\epsfxsize=\hsize\epsffile{./btag/top-sd0.eps}}
  \end{center}
  \caption{Using an ISAJET $\ttbar$ sample,
	the plot shows the fraction of jets, associated with either a
	$W$ or a $b$ hadron, which have two (dashed curves) or three
	(solid curves) significantly displaced tracks for increasing
	$\left |d0\right |/ \sigma (d0)$ cuts.}
  \label{f:rate}
\end{figure}

The selection criteria for the tight and loose algorithms, PASS1 and PASS2,
are enumerated below:

\begin{itemize}
\item[$\bullet$] PASS1 cuts:
\begin{itemize}
\item [-]track $p_T \geq$ 0.5 GeV/c with at least 4 (axial + stereo)
SMT hits
\item [-]require $d0 < 0.15$ cm to remove tracks consistent with $\gamma$
conversions and $K_S$ and $\Lambda$ decays originating from the primary vertex
\item [-]impact parameter significance, $d0/\sigma (d0) = S_{d0} \geq 2.5$
\item [-]give list of at least 3 candidate tracks to vertex finder
\item [-]require at least 3 tracks associated to the vertex
\end{itemize}
\item[$\bullet$] PASS2 cuts:
\begin{itemize}
\item [-]track $p_T \geq$ 1.5 GeV/c
\item [-]impact parameter significance, $d0/\sigma (d0) = S_{d0} > 3.0$
\item [-]give list of at least 2 candidate tracks to vertex finder
\item [-]require at least 2 tracks associated to the vertex
\end{itemize}
\item[$\bullet$] Secondary Vertex:
\begin{itemize}
\item [-]candidate tracks are passed to a 3D constrained vertex fit
\item [-]find a set of pairwise compatible tracks and
\item [-]reject tracks whose contribution to the overall $\chi^2$
of the fit $>$ 16
\item [-]$\chi^2$ per degree of freedom of vertex fit $< 10$
\item [-]calculate the 2D decay length, $L_{xy}$, and its error $\sigma (L_{xy})$
\item [-]sign of $L_{xy}$ = dot product of $L_{xy}$ and the jet direction
\item [-]require $\frac{\left | L_{xy} \right |}{\sigma (L_{xy})} > 3$ and $\left | L_{xy} \right | < 2.5$ cm
\end{itemize}
\end{itemize}

Candidate tracks are constrained to a common vertex to determine the
transverse decay length $L_{xy}$.  A vertex position is calculated and tracks
with a large contribution to the overall fit $\chi^2$ are excluded.  There
should be at least two tracks used in the vertex fit.
The reconstructed vertex is required to be within the innermost
layer of the SMT, with $L_{xy} <$ 2.5 cm, and that the decay length significance
be $\left |L_{xy}\right |/\sigma (L_{xy}) >$ 3.

\begin{figure}
  \begin{center}
    \parbox{5.0in}{\epsfxsize=\hsize\epsffile{./btag/ptfit-comprsn.eps}}
  \end{center}
  \caption{Tag efficiency per jet as a function of the $b$-jet $p_T$
	for two different physics samples.  The rates are parameterized
	using the functional form $\epsilon \ \tanh \ (p_T/c)$.}
  \label{f:tagged}
\end{figure}

\begin{figure}
  \begin{center}
    \parbox{5.0in}{\epsfxsize=\hsize\epsffile{./btag/tagged.eps}}
  \end{center}
  \caption{Mistag efficiency per jet as a function the jet $p_T$ for
  	$c$-jets (top plot) and $u,d,s,g$ jets (bottom plot).}
  \label{f:mistags}
\end{figure}

\vspace{0.2in}
{\bf Tag Rates} \\ \nopagebreak

The tag rates obtained after applying the selection criteria enumerated above
are shown in Figs.~\ref{f:tagged} and ~\ref{f:mistags}.  The plot in
Fig.~\ref{f:tagged} shows a comparison of the $b$-tag rates for two
different samples, a $\ttbar$ sample generated using ISAJET shown as
the points and a Higgs sample ($\qq \to WH$) generated using PYTHIA
shown as the dashed histogram.  The tag rates are parameterized in terms of
the $b$-jet $p_T$ using the functional form $\epsilon \ \tanh \ (p_T/c)$,
where $\epsilon = 0.57 \pm 0.012$ and  $c=36.05 \pm 1.42$ are the constants
obtained from the average of the two fits.  This plot also shows that
the difference between the rates obtained assuming different fragmentation
and hadronization schemes is small.

Fig. ~\ref{f:mistags} shows the efficiency of mistagging light quark
or gluon jets as $b$-jets.  The tag rate for $c$-jets, shown in the
top plot, is roughly one third of the $b$-jet tag rate. Tag rates for
$u,d,s$ and $g$ jets, shown in the bottom plot, are less than 1\%
for $p_T < 70$ GeV.  However, this still presents a serious problem
when dealing with background QCD multi-jet events with cross sections
which are orders of magnitude larger than the $\ttbar$ or the
expected Higgs production cross sections.


