
CDF has recently published~\cite{juano} a search for Higgs bosons
produced in association with vector bosons in $91 \pm 7$ pb$^{-1}$ of
Run 1 data.  Observations are consistent with background
expectation. 95\% confidence level upper limits are set as a function
of the Higgs mass on $\sigma(p\bar{p} \rightarrow HV) \cdot \beta$ ,
with $\beta$ the branching ratio of the Higgs decays to
$b\bar{b}$. The sensitivity of the search is limited by statistics to
a cross section approximately two orders of magnitude larger than the
predicted cross section for standard model Higgs production. In this
paper we extrapolate the results from~\cite{juano} to the next
Tevatron Run 2 were we hope for an approximately twenty-fold increase
in the total integrated luminosity as well as a significant
improvement in the total acceptances.

\vspace{0.2in}
{\bf Selection Requirements} \\ \nopagebreak

The results from~\cite{juano} 
are based on a data sample recorded with a trigger
which requires four or more clusters of contiguous calorimeter towers,
each with transverse energy $E_T \ge 15$ GeV, and a total
transverse energy $\sum E_T \ge 125$ GeV. Further steps in the data reduction 
are: 1) four or more reconstructed jets with uncorrected
$E_T > 15$ GeV and $|\eta| < 2.1$. Jets are defined as localized 
energy depositions in the calorimeters and are reconstructed 
using an iterative clustering algorithm with a fixed cone of radius
$\Delta R = \sqrt{\Delta\eta^2 + \Delta\phi^2} = 0.4$ in $\eta - \phi$ space.
After this initial selection the sample
contains 207,604 events; 2) at least two among the four highest-$E_T$ 
jets in the event are identified (tagged) as $b$ quark 
candidates~\cite{secvtx}. There are 764 events with four or more jets and two 
or more $b$-tags; 3) a cut on the transverse momentum of the $b\bar{b}$ 
system, $p_T({b\bar{b}})\ge 50$ GeV/c$^2$ strongly discriminate against QCD 
direct production and flavor excitation of heavy quarks. A total of 589 
events remain after this cut.

\vspace{0.2in}
{\bf Backgrounds and efficiencies} \\ \nopagebreak

The main source of background events is QCD heavy flavor production. The 
heavy flavor content of QCD hard processes are conventionally classified in 
three groups: direct production, gluon splitting, and flavor excitation.
The relative contributions have been modelled with the PYTHIA Monte Carlo 
program~\cite{PYTHIA}. 
No attempt to estimate the normalization of this source of background directly 
from Monte Carlo is made due to the large number of uncertainties in the 
predictions. 

Other backgrounds are $t\bar{t}$ production, $Z$ + jets 
events with $Z\rightarrow b\bar{b}/c\bar{c}$
and fake double-tags. The first two are estimated from Monte Carlo and the 
last one from data.
 After trigger, kinematic and $b$-tag requirements 
we expect $26 \pm 7$ and $17 \pm 4$ $t\bar{t}$ and $Z$ + jets background 
events, respectively.

The combined trigger and acceptance efficiency is determined using PYTHIA 
followed by detector and trigger simulations. It depends on the Higgs mass 
and increases from $8\pm 1\%$ for $m_H = 70$ GeV/$c^2$ to $31 \pm 3\%$ 
for $m_H = 140$ GeV/$c^2$.
The SVX double $b$-tagging efficiency is calculated with a combination 
of data and Monte Carlo samples and is $14\pm 3\%$ with 
a small dependence on the Higgs mass.
The total efficiency increases linearly from 
$0.6\pm 0.1\%$ to $2.2\pm 0.6\%$ for Higgs masses ranging from 70 GeV/$c^2$ 
to 140 GeV/$c^2$.

\vspace{0.2in}
{\bf CDF Run 1 Results} \\ \nopagebreak

The shape of the observed $b$-tagged dijet invariant mass distribution is fit,
using a binned maximum-likelihood method, to a combination of  
signal, fake double-tag events, and QCD, $t\bar{t}$ and 
$Z$ + jets backgrounds. 
The QCD and signal normalizations are left free in the fit while
the normalizations of the $t\bar{t}$, $Z$ + jets and fakes are 
constrained by Gaussian functions to their expected values and 
uncertainties. 
The fit yields $\sigma_{VH^0} \cdot \beta = 44 \pm 42$ pb 
for $m_H = 70$ GeV/$c^2$, statistically compatible with zero signal.
For larger masses, zero signal contribution is preferred.
Table~\ref{tab1} shows the result of the fits  
as a function of the Higgs mass and Figure~\ref{fig1} shows the 
$b$-tagged dijet invariant mass distribution for the data 
compared to the results of the fit for $m_H \ge 80$ GeV/$c^2$.

