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%% "low subscript" %%
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%%
\vspace{0.1in}
\small
\begin{center}
{\it M. Carena, H.E. Haber, S. Mrenna and C.E.M. Wagner}
\end{center}
\normalsize\nopagebreak


\def\tightenlines{\def\baselinestretch{2}\small\normalsize}
%\tightenlines

\subsubsection{Reinterpretation of the Standard Model Analysis in
Extended Higgs Sector Theories} 

The Standard Model Higgs boson search simulations presented in this
report can be interpreted more generally in the framework of an
extended Higgs sector.  Generically, extended Higgs sectors contain
additional scalar states beyond $\hsm$ with different couplings from
those found in the Standard Model.  The Higgs boson search strategies
of Sections~II.C and II.D can be applied to a CP-even neutral Higgs
boson, and those of Section~II.F to both CP-even and CP-odd neutral Higgs
bosons.  If CP is not conserved, then the neutral scalar states can be
considered to be mixtures of CP-even and CP-odd scalars, and the
discussion that follows must be modified accordingly
\cite{cpcarlos,cpcarlos2}.  Charged Higgs
boson states may also be present; the corresponding search strategies
were discussed in Section~II.G.  Finally, signatures not considered in
this report are also possible, such as the associated production of
CP-even and CP-odd neutral scalars and Higgs boson decays into $\tau$ leptons.
These additional signatures deserve a closer examination and might
improve the Higgs discovery reach beyond the results given in this section.

Before considering specific extensions of the Higgs sector, it is
useful to express the Standard Model results in a more  
model-independent fashion.  We focus on the analyses based on
Higgs processes in
which the production and decay depend primarily on a single Higgs
boson coupling.  To simplify the presentation, several
fixed values of integrated luminosity ($1,2,5,10,20,30$~fb$^{-1}$)
were chosen as well as two statistical measures (95\% Confidence Level
exclusion, 5$\sigma$ discovery) for the significance of a Higgs boson
signal.  


As a function of the Higgs boson mass, the experimental
sensitivity can be described by the ratio of the chosen level of
statistical significance and the actual statistical significance for a
fixed value of integrated luminosity.  This ratio is denoted by \rexp\
and depends on the assumed integrated luminosity,\footnote{Following
the conventions of this report, the integrated luminosity is taken to
be {\it per experiment}, corresponding to the total
luminosity delivered to the collider.  All plots shown in this
section assume that equal data sets from the CDF and D\O\ detectors are
combined.} 
the chosen statistical measure and the Higgs boson mass.
%This calculation defines a
%\rexp-curve value.
%(\rexp~value as a function of Higgs boson mass).
For example, if the discovery criterion is $5\sigma$, and the expected
significance of a Standard Model Higgs boson with mass 125 GeV is
$3.3\sigma$ with 10~fb$^{-1}$ of data, then \rexp$=1.5$ at 125 GeV.
This means that a 125 GeV Higgs boson from some extended Higgs sector
could be discovered with 10~fb$^{-1}$ of data if the production cross
section times branching ratio were 1.5 times the Standard Model value.
Alternatively, $(1.5)^2=2.25$ times more data is required for a
$5\sigma$ discovery of a Standard Model Higgs boson at this mass 
(in the limit of Gaussian statistics, for a background  
limited measurement, when the significance scales
as the square-root of integrated luminosity).

Once the \rexp-values are obtained, they can be applied to a particular
model of an extended Higgs sector.  For a given extended Higgs model,
a {\it theoretical} value \rth~at a fixed $\mphi$ 
can be computed for the set of
parameters that define the model.  {The Higgs mass, 
$\mphi$, may be one of the defining input parameters of the extended
Higgs model.}  
To be more precise, at a fixed Higgs boson mass,
for a given Higgs boson production process
$X$ ({\it e.g.}, $q\bar q' \to W\phi$ or $gg\to\phi$)
with the subsequent decay $\phi\to Y$, we define
\beq \label{ratiodef}
R_{\rm th}(X\,;\,Y)\equiv {\sigma_X(\phi) {\rm BR}(\phi\to Y)\over
\sigma_X(\hsm){\rm BR}(\hsm\to Y)}\,,
\eeq
where the numerator corresponds to the Higgs cross section times
branching ratio predicted in the extended Higgs sector model, and the
denominator corresponds to the same quantity in the Standard Model.
%
If the theoretical value \rth~lies below \rexp~for a particular $\mphi$,
then there is no experimental
sensitivity to this point of the model parameter space for
the assumed luminosity and statistical significance.
%
Clearly, if \rth=1, then the Higgs mass reach is precisely that of the
Standard Model, whereas if \rth$>1$ [\rth$<1$] the corresponding Higgs boson mass
reach is increased [decreased] with respect to that of the Standard Model.
%Thus, using the simulations of
%the Standard Model Higgs search, one can obtain curves at fixed
%integrated luminosity of $R$ {\it vs.} the Higgs mass reach 
%(an ``$R$-curve'') in an arbitrary extended Higgs model.  Thus, one 
%gets a family of $R$-curves by considering different integrated
%luminosities (and/or different definitions of Higgs mass reach as
%described above).  


%Thus by scanning points of the model parameter space, one can use the
%$R$-curves to trace out contours in the parameter space corresponding
%to discovery reach or the 95\% exclusion limits for the detection of
%$\phi$ for various choices of integrated luminosity.

Consider first the
production of a neutral scalar, $\phi$,
in association with a vector boson ($V=W$ or $Z$), where $\phi\to
b\bar b$.  For the Tevatron Standard Model Higgs search, 
this mechanism is relevant
in the mass range $90~{\rm GeV}\lsim\mphi\lsim 130$~GeV.  In this case, the
Higgs search simulations of Sections~II.C.1--4 apply.  
The resulting \rexp-curves corresponding to a 95$\%$ CL exclusion
and discovery at the 5$\sigma$ level are shown 
in \fig{rplot95}.
%
\begin{figure}[ht]
\centering
\centerline{\psfig{file=mssm/rvalues95_2000.eps,width=9cm,angle=0}
\hfill 
\psfig{file=mssm/rvalues5_2000.eps,width=9cm,angle=0}}
\vskip2pc
\caption[0]{\label{rplot95} The ratio \rexp
$\equiv \sigma(q\bar q'\to V\phi){\rm BR}(\phi\to b\bar b)/\sigma(q\bar q'\to V\hsm){\rm BR}(\hsm\to b\bar b)$ 
as a function of Higgs boson mass in GeV.
Each shaded region represents the minimal value of \rexp~necessary at
a given Higgs boson
mass to (a)~exclude a Higgs boson signal at $95\%$ CL or (b)~discover
a Higgs boson at the $5\sigma$ level, 
for the integrated luminosity per detector as
indicated (combining the statistical power of both experiments). 
As in \fig{f:combined-final}, the thickness of the bands,
extending upwards from
the estimated thresholds, indicate the uncertainties
in $b$-tagging efficiency, background, mass resolution, and other
effects.}

\end{figure}
The cross section is
governed by the $VV\phi$ coupling $g_{VV\phi}$,\footnote{It is assumed 
that the ratio of $\phi W^+ W^-$ and $\phi ZZ$ couplings is
the same as in the Standard Model.  This assumption is correct in
multi-doublet Higgs models, but can be violated in models with higher
Higgs representations \cite{wudka}.}  
while the rate for the observed final state  
($b\bar b V$, where $V$ decays leptonically) depends on
the branching ratio of $\phi\to b\bar b$.  The relevant 
\rth-curves are obtained by taking $X=q\bar q'\to V\phi$ and $Y=b\bar b$ in
\eq{ratiodef}:
\beq 
R_{\rm th}(q\bar q'\to V\phi\,;\,b\bar b)\equiv 
\left({g^2_{VV\phi} \over g^2_{VV\hsm}}\right)
\left[{{\rm BR}(\phi\to b\bar b)\over {\rm BR}(\hsm\to b\bar b)}\right]\,.
\eeq
These curves will be used in the next section to examine the Higgs
mass reach of the upgraded Tevatron in the framework of the MSSM.

In the Higgs boson mass range of $130~{\rm GeV}\lsim\mphi\lsim 200$~GeV,
the main Tevatron discovery signature for a Standard Model Higgs boson 
comes from two channels: $(i)$ $gg\to\phi\to WW^*$ (where both $W$'s decay
leptonically) and $(ii)$ $q\bar q'\to V\phi\to VWW^*$ [$V=W^\pm$ or
$Z$] (where two of the three vector bosons decay leptonically and the
third decays hadronically; the same-sign dilepton final states
provide the largest statistical weight).   The two production mechanisms
depend upon different Higgs boson couplings and are treated separately.
The simulations of Section~II.D are used to obtain the separate
\rexp-curves shown in \figs{ratwwi}{ratwwii}.
Separate theoretical \rth-curves are defined for cases $(i)$ and $(ii)$: 
\beqa
%\begin{eqnarray}
&\phantom{i}(i)\qquad \phantom{V}R_{\rm th}(gg\to\phi\,;\,WW^*)\, \equiv \,  
\left({\Gamma(\phi\to gg) \over \Gamma(\hsm\to gg)}\right)
\left[
{{\rm BR}(\phi\to WW^*)\over{\rm BR}(\hsm\to WW^*)}\right]\,, \nonumber \\
&(ii) \qquad R_{\rm th}(q\bar q'\to V\phi\,;\,WW^*) \,\equiv \, 
\left({g^2_{VV\phi} \over g^2_{VV\hsm}}\right)
\left[{{\rm BR}(\phi\to WW^*)\over{\rm BR}(\hsm\to WW^*)}\right]\,.
\label{ratiowwdef}
\eeqa

\begin{figure}
\centering
%\centerline{\psfig{file=mssm/ratwwi5.ps,width=8cm,angle=0}
\centerline{\psfig{file=mssm/howie2.ps,width=8cm,angle=90}
\hfill
%\psfig{file=mssm/ratwwi95.ps,width=8cm,angle=0}}
\psfig{file=mssm/howie1.ps,width=8cm,angle=90}}
\vskip2pc
\caption[0]{\label{ratwwi} The ratio \rexp$\equiv 
\sigma(gg\to \phi){\rm BR}(\phi\to WW^*)/\sigma(gg\to \hsm){\rm
BR}(\hsm\to WW^*)$ as a function of Higgs boson mass in GeV
necessary for
(a) excluding a Higgs boson signal at $95\%$ CL or (b) discovering
a Higgs boson at the $5\sigma$ level, 
for the integrated luminosity per detector as
indicated (combining the statistical power of both experiments). }
\end{figure}

\begin{figure}
\centering
%\centerline{\psfig{file=mssm/ratwwii5.ps,width=7cm,angle=0}}
\centerline{\psfig{file=mssm/howie4.ps,width=8cm,angle=90}
%\centerline{\psfig{
\hfill
\psfig{file=mssm/howie3.ps,width=8cm,angle=90}}
%\psfig{bbllx=0pt,bblly=0pt,bburx=200pt,bbury=200pt}}
\vskip2pc
\caption[0]{\label{ratwwii} Same as \fig{ratwwi}, but for \rexp$\equiv 
\sigma(q\bar q' \to V\phi){\rm BR}(\phi\to WW^*)/\sigma(q\bar q'\to V\hsm){\rm
BR}(\hsm\to WW^*)$.}
\end{figure}

Clearly, a combination of both channels $(i)$ and $(ii)$ yields
the maximal sensitivity for Higgs discovery reach (or exclusion).
However, this must be done on a model-by-model basis,
since different couplings may be involved in the calculation of
the two \rth~values in any given extension of the Standard Model.
Note that in the MSSM, the $VV\phi$ coupling is
significantly suppressed for $\mphi>130$~GeV (in this mass range, 
$\phi$ is the heavier CP-even Higgs boson,
$\hh$).  Thus, in the MSSM, the predicted values of \rth$(i)$ and
\rth$(ii)$ are substantially smaller than 1.\footnote{$\sigma(gg\to\phi)$ 
is suppressed [enhanced] relative to the Standard
Model for moderate [large] values of $\tan\beta$.  But, at large
$\tan\beta$,  the suppression of ${\rm BR}(\phi\to WW^*)$ 
is more significant than the enhancement of
the $gg\to\phi$ rate, resulting in \rth$(i)\ll 1$.} 
As a result, the high mass Tevatron Standard Model
Higgs search has no relevance for
the MSSM analysis, and it will not be employed in the next section.  
In models of extended Higgs sectors with exotic
Higgs representations, parameter ranges in which \rth$(i)$ and/or \rth$(ii)$
are larger than one can occur.  As an extreme example, 
models of non-minimal Higgs sectors have been constructed in which the
tree-level coupling of the Higgs bosons to fermions is absent (or
highly suppressed).  In such ``fermiophobic'' Higgs 
models \cite{fermiophobe}, the
Higgs coupling to gluons (which normally occurs at one-loop via
the fermion loop graph) is approximately zero and \rth$(i)$ vanishes.

