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\section{Physics benefits\label{s-benefits}}

\subsection{Introduction}
The proposed trigger processor will perform the following tasks:
\begin{itemize}
\item find all hit clusters in the SMT 
\item determine the cluster centroid (by forming pulseheight-weighted averages)
and the total pulseheight of every cluster;
\item provide L3 with the cluster and pulseheight information
\item find tracks in the roads determined from track candidates found by the L1
CFT trigger
\item determine the track parameters (transverse momentum, impact parameter) and
send them to the L2 global processor 
\item find the z-positions of the primary vertices of the \pbp\ interaction(s)
during the beam crossing, using the clusters found in the 90\Deg stereo strips,
and send the z-values and the corresponding track multiplicity to L3.
\end{itemize}
The additional information, namely $p_T$ with a better resolution than would be
available at L2 without STT, the impact parameter, and the z-position of the
vertices, provide powerful  handles that can be used in the trigger decision to
reduce background trigger rate and/or increase signal efficiency. In this
section, we describe some of the studies that we have done to assess and
quantify the benefits that the STT could bring to our physics program.
 
\subsection{Trigger simulation}
\subsubsection{Data Samples}
%\par

A number of studies were done to assess the physics benefits and understand 
the potential performance of the STT
\cite{fs-ttbar, fs-ttbar-upd, dn-susy,
jh_higgs, mn-zvertex, mn-zbb, fs-bphysics,th-singletop,th-sttperf}. 
For this purpose  
Monte Carlo events were generated
with various generators (ISAJET, PYTHIA, COMPHEP) and then underwent 
full detector simulation using
UPG\_GEANT. After that, they were converted to the standard run
II trigger simulation ntuple\cite{ntuple}. Three types of  data 
samples were used in these studies:
\begin{itemize}
\item single track events, with and without additional minimum 
bias events overlayed;
\item signal events, with and without additional minimum bias events 
overlayed;
\item background events.
\end{itemize}


All of the samples were generated with the
 mean primary vertex position at (0.,0.,0.). The  $z$ co-ordinate
of the primary vertex was smeared with a Gaussian of width
25 cm. The effect of lateral smearing of the primary vertex position due to the
finite beam size was found to have a small effect on the conclusions about the
efficiency vs. background rejection power of the trigger~\cite{fs-ttbar-upd}
(see also subsection~\ref{ss-ttbar}.

A summary of the signal events used for these studies is given in Table \ref{tab:sigevt}.
More details about these samples can be found in the respective 
D\O\ notes~\cite{fs-ttbar, dn-susy,
jh_higgs, mn-zvertex, mn-zbb, fs-bphysics,th-singletop,th-sttperf}.

\begin{table}[htb]
\caption{Signal events}
\label{tab:sigevt}
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
reaction & generator & add'l       & number        \\
         &           & min. bias   & of events     \\
\hline 

\ttaj  & ISAJET       & 0        & 2126   \\
\ttaj  & ISAJET       & 3        & 500   \\
\ttmj  & ISAJET       & 0        & 1175   \\
\Zbb   & ISAJET       & 0        & 9979   \\
\Zbb   & ISAJET       & 2        & ~995   \\
\Zbb   & ISAJET       & 5        & ~995   \\
$\ppWHqq$& Pythia V5.7  & 0        & 1700   \\
$t \to W \to e \nu$ & COMPHEP&0    & 20000    \\
$t \to W \to \mu \nu$ & COMPHEP&0    & 20000    \\
$t \to W \to j j$ & COMPHEP&0    & 5000        \\
$ Q \bar Q \to \mu \mu X$        & NLO   & 0 & 45000   \\
$ Q \bar Q \to e^+e^- X$        & NLO   & 0  & 45000   \\
$b \to J/\psi + X, J/\psi \to \mu^+ \mu^- $             & ISAJET & 0      &         \\
$b \to J/\psi + X, J/\psi \to e^+ e^- $             & ISAJET & 0      &         \\
direct $ J/\psi + X $          & ISAJET & 0      & 9760     \\
 \bkspsi                       & ISAJET & 0      &         \\
$B_s \to D_s^- + \mu ^+ + \nu_{\mu} $ & ISAJET & 0      & 2275         \\
$B_s \to D_s^- + 3\pi^{\pm} $ & ISAJET & 0      & 2323         \\
 \hline 
\end{tabular}
\end{center}
\end{table}

\par 
The QCD background sample was the standard run II trigger studies
sample\cite{trigsimII}.  The sample consists of dijet events generated in six
hard-scatter $E_T$ bins.  For each $E_T$ bin, samples were created with 1, 3, 5, or 7 additional 
minimum bias interactions.    Trigger rates and pass fractions used in this
study were obtained by a weighted sum of the results from each of the 24
samples.  The weight for a given sample was the ISAJET cross section times the
probability of seeing the number of interactions corresponding to the generated
file.  The results for +0, +2, +4, +6 and +8 interactions (where no events were
generated) were found by interpolating the results from the generated files.
The weighted sum is performed automatically within the run II trigger
package\cite{trigsimII}.

\subsubsection{Simulation of trigger performance}
The trigger ntuple maker package is an attempt to simulate L1 and L2 
trigger conditions for run 2.
For every detector foreseen for run 2, that information is calculated from the 
raw GEANT data which will be available at the
trigger level. Therefore it allows the study of the effects of additional 
trigger conditions beyond the standard 
trigger menu which does not use the SMT information.
The  track finding algorithm used in the trigger simulation code  for the CFT at L1 
is the algorithm foreseen for the actual
trigger implementation in run 2 \cite{fb-cft-baseline,fb-cft-impact, ks-ky-cftcode}. 

The track finding algorithm for the STTpp at L2 used in the code\cite{pg-track} is such 
that it could be
implemented in the hardware that we anticipate using.
It assumes that the L1 CFT trigger provides information about track candidates 
found at L1. The roads defined by the L1 CFT 
information are then extrapolated into the SMT and the hits within the road are 
used in the track fit.
The  equation used for the extrapolation is $\varphi(r)=\kappa r + \varphi_0$, where 
$\kappa= \frac{eB}{2p_T}$ is the curvature, $\varphi_0=\varphi(r=0)$, 
and  $r$ is the distance from nominal beam line to the 
              detector element.
  Using fiber tracker (CFT) and silicon (SMT) hits, the track  
parameters are then determined by a $\chi^2$ minimization, fitting
  the hits to 
the equation $\varphi(r) = \frac{b}{r}+  \kappa r + \phi_0$, where
$b$ is the impact parameter.
% and $\phi_0 $ is $\phi $at the point
%   of closest approach of the track to the nominal beam line. 
This is a 
good approximation as long as the track length is small compared to the 
radius of the track helix. 
   The  equations can be linearized by expanding parameters 
          about a central track  
%(Note that the expansion is exact
%          since it comes from a linear least squares fit):
$$p_j = p_j^c+\sum_{i=1}^n \left(\frac{\partial p_j}
            {\partial \varphi_i}\right)_c (\varphi_i - \varphi_i^c).$$
This way, the parameter fits are reduced  to products and sums.
Derivatives can be calculated ahead of time and stored in a LUT.
In the studies done so far, the central track was   taken as the center of 
the CFT road  
projection into the SMT,
and the SMT hits considered in the fit were those closest to the center of 
the CFT road. It turned out that this choice of hits to be used in the fit 
was not always the correct one, in particular for tracks with large impact 
parameter. To remedy for the resulting inefficiency, the STT tracks with 
reduced $\chi^2 > 4$ were refitted after having 
dropped the SMT hit giving the largest contribution to the
$\chi^2$~\cite{mn-refit}.
Only those tracks 
associated with hits in at least 3 layers of the SMT and with 
reduced $\chi^2 \leq 4$ were considered ``good'' STT tracks.
All of the performance and physics benefit studies reported 
here used these track quality cuts, and the signal efficiencies quoted include 
the inefficiency due to these cuts. It should be noted that it is conceivable
that the tracking algorithms that will be used in the final implementation 
may well be more sophisticated than the one used in the simulation, thus
possibly improving the efficiency beyond the values found in our studies.

