\vspace{0.1in}
\small
\begin{center}
{\it J. Conway}
\end{center}
\normalsize\nopagebreak

As the results of Section II show, there is no single search channel
for the Higgs which one might call ``golden''; to maximize the
sensitivity of the Higgs search it is necessary to combine the results
of all the channels.  This section presents the results of combining
all Standard Model Higgs search channels, from both experiments, in
terms of the integrated luminosity needed to exclude the Higgs at 95\%
CL (assuming it is not there) or discover it at the $3\sigma$ or
$5\sigma$ level if it is.

\vspace{0.2in}
{\bf Statistical Method} \\ \nopagebreak

The statistical method employed here for combining channels is
discussed in the Appendix.  Briefly, the result of each search channel
is treated as a counting experiment, and for a given outcome there is
some Poisson probability.  For all channels in both experiments, these
probabilities are multiplied together to form a joint likelihood.  This
likelihood can be expressed as a function of the Higgs signal cross
section, and can be used to set 95\% CL limits or discovery 
significances.  To take into account all possible experimental outcomes,
the integrated luminosity threshold quoted below represent those 
values for which the desired statistical threshold is met in 50\%
of all possible outcomes.

\vspace{0.2in}
{\bf Overview of results} \\ \nopagebreak


The results of all the channels studied are summarized in Tables
\ref{t:low} and \ref{t:high}.  The tables show
the expected signal $S$, the expected background $B$, and the
sensitivity $S/\sqrt{B}$ in each channel as a function of the assumed
Higgs mass.  In all the low-mass channels, we have taken the 
numbers from assuming a 10\% resolution in $m_{\bb}$.

The tables indicate that of the low mass channels, the $\ell\nu\bb$
and $\nn\bb$ have the most sensitivity.  Also, while the dilepton mode
adds significantly to the sensitivity, the all-hadronic channel
brings little information to the final combination.

In comparing the different analyses, it is clear that the neural
network technique results in significantly enhanced sensitivity in the
three channels where it has been studied.  Note that the NN- and
SHW-based channel analyses do not take into account trigger
inefficiency for events which otherwise pass the selection; this
should be no problem in the $\ell\nu\bb$ and $\lpm\bb$ cases but may
be a slight overestimate at low masses in the $\nn\bb$ case.

For the high-mass channels, the $\lpm\nn$ channel has the most sensitivity,
whereas the $\lsdilep$ channel has nearly as good sensitivity over a 
broader mass range.  

%The $\trilep$ channel has competitive sensitivity,
%but with its very low expected signal it contributes significantly only

%at the highest integrated luminosities.\footnote{This underscores
%dramatically the need to use true Poisson statistics rather than a
%Gaussian approximation!}

 \begin{table}
   \begin{center}
   \caption{Summary of low-mass Standard Model Higgs search
                 channel sensitivities used in the combined integrated
                 luminosity threshold calculations.  The values of $S$
                 and $B$ are expressed as the number of
                 events expected per detector in 1 fb$^{-1}$.
                 Here we assume an improved Run 2
                 $m_{\bb}$ resolution of 10\%.  ``SHW'' indicates the
                 analyses based on the SHW simulation, ``NN''
                 indicates the SHW neural-network-based analyses, and
                 ``CDF'' indicates the analyses based on
                 extrapolations from the CDF Run 1 conditions to Run 2
                 detector geometry and efficiencies.  Note that in the
                 $\nn\bb$ analyses the background from $\bb$ dijet
                 production is included by doubling the non-dijet
                 background, as discussed in the text.}