The last search channel considered here is Higgs production 
via $gg$, $q\bar q\to b\bar
b\phi$, followed by the decay $\phi\to b\bar b$.  
The ratio \rth~is
defined using \eq{ratiodef} with $X=q\bar q,gg\to b\bar b\phi$
and $Y=b\bar b$.
Since $\sigma(gg,\,q\bar q\to b \bar b\phi)$ is proportional to the 
square of the $\phi b\bar b$ Yukawa coupling, $g_{\phi b\bar b}^2$, it
follows that:
\beq \label{rbbbar}
R_{\rm th}(q\bar q,gg\to b\bar b\phi\,;\, b\bar b)= 
\left( g_{\phi b\bar b}^2\over g_{\hsm b\bar b}^2 \right)
\left[{{\rm BR}(\phi\to b\bar b)\over {\rm BR}
(\hsm\to b\bar b)}\right]  \,.
\eeq
The resulting curves corresponding to a 95$\%$ CL Higgs mass exclusion
and Higgs discovery at the 5$\sigma$ level, based on the CDF and
D\O\ analyses
of  Section~II.F, are shown in \fig{rplotmaria}.
Note that the D\O\ analysis has greater sensitivity at large values of $\mphi$
than the CDF analysis, whereas the CDF analysis has greater sensitivity at
low values of $\mphi$.  The case of Standard Model Higgs boson
production, $b\bar b\hsm\to b\bar b b \bar b$, would correspond to 
\rexp$=1$ in \fig{rplotmaria}.  Clearly, neither analysis shows
any sensitivity to the production of a Standard
Model Higgs boson in the $b\bar b\phi\to b\bar b b\bar b$ channel for
any reasonable luminosity.
\begin{figure}[ht]
\centering
\centerline{\psfig{file=mssm/rvalueb95.eps,width=9cm,angle=0}
\psfig{file=mssm/rvalueb5.eps,width=9cm,angle=0}}
\vskip2pc
\caption[0]{\label{rplotmaria} The same as \fig{ratwwi}, but for
\rexp$\equiv 
\sigma(gg,\,q\bar q\to b\bar b\phi){\rm BR}(\phi\to b\bar b)/
\sigma(gg,\,q\bar q\to \hsm){\rm BR}(\hsm\to b\bar b)$ and
for the CDF (dashed lines) and D\O\ (solid lines) analyses
described in Section~II.F.}


\end{figure}


Although the values of \rth~must be quite large in order to have any
sensitivity to the $b\bar b\phi\to b\bar b b\bar b$ channel, such
enhancements can readily occur in extensions of the
Standard Model.  The main limitation of the SM searches in this
channel, is that ${\rm BR}(\hsm\to b\bar b)$ is greatly
suppressed
once $\mhsm \gsim$ 135 GeV and decays to W or Z boson pairs become
important.
In the MSSM at large $\tan\beta$, \rth~can be significantly enhanced.
In Section I.C.1 [below
\eq{hpmqq}], we showed that the tree-level
$\phi b\bar b$ coupling is enhanced 
(relative to the Standard Model value) by a factor of $\tan\beta$
for either $\phi=\hl$, $\ha$ or $\phi=\hh$, $\ha$ (depending on the
value of $\mha$).  
Evaluating \eq{rbbbar} for the case where $\phi$ is one of the neutral Higgs
bosons with an enhanced coupling to $b\bar b$, and taking
${\rm BR}(\phi\to b\bar b)\simeq 1$, one finds~\footnote{The value of
\rth~can be modified at large $\tan\beta$ in some regions of
the MSSM parameter space.}
\beq
R_{\rm th}(q\bar q,gg\to b\bar b\phi\,;\, b\bar b)\simeq 
\left[{\rm BR}(\hsm\to b\bar b)\right]^{-1}
\tan^2\beta\,.
\eeq
Since $[{\rm BR}(\hsm\to b\bar b)]\ll 1$ for $\mphi\gsim 150$~GeV,
and sensible parameter regimes exist where $\tan\beta\gg 1$ is possible,
there is sensitivity to the $b\bar b\phi\to b\bar b b\bar b$ channel 
for an interesting range of Higgs masses and $\tan\beta$ for
integrated luminosities as low as 2${\rm~fb}^{-1}$.  This will be
discussed further in Section III.B.3.
Whereas the \rexp-curves in \fig{rplotmaria} are based on the production 
of a CP-even
neutral scalar, they also apply to the CP-odd Higgs boson,
because the corresponding cross-section and decay branching ratios
are very similar [see \figs{fg:2}{fg:5}].
Thus, one can reliably employ
the \rexp-curves obtained above to $b \bar b \ha$ production in the MSSM.


\subsubsection{MSSM Higgs Sector Results:
$q\bar q\to V\phi$ [$\phi=\hl$, $\hh$], $\phi\to b\bar b$}


%In this section, the Standard Model Higgs boson searches are interpreted
%in the context of the MSSM for several illustrative choices of 
%MSSM parameters.
The general MSSM parameter space involves many {\it a priori} unknown
parameters.  
%The upper limits of the supersymmetric
%mass parameters are limited by arguments
%based on naturalness, but the bounds are not very precise.
%The lower limits of supersymmetric masses
%are constrained by the non-observation of
%supersymmetric particles in present day colliders.  However, 
%still leaves a significant 
%There are otherwise few constraints on those parameters
%not excluded by experiment.
In practice, only a small subset of these parameters govern the 
properties of the Higgs sector.
A full scan of this reduced subset is a formidable task.
However, a detailed study of a few appropriately chosen points
of the parameter space can help determine the ultimate MSSM Higgs
discovery reach of the upgraded Tevatron.

In order to select a representative set of parameters, it is useful to
study approximate analytic expressions for
the elements of the CP-even Higgs boson squared-mass matrix.
At tree-level, all MSSM Higgs masses and couplings are functions of
$\mha$ and $\tan\beta$.  As shown in Section~I.C.2 and I.C.3, the
radiative corrections to Higgs masses and couplings depend primarily 
on the third generation squark mass and mixing parameters 
(with a somewhat weaker dependence on the gluino mass).
%and certain relationships between them
%yield interesting patterns of masses and couplings.  These patterns
%are revealed by studying the analytic, but approximate, expressions for
%the elements of the CP-even Higgs boson mass squared matrix.
%Since radiative corrections to Higgs masses and couplings depend primarily 
%on the third generation squark mass and mixing parameters and the
%gluino mass, 
In the ``generic'' MSSM analysis, we derive exclusion and discovery
regions in the $\mha$--$\tan \beta$ plane for various
benchmark values of these parameters.

In the generic MSSM analysis, all the relevant MSSM parameters that
govern the Higgs sector masses and couplings are regarded as
independent.   These parameters arise primarily from the 
supersymmetry-breaking sector of the theory.
However, there is no underlying theoretical model of
supersymmetry-breaking that yields precisely the set of benchmark
parameters that we have employed in the generic MSSM analysis.  An
alternative approach is to choose a specific model of
supersymmetry-breaking, and then derive the values of the
relevant supersymmetric parameters (as a function of more fundamental
model parameters).  We shall illustrate this procedure in the 
minimal supergravity (mSUGRA) model and in simple
models of gauge-mediated supersymmetry-breaking.  
%These results are
%more model-dependent than our generic analysis, although they provide
%concrete examples of interesting relations among the supersymmetry




\vskip6pt\noindent
\underline{a. Generic MSSM Analysis} \\[0.2cm]


The results of Sections~II.C.1--4, based on the SHW simulation improved by
neural network techniques, 
show that a Standard Model Higgs boson with $\mhsm\lsim
130$~GeV can be excluded at the 95\% CL [discovered
at the 5$\sigma$ level] 
for an integrated luminosity of 
5~fb$^{-1}$ [30~fb$^{-1}$].
In Section I.C.2, it was noted that there is an upper bound
to the value of the lightest CP-even neutral Higgs boson, $\hl$, of
the MSSM.  Assuming an average squark mass parameter of $M_S\lsim
2$~TeV,  then $\mhl\leq \mhmax\simeq 130$~GeV.
This suggests that a
significant region of the MSSM parameter space can be probed by the
Tevatron Higgs search with enough integrated luminosity.
This can be understood by
considering limiting cases of the MSSM in which some or all of the
properties of one of the neutral CP-even Higgs bosons approach those
of $\hsm$.  For example, the limit $\mha\gg m_Z$ corresponds to the
case in which the heavy CP-even, the CP-odd and the charged Higgs
 decouple from the low energy effective theory and $\hl\simeq\hsm$.  
In this
``decoupling'' region of MSSM parameter space, the Standard Model
simulations apply directly.  
%In particular, if one has sensitivity to $\hsm$
%with $\mhsm\lsim 130$~GeV, then one has sensitivity to the decoupling
%region of the MSSM parameter space (where $M_S\lsim 1$~TeV).  
Other
regions of MSSM parameter space exist in which $\hh$ behaves in part
like a Standard Model Higgs boson.  As discussed in Section I.C.2, in
the region of $\tan\beta\gg 1$ and $\mha<\mhmax$, one finds that
$\mhh\simeq \mhmax\lsim 130$~GeV and the coupling of $\hh$ to $VV$
pairs is nearly equal to the corresponding $\hsm$ coupling.  As
long as ${\rm BR}(\hh\to b\bar b)$ is not significantly suppressed,
the Standard Model Higgs simulation can be applied to $\hh$ in this
MSSM parameter regime.

In the rest of the MSSM parameter space, both $\hl$ and 
$\hh$ couplings to vector bosons and fermions differ from those of $\hsm$.
For the generic MSSM analysis, 
\rth$(q\bar q'\to V\phi\,;\,\phi\to b\bar b)$ 
[$\phi=\hl$, $\hh$] is computed as
a function of $\mha$ and $\tan\beta$.  Applying \eq{ratiodef} to the MSSM
yields
\beqa \label{rvhsusy}
R_{\rm th}(q\bar q'\to V\hl\,;\,\hl\to b\bar b) 
&= \left[\frac{{\rm BR}(\hl\to b\bar b)}{{\rm
BR}(\hsm\to b\bar b)}\right] \sin^2(\beta-\alpha)
\, , \nonumber \\[6pt]
R_{\rm th}(q\bar q'\to V\hh\,;\,\hh\to b\bar b) 
&= \left[\frac{{\rm BR}(\hh\to b\bar b)}{{\rm
BR}(\hsm\to b\bar b)}\right] \cos^2(\beta-\alpha)\,.
\eeqa
The above equations demonstrate the complementarity of the 
$V\hl \to V b\bar b$ and  $V\hh \to V b\bar b$
 channels in
the coverage of the $\mha$--$\tan \beta$ plane coming from 
the dependence on $\sin^2(\beta-\alpha)$ and $\cos^2(\beta-\alpha)$
 of the ratio of MSSM to SM
cross sections in each case.  The ratio of branching ratios has a more
complicated behavior as a function of the supersymmetric parameters.   
In a large region of the MSSM parameter space,\footnote{This is
especially true in the region of large $\tan\beta$  
where $g_{\phi b\bar b}$  [$\phi=\hl$ or $\hh$ depending on $\mha$]
may be enhanced relative to the Standard Model.} 
${\rm BR}(\phi\to b\bar b)/{\rm BR}(\hsm\to b\bar b)$ with  
[$\phi=\hl$, $\hh$] is 
slightly above 1 (over the Higgs mass range of interest).
Thus, \rth$(q\bar q'\to V\phi\,;\,\phi\to b\bar b)$ may be slightly above 1 if
$\sin^2(\beta-\alpha)\simeq 1$ [for $\phi=\hl$] or
$\sin^2(\beta-\alpha)\simeq 0$ [for $\phi=\hh$].
In contrast, in the region where $\sin^2(\beta-\alpha)\simeq
\cos^2(\beta-\alpha)\simeq 0.5$, 
the~\rth$(q\bar q'\to V\phi\,;\,\phi\to b\bar b)$ values for
both $\phi=\hl$ and $\hh$ are significantly below 1, which severely limits
the sensitivity of the Tevatron Higgs search.  Finally, 
small regions of MSSM parameter space exist (as discussed in
Section I.C.3), in which $g_{\phi b\bar b}$  
[$\phi=\hl$ or $\hh$] is strongly suppressed relative to the Standard Model,
even though $g_{VV\phi}$ is near the Standard Model value.
In these regions, \rth$(q\bar q'\to V\phi\,;\,\phi\to b\bar b)\ll 1$ 
and the $V\phi\to Vb\bar b$ channel is not a promising search mode.