\subsubsection{Performance Studies}
\noindent\underline{\sl $p_T$ resolution}:\\
Using single track events containg a  $\mu$ or $e$ with $|\eta| \leq 1.0$ and 
$1.5 \leq p_T \leq 15$ GeV, the 
CFT momentum was computed three ways:
    \begin{itemize}
    \item ``CFT(a)'': use CFT fibers in inner and outer layer only;
(this is the information about L1 CFT track candidates broadcast to L2)
    \item ``CFT(b)'': use $p_t^{min}$ and  $p_t^{max}$, the momentum limits
               from the CFT track equation, i.e.\\
   ${1\over p_t} = {1\over 2} ({1\over p_t^{min}} + {1\over p_t^{max}}) $
 (This is the momentum as determined and used for a decision at L1.)
    \item ``STT'': use both SMT and CFT(a) information in fit 
    \end{itemize}

\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=dk_momentum.eps,height=10cm}}
\vspace{-5mm}
\caption{Transverse momentum resolution for muons in single track events; 
transverse momentum calculated three different ways:
CFT momentum (a) (squares), CFT momentum (b) (circles), 
momentum calculated using SMT and CFT information (triangles)}
\label{fig:ptresol}
\end{center}
\end{figure}

Figure \ref{fig:ptresol} shows that the resolution using STT improves compared 
to that obtained using only the CFT information.
\begin{itemize}
\item for $p_T$ = 3 GeV:~~
${\sigma (STT)\over \sigma (CFT (a))} \approx 0.5$, 
\hspace{1cm} ${\sigma (STT)\over \sigma (CFT (b))} \approx 0.8$
\item for $p_T$ = 15 GeV:~~
${\sigma (STT)\over \sigma (CFT (a))} \approx 0.3$, 
\hspace{1cm} ${\sigma (STT)\over \sigma (CFT (b))} \approx 0.6$
\end{itemize}
\noindent\underline{\sl $E_T/p_T$ for electrons at L2:}\\
Fig.~\ref{fig:Et_over_pt} shows a comparison of $E_T/p_T$ for electron tracks, 
with $p_t$ calculated in the three different ways
mentioned above.
\begin{figure}[htb]
\begin{center}
\mbox{\hspace{-2cm}\epsfig{figure=mn_e_over_p5.eps,width=8cm}
\hspace{1cm}\epsfig{figure=mn_e_over_p20.eps,width=8cm}}
\vspace{-5mm}
\caption{$E_t/p_t$ for electrons with $p_t = 5{\rm GeV}$ (left) and $p_t = 20{\rm GeV}$
(right), assuming a calorimeter energy resolution of $13\%/E$.}
\label{fig:Et_over_pt}
\end{center}
\end{figure}
It can be seen that using the STT improves the $E_T/p_T$ resolution  
by a factor of 2 for high $p_T$
electrons compared to CFT(a).
%(Note : currently not possible to use CFT $p_T$ (b) at L2 ??).
This leads to improved rejection and efficiency relative to the 
`standard isolated electron'
definition, summarized in Table~\ref{tab:Etpt}.
\begin{table}[htb]
\caption{Background rejection and efficiency of $E_T/p_T$ matching relative 
to the standard isolated electron definition}
\label{tab:Etpt}
\begin{center}
\begin{tabular}{|l||c|c||c|c|} \hline \hline
 & \multicolumn{2}{c||}{ $p_T(ele)$ = 20 GeV} &
   \multicolumn{2}{c|}{ $p_T(ele)$ = 5 GeV}   \\
 $p_T$ type & Eff. (\%)  & Bkg Rej. & Eff. (\%)  & Bkg Rej. \\
\hline \hline
CFT(a) &  {90}  & {x3} &  {  99}  & {  x2} \\
STT+CFT &  {87}  & {x5} &  {  99}  & {  x8} \\
 \hline \hline
\end{tabular}
\end{center}
\end{table}

\noindent \underline{\sl Impact Parameter Resolution:}\\
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=fs_impact.eps,width=8cm}}
\vspace{-5mm}
\caption{Impact parameter resolution vs $p_t$, for $\mu$'s in single track events}
\label{fig:impactres}
\end{center}
\end{figure}
The dependence of the impact parameter resolution on the track $p_t$ was also studied with single
         pion samples ($p_t$ from 2 to 15 GeV).
It was found that the resolution is 
$\sim$ 15 $\mu$m at $p_t$ = 15 and $\sim $35 $\mu$m at 2 GeV.
The $p_t$ dependence of the resolution was parameterized as
    $$\sigma_B^2 = \sigma_0^2 + (\frac{\alpha_{ms}}{P_T})^2, $$
with $\sigma_0 = 12.6 \ \mu$m, and $\alpha_{MS} = 49.7$ (GeV/$c)\cdot \mu$m.
%
It was found that this same parameterization of the impact parameter resolution vs $p_t$ 
is also valid for tracks from the primary vertex in \bb\ and \tt\ events.\\

\noindent The \underline{\sl Impact parameter significance,}
    $$S_B = \frac{B}{\sigma_B(p_T)}, $$
was determined in the trigger simulation algorithm for every track, using the 
 parameterization of $\sigma_B(p_T)$ mentioned above.

\begin{figure}[htb]
\begin{center}
\mbox{\hspace{-1cm}\epsfig{figure=fs_svt_tksel_bb.eps,width=8cm}
\vspace{-5mm}
\hspace{0.1cm}\epsfig{figure=fs_svt_tksel_tt.eps,width=8cm}}
\vspace{-5mm}
\caption{Impact parameter significance distribution for tracks matched to MC tracks 
coming from primary vertex, 
for $b \bar b$ events (left), and for $t \bar t $ events (right)}
\label{fig:ipsig_dist}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\mbox{\hspace{-1.0cm}\psfig{figure=fs_svt_sip_bks.eps,width=8cm}
\hspace{-0.1cm}\psfig{figure=fs_svt_sip_ttaj.eps,width=8cm}}
\vspace{-1.0cm}
\caption{Impact parameter significance distribution for matched tracks, for 
\bks \ events (left) and \ttaj \ events (right); the shaded histogram corresponds to 
tracks from the primary vertex.}
\label{fig:ipsig_dist2}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=fs_svt_siptop_bjet2.eps,height=10cm}}
\vspace{-5mm}
\caption{Impact parameter significance distribution for tracks matched to MC tracks
in \ttaj\ events; the shaded histogram corresponds to tracks coming from $b$-quark decays.}
\label{fig:ipsig_dist3}
\end{center}
\end{figure}

As can be seen in Fig.~\ref{fig:ipsig_dist}, the distribution of impact 
parameter significance for tracks coming from the primary vertex  
is satisfactorily described by a Gaussian for both \bb\ and \tt events, 
although there are non--Gaussian tails. The width is very close to 1 for \bb\
events, with a slight degradation and more pronounced tails in \tt\ events,
due to the high track multiplicity.
Furthermore, in both \bb\ and \tt\ events, the impact parameter significance 
distribution 
for tracks from the primary vertex is significantly narrower than that for all 
tracks 
(see Fig.\ref{fig:ipsig_dist2}), while it is wide for tracks from $b$ decays 
(Fig.\ref{fig:ipsig_dist3}).

\bigskip

\subsection{Physics benefit studies}
%\subsubsection{Trigger optimization}
In order to assess the physics benefit of the STT trigger for a given physics signal of interest,
we first use a set of trigger conditions
(both L1 and L2) which do not use the SMT information, i.e. are only based on the calorimeter, 
muon and CFT triggers.
We then vary  these trigger conditions to optimize signal efficiency and background rejection, 
while trying to maintain trigger rates
which are compatible with the available bandwidth. Then we check whether  additional use of SMT information at 
L2 allows better signal efficiency for the same total trigger rate,
 or allows a reduction of the trigger rate while maintaining the efficiency.

The additional handles made available due to an STTpp that can be used in a L2 trigger decision include:\\
\tab \Bul maximum $p_t$\\
\tab \Bul number of tracks with $p_t$ above some threshold\\
\tab \Bul maximum impact parameter\\
\tab \Bul maximum impact parameter significance\\
\tab \Bul number of tracks with impact parameter above some threshold\\
\tab \Bul number of tracks with impact parameter significance above some threshold.

In the following we describe briefly the physics benefits that could be derived
from using the additional handles provided by the STT. It will be shown that the
STT as proposed could bring improvements for \tt\ and single top studies, for
Higgs searches and for $B$ physics. In addition it will allow to record a sample
of \Zbb\ events which can be used for  $b$ jet energy calibration. Finally, it
offers the possibility of determining the $z$ position of the primary vertices 
at L2, which can be used at L3 to sharpen the $E_t$ thresholds when triggering 
on electrons and jets.
 