    \label{t:low}
%%   \begin{tabular}{|c|c|ccccc|} \cline{3-7}
%%    \multicolumn{2}{l}{ }           & \multicolumn{5}{|c|}{Higgs mass (GeV/$c^2$)}  \\  \cline{1-2}
%\setlength{\tabcolsep}{1pc}
     \begin{tabular}{llccccc}
                      &              & \multicolumn{5}{c}{Higgs Mass (GeV/$c^2$)} \\ \cline{3-7}
\noalign{\vskip3pt}
\multicolumn{1}{c}{Channel} &\multicolumn{1}{c}{Rate}& 
                                        90   &  100  &  110  &  120  &  130 \\ \tableline \tableline
%--------------------------------------------------------------------------------------
                      & $S$          &  8.4  &  6.6  &  5.0  &  3.7  &  2.2  \\   
 $\ell\nu\bb$ (CDF)   & $B$          &  48   &  52   &  48   &  49   &  42   \\  
                      & $S/\sqrt{B}$ &  1.2  &  0.9  &  0.7  &  0.5  &  0.3  \\  \tableline
%--------------------------------------------------------------------------------------
                      & $S$          & 10    &  8    &  5    &  4    &  3    \\  
 $\ell\nu\bb$ (SHW)   & $B$          & 75    & 68    & 57    & 58    & 52    \\ 
                      & $S/\sqrt{B}$ &  1.1  &  1.0  &  0.7  &  0.5  &  0.4  \\  \tableline 
%--------------------------------------------------------------------------------------
                      & $S$          &  8.7  &  9.0  &  4.8  &  4.4  &  3.7  \\  
 $\ell\nu\bb$ (NN)    & $B$          &  28   &  39   &  19   &  26   &  46   \\ 
                      & $S/\sqrt{B}$ &  1.6  &  1.4  &  1.1  &  0.9  &  0.5  \\  \hline  \hline
%--------------------------------------------------------------------------------------
%--------------------------------------------------------------------------------------
                      & $S$          &  2.5  &  2.2  &  1.9  &  1.2  &  0.6  \\  
 $\nn\bb$ (CDF)       & $B$          & 20.0  & 18.6  & 16.0  & 13.0  &  9.6  \\ 
                      & $S/\sqrt{B}$ &  0.6  &  0.5  &  0.5  &  0.3  &  0.2  \\  \tableline 
%--------------------------------------------------------------------------------------
                      & $S$          &  8.9  &  6.7  &  4.6  &  3.2  &  2.1  \\   
 $\nn\bb$ (SHW)       & $B$          &  77   &  69   &  56   &  39   &  30   \\  
                      & $S/\sqrt{B}$ &  1.0  &  0.8  &  0.6  &  0.5  &  0.4  \\  \tableline 
%--------------------------------------------------------------------------------------
                      & $S$          &  12   &   8   &  6.3  &  4.7  &  3.9  \\   
 $\nn\bb$ (NN)        & $B$          & 123   &  70   &  55   &  45   &  47   \\  
                      & $S/\sqrt{B}$ &  1.1  &  1.0  &  0.8  &  0.7  &  0.6  \\  \hline \hline
%--------------------------------------------------------------------------------------
%--------------------------------------------------------------------------------------
                      & $S$          &  1.0  &  0.9  &  0.8  &  0.5  &  0.3  \\    
 $\lpm\bb$ (CDF)      & $B$          &  3.6  &  3.1  &  2.5  &  1.8  &  1.1  \\   
                      & $S/\sqrt{B}$ &  0.5  &  0.5  &  0.5  &  0.4  &  0.3  \\  \tableline 
%--------------------------------------------------------------------------------------
                      & $S$          &  1.5  &  1.2  &  0.9  &  0.6  &  0.4  \\    
 $\lpm\bb$ (SHW)      & $B$          &  4.9  &  4.3  &  3.2  &  2.3  &  1.9  \\   
                      & $S/\sqrt{B}$ &  0.7  &  0.6  &  0.5  &  0.4  &  0.3  \\  \tableline
%--------------------------------------------------------------------------------------
                      & $S$          &  1.2  &  0.9  &  0.8  &  0.8  &  0.6  \\    
 $\lpm\bb$ (NN)       & $B$          &  2.9  &  1.9  &  2.3  &  2.8  &  1.9  \\   
                      & $S/\sqrt{B}$ &  0.7  &  0.7  &  0.5  &  0.5  &  0.4  \\  \hline \hline
%--------------------------------------------------------------------------------------
%--------------------------------------------------------------------------------------
                      & $S$          &  8.1  &  5.6  &  3.5  &  2.5  &  1.3  \\     
 $\qq\bb$ (SHW)       & $B$          & 6800  & 3600  & 2800  & 2300  & 2000  \\    
                      & $S/\sqrt{B}$ & 0.10  & 0.09  & 0.07  & 0.05  & 0.03  \\  \tableline 
%--------------------------------------------------------------------------------------
    \end{tabular} 
      \end{center} 
  \end{table}