Once the values of the squark
mass and mixing parameters are fixed, the two ratios
\rth$(q\bar q'\to V\phi\,;\,\phi\to b\bar b)$ [$\phi=\hl$ and $\hh$] can
be computed as a function of $\mha$ and $\tan\beta$.  If the
\rth~value is larger than \rexp~at the same $\mphi$ in
\fig{rplot95}(a) (for a given integrated luminosity), then this
point of parameter space can be excluded at the 95\% CL.  If
\rth$(q\bar q'\to V\phi\,;\,\phi\to b\bar b)$ is larger than \rexp~at
the same $\mphi$ in \fig{rplot95}(b), then $\phi$ can be discovered
at the 5$\sigma$ level.  Whereas in most of the $\mha$--$\tan \beta$
plane, either $h$ or $H$ is the relevant Higgs boson,
there are some regions of MSSM parameter space where 
where both Higgs bosons have comparable rates.
If $\mhh-\mhl\lsim 10$~GeV, then the individual \rth~values of the $\hl
V$ and $\hh V$ signals are added in quadrature before comparing to
\rexp.  This procedure is meant to crudely represent the fact that the
two signals will add constructively as long as the mass difference is
within the experimental resolution.\footnote{One would expect some
loss of significance, depending on the actual mass splitting, since some of
the signal will leak out of the {\it a priori} chosen mass window.}
Note that the near degeneracy of $\hl$ and $\hh$ occurs only in the
narrow region of MSSM parameter space when $\mha\simeq m_{h}^{max}$.
%This combination of Higgs signals
%is necessary near the region where $m_A\simeq m_h^{\rm max}\pm{\rm
%few~GeV}$ to obtain coverage in the $\mha$ {\it vs.} $\tan\beta$
%plane.

\begin{table}
\caption{\label{benchmarks}
Description of the values of the MSSM parameters for the benchmark scenarios 
considered in this report \protect\cite{benchmark2}.  
In all cases, the $\widetilde W$-gaugino (``wino'') mass parameter
has been taken to be $M_2=200$~GeV.}
%\vskip6pt
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lcccccc} 
& \multicolumn{5}{c}{Mass parameters [TeV]} & (GeV) \\ \cline{2-6}
\noalign{\vskip3pt}
Benchmark   & $\mu$ &$X_t\equiv A_t-\mu\cot\beta$ & $A_b$ & $M_{SUSY}$ 
   & $M_{\tilde g}$\quad &  $m_h^{max}$ \\ [2pt]
\tableline
%
maximal mixing &  $-0.2$ & $\sqrt{6}$ & $A_t$ & 1.0 & 1.0 & 129
\\
no-mixing   &  $-0.2$ & $0$ & $A_t$ & 1.0 & 1.0 & 115 \\
suppressed $V \phi \to V b\bar b$ production
   &    1.5 & $-1.5 (1+\cot\beta)$  &0& 1.0 & 1.0 &  120 \\[3pt]
%Suppressed $\phi b\bar b$ couplings  
%&    1.2 & $-1.2 (1+\cot\beta)$  &0& 1.0 & 0.5 &   \\
%Enhanceded $A b\bar b$ production   
%&   $-1.2$ & $1.2 (1+\cot\beta)$  &0& 1.0 & 0.5 &  \\
\end{tabular}
\end{table}

Table~\ref{benchmarks} defines the set of benchmark parameters for the
relevant supersymmetric parameters that govern the Higgs radiative
corrections.  These include the supersymmetric Higgs mass parameter
$\mu$, the third generation squark mixing parameters, $A_t$ and $A_b$,
the gluino mass $M_{\tilde g}$, and the diagonal
soft-supersymmetry-breaking third generation squark squared-masses
(which we take for simplicity to be degenerate and equal to $\MSUSY$).
The maximal mixing benchmark scenario, as the name suggests, is
defined as the one in which the squark mixing parameters are such that
they maximize the value of the lightest CP-even Higgs boson mass for
fixed values of $\mha$, $\tan \beta$ and $\MSUSY$.
%maximal squark mixing ({\it i.e.}, fixed $\overline X_t\equiv \overline A_t-
%\overline\mu\cot\beta\simeq \sqrt{6}$),
%with $A_t\mu < 0$ and $M_2 = -\mu = 200 GeV$ 
%and $M_{\tilde{g}}= 1$ TeV.
In contrast, in the no-mixing scenario, the parameters $\mu$ and $A_t$
are chosen such that the combination $X_t\equiv A_t-\mu\cot\beta$
vanishes.  In this limit, the value of the lightest CP-even Higgs boson
is at its minimum for fixed values of 
$\mha$, $\tan \beta$ and $\MSUSY$.  
This corresponds to the minimal mixing case studied in Section~I.C.2.
[See \fig{mhxt} for the dependence of
$\mhl$ on the squark mixing parameter $X_t$.]

One might conclude that the maximal mixing scenario, in
which $\mhl$ can attain its maximal value for a given set of 
MSSM parameters, is the most conservative case for ascertaining the
Higgs mass discovery reach of the upgraded Tevatron.
However, a scan through the MSSM
parameters away from the maximal squark mixing point reveals
regions of parameter space in which the CP-even neutral Higgs boson with
Standard-Model-like couplings to $VV$ has suppressed couplings to
$b\bar b$. The benchmark scenario denoted by
``suppressed $V \phi \to V b\bar b$ production''  
%``suppressed $\phi b\bar b$ couplings'' and 
%``enhanceded $Ab\bar b$ production'' 
in table~\ref{benchmarks} is an example of this behavior.  
% as shown in \fig{brhbb}.  
The regions of strongly
suppressed ${\rm BR}(\phi\to b \bar b)$ correspond to a suppressed
$\hh b\bar b$ coupling at lower $\mha$ and a suppressed
$\hl b\bar b$ coupling at larger $\mha$.  In particular,
the suppression for large $\tan\beta$ extends to
relatively large values of $\mha$, indicating a delay in the onset of
the decoupling limit (where the $\hl b\bar b$ coupling approaches the
Standard Model value). 
This is the case when one of the Higgs bosons
becomes mostly $\Phi^0_u$, with no coupling to down-type quarks except those
induced by supersymmetry-breaking effects [see \eq{couplings}].  
Such a Higgs boson has SM--like couplings
to the $W,Z$ and $t$, but vanishing couplings to $b\bar b$ and
$\tau^+\tau^-$.  The necessary condition for this limit to
arise is explained in Section~I.C.3.  
In fact, in the suppressed $V\phi\to Vb\bar b$ benchmark 
scenario, {\it all} the Higgs couplings to $b\bar b$ are generally
suppressed, since $0<\Delta_b\ll 1$ and $\sin 2\alpha\simeq 0$ 
[see eqs.~(\ref{bhCP}), (\ref{barhb}) and (\ref{tildehb})].
From the analytic formulae, it can be deduced that $\mu A_t<0$ and 
large values of $|A_t|$, $|\mu|$ and $\tan\beta$ are needed.  

%Finally, we distinguish between the two scenarios labeled
%``suppressed $\phi b\bar b$ couplings'' and 
%``enhanced $Ab\bar b$ production'' in table~\ref{benchmarks}.
%In both cases, the sign of $A_t\mu$ is negative; the two cases are
%distinguished by the sign of $\mu$ (in the convention where the gluino
%mass parameter is positive).  If $\mu>0$ one finds that the $Ab\bar b$
%coupling is suppressed (relative to its tree-level value) by radiative
%corrections, while for $\mu<0$ the $Ab\bar b$ coupling is enhanced.

\begin{figure}[ht!]
\begin{center}
\centerline{\psfig{file=mssm/mhmax95_final.eps,width=8cm,angle=0}
\hfill
\psfig{file=mssm/mhmax_final.eps,width=8cm,angle=0}}
\end{center}
\caption[0]{\label{amaxnn} Regions of MSSM parameter space, 
for the maximal mixing benchmark scenario of table~\ref{benchmarks},
that correspond to 
(a) the exclusion of a Higgs boson signal at $95\%$ CL or (b) the discovery
a Higgs boson at the $5\sigma$ level. 
As one increases the integrated luminosity,
the corresponding shaded areas successively cover the plane, as
indicated (combining the statistical power of both experiments),
based on the improvement of the SHW simulation employing neural
network techniques.  The darker shading of a given color corresponds
to  a degradation in the coverage of the MSSM parameter space due to 
the uncertainties in  $b$-tagging efficiency, background, mass
resolution and other effects.}
\end{figure}

\Fig{amaxnn} shows
contour plots demonstrating regions of 95$\%$ exclusion and 
$5\sigma$ discovery for
the maximal mixing benchmark scenario  
as defined in table~\ref{benchmarks}.
This illustrates more
generally the fact that the MSSM parameter space coverage 
in the $V\phi\to Vb\bar b$ channel depends
sensitively on the maximal value of the lightest Higgs boson 
mass for a particular set of MSSM parameters.
The darker shading of a given color illustrates how a net $30 \% $
reduction in the detection efficiency, due to   
the uncertainties in $b$-tagging efficiency, background, mass
resolution and other effects,  
 translates in a the coverage of the MSSM 
$\tan \beta$--$m_A$ plane. 


 
A closer examination of \fig{amaxnn} shows three regions of the MSSM
parameter space that raise difficulties for the Tevatron 
$V\phi\to Vb\bar b$ search.  In the region of large $\tan\beta$ and
$\mha\lsim \mhmax$ (where $\mhh\simeq\mhmax$), 
the $\hl VV$ coupling is highly suppressed, 
so that $V\hh \to V b\bar b $ is the relevant process. 
%The difficulty in coverage for $\mha\lsim 100$~GeV
%is mainly due to the fact that the heavy CP-even Higgs has 
%couplings to the bottom quarks that become more SM-like 
%and its mass is very close to $\mhmax$, hence,  
%as already shown for the SM case  very high luminosity is required.
%
%However, for $\mha\lsim 100$~GeV the $\hh b\bar b$ coupling is somewhat
%suppressed  with respect to that of the
%Standard Model (for the case of $A_t\mu>0$).  This latter suppression
%is enough to explain the observed higher luminosity requirement for
%coverage of that region of the MSSM parameter plane.
%  
%This area of parameter space can be covered by LEP searches for 
%$\ha \hl$ associated production.  Moreover, as we will demonstrate
%later in this section, the Tevatron search for Higgs production in
%association with $b\bar b$ (followed by Higgs decay to $b\bar b$)
%is also sensitive to the region of $\mha\lsim 100$~GeV.
For large $\mha$ and large $\tan\beta$ the incomplete parameter space
coverage is
simply due to the fact that the lightest Higgs has SM like couplings 
and its mass is at the maximal value.
Another problematical region of MSSM parameter space corresponds to
the case of $\mha\simeq\mhmax$ 
and moderate $\tan\beta$, where $\sin^2(\beta - \alpha) \simeq
\cos^2(\beta - \alpha) \simeq 0.5$. 
If $\tan\beta$ is sufficiently large,
the two CP--even Higgs bosons tend to have similar masses, and both
states contribute to the Higgs signal in the same mass bin.
However, a hole in the coverage of
the $\mha$ {\vs} $\tan\beta$ plane persists in a small region  
for $\tan\beta$ between 5 and 20, since in this case the
mass difference is large and $\sin^2(\beta-\alpha)\simeq 0.5$. 
%Finally, 
In contrast, 
a signal can be observed with less integrated luminosity
in the region of $\mha \simeq 150$ GeV,
since $\sin^2(\beta - \alpha)\simeq 1$ 
%in this region 
and the $\hl b\bar b$ coupling
is strongly enhanced with respect to the Standard Model
case, implying an increase in the branching ratio of this Higgs boson
into bottom quarks.


