% ----------- collect new commands here ------------------
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\makeatletter
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{IJCGA warning: optional citation argument
ignored: `#2'} \fi}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% are handled.  No sorting is done.  E.g., 1,3,2,3,4,5,foo,1,2,3,?,4,5
% gives 1,3,2-5,foo,1-3,?,4,5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \citenum emits the plain citation number without ornament
% \citea puts it's argument into the ornamentation for citations
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%\def\citenum#1{{\def\@cite##1##2{##1}\cite{#1}}}
%\def\citea#1{\@cite{#1}{}}

% Collapse citation numbers to ranges.  Non-numeric and undefined labels
% are handled.  No sorting is done.  E.g., 1,3,2,3,4,5,foo,1,2,3,?,4,5
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\def\mpl{M_{\rm PL}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\crr{\crcr\noalign{\vskip .1in}}
%\newenvironment{Eqnarray}%
%     {\arraycolsep 0.14em\begin{eqnarray}}{\end{eqnarray}}
\newcommand{\mathbold}[1]{\mbox{\boldmath $\bf#1$}}
\def\beqa{\begin{eqalignno}}
\def\eeqa{\end{eqalignno}}
\def\beqno{\begin{eqalignno}}
\def\eeqno{\end{eqalignno}}
\def\ifmath#1{\relax\ifmmode #1\else $#1$\fi}
\def\half{\ifmath{{\textstyle{1 \over 2}}}}
\def\lsim{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\gsim{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\eq#1{eq.~(\ref{#1})}
\def\fig#1{fig.~\ref{#1}}
\def\Fig#1{Fig.~\ref{#1}}
\def\figs#1#2{figs.~\ref{#1} and \ref{#2}}
\def\Figs#1#2{Figs.~\ref{#1} and \ref{#2}}
\def\Ref#1{ref.~\cite{#1}}
\def\Refs#1#2{refs.~\cite{#1} and \cite{#2}}
\def\Rref#1{Ref.~\cite{#1}}
\def\Rrefs#1#2{Refs.~\cite{#1} and \cite{#2}}
\def\eqs#1#2{eqs.~(\ref{#1})--(\ref{#2})}
\def\Eq#1{Eq.~(\ref{#1})}
\def\Eqs#1#2{Eqs.~(\ref{#1})--(\ref{#2})}
\def\eqns#1#2{eqs.~(\ref{#1}) and (\ref{#2})}
\def\tanb{\tan\beta}
\def\sinb{\sin\beta}
\def\cosb{\cos\beta}
\def\sina{\sin\alpha}
\def\cosa{\cos\alpha}
\def\sinbma{\sin(\beta-\alpha)}
\def\cosbma{\cos(\beta-\alpha)}
\def\sinbmaii{\sin^2(\beta-\alpha)}
\def\cosbmaii{\cos^2(\beta-\alpha)}
\def\hsm{h_{\rm SM}}
\def\mhsm{m_{h_{\rm SM}}}
\def\hl{h}
\def\ha{A}
\def\hh{H}
\def\hpm{H^\pm}
\def\mha{m_{\ha}}
\def\mhl{m_{\hl}}
\def\mhh{m_{\hh}}
\def\mhpm{m_{\hpm}}
\def\mhmax{m_h^{\rm max}}
\def\mz{m_Z}
\def\mw{m_W}
\def\mww{m_W^2}
\def\mzz{m_Z^2}
\def\mt{m_t}
\def\mb{m_b}
\def\mstopa{M_{\widetilde t_1}}
\def\mstopb{M_{\widetilde t_2}}
\def\msusy{M_{\rm S}}
\def\msusyy{M_{\rm S}^2}
\def\MSUSY{M_{\rm SUSY}}
\def\MSUSYY{M^2_{\rm SUSY}}
\def\mgut{M_{\rm X}}
\def\ie{{\it i.e.}}
\def\eg{{\it e.g.}}
\def\etc{{\it etc.}}
\def\vs{{\it vs.}}
\def\opcit{{\it op.~cit.}}
\def\SM{Standard Model}
\def\phm{\phantom{-}}
%% "low subscript" %%
\def\ls#1{\ifmath{_{\lower1.5pt\hbox{$\scriptstyle #1$}}}}
%%
\def\app#1#2#3{{\sl Act. Phys. Pol. }{\bf B#1} (#2) #3}
\def\apa#1#2#3{{\sl Act. Phys. Austr.}{\bf #1} (#2) #3}
\def\ppnp#1#2#3{{\sl Prog. Part. Nucl. Phys. }{\bf #1} (#2) #3}
\def\npb#1#2#3{{\sl Nucl. Phys. }{\bf B#1} (#2) #3}
\def\jpa#1#2#3{{\sl J. Phys. }{\bf A#1} (#2) #3}
\def\plb#1#2#3{{\sl Phys. Lett. }{\bf B#1} (#2) #3}
\def\prd#1#2#3{{\sl Phys. Rev. }{\bf D#1} (#2) #3}
\def\pR#1#2#3{{\sl Phys. Rev. }{\bf #1} (#2) #3}
\def\prl#1#2#3{{\sl Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\sl Phys. Reports }{\bf #1} (#2) #3}
\def\cpc#1#2#3{{\sl Comp. Phys. Commun. }{\bf #1} (#2) #3}
\def\nim#1#2#3{{\sl Nucl. Inst. Meth. }{\bf #1} (#2) #3}
\def\pr#1#2#3{{\sl Phys. Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\sl Sov. J. Nucl. Phys. }{\bf #1} (#2) #3}
\def\jl#1#2#3{{\sl JETP Lett. }{\bf #1} (#2) #3}
\def\jet#1#2#3{{\sl JETP Lett. }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\sl Z. Phys. }{\bf C#1} (#2) #3}
\def\ptp#1#2#3{{\sl Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\sl Nouvo~Cim.~}{\bf#1A} (#2) #3}
\def\hpa#1#2#3{{\sl Helv.~Phys.~Acta~}{\bf #1} (#2) #3}
\def\aop#1#2#3{{\sl Ann.~of~Phys.~}{\bf #1} (#2) #3}
\def\fP#1#2#3{{\sl Fortschr.~Phys.~}{\bf #1} (#2) #3}
\def\9{\phantom 0}     %%  for lining up numbers in table columns

% --------------------------------------------------------

\subsection{Introduction: the Quest for the Origin of Electroweak
Symmetry Breaking}

With the discovery of the top quark at the Tevatron \cite{Tevatron},
the Standard Model of particle physics appears close to final
experimental verification.  Ten years of precision measurements of
electroweak observables at LEP, SLC and the Tevatron have failed to
find any definitive departures from Standard Model predictions
\cite{ELW,Arno,strom}.  In some cases, theoretical predictions have been
checked with an accuracy of one part in a thousand or better.  The
consistency of these calculations is evidenced by the excellent
agreement between the value of the top quark mass measured directly at
the Tevatron, and the corresponding value deduced from precisely
measured electroweak observables at LEP and SLC that are sensitive to
top-quark loop radiative corrections.

Although the global analysis of electroweak
observables provides a superb fit to the Standard Model predictions,
there is still no {\it direct}
experimental evidence for the underlying dynamics responsible for
electroweak symmetry breaking.  The observed masses of the $W^\pm$
and $Z$ bosons can be understood as a consequence
%(as well as the masses for quarks and charged leptons)
%imply the existence
of three Goldstone bosons ($G^\pm$ and $G^0$) that
end up as the longitudinal components of the gauge bosons.
%still demands an explanation.  The masses of the $W^\pm$
%and the $Z$ must be a consequence of the spontaneous
%breaking of the SU(2)$\times$U(1) electroweak symmetry.
%In particular, the gauge boson masses arise when
%the three Goldstone bosons ($G^\pm$ and
%$G^0$) that are  generated are absorbed by the $W^\pm$ and $Z$.
However, the origin of the
Goldstone bosons still demands an explanation.  The electroweak symmetry
breaking dynamics that is employed by the Standard Model posits a
self-interacting complex doublet of scalar fields, which consists of four
real degrees of freedom.  Renormalizable interactions are
arranged in such a way that the neutral component of the scalar doublet
acquires a vacuum expectation value, $v=246$~GeV, which sets the scale
of electroweak symmetry breaking.
%(since the minimum of the scalar potential is not at zero scalar
%field).
Consequently, three massless Goldstone bosons are generated, while
%which become the longitudinal components
%of the $W^\pm$ and $Z$ as a result of the gauge-invariant interactions
%of the scalars to the vector bosons.
the fourth scalar degree of freedom that remains in the physical spectrum
is the CP-even neutral Higgs boson of the Standard Model.
It is further assumed in the SM that the scalar doublet also couples to fermions through
Yukawa interactions.  After electroweak symmetry breaking, these interactions
are responsible for the generation of quark and charged
lepton masses.

The self-interacting scalar field is only one  model of
electroweak symmetry breaking; other approaches,
based on very different dynamics, are also possible.
For example, one can introduce new fermions and new dynamics
({\it i.e.}, new forces), in
which the Goldstone bosons are a consequence of the strong
binding of the new fermion fields \cite{techni}.  Present
experimental data are not sufficient to identify with certainty the
nature of the dynamics responsible for electroweak symmetry breaking.
The quest to understand electroweak symmetry breaking requires continued
experimentation at present and future
colliders: the upgraded Tevatron, the
LHC and proposed lepton colliders under development.

As described above, the Standard Model is clearly a very good
approximation to the physics of elementary particles and their
interactions at an energy scale of ${\cal O}(100)$~GeV and below.  However,
theoretical considerations teach us that the Standard Model is not the
ultimate theory of the fundamental particles and their interactions.
At an energy scale above the Planck scale, $\mpl\simeq 10^{19}$~GeV, quantum
gravitational effects become significant
and the Standard Model must
be replaced by a more fundamental theory that incorporates
gravity.\footnote{Similar conclusions
also apply to recently proposed extra-dimensional theories in which
quantum gravitational effects can become
significant at energies scales as low as ${\cal{O}}$(1 TeV) \cite{ED}.}
It is also possible that the Standard Model breaks down at some energy scale
(called $\Lambda$) below the Planck scale.
In this case, the Standard Model degrees of freedom are no longer
adequate for describing the theory above $\Lambda$ and new physics
must become relevant.  One possible signal of this occurrence lies in
the behavior of the Standard Model couplings.
The Standard Model is not an
asymptotically free theory since some of the couplings ({\it e.g.}, the
U(1) gauge coupling, the Higgs--top-quark Yukawa coupling, and the Higgs
self-coupling) eventually blow up at some high energy scale.   Among
these couplings, only the Higgs self-coupling may blow up at an energy
scale below $\mpl$.  Of course, there may be other
experimental or theoretical hints that
new degrees of freedom exist at some high energy scale below $\mpl$.
For example, the recent experimental evidence for neutrino
masses of order $10^{-2}$~eV or below cannot be strictly explained in
the Standard Model.  Yet, one can easily write down a dimension-5
operator that is suppressed by $v/\Lambda$, which is responsible for
neutrino masses.  If $m_\nu=10^{-2}$~eV, then one obtains as a rough estimate
$\Lambda\sim 10^{15}$~GeV.

%On the theoretical side, the unification
%of gauge couplings (in models of low-energy supersymmetry) suggest the
%existence of a grand unification scale of order $10^{16}$~GeV.

It is clear from the above discussion that
the Standard Model is not a {\it fundamental} theory;
at best, it is an {\it effective field theory}~\cite{EFT}.
At an energy scale below $\Lambda$, the Standard Model (with
higher-dimension operators to parameterize the physics generated at
$\Lambda$) provides an extremely good description of all observable
phenomena.  Therefore, an essential question that future experiments
must address is: what is the minimum scale $\Lambda$ at which new
physics beyond the Standard Model must enter?
The search for the origin of electroweak symmetry breaking and the quest
to identify $\Lambda$ are intimately tied together.  We can consider two
scenarios.  In the first scenario, electroweak symmetry breaking
dynamics results in the existence of a single Higgs boson as posited by
the Standard Model.  In this case, one would ask whether new
phenomena beyond the Standard Model must enter at an energy scale $\Lambda$
that is accessible to experiment.  In the second scenario, electroweak
symmetry breaking dynamics does not result in a weakly-coupled Higgs
boson as assumed in the Standard Model.  In this case, the effective
theory that describes current data is a theory that contains the
Standard Model fields {\it excluding} the
Higgs boson.  In such an approach, the latter effective field theory
{\it must} break down at $\Lambda\simeq{\cal O}(1~{\rm TeV})$
in order to restore the unitarity of the theory, and new
physics associated with the electroweak symmetry breaking dynamics must
enter.

Although current data provides no {\it direct} evidence to distinguish
between the two scenarios just described, there is
indirect evidence that could be interpreted as favoring the first
approach.  Namely, the global Standard Model fit to electroweak data
takes the Higgs mass as a variable to be fitted.  The results of the LEP
Electroweak Working Group analysis yields \cite{Arno}:
\beq
\mhsm=67^{+60}_{-33}~{\rm GeV}\,.
\eeq
In fact, direct searches at LEP show no evidence for the Higgs boson,
and imply that $\mhsm> 107.9$~GeV \cite{LEPHiggs}
at the 95\% CL.\footnote{Preliminary
results from the 2000 LEP run show no clear evidence of the Higgs
boson, with a corresponding Higgs mass lower limit of 113.2~GeV (at
95\% CL) \cite{LEPHiggsprelim,tomjunk}.}
Thus, it probably is more useful
to quote the 95\% CL upper limit that is obtained in the global Standard
Model fit \cite{Arno}:~\footnote{The 95\% CL upper limit can change
by as much as $\sim 20$~GeV depending on the analysis.  
More recent upper Higgs mass limits, ranging between 170~GeV and 210~GeV, were
presented at the ICHEP 2000 meeting in Osaka, Japan and reviewed
in \Ref{strom}.}
\beq \label{smhiggslim}
\mhsm<188~{\rm GeV}\quad {\rm at}~95\%~{\rm CL}\,.
\eeq
These results reflect the logarithmic sensitivity to the Higgs mass via
the virtual Higgs loop contributions to the various electroweak
observables.  The Higgs mass range above is consistent
with a weakly-coupled Higgs scalar that is expected to emerge from the
Standard Model scalar dynamics (although the Standard Model does not
predict the mass of the Higgs boson; rather it relates it to the
strength of the scalar self-coupling).