\subsection{$t \bar t$\label{ss-ttbar}}

In the Standard Model with three generations, existing experimental constraints
and the unitarity of the CKM matrix require $\mid V_{tb}|\simeq 1$, predicting
that the weak decay of the top  proceeds almost exclusively through W + b.
The $t\rightarrow Wb$ decay vertex is completely fixed by the universal
$\rm{V}-\rm{A}$ coupling to the SU(2) bosons. The decay width is given by
\cite{bigi}
\begin{eqnarray*}
\Gamma(t\rightarrow bW^+) & = &\frac{G_F m_t^3}{8\sqrt{2}\pi} |V_{tb}|^2
  [1-\frac{m_W^2}{m_t^2}]^2[1+2\frac{m_W^2}{m_t^2}]
\end{eqnarray*}

For $\mtop$ = 170 GeV/$c^2$, this partial  decay
width is $\sim$ 1.8~GeV, corresponding to a lifetime of
$\sim 0.4 \times 10^{-24}$ seconds. This rapid decay cuts off the long distance
part of the strong interaction; there is no hadronization, and all strong
interaction issues for the top quark should be well described by
perturbative QCD. 
Nature provides a tool
which does not exist for
the light quarks: a two body weak decay which precedes hadronization
and therefore carries helicity information related to the fundamental
couplings.
Because the
top is heavy, it is possible that the physics of an underlying theory at a
high mass scale may manifest itself via new non-universal top interactions
\cite{Chivukula}. 
The top quark provides the first opportunity to study the
decays of a naked quark, with experimental techniques and advantages
familiar from muon decay.

    The top quark has been discovered at the Tevatron
\cite{topfind1,topfind2} in its dominant production mode, strong
production of $t \bar{t}$ pairs \cite{twotopth}. In addition, a single
top quark can be produced via the electroweak interaction
\cite{onetopth}.
The event 
yield with and without a STT strongly depends on the bandwidth
allocation for top physics during Run II.  If the bandwidth for top physics is
significantly restricted, the gain from a STT will be larger, 
since it will help fit the top events into the available bandwidth. 

To study the benefits of the STT for the selection of \tt{} events, two 
samples of events generated by ISAJET and processed with the FEB97 version 
of UPG\_GEANT were used:
\begin{itemize}
   \item a sample of 2126 events in which both W's decay into hadrons,
         denoted as \ttaj{};
   \item a sample of 1175 events in which one of the W's is forced to
         decay into a muon+neutrino final state, and denoted as \ttmj{}.
\end{itemize}



We start out by looking for ``standard'' triggers (i.e. without STT) 
which have a high efficiency for \tt\ events.
The most promising ones with their definitions, background rates, 
as well as efficiencies for \tt\ events are shown in Table \ref{tab:ttbar}.


\begin{table}[htb]
\begin{center}
\caption{Background rates and \tt{} trigger efficiencies}
\label{tab:ttbar}
\begin{tabular}{|l||c|c|c||c|c||c|c|} \hline \hline
 & \multicolumn{3}{c||} {Background (Hz)}    
 & \multicolumn{2}{c||} {\ttmj{} (\%)}    
 & \multicolumn{2}{c|}  {\ttaj{} (\%)}    \\ \hline
 Trigger
 & L1 & L2     & L2+STT
 & L2 & L2+STT
 & L2 & L2+STT \\ \hline
 1--JMULT   &  49.1 & 2.1 &  0.8 & 27.4 & 22.3 & 69.8 & 58.5 \\
 2--MJTMET  &  55.2 & 3.3 &  1.0 & 68.5 & 55.1 & 48.7 & 38.6 \\
 3--JMAX    &  49.7 &13.0 &  6.3 & 68.3 & 55.7 & 84.2 & 69.7 \\
 4--MUJET   &  64.1 &41.6 &  8.1 & 74.3 & 58.5 & 53.2 & 42.4 \\
 5--JET2MED & 186   &85.2 & 24.2 & 88.3 & 70.9 & 98.3 & 80.2 \\ 
 6--MUMAX   &  13.0 & 6.5 &  3.4 & 51.6 & 42.0 & 14.3 & 10.9 \\ \hline
 7--$\Sigma\, (1\rightarrow 3)$ & 82.5  & 17.2 &  7.2 & 86.0 & 69.1 & 96.2 & 78.3 \\
 8--$\Sigma\, (1\rightarrow 5)$ & 239   &113  & 28.5  & 96.1 & 76.0 & 99.8 & 81.1 \\
 9--$\Sigma\, (1\rightarrow 6)$ & 248   &118  & 31.3  & 97.2 & 76.8 & 99.8 & 81.1 \\ \hline
 10--$\Sigma\, (1\rightarrow 3)+$ & & & & & & & \\
 \verb+   +$\Sigma (4\rightarrow 5)\times STT$ & 239  &113   & 38.5 & 96.1 & 92.9
                                     & 99.8 & 98.6  \\
 11--$\Sigma\, (1\rightarrow 3)+$ & & & & & & & \\
 \verb+   +$\Sigma (4\rightarrow 6)\times STT$ & 248  &  118 & 41.3 & 97.2 & 93.7
                                     & 99.8 & 98.6 \\ \hline \hline
\end{tabular}
\begin{tabular}{|l|l|}
\hline
JMULT   &  $ \ge$ 4 jets with $E_T \ge 15$ GeV \\
MJTMET  &  $ \ge 3\ E_T^{jet} \ge 10$ GeV + missing $E_T \ge 20$ GeV \\
JMAX    &  $ \ge$ 1 jet with $E_T \ge 45$ GeV \\
MUJET   &  $ \ge 1\ p_T^{\mu} \ge 6$, $E_T^1 \ge 20$ GeV and
                    $E_T^2 \ge 10$ GeV \\
JET2MED &  $ \ge$ 2 jets with $E_T \ge 20$ GeV \\
STT     &  at least 2 tracks with $\mid S \mid \ge 2.5$\\
\hline
\end{tabular}
\end{center}
\end{table}

From the observed differences in the $S_B$ distributions between primary tracks 
and tracks from $b$ decays (see Figs.~\ref{fig:ipsig_dist2} and 
\ref{fig:ipsig_dist3})
 we expect that imposing conditions on $S_B$ will  
reduce the background more than the signal. This is demonstrated in Fig.{~\ref{fig:stt_ttbar1} 
for one of the triggers (JET2MED) and for the \ttaj{} sample. Very similar 
results were obtained for the \ttmj{} events. About 8\% of the events are
lost when requiring at least 1 STT track due to the limited acceptance
of the SMT along the $z$ axis. Efficiency well above 80\% is obtained
when requiring only one track with a large impact parameter significance.


\begin{figure}[htb]
\begin{center}
\mbox{\hspace{-2cm}\epsfig{figure=stt_eff_ttaj1.eps,width=8.0cm}
\hspace{-0.2cm}\epsfig{figure=stt_bgd_jet2med.eps,width=8.0cm}}
\vspace{-5mm}
\caption{\tt{} : Signal efficiency for \ttaj{} events (left) and QCD background rate for 
the events satisfying the L2 JET2MED trigger
         together with a STT trigger (condition: 2 tracks with $|S_B| \geq S_{min}$)}
\label{fig:stt_ttbar1}
\end{center}
\end{figure}



Fig.~\ref{fig:stt_ttbar1} also shows that the trigger 
rate can be reduced by a factor of $\sim 2$ when requiring at least 1 track 
with a significance greater than 3, and  a reduction factor of 4 is reached when at 
least 2 tracks with $\mid S_B \mid > 3$ are requested. The \ttaj{} trigger
efficiencies corresponding to these two criteria are 86.9\% and 77.7\%, 
respectively.

A 30 $\mu$m uncertainty on the primary vertex position leads to a 5\%
reduction of the signal efficiency for a fixed JET2MED background rate
of 20 Hz. A systematic shift of the primary vertex position in the 
transversal plane limited to 100 $\mu$m does not introduce a signal 
efficiency loss as long as the STT criterion is adapted to maintain
the background rate at a constant level (see ref.~\cite{fs-ttbar-upd}).