 \begin{table}
   \begin{center}
   \caption{Summary of high-mass Standard Model Higgs search channel
            sensitivities; all results are based on SHW studies.
            The values of $S$ and $B$ are expressed as the number of 
            events expected per detector in 1 fb$^{-1}$.}
   \label{t:high}
%  \begin{tabular}{|c|c|ccccccc|} \cline{3-9}
%     \multicolumn{2}{l}{ }           & \multicolumn{7}{|c|}{Higgs mass (GeV/$c^2$)}  \\
\setlength{\tabcolsep}{9pt}
  \begin{tabular}{llccccccc}
                      &              & \multicolumn{7}{c}{Higgs Mass (GeV/$c^2$)}  \\ \cline{3-9}
\noalign{\vskip3pt}
\multicolumn{1}{c}{Channel} & \multicolumn{1}{c}{Rate}  &
                                         120  &  130  &  140  &  150  &  160  &  170  &  180  \\  \tableline
%------------------------------------------------------------------------------------------------------
%                      & $S$          & 0.011 & 0.025 & 0.039 & 0.050 & 0.057 & 0.033 & 0.033 \\      
% $\trilep$            & $B$          & 0.025 & 0.025 & 0.025 & 0.025 & 0.025 & 0.025 & 0.025 \\    
%                      & $S/\sqrt{B}$ & 0.07  &  0.16 &  0.25 &  0.32 &  0.36 &  0.21 &  0.21 \\  \tableline  
%------------------------------------------------------------------------------------------------------
                      & $S$          &   -   &   -   &  2.6  &  2.8  &  1.5  &  1.1  &  1.0  \\      
 $\lpm\nn$            & $B$          &   -   &   -   &  44   &  30   &  4.4  &  2.4  &  3.8  \\    
                      & $S/\sqrt{B}$ &   -   &   -   &  0.39 &  0.51 &  0.71 &  0.71 &  0.51 \\  \tableline  
%------------------------------------------------------------------------------------------------------
                      & $S$          &  0.08 &  0.15 &  0.29 &  0.36 &  0.41 &  0.38 &  0.26 \\      
 $\lsdilep$           & $B$          &  0.58 &  0.58 &  0.58 &  0.58 &  0.58 &  0.58 &  0.58 \\    
                      & $S/\sqrt{B}$ &  0.11 &  0.20 &  0.38 &  0.47 &  0.54 &  0.50 &  0.34 \\
%------------------------------------------------------------------------------------------------------
   \end{tabular}
   \end{center}
 \end{table}


\vspace{0.2in}
{\bf Combined channel integrated luminosity thresholds} \\ \nopagebreak

We perform determinations of the integrated luminosity thresholds
combining all low-mass and high-mass channels.  In the combination we
assume that both experiments' data is used by doubling each channel: we
generate separate pseudoexperimental outcomes for each channel in each
experiment, and combine all the results together in the final
likelihood.