\begin{figure}[hp!]
\begin{center}
\centerline{\psfig{file=mssm/mixtev95_final.eps,width=8cm,angle=0}
\hfill
\psfig{file=mssm/mixtev_final.eps,width=8cm,angle=0}}
\end{center}
\caption[0]{\label{holenn}  The same as \fig{amaxnn} but for the
``suppressed $V \phi \to V b\bar b$ production'' benchmark scenario of
table~\ref{benchmarks}.}
\end{figure}


\begin{figure}[htb]
\begin{center}
\centerline{\psfig{file=mssm/nomixing95_final.eps,width=8cm,angle=0}
\hfill
\psfig{file=mssm/nomixing_final.eps,width=8cm,angle=0}}
\end{center}
\caption[0]{\label{aminnn}   The same as \fig{amaxnn} but for 
the no-mixing benchmark scenario of table~\ref{benchmarks}.}
\end{figure}


\Fig{holenn} illustrates the effect of the suppressed $\phi b\bar b$
couplings for the Tevatron MSSM Higgs search for the ``suppressed
$V\phi\to Vb\bar b$''
benchmark scenario of table~\ref{benchmarks}.  
The small
hole in the coverage of the $\mha$ {\vs} $\tan\beta$ plane for
$\tan\beta$ between 5 and 15 corresponds to the same difficult region of MSSM
parameter space noted in the maximal mixing case above.  However, in the
present case, two new larger holes open up corresponding to the
regions of strongly suppressed $\phi b\bar b$ coupling noted above.
Since in this case one has $A_t  \mu < $0, there is a suppression
of the  $\phi b\bar b$ coupling via a suppression of the radiatively
corrected value of $\sin \alpha$. On the other hand, although the
gluino mass and $\mu$ are large, the  vertex
corrections to the bottom Yukawa coupling have the effect of
suppressing the  $\phi b\bar b$ coupling, 
since $\mu > $0 and, hence, $\Delta_b$ is also positive. 
Again, as explained above, the darker shading of a given color 
shows the uncertainty in the discovery and exclusion potential 
due to uncertainties in the experimental simulations.

%

\Fig{aminnn} shows contours of $5\sigma$ discovery and 95\% CL exclusion
for the case of the ``no-mixing'' benchmark scenario defined in 
table~\ref{benchmarks}, which yields the smallest value for the
mass of the Higgs boson that has SM--like couplings to
the $W$ and $Z$ bosons as a function of the squark mixing effects.
This is an optimistic
scenario for the Tevatron, in which only modest values of integrated 
luminosity are necessary 
to cover a significant portion of the $\mha$--$\tan \beta$
plane.  Note, however, that 
the maximal mass for the lightest Higgs boson is about 115 GeV.
Much of this parameter regime will have been probed at LEP by the end of
its final run.


\break
\vskip6pt\noindent
\underline{b. Model-Dependent MSSM Analysis}

\vspace{0.1in}
\small
\begin{center}
{\it  H. Baer, B.W. Harris and X. Tata}
\end{center}
\normalsize\nopagebreak


%\phantom{0.2cm}

%In a generic version of the MSSM, the number of unknown parameters
%that governs the complete description of the low-energy supersymmetric
%model is very large.   However, many of these parameters do not
%directly affect the properties of the Higgs sector in a significant
%way.  Nevertheless, we have seen that Higgs sector masses and
%couplings can be affected in a non-trivial way by loop-corrections
%involving supersymmetric particles.  Thus, a complete description of
%the MSSM Higgs sectors requires  specification of the supersymmetric masses
%and couplings.   These 
The Higgs sector masses and couplings 
are determined from a set of
soft-supersymmetry-breaking parameters that include 
three gaugino mass parameters, scalar
squared-mass parameters for squarks, sleptons and Higgs bosons, 
the $A$ parameters that govern cubic scalar interactions, and 
the $B$ parameter that mixes the two Higgs scalars.
In addition, there is also dependence on the 
supersymmetric Higgsino mass parameter $\mu$.  The latter two
parameters can be exchanged for the two Higgs vacuum expectation
values, under the assumption that the electroweak symmetry is broken
in the usual manner.
The origin of the soft-supersymmetry-breaking parameters is not
presently known.  Nevertheless, one may consider specific models of
supersymmetry breaking, where the soft-supersymmetry-breaking
parameters are specified.  In such approaches, one often makes 
simplifying assumptions in order to reduce the total number of free
parameters.  Although such assumptions are ad-hoc, the resulting
models do provide examples of how the soft-supersymmetry-breaking
parameters could be correlated.  The results of such correlations
would constrain the Higgs sector of a generic MSSM.  In particular, the
discovery of the Higgs bosons of the MSSM would place interesting
constraints on the underlying supersymmetric parameters.

In the {\it minimal} supergravity (mSUGRA)
framework \cite{Nilles85,sugra,sugrawork}, the
supersymmetry breaking takes place in a hidden sector, and 
supersymmetry-breaking effects are communicated to the observable sector via
Planck-scale interactions. In the minimal rendition of this model
the soft-supersymmetry-breaking parameters take a particularly
simple form when evaluated at a very high energy scale (usually taken
to be either the Planck scale or the grand unification scale).
At this high-energy scale, all scalars have a common mass $m_0$,  
all gauginos have a common mass $m_{1/2}$, and all trilinear scalar couplings
unify to $A_0$.  The low-energy Higgs and supersymmetric parameters that
govern the observed Higgs and supersymmetric phenomenology
are obtained via renormalization group running of soft-supersymmetry-breaking
masses and couplings from the high-energy scale down to the
electroweak scale.  In a successful mSUGRA model, electroweak symmetry 
breaking is radiatively generated, since at least one of the Higgs
squared-masses is driven negative in the renormalization-group evolution.

One can count the number of independent parameters in the
mSUGRA framework.  In addition to usual Standard Model parameters
(excluding the Higgs mass), one must specify
$m_0$, $m_{1/2}$, $A_0$, and Planck-scale values for the $\mu$ and
$B$-parameters (denoted by $\mu_0$ and $B_0$).  In principle,
$A_0$, $B_0$ and $\mu_0$ can be complex, although in
the mSUGRA approach, these parameters are taken (arbitrarily)
to be real.  As previously noted,
renormalization group evolution is used to compute the low-energy
values of the mSUGRA parameters, which then fixes all the
parameters of the low-energy MSSM.  In particular, the two Higgs
vacuum expectation values (or equivalently, $m_Z$ and $\tan\beta$)
can be expressed as a function of the Planck-scale supergravity
parameters.  The simplest procedure is to express $m_Z$ and $\tan\beta$ 
in terms of the low-energy $\mu$ and $B$ parameters and the other
free mSUGRA parameters (the sign of $\mu$ is not fixed in this
process).  The resulting MSSM spectrum
and its interaction strengths are then determined by five parameters:
$m_0$, $A_0$, $m_{1/2}$, $\tan\beta$, and the
sign of $\mu$, in addition to the usual parameters of the Standard Model.

Recently, attention has been given to a class of supergravity models in
which the common tree-level gaugino mass is absent.  In such models,
one finds a model-independent contribution to the
gaugino mass \cite{anomalymed},
whose origin can be traced to the super-conformal
(super-Weyl) anomaly which is common to all supergravity models.
This approach has been called {\it anomaly-mediated} supersymmetry
breaking.  In the simplest scenario, all supersymmetric particle
masses are determined by one overall mass parameter 
(the gravitino mass, $m_{3/2}$).
However, anomaly-mediation cannot be the sole
source of supersymmetry-breaking in the MSSM (since in the simplest of such
models, slepton squared-masses are negative).  Moreover,
one still needs a model for the origin of the $\mu$ and $B$
parameters in order to generate electroweak symmetry breaking and a
realistic Higgs sector.  One simple approach is to add to the theory a
universal scalar mass parameter $m_0$ and $\mu$ and $B$ by hand.
The resulting model depends on four parameters: $m_{3/2}$, $m_0$,
$\tan\beta$ and ${\rm sign}(\mu)$.  A preliminary study of the Higgs boson
discovery reach of the upgraded Tevatron in this scenario has been
presented in \Ref{shufang}.  

In models of {\it gauge-mediated} super\-symmetry breaking \cite{dine},
super\-symmetry breaking is transmitted to the MSSM via gauge forces.
A typical structure of such models involves a hidden sector
where supersymmetry is broken, a ``messenger sector'' consisting of
particles (messengers) with SU(3)$\times$SU(2)$\times$U(1) quantum
numbers, and the visible sector consisting of the fields of the
MSSM \cite{gmsbreview}.
The direct coupling of the messengers to the hidden sector generates a
super\-symmetry breaking spectrum in the messenger sector.
Finally, super\-symmetry
breaking is transmitted to the MSSM via the virtual exchange of the
messengers.

In the gauge-mediated supersymmetry-breaking (GMSB) approach,
the low-energy scalar and gaugino mass parameters are generated
through loop-effects (while the resulting $A$-parameters are
suppressed).  In minimal GMSB models,
the supersymmetric particle masses are proportional to the parameter
$\Lambda = F/M$, where $\sqrt{F}$ is the supersymmetry-breaking
scale and $M$ the mass scale for the messenger particles.  The origin
of the $\mu$ and $B$-parameters is model-dependent and lies somewhat
outside the ansatz of gauge-mediated supersymmetry-breaking and must
be taken to be additional free parameters.  The minimal gauge-mediated
(mGMSB) models of this type are parameterized in terms
of $\Lambda$, $M$, $n_5$, $C_{\rm grav}$, $\mu$ and
$B$, where $n_5$ is the number of complete $SU(5)$ messenger
multiplets ($n_5
\leq 4$ if $M\lsim 1000$~TeV to avoid the presence of Landau poles
below $M_{GUT}$) and 
$C_{\rm grav}$ is the ratio of hidden sector to messenger sector vacuum 
expectation values of auxiliary fields \cite{pedro}.   
As in the case of mSUGRA,
one can re-express $\mu$ and $B$ in terms of $m_Z$ and $\tan\beta$
after the electroweak symmetry breaking conditions are imposed.  

The mSUGRA and mGMSB models are incorporated in the event generator
ISAJET \cite{ISAJET}.  The Higgs boson mass spectrum is obtained in
ISAJET by minimizing the renormalization-group-improved one-loop
effective potential. 
The minimization is performed at an optimized
scale choice $Q\simeq (M_{\widetilde t_L}M_{\widetilde t_R})^{1/2}$,
which effectively includes the dominant two-loop
contributions  to the Higgs mass computations.  
The latest improvements including finite two loop
effects through vertex corrections to the top and bottom
Yukawa couplings are not taken into account in these analyses.
Supersymmetric parameter
values are used by ISAJET to calculate the various  
supersymmetric particle masses and mixing angles, and the Higgs boson 
masses and branching fractions \cite{bisset}. The calculations
presented here are an update of work previously presented in
\Ref{bht}.



\begin{figure}[hp!]
%\vskip-4pc
\begin{center}
\centerline{\psfig{file=mssm/sugra_04_p_exclusion.eps,width=9cm}
\psfig{file=mssm/sugra_04_p_5sigma.eps,width=9cm}}
\end{center}
\vskip-2pc
\caption[0]{\label{sugra:1} 
Regions of mSUGRA model parameter space corresponding to 
(a) 95\% CL exclusion and (b)
5$\sigma$ discovery that are accessible to 
the upgraded Tevatron with different integrated luminosities.
In this case $A_0=0$, $\tan\beta =4$ and $\mu > 0$.  The TH region is 
excluded either due to lack of proper 
electroweak symmetry breaking or because the lightest neutralino 
is not the lightest supersymmetric particle.  
The EX region is excluded from the experimental bound on the 
lightest chargino mass, and the marked LEP2 region is explorable by
direct Higgs boson searches at LEP.}
%\end{figure}
\vskip 2pc
%\begin{figure}[ht!]
\vskip-6pc
\begin{center}
\centerline{\psfig{file=mssm/sugra_35_p_exclusion.eps,width=9cm}
\psfig{file=mssm/sugra_35_p_5sigma.eps,width=9cm}}
\end{center}
\vskip-2pc
\caption[0]{\label{sugra:3} 
As in \fig{sugra:1}, but for
the case $A_0=0$, $\tan\beta =35$ and $\mu > 0$.}
\end{figure}

The 95\% exclusion and 5$\sigma$ discovery regions for the mSUGRA model are 
shown in \fig{sugra:1} for $\tan\beta =4$,  $A_0=0$ and  
$\mu>0$. The TH region is excluded either due to a lack of 
appropriate electroweak symmetry breaking, or because the lightest
neutralino is not the lightest supersymmetric particle.
The EX region is excluded by the negative results from LEP2 searches
for chargino pair production, which require 
$m_{\widetilde\chi^\pm_1} > 100$ GeV. 
The LEP reach is shown here as indicative, assuming a maximal Higgs
mass discovery potential of 108 GeV.
For the region of mSUGRA parameters considered in \fig{sugra:1}, the 
 Fermilab Tevatron experiments should be 
able to explore the entire plane shown at 95\% CL with 2~fb$^{-1}$ or
at $5\sigma$ with 20~fb$^{-1}$.