The background rates and signal efficiencies obtained for various trigger 
configurations with and without STT are compiled in Table~\ref{tab:ttbar}. 
The STT condition consists in requiring at least two tracks with 
$\mid S_B \mid > 2.5$. 
The combination $\Sigma\, (1\rightarrow 3)$ based on calorimeter-only 
triggers gives efficiencies of 86\% and 96.2\% for the signal samples
with a background rate of 7.2 Hz at L2.
To reach efficiencies close to 100\% for both the \ttmj{} and \ttaj{} events,
it is necessary to use a combination of 5 or 6 of the base triggers but
the corresponding background rates increase to about 110 Hz. 
The trigger combination
\mbox{$\Sigma\, (1\rightarrow 3)+\Sigma (4\rightarrow 6)\times STT$}
allows then a reduction by a factor 3 of this background rate
without significantly affecting the signal efficiencies. This combination
allows to reach better signal efficiencies than the combination
\mbox{$\Sigma\, (1\rightarrow 3)$} while keeping background rates well below 
100 Hz.




\subsection{Single top production}

The single top process presents an interesting laboratory for study of the
top quark. During Run I this very important production mode was inaccessible 
since
the integrated luminosity was not high enough~\cite{sngtop-I}.
At $\mtop$ = 170 GeV/$c^2$ the Tevatron Run II with integrated luminosity of
2 \ifb will produce approximately 4K single top events \cite{onetopth}.



The coupling between a $W$-boson and top and bottom quarks has not yet been
studied directly. Top quark pair production is not a good process for probing
the CKM matrix element $V_{tb}$ since this production proceeds via a $gtt$
vertex.  Information about $Wtb$ coupling from top quark decay is relatively
inaccessible in \tt production because the width of the decay into $Wb$ is
proportional to the branching fraction of $t \to W b $, which is $\approx 1$  
in the standard model ($0.9991 \le \mid V_{tb} \le 0.9994$ at
90\%C.L.~\cite{PDG98}). Single top quarks are produced at hadron colliders
mainly from a $W t b $ vertex, and thereby provide a direct probe of the nature
of the $W t b $ coupling and of the CKM matrix element $V_{tb}$.

The important measurements to be done with the sample of single
top events include:
\begin{itemize}
    \item The single top cross section, which is proportional
          to $\Gamma (t \rightarrow W b)$.
    \item Branching ratio of top to left-handed vs. longitudinally
          polarized W's. They depend on the top quark a mass and the
          W boson mass.  Non-universal top couplings will, in
          many cases, appear as a departure from the expected value.
    \item Measurement of $|V_{tb}|$.
    \item A single top will be produced with almost 100\% left-handed
          polarization.  This polarization allows one to search for
          CP violation in the decay of the top quark.
\end{itemize}


The single top production events generated using COMPHEP~\cite{comphep}
were processed through the Feb97 version of UPG\_GEANT. 
A combination of electron, muon and jet triggers with high efficiency for single top events 
(see ref. \cite{th-singletop}) was
chosen from the standard trigger menu, and the effect of additonal STT conditions on signal 
efficiency and background rejection was examined. 


The main findings are summarized in Tables
\ref{tab:sgt1} and \ref{tab:sgt2}.
It is found that in both leptonic and hadronic decay modes of the $W$, 
the additional handle provided by the STT
allows a reduction of the L2 trigger rate, with little loss in signal efficiency.
Using the STT may become indispensible if it turns out that the background rates are 
underestimated by the simulation.