Henceforth, we shall take the above result as an indication that
the Standard Model (with a weakly-coupled Higgs boson as suggested
above) is the appropriate effective field theory at the 100~GeV scale.
If this is the case, then the eventual discovery of the Higgs boson will
have a profound effect on the determination of $\Lambda$, the scale at
which the Standard Model must break down.  The key parameter for
constraining $\Lambda$
is the Higgs mass, $\mhsm$.  If $\mhsm$ is too large, then the
Higgs self-coupling blows up at some scale $\Lambda$ below the
Planck scale \cite{hambye}.  If $\mhsm$ is too small, then the Higgs
potential develops
a second (global) minimum at a large value of the scalar field of order
$\Lambda$ \cite{quiros}.  Thus new physics must enter at a scale
$\Lambda$
or below in order that the true minimum of the theory correspond to the
observed SU(2)$\times$U(1) broken vacuum with $v=246$~GeV.
Thus, given a value of
$\Lambda$, one can compute the minimum and maximum Higgs mass allowed.
The results of this computation (with shaded bands indicating the
theoretical uncertainty of the result) are illustrated in \fig{trivial}.

\begin{figure}
  \begin{center}
\centerline{\psfig{file=theory/triviality.ps,width=9cm}}
  \end{center}
  \caption[0]{\label{trivial} The lower \protect\cite{quiros} and the
upper \protect\cite{hambye} Higgs mass bounds as a function of the
energy scale $\Lambda$ at which the Standard Model breaks down,
assuming $M_t=175$~GeV and $\alpha_s(m_Z)=0.118$.  The shaded
areas above reflect the theoretical uncertainties in the
calculations of the Higgs mass bounds.  This figure is taken from
\protect\Ref{Riesselmann}.}
\end{figure}



Three Higgs boson mass ranges are of particular
interest:
\begin{enumerate}
\item 110~GeV~$\lsim \mhsm\lsim$~130~GeV
\item 130~GeV~$\lsim \mhsm\lsim$~180~GeV
\item 180~GeV~$\lsim \mhsm\lsim$~190~GeV
\end{enumerate}
In mass range 1, the Higgs boson mass lies above the present
direct LEP search limit.
Moreover, if the Higgs boson were discovered in this range, then
\fig{trivial} would imply that $\Lambda<\mpl$.  Finally, as we shall explain
in Section~I.C, this is the mass range expected in the minimal
supersymmetric model. Mass range 2 corresponds to a range of Higgs
masses which would be consistent with $\Lambda=\mpl$.
%That is, if the
%Higgs boson were discovered in this mass range, with no new physics
%beyond the Standard Model observed at the TeV scale or below, one would
In such a scenario, the Standard Model could in principle remain viable
all the way up to the Planck scale.
%The reader should bear in mind
%that this is a logical possibility.
Finally, in mass range 3, we are
still consistent with the 95\% CL Higgs mass limit quoted
in \eq{smhiggslim}.
Again, a Higgs boson discovered in this mass range would imply that
$\Lambda<\mpl$.
%However, most of
%this mass range is beyond the expected reach
%of the upgraded Tevatron, and will require experimentation at the LHC
%for Higgs discovery and elucidation.

\subsection{The Standard Model Higgs Boson}

In the Standard Model, the Higgs mass is given by: $\mhsm^2=\lambda
v^2$, where $\lambda$ is the Higgs self-coupling parameter.  Since
$\lambda$ is unknown at present, the value of the Standard Model Higgs
mass is not predicted.  However, other theoretical considerations,
discussed in Section~I.A, place constraints on the Higgs mass
as exhibited in \fig{trivial}.  In contrast, the Higgs couplings to
fermions and gauge bosons are predicted by the theory.  In particular,
the Higgs couplings are proportional to the corresponding particle
masses, as shown in \fig{smcouplings}.  The vertices of
\fig{smcouplings}
govern the most important features of Higgs phenomenology at colliders.
In Higgs production and decay processes, the dominant mechanisms involve
the coupling of the Higgs boson to the $W^\pm$, $Z$ and/or
the third generation quarks and leptons.
It should be noted that a $\hsm gg$ ($g$=gluon)
coupling is induced by virtue of a one-loop graph
in which the Higgs boson couples to a virtual $t\bar t$ pair.
Likewise, a $\hsm\gamma\gamma$ coupling is generated, although in this
case the one-loop graph in which the Higgs boson couples to
a virtual $W^+W^-$ pair is the dominant contribution.   Further details
of Standard Higgs boson properties are given in \Ref{hhg}.
\vskip 0.2in
\begin{figure}[htb]
\centering
\centerline{\psfig{file=theory/hunt10a.ps,width=5cm,angle=90}
\hskip1pc \psfig{file=theory/hunt10b.ps,width=5cm,angle=90}
\hskip1pc
\psfig{file=theory/hunt10c.ps,width=5cm,angle=90}}
\vskip1pc
\caption[0]{\label{smcouplings}Standard Model Higgs boson
interactions at tree-level.}
\end{figure}

%%%
\subsubsection{Present Status of the Standard Model Higgs Boson Search}

Before turning to the relevant \SM\ Higgs production processes and
decay modes at the Tevatron,
we briefly comment on the expected status of the Higgs search
at the end of the final year of the LEP2 collider run in 2000.
In 2000, the LEP2 collider operated at a variety of
center-of-mass energies between 203 and 208 GeV (with most data taken at
204.9 GeV and 206.8 GeV).  The total integrated luminosity 
exceeds
100~pb$^{-1}$ per experiment.  If no Higgs signal is seen, then the
95\%~CL lower limit on the \SM\ Higgs mass is projected to be
$\mhsm\gsim 115$~GeV.  In order to discover the Higgs boson at the
$5\sigma$ level, one must have $\mhsm\lsim 112$~GeV.
These results are based on \fig{SM95}~$(a)$ and $(b)$, taken from
\Ref{egross}.

\begin{figure}[!ht]
\centering
\centerline{\psfig{file=theory/smhiggs95.eps,width=8cm}
\hfill
\psfig{file=theory/smhiggs5s.eps,width=8cm}}
\vskip1pc
\caption[0]{ \label{SM95} Projected LEP2 limits on the Standard Model Higgs boson
mass (in GeV) as a function of integrated luminosity per experiment,
for different assumptions concerning the operation of
the machine: (a)  95\% CL exclusion and
             (b)  $5\sigma$ discovery.
This figure is taken from \Ref{egross}.}
\end{figure}
%%%
One of the goals of the Higgs Working Group is to
examine the potential for the upgraded Tevatron to extend the LEP2
Higgs search.
We now briefly survey the dominant Higgs decay and production processes
most relevant to the Higgs search at the upgraded Tevatron.

\subsubsection{Standard Model Higgs Boson Decay Modes}

The branching ratios for the dominant decay modes of a Standard Model
Higgs boson are shown as a function of Higgs boson mass in \fig{fg:1}
and table~\ref{tableBR} \cite{9}.

\vskip6pt\noindent
\underline{$\hsm\to f\bar f$} \\[0.2cm]
%              =================
For Higgs boson masses below about 130 GeV, the
decay $\hsm\to b\bar b$
dominates, while the decay $\hsm\to \tau^+\tau^-$ can also be
phenomenologically relevant.
\begin{figure}
\begin{center}
%\centerline{\psfig{file=theory/brsm.ps,width=9cm,angle=-90}}
\centerline{\psfig{file=theory/brsm_new.eps,width=9cm}}
\end{center}
\caption[0]{\label{fg:1}  Branching ratios of the dominant decay
modes of the \SM\ Higgs boson \cite{9}.  These results
have been obtained with the program HDECAY \cite{hdecay}, and include
QCD corrections beyond the leading order. The shaded bands represent the
variations due to the uncertainties in the input parameters:
$\alpha_s(M_Z^2) = 0.120 \pm 0.003$, $\overline{m}_b(M_b) = 4.22
\pm 0.05$~GeV, $\overline{m}_c(M_c) = 1.22 \pm 0.06$~GeV, $M_t =
174 \pm 5$~GeV.}
\end{figure}

\begin{table}[htbp]
\caption [0]
{\label{tableBR} Table of branching ratios of the dominant decay
modes of the \SM\ Higgs boson \cite{9}. The branching ratios listed
above correspond to the central values of $\alpha_s$ and the quark
mass parameters given in the caption of \fig{fg:1}.}
\begin{center}
\begin{tabular}{c c c c c c c c}
$\mhsm$ [GeV] & $b \bar{b}$ & $\tau^+ \tau^-$ & $ gg$ & $c \bar{c}$ &
$WW^{(*)}$ & $ZZ^{(*)}$ & $\gamma \gamma$ \\[1ex]
\tableline
\980 &  0.841 &  $7.94\times 10^{-2}$ & $4.00\times 10^{-2}$ & $3.73
\times 10^{-2}$  & $5.71 \times 10^{-4}$ & $1.58 \times 10^{-4}$ & $9.01
\times 10^{-4}$  \rule{0in}{3ex} \\
\990 &  0.829 &  $8.02\times 10^{-2}$ & $4.96\times 10^{-2}$ & $3.67
\times 10^{-2}$  & $1.88 \times 10^{-3}$ & $3.95 \times 10^{-4}$ & $1.20
\times 10^{-3}$   \\
100 &  0.811 &  $8.00\times 10^{-2}$ & $5.99\times 10^{-2}$ & $3.58
\times 10^{-2}$  & $1.01 \times 10^{-2}$ & $1.06 \times 10^{-3}$ & $1.55
\times 10^{-3}$  \\
110 &  0.769 & $7.73\times 10^{-2}$ & $6.89\times 10^{-2}$  &
$3.39\times 10^{-2}$ & $4.41 \times 10^{-2}$ &$ 4.12 \times 10^{-3}$ &
$1.92 \times 10^{-3}$\\
120 &  0.677 &  $6.93 \times 10^{-2}$ & $7.27\times 10^{-2}$ & $2.99\times
10^{-2}$ &  0.132 & $1.52\times 10^{-2}$ & $2.21 \times 10^{-3}$ \\
130 &  0.526 & $5.46 \times 10^{-2}$  & $6.67 \times 10^{-2}$  &
$2.32\times 10^{-2}$ & 0.287 &$3.88\times 10^{-2}$ & $2.25\times 10^{-3}$ \\
140 &  0.342 & $3.61 \times 10^{-2}$  &$5.08 \times 10^{-2}$ &
$1.51\times 10^{-2}$ & 0.483 &$6.81 \times 10^{-2}$ &$ 1.96 \times 10^{-3}$ \\
150 &  0.175 & $1.87\times 10^{-2}$  & $3.01 \times 10^{-2}$& $7.71\times
10^{-3}$ & 0.681 & $8.35\times 10^{-2}$ & $1.40\times 10^{-3}$ \\
160 &   $3.97 \times 10^{-2}$& $4.29\times 10^{-3}$ &$7.85 \times
10^{-3}$  & $1.75\times 10^{-3}$ &0.901 &$4.35 \times 10^{-2}$
&$5.54\times 10^{-4}$ \\
170 &   $8.35 \times 10^{-3}$ &$9.12\times 10^{-4}$  &$1.89\times
10^{-3}$   &$3.68 \times 10^{-4}$ &0.965 &$2.25\times 10^{-2}$ & $1.50
\times 10^{-4}$\\
180 &  $5.35 \times 10^{-3}$  &$5.90\times 10^{-4}$  &$1.37\times
10^{-3}$   & $2.35 \times 10^{-4}$ & 0.935 &$5.75\times 10^{-2}$ &
$1.02\times 10^{-4}$ \\
190 &  $3.38 \times 10^{-3}$  &$3.77  \times 10^{-4}$ &$9.82\times
10^{-4}$   & $1.49\times 10^{-4}$ &0.776 &0.219 &$6.71 \times 10^{-5}$\\
200 &  $2.57 \times 10^{-3}$ &$2.89 \times 10^{-4}$  &$8.40 \times
10^{-4}$   &$1.13 \times 10^{-4}$ &0.735 &0.261 &$5.25 \times 10^{-5}$
\\%[1ex] %\hline
\end{tabular}
\vspace{.1in}
\end{center}
\end{table}


These results
have been obtained with the program HDECAY \cite{hdecay}, and include
QCD corrections beyond the leading
order \cite{DSZ}.\footnote{The leading electroweak
corrections are small in the Higgs mass range of interest and may be
safely neglected.}
The QCD corrections are significant for the $\hsm\to b\bar b, 
c\bar c$ decay widths due
to large logarithmic contributions.  The dominant part of these
corrections can be absorbed by evaluating
the running quark mass at a scale equal to the Higgs mass. In
order to gain a
consistent prediction of the partial decay widths one has to use
$\overline{\rm MS}$ masses, $\overline{m}_Q(M_Q)$ [where $M_Q$ is the
corresponding quark pole mass], obtained by fits to
experimental data. The evolution of $\overline{m}_Q(M_Q)$ to
$\overline{m}_Q(\mhsm)$ is controlled by the renormalization
group equations for the running $\overline{\rm MS}$ masses.
A recent analysis of this type can be found in \Ref{quarkmasses}.
For example, for $\alpha_s(\mz)=0.118\pm 0.003$, 
the following $\overline{\rm MS}$ quark masses are obtained:
$\overline{m}_b(\mz)=3.00\pm 0.11$~GeV and $\overline{m}_c(\mz)=
0.677^{+.056}_{-.061}$~GeV.  Using
$m_\tau=1.777$~GeV, it follows that the following hierarchy of Higgs
branching ratios is expected for Higgs masses of order 100~GeV:
${\rm BR}(\tau^+\tau^-)/{\rm BR}(b\bar b)\simeq 0.1$ and
${\rm BR}(c\bar c)/{\rm BR}(b\bar b)\simeq 0.04$.  Note that the
large decrease in the charm quark mass due to QCD running is responsible
for suppressing ${\rm BR}(c\bar c)$ relative to ${\rm BR}(\tau^+\tau^-)$,
in spite of the color enhancement of the former, thereby reversing the
naively expected hierarchy.
In \fig{fg:1}, the shaded bands indicate the theoretical uncertainty in
the predicted branching ratios.  These arise primarily from the
uncertainty in $\alpha_s$, and to a lesser extent the uncertainty in the
quark masses.

%For Higgs bosons with $\mhsm\gsim 2m_t$, the decay $\hsm\to t\bar t$,
%where one of the top quarks can be off--shell, should also be
%considered.
%The branching ratio for this decay never exceeds about 20\% in this
%mass
%range.  This mass range is of no relevance to the Tevatron searches,
%and will not be discussed further.


\vskip6pt\noindent
\underline{$\hsm\to gg$} \\[0.2cm]
%
Though one--loop suppressed, the decay $\hsm\to gg$ is competitive
with other decays in the relevant Higgs mass region because of the
large top Yukawa
coupling and the color factor.  The partial width for this decay is
primarily of interest because it determines the $gg\to\hsm$ production
cross-section.

\vskip6pt\noindent
\underline{$\hsm\to WW,ZZ$} \\[0.2cm]
%          ============
For Higgs boson masses above about 110 GeV, the decay mode
%The Higgs branching ratios to vector boson pairs are also shown in
%\fig{fg:1}.  In the \SM\ the decays $\hsm\to WW$ is dominant for
%$\mhsm\gsim
%2\mw$ since the corresponding partial width increases with the third
%power of the Higgs mass for large Higgs masses.
%Higgs decay branching ratio
$\hsm\to WW$, where (at least) one of the $W$ bosons is
off-shell (denoted henceforth by $WW^*$) becomes relevant.
Above 135 GeV, this is the dominant decay mode
%even though it is significantly below the $WW$ threshold
\cite{13,16}.
The corresponding Higgs branching ratio to $ZZ^{(*)}$ is less useful
for the Tevatron Higgs search, while constituting the gold-plated mode
for the Higgs search at the LHC \cite{lhchiggs} when both $Z$ bosons
decay to electrons or muons.




\subsubsection{Standard Model Higgs Boson Production at the Tevatron}

This section describes the most important Higgs production processes at
the Tevatron.  The relevant cross-sections are depicted in
\fig{fg:4} (based on computer programs available from M.~Spira
\cite{hxsec}), and the corresponding numerical results are given in 
table~\ref{tableSIG} \cite{9,DSSW,private}.
 