To take into account reasonable systematic errors, we incorporate into
the likelihood a relative uncertainty on the background for each
channel which is the smaller of 10\% of the expected background or
$1/\sqrt{LB}$, where $L$ is the integrated luminosity and $B$ is the
expected number of background events in 1 fb$^{-1}$.  Such an
assumption is typical of the level of uncertainty experienced in new
particle searches at the Tevatron.  Note that if one does not let
the systematic error decrease with integrated luminosity, numerical
instability can result.  More importantly, in the real experiments as
the integrated luminosity increases the experimenters will have better
control of the systematic errors, and will in all likelihood harden
the selection criteria to improve the sensitivity while maintaining
tolerable systematic uncertainties.  Without the inclusion of these
systematic errors, the integrated luminosity thresholds are
approximately 30-50\% smaller.

\vspace{0.2in}
{\bf Reliability of the results} \\ \nopagebreak

The nature of a study like this is such that the final results (here
the integrated luminosity thresholds) depend on several unknown
performance factors.  The main factors affecting the result of this
study, in particular the results in the low-mass channels, are
\begin{itemize}
  \item the $b$-tagging efficiency,
  \item the $\bb$ mass resolution, and
  \item the backgrounds in the various channels.
\end{itemize}

All the low mass channels require two $b$-tagged jets, which makes
the result very sensitive to this efficiency.  The signal efficiency 
increases as the square of the $b$-tag efficiency, whereas only the
$W\bb$, $Z\bb$, and QCD $\bb$ dijet backgrounds follow the same 
dependence.  In the gaussian limit, the integrated luminosity
thresholds scale with $B/S^2$, so even if the backgrounds are 
dominated by real $\bb$, increasing the $b$ tag efficiency by 10\%
for example while maintaining the mistag rate would result in a 
20\% reduction in the required integrated luminosity threshold.

Both CDF and D\O\ will have excellent vertex tracking, and new
algorithms and solid understanding of vertex tracking systematics
could result in efficiencies somewhat higher than assumed in this
report.  However, physics effects such as multiple interactions,
and/or unforeseen detector performance issues could adversely affect the
$b$ tag efficiency and purity.  The assumptions made here are
reasonably conservative.

The mass resolution is directly related to the background - indeed the
main effect of improving the resolution is of course to reduce the
number of background events in the window around the Higgs mass.  As
noted above, since the integrated luminosity thresholds scale with
$B/S^2$, increasing the background (or worsening the resolution) by
some relative amount has a the same relative effect on the thresholds.  

As the careful reader will note, in the $\nn\bb$ channel analyses no
estimates were made of the generic $\bb$ dijet background.  This
process has a very large cross section but tiny acceptance, and is
thus not modeled reliably.  In the CDF Run 1 analysis, this
background was about half of the total, and estimated from the data.
With better understanding, this fraction could potentially be reduced.
However, we conservatively assume in the final combination that this
background is equal to the rest of the background, that is, half of
the total.  This is reflected in Table~\ref{t:low} by doubling the 
non-dijet background in the $\nn\bb$ channel analyses.

Perhaps of more concern is the assumption regarding the mass
resolution ultimately attainable.  In the low mass region this has
been assumed to be 10\% (of the Higgs mass).  Even with a perfect
detector the physics of the jet fragmentation and gluon radiation,
along with the missing neutrinos from the $b$ quark decay limit the
$\bb$ mass resolution attainable to a lineshape with a width of about
8\% with a tail on the low side.  If detector-related effects can be
understood and controlled well, and if all the available detector
information is brought to bear, the estimate of 10\% may be reasonable.
Even if the mass resolution is closer to 12\%, though, the resulting
integrated luminosity thresholds will only be 20\% larger, well within
the other uncertainties inherent in this work.  

Furthermore, at this level the relative strengths and weaknesses of
the two detectors will ultimately determine the sensitivity to the
Higgs.  With more elaborate tracking, CDF will most likely excel in
terms of $b$ tagging efficiency, whereas with its more hermetic
calorimeter the D\O\ experiment may attain the best mass resolution.