\begin{table}[htb]
\begin{center}
\caption{Single top production: Summary of background rejection and single 
top signal efficiency for three 
trigger classes, augmented by an STT condition at L2; the STT condition is at least one track with 
$|S_B| \ge S_{min}$.}
\label{tab:sgt1}
\begin{tabular}{|l|c|c|c|c|} 
\hline
          &\multicolumn{2}{c|}{rejection} & \multicolumn{2}{c|}{efficiency} \\
          &\multicolumn{2}{c|}{beyond L2} & \multicolumn{2}{c|}{relative to L2}\\ 
          & $S_{min} = 2$  & $S_{min} = 4$ &$S_{min} = 2$  & $S_{min} = 4$ \\ 
\hline
electrons &  1.6           &  3.0          &   87\%        &  74 \%        \\
muons     &  1.4           &  7.0          &   90\%        &  78 \%        \\
jets      &  1.4           &  3.0          &   94\%        &  80 \%        \\ 
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[htb]
\begin{center}
\caption{Background rates at L1 and at L2 for single top triggers, 
with and without additonal STT condition 
($|S_B| \ge S_{min}$) at L2; ``TT'' means ``trigger tiles''.}
\label{tab:sgt2}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Trigger} & L1    & L2     &  \multicolumn{2}{c|}{L2 rates with STT} \\
~~~L1        & ~~~L2         &rates  & rates  & $S_{min} = 2$  & $S_{min} = 4$ \\
\hline
cent. $e$ (7 GeV)    & cent. $e$(7 GeV)     & 105   & 45     & 21             &  17           \\
$e$ (7 GeV) + 2 TT (7 GeV) & $e$ (7 GeV) + 2 jets (7 GeV)     & 136   & 36     & 22             &  11           \\
single $\mu$ (7 GeV) & single $\mu$ (7 GeV)  &  13   &  7     &  5             &   1           \\
5 TT $\ge$ 3 GeV& 2 jets $\ge$ 15 GeV & 180   & 117    & 73             &  40           \\
6 TT $\ge$ 3 GeV& 2 jets $\ge$ 15 GeV &  66   &  60    & 43             &  25           \\
2 TT $\ge$ 5 GeV& 2 jets $\ge$ 15 GeV & 620   & 265    &152             &  76           \\
\hline
\end{tabular}
\end{center}
\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  *********  Higgs ****
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Higgs search}
A major question facing particle physics is understanding the mechanism for
electroweak symmetry breaking.  The current, unconfirmed, view is that the
neutral Higgs boson plays a significant role.  Given this, it is obvious that a
Higgs search should be a significant component of an experimental program if at
all possible.  The current lower bound on the Higgs mass is 89~GeV\cite{LEP-ICHEP98a}
and is expected to reach approximately 105~GeV~\cite{LEP-ICHEP98b}
 by the end of the LEP II running.
Theoretical constraints on the Higgs are weak, but favor masses below 150~GeV.

The LHC experiments are being designed specifically for Higgs searches with
mass sensitivity between 90~GeV and 1~TeV.  However, these experiments have
sensitivity to Higgs of mass less than 130~GeV only through the experimentally
challenging $H\rightarrow\gamma\gamma$ signature.  Even if an excess is
observed in this channel, interpreting these results will be very difficult.

The prospect for Higgs searches during the Tevatron run II and TeV33 eras has
been outlined elsewhere\cite{tev2000}\cite{tev33}.  For run II a factor of 10
improvement over that discussed in the studies would be needed to find a
100~GeV higgs boson.  These studies considered only the canonical discovery
mode, $\ppWHln$, but the bulk of the cross section times branching ratio for
$WH^0$ and $ZH^0$ lies in purely hadronic decay modes.  For $WH^0$ production,
67\% of the decays are to $\qqb\bbb$.  For $ZH^0$ production 70\% of the events
decay to $\qqb\bbb$.  It is clear that sensitivity in these channels would
significantly enhance the viability of Higgs searches in run II.

In the past, specific channels were allotted trigger bandwidth using a priority
determined by a D\O\ trigger board.  For a speculative but important analysis
like the higgs search, a trigger rate of 2-3\% of the total available bandwidth
has arbitrarily been chosen as the goal.

For this study~\cite{jh_higgs},
a sample of  1700  $\ppWHqq$ ~events were generated using
Pythia V5.7 with $M_H=90$~GeV.  The events were passed through the August 
'96 version of UPG\_GEANT and the standard run II trigger--simulation
package\cite{ntuple,trigsimII}.  
No additional min-bias interactions were added to this sample.
As in the \tt\ case, we first look for a set of standard triggers so as to 
maximize efficiency for the signal, the hadronic decay of the Higgs Boson. 
Some of the triggers that were considered are shown in 
Table~\ref{tab:higgscaldef}.

\begin{table}
\begin{center}
 \caption{Higgs search: Trigger definitions, background rates(r) and signal
   efficiencies(\effi)
   for the pure calorimeter triggers. The total bandwidth is expected to
   be 10 kHz for L1 and 1 kHz for L2.\label{tab:higgscaldef}}
  \begin{tabular}{|l|c|c||r|r||r|r|} \hline
         &  Calorimeter & Cal. Thresholds  & \multicolumn{2}{c||}{L1} &
                                  \multicolumn{2}{c|}{L2}  \\
  Trigger & Multiplicity & (L1,L2), GeV & r (Hz) &\effi(\%)& r(Hz) &
                                                            \effi(\%) \\
  \hline
  \JNX{3}{LOW}  &  3  & (3,5) & 2000 & 100 & 140 & 78 \\
  \JNX{3}{MED}  &  3  & (5,10)& 200  &  91 &  63 & 74 \\
  \JNX{3}{HIGH} &  3  & (7,15)& 52   &  71 &  15 & 52 \\
  \hline
  \JNX{4}{LOW}  &  4  & (3,5) & 670  &  98 &  13 & 36 \\
  \JNX{4}{MED}  &  4  & (5,10)& 58   &  78 &   6 & 34 \\
  \JNX{4}{HIGH} &  4  & (7,15)& 17   &  51 &   1 & 17 \\
  \hline
  \JNX{5}{LOW}  &  5  & (3,5) & 250  &  93 & 2.0 & 8 \\
  \JNX{5}{MED}  &  5  & (5,10)& 24   &  62 & 0.1 & 8 \\
  \JNX{5}{HIGH} &  5  & (7,15)& 5    &  32 & 0.04& 3 \\
  \hline
 \end{tabular}
\end{center}
\end{table}
\par
From this table, one sees that all of the triggers which meet the target
bandwidth of 2-3\% for L1 and L2 have signal efficiencies below 50\%.  For
searches it is critical to maintain as much efficiency as possible, so these
triggers are not particularly desirable.  The trigger \JNX{3}{MED} has
acceptable L1 rates and significantly better signal efficiency, but the L2 rate
is somewhat high.  
Additional trigger conditions using track information 
 offer a possibility to reduce the trigger rate 
while maintaining acceptable signal efficiency. This is summarized in
Table~\ref{tab:higgs_summ} and Fig.~\ref{fig:hgs_last}.
 It can be seen  that for relative trigger efficiencies of 90\%, the raw
calorimeter trigger rate can be reduced by a factor of two to three, and 
that
the use of STT information increases the rejection by 25\% over the best
non-STT trigger.  As the trigger efficiency drops, the effect of STT
information becomes much more significant.  At a relative efficiency of 80\%,
the STT triggers have rejection which is a factor of 1.5 better than the
CFT--based triggers.  For a relative efficiency of 50\%, the STT triggers have
rejection factors of up to 40, while the CFT trigger rejection is only a
factor of 10.  The best algorithm is that which requires at least N STT
tracks having $p_T\ge3'~GeV$.  
Triggers which use
    STT information have significantly better performance than those which
    use only CFT information.


%\begin{figure}
%  \insertfig{jh_evtb.eps}
%  \caption{Relative signal trigger efficiency as a function of trigger rate
%    for the trigger based on requiring either (a) the event impact parameter
%    or (b) the event impact parameter significance to be greater than some
%    threshold. Each curve corresponds to a different threshold.
%    \label{fig:hgs_evtb}}
%\end{figure}

\begin{table}[htb]
\begin{center}
  \caption{Higgs search: Representative background rates and rejections for four
  (arbitrarily chosen)  signal
     efficiencies for the different trigger types.
The lower part of the Table gives a definition of the different STT conditions 
considered. The good tracks are
   ordered separately by $b$ and $S$, and an index $i$ or $j$ denotes the track
   rank.  The ranking is by $p_T$, with the highest $p_T$ track first.
\label{tab:higgs_summ}}
  \begin{tabular}{|c||c|c|c|c|} \hline
  Trigger & \multicolumn{4}{c|}{Background Rejection} \\
   Type   & $\epsilon\approx90$\% & $\epsilon\approx80$\%
              &  $\epsilon\approx50$\%  & $\epsilon\approx10$\% \\ \hline
 N CFT tracks & & & &\\
     w/$p_T\ge 3$~GeV   &  2.5  &  4.7   &  10   &     35      \\
 N STT tracks & & & &\\
     w/$p_T\ge 3$~GeV   &  3.1  &  7.5   &  40   &   1000      \\
 N STT tracks & & & &\\