Combining these Higgs production mechanisms with the decays discussed
in the previous section, one obtains the most promising signatures.

\begin{figure}[htb]
\begin{center}
\centerline{\psfig{file=theory/cxnsmnew.ps,width=9cm,angle=-90}}
\end{center}
\vskip-1.5pc
\caption[0]{\label{fg:4} Higgs production cross-sections 
(in units of pb) at the Tevatron
[$\sqrt{s}=2$ TeV] for the various production mechanisms as a function of the
Higgs mass, taken from \Refs{9}{private}.
The full NLO QCD-corrected results are shown for the gluon 
fusion $gg \to \hsm$, vector boson fusion $qq\to V^*V^*qq \to \hsm qq$
[here, $qq$ refers to both $ud$ and $q\bar q$ scattering, although the
latter dominates at the Tevatron],
Higgs-strahlung processes $q\bar q \to V^* \to V\hsm$ (where
$V=W^\pm$, $Z$) and $b\bar b\to\hsm$.  
Tree-level cross-sections are exhibited for
Higgs production processes in association with heavy quark pairs:
$gg,q\bar q \to \hsm t\bar t, \hsm b\bar b$.  In the latter case,
the cross-section has been computed with the running 
Higgs--bottom-quark Yukawa
coupling evaluated at the corresponding Higgs mass.  See
text for further discussion.}
\end{figure}

\begin{table}[htbp]
\vspace*{1pc}
\caption [0]
{\label{tableSIG} Table of Higgs production cross-sections
(in units of pb) at the Tevatron
[$\sqrt{s}=2$ TeV] for the various production mechanisms as a function of the
Higgs mass.  For further details, see the caption of \fig{fg:4}.
}
\begin{center}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{lc c c c c c c }
\multicolumn{1}{c}{$\mhsm$ [GeV]} & $gg\to\hsm$ & $q\bar q'\to\hsm W$ &  $q\bar q\to\hsm Z$
& $qq\to \hsm qq$ & $gg$, $q\bar q\to\hsm t\bar t$
&  $gg$, $q\bar q\to\hsm b\bar b$ & $b\bar b\to\hsm$
%\rule{0in}{3ex}
\\[1ex] \tableline
\noalign{\vskip1ex}
\980 &  2.132 &  0.616 & 0.334 & 0.163 & $1.11 \times 10^{-2}$ & $1.08
\times 10^{-2}$ & $8.67\times 10^{-2}$  \\
\990 &  1.557 &  0.431 & 0.238 & 0.138 & $8.31 \times 10^{-3}$ & $6.60
\times 10^{-3}$ & $5.40\times 10^{-2}$  \\
100 &  1.170 &  0.308 & 0.173 & 0.117 & $6.29 \times 10^{-3}$ & $4.17
\times 10^{-3}$ & $3.47\times 10^{-2}$   \\
110 &  0.900 & 0.224 & 0.128  & 0.100 & $4.81 \times 10^{-3}$
&$ 2.71 \times 10^{-3}$ & $2.30\times 10^{-2}$ \\
120 &  0.704 &  0.165 & $9.65\times 10^{-2}$ & $8.54\times
10^{-2}$ &  $3.71\times 10^{-3}$ & $1.80 \times 10^{-3}$ 
& $1.55\times 10^{-2}$ \\
130 &  0.558 & 0.124  & $7.35 \times 10^{-2}$  & $7.32\times 10^{-2}$
& $2.88\times 10^{-3}$ & $1.22\times 10^{-3}$ & $1.07\times 10^{-2}$ \\
140 &  0.448 & $9.37 \times 10^{-2}$  &$5.65 \times 10^{-2}$ &
$6.29\times 10^{-2}$ & $2.24 \times 10^{-3}$ &$ 8.45 \times 10^{-4}$ 
& $7.53 \times 10^{-3}$ \\
150 &  0.364 & $7.18\times 10^{-2}$  & $4.40 \times 10^{-2}$&
$5.42\times 10^{-2}$ & $1.76\times 10^{-3}$ & $5.92\times 10^{-4}$ 
& $5.38\times 10^{-3}$ \\
160 &  0.298 & $5.54\times 10^{-2}$ &$3.45 \times
10^{-2}$  & $4.69\times 10^{-2}$ &$1.39 \times 10^{-3}$
&$4.21\times 10^{-4}$  & $3.89\times 10^{-3}$ \\
170 & 0.247 & $4.32\times 10^{-2}$  &$2.73\times
10^{-2}$   & $4.06\times 10^{-2}$ &$1.10\times 10^{-3}$ &
$3.03\times 10^{-4}$ & $2.86\times 10^{-3}$ \\
180 & 0.205 &$3.39\times 10^{-2}$  &$2.17\times
10^{-2}$   & $3.52 \times 10^{-2}$ & $8.77\times 10^{-4}$ &
$2.20\times 10^{-4}$ & $2.12\times 10^{-3}$ \\
190 &  0.172  &$2.68  \times 10^{-2}$ &$1.74\times
10^{-2}$   & $3.06\times 10^{-2}$  & $7.00\times 10^{-4}$ &
$1.62 \times 10^{-4}$ & $1.59\times 10^{-3}$ \\
200 &  0.145  &$2.13 \times 10^{-2}$  &$1.40 \times 10^{-2}$ &
$2.66 \times 10^{-2}$ & $5.62\times 10^{-4}$ &$1.20 \times 10^{-4}$
& $1.20\times 10^{-3}$ \\[1ex] %\hline
\end{tabular}
\vspace{.1in}
\end{center}
\end{table}

\break
\vskip6pt\noindent
\underline{$q\bar q\to V^*\to V\hsm$ [$V=W^\pm$ or $Z$]} \\[0.2cm]
%          ================================
Given sufficient luminosity, the most promising
\SM\ Higgs discovery mechanism at the Tevatron for $\mhsm\lsim 130$~GeV
consists of $q\bar q$ annihilation into a virtual $V^*$ ($V=W$ or $Z$),
where the virtual $V^*\to V\hsm$ followed by $\hsm\to b\bar b$ and
the leptonic decay of the $V$ \cite{Stange:1994ya}.
The cross-section for $q\bar q\to W^\pm\hsm$ (summed over both $W$
charge states) reaches values of $0.3$---$0.02$~pb for
$100~{\rm GeV}\lsim\mhsm\lsim 200$~GeV as shown in \fig{fg:4},
taken from \Ref{9}.
The corresponding $q\bar q\to Z\hsm$ cross-section is roughly a
factor of two lower over the same Higgs mass range.  The QCD corrections
to $\sigma(q\bar q\to V\hsm)$ coincide with those
of the Drell-Yan process and increase the cross-sections by about 30\%
\cite{23,23a0,23a}. The
theoretical uncertainty is estimated to be about $15\%$ from the
remaining scale dependence. The dependence on different sets of parton
densities is rather weak and also leads to a variation of the
production cross-sections by about 15\%.


In order to discover a Higgs signal in $q\bar q\to V\hsm\to V
b\bar b$ at the Tevatron, one must be able to separate the signal from an
irreducible Standard Model $Vb\bar b$ background.  The kinematic
properties of the signal and background are not identical, so by
applying appropriate cuts, a statistically significant signal can be
extracted given sufficient luminosity.  Although the Standard Model
$Vb\bar b$ signal can be studied experimentally, a reliable theoretical
computation of the predicted $Vb\bar b$ differential cross-section is
an essential ingredient of the Tevatron Higgs search.

%We believe that the
%present understanding of the $Wb\bar b$ background estimate used in
%our studies is reliable, based on the CDF
%analysis of the $Wb\bar b$ component of the $t\bar t$ background.
%In particular, the $W+2$ jet rate is a measured quantity.
%First, the predicted $W+2$ jet rate from the Monte Carlo is normalized
%to the measured Run I data corrected for the increased beam energy.
%Then, the fractional contributions of $Wb\bar b$ and $Wc\bar c$ final
%states are estimated from the Monte Carlo simulations,
%and an absolute prediction is made for the $W+$
%heavy quark backgrounds.

At lowest order in QCD perturbation theory, the rate for $Vb\bar b$
production is of ${\cal O}(\alpha_s^2)$.  The appropriate energy scale
of the strong coupling constant is not fixed at lowest order, leading
to a significant ambiguity in the theoretical predictions.  Recently,
the QCD next-to-leading order (NLO) corrections to the $Wb\bar b$
differential cross-section were calculated \cite{kellis}.  The
higher order corrections greatly reduce the scale dependence of the
predicted rate. Therefore, we now have a better absolute prediction
from theory, and one might be tempted to simply rescale the
tree-level cross-section used above to match the NLO result for the
overall rate.  However, the NLO calculation may also change the shape of
kinematic distributions from the lowest order results.  In particular,
the NLO calculation determines more accurately the kinematics of the
$b\bar b$ pair, and thus permits an extrapolation of the $M_{b\bar b}$
distribution to higher values than have been measured.  Unfortunately,
the NLO cross-sections are generally not reliable for describing {\it
details} of the kinematic distributions, since these are sensitive to
the details of the hadronization and fragmentation of the final state.
In order to overcome this deficiency, a consistent treatment
that combines the NLO calculations with the Monte Carlo simulations is
required.   The analyses presented in Section II of this report have
not yet taken advantage of the new information provided by the NLO
result.

The signatures of Higgs
production in the $V\hsm$ channel are governed by the corresponding
decays of the Higgs and vector boson.  The dominant decay mode of the
Higgs boson in the mass range of $\mhsm\lsim 135$~GeV is $\hsm\to b\bar
b$; in this case, the leptonic decays of
the final state $W$ and $Z$ (these include the missing energy signature
associated with $Z\to \nu\bar\nu$) serve as a trigger for the $V\hsm$
events and significantly reduce QCD backgrounds.
The detection of the Higgs signal via
%although one suffers the attendant branching ratio reduction of
%signal events.
the more copious four-jet final states
resulting from hadronic decays of the $W$ and $Z$ is severely
hampered by huge irreducible backgrounds.

For $\mhsm\gsim 135$~GeV, the
Higgs decay mode $\hsm\to W^+W^-$ (where one $W$ is off-shell if
$\mhsm<2\mw$) becomes dominant.  In this case, the final state consists
of three gauge bosons, $VW^+W^-$ ($V=W^\pm$ or $Z$), and the like-sign
di-lepton signature becomes the primary signature for Higgs discovery.



\vskip6pt\noindent
\underline{$gg\to \hsm$} \\[0.2cm]
%          ============
The gluon fusion processes proceeds primarily through
a top quark triangle loop \cite{29a,sally,19}, and
is the dominant neutral Higgs
production mechanism at the Tevatron, with cross-sections 
of roughly $1.0$---$0.1$~pb for
$100~{\rm GeV}\lsim\mhsm\lsim 200$~GeV,
as shown in \fig{fg:4}.
The two-loop QCD corrections enhance the gluon fusion cross-section by
about 60---100\% \cite{19}. These are dominated by
soft and collinear gluon radiation in the \SM\
\cite{29}.\footnote{Multiple soft-gluon emission also has a large effect
on the Higgs boson transverse momentum distribution \cite{balazs}.}
The remaining scale dependence results in a theoretical
uncertainty of about $20\%$. The dependence of the gluon fusion 
cross-section on different parton densities yields roughly an additional
15\% uncertainty in the theoretical prediction.
The analytical QCD corrections to Higgs boson plus one jet production
have recently
been evaluated in the limit of heavy top quarks, but there is no
numerical analysis so far \cite{30}.

The signature $gg\to\hsm\to b\bar b$ is not promising
at the Tevatron due to the
overwhelming QCD background of $b\bar b$ production.
The $gg\to\hsm\to\tau^+\tau^-$ signature, although not thoroughly
studied, probably requires a missing $E_T$ resolution beyond the
capabilities of the upgraded CDF and D\O\ detectors.
For $\mhsm\gsim 135$~GeV, the $\hsm\to WW^*$ decay channel
becomes dominant and provides a potential Higgs discovery mode for the
Tevatron.  The strong
angular correlations of the final state leptons resulting from
$WW^*$ is one of the crucial ingredients for this
discovery channel \cite{5,dreiner}.


%\newpage
\vskip6pt\noindent
\underline{$V^*V^*\to \hsm$ [$V=W^\pm$ or $Z$]} \\[0.2cm]
%          ============
Vector boson fusion is a shorthand notation for the full
$q\bar q\to
q\bar q\hsm$ process, where the quark and anti-quark both radiate 
virtual vector bosons ($V^*$)
which then annihilate to produce the Higgs boson.  Vector
boson fusion via $ud\to du\hsm$ (and its charge-conjugate process)
is also possible, although the
relative contribution is small at the Tevatron.  In \fig{fg:4},
all contributing
processes are included, labeled $qq\to qq\hsm$ for simplicity.
The resulting \SM\ cross-sections are in the range $0.1$---$0.03$~pb for
$100~{\rm GeV}\lsim\mhsm\lsim 200$~GeV.
The QCD corrections enhance the cross-section by about 10\% \cite{23a,32}.

The modest $V^*V^*$ fusion cross-section precludes observation of any of
the rare SM Higgs decay modes in $qq\to qq\hsm$ events
at the Tevatron. For example, for
$\mhsm = 120$~GeV and 30~fb$^{-1}$ of data, only six
$\hsm\to \gamma\gamma$ events are expected
from the production process $qq\to qq\hsm$.
Similarly, under the same assumptions, only eleven di-lepton events
resulting from $\hsm\to\tau^+\tau^-$ are expected from the same data
sample prior to any acceptance cuts that
%Numbers for the most copious sub-dominant decay mode,
%$\hsm\to \tau\tau$, are similar: after taking into account the
%branching ratios $B(\hsm\to\tau\tau)\approx 0.07$ and leptonic tau
%branching ratios $2B(\tau\to
%e\nu\bar\nu)B(\tau\to\mu\nu\bar\nu)\approx
%0.06$,  only  12 final events are left,
%before any of the acceptance cuts
are required to reduce the
large $Z\to\tau^+\tau^-$, $W^+W^-$ or $t\bar t$ backgrounds. Typically
these cuts reduce the Higgs signal by another order of magnitude or
more~\cite{wbfLHC}.
%and thus make rare Higgs decay modes unobservable in weak boson fusion at
%the Tevatron.

For $\mhsm\lsim 135$~GeV, the dominant Higgs decay channel is $\hsm\to
b\bar b$.  In $qq\to qq\hsm$ events, the Higgs boson would appear
as a $\bar bb$ invariant mass peak in 4-jet events in which two of the jets
are identified as $b$-quarks.  Note that the $qqb\bar b$ final state
can also arise from $W\hsm$ and $Z\hsm$ events.  These channels are
very difficult to detect due to the very large QCD backgrounds.  In
the case of $V^*V^*$ fusion, because
there are two forward jets in the full
process $qq \to V^*V^*qq \to \hsm + qq$, one may hope to
be able to somewhat suppress the QCD backgrounds by appropriate cuts.
A preliminary analysis is presented in Section~II.B.4c.
The initial conclusion is that
this channel does not appear to be promising at
the upgraded Tevatron.


\vskip6pt\noindent
\underline{$q\bar q, gg\to \hsm Q\overline Q$, $Q=t,b$} \\[0.2cm]
%          ==============================================
The theoretical predictions for the cross-section of associated
production of a Higgs boson and a heavy quark pair, 
$Q\bar Q$, ($Q=t,b$)\footnote{At Tevatron energies,
the $q\bar q$ annihilation contribution
dominates over $gg$-fusion for $t\bar t\hsm$ production, whereas
the reverse is true for $b\bar b\hsm$ production.  
Both mechanisms are included in the results exhibited in \fig{fg:4}\
and in table~\ref{tableSIG}.} are shown in \fig{fg:4}\
and in table~\ref{tableSIG}.