Given the above discussion, the basic conclusions arrived at below are
not unreasonably aggressive.  Breakthroughs in technique are always
possible, and have indeed been the norm in the past.  For example both
the Higgs search in LEP1 and the top quark search in Run 1 at the
Tevatron exceeded the expectations of studies prior to machine
turn-on.  The studies presented here should be taken as cautiously
optimistic: Using full mass spectrum fits, applying neural network
techniques, improving the trigger efficiencies, adding other search
mode, and improving the mass resolution and tagging efficiency beyond
that projected here may all serve to dramatically improve the
discovery potential for the Higgs at the Tevatron.


\begin{figure}[th]
  \begin{center}
    \parbox{6.0in}{\epsfxsize=\hsize\epsffile[0 200 500 540]
           {combine/final.ps}}
  \end{center}
  \caption{The integrated luminosity required per experiment, to
            either exclude a SM Higgs boson at 95\% CL or discover it at the
            $3\sigma$ or $5\sigma$ level, as a function of the Higgs mass.  
            These results are based on the combined
            statistical power of both experiments.  The curves shown  
            are obtained by combining the $\ell\nu b\bar b$,
            $\nu\bar\nu b\bar b$ and $\ell^+\ell^-b\bar b$ channels
            using the neural network selection in the low-mass Higgs region
            ($90~{\rm GeV}~\lsim\mhsm\lsim 130$~GeV), and the
            $\lsdilep$ and $\lpm\nn$ channels in the high-mass Higgs region
            ($130~{\rm GeV}~\lsim\mhsm\lsim 200$~GeV).  The lower edge of 
            the bands is the calculated threshold; the bands extend upward 
            from these nominal thresholds by 30\% as an indication of the
            uncertainties in $b$-tagging efficiency, background rate,
            mass resolution, and other effects.}
  \label{f:combined-final}
\end{figure}

\vspace{0.2in}
{\bf Combined channel integrated luminosity thresholds} \\ \nopagebreak


Fig.~\ref{f:combined-final} shows the integrated luminosity required
to either exclude the SM Higgs boson at 95\% CL or discover it at the
$3\sigma$ or $5\sigma$ level of significance, as a function of Higgs
mass, for the SHW analyses with the neural net selection.  The
integrated luminosity displayed in the plot is the delivered
integrated luminosity {\it per} experiment; whereas the shaded bands shown are
the results obtained by combining the statistical power of {\it both}
experiments.  The required integrated luminosity thresholds for a
single experiment are very close to a factor of two higher than those
for the two combined experiments.  
The bands extend from the calculated threshold
on the low side upward in required integrated luminosity by 30\% to
the high side, as an indication of the range of uncertainty in the
various factors discussed above.
As the plots show, the required integrated luminosity increases
rapidly with Higgs mass to 140 GeV, beyond which the high-mass
channels play the dominant role.  At the upgraded Tevatron with
2~fb$^{-1}$ per detector, the 95\% CL limits will barely extend the
expected LEP2 limits, but with 10 fb$^{-1}$, the SM Higgs boson can be
excluded up to 180 GeV in the absence of a Higgs signal.
%if it does not exist in that mass range.

If a SM Higgs boson exists with mass less than 180 GeV, the combined
sensitivity of CDF and D\O\ will yield evidence for the Higgs boson at
the $3\sigma$ level with 20 fb$^{-1}$.  (The Higgs mass region around
140 GeV may require slightly more luminosity depending on the
magnitude of systematic errors due to uncertainties in $b$-tagging
efficiency, background rate, the $b\bar b$ mass resolution, {\it
etc.})  With 30~fb$^{-1}$ integrated luminosity delivered per
detector, a $5\sigma$ Higgs boson discovery may be possible for Higgs
masses up to about 130 GeV, a significant extension of the LEP2
Standard Model Higgs search.  The latter figure of merit is
particularly significant when applied to the search for the lightest
Higgs bosons of the minimal supersymmetric model.  We address this
case in detail in the following section.