\begin{table*}[t!]
\caption{Summary of the CDF Run 1 hadronic analysis fit results,
standard model predictions for $\beta\sigma$, and 95\% C.L. upper limits.}
\label{tab1}
\begin{tabular}{lcccccccc} 
 & \multicolumn{8}{c}{$m_H$ (GeV/$c^2$)} \\ [0.1cm] \cline{2-9}
\multicolumn{9}{c}{} \\ [-0.2cm]
 & 70 & 80 & 90 & 100 & 110 & 120 & 130 & 140 \\ [0.1cm] \tableline
\multicolumn{9}{c}{} \\ [-0.2cm]
Fit $\beta\sigma$ (pb) & $44 \pm 42$ & $0^{+19}_{-0}$ & $0.0^{+9.7}_{-0.0}$
&$0.0^{+7.6}_{-0.0}$ &$0.0^{+6.3}_{-0.0}$ & $0.0^{+5.9}_{-0.0}$ 
& $0.0^{+5.5}_{-0.0}$ & $0.0^{+5.1}_{-0.0}$ \\ [0.15cm]
SM $\beta\sigma$ (pb) & $1.13$ & $0.76$ & $0.55$ & $0.41$ & $0.30$ & 
$0.20$ & $0.12$ & $0.06$ \\ [0.1cm]
Limit $\beta\sigma$ (pb) & $117.3$& $53.2$& $28.9$& $22.8$& $18.7$
& $17.6$& $16.7$& $15.3$ \\ [0.15cm]
\end{tabular}
%\caption{Summary of the CDF Run 1 hadronic analysis fit results,
%standard model predictions for $\beta\sigma$, and 95\% C.L. upper limits.}
%\label{tab1}
\end{table*} 

\begin{figure}[t]
\centerline{\epsfxsize3.2in\epsffile{whjj/figure1.eps}}
\caption[0]{Invariant mass distribution, $M_{b\bar{b}}$, for 90.6 pb$^{-1}$
of CDF data (points) compared to the fit prediction~\cite{juano}. The solid line is the 
sum of the QCD, fakes, $t\bar{t}$, and $Z$ + jets components.}
\label{fig1}
\end{figure}

\begin{figure}[t]
\hspace*{1.26cm}
\begin{minipage}[t]{2.5in}
\hspace*{0.26cm}
\centerline{\epsfxsize3.6in\epsffile{whjj/figure2.eps}}
\caption[0]{The Run 1 CDF 95\% C.L. upper limits on $\sigma(p\bar{p} 
\rightarrow VH^0) \cdot \beta$ where $\beta = {\cal B}(H^0 \rightarrow 
b\bar{b})$. Also shown is the extrapolation for three different enhanced
luminosity scenarios and Run 2 acceptances.}
\label{fig2}
\end{minipage}
\hspace*{1.5cm}
\begin{minipage}[t]{2.5in}
\centerline{\epsfxsize3.6in\epsffile{whjj/figure3.eps}}
\caption{Luminosity as a function of the Higgs mass for a 95\% CL limit reach
in Run 2.}
\label{fig3}
\end{minipage}
\end{figure}

\vspace{0.2in}
{\bf CDF Run 2 Extrapolation} \\ \nopagebreak

Figure~\ref{fig2} shows the CDF Run 1 results for the 95\% CL upper
limits on $\sigma\beta$ as a function of the Higgs mass. In general,
the obtained 95\% CL statistical limits agree very well with the
expected ones, except for the region of very low masses were the
magnitude of the limits is strongly influenced by the fluctuation
structure of the~data.

A simple extrapolation to account for the expected increase in
luminosity and in the total acceptances lead to improved limits by, at
least, a factor $1/\epsilon\sqrt{{\cal L}}$.\footnote{No account is
taken here of the increase in center of mass energy from 1.8 TeV to
2.0 TeV.}  Here $\epsilon$ and ${\cal L}$ represent the increased
acceptances and luminosity factors with respect to Run 1. We consider
three different luminosity scenarios for Run 2 and beyond: 1, 20 and
100 fb$^{-1}$. In addition we consider the increased geometrical
acceptance of the new Run 2 CDF silicon vertex detector SVX II, which
will extend the $b$-tagging capabilities up to the range $|\eta|<2$. A
Monte Carlo study for the process $p\bar{p} \rightarrow V +
H^0\rightarrow jjb\bar{b}$ indicates a total signal improvement of
$\sim 80\%$ in double $b$-tag efficiencies (assuming Run 1 central
$b$-tag intrinsic efficiencies per jet). We finally consider an
optimized multijet trigger for associated Higgs production in Run
2. The requirement of displaced tracks with large impact parameter at
the trigger level makes it possible to relax the Run 1 multijet
criteria to only 3 jets with uneven $E_T$ cuts and a lower $\sum E_T$
threshold. A preliminary study~\cite{petar} shows that these cuts
result on a few-fold increase in signal efficiencies without
compromise the total trigger rates.

Figure~\ref{fig2} shows the CDF Run 1 95\% CL extrapolated limits for
the different luminosity scenarios combined with the roughly four times
better acceptance for Run 2. Figure~\ref{fig3} shows the 95\% CL
upper limit reach as a function of the required luminosity and the
Higgs mass for three different improved acceptances.