The tree-level cross-section for $gg$, $q\bar q\to t\bar t\hsm$
is displayed in \fig{fg:4} and in 
table~\ref{tableSIG}.  Although the full NLO
QCD result is not presently known, the NLO QCD corrections are known
in the limit of $\mhsm\ll m_t$ \cite{24}. In this limit the cross-section 
factorizes into the production of a $t\bar t$ pair, which is
convolved with a splitting function for Higgs radiation $t\to t\hsm$,
resulting in an increase of the cross-section by about 20--60\%.
However, since this equivalent Higgs approximation is only valid to
within a factor of two, this result may not be sufficiently reliable.
%Also, the falling parton luminosity
%at high invariant masses preclude any further discussion of this channel at
%the Tevatron.

The case of associated production of a Higgs boson and a bottom
quark pair ($b\bar b$) is more subtle and requires a separate discussion.
For values
of $\mhsm\gg m_b$ (which is satisfied in practice), the total
inclusive cross-section is known at next-to-leading order
\cite{DSSW}.  The leading-order process is $b\bar b\to \hsm$,
where the initial $b$ quarks reside in the proton sea.  Since the
$b$ quark sea arises from the splitting of gluons into $b\bar b$
pairs, the final state is actually $\hsm b\bar b$, but the final
state $b$ quarks tend to reside at low transverse momentum. The
next-to-leading-order correction to this process is modest,
indicating that the perturbative expansion is under control.
Large logarithms, $\ln(\mhsm^2/m^2_b)$, which arise in the
calculation are absorbed into the $b$ distribution functions.  The
calculation also makes evident that the Yukawa coupling of the
Higgs to the bottom quarks should be evaluated from the running
mass evaluated at the Higgs mass scale, $\overline m_b(\mhsm)$
[as in the case of the QCD-corrected decay rate for $\hsm\to
b\bar b$], rather than the running mass evaluated at the $b$
quark mass or the physical (``pole'') mass.  

The relation between the $b\bar b\to\hsm$ cross-section and
the $gg\to b\bar b\hsm$ cross-section requires some
clarification.  For example, it is not correct to simply
add the results of the $b\bar b\to\hsm$ cross-section and the
cross-section for $b\bar b\hsm$ associated production, since
logarithmic terms [proportional to $\ln(\mhsm^2/m^2_b)$] in the latter
have been incorporated into the $b$-quark distribution functions that
appear in the former.  However, one can subtract out these logarithmic
terms.  The resulting {\it subtracted} $b\bar b\hsm$ cross-section
(which can be negative) can be added to the previously obtained $b\bar
b\to\hsm$ cross-section without double counting \cite{soper,25}.
Likewise, the processes $gb\to b\hsm$ and $g\bar b\to\bar b\hsm$ also
contribute and the relevant logarithms must be subtracted before
adding the corresponding cross-sections \cite{25}.  Nevertheless, we
have checked that for $\mhsm\gg m_b$ the correction due to the
subtracted $b\bar b\hsm$ cross-section is small \cite{DSSW,private}.  Thus,
the NLO QCD-corrected $b\bar b\to\hsm$ cross-section should provide a
fairly reliable estimate to the total inclusive $\hsm b\bar b$
cross-section.\footnote{For a completely consistent ${\cal
O}(\alpha_s^2)$ computation, one would need to include two-loop (NNLO)
QCD corrections to $b\bar b\to \hsm$ and the one-loop (NLO)
QCD corrections to $gb\to b\hsm$ (and its charge-conjugate process). 
[Note that for $\mhsm\gg m_b$, the $b$-quark parton distribution
function is of order $\alpha_s\ln(\mhsm^2/m^2_b)\sim {\cal O}(1)$.]
The modest size of the NLO corrections
to $b\bar b\to\hsm$ suggest that the ${\cal O}(\alpha_s^2)$ 
corrections are probably small.}

%There are additional sources of large logarithms, \eg\
%%Sudakov logarithms from soft gluon radiation and
%large logarithms associated with the QCD-corrected Yukawa coupling,
%which will appear in the next-to-leading (NLO) cross-section.
%Naively, one might expect that the dominant
%correction of this type can be
%estimated by replacing the physical (``pole'') mass, $M_b$, by the
%running mass evaluated at the Higgs mass scale, $\overline m_b(\mhsm)$
%(as in the case of the
%QCD-corrected decay rate for $\hsm\to b\bar b$).
%If one simply replaces $M_b$ with $\overline m_b(\mhsm)$
%in the expression for the
%$2\to 3$ body scattering cross-section, the resulting cross-section
%is reduced by about a factor of two.  
%However, $2\to 3$ body
%scattering is a multi-scale process; thus a full NLO calculation
%is needed in order to gain a more complete and satisfactory theoretical
%prediction of these corrections.

In practice, the {\it total} inclusive cross-section for Higgs
production in association with a bottom quark pair is not 
measurable at the Tevatron, since one must observe one or both of the
final-state $b$ quarks in order to isolate the signal above other
Standard Model
backgrounds.  Hence one or both of the final state $b$ quarks must
be produced at large transverse momentum.  This is a fraction of
the total inclusive cross-section, depending on the minimum
transverse momentum required on one or both $b$ quarks.  The
leading-order process for Higgs production in association with a
pair of high transverse momentum $b$ quarks is $q\bar q, gg \to
\hsm b\bar b$.  Since this process is known only at leading order,
the theoretical uncertainty in the cross-section is large.
%similar
%to the case of $\hsm t\bar t$ production.  
Clearly, it is very important to compute the NLO QCD corrections for the
differential cross-section (as a function
of the final state $b$ quark transverse momenta) for $b\bar b\hsm$
production.  This computation is not yet available in the literature.


%For values of $\mhsm\gg m_b$,
%large logarithms, $\ln(\mhsm^2/m^2_b)$, arise due to quasi-on-shell
%$t$-channel quark exchanges, which can be resummed by absorbing them
%into the $b$-quark parton densities.
%After adding the processes $q\bar q,gg\to b\bar b
%\hsm$, $gb \to b\hsm$ and $b\bar b\to \hsm$ and subtracting the
%logarithms accordingly, the resulting {\it inclusive} cross-section for
%$\hsm b\bar b$ production is roughly a factor of 2 larger than the
%tree-level $q\bar q$, $gg \to b\bar b\hsm$ cross-section.  Note
%that in the case of $b\bar b\to\hsm$ and $gb\to b\hsm$, the
%initial $b$-quarks have actually resulted from gluon splitting ($g\to
%b\bar b$).  This implies that for each $b$-quark that participates
%in the hard scattering, there is an associated $b$-quark that is
%produced in the forward direction (parallel to the initial beam)
%\cite{25}.  The factor of 2 to 3 enhancement is a result of
%the resummation of the collinear logarithms and therefore corresponds
%primarily to the kinematic region in which the final state $b$-quarks
%are close to the beam direction.  In the Tevatron Higgs search via the
%$\hsm b\bar b$ channel, at least one of the $b$ quarks ({\it not}
%resulting from the $\hsm\to b\bar b$ decay) is used as a tag for the
%search.  Hence, the QCD-corrected signal cross-section
%(with a realistic $p_T$ cut on the final state $b$-quarks) will not
%gain the full factor of 2 to 3 enhancement.


The tree-level $gg$, $q\bar q\to b\bar b\hsm$ 
cross-section (as a function of $\mhsm$) shown in \fig{fg:4}
has been computed by fixing
the scales of the initial (CTEQ4M \cite{CTEQ}) parton densities, 
the running coupling $\alpha_s$ and the running Higgs--bottom-quark
Yukawa coupling (or equivalently, the running $b$-quark mass) at the
value of the corresponding $\hsm$ mass.  In particular, by employing
the running $b$ quark mass evaluated at $\mhsm$, we are implicitly
resumming large logarithms associated with the QCD-corrected Yukawa
coupling.  Thus, the tree-level $b\bar b\hsm$ cross-section
displayed in \fig{fg:4} implicitly includes a part of the QCD
corrections to the full inclusive cross-section.\footnote{The effect
of using the running $b$-quark mass [$\overline m_b(\mhsm)$]  
as opposed to the $b$-quark pole mass
[$M_b\simeq 5$~GeV] is to {\it reduce} the cross-section by roughly a 
factor of two.}  Nevertheless, as emphasized above,
the most significant effect of the QCD
corrections to $b\bar b\hsm$ production arises from the kinematical
region where the  $b$ quarks are emitted near the forward
direction.  In fact, large logarithms arising in
this region spoils the convergence of the 
QCD perturbation series since $\alpha_s\ln(\mhsm^2/m_b^2)\sim {\cal
O}(1)$.  These large logarithms (already present at lowest order)
must be resummed to all orders, and this resummation
is accomplished by the generation of the $b$-quark distribution
function as described above.  Thus, the QCD-corrected {\it fully} inclusive 
$b\bar b\hsm$ cross-section is well approximated by $b\bar b\to\hsm$ and its
QCD corrections.  The latter is also exhibited in \fig{fg:4} 
and is seen to be roughly an order of magnitude larger than
the tree-level $b\bar b\hsm$ cross-section.    
Of course, this result is not very relevant for the Tevatron
experimental searches in which transverse momentum cuts on the
$b$-jets are employed.  Ultimately, one needs the QCD-corrected
{\it differential} cross-section for $b\bar b\hsm$ (as a function of the final
state $b$-quark transverse momenta) in order to do realistic simulations of
the Higgs signal in this channel.

Based on our best estimates for the $\hsm Q\bar Q$ production
cross-sections shown in \fig{fg:4}, we conclude that in
the \SM\ both processes of Higgs radiation off top and bottom quarks
%are unimportant due to the
have very  small event rates.  However, in some
extensions of the \SM, the coupling of the Higgs boson to $b\bar b$
can be significantly enhanced.  This leads to enhanced $\hsm b\bar b$
production which may be observable at an upgraded Tevatron with
sufficient luminosity.  A concrete example (the minimal supersymmetric
model at large $\tan\beta$) will be discussed further in Section~I.C.6a.

To detect enhanced $b\bar b\hsm$ production, we assume that the dominant
decay of the Higgs boson is into $b\bar b$ pairs.  Thus, the signal must
be extracted from events with four $b$-quarks (suggesting the need for
tagging at least three of the $b$-jets).
Theoretical estimates of the $b\bar b b\bar b$ Standard Model
background rates are less reliable than the corresponding $W b\bar b$
rates discussed above.  Heavy quark production rates are typically
difficult to estimate, and the situation is
complicated by demanding 3 or 4 $b$--tagged jets.
Improved theoretical modeling and detailed comparisons with Run 2
Tevatron data will be essential to improve the present background
estimates.


\subsection{Higgs Bosons in Low-Energy Supersymmetry}

Although the Higgs mass range 130~GeV~$\lsim\mhsm\lsim 180$~GeV appears
to permit an effective Standard Model
that survives all the way to the Planck scale, most theorists consider
such a possibility unlikely.  This conclusion is based on
the ``naturalness'' \cite{natural} argument as follows.  In
an effective field theory, all parameters of the low-energy theory
({\it i.e.} masses and couplings) are calculable in terms of parameters
of a more fundamental, renormalizable theory that
describes physics at the energy scale $\Lambda$.  All
low-energy couplings and fermion masses are logarithmically sensitive to
$\Lambda$.  In contrast, scalar squared-masses are {\it quadratically}
sensitive to $\Lambda$.  Thus, in this framework, the observed Higgs
mass (at one-loop) has the following form:
\beq \label{natural}
m_h^2= (m_h^2)_0+cg^2\Lambda^2\,,
\eeq
where $(m_h)_0$ is a parameter of the fundamental theory and $c$ is a
constant, presumably of ${\cal O}(1)$, which is calculable within the
low-energy effective theory.  The ``natural'' value
for the scalar squared-mass is $g^2\Lambda^2$.  Thus, the expectation
for $\Lambda$ is
\beq \label{tevscale}
\Lambda\simeq {m_h\over g}\sim {\cal O}(1~{\rm TeV})\,.
\eeq
If $\Lambda$ is significantly larger than 1~TeV (often called the
hierarchy problem in the literature), then the only way
to find a Higgs mass associated with the scale of electroweak symmetry
breaking is to have an ``unnatural'' cancellation between the two terms
of \eq{natural}.
This seems highly unlikely given that the two terms of
\eq{natural} have completely different origins.  As a cautionary note,
what appears to be unnatural today might be shown in the future to be
a natural consequence of dynamics or symmetry at the Planck scale.
Only further experimentation that tests our present
understanding can determine if the quest for new physics at
scales $\Lambda\ll\mpl$ is ultimately fruitful.

A viable theoretical framework that incorporates weakly-coupled Higgs
bosons and satisfies the constraint of \eq{tevscale} is that of
``low-energy'' or ``weak-scale'' supersymmetry \cite{Nilles85,Haber85,smartin}.
In this framework, supersymmetry is
used to relate fermion and boson masses (and interactions).  
Since fermion masses are only
logarithmically sensitive to $\Lambda$, boson masses will exhibit the
same logarithmic sensitivity if supersymmetry is exact.  Since no
supersymmetric partners of Standard Model particles have been
found, we know that supersymmetry cannot be an exact symmetry of nature.
Thus, we identify $\Lambda$ with the supersymmetry-breaking scale.  The
naturalness constraint of \eq{tevscale} is still relevant, so in the
framework of low-energy supersymmetry, the scale of supersymmetry
breaking should not be much larger than about 1~TeV in order that the
naturalness of scalar masses be preserved.  The supersymmetric extension
of the Standard Model would then replace the Standard Model as the
effective field theory of the TeV scale.  One could then ask: at what
scale does this model break down?  The advantage of the supersymmetric
approach is that the effective low-energy supersymmetric theory {\it
can} be valid all the way up to the Planck scale, while still being
natural!

In order to begin our study of Higgs bosons in low-energy supersymmetry,
we need a specific model framework.  The simplest realistic model of
low-energy supersymmetry is a minimal supersymmetric extension of the
Standard Model (MSSM), which employs the minimal supersymmetric particle
spectrum \cite{habermssm}.  Higgs phenomenology in the MSSM and the
corresponding
discovery reach at the Tevatron was a primary focus of the Higgs
Working Group.  Non-minimal supersymmetric approaches are also of interest and
have been addressed in part by the Beyond the MSSM Working Group of this
workshop \cite{btmssm}.
%Models of strongly-coupled physics are being investigated
%in the Strong Dynamics Workshop.
%The phenomenology
%of non-minimal supersymmetric Higgs sectors will be addressed
%elsewhere.



\subsubsection{The Tree-Level Higgs Sector of the MSSM}



Both hypercharge $Y=-1$ and $Y=+1$ Higgs doublets are
required in any Higgs sector of
an anomaly-free supersymmetric extension of the Standard Model.
The supersymmetric structure of the theory also requires (at least) two
Higgs doublets to generate mass for both ``up''-type and ``down''-type
quarks (and charged leptons) \cite{Inoue82,Gunion86}.
Thus, the MSSM
contains the particle spectrum of a two-Higgs-doublet extension of the
Standard Model and the corresponding supersymmetric partners
\cite{Haber85,smartin}.