The corresponding mSUGRA plots for $\tan\beta =35$ are shown in
\fig{sugra:3}.  In contrast to the small $\tan\beta$ case just discussed,
at large $\tan\beta$ the mass of $\hl$ lies above
the LEP2 reach.  The Tevatron experiments should
be able to probe the entire plane shown at 95\% CL with 5~fb$^{-1}$ of
data.  The discovery potential is more limited and 
requires an integrated luminosity as 
high as 30~fb$^{-1}$ (for very large $m_{1/2}$)
to ensure Higgs boson discovery.

%
\begin{figure}[htp]
\vskip-5pc
\begin{center}
\centerline{\psfig{file=mssm/gmsb_m_exclusion.eps,width=9cm}
\psfig{file=mssm/gmsb_m_5sigma.eps,width=9cm}}
\end{center}
\vskip-2pc
\caption[0]{\label{gmsb:5} 
As in \fig{sugra:1}, but 
for a mGMSB model  with the parameter choice
$M=10^6$ GeV, $n_5 =1$ and $\mu < 0$.  The EX 
regions are excluded by experimental bounds on the mass of the 
lightest neutralino or the stau.}
\end{figure}



The 95\% exclusion and 5$\sigma$ discovery regions for the minimal
gauge mediated supersymmetry breaking model, in the
$\Lambda$--$\tan\beta$ plane, are shown in \fig{gmsb:5}, for $n_5=1$
and messenger scale $M=10^6$ GeV.  The TH region is excluded due to a
lack of appropriate electroweak symmetry breaking.  The EX regions are
excluded by experimental bounds on the mass of the lightest neutralino
($m_{\widetilde\chi^0_1} > 84$ GeV).  In these cases, the whole plane
can be probed by Tevatron experiments at the 95\% CL with 5~fb$^{-1}$
of data.  At the $5\sigma$ discovery level, the Tevatron experiments
would need 20~fb$^{-1}$ per experiment to be able to probe this model.



\goodbreak

\subsubsection{MSSM Higgs Sector Results:
$gg$, $q\bar q\to b\bar b\phi$ [$\phi=\hl$, $\hh$, $\ha$]}
%\subsubsection{Production of non--Standard Model--like Higgs bosons in
%association with heavy flavor }
%[$\phi=\hl$, $\hh$, $\ha$]}

In many extensions of the Standard Model (including the MSSM), there exists
at least one non--Standard Model--like Higgs boson: its couplings to gauge
bosons may vanish at tree--level or be suppressed through mixing, 
and their relative couplings to different fermions are not strictly 
set by the corresponding fermion masses.
Such Higgs bosons will exist in models with many Higgs doublets when 
one of the doublets is mostly responsible for electroweak symmetry breaking.
In the MSSM,  
the CP-odd scalar $\ha$ differs from the Standard Model Higgs boson,
with enhanced couplings at large $\tan\beta$
to down--type fermions and suppressed couplings to
up--type ones.  Additionally, when one of the CP-even Higgs bosons 
(either $\hl$ or $\hh$) has Standard Model--like couplings to the gauge
bosons, the other one has a pattern of enhanced and suppressed
couplings to fermions similar to that of the $\ha$.

We consider the prospects for discovering such a non--Standard Model--like
Higgs boson at the Tevatron, based on the
$b\bar b\phi\to b\bar b b\bar b$ signature studied in Section~II.F.
%To find a viable signature, it is necessary to consider the relative
%sizes of signal and background and to find a suitable trigger.  The
%total production cross section for $b \bar b$ pairs at the Tevatron is
%on the order of $\mu b$, which is far larger than the total inclusive
%cross section for $b\bar b\to\ha$ [not a fair comparison yet, since
%one should restrict the invariant mass range, etc].  On the other
%hand, the exclusive partonic processes $gg$, $q\bar q\to b\bar b\phi$
%can yield several $b$ jets with some distinctive kinematics.  The
%scalar sum of jet $E_T$ in such events is large and can be used as a
%trigger.  This signature is the focus of the present study.
In the case of the MSSM, $\phi$ can in principle be one of three
neutral Higgs bosons: $\hl$, $\hh$ and $\ha$.
If $\tan\beta$ is large, two of the Higgs bosons are
produced with enhanced rates.\footnote{As emphasized
below \eq{hpmqq}, at large $\tan\beta$ the production rate of 
$b\bar b\phi$ is enhanced for two of the neutral MSSM Higgs bosons: 
$\phi=\ha$ and $\phi=\hl$ [$\phi=\hh$] if $\mha\lsim\mz$
[$\mha\gg\mz$].}  
If these two Higgs bosons are very
close in mass, then the corresponding signals should be added as they
will not be distinguishable within the experimental resolution.  
However, over some range of $\mha$ and $\tan\beta$ values,
the two Higgs bosons with enhanced production rates are
significantly split in mass.  For example, for
moderate values of $\tan\beta$, the corresponding mass splitting 
can be as much as $10$--$20$ GeV, and the
procedure for combining the signals must reflect this fact.
%In particular, when both 
%$|\bar{\mu}|$ and $| \bar X_t | \gsim {\cal{O}}(1)$ (barred quantities
%are defined with respect to the scale $M_S$), it might be
%a poor approximation to simply 
%double the signal from $b\bar b\ha$ production~\cite{Hempfling} 
%to include the signal from the other non--Standard Model--like Higgs boson.
%The CP--even Higgs boson with similar properties to the CP--odd one
%has a mass $M_{CP}$ approximately equal to ${\cal M}_{11}$, with the 
%latter given approximately by
%\begin{equation}
%{\cal M}_{11}^2 \simeq m_A^2 - \frac{m_t^4}{16 \pi^2 v^2}\bar{\mu}^2
%{\bar X_t}^2. 
%\end{equation}
%{(\rm under what conditions will $m_H>m_A$)?}
%In the case of maximal mixing case, ${\bar X_t}^2 = 6$, when
%the radiative corrections to $m_{\hsm}$  are maximized and 
%${\cal M}_{12}^2$ receives only small radiative corrections for moderate or 
%large values of $m_A$, one gets approximately
%\begin{equation}
%M_A - M_{CP} \simeq (3.3~{\rm GeV})\frac{m_t}{m_A}\bar{\mu}^2
%\end{equation}
%For $|\bar{\mu}| \gsim 1$, 
%both CP--even Higgs boson masses can be significantly different from
%the CP--odd one, 

%The complications near $\mha\simeq m_h^{max}$ for the $V\phi$ studies
%and for almost all choices of parameters for the $b\bar b\phi$ studies
The previous considerations
illustrate one of the limitations of encoding all the analysis
information in a \rexp-curve.  In some cases, knowledge of the signal
size is also necessary.  The expected signal in a given model is
simply obtained by multiplying the Standard Model signal by the
\rth-value for the mass $m_\phi$.  Of course,
the expected background in a mass
window centered on the mass $m_\phi$ is unchanged by
choosing a different model.  The experimental sensitivity in the
$b\bar b\phi$ channel to a given set of MSSM parameters is estimated
using the following algorithm.  First, the CP even Higgs boson $\phi$
with the smallest value of $g^2_{VV\phi}$ is chosen as the
non--Standard Model--like one.  The expected signal for the CP even
Higgs boson $S_{CP}$ is calculated by multiplying the Standard Model
signal by the \rth-value for the CP-even Higgs boson, whereas the
expected background $B_{CP}$ in the mass window centered on the CP
even Higgs boson mass $M_{CP}$ is the same as in the Standard Model.
The signal $S_A$ and background $B_A$ for $\ha$ is determined in a
similar fashion.

The invariant mass distributions of the individual signals $S_{CP}$
and $S_A$ are assumed to be well described by Gaussian distributions.
To determine if the signals overlap, the mass splitting $\Delta
M=|M_A-M_{CP}|$ is compared to the quantity
$\sigma_4=(M_A+M_{CP})\times 0.15\times 2$.  The latter is the sum of
two standard deviations in the invariant mass distribution of each
signal assuming a $b\bar b$ mass resolution of 15\%.  If $\Delta
M<\sigma_4$, then the two signals overlap within the experimental
resolution.  In this case, the mass window is expanded to contain both
signals ($2\sigma$ below the lightest Higgs boson mass and $2\sigma$
above the heavier one), and the number of background events within the
window is calculated: $B=(B_{CP}+B_A)[1-\frac{1}{2}(1-\Delta
M/\sigma_4)]$.  The systematic error on the background is assumed to
be ${\rm min}(0.1 B,\sqrt{B})$.  This choice represents the Run 1
experience that systematic errors are roughly 10\% of the number of
background events, but also that the systematic errors do not increase
without bound and tend to saturate at a level comparable to the
statistical errors.  The full error on the background, $\Delta B$, is
the quadrature of statistical and systematic errors, and the
significance is defined as $(S_A+S_{CP})/\Delta B$.  If the two
signals are well separated within the experimental resolution, then
the significance of {\it each} signal is calculated separately:
$S_A/\Delta B_A$ and $S_{CP}/\Delta B_{CP}$.  These two quantities are
added in quadrature to determine the full significance.  This
algorithm is somewhat biased.  For the case when both signals overlap
(which coincides with the range of $\tan\beta$ where the experiments
might observe a signal), one must ``know'' that the best mass window
is a larger one.  It may be more realistic to choose the average Higgs
boson mass as the center of the mass window of size $\pm
2\sigma$.\footnote{In the case when the signals do not overlap (which
is not of practical interest since $\tan\beta$ must be so small that
the production cross section is highly suppressed), one should
consider background fluctuations over all reasonable mass windows that
do not overlap.  Given the experimental resolution, and the maximal
mass splitting possible, this is not a serious concern.}

%
\begin{figure}[ht!]
\vskip-2pc
\begin{center}
\centerline{\psfig{file=mssm/b4mhmax.eps,width=14cm}}
\end{center}
\vskip-2pc
\caption[0]{\label{b4max5} The $5\sigma$
discovery reach for the maximal-mixing benchmark scenario 
of table~\ref{benchmarks} for the
signature $q\bar q,gg\to b\bar b\phi\to b\bar b b\bar b$ for different total
integrated luminosities: (a) CDF analysis and  (b) D\O\ analysis.}
%\end{figure}
%
%\begin{figure}[h]
\begin{center}
\centerline{\psfig{file=mssm/b4mhmax95.eps,width=14cm}}
\end{center}
\vskip-2pc
\caption[0]{\label{b4max95} Same as \fig{b4max5}, but for 95\%
CL exclusion contours.}
\vskip-2pc
\end{figure}

\begin{figure}[htbp]
\vskip-2pc
\begin{center}
\centerline{\psfig{file=mssm/b4nomixing.eps,width=14cm}}
\end{center}
%\vskip-2pc
\caption[0]{\label{b4nomix5} Same as \fig{b4max5}, but for 
the no-mixing benchmark scenario of table~\ref{benchmarks}.}
%\end{figure}
%
%\begin{figure}[htbp]
\begin{center}
\centerline{\psfig{file=mssm/b4mixtev_final.eps,width=14cm}}
\end{center}
%\vskip-2pc
\caption[0]{\label{b4mix5} Same as \fig{b4max5}, but for the  
``suppressed $V \phi \to V b\bar b$ production'' benchmark scenario
of table~\ref{benchmarks}.}
\end{figure}