    w/$b\ge 30\ \mu$m   &  2    &  3.8   &  27   &    900      \\
 N STT tracks & & & &\\
      w/$S\ge 2$        &  2    &  5.5   &  30   &    300      \\
 Event $|b|_{25}\ge X$  &  2.2  &  3.8   &  21   &    300      \\
 Event $S_{25}\ge X$    &  2.0  &  5     &  27   &    200      \\
 N CFT w/$p_T\ge 3$ & & & &\\
 and $\overline{b_{25}}\ge X$&  3.0  &  6.4   &  26   &    500      \\
 N w/$p_T\ge 3$ & & & &\\
 and $\overline{S_{25}}\ge X$ & 2.8  &  7.1   &  33   &    650      \\
  \hline
  \end{tabular}
  \begin{tabular}{|l|l|} \hline
   Name & Description \\ \hline
   $|b|_i,\ i=1,5$ & The impact parameter  of the $i$-th track \\
   $S_i,\ i=1,5$  & The impact parameter significance of the $i$-th track \\
   $\overline{|b|}_{ij}$
      & The average impact parameter of tracks $i$ through $j$. \\
   $\overline{S}_{ij}$
      & The average impact parameter significance of tracks $i$ through $j$. \\
  \hline
  \end{tabular}
\end{center}
\end{table}

\begin{figure}[htb]
  \insertfig{jh_effrejbest.eps}
  \caption{Efficiency versus rejection for the best result from each of the
    trigger algorithms considered here. \label{fig:hgs_last}}
\end{figure}
In conclusion, we find that for a  the dominant decay
signature for associated production of a 90~GeV Higgs Boson in the modes $WH^0$ and
$ZH^0$, a calorimeter plus CFT trigger could have 68\%
overall efficiency with a L1 rate of 200 Hz and a L2 rate of 20
Hz.  The efficiency of the calorimeter portion of the trigger is 
74\%, while
the track portion has an efficiency of 90\%.  With the help of the STT,
the rejection can be improved by an additional
25\%.  For tracking--only efficiencies in the range 50\% to 85\%, the
STT trigger gives factors of 4$\times$ to 1.5$\times$ better
rejection than the best trigger without using SMT information.
From these results, it is clear that a silicon track trigger can 
significantly help in the $W/Z+H^0$ search.

\subsection{\Zbb\ }
%\subsubsection{Motivation}
Reconstructing a $Z\to b\overline b$ signal in Run II could be quite 
important.
It would provide proof that a resonance decaying to $b \overline b$ can be
reconstructed and give a measurement of the $b\overline b$ invariant mass
resolution, thereby giving credibility to searches for new particles that
decay to $b\overline b$. This is the primary decay mode of the Higgs boson in
the mass range accessible to the Tevatron \cite{tev2000}. In SUSY and
technicolor theories \cite{tc} there may also be particles that decay to $b\overline
b$. A measurement of the position of the invariant mass peak also would cross
check the jet energy scale calibration, which is the largest systematic
uncertainty in the top quark mass measurement. The decay $Z\to b\overline b$
provides the best chance to reconstruct a narrow resonance decaying to jets, 
because the $b$ quark decays allow flavor tagging for the final state jets and
therefore rejection against the overwhelming background from continuum gluon
jet production.
As we will show the large gluon jet background makes it impossible to
trigger efficiently on $Z\to b\overline b$ decays without flavor tagging. 


For this study~\cite{mn-zbb} we use  \Zbb\ events generated with ISAJET, 9979
of which have no additional minumum bias events, 995 
are overlayed with 2, and 992 with 5 additional  minimum bias events.
These events were processed by the UPG\_GEANT program (release version
FEB97).

Using the fact  that most of the $\Zbb$ events have
at least two jets with $E_T$ above 20 GeV,
 we define a set of calorimeter-based triggers
 which require the presence of at least two hadronic 
towers in the calorimeter at Level 1 and
and also two clustered jets at Level 2 above 
a certain $E_T$ threshold. The definitions of these triggers, as well as 
rates and efficiencies for \Zbb\ are
summarized  in table~\ref{tab:zbb-cal}. 
One sees that for 40\% L2  efficiency 
the L2 rate  is about  25\% of the available bandwidth,
which is unacceptably high, and  even for 20\% efficiency the L2 rate
corresponds to an uncomfortably high 8.5\% of the bandwidth.

\begin{table}[htb]
\caption{Trigger rates and efficiencies for \Zbb\ using calorimeter based
 triggers only.
The trigger
definition is given in the second column where n(L1,L2) represents
``n'' towers in Level 1 and and ``n'' jets in Level 2 with $E_T$
greater than (L1, L2) GeV respectively. The efficiencies \effi$_1$, \effi$_3$ 
and \effi$_6$ correspond to 
\Zbb\ events with 0, 2, or 5 additional minbias events overlayed,
respectively.
}
\label{tab:zbb-cal}
\begin{center}
\begin{tabular}{|l|c||r|rrr||r|rrr|} \hline
         &  Cal & \multicolumn{4}{c|}{L1} &
                                  \multicolumn{4}{c|}{L2}  \\
  Trigger & n(L1,L2)
  & r(Hz) & \effi$_1$(\%) & \effi$_3$(\%) & \effi$_6$(\%)
  & r(Hz) & \effi$_1$(\%) & \effi$_3$(\%) & \effi$_6$(\%) \\
  \hline
  \JNX{2}{MIN}  &  2(3,10) & 8028 &  82 &  85 &  85 & 1390 & 67 & 72 & 75 \\
  \JNX{2}{MED}  &  2(5,15) &  621 &  56 &  59 &  58 &  264 & 41 & 47 & 49 \\
  \JNX{2}{MAX}  &  2(7,20) &  186 &  32 &  34 &  34 &   85 & 20 & 23 & 27 \\
  \JNX{3}{MIN}  &  3(3,10) & 1502 &  68 &  72 &  74 &  132 &  9 & 11 & 16 \\
  \JNX{3}{MED}  &  3(5,15) &  167 &  31 &  34 &  34 &   25 &  4 &  5 &  6 \\
  \hline
 \end{tabular}
\end{center}
\end{table}

To control the rate, tracking information at the trigger level can be   used. 
Table~\ref{tab:zbb-calcft} and Fig.~\ref{fig:zbb-effrej} demonstrates the effect
of using CFT information in addition to the calorimeter.
From the
curve as well as the table we note that for an acceptable efficiency
of 35\% out of Level 2, the background rate still remains at about
20\% of the bandwidth. By comparing \JNX{2}{MED}\_T3 from 
table~\ref{tab:zbb-calcft}
to \JNX{2}{MED} in table~\ref{tab:zbb-cal} we note that the CFT requirement
of  two tracks with $p_T$ above 3 GeV gives a factor of 1.4
background rejection for a 15\% loss of efficiency. For a 50\%
loss  in efficiency we expect a background rejection
factor of 2.5.


\begin{table}[h]
\caption{Trigger rates and efficiencies using Calorimeter and CFT.}
\label{tab:zbb-calcft}
\begin{center}
  \begin{tabular}{|l|c||r|rrr||r|rrr|} \hline
         &  Cal/CFT & \multicolumn{4}{c|}{L1} &
                                  \multicolumn{4}{c|}{L2}  \\ 
  \hline 
  Trigger &n(L1,L2)+(n,$p_T$)  
  & r(Hz) & \effi$_1$(\%) & \effi$_3$(\%) & \effi$_6$(\%)
  & r(Hz) & \effi$_1$(\%) & \effi$_3$(\%) & \effi$_6$(\%) \\
  \hline 
  \JNX{2}{MIN}\_T2  &  2(3,10)+(2,3) & 2389 &  67&  72 &  75& 866 & 57 & 62 & 66 \\ 
  \JNX{2}{MIN}\_T3  &  2(3,10)+(3,3) & 1319 &  57&  62 &  69& 541 & 50 & 55 & 62 \\ 
  \JNX{2}{MED}\_T2  &  2(5,15)+(2,3) & 450  &  48&  51 &  52& 201 & 35 & 41 & 43 \\ 
  \hline
 \end{tabular}
\end{center}
\end{table}


It is clear from the above discussion that given the present trigger
scheme the $\Zbb$ trigger will be heavily prescaled and it is
questionable if we can record enough events 
to see a signal. 

\begin{figure}[h]
\begin{center}
\mbox{\hspace{-2cm}\epsfig{figure=zbb_bimp_refit-csq4.eps,width=8.0cm}
\hspace{-0.2cm}\epsfig{figure=zbb_simp_refit-csq4.eps,width=8.0cm}}
\vspace{-5mm}
\caption{\Zbb : Impact parameter (left) and impact parameter significance (right)
 distribution of all tracks fitted
by the Silicon Track Trigger. The shaded region corresponds
to $b$-quark descendant tracks in the event. 
\label{fig:zbb_sttimp}}
\end{center}
\end{figure}

Using the impact parameter information provided by the STT provides a much more
powerful handle for background rejection, as shown in 
 Figs.~\ref{fig:zbb_sttimp}~and~\ref{fig:zbb-effrej} and Table~\ref{tab:zbb-calstt}. 
We find, e.g.,  that by using the STT one gains
a factor of 2  in background rejection compared to the CFT based 
\JNX{2}{MED}\_T2 trigger, for the same overall efficiency of 35\%. 
For a background rate of as low as 20 Hz, we still have a final efficiency of 
around 20\%. This is a factor of 5 increase in efficiency compared to
the calorimeter based \JNX{3}{MED} trigger which gave the same low
background rate. Thus, adding the Silicon Track Trigger at Level 2
gives us the capability of triggering on \Zbb\ events at a low
rate while maintaining an acceptable efficiency.


\begin{table}[htb]
\caption{L2 trigger rates and efficiencies for \Zbb , with CFT and with  STT; 
''STT(a)'' corresponds to STT conditions adjusted so as to keep the overall 
signal efficiency 35\%,
while ''STT(b)'' corresponds to conditions which keep the background rate