The two-doublet Higgs sector \cite{hhgchap4} contains eight scalar
degrees of freedom:
one complex $Y=-1$ doublet, {\boldmath $\Phi_d$}$=(\Phi_d^0,\Phi_d^-)$
and one complex $Y=+1$ doublet, {\boldmath
$\Phi_u$}$=(\Phi_u^+,\Phi_u^0)$.  The notation reflects the
form of the MSSM Higgs sector coupling to fermions: $\Phi_d^0$
[$\Phi_u^0$] couples exclusively to down-type [up-type] fermion pairs.
When the Higgs potential is minimized, the neutral components of the
Higgs fields acquire vacuum expectation values:\footnote{The
phases of the Higgs fields can be chosen such that the vacuum
expectation values are real and positive.  That is, the tree-level MSSM
Higgs sector conserves CP, which implies that the neutral Higgs mass
eigenstates possess definite CP quantum numbers.}
\beq
\langle {\mathbold{\Phi_d}} \rangle={1\over\sqrt{2}} \left(
\begin{array}{c} v_d\\ 0\end{array}\right), \qquad \langle
{\mathbold{\Phi_u}}\rangle=
{1\over\sqrt{2}}\left(\begin{array}{c}0\\ v_u
\end{array}\right)\,,\label{potmin}
\eeq
where the normalization has been chosen such that
$v^2\equiv v_d^2+v_u^2={4\mw^2/ g^2}=(246~{\rm GeV})^2$.
Spontaneous electroweak symmetry breaking results in
three Goldstone bosons, which
are absorbed and become the longitudinal components of
the $W^\pm$ and $Z$.  The remaining five physical Higgs particles
consist of a charged Higgs pair
\vspace*{-1pc}
\beq \label{hpmstate}
\hpm=\Phi_d^\pm\sinb+ \Phi_u^\pm\cosb\,,
\eeq
one CP-odd scalar
\vspace*{-1pc}
\beq \label{hastate}
\ha= \sqrt{2}\left({\rm Im\,}\Phi_d^0\sinb+{\rm Im\,}\Phi_u^0\cosb
\right)\,,
\eeq
and two CP-even scalars:
\vspace*{-1pc}
\beqno
\hl &=-(\sqrt{2}\,{\rm Re\,}\Phi_d^0-v_d)\sin\alpha+
(\sqrt{2}\,{\rm Re\,}\Phi_u^0-v_u)\cos\alpha\,,\nonumber\\
\hh &=(\sqrt{2}\,{\rm Re\,}\Phi_d^0-v_d)\cos\alpha+
(\sqrt{2}\,{\rm Re\,}\Phi_u^0-v_u)\sin\alpha\,,
\label{scalareigenstates}
\eeqno
(with $\mhl\leq \mhh$).
The angle $\alpha$ arises when the CP-even Higgs
squared-mass matrix (in the $\Phi_d^0$---$\Phi_u^0$ basis) is
diagonalized to obtain the physical CP-even Higgs states (explicit
formulae will be given below).


The supersymmetric structure of the theory imposes constraints on the
Higgs sector of the model.  For example, the Higgs
self-interactions are not independent parameters; they can be expressed
in terms of the electroweak gauge coupling constants.  As a result,
all Higgs sector parameters at tree-level
are determined by two free parameters: the ratio of the two
neutral Higgs field vacuum expectation values,
\beq \label{tanbdef}
\tanb\equiv {v_u\over v_d}\,,
\eeq
and one Higgs mass, conveniently chosen to be $\mha$. In particular,
\beq
\mhpm^2 =\mha^2+\mw^2\,,
\label{susymhpm}
\eeq
and the CP-even Higgs bosons $\hl$ and $\hh$ are eigenstates of the
following squared-mass matrix
\beq
{\cal M}_0^2 =    \left(
%\begin{array}{ll}
\matrix{\mha^2 \sin^2\beta + m^2_Z \cos^2\beta&
           -(\mha^2+m^2_Z)\sin\beta\cos\beta \cr
  -(\mha^2+m^2_Z)\sin\beta\cos\beta&
  \mha^2\cos^2\beta+ m^2_Z \sin^2\beta }
%\end{array}
   \right)\,.\label{kv}
\eeq
The eigenvalues of ${\cal M}_0^2$ are
the squared-masses of the two CP-even Higgs scalars
\beq
  m^2_{H,h} = \half \left( \mha^2 + m^2_Z \pm
                  \sqrt{(\mha^2+m^2_Z)^2 - 4m^2_Z \mha^2 \cos^2 2\beta}
                  \; \right)\,,\label{kviii}
\eeq
and $\alpha$ is the angle that diagonalizes the CP-even Higgs
squared-mass matrix.
From the above results, one obtains:
\beq
\cos^2(\beta-\alpha)={\mhl^2(\mz^2-\mhl^2)\over
\mha^2(\mhh^2-\mhl^2)}\,.
\label{cbmasq}
\eeq
In the convention where $\tanb$ is positive ({\it i.e.},
$0\leq\beta\leq\pi/2$), the angle $\alpha$ lies in the range
$-\pi/2\leq\alpha\leq 0$.

An important consequence of \eq{kviii} is that there is an upper bound
to the mass of the light CP-even Higgs boson, $\hl$.  One finds that:
\beq \label{kx}
\mhl^2\leq\mz^2\cos 2\beta\leq\mz^2\,.
\eeq
This is in marked contrast to the Standard Model, in which the theory
does not constrain the value of $\mhsm$ at tree-level.  The origin of
this difference is easy to ascertain.  In the \SM, $\mhsm^2=\lambda v^2$
is proportional to the Higgs self-coupling $\lambda$, which is a free
parameter.  On the other hand, all Higgs self-coupling
parameters of the MSSM are related to the squares of the electroweak
gauge couplings.

Note that the Higgs mass inequality [\eq{kx}] is saturated in the limit
of large $\mha$.  In the limit of $\mha\gg\mz$, the expressions for the
Higgs masses and mixing angle simplify and one finds
\beqno
\mhl^2 &\simeq\ \mz^2\cos^2 2\beta\,,\nonumber \\[3pt]
\mhh^2 &\simeq\ \mha^2+\mz^2\sin^2 2\beta\,,\nonumber \\[3pt]
\mhpm^2& = \ \mha^2+\mw^2\,,\nonumber \\[3pt]
\cos^2(\beta-\alpha)&\simeq\ {\mz^4\sin^2 4\beta\over 4\mha^4}\,.
\label{largema}
\eeqno
Two consequences are immediately apparent.
First, $\mha\simeq\mhh
\simeq\mhpm$, up to corrections of ${\cal O}(\mz^2/\mha)$.  Second,
$\cos(\beta-\alpha)=0$ up to corrections of ${\cal O}(\mz^2/\mha^2)$.
This limit is known as the {\it decoupling} limit \cite{decoupling}
because when $\mha$ is
large, one can focus on an effective low-energy theory below the scale
of $\mha$ in which the effective Higgs sector consists only of one
CP-even Higgs boson, $\hl$.  As we shall demonstrate below, the
tree-level couplings of $\hl$ are precisely those of the Standard
Model Higgs boson when $\cos(\beta-\alpha)=0$.
From \eq{largema}, one can also derive:
\begin{equation}  \label{cotalf}
\cot\alpha = -\tan\beta - \frac{2\mz^2}{\mha^2} \tan\beta \cos
2\beta + {\cal O}\left(\frac{\mz^4}{\mha^4}\right)\,.
\end{equation}
This result will prove useful in evaluating the CP-even Higgs boson
couplings to fermion pairs in the decoupling limit.


The phenomenology of the Higgs sector depends in detail on
the various couplings of the Higgs bosons to gauge bosons, Higgs
bosons and fermions.  The couplings of the two CP-even Higgs bosons
to $W$ and $Z$ pairs
are given in terms of the angles $\alpha$ and $\beta$ by
\beqno
g\ls{\hl VV}&=g\ls{V} m\ls{V}\sinbma \nonumber \\[3pt]
           g\ls{\hh VV}&=g\ls{V} m\ls{V}\cosbma\,,\label{vvcoup}
\eeqno
where
\beq
g\ls V\equiv\begin{cases}
g,& $V=W\,$,\\ g/\cos\theta_W,& $V=Z\,$. \end{cases}
\label{hix}
\eeq
There are no tree-level couplings of $\ha$ or $\hpm$ to $VV$.
Next, consider the couplings of one gauge boson to two
neutral Higgs bosons:
\beqno g\ls{\hl\ha Z}&={g\cosbma\over 2\cos\theta_W}\,,\nonumber \\[3pt]
           g\ls{\hh\ha Z}&={-g\sinbma\over 2\cos\theta_W}\,.
           \label{hvcoup}
\eeqno
From the expressions above, we see
that the following sum rules must hold separately for $V=W$ and $Z$:
\beqno
g_{\hh V V}^2 + g_{\hl V V}^2 &=
                      g\ls{V}^2m\ls{V}^2\,,\nonumber \\[3pt]
g_{\hl\ha Z}^2+g_{\hh\ha Z}^2&=
             {g^2\over 4\cos^2\theta_W}\,, \nonumber \\[3pt]
 g^2_{\phi ZZ} + 4m^2_Z g^2_{\phi \ha Z}& = {g^2m^2_Z\over
        \cos^2\theta_W}\,,\qquad \phi=\hl, \hh \,.\label{hxi}
\label{sumruletwo}
\eeqno
Similar considerations also hold for
the coupling of $\hl$ and $\hh$ to $W^\pm H^\mp$.  We can summarize
the above results by noting that the coupling of $\hl$ and $\hh$ to
vector boson pairs or vector--scalar boson final states is proportional
to either $\sin(\beta-\alpha)$ or $\cos(\beta-\alpha)$ as indicated
below \cite{hhg}:
\beq
\renewcommand{\arraycolsep}{2cm}
\let\us=\underline
\begin{array}{ll}
% \matrix{%
\us{\cos(\beta-\alpha)}&  \us{\sin(\beta-\alpha)}\\ [3pt]
\noalign{\vskip3pt}
       \hh W^+W^-&        \hl W^+W^- \\
       \hh ZZ&            \hl ZZ \\
       Z\ha\hl&          Z\ha\hh \\
       W^\pm H^\mp\hl&  W^\pm H^\mp\hh \\
       ZW^\pm H^\mp\hl&  ZW^\pm H^\mp\hh \\
       \gamma W^\pm H^\mp\hl&  \gamma W^\pm H^\mp\hh
%}
\end{array}
\label{littletable}
\eeq
Note in particular that {\it all} vertices
in the theory that contain at least
one vector boson and {\it exactly one} non-minimal Higgs boson state
($\hh$, $\ha$ or $\hpm$) are proportional to $\cos(\beta-\alpha)$.
This can be understood as a consequence of unitarity sum rules which
must be satisfied by the tree-level amplitudes of the
theory \cite{cornwall,wudka}.

In the MSSM, the Higgs tree-level couplings to fermions obey the
following
property: $\Phi_d^0$ couples exclusively to down-type fermion pairs and
$\Phi_u^0$ couples exclusively to up-type fermion pairs. This pattern
of Higgs-fermion couplings defines the
Type-II two-Higgs-doublet model \cite{wise,hhg}.
The gauge-invariant Type-II Yukawa interactions (using 3rd family
notation) are given by:
\begin{equation} \label{typetwo}
-{\cal L}_{\rm Yukawa}= h_t\left[\bar t P_L t \Phi^0_u-\bar t P_L b
\Phi^+_u\right] + h_b\left[\bar b P_L b \Phi^0_d-\bar b P_L t
\Phi^-_d\right] + {\rm h.c.}\,,
\end{equation}
where $P_L\equiv\half(1-\gamma_5)$ is the left-handed projection
operator.  [Note that ($\overline\Psi_1 P_L \Psi_2)^\dagger=
\overline\Psi_2 P_R \Psi_1$, where $P_R\equiv\half(1+\gamma_5)$.]
Fermion masses are generated when the neutral Higgs components acquire
vacuum expectation values.  Inserting \eq{potmin} into \eq{typetwo}
yields a relation between the quark masses and the Yukawa couplings:
\beqa
h_b =& {\sqrt{2}\,m_b\over v_d}={\sqrt{2}\, m_b\over
v\cos\beta}\,,\label{hbdef} \\
h_t =& {\sqrt{2}\,m_t\over v_u}={\sqrt{2}\, m_t\over v\sin\beta}\,.
\label{htdef}
\eeqa
Similarly, one can define the Yukawa coupling of the Higgs boson to
$\tau$-leptons (the latter is a down-type fermion).  The couplings
of the physical Higgs bosons to the third generation fermions is
obtained from \eq{typetwo} by using \eqs{hpmstate}{scalareigenstates}.
In particular, the couplings of the neutral Higgs bosons to $f\bar f$
relative to the Standard Model
value, $gm_f/2\mw$, are given by
\begin{eqaligntwo}
 \label{qqcouplings}
\hl b\bar b \;\;\; ({\rm or}~ \hl \tau^+ \tau^-):&~~~ -
{\sin\alpha\over\cos\beta}=\sin(\beta-\alpha)
-\tan\beta\cos(\beta-\alpha)\,,\nonumber\\[3pt]
%
\hl t\bar t:&~~~ \phm{\cos\alpha\over\sin\beta}=\sin(\beta-\alpha)
+\cot\beta\cos(\beta-\alpha)\,,\nonumber\\[3pt]
%
\hh b\bar b \;\;\; ({\rm or}~ \hh \tau^+ \tau^-):&~~~
\phm{\cos\alpha\over\cos\beta}=
\cos(\beta-\alpha)
+\tan\beta\sin(\beta-\alpha)\,,\nonumber\\[3pt]
%
\hh t\bar t:&~~~ \phm{\sin\alpha\over\sin\beta}=\cos(\beta-\alpha)
-\cot\beta\sin(\beta-\alpha)\,,\nonumber\\[3pt]
%
\ha b \bar b \;\;\; ({\rm or}~ \ha \tau^+
\tau^-):&~~~\phm\gamma_5\,{\tan\beta}\,,
\nonumber\\[3pt]
%
\ha t \bar t:&~~~\phm\gamma_5\,{\cot\beta}\,,
\end{eqaligntwo}
(the $\gamma_5$ indicates a pseudoscalar coupling), and the
charged Higgs boson couplings to fermion pairs
(with all particles pointing into the vertex) are given by
\begin{eqaligntwo}
\label{hpmqq}
g_{H^- t\bar b}= &~~~ {g\over{\sqrt{2}\mw}}\
\Bigl[m_t\cot\beta\,P_R+m_b\tan\beta\,P_L\Bigr]\,,
\nonumber\\[3pt]
g_{H^- \tau^+ \nu}= &~~~ {g\over{\sqrt{2}\mw}}\
\Bigl[m_{\tau}\tan\beta\,P_L\Bigr]\,.
\end{eqaligntwo}
%where $P_{R,L}\equiv\half(1\pm\gamma_5)$ are the right and left-handed
%projection operators, respectively.