\Figs{b4max5}{b4max95} are
contour plots demonstrating regions of $5\sigma$ discovery and 95\%
C.L. exclusion for
the maximal mixing benchmark scenario. 
%of $M_S=1$~TeV and
%maximal squark mixing.
% with $M_2= - \mu = 200$ GeV and 
%$M_{\tilde{g}}= 1$ TeV.
Clearly, large values of $\tan\beta$, for which the CP-odd Higgs
boson and one of the CP-even Higgs bosons has enhanced couplings to
the bottom quarks, are necessary to render  
the  $b\bar b\phi$ channel experimentally viable.  In \fig{b4nomix5},
we show the parameter space coverage for $5\sigma$ discovery
in the no-mixing benchmark scenario.  Comparison with \fig{b4max5}
shows that the distinction between the maximal squark mixing and
no squark mixing cases is small.  This can be attributed to the fact
that in both cases, the
parameter $\mu$ was taken to be rather small; hence, the effects of 
the vertex corrections to the bottom Yukawa coupling are small. 
%for these two sets of MSSM parameters.
However,  this does not mean
that the $b\bar b\phi$ channel is, in general, 
robust under variation of the MSSM parameters.
Precisely at large values of $\tan\beta$, for which 
the  $b\bar b\phi$ channel is 
promising, the vertex corrections to the bottom Yukawa coupling may
become quite relevant. 
As explained in Section~I.C., depending 
on the sign and magnitude of $\Delta_b$ [\eq{deltamb}], 
large vertex corrections to the bottom Yukawa 
coupling may be induced  varying drastically the coverage  of the 
$\mha$--$\tan \beta$ plane through the 
$q\bar q,gg\to b\bar b\phi \to b\bar b b\bar b$ channel.
%
%Contours of 95\% CL exclusion are shown in \fig{b4mix95} for the case 
%of $A_t=-\mu=1.5$ TeV ($M_S$=1 TeV),
%which produced large holes in the $\mha$ {\it vs.} 
%$\tan\beta$ plane for the $q\bar q'\to V\phi$ analysis.
For example, in the suppressed $V\phi\to Vb\bar b$ benchmark 
scenario, since all the Higgs couplings to $b\bar b$ are generally
suppressed, the MSSM parameter coverage is reduced as shown in 
\fig{b4mix5}.

%
\begin{figure}[htbp]
\vskip-2pc
\begin{center}
\centerline{\psfig{file=mssm/b4mix_minus.eps,width=14cm}}
\end{center}
\caption[0]{\label{b4susy1}
The $5\sigma$ discovery reach for $b\bar b\phi$ production, for MSSM
parameters which induce radiative corrections that enhance the coupling
of the CP-odd Higgs and one of the CP-even Higgs to the bottom quarks:
(a) CDF analysis, (b) D\O\ analysis}
%\end{figure}
%
%\begin{figure}[h]
\begin{center}
\centerline{\psfig{file=mssm/b4mix_plus.eps,width=14cm}}
\end{center}
%\vskip-2pc
\caption[0]{\label{b4susy2}
The $5\sigma$ discovery reach for $b\bar b\phi$ production, for MSSM
parameters which induce radiative corrections that suppress the coupling
of the CP-odd Higgs and one of the CP-even Higgs to the bottom quarks:
(a) CDF analysis, (b) D\O\ analysis}
\end{figure}

As previously noted, 
the leading one-loop corrections to the bottom Yukawa coupling
suppress [enhance] the $b\bar b\phi$ coupling 
for $\Delta_b>0$
[$\Delta_b <0$], thus degrading [improving] 
the Higgs discovery potential in the 
$q\bar q,gg\to b\bar b\phi\to b\bar b b\bar b$ channel.
In Section~I.C.3, the dependence of $\Delta_b$ on the supersymmetric
parameters is exhibited [\eq{deltamb}] and its phenomenological
implications are discussed.  Here, we remind the reader that the sign
of $\Delta_b$ is governed by the sign of $\mu$ (in the convention
where $M_{\tilde{g}}$ is positive), as the dominant contribution to
$\Delta_b$ from bottom-squark loops is proportional to $M_{\tilde{g}}\mu$.
The top-squark loop contribution to $\Delta_b$ adds a sub-dominant
contribution proportional to $A_t\mu$, which increases [decreases] the
overall magnitude of $\Delta_b$ for $A_t>0$ [$A_t<0$].
The dependence of the parameter space coverage on the sign of $\mu$ is
illustrated in
\figs{b4susy1}{b4susy2}, which exhibit discovery contours for the
case  $A_t = |\mu|$ = 1 TeV, $M_S$= 1 TeV
and  $M_{\tilde{g}}=0.5$ TeV.
% which correspond to the two
%benchmark scenario denoted in Table
%\ref{benchmarks} by
%Suppressed $V \phi \to V b\bar b$ production. 
%
%``suppressed $\phi b\bar b$ couplings'' and 
%``enhanceded $Ab\bar b$ production''.
These plots illustrate the two possible choices 
for the sign of $\mu$, which
correspond to different signs for the loop corrections to the 
Higgs--bottom-quark
Yukawa coupling.  Moreover, since $A_t>0$, 
the leading one-loop corrections are further enhanced for
$\mu <0$, thus improving the Higgs discovery potential in the 
$q\bar q,gg\to b\bar b\phi\to b\bar b b\bar b$ channel.
Thus, we see that there can be strong dependence of the potential for
$q\bar q,gg\to b\bar b\phi\to b\bar b b\bar b$ discovery on
the specific choice of MSSM parameters.
Observation of a signal in this channel will clearly suggest that
$\tan \beta$ is large, and can place some useful constraints on the
allowed regions of MSSM parameter space.
%interpret in terms of the MSSM parameters of the model,
%unless further experimental information is available, 
%perhaps from the direct observation of  supersymmetric particles as 
%gluinos, charginos, top-- or bottom--squarks.



\subsubsection{MSSM Higgs Sector Summary}

We may combine the results for the various channels
obtained in the generic MSSM analysis  to provide summary plots
of the MSSM Higgs discovery reach of the upgraded Tevatron collider.
For comparison, we show in each case the expected LEP final
coverage of the $\mha$--$\tan\beta$
plane obtained from the search mode
$e^+ e ^- \to Z \phi$ with the subsequent decay of $\phi = \hl$ or
$\hh$ into $b\bar b$ or $\tau^+\tau^-$. 
Moreover, although not 
explicitly shown in the final combined plots presented here, 
the absence of a Higgs signal at LEP2 in the associated production process
$e^+ e ^- \to \ha \phi$ would rule out the parameter space region
defined by $\mha\lsim 95$~GeV and $\tan\beta\gsim 5$ as shown in
\fig{mh200} in the maximal mixing scenario.  
%coverage of the  $\mha$ vs $\tan \beta$ plane by LEP is possible 
%via the  production process $e^+ e ^- \to \ha \phi$.
The reach of the latter channel has not yet been presented in a model
independent way by the LEP collaborations.  Nevertheless, for the
three benchmark scenarios of table~\ref{benchmarks}, the excluded
region at low $\mha$ and large $\tan\beta$ will not be appreciably
different from the one quoted above.


\begin{figure}[hb!]
\centering
\centerline{\psfig{file=mssm/fullmhmax95_final.eps,width=8cm}
\hfill
\psfig{file=mssm/fullmhmax_final.eps,width=8cm}}
\vskip2pc
\caption[0]{\label{fullmhmax95} (a) 95$\%$ CL exclusion region
and (b) $5\sigma$ discovery region on the $\mha$--$\tan \beta$
plane, for the maximal mixing benchmark scenario of table~\ref{benchmarks}.  
and two different search channels:
$q\bar q\to V\phi$ [$\phi=\hl$, $\hh$], $\phi\to b\bar b$
(shaded regions) and 
$gg$, $q\bar q\to b\bar b\phi$ [$\phi=\hl$, $\hh$, $\ha$],
$\phi\to b\bar b$ (region in the upper left-hand corner bounded by the
solid lines).  Different integrated 
luminosities are explicitly shown by the color coding.
The two sets of lines (for a given color) 
correspond to the CDF and D\O\ simulations, 
respectively.  The region below the solid black line near the bottom
of the plot is excluded by the absence of observed $e^+e^-\to Z\phi$
events at LEP2.
}
\end{figure}

\begin{figure}[hp!]
\begin{center}
\centerline{\psfig{file=mssm/fullmixtev95_final.eps,width=8cm}
\hfill
\psfig{file=mssm/fullmixtev_final.eps,width=8cm}}
\end{center}
\caption[0]{\label{fullmix95}
The same as \fig{fullmhmax95} but for the ``suppressed 
$V \phi \to V b\bar b$ production'' benchmark scenario
of table~\ref{benchmarks}. 
}
%\end{figure}
%
           \vskip2pc
%
%\begin{figure}[ht]
\begin{center}
\centerline{\psfig{file=mssm/fullnomixing95_final.eps,width=8cm}
\hfill
\psfig{file=mssm/fullnomixing_final.eps,width=8cm}}
\end{center}
\caption[0]{\label{fullnomix95}
The same as \fig{fullmhmax95} but for the the no-mixing benchmark 
scenario of table~\ref{benchmarks}.  In (a), 
the region between the two solid black lines
is {\it not} excluded by the absence of observed $e^+e^-\to Z\phi$
events at LEP2, while in (b) the region unexcluded by LEP2 lies above
the solid black line.  
}
\end{figure}


In figs.~\ref{fullmhmax95}--\ref{fullnomix95}, 
we show the regions of 95$\%$ CL Higgs exclusion  
and $5\sigma$ Higgs discovery on the $\mha$--$\tan \beta$
plane, for representative MSSM parameter choices,
via the search of neutral Higgs bosons in the 
channels: $q\bar q\to V\phi$ [$\phi=\hl$, $\hh$], $\phi\to b\bar b$
(shaded regions) and 
$gg$, $q\bar q\to b\bar b\phi$ [$\phi=\hl$, $\hh$, $\ha$],
$\phi\to b\bar b$ (region in the upper left-hand corner bounded by
the solid lines), for different integrated 
luminosities as indicated by the color coding.
The two sets of lines (for a given color) bounding the regions accessible
by the $b\bar b\phi$ search correspond to the CDF and D\O\ simulations, 
respectively.  Finally, the solid black line (usually near the bottom
of each plot) reflects the upper limit of $\tan\beta$ (as a function
of $\mha$) deduced from the absence of observed $e^+e^-\to Z\phi$
events at LEP2.
The shaded regions presented in these figures reflects the results
of the SHW simulation of $q\bar q\to V\phi$
improved by neutral network techniques, as described in the generic
MSSM analysis presented in Section~III.B.2.  We do not display here
the effects of systematic error due to uncertainties in $b$-tagging 
efficiency, background, mass resolution and other effects.  The
implications of a 30\% reduction in detection efficiency has already
been illustrated by comparing lighter and darker color shading in
figs.~\ref{amaxnn}--\ref{aminnn}.

From our results, it follows that in the most demanding 
scenarios, (shown in \figs{fullmhmax95}{fullmix95}),
5~fb$^{-1}$ of integrated luminosity per experiment will allow one to
test nearly all of the MSSM Higgs parameter space at 95\% CL. 
Note the importance of the complementarity between the
$q\bar q\to V\phi$ and $q\bar q\to b\bar b\phi$ channels 
for improving the coverage of the
MSSM parameter space in the low $\mha$ region in \fig{fullmix95},
where there is a strong
reduction of ${\rm BR}({\phi \to b \bar b})$ for the  
CP-even Higgs boson $\phi$ with strongest coupling to the vector 
gauge bosons. 