at 20 Hz.}
\label{tab:zbb-calstt}
\begin{center}
  \begin{tabular}{|l|r|r|r|r|r|r|} \hline
% & \multicolumn{2}{c|}{ Cal/CFT} & \multicolumn{4}{c|}{Silicon Trigger} \\
 & \multicolumn{2}{c|}{ Cal/CFT} & \multicolumn{2}{c|}{STT(a)} 
 &\multicolumn{2}{c|}{STT(b) } \\
  Trigger  & r (Hz) &\effi(\%)& r(Hz) & \effi(\%)& r(Hz) & \effi(\%) \\
  \hline
  \JNX{2}{MIN}      &  1390  & 67 &  60  & 35   &   20  &   19 \\
  \JNX{2}{MED}      &  264   & 41 & 110  & 35   &   20  &   23 \\
  \JNX{2}{MIN}\_T2  &  866   & 57 &  62  & 35   &   20  &   20 \\
  \JNX{2}{MIN}\_T3  &  541   & 50 &  77  & 35   &   20  &   19 \\
  \JNX{2}{MED}\_T2  &  201   & 35 &      &      &       &  \\
  \JNX{3}{MED}      &   25   &  4 &      &      &       &  \\
  \hline
\end{tabular}
\end{center}
\end{table}


\begin{figure}[h]
\begin{center}
\mbox{\hspace{-2cm}\epsfig{figure=zbb_svt_bkgrej_eff_sip.eps,height=8cm}
\hspace{-0.2cm}\epsfig{figure=zbb_svt_eff_sip_multint.eps,height=8cm}}
\caption{\Zbb :  Efficiency and  background rejection
for various minimum STT track multiplicities as a function
of the impact parameter significance threshold $S_{min}$. Efficiencies and
background rejections are relative to the $\JNX{2}{MED}$ trigger term;
left: \Zbb\ without additional interactions; right: with  additonal minbias
interactions.}
\label{fig:zbb_smin_effrej}
\end{center}
\end{figure}





\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=zbb_rejeff.eps,height=9cm}}
\vspace{-5mm}
\caption{\Zbb : Efficiency vs. background rates for various 
trigger configurations using tracking information in additon to the \JNX{2}{MED}
trigger. 
The efficiencies are computed relative to the \JNX{2}{MED} term.}
\label{fig:zbb-effrej}
\end{center}
\end{figure}


In conclusion, we have shown that we can reduce the trigger rate at
Level 2 to 20 Hz at an efficiency of 20\% for $Z\to b \overline b$. Assuming this
trigger efficiency, we estimate the expected signal and background yields for a 1
fb$^{-1}$ data set:
The number of reconstructed events is given by
\begin{equation}
N = \sigma \epsilon_1 \epsilon_2\int{\cal L} dt,
\end{equation}
where $\sigma$ is the cross section for the process considered, $\epsilon_1$
the trigger efficiency, $\epsilon_2$ the reconstruction efficiency, and $\cal
L$ the luminosity. Table \ref{tab:zbb-yield} lists the relevant numbers for signal and
background. The value for $\epsilon_2$ is a guess of the additional efficiency
of an offline double $b$ tag. Even with the pessimistic assumption of 25\% 
efficiency for the signal and only a factor 10 rejection for background we get
$\hbox{signal}/\sqrt{\hbox{background}}\approx 17$. 

\begin{table}[h]
\caption{\Zbb : Signal and Background Yields for $\int{\cal L}dt=1\ \hbox{fb}^{-1}$.}
\label{tab:zbb-yield}
\begin{center}
\begin{tabular}{|l|r|r|r|r|}\hline
\hline
process & $\sigma$ & $\epsilon_1$ & $\epsilon_2$ & $N$ \\
\hline
$Z\to b\overline b$ & 1 nb & 20\% & 25\% & $5\times10^4$ \\
background          & 100 nb & 100\% & 10\% & $10^7$ \\
******              &  ?     &  ?    &      &        \\
\hline
\end{tabular}
\end{center}
\end{table}




\subsection{$B$-physics}

With the commissioning of the B Factory at SLAC and the upgraded CLEO detector,
$B$ physics will be intensively studied during the years of Run II.  Hadron 
colliders can play a role in these investigations.
 The areas of $B$ physics that
will be of intense interest are:
\begin{itemize}
      \item $B_s$ mixing
      \item CP Violation
       \begin{itemize}
            \item Measure of sin(2$\alpha$) with the 
               $B \rightarrow \pi^+ \pi^-$ decay mode
            \item Measure of sin(2$\beta$) with $B \rightarrow J/\psi K_s$
            \item Search for CP violation in $B_s \rightarrow J/\psi \phi$
       \end{itemize}
       \item Search for rare decay modes
        \begin{itemize}
                \item $B \rightarrow \ell \ell X_s$
                \item $B_s \rightarrow \ell^+ \ell^-$ 
		or $B_d \rightarrow \ell^+ \ell^-$
        \end{itemize}
        \item Measure of $\Delta \Gamma_s / \Gamma_s$ with  
                $B_s \rightarrow J/\psi \phi$
              and $B_s \rightarrow  D_s \ell \nu$.
        \item Radiative B decays ({\em e.g.} $B \rightarrow \rho \gamma$)
        \item Measures of $V_{ub}$ and $V_{cb}$ with semileptonic decay channels
\end{itemize}
The $B$ production cross section at the Tevatron is large.
The challenge at a hadron
collider is isolating the most interesting $B$ physics from the copious 
backgrounds.  Due to the event rate and bandwidth limitations in the DAQ
system, it is desirable to achieve as high as possible signal-to-background
at the trigger level.  For $B$ physics, this is best accomplished using 
silicon tracking information to exploit the long lifetime of $B$ hadrons.

These physics topics must be considered within the context of the upgraded
trigger and data aquisition system.  Since an impact parameter device would
reside at the second level of the trigger, any physics process must emerge 
through the Level 1 bandwidth ($\approx$10 kHz) before improving the 
signal-to-background 
with the silicon tracking information.  This constraint eliminates
the  possibility of triggering on all hadronic decay modes of the $B$ hadrons.  
However,many of the
processes depend on a  leptonic decay  mode such as a single muon or
dimuon.  Therefore, to understand the impact on $B$ physics, we 
concentrate on the effect of an impact  parameter trigger on the dimuon and
single muon events at Level 2. 

A very detailed study of triggering for $B$-physics~\cite{fs-bphysics}
considered various samples:
      \begin{enumerate}   
        \item     \bkspsi{}, $J/\psi \rightarrow\,\mu^+\mu^-$  ($CP$ asymmetry)
        \item     \bsds3pi{}, \dskkp{} ($B_s$ mixing)
        \item     \bsdsphi{}, \dskkp{} ($B_s$ mixing),
      \end{enumerate}

where in the $B_s$ samples the $B$ on the other side was forced to decay
           into $\mu + X$, giving at least one muon to trigger on.
 
%\vspace{-5mm}
%\begin{figure}[htb]
%{\mbox{
%    \epsfig{figure=fs_svt_tksel_bks.eps,width=8.0cm}
%    \epsfig{figure=fs_svt_sip_bks.eps,width=8.0cm,height=8cm} } }
%\end{figure}

%\begin{table}
%\label{tab:mu-eff} 
%\begin{center}
%\begin{tabular}{|l||c||c|c|c|} \hline \hline
% Trigger & rate & \psiks{} & \dsphi{} & \ds3pi{} \\ \hline 
% Single mumuonon & 38.6 Hz & 23.9\% & 11.8\% & 4.6\%  \\ \hline
% Dimuon +STT &  7.6 Hz & 14.9\% &  5.5\% & 0.3\%  \\ \hline
% Single + di  & { 45.0 Hz} & { 31.9\%} 
%                                  & { 15.6\%}  & { 4.9\%}  \\
 %\hline \hline
%\end{tabular}
%\end{center}
%\end{table}

 %   $\epsilon_{trig}(B_d^{\circ} \rightarrow J/\psi \, K_s) = 32$\% 
%$\Rightarrow$ 8.5k ---  13k fully reconstructed events
%
%\begin{table}
%\begin{center}
%\caption{$B$ Physics: Rates and efficiencies for various single muon triggers.}
%\label{tab:sgmu-eff}
%\begin{tabular}{|l||c||c|c|} \hline \hline
% Trigger Conf.   & L2 rate &  \psiks{} & \ds3pi{}  \\ \hline
% $ \ p_T > 6$, $\mid \eta \mid < 2$ & 13.5 Hz & 10.8\% & 1.4\%  \\
%{$ \ p_T > 4$, $\mid \eta \mid < 2$ } 
%          & { 38.6 Hz} & { 23.9\%} & { 4.6\%}  \\ \hline
% $ \ p_T > 4$ + CFT & 21.0 Hz       & 13.5\%  & 2.0\%  \\
% $ \ p_T > 4$ + STT & 13.2 Hz       & 17.3\%  & 2.5\%  \\
% \hline \hline
%\end{tabular}
%\end{center}
%\end{table}

   \vspace{2mm}
\begin{figure}[htb]
\begin{center}
\mbox{\hspace{-2cm}\epsfig{figure=fs_svt_effbgd_bks.eps,height=8cm}
\hspace{-0.2cm}\epsfig{figure=fs_svt_effbgd_2mu.eps,height=8cm}}
\vspace{-0.5cm}
\caption{Background rate vs signal efficiency for (a) (left): \QQ\ events using a single muon trigger with STT,
and (b) (right) for \bks\ events using a dimuon trigger, with and without STT, and with various $p_t$ thresholds
as indicated.}
\label{fig:mutrig}
\end{center}
\end{figure}

%\begin{table}
%\caption{Rates and efficiencies for various dimuon triggers.}
%\label{tab:dimu-eff}
%\begin{center}
%\begin{tabular}{|l||c||c|c|} \hline \hline
%   Trigger                    & L2 rate        & \psiks{}      & \dsphi{}\\ \hline
% (2 GeV, 4 GeV)      &  1.1 Hz        &  8.8\%        & 2.5\%   \\ 
% (2 GeV, 2 GeV)      &  272 Hz        & 20.1\%        & 8.2\%   \\
% (2,2) + STT  &  { 7.6 Hz} & {14.9\%} & {5.5\%}\\
% (2,2)+ $E_T^{jet} > 10$ GeV & 102 Hz & 0.9\%         & 0.5\%  \\ 
% \hline \hline
%\end{tabular}
%\end{center}
%\end{table}
%
\begin{table}[htb]
\begin{center}