We next consider the behavior of the Higgs couplings at large
$\tan\beta$.  This limit is of particular interest since
at large $\tan\beta$, some of the Higgs couplings to down-type
fermions can be significantly enhanced.\footnote{In models of low-energy
supersymmetry, there is some theoretical prejudice that suggests that
$1 < \tanb \lsim m_t/m_b$, with the fermion running masses evaluated
at the electroweak scale.
%In this case, it is possible to generate
%radiative electroweak symmetry breaking as a consequence of the large
%top-quark mass.  Moreover,
For example, $\tanb\lsim 1$
[$\tanb > \mt/\mb$] is disfavored
since in this case, the Higgs--top-quark [Higgs--bottom-quark]
Yukawa coupling blows up at an
energy scale significantly below the Planck scale.}
%Thus, the limit of
%large $\cot\beta$ in which the Higgs couplings to up-type fermions are
%enhanced, is theoretically disfavored.}
Let us examine two
particular large $\tan\beta$ regions of interest.  (i) If $\mha\gg\mz$,
then the decoupling limit is reached, in which $|\cos(\beta-\alpha)|\ll
1$ and $\mhh\simeq\mha$.
From \eqs{largema}{qqcouplings}, it follows that the $b\bar b\hh$
and $b\bar b\ha$ couplings have equal strength and are significantly
enhanced (by a factor of
$\tanb$) relative to the $b\bar b\hsm$ coupling, whereas the $VV\hh$
coupling is negligibly
small. In contrast, the values of the $VV\hl$ and $b\bar b\hl$ couplings
are equal to the corresponding couplings of the \SM\ Higgs boson.
To show that the value of the $b\bar b\hl$ coupling
[see \eq{qqcouplings}]
reduces to that of $b\bar b\hsm$ in the decoupling limit,
note that \eq{largema} implies that $|\tan\beta\cos(\beta-\alpha)|\ll 1$
when $\mha\gg \mz$ even when $\tan\beta\gg 1$.
(ii) If $\mha\lsim \mz$ and $\tanb\gg 1$, then $|\sin(\beta-\alpha)|\ll
1$ [see \fig{cosgraph}] and $\mhl\simeq\mha$.
In this case, the $b\bar b\hl$
and $b\bar b\ha$ couplings have equal strength and are significantly
enhanced (by a factor of
$\tanb$) relative to the $b\bar b\hsm$ coupling, while the $VV\hl$
coupling is negligibly small.  Using \eq{sumruletwo} it follows that
the $VV\hh$ coupling is equal in strength to the $VV\hsm$ coupling.
However, the value of the $b\bar b\hh$ coupling can differ
from the corresponding $b\bar b\hsm$ coupling when $\tanb\gg 1$
[since in case (ii), where $|\sin(\beta-\alpha)|\ll 1$,
the product $\tan\beta\sin(\beta-\alpha)$ need not
be particularly small]. Note that in both cases above,
%it is found that the $b\bar b\hh$ coupling is close in
%magnitude to the corresponding \SM\ Higgs coupling, so in as in the
%previous case,
only two of the three neutral Higgs bosons have enhanced
couplings to $b\bar b$.
\begin{figure}[!htb]
\centering
\centerline{\psfig{file=theory/hsbmatb3.ps,width=8cm}
\hfill
\psfig{file=theory/hsbmatb30.ps,width=8cm}}
\vskip1pc
\caption[0]{\label{cosgraph} The value of $\sin^2(\beta-\alpha)$
is shown as a function of
$\mha$ for two choices of $\tan\beta = 3$ and $\tan\beta = 30$.
When radiative-corrections are included, one can define an approximate
loop-corrected angle $\alpha$ as a function of $\mha$, $\tan\beta$ and
the MSSM parameters.  In the figures above, we
have incorporated radiative corrections, assuming that
$\MSUSY\equiv M_Q=M_U=M_D=1$~TeV.  In addition,
two extreme cases for the squark mixing parameters
are shown (see Section~I.C.2 for further discussion of the
radiative corrections and their dependence on the supersymmetric
parameters). The decoupling effect expected from
\eq{largema}, in which $\sin^2(\beta-\alpha)\simeq 1$ for $\mha\gg m_Z$,
continues to hold even when radiative corrections are included.
}
\end{figure}


The decoupling limit of $\mha\gg\mz$ is effective for all values of
$\tan\beta$.  It is easy to check that the pattern of all Higgs
couplings displayed in \eqs{vvcoup}{qqcouplings}
respects the decoupling limit.  That is, in the limit where $\mha\gg\mz$,
$\cos(\beta-\alpha)={\cal O}(\mz^2/\mha^2)$, which means that
the $\hl$ couplings to
Standard Model particles approach values corresponding precisely
to the couplings of the Standard Model Higgs boson.
The region of MSSM Higgs sector parameter space in which the decoupling
limit applies is large, because
$\sin(\beta-\alpha)$ approaches one quite rapidly once
$\mha$ is larger than about 200~GeV, as shown in \fig{cosgraph}.
As a result, over a significant region of the MSSM parameter space, the
search for the lightest CP-even Higgs boson
of the MSSM is equivalent to the search for the \SM\
Higgs boson.  This result is more general; in
many theories of non-minimal Higgs sectors, there is a significant
portion of the parameter space that approximates the decoupling limit.
Consequently, simulations of the \SM\ Higgs signal are also relevant for
exploring the more general Higgs sector.

\subsubsection{The Radiatively-Corrected MSSM Higgs Sector: (a) Higgs
masses}
\label{RCMSSM}

So far, the discussion has been based on a tree-level analysis of the
Higgs sector.  However, radiative corrections can have a significant
impact on the predicted values of Higgs masses and couplings.
The radiative corrections involve both loops of Standard Model particles
and loops of supersymmetric partners.  
The dominant effects arise from
loops involving the third generation quarks and squarks
and are proportional to the corresponding Yukawa couplings.  
Thus, we first review the parameters that control the masses
and mixing of the third-generation squarks.  (We shall neglect
intergenerational mixing effects, which have little impact on the
discussion that follows.)

For each left-handed and right-handed quark of fixed flavor, $q$,
there is a corresponding supersymmetric partner
$\widetilde q_L$ and $\widetilde q_R$, respectively.  These are the
so-called interaction eigenstates, which mix according to the squark
squared-mass matrix. The mixing angle that diagonalizes the squark mass
matrix will be denoted by $\theta_{\widetilde q}$.   The
squark mass eigenstates, denoted by $\widetilde q_1$ and $\widetilde q_2$,
are obtained by diagonalizing the following $2\times 2$ matrix
\begin{equation}
 \left(  %\begin{array}{cc}
\matrix{M_{Q}^2+m_f^2+D_L & m_f X_f \crr
   m_f X_f & M_{R}^2+m_f^2+D_R}
%\end{array}
\right)  \,,
\label{stopmatrix}
\end{equation}
where $D_L\equiv (T_{3f}-e_f\sin^2\theta_W)\mz^2\cos2\beta$ and
$D_R\equiv e_f\sin^2\theta_W\mz^2\cos2\beta$.  In addition,
$f=t$, $M_R\equiv M_U$, $e_t=2/3$ and
$T_{3f}=1/2$ for the top-squark (or {\it stop}) mass matrix, while
$f=b$, $M_R\equiv
M_D$, $e_b=-1/3$ and $T_{3f}=-1/2$ for the bottom-squark
(or {\it sbottom}) mass matrix.
The squark mixing parameters are given by
\beqa
 X_t & \equiv A_t-\mu\cot\beta\,,\nonumber \\
 X_b & \equiv A_b-\mu\tan\beta\,.
\eeqa
Thus, the top-squark and bottom-squark masses and mixing angles
depend on the supersymmetric Higgs mass parameter $\mu$ and
the soft-supersymmetry-breaking parameters: $M_Q$, $M_U$, $M_D$,
$A_t$ and $A_b$.\footnote{For simplicity, we shall take $A_t$,
$A_b$ and $\mu$ to be real parameters.  That is, we are neglecting
possible CP-violating effects that can enter the MSSM Higgs sector
via radiative corrections. For a discussion of the implications of
such effects, see \Refs{cpcarlos}{cpcarlos2}.}

The radiative corrections to the Higgs squared-masses have been computed
by a number of techniques, and using a variety of approximations such as
the effective potential at one-loop
\cite{early-veff,veff,berz,erz} and
two-loops \cite{zhang,espizhang,EZ2} [only the
${\cal O}(m_t^2 h_t^2\alpha_s)$ and ${\cal O}(m_t^2 h_t^4)$
two-loop results are known], and diagrammatic methods
\cite{turski,hhprl,brig,madiaz,1-loop,completeoneloop,hempfhoang,weiglein}.
Complete one-loop diagrammatic computations of the MSSM Higgs masses
have been presented by a number of groups \cite{completeoneloop};
the resulting expressions are quite complex,
and depend on all the parameters of the MSSM.
Partial two-loop diagrammatic
results are also known~\cite{hempfhoang,weiglein}.
These include the ${\cal O}(m_t^2 h_t^2 \alpha_s)$ contributions to
the neutral CP-even Higgs boson squared-masses in the on-shell
scheme~\cite{weiglein}.


One of the most striking effects of the radiative corrections to the
MSSM Higgs sector is the modification of the upper bound of the
light CP-even Higgs mass, as first noted in \Refs{early-veff}{hhprl}.  
Consider the region of parameter space
where $\tanb$ is large and $\mha\gg\mz$.  In this limit, the
{\it tree-level} prediction for $\mhl$ corresponds to its theoretical
upper bound, $\mhl=\mz$.  Including radiative corrections, the theoretical
upper bound is increased.  The dominant effect arises from an incomplete
cancellation of the top-quark and top-squark loops (these
effects actually cancel in the exact supersymmetric limit).\footnote{In
certain regions of parameter space (corresponding to large $\tanb$ and
large values of $\mu$), the
incomplete cancellation of the bottom-quark and bottom-squark loops can
be as important as the corresponding top sector contributions.  For
simplicity, we ignore this contribution in \eq{deltamh} below.}
The qualitative behavior of the radiative corrections can be most easily
seen in the large top squark mass limit, where in addition, the
splitting of the two diagonal entries and the off-diagonal entry
of the top-squark squared-mass matrix are both small in comparison to
the average of the two stop squared-masses:
\beq \label{mstwo}
\msusyy\equiv\half(\mstopa^2+\mstopb^2)\,.
\eeq
In this case, the upper bound on the lightest CP-even Higgs
mass is approximately given by
\beq \label{deltamh}
\mhl^2\lsim \mz^2+{3g^2\mt^4\over
8\pi^2\mw^2}\left[\ln\left({M_S^2\over\mt^2}\right)+{x_t^2}
\left(1-{x_t^2\over 12}\right)\right]\,,
\eeq
where $x_t\equiv X_t/M_S$.

The more complete treatments of the radiative corrections
cited above show that
\eq{deltamh} somewhat overestimates the true upper bound of $\mhl$.
Nevertheless, \eq{deltamh} correctly reflects some noteworthy features
of the more precise result.  First, the increase of the
light CP-even Higgs mass bound beyond $\mz$ can be significant.  This is
a consequence of the $m_t^4$ enhancement of the one-loop radiative
correction.
Second, the dependence of the light Higgs mass on the stop mixing
parameter $X_t$ implies that (for a given value of
$\msusy$) the upper bound of the light Higgs mass
initially increases with $X_t$ and reaches its {\it maximal} value
$X_t=\sqrt{6}\msusy$.  This point is referred to as the {\it maximal
mixing} case (whereas $X_t=0$ corresponds to the {\it
minimal mixing} case).  In a more complete computation that includes
both two-loop logarithmic and non-logarithmic corrections, the $X_t$
values corresponding to maximal and minimal mixing are 
shifted and exhibit an asymmetry under $X_t\to -X_t$ as shown in
\fig{mhxt}.
In the numerical
analysis presented in this and subsequent figures in this
section, we assume for simplicity that the third generation
diagonal soft-supersymmetry-breaking 
squark squared-masses are degenerate: $\MSUSY\equiv
M_Q=M_U=M_D$, which defines the parameter $\MSUSY$.\footnote{We also
assume that $\MSUSY\gg\mt$, in which case it follows that
$M_S^2\simeq\MSUSY^2$ up to corrections of ${\cal O}(\mt^2/\MSUSY^2)$.}


\begin{figure}
  \begin{center}
\centerline{\psfig{file=theory/mhxt.ps,height=6.5cm}}
  \end{center}
  \caption[0]{\label{mhxt} The radiatively corrected
light CP-even Higgs mass is plotted
as a function of $X_t$, where $X_t\equiv A_t-\mu\cot\beta$,
for $\mha=1$~TeV and two choices of $\tanb=3$ and 30.
%
%Three values of $\mu$ are plotted in each case: $-1$~TeV [dashed], 0
%[solid] and 1~TeV [dotted].
Here, we have taken $M_t=174.3$~GeV, and we have assumed that
the diagonal soft squark squared-masses are degenerate:
$\MSUSY\equiv M_Q=M_U=M_D=1$~TeV.
}
\end{figure}

Third, note the logarithmic sensitivity to the top-squark
masses.  Naturalness arguments that underlie
low-energy supersymmetry imply that the supersymmetric particles masses
should not be larger than a few TeV.  Still, the precise upper bound on
the light Higgs mass depends on the specific choice for the upper limit
of the stop masses.  The dependence of the light Higgs mass obtained by
the more complete computation (to be discussed further below) as a
function of $\MSUSY$ is shown in \fig{mhmsusy}.
\begin{figure}[!ht]
\centering
\centerline{\psfig{file=theory/hmhmstb3.ps,width=8cm}
\hfill
\psfig{file=theory/hmhmstb30.ps,width=8cm}}
\vskip1pc
  \caption[0]{\label{mhmsusy} The radiatively corrected light CP-even
Higgs mass is plotted as a function of $\MSUSY\equiv M_Q=M_U=M_D$,
for $\mha= 1$~TeV and two choices of $\tanb=3$ and
$\tanb=30$.  Maximal mixing and minimal mixing are defined according
to the value of $X_t$ that yields the maximal and minimal Higgs mass
as shown in \fig{mhxt}.  Here, we have chosen $M_t=174.3$~GeV,
while the supersymmetric parameters in the maximal and minimal mixing
cases are chosen according to the first two benchmark scenarios of
\Ref{benchmark} (displayed in table~\ref{benchmarks} of Section~III.B.2).
}
\end{figure}

As noted above, the largest contribution to the one-loop radiative
corrections is enhanced by a factor of $m_t^4$ and grows
logarithmically with the top squark mass.  Thus, higher order
radiative corrections can be non-negligible for large top
squark masses, in which case the large logarithms must be resummed.
The renormalization group (RG) techniques for resumming the leading
logarithms have been developed by a number of
authors \cite{rge,2loopquiros,llog,carena}.
%This procedure involves integrating a set of coupled partial
%differential equations.
%As a result, the numerical evaluation of the one-loop Higgs masses
%is time consuming and not very suitable for detailed phenomenological
%analyses.
The computation of the RG-improved
one-loop corrections requires numerical integration of a coupled set of
RG equations~\cite{llog}.
Although this procedure has been
carried out in the literature, the analysis is unwieldy
and not easily amenable to large-scale Monte-Carlo studies.
It turns out that over most of the parameter range, it is sufficient to
include the leading and sub-leading logarithms at two-loop order.
(Some additional non-logarithmic terms, which cannot be ascertained by
the renormalization group method, must also be included \cite{chhhww}.)
Compact analytic expressions have been obtained for the dominant one and
two-loop contributions to the matrix elements of the
radiatively-corrected CP-even Higgs squared-mass matrix:
\beq \label{calmatrix}
{\cal M}^2\equiv \left(
\matrix{{\cal M}_{11}^2 &  {\cal M}_{12}^2
\crr {\cal M}_{12}^2 &  {\cal M}_{22}^2 } \right)
={\cal M}_0^2+\delta {\cal M}^2\,,
\eeq
where the tree-level contribution ${\cal M}_0^2$ was given in \eq{kv}
and $\delta {\cal M}^2$ is the contribution from the radiative
corrections.
% We denote by  [the tree-level
%contribution to ${\cal M}^2$ was given in \eq{kv}]:
%\beq \label{calmatrix}
%{\cal M}^2\equiv\left(
%\matrix{{\cal M}_{11}^2 &  {\cal M}_{12}^2
%\cr {\cal M}_{12}^2 &  {\cal M}_{22}^2 } \right)\,.
%\eeq
Diagonalizing this matrix yields radiatively-corrected values for
$\mhl^2$, $\mhh^2$ and the CP-even Higgs mixing angle $\alpha$.
Explicit expressions for the ${\cal M}_{ij}^2$, given in
\Refs{carena}{hhh}, include the dominant leading and sub-leading
logarithms at two-loop order (the latter are generated by an
iterative solution to the RG-equations).  Also included are the
leading effects at one loop of the supersymmetric thresholds and the
corresponding two-loop logarithmically enhanced terms, which can again
be determined by iteration of the RG-equations.
The most important
effects of this type are squark mixing effects in the third
generation.   The procedures described above produce a
prediction for the Higgs mass in terms of running parameters in the
$\overline{\rm MS}$ scheme.  It is
a simple matter to relate these parameters to the corresponding
on-shell parameters used in the diagrammatic
calculations~\cite{espizhang,chhhww}.
%Once the latter is taken into account, the numerical results of
%the two-loop-improved analytic approximations described above
%should yield a reliable prediction for the neutral Higgs mass spectrum.