To assure discovery of a CP-even Higgs boson at the 5$\sigma$ level,
the luminosity requirement becomes very important.
Figs.~\ref{fullmhmax95}(b), \ref{fullmix95}(b) and \ref{fullnomix95}(b) 
show that 
a total integrated luminosity of about 20~fb$^{-1}$ per experiment is 
necessary in order to assure a significant, although not exhaustive,
coverage of the MSSM parameter space. In general, we observe that
the complementarity between the two channels,  
$q\bar q\to V\phi$ and  $q\bar q\to b\bar b\phi$,   
is less effective in assuring discovery of a Higgs boson as compared
with a 95\% CL Higgs exclusion. 
This is due to 
the much higher requirement of total integrated  
luminosity combined with the existence of MSSM parameter regimes
which can independently suppress both Higgs production channels.
%
%effect of the 
%  relative signs   and magnitudes of
%$A_t \times\mu$ and $M_{\tilde{g}} \times \mu$ 
%on the production and decay of the search modes under study.
%The product $A_t \times\mu$ principally governs the strength of 
%the bottom  Yukawa coupling to the Higgs boson which has the 
%strongest couplings to the vector bosons, and hence it governs the 
% reach in the $q\bar q\to V\phi$ [$\phi=\hl$,
%$\hh$] channel. The product $M_{\tilde{g}} \times \mu$ principally 
%governs the reach in the $gg$, $q\bar q\to b\bar b\phi$ [
%$\phi=\hl$, $\hh$, $\ha$] channel.
%In the case of the maximal mixing scenario,   
%the sign of  $A_t\mu$ is defined to be negative, and this implies, 
%for $A_t$ of maximal mixing, that the  
%resulting coverage in the $\mha$ {\vs} $\tan\beta$ plane is
%somewhat enhanced  with respect to the opposite sign combination.
%The choice of $A_t\times \mu>0$ would lead to a slight   suppression
%of the bottom quark coupling to the Standard Model--like Higgs boson.
%The main effect of the reach of the Tevatron for this benchmark is,
%however,  due to the large values of the Higgs masses involved.
%On the other hand the sign of $M_{\tilde{g}} \times \mu$ is negative,
%and the main effect of this translates in a much better coverage in
%the  4b's final channel. Therefore, the complementarity between 
%the two channels works well in this case, even for the 
%$5 \sigma$ discovery reach.
%For the $-A_t = \mu =$ 1.5, $M_{\tilde{g}} = 1$ TeV 
%benchmark scenario, the sign and magnitude of  $A_t \times\mu$ was
%defined  to show the problematic region where the coupling of the
%relevant Higgs boson to bottom quarks is suppressed and coverage may
%become difficult. At the same time, the sign
%of  $M_{\tilde{g}} \times \mu$ is such that the couplings of 
%the relevant Higgs to bottom quarks in the 4b's final state
%are only moderately enhanced over the SM one, implying that the 
%coverage of the  $\mha$ {\vs} $\tan\beta$ plane due to this channel is
%not spectacular. For example, the opposite 
%sign of  $M_{\tilde{g}}$ will enhance the covered region  through the 
%4b's final state in a very relevant way. 
%
%JUST TO REMEMBER
% A similar change will occur if we leave the gluino mass positive
% defined and interchange the signs  of $\mu$ and $A_t$. The latter
% will still yield holes in the $V \phi$ channel,  due to suppressed
%values of the SM-like Higgs coupling to $b \bar b$, but the sign of
%$M_{\tilde{g}} \times \mu$ will be negative, therefore enlarging 
%the reach
%in the 4 b final state. In this case however, the gluino mass will be
%contrained to be below 300 GeV, or else a very large, non
%perturbative values of the bottom Yukawa coupling will occur for
%relatively low values of $\tan \beta$.
%
\Fig{fullmix95} exhibits one of the most difficult regions of
MSSM parameter space for Higgs searches at the Tevatron collider.
Nevertheless, even in this case,
a very high luminosity experiment can cover a significant fraction
of the available MSSM parameter space.

In conclusion, Run 2 of the Tevatron (with an integrated luminosity of
2~fb$^{-1}$) cannot significantly improve the MSSM Higgs mass limits
obtained at LEP.  However, if the integrated luminosity can be
increased to 10~fb$^{-1}$ and beyond, then substantial improvements
beyond the LEP reach are achievable.  Although 
the upgraded Tevatron cannot probe the entire MSSM Higgs parameter
space, integrated luminosities of 20--30~fb$^{-1}$ would provide close
to complete coverage for a 5$\sigma$ discovery of the Higgs boson.



%\Fig{combmax5} and {cpmbmax95} show the *******

%For minimal mixing?  Maximal mixing?  Have the exclusion regions already
%included on the plots and only comment on them now?  Do it earlier in
%the text?


%Note that the relevance of the also holds for signatures
%of charged Higgs bosons, though to a smaller extent, since the 
%dominant decay at
%large $\tan\beta$ is $H^+\to\tau^+\nu_\tau$ below the top decay threshold.
%This was first noted by Sola, as discussed in section (*theory*).




\subsection{New Higgs Physics Beyond the Standard Model/MSSM}

\vspace{0.1in}
\small
\begin{center}
{\it M.C. Gonzalez-Garcia, S.M. Lietti and S.F. Novaes}
\end{center}
\normalsize\nopagebreak

Effective Lagrangians provide a formalism for a model-independent
description of new TeV-scale physics beyond the Standard Model.
An effective Lagrangian for gauge boson--Higgs boson
interactions was presented in \eqs{lagrangian}{H}.  
%Such a Lagrangian parameterizes TeV-scale
%physics, whose nature is presently unknown, and allows for a
%model-independent study of new physics beyond the Standard Model.
A set of new effective Higgs couplings are thus introduced
that in principle can be measured in future
experiments.  In the Standard Model, these effective couplings are
generated at the loop level.  Thus, to discover physics beyond the Standard
Model, one would have to measure these couplings with sufficient
accuracy to detect deviations from Standard Model predictions.

In this section, we discuss the present limits on anomalous Higgs
couplings and evaluate the improved limits attainable from searches at
the upgraded Tevatron.  Since the Higgs boson in the model-independent
approach is not strictly the Standard Model Higgs boson (although the
deviation from Standard Model behavior may be small), we shall
henceforth denote the Higgs boson by $H$ for the remainder of this
section. 

\vskip6pt\noindent
{\bf {Present Bounds from Searches at Tevatron Run 1 and LEP2}}

\vskip3pt
Anomalous Higgs boson couplings have been studied in Higgs and
$Z$ boson decays in $e^+ e^-$
\cite{ee,our:lep2aaa,our:NLCWWA,our:NLCZZA} and in $p \bar{p}$
collisions
\cite{our:tevatronjj,our:tevatronmis,our:tevatron3a,our:combined}.
The combined bounds on anomalous Higgs
boson interactions are derived from the following data:
\begin{equation}
\begin{array}{lll}
~~~ \mbox{Process  } &
~~~~~ \mbox{Anomalous Higgs Contribution}  &  ~~~ \mbox{Exp.\ Search}\\
&&\\
p\, \bar p  \rightarrow j \, j \, \gamma\, \gamma &  ~~~~~
p \bar{p} \to  W (Z) (\to j\, j) + H (\to \gamma \gamma) & ~~~~~
\mbox{D\O \protect\cite{d0jj}}
\\
p\, \bar p \rightarrow  \gamma \,\gamma \,+ \,\not \!\! E_T \nonumber &~~~~~
p \bar{p} \to  Z^0 (\to \nu \bar{\nu}) + H (\to \gamma \gamma) & ~~~~~
\mbox{D\O\protect\cite{d0miss}}  \\
& ~~~~~ p \bar{p} \to  W (\to [\ell] \nu) + H (\to \gamma \gamma) & ~~~~~
\nonumber \\
p\, \bar p  \rightarrow  \gamma\, \gamma\, \gamma & ~~~~~
p \bar p \rightarrow \gamma +H(\rightarrow \gamma \gamma) & ~~~~~
\mbox{CDF\protect\cite{cdf}} \\
e^+\, e^-  \rightarrow \gamma\, \gamma\, \gamma & ~~~~~
e^+ e^-  \rightarrow \gamma +H(\rightarrow \gamma \gamma) & ~~~~~
\mbox{OPAL  \protect\cite{opal:ggg} }
\\
\label{proc}
\end{array}
\end{equation}
Events containing two photons plus missing transverse energy 
($\not \!\! E_T$), additional photons or charged fermions represent 
a signature for several theories involving physics beyond the SM.
% and they have
%been extensively searched for \cite{d0jj,d0miss,cdf,opal:ggg}.
In the framework of anomalous Higgs couplings presented 
in \eqs{lagrangian}{H}, they
can arise from the production of a Higgs boson which
subsequently decays in two photons [second column in
Eq.(\ref{proc})].  
%In the SM, the  decay width $H \to \gamma
%\gamma$ is very small since it occurs just  at one--loop level
%but  the existence of the new interactions (\ref{H})  can enhance
%this width in a significant way. 
Recent analyses of these
signatures showed a good agreement with  the expectations from
the SM. Thus, these negative experimental  results can be used
to constrain new anomalous couplings in the bosonic sector of the
SM.
\begin{figure}
\centering
\centerline{\psfig{file=mssm/fig01_new.ps,width=8cm}
\hfill
\psfig{file=mssm/fig02_new.ps,width=8cm}}
%\centerline{\epsfig{file=fig01_new.ps,width=0.5\textwidth}
%            \epsfig{file=fig02_new.ps,width=0.5\textwidth}}
\caption[0]{\label{conchafig1} (a) Exclusion region outside the curves in the
$f_{BB} \times f_{WW}$ plane, in TeV$^{-2}$, based  on the D\O
~analysis \protect\cite{d0jj} of $\gamma\gamma j j$ production,
on  the D\O ~analysis \protect\cite{d0miss} of $\gamma\gamma \not
\!\! E_T$, on the CDF analysis \protect\cite{cdf} of
$\gamma\gamma\gamma$ production, and on the OPAL analysis
\protect\cite{opal:ggg} of $\gamma\gamma\gamma$ production,
always assuming $ M_H = 100$ GeV.  The curves show the 95\% CL
deviations from the SM total cross section. (b) Same as
(a) for the combined analysis.}
\end{figure}

%All processes listed in (\ref{proc}) have been the object of
%direct experimental searches.

D\O\ has searched for high invariant--mass 
photon pairs in $p \bar{p} \to \gamma
\gamma j j$ events \cite{d0jj} at $\sqrt{s}=1.8$ TeV and $100$
pb$^{-1}$ of integrated luminosity.
No events were observed with two--photon invariant mass  
in the range $100 < M_{\gamma\gamma}\lesssim 220$,
which constrains the anomalous Higgs boson couplings defined in
\eq{H} that lead to the process $p\bar{p} \to
W (Z) (\to j\, j)  + H (\to \gamma \gamma)$ \cite{our:tevatronjj}.
In the analysis of \Ref{concha},
the cuts of \Ref{d0jj} were applied and included the particle
identification and  trigger efficiencies. The searches
discussed here are limited to a
Higgs boson with mass in the range $100 < M_H \lesssim 220$,
since after the $WW(ZZ)$ threshold is reached the diphoton
branching ratio of Higgs is quite reduced.

D\O\ has also searched for events with two photons and large missing
transverse energy \cite{d0miss}.  No events were observed in the invariant
mass range $100 < M_{\gamma\gamma} \lesssim 2 M_W$.  This can
be used to constrain operators leading to the processes
$p\bar p\to Z(\to\nu\bar\nu) H(\to\gamma\gamma)$ {\it and}
$p\bar p\to W(\to\ell\nu) H(\to\gamma\gamma)$, where the
lepton is not detected \cite{our:tevatronmis}.
To compare with the D\O\ results 
\cite{d0miss}, the same cuts as in the last article in
\Ref{d0miss} were applied,
including the particle identification and trigger efficiencies
which vary from 40\% to 70\% per photon.
After these cuts, 80\% to 90\%
of the signal comes from associated $ZH$ production while
10\% to 20\% arises from $WH$ production. 