\caption{$B$ physics:Background rate and signal efficiency
         for various muon triggers. Muon trigger code has two letters and one
         number; 1st letter: region covered (C -- $\mid \eta \mid \leq 1$,
         X $\mid \eta \mid \leq 2$), 2nd letter: tag condition (L -- loose, M --
         medium, T -- tight), number: $p_t$ threshold (1 -- 2GeV, 2 -- 4GeV,
         3 -- 7GeV). ``STT(2.5)'' means at least one track found by STT with 
	 $\mid S_B \mid \geq 2.5$.}
\label{tab:b-trigeffsumm}
\begin{tabular}{|l||c|c|c|c|} \hline \hline
 Trigger & Bgd. (Hz) & \kspsi{} (\%) & \dsphi{} (\%) & \ds3pi{} (\%)\\ 
 \hline
 dimuon      &&&&                                                  \\
% DIMU2                             &  6.2 &  7.4 &  2.1 & 0.0 \\ 
% \Bul (XM1,XM2)                         &  6.4 & 13.9 &  5.1 & 0.0 \\
 \Bul (CM1,CM2)                         &  1.3 & 10.2 &  3.3 & 0.0 \\
  \Bul (CT1,CT2)                         &  272 & 20.1 &  8.2 & 0.5 \\
 \Bul (CT1,CT1)                         &      &      &      &     \\
    \tab + $E_T^{jet} > 10$GeV&  102 & 0.9 &  0.5 & 0.0 \\ 
\Bul (CT1,CT1) 
    +STT(2.5) &  7.6 & 14.9 &  5.5 & 0.3 \\ \hline
 single muon   &&&&                                                \\   
\Bul XM2                               & 38.6 & 23.9 & 11.8 & 4.6 \\
\Bul XM3                               & 13.5 & 10.8 &  4.4 & 1.4 \\
\Bul XM2+CFT                           & 21.0 & 13.5 &  5.7 & 2.0 \\
\Bul XM2+STT(2.5) & 13.2 & 17.3 &  7.3 & 2.5 \\
\Bul XM2+STT(3.5) &  5.6 & 16.0 &  6.2 & 2.0 \\ \hline
 $\mu \mu $ + $\mu $ trigger combinations &&&&                     \\
\Bul XM2.OR.                           &      &      &      &     \\
  \tab (CT1,CT1)+STT(2.5) 
                                   & 45.0 & 31.9 & 15.6 & 4.9 \\
\Bul (XM2.OR.(CT1,CT1))                &      &      &      &     \\
 \tab +STT(2.5)    & 19.6 & 25.2 & 11.0 & 2.7 \\
\Bul (XM2.OR.(CT1,CT1))                &      &      &      &     \\
 \tab +STT(3.5)    & 11.5 & 23.2 &  9.2 & 2.2 \\
 \hline \hline
\end{tabular}
\end{center}
\end{table}



The benefits that would result from the use of a
Silicon Track Trigger at L2 are summarized in Table~\ref{tab:b-trigeffsumm} and 
demonstrated in Fig.~\ref{fig:mutrig}. The main conclusions 
are:
%*****
\begin{itemize}
\item  The STT provides a better background rejection than the CFT while
preserving a
better signal efficiency.
\item An STT would allow to reduce the XM2 (see the caption of
Table~\ref{tab:b-trigeffsumm} for definition) 
background rate by a factor 10 while preserving 70\% of the reconstructable
\QQ{} events. This result is marginally affected when a 30 $\mu$m smearing
of the primary vertex position is taken into account.
\item For the dimuon triggers, it would permit a reduction to 2
GeV/$c$ of the \pt{} threshold on both muons, enriching the sample of
signal events by 75\% (\bks{}) to 95\% (\bsdsphi{}) compared to the
(CM1,CM2) trigger and keeping the background rate as low as 10 Hz.
If the XM2 background rate turns out  to be acceptable,
the trigger combination \mbox{XM2.OR.\verb+[+(CT1,CT1)+STT\verb+]+}
allows a 33\%
increase of the trigger efficiency for \bks{} and \bsdsphi{} events
with respect to XM2 alone. This efficiency gain is accompanied
by a moderate 16\% increase of the background rate.
The best compromise between background rate and efficiency is
probably obtained with the combination
\mbox{\verb+[+XM2.OR.(CT1,CT1)\verb+]+ +STT(2.5)}, which
provides a 25.2\% efficiency for \kspsi{} events and
leads to a background rate of 19.6 Hz, representing 2\% of the L2 bandwith.
\item The STT would also be very useful to control the rate of a low threshold
dielectron trigger, allowing the collection of low \pt{}
$J/\psi \rightarrow ee$ events for calibration and physics analysis purposes.
\item An efficient single muon trigger with $p_T \geq 4$
GeV/$c$ is our only chance to search for $B_s$ mixing up to $x_s = 20$
and to determine $\sin 2\beta$ with a competitive precision;
an STT would offer the possibility to have such a trigger, with acceptable
background rates.
\end{itemize}
In summary, for all trigger configurations of interest for $B$-physics,
a Silicon Track Trigger would represent a powerful tool to reduce the
background rates. From the physics analysis point of view, it would
be more effective than a simple blind prescaling.


%\subsection{SUSY $\to b b \bar b \bar b$}

\subsection{$z_{vertex}$ determination}

Currently the Level 3 system for Run II is allocated a
25 ms time budget for reconstructing an event in a L3 node.
It is not clear whether there is enough time to unpack
the hits from the Silicon Vertex Detector and use them to implement a
fast $z$-vertex finding. If this is not feasible then
it would be advantageous to compute the $z$-vertex at Level 2 in hardware and 
pass it along to
Level 3 for use in further computations of global quantities, e.g. the $E_T$ of
jets and electrons. 
Knowledge of the $z$-vertex position at Level 3 is extremely 
beneficial. During Run I, the Level 2 electron algorithm used
the ``slow $z$'' position from Level 0 in order to get a better
estimate of the electron $E_T$. 


Using the hits in the
90$^\circ$ stereo view of silicon ladders in layers 1 and 3 of the inner four
barrel segments of the SMT offers the possibility to determine the $z$ position of the
primary event vertex  Level 2. 
An algorithm was found~\cite{mn-zvertex} which could be implemented in hardware at L2, and provide
the $Z_V$ information at L2 time, to be used either in the global L2 decision
or at L3.


The resolution of the $z$-vertex measurement is
found to be 550 $\mu$m for events with one high-$p_T$ interaction.
For events with 2 additional min-bias events overlayed, the
vertex resolution is determined to be 610 $\mu$m.
For a sample of  $Z\to b \bar b$ events without additional interactions and 
 in which the
event vertex is within $\pm$ 25 cm,  the
efficiency for reconstructing a vertex in the $Z\to b \bar b$ events
is found to be 98\%. Since  there is only one vertex in the event,
we always identify the correct vertex as the high-$p_T$ vertex.
If we superpose 2 min-bias interactions on these $Z\to b \bar b$ events, the efficiency
of reconstructing at least one vertex in these events is 99\%.
The efficiency for identifying the correct vertex as the high-$p_T$ vertex
is 78\%. In these events there maybe one or more vertices
within the acceptance region and we find that the efficiency to
reconstruct a vertex in this environment is about 85\%.


Due to the limited acceptance of the silicon detector elements with
90$^\circ$ strips, (only four inner 
barrels out of six), the reconstruction of $z$-vertices with 
this method is limited to $\pm$30 cm around the center of the
detector. The rms width of the luminous region during Run II 
is expected to be 25 cm. About 30\% of the events have the 
high-$p_T$ vertex outside this region.
For these either no vertex is found or a 
min-bias vertex inside this acceptance region 
(reconstructed with 97\% efficiency) is incorrectly 
identified  as the high-$p_T$ vertex.


The influence of the knowledge of the vertex position on the
$E_T$ trigger turn-on is shown in Fig.~\ref{fig:turnon_ele}.
Even if the correct vertex position is only found for $|z|<30$cm (dash-dotted
curve in the figure),
 there still is a significant  improvement in the $E_T$ threshold due to 
 the knowledge of
the position of the $z$-vertex.
This improvement implies that we can trigger more efficiently
with lower rates on single electrons at Level  3.

\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=ztrig_pt10.eps,width=2.5in}
            \epsfig{figure=ztrig_pt20.eps,width=2.5in } }
\vspace{-0.5cm}
\caption {Turn-on curves for electrons with $E_T$ = 10 GeV (left plot) and
20 GeV (right plot). The solid curve is with $z$-vertex assumed to be = 0, the
dashed curve is with a reconstructed $z$-vertex with 100\% reconstruction
efficiency;  the dotted curve takes into account the $z$-vertex
reconstruction efficiency (78\%), and the dash-dotted curve in addition accounts
for the  limited detector acceptance. The electron energy resolution is assumed
to be 13\%/$\sqrt{E}$, and the $z_V$ resolution is taken to be 600$ \mu $m. }

\label{fig:turnon_ele}
\end{center}
\end{figure}


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%%%%%%%%  bibliography
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\newpage
\input{physbiblio}


\end{document}