Additional non-logarithmic two-loop
contributions, which can generate a non-negligible shift in
the Higgs mass (of a few GeV), must also be included.\footnote{An 
improved procedure for computing the 
radiatively-corrected neutral Higgs mass matrix and the charged Higgs 
mass in a self-consistent way (including possible CP-violating
effects), which incorporates
one-loop supersymmetric threshold corrections to the 
Higgs--top-quark and Higgs--bottom-quark Yukawa couplings,
can be found in \Ref{cpcarlos2}.} 
A compact analytical expression that incorporates these effects
at ${\cal O}(m_t^2 h_t^2 \alpha_s)$ was given in \Ref{compact}.
An important source of such contributions are the
one-loop supersymmetric threshold
corrections to the relation between the Higgs--top-quark and 
Higgs--bottom-quark Yukawa couplings and the corresponding quark masses
[see, {\it e.g.}, \eq{yukbmass}].  These
generate a non-logarithmic two-loop shift of the radiatively corrected
Higgs mass proportional to the corresponding squark mixing parameters.
One consequence of these contributions \cite{chhhww} is the asymmetry
in the predicted value of $\mhl$ under $X_t\to -X_t$ as noted 
in \fig{mhxt}.
Recently, the computation of $\mhl$ has been
further refined by the inclusion of genuine two-loop corrections of
${\cal O}(m_t^2 h_t^4)$ \cite{EZ2}.
These non-logarithmic corrections, which depend on the stop
mixing parameters, can slightly increase the Higgs mass. This
improvement is not yet implemented in the figures shown in this section.

The numerical results displayed in figs.~\ref{cosgraph}--\ref{massvsma}
are based on the calculations of \Refs{carena}{hhh},
with improvements as described in \Refs{weiglein}{chhhww}.
The supersymmetric parameters in the maximal and minimal mixing
cases have been chosen according to the first two benchmark scenarios of
\Ref{benchmark} (displayed in table~\ref{benchmarks} of Section~III.B.2).
Of particular interest is the
upper bound for the lightest CP-even Higgs mass ($\mhl$).  At fixed
$\tan\beta$, the maximal value of
$\mhl$ is reached for $\mha\gg\mz$ (see \fig{massvsma}).
Taking $\mha$ large, \fig{mhtanb} illustrates that the
maximal value of the lightest CP-even Higgs mass
bound is realized at large $\tanb$ in the case of maximal mixing.
For each value of $\tan\beta$,
we denote the maximum value of $\mhl$
by $\mhmax(\tan\beta)$ [this value also depends on the
third-generation squark mixing parameters].
Allowing for the
uncertainty in the measured value of $\mt$ and the
uncertainty inherent in the theoretical analysis,
one finds for $\MSUSY\lsim 2$~TeV that $\mhl\lsim
\mhmax=\mhmax(\tan\beta\gg 1)$,
where
\beqa \label{mhmaxvalue}
\mhmax&\simeq  122~{\rm GeV}, \quad
\mbox{if top-squark mixing is minimal,} \nonumber \\
\mhmax&\simeq  135~{\rm GeV}, \quad
\mbox{if top-squark mixing is maximal.}
\eeqa
In practice, parameters leading to maximal
mixing are not expected in typical models of
supersymmetry breaking. Thus, in general, the  upper bound
on the lightest Higgs boson mass is expected to be
somewhere between the two extreme limits quoted above.
Cross-checks among various programs \cite{cwprog,hehprog,feynhiggs}
and rough estimates of higher order corrections not yet computed
suggest that the results for Higgs masses should be accurate to within
about 2 to 3 GeV over the parameter ranges displayed
in figs.~\ref{mhxt}---\ref{massvsma}.

\begin{figure}
  \begin{center}
\centerline{\psfig{file=theory/hmhtbmt.ps,width=9cm}}
  \end{center}
  \caption[0]{\label{mhtanb} The radiatively corrected
light CP-even Higgs mass is plotted
as a function of $\tanb$, for $\MSUSY\equiv M_Q=M_U=M_D=1$~TeV and
$\mha=1$~TeV,
for the maximal mixing [upper band] and minimal mixing [lower band]
benchmark cases [see caption to \fig{mhmsusy}].  The impact of the top
quark
mass is exhibited by the shaded bands; the central value corresponds
to $M_t=175$~GeV, while the upper [lower] edge of the bands
correspond to increasing [decreasing] $M_t$ by 5~GeV. }
\end{figure}

%In \fig{mhtanb} a sharp decrease of the value of $\mhl$ is
%observed for maximal mixing and very large values of $\tan\beta$.
%This behavior is due to  large negative corrections
%induced by the bottom Yukawa coupling, which,  due to the
%vertex corrections to be discussed below, becomes larger
%for $X_t = 2.4 \; M_S$ than for $X_t = 0$.

In \fig{massvsma}, we exhibit the masses of
the CP-even neutral and the charged Higgs masses as a function of
$\mha$.  The squared-masses of the lighter and heavier neutral CP-even
Higgs are related by
\beq \label{mhmaxtb}
\mhh^2\cosbmaii+\mhl^2\sinbmaii=[\mhmax(\tan\beta)]^2\,.
\eeq
Note that $\mhh\geq\mhmax$ for all values of
$\mha$ and $\tan\beta$ [where $\mhmax$ is to be evaluated depending on
the top-squark mixing, as indicated in \eq{mhmaxvalue}].
It is interesting to consider the behavior of the
CP-even Higgs masses in the large $\tan \beta$ regime.
For large values of $\tanb$ and for $\mha/\tan\beta \ll
\mhmax(\tan\beta)$, the off-diagonal elements of
the Higgs squared-mass matrix ${\cal M}^2$
become small compared to the diagonal elements
$|{\cal M}_{12}^2| \ll    {\cal M}_{11}^2 + {\cal M}_{22}^2 $;
${\cal M}_{12}^4 \ll    {\cal M}_{11}^2 {\cal M}_{22}^2 $.
Hence the two CP-even Higgs squared-masses
are approximately given by the diagonal elements of ${\cal M}^2$.
As before, we employ the notation where
$\mhmax$ refers to the asymptotic
value of $m_h$ at large $\tanb$ and $\mha$ (the actual numerical
value of $\mhmax$ depends primarily on the assumed values of
the third generation squark mass and mixing parameters).
If $\mha>\mhmax$, then $\mhl\simeq\mhmax$ and
$\mhh\simeq\mha$, whereas if $\mha<\mhmax$, then
$\mhl\simeq\mha$ and $\mhh\simeq\mhmax$.
This behavior can be seen in \fig{massvsma}.

\begin{figure}
%\vskip6.5cm
  \begin{center}
\centerline{\psfig{file=theory/hhmasses.ps,height=6.5cm}}
  \end{center}
  \caption[0]{\label{massvsma} Lightest CP-even Higgs mass ($\mhl$),
heaviest CP-even Higgs mass ($\mhh$) and charged Higgs mass ($\mhpm$) as
a function of $\mha$ for two choices of $\tan\beta=3$ and
$\tan\beta=30$.  Here, we have taken $M_t=174.3$~GeV, and we have assumed
that the diagonal soft squark squared-masses are degenerate:
$\MSUSY\equiv M_Q=M_U=M_D=1$~TeV.  In addition, we choose the
other supersymmetric parameters
corresponding to the maximal mixing benchmark scenario [see caption to
\fig{mhmsusy}].  The slight increase in the charged Higgs mass as
$\tan\beta$ is increased from 3 to 30 is a consequence of the
radiative corrections.
}
\end{figure}


%Recently, a diagrammatic computation of the dominant
%contributions in ${\cal O}( \alpha \alpha_s)$ to the neutral
%CP-even Higgs-boson masses, on the on shell scheme,
%has been combined~\cite{weiglein}
%with the complete diagrammatic one-loop on-shell
%result~\cite{mhiggsOL2} and the leading
%two-loop Yukawa corrections of ${\cal O}(G_F^2 m_t^6)$ obtained within
%the effective potential approach~\cite{carena}.
%After re-expressing the on-shell result in terms of the $\overline{MS}$
%parameters,  it is shown that\cite{chhhww},
%the maximal value of the lightest Higgs boson mass
%obtained via the diagrammatic two loop
%calculation \cite{weiglein} lies about 3 GeV higher than the maximal value
%obtained form the  effective
%potential-renormalization
%group methods \cite{hhh,carena}. This is effect is more important in
%the  low $\tan\beta$ region.
%The non-logarithmic contributions in particular give rise to
%an asymmetry in $\pm X_t$, while
%the RG result is symmetric in $\pm X_t$.
%It is furthermore shown (see \fig{georg}, that, by choosing the
%$\overline{MS}$  parameterization appropriately the
%dominant diagrammatic two-loop
%corrections can be absorbed into the one-loop result \cite{chhhww}.
%
%\begin{figure}
%\vskip6.5cm
%  \begin{center}
%\centerline{\psfig{file=theory/georg.eps,height=6.5cm}}
%  \end{center}
%  \caption[0]{\label{georg} Lightest CP-even Higgs mass ($\mhl$) as a
%    function of the stop mixing parameter $X_t$ evaluated in the
%    $\overline{MS}$
%scheme. Comparison of the diagrammatic two loop and the RG one loop
%    results  once the
%    renormalization scale in the latter has been chosen appropriately.
%%for  $\tanb= 30$ and $\mha = \msusy=1$~TeV.
%}
%\end{figure}


\subsubsection{The Radiatively-Corrected MSSM Higgs Sector: (b) Higgs
couplings}

Radiative corrections also significantly modify the tree-level values
of the Higgs boson couplings to fermion pairs and to vector boson pairs.
As discussed above, the tree-level Higgs couplings depend crucially on
the value of $\sinbma$.  In first approximation,
when radiative corrections of the Higgs
squared-mass matrix are computed, the diagonalizing angle $\alpha$ is
renormalized.  Thus, one may compute a radiatively-corrected value for
$\sinbma$.  This provides one important source of the radiative
corrections of the Higgs couplings.   In \fig{cosgraph}, we show the
effect of radiative corrections on the
value of $\sinbma$ as a function of $\mha$  for different values of the
squark mixing parameters and $\tanb$.  One can then
simply insert the radiatively corrected value of $\alpha$ into
eqs.~(\ref{vvcoup}), (\ref{hvcoup}) and
(\ref{qqcouplings}) to obtain radiatively-improved couplings of
Higgs bosons to vector bosons and to fermions.

To better understand the behavior of the Higgs boson couplings
to fermion pairs, consider the radiatively corrected CP-even Higgs
squared-mass matrix ${\cal M}^2$ [\eq{calmatrix}].  The complete
expressions for the individual matrix elements (even after a
series of approximations in which many sub-leading terms are
dropped) are quite involved, and we will not display them here.
However, it is useful to examine the leading contributions to the
off-diagonal element, ${\cal{M}}^2_{12}$. After including the
dominant one--loop corrections induced by the top-squark
sector\footnote{If $\tanb$ is large, bottom-squark sector effects
may become important.  For simplicity, these effects will be
neglected in the formulae exhibited in this section.}, together
with the two--loop, leading--logarithm effects\footnote{This
result is only valid if the splitting between the two top-squark
squared-masses is small compared to $\msusyy$. In addition, the
conditions $2m_t|A_t|$, $2m_t|\mu|< \msusyy$ must also be
fulfilled.} \cite{CMW1,CMW2},
\beqa
{\cal M}^2_{12} & \simeq  -\left[\mha^2 + \mz^2 \right]
\sin\beta \cos\beta
\nonumber\\
& -  \left[
\frac{3h_t^4 v^2}{48\pi^2 M_S} {\mu x_t}\left(6-{x_t
a_t}\right)\sin^2\beta - \frac{3 h_t^2 \mzz}{32 \pi^2}
{\mu x_t} \right]
\left[ 1 +{9 h_t^2 - 32 g_s^2\over
32\pi^2}\ln\left({M_S^2\over\mt^2}\right)\right],
\label{matel}
\eeqa
where $a_t\equiv A_t/M_S$, $x_t\equiv X_t/M_S$, $g_s$ is the QCD
coupling constant and  $h_t$ is the Higgs--top-quark Yukawa coupling
[\eq{htdef}].  
%Higgs-bottom quark Yukawa coupling effects have been
%neglected (these may be important only if $\tanb$ is large).
%In the above, we have assumed, for simplicity, that
%$A_b \simeq 0$ (an assumption we shall always make in the following
%analysis)\footnote{While $A_b$ can be safely ignored in the sbottom
%squark mixing, it can become relevant at large $\tan\beta$ in models
%with universal boundary conditions at the GUT scale.  Since we are
%performing a more generic analysis, we choose to fix $A_b=0$. Non-zero
%values of $A_b$ would modify slightly the Higgs masses and mixing
%angles only for large values of $\tan\beta$ and small values of $m_A$.}
%and retained only the leading terms in powers of $h_b$ and $\tan\beta$.
%
%The leading, two--loop, logarithmic corrections to the squared Higgs
%mass matrix elements included above can
%be as large as $10-20\%$ when supersymmetric particles are heavy,
%and are very relevant in determining
%the Higgs boson mass eigenvalues
%and mixing angles. Observe that Eq.~(\ref{higgsm2l}) may be easily
%obtained from the above expression,\footnote{There is a slight
%discrepancy in the subdominant, Yukawa--dependent, two--loop,
%leading--logarithmic
%corrections, which is due to the fact that the expressions written
%above
%are only strictly valid for values of the CP--odd mass of the order
%of the weak scale.} by computing the determinant of
%the Higgs boson mass matrix and
%setting the heavy CP--even Higgs boson mass approximately equal to
%$m_A$.
The mixing angle $\alpha$ can be determined by diagonalizing the CP-even
Higgs squared-mass matrix [\eq{calmatrix}]:
\beq
\sin\alpha\cos\alpha = {{\cal M}_{12}^2 \over
\sqrt{({\rm Tr}{\cal M}^2)^2-4\det{\cal M}^2}}\,,
\qquad\qquad
\cos^2\alpha-\sin^2\alpha = {{\cal M}_{11}^2-{\cal M}_{22}^2 \over
\sqrt{({\rm Tr}{\cal M}^2)^2-4\det{\cal M}^2}}\,.
\label{sin2alpha}
\eeq
It follows that in the limit where ${\cal M}_{12}^2\to 0$,
either $\sin\alpha\to 0$ (if ${\cal M}_{11}^2>{\cal M}_{22}^2$) or
$\cos\alpha\to 0$ (if ${\cal M}_{11}^2<{\cal M}_{22}^2$).
As a result, some of the Higgs boson couplings
to quark and lepton pairs [\eq{qqcouplings}] can be strongly
suppressed, if radiative corrections suppress the value of
${\cal M}_{12}^2$.\footnote{Although ${\cal M}_{12}^2$ is {\it negative}
at tree level (implying that $-\pi/2\leq\alpha\leq 0$), it is possible
that radiative corrections flip the sign of ${\cal M}_{12}^2$ [see
\eq{matel}].  Thus, the range of the radiatively corrected angle
$\alpha$ can be taken to be $-\pi/2\leq\alpha\leq\pi/2$.}