CDF has searched for events with three photons, requiring two photons in the
central region of the detector, with a minimum transverse energy
of 12 GeV, plus an additional photon with $E_T > 25$ GeV, and
a photon separation of at least $15^\circ$ \cite{cdf}.  These
results have been used to constrain operators leading to
$p\bar p\to \gamma H(\to\gamma\gamma)$.
These same operators can be constrained by results 
for electron--positron collisions \cite{our:lep2aaa}, in particular
the recent OPAL results \cite{opal:ggg}, where data
taken at several energy points in the range $\sqrt{s}=130$--$172$ GeV 
were combined.

The calculations have included all SM (QCD plus
electroweak), and anomalous contributions that lead to these
final states. The SM one-loop contributions to the
$H\gamma\gamma$ and $H Z\gamma$ vertices were introduced through
the use of the  effective operators with the corresponding form
factors in the coupling.  Neither the narrow--width approximation
for the Higgs boson  contributions, nor the effective $W$ boson
approximation were employed. The interference effect
between the anomalous signature and the SM
background have been included consistently. 
For all $p\,\bar p$ analyses, the  MRS (G)
\cite{mrs} set of proton structure functions were employed with the scale
$Q^2=\hat{s}$.

\begin{figure}
\centerline{\psfig{file=mssm/kappa.ps,width=8cm}}
%\centerline{\epsfig{file=kappa.ps,width=0.5\textwidth}}
\caption{Excluded region in the $f \times M_H$
plane from the combined analysis of the LEP2 and Tevatron
searches.}
\label{kappa}
\end{figure}

The coupling $H\gamma\gamma$ [see \eq{g}] involves $f_{WW}$ and
$f_{BB}$ \cite{hagiwara2}.  As a consequence, the anomalous
signature $f\bar f \gamma\gamma$ is only possible when those
couplings are not vanishing. The couplings $f_B$ and $f_W$, on
the other hand, affect the production mechanisms for the Higgs
boson.  \Fig{conchafig1} (a) shows the
excluded  region in the $f_{WW}$, $f_{BB}$ plane from the
different channels  studied for $M_H = 100$ GeV, assuming that
these are the only non--vanishing couplings. Since the anomalous
contribution to  $H\gamma\gamma$ is zero for $f_{BB} = - f_{WW}$
(see \eq{g}), the bounds become very weak close to this
line, as is clearly shown in \fig{conchafig1}. 
Note that these bounds are based
%, we imposed an upper limit on the number
%of signal events  based 
on Poisson statistics. 
%In the absence of
%background  this implies $N_{\mbox{signal}} < 1 \,(3)$ at 64\%
%(95\%) CL. In the presence of background events, we employed the
%modified Poisson analysis.

The results obtained from the analysis of the four reactions
(\eq{proc}) can be statistically  combined in order to constrain
the value of the coefficients $f_i$, $i=WW, BB,W, B$ of
(\eq{lagrangian}) \cite{our:combined}. 
\Fig{conchafig1} (b) exhibits the 95\% CL exclusion region in the plane $f_{BB}
\times f_{WW}$ obtained from combined results.

In order to reduce the number of free parameters, one can make the
assumption that all blind operators affecting the Higgs
interactions have a common coupling $f$.
%, {\it i.e.} $f = f_W =
%f_B = f_{WW} = f_{BB}$ \cite{hisz,hagiwara2}.
\Fig{kappa} shows the combined limits for the coupling
constant $f = f_{BB}= f_{WW}= f_{B} = f_{W}$ for Higgs boson masses
in the range of $100 \leq M_H \leq 220$ GeV.

\vskip6pt\noindent
{\bf {Attainable Bounds at the Upgraded Tevatron}}

\vskip3pt
The effect of the anomalous operators becomes more evident with
the increase of energy, and therefore, higher sensitive to
smaller values of the anomalous coefficients can be achieved by
studying their contribution to different processes at the
upgraded Tevatron collider. 
Estimates for the upgraded Tevatron, 
with 1~fb$^{-1}$ and 10 fb$^{-1}$ respectively are considered here. 
For the reactions $p \bar{p} \to\gamma \gamma \not
\!\! E_T$ and $p \bar{p} \to \gamma \gamma j j$, Run 1 cuts
and efficiencies are used.
For the $\gamma\gamma\gamma$ final state, the effects of
additional kinematic cuts to
improve the sensitivity have been considered \cite{our:tevatron3a}.
The best results in this case are obtained for the following set of cuts:
$E_{T_{1}} > 40$ GeV, with $E_{T_{2,3}} > 12$ GeV, where 
the three photons are ordered according to their transverse energy,
{\it i.e.\/} $E_{T_{1}} > E_{T_{2}} > E_{T_{3}}$, and
$|\eta_{i}| < 1$.
For these estimates, the CDF Run 1 detection \cite{cdf}
efficiency for photons has been assumed.


\begin{table*} [h!]
\caption{95\% CL allowed range for $f/\Lambda^2$,  from
$\gamma\gamma\gamma$, $\gamma\gamma + E_T $,  $\gamma\gamma
j j $ production at the upgraded Tevatron 
with an integrated luminosity of 1~fb$^{-1}$ [10~fb$^{-1}$]
assuming all $f_i$
to be equal.  Limits worse than $|f|=100$
TeV$^-2$ are denoted by ---.}
\label{tab:run2}
\vskip6pt
\renewcommand{\arraystretch}{1.1}
%\setlength{\tabcolsep}{9pt}
\begin{tabular}{ccccc}
  &\multicolumn{4}{c}{$f/\Lambda^2$(TeV$^{-2}$)} \\ \cline{2-5}
\noalign{\vskip3pt}
$\;\;M_H$(GeV)\qquad & $p \bar{p} \to \gamma \gamma \gamma$  & $p \bar{p}
  \to \gamma \gamma +  E_T $  &
  $p \bar{p} \to \gamma \gamma j j$  & Combined\\[3pt]
\tableline
\noalign{\vskip3pt}
100     & ($-$24, 24)  [$-$13, 15]
        & ($-$16, 36)  [$-$9.4, 26]
        & ($-$9.2, 22) [$-$3.3, 5.6]
        & ($-$7.6, 19) [$-$3.0, 5.6]
\\
120     & ($-$26, 26)  [$-$14, 14]
        & ($-$20, 39)  [$-$15, 27]
        & ($-$8.6, 21) [$-$3.4, 5.9]
        & ($-$7.4, 18) [$-$3.3, 5.9]
\\
140     & ($-$30, 31)  [$-$15, 16]
        & ($-$25, 44)  [$-$14, 30]
        & ($-$10, 23)  [$-$4.5, 8.9]
        & ($-$9.1, 20) [$-$4.0, 8.7]
\\
160     & ($-$36, 38)  [$-$17, 19]
        & ($-$29, 50)  [$-$14, 33]
        & ($-$11, 24)  [$-$6.0, 14]
        & ($-$9.9,22)  [$-$5.1, 13]
\\
180     & (---, ---)  [---, ---]
        & ($-$63, 72) [$-$46, 53]
        & ($-$26, 34) [$-$16, 24]
        & ($-$24, 33) [$-$16, 24]  \\
200     & (---, ---)  [---, ---]
        & ($-$87, 90) [$-$50, 53]
        & ($-$33, 40) [$-$17, 23]
        & ($-$32, 39) [$-$17, 23] \\
220     & (---, ---)  [---, ---]
        & (---, ---)  [---, ---]
        & ($-$42, 45) [$-$19, 26]
        & ($-$42, 45) [$-$19, 26] \\[3pt]
\end{tabular}
\end{table*}

Table \ref{tab:run2} contains the 95\% CL  limit on the
anomalous couplings for the upgraded Tevatron 
for each individual process. All couplings are assumed equal  ($f =
f_{BB}= f_{WW}= f_{B} = f_{W}$) and the Higgs boson mass is
varied in the range $100 \leq M_H \leq 220$ GeV. Combination
of the results obtained from the analysis of the first three
reactions in \eq{proc} leads to the improved bounds given
in the last column of   Table \ref{tab:run2}. Comparing these
results with those in \fig{kappa}, we observe an improvement
of about a factor $\sim$ 2--3 [$\sim$ 4--6] for the combined
limits at the upgraded Tevatron with 1~fb$^{-1}$ [10~fb$^{-1}$]  
integrated luminosity.

As mentioned previously for linearly realized effective Lagrangians,
the modifications introduced in the Higgs boson and in the vector
boson sectors are related to each other.  Consequently, the
bounds on the new Higgs couplings should also restrict the
anomalous gauge--boson self interactions.  Under the assumption
of equal coefficients for all anomalous Higgs operators, 
the common Higgs boson anomalous coupling $f$ can
be related to the
conventional parameterization of the vertex $WWV$ ($V = Z^0$,
$\gamma$) \cite{hhpz},
\begin{equation}
\Delta \kappa_\gamma =\frac{M_W^2}{\Lambda^2}~ f =
\frac{2 \cos^2 \theta_W }{1 - 2 \sin^2 \theta_W}~
\Delta \kappa_Z= 2 \cos^2 \theta_W ~\Delta g^Z_1
\label{trad}
\end{equation}
A different set of parameters has been also used by the LEP
Collaborations in terms of  three independent couplings,
$\alpha_{B \Phi}$, $\alpha_{W \Phi}$, and $\alpha_{W}$, These
parameters are related to the parameterization of Ref.\
\cite{hhpz} through $\alpha_{B \Phi} \equiv \Delta \kappa_\gamma
- \Delta g^Z_1 \cos^2 \theta_W$, $\alpha_{W \Phi} \equiv \Delta
g^Z_1 \cos^2 \theta_W$, $\alpha_{W} \equiv \lambda_\gamma$.
Under  the relations given in \eq{trad} they verify
$\alpha_{B \Phi}=\alpha_{W \Phi}\equiv \alpha= \Delta \kappa_\gamma/2$.

The current experimental limit on these couplings
from combined results on double gauge
boson production at Tevatron and LEP2 \cite{vancouver}
is $-0.15\;<\;\Delta\kappa_\gamma\;=\;2\alpha \;<\;0.41$
at 95 \% CL. This limit is derived under the relations given in
\eq{trad} \cite{hisz} and it is plotted in \fig{kappa}
for comparison with the bounds obtained from this analysis.

Table \ref{tab:f5} contains the 95\% CL limit on the
anomalous coupling $\Delta\kappa_\gamma$ using the limits on
$f/\Lambda^2$ obtained  through the analysis of the processes
in \eq{proc}.  The expected bounds that are attainable
at the upgraded Tevatron are also shown. These 
results show that the
present combined limit from the Higgs production analysis
obtained here is comparable with the existing bound
from gauge boson production for $M_H \leq 170$ GeV.
\begin{table*}[t!]
\caption{95\% CL allowed range for the anomalous triple gauge boson
couplings derived from the limits obtained for the anomalous Higgs boson
coupling $f$ for $M_H= 100$ GeV.}
\label{tab:f5}
\vskip6pt
\setlength{\tabcolsep}{1.85pc}
\begin{tabular}{lc}
\multicolumn{1}{c}{Process}  & $ \Delta\kappa_\gamma = 2 \,\alpha= 2 \,\alpha_{B \Phi} =
2 \,\alpha_{W \Phi}$ \\ [3pt]
\tableline
$\;$Combined Tevatron Run 1 + LEP2         & ($-$0.084, 0.204) \\[0.1cm]
$\;$Combined Tevatron Run 2 (1 fb$^{-1}$)  & ($-$0.048, 0.122) \\ [0.1cm]
$\;$Combined Tevatron Run 2 (10 fb$^{-1}$) & ($-$0.020, 0.036) \\ [3pt]
\end{tabular}
\end{table*}

%Summarizing, we have estimated the limits on anomalous
%dimension--six Higgs boson interactions that can be derived from
%the investigation of three photon events at LEP2 and Tevatron and
%diphoton plus missing transverse energy events or dijets at
%Tevatron. Under the assumption that the coefficients of the four
%``blind'' effective operators contributing to Higgs--vector boson
%couplings are of the same magnitude, the study can give rise to a
%significant indirect limit on anomalous $WWV$ couplings.  We have
%also studied the expected improvement on the sensitivity  to
%Higgs anomalous couplings at the  Fermilab Tevatron upgrades.

