If ${\cal M}_{12}^2\simeq 0$ and $\tanb$ is large (values of
$\tan\beta \gsim 5$ are sufficient), the resulting pattern of Higgs
couplings is easy to understand.  In this limit,
${\cal M}^2_{11}\simeq\mha^2$ and ${\cal M}^2_{22}\simeq \mhmax$,
as noted at the end of section~I.C.2.  Two cases must be treated
separately depending on the value of $\mha$.
First, if $\mha < \mhmax$, then
$\sin\alpha \simeq - 1$ (assuming ${\cal M}_{12}^2 <0$ as in tree-level),
$\cos\alpha \simeq 0$ and $\sinb\simeq \cos(\beta - \alpha) \simeq 1$.
In this case, the lighter CP-even Higgs boson
$\hl$ is roughly aligned along the $\Phi_d^0$ direction and
the heavier CP-even Higgs boson
$\hh$ is roughly aligned along the $\Phi_u^0$ direction [see
\eq{scalareigenstates}].  In particular, the coupling of $\hh$ to
$b\bar b$ and $\tau^+\tau^-$ is significantly diminished (since
down-type fermions couple to $\Phi_d^0$), while the $\hh VV$ couplings
[\eq{vvcoup}] are approximately equal to those of the Standard Model
[since $\cosbmaii \simeq 1$].
Consequently, the branching ratios of $\hh$ into
$gg$, $\gamma\gamma$, $c\bar c$, and $W^{+}W^{-}$ can be
greatly enhanced over \SM\ expectations~\cite{CMW1,CMW2,Wells,bdhty}.
Second, if $\mha > \mhmax$ then $\sin\alpha \simeq 0$ and
$\sinb\simeq\cosa\simeq \sin(\beta - \alpha) \simeq 1$ and
the previous considerations for $\hh$ apply now to $\hl$.

For moderate or large values of
$\tan\beta$, the vanishing of ${\cal M}_{12}^2$ leads
to the approximate numerical relation \cite{CMW1}:
\begin{equation}
\left[{\mha^2\over \mz^2}
+ 1 \right] \simeq
{\mu x_t \tanb \over 100M_S} \left(
{2 a_t x_t} -
11 \right)
\left[ 1 -\frac{15}{16 \pi^2}  \log\left({M_S^2\over\mt^2}\right) \right],
\label{suppression}
\end{equation}
where we have replaced $h_t$, $g_s$
and the weak gauge couplings by their approximate numerical values at the
weak scale.
For low values of $\mha$, or large values of the
mixing parameters, a cancellation can easily
take place for large values
of $\tan\beta$.  For instance, if $M_S \simeq
1$ TeV,
$\mu=- X_t=M_S$,
and $m_A  \simeq m_h \simeq 80$ GeV, a cancellation can
take place for $\tan\beta \simeq 28$, with
$m_H \simeq 117$ GeV. The heaviest
CP--even Higgs boson has Standard Model--like couplings
to the gauge bosons
[$\cos^2(\beta-\alpha)\simeq 1$], but the branching ratios for
decays into $W^{\pm}$ bosons, gluons and charm quarks are
enhanced with respect to the SM case:
${\rm BR}(W^+W^-)=0.34$, ${\rm BR}(gg)=0.27$ and ${\rm BR}(c\bar c)=0.11$.


Although it is difficult to have exact cancellation of the off-diagonal element
${\cal M}_{12}^2$, in many regions of the supersymmetric parameter
space,  important suppressions may be present.
Generically, the $m_t^4$--dependent radiative corrections
to ${\cal M}_{12}^2$ depend strongly on the sign of the product
$\mu X_t$ ($A_t \simeq X_t$ for large $\tan\beta$
and moderate $\mu$) and on the value of $|A_t|$.
For the same value of $X_t$, a change in the
sign of $\mu$ can lead to observable variations in the branching
ratio for the Higgs boson decay into bottom quarks. If $|a_t|\lsim
\sqrt{11/2}$, the absolute value of the off--diagonal matrix
element, and hence, the coupling of bottom
quarks to the Standard Model--like Higgs boson tends to be suppressed
(enhanced) for values of
$\mu A_t < 0$  ($\mu A_t > 0$).
For larger values of $|a_t|$, the suppression (enhancement)
occurs for the opposite sign of $\mu A_t$.

In addition to the radiative corrections to couplings that
enter via the renormalization of the CP-even Higgs mixing angle
$\alpha$, there is another source of radiative corrections to Higgs
couplings that is potentially important at large $\tanb$.  Such
corrections depend on the details of the MSSM spectrum (which enter via
loop-effects).  The corrections we wish to explore now are those that
arise in the relation between $m_b$ and $\tanb$.  At tree-level, the
Higgs couplings to $b\bar b$ are proportional to the
Higgs--bottom-quark Yukawa coupling $h_b$ [\eq{hbdef}].
Deviations from the tree-level relation, \eq{hbdef},
due to radiative corrections are calculable
and finite \cite{hffsusyqcd,deltamb,deltamb1,deltamb2}.  One of the
fascinating properties of such corrections is that in certain cases
the corrections do {\it not} vanish in the limit of large supersymmetric
mass parameters.  These corrections grow with $\tanb$ and therefore can
be significant in the large $\tanb$ limit.
%
%Some of the phenomenological implications of these corrections
%have been considered for MSSM Higgs boson decays.
%The main reason why these one--loop corrections
%are particularly important
%is that they do not decouple in the limit of a heavy supersymmetric
%spectrum.
%As mentioned above,
In the supersymmetric limit, bottom quarks only couple to $\Phi_d^0$.
However, supersymmetry is broken and
the bottom quark will receive a small coupling
to $\Phi_u^0$ from radiative corrections,
\begin{equation}
-{\cal L}_{\rm Yukawa} \simeq h_b \Phi_d^0 b \bar{b} + (\Delta h_b)
\Phi_u^0 b\bar{b}\,.
\label{couplings}
\end{equation}
Because the Higgs doublet acquires a vacuum
expectation value, the bottom quark mass receives
an extra contribution equal to
$(\Delta h_b) v_u$.  Although $\Delta h_b$ is
one--loop suppressed with respect to $h_b$,
for sufficiently large values of $\tan\beta$ ($v_u \gg v_d$)
the contribution to the bottom quark mass of both terms in
\eq{couplings} may be comparable in size. This induces a
large modification in the tree level relation [\eq{hbdef}],
\beq
m_b = {h_b v_d\over\sqrt{2}} (1+\Delta_b)\,, \qquad
\label{yukbmass}
\eeq
where $\Delta_b \equiv (\Delta h_b)\tan\beta/h_b$.
The function $\Delta_b$ contains two main
contributions, one from a bottom squark--gluino loop
(depending on the two bottom squark masses $M_{\tilde b_1}$
and $M_{\tilde b_2}$ and the gluino mass $M_{\tilde g}$) and another one
from a
top squark--higgsino loop (depending on the two top squark masses
$M_{\tilde t_1}$ and $M_{\tilde t_2}$ and the higgsino mass parameter
$\mu$).  The explicit form of $\Delta_b$ at one--loop in the limit of
$M_S \gg m_b$ is given by \cite{deltamb,deltamb1,deltamb2}:
%\begin{center}
\beq
\Delta_b \simeq {2\alpha_s \over 3\pi} 
M_{\tilde g}\mu\tan\beta~I(M_{\tilde b_1},
M_{\tilde b_2},M_{\tilde g})
 + {Y_t \over 4\pi} A_t\mu\tan\beta~I(M_{\tilde t_1},M_{\tilde t_2},\mu),
\label{deltamb}
\eeq
%\end{center}
where $\alpha_s=g_s^2/4\pi$,
$Y_t\equiv h_t^2/4\pi$, and contributions proportional to the
electroweak gauge couplings have been neglected.  In addition,
the function $I$ is defined by
\beq
I(a,b,c) = {a^2b^2\ln(a^2/b^2)+b^2c^2\ln(b^2/c^2)+c^2a^2\ln(c^2/a^2) \over
(a^2-b^2)(b^2-c^2)(a^2-c^2)},
\eeq
and is manifestly positive.
%Smaller contributions to $\Delta_b$ have been neglected for the
%purpose of this discussion.
Note that the Higgs coupling proportional to $\Delta h_b$ is a
manifestation of the broken supersymmetry in the low energy theory;
hence, $\Delta_b$ does not decouple
in the limit of large values of the supersymmetry breaking masses. Indeed,
if all supersymmetry breaking mass parameters (and $\mu$)
are scaled by a common factor, the correction
$\Delta_b$ remains constant.

Similarly to the case of the bottom quark, the relation between $m_\tau$ and
the Higgs--tau-lepton Yukawa coupling $h_\tau$ is modified:
\beq
m_\tau = {h_\tau v_d\over\sqrt{2}} (1+\Delta_\tau).
\eeq
The correction $\Delta_\tau$ contains a contribution from a
tau slepton--neutralino loop (depending on the two stau masses
$M_{\tilde \tau_1}$ and $M_{\tilde \tau_2}$ and the
mass parameter of the $\widetilde B$ (``bino'') component
of the neutralino, $M_{1}$) and a
tau sneutrino--chargino loop (depending on the tau sneutrino mass
$M_{\tilde \nu_\tau}$, the mass parameter of the $\widetilde W^\pm$
component of the chargino, $M_{2}$, and $\mu$).
It is given by \cite{deltamb1,deltamb2}:
\beq
\Delta_\tau = {\alpha_1 \over 4\pi} M_1\mu\tan\beta\,
I(M_{\tilde\tau_1},
M_{\tilde\tau_2},M_1) + {\alpha_2 \over 4\pi} M_2\mu\tan\beta \,
I(M_{\tilde\nu_\tau},M_2,\mu),
\eeq
where $\alpha_2\equiv g^2/4\pi$ and $\alpha_1\equiv g^{\prime 2}/4\pi$
are the electroweak gauge couplings.
%=\frac{g_1^2}{ 4\pi}$, $g_1$ is the $U(1)$ hypercharge coupling,
%$\alpha_2=\frac{g_2^2}{4\pi}$, $g_2$ is the $SU(2)$ weak isosopin coupling.
Since corrections to $h_\tau$ are proportional to $\alpha_1$ and
$\alpha_2$, they are  expected to be smaller than
the corrections to $h_b$.  For example,
for $\tan\beta \lsim 50$, one finds
$\Delta_\tau < 0.15$.


From \eq{couplings} we can obtain the couplings of the physical neutral Higgs
bosons to $b\bar b$.  First, consider the CP-odd Higgs boson
[\eq{hastate}].  From \eq{couplings}, we obtain for the $\ha b\bar b$
coupling:
\beq \label{lintabb}
{\cal L}_{\rm int} = - i g_{\ha b\bar b} A \bar{b} \gamma_5 b
\eeq
with
\beq
g_{\ha b\bar b}
= h_b \sin\beta + \Delta h_b \cos\beta \simeq h_b
\sin\beta = {\sqrt{2}m_b\over (1 + \Delta_b) v}\tan\beta\,,
\label{bhCP}
\eeq
where we have used the result of \eq{yukbmass} for $h_b$, and we have
discarded a term of ${\cal O}(\Delta_b/\tan^2\beta)$.
Similarly, for the CP-even Higgs bosons [\eq{scalareigenstates}],
we obtain for the $\hl b\bar b$ and $\hh b\bar b$ couplings:
%Using Eqs. (\ref{couplings}) and (\ref{mixings}), together
%with the relation of the bottom Yukawa coupling to the bottom mass,
%Eq. (\ref{yukbmass}), it is easy to show that the effective
%couplings of the CP--even Higgs bosons, $\bar{h}_b$ and $\tilde{h}_b$,
\begin{equation}
{\cal L}_{\rm int} =  g_{\hl b\bar b}\hl b \bar{b} +
 g_{\hh b\bar b}\hh b \bar{b}
\end{equation}
with
\begin{equation}
 g_{\hl b\bar b}  \simeq -\frac{\sqrt{2}m_b \sin\alpha}{v \cos\beta}\
{1\over 1+\Delta_b}
\left[ 1 - \frac{\Delta_b}{\tan\alpha \tan\beta} \right]\,,
\label{barhb}
\end{equation}
\begin{equation}
 g_{\hh b\bar b} \simeq \frac{\sqrt{2}m_b \cos\alpha}{v \cos\beta}
{1\over 1+\Delta_b}
\left[ 1 + \frac{\Delta_b\tan\alpha}{\tan\beta} \right]\,.
\label{tildehb}
\end{equation}
~\\
%The value of $\Delta(m_b)$
%in Eq. (\ref{deltamb}) is defined
%at the scale $M_{S}$,
%where the sparticles are decoupled. The $ h_b$ and $\Delta h_b$  couplings
%should be computed at that scale, and run down with their respective
%renormalization group equations to the scale $m_A$, where the
%relations between the couplings of the bottom quark  to the neutral Higgs
%bosons and the running bottom quark mass, Eqs. (\ref{bhCP}), (\ref{barhb}),
%and (\ref{tildehb}) are defined.
%In the present study we have
%defined the running bottom mass at the scale $m_A$ as a function of
%$m_b(m_b) \simeq$ 4.25 GeV while using two--loop renormalization group
%equations in the effective standard model theory at scales $Q < m_A$.
%The above procedure leads to
% a consistent definition of the bottom quark couplings to
%the Higgs bosons when the three neutral Higgs boson
%masses are of the same order.
%For large values of the CP--odd Higgs boson mass, instead, $\bar{h}_b$ must be
%evolved with SM renormalization group equations from $m_A$ down to $m_h$.
%The definition of the couplings of the bottom quark to the Higgs bosons
%at the scale of the
%corresponding Higgs boson mass
%is  chosen to take into account the bulk of the QCD correction.
%Indeed, it
%is known that this choice of scale represents well
%the bulk of the QCD corrections to the Higgs boson decay into quarks
%and gluons~\cite{CARZER}. However,
%for the production process $b \bar{b} \phi$,
%a complete study of the NLO effects remains necessary to get
%a definitive estimate of the Tevatron reach in this production channel.

The sign of  $\Delta_b$
is governed by the sign of $M_{\tilde{g}}  \mu$,
since the  bottom-squark gluino loop  gives the dominant contribution
to Eq. (\ref{deltamb}). Henceforth, we define  $M_{\tilde{g}}$ to be
positive.  Then for $\mu >0$ ($\mu<0$), the radiatively
corrected coupling  $g_{\ha b\bar b}$ in \eq{bhCP}
is suppressed (enhanced) with respect to
its  tree level value.  In contrast, the radiative corrections to
$g_{\hl b\bar b}$ and $g_{\hh b\bar b}$ [\eqns{barhb}{tildehb}]
have a more complicated
dependence on the supersymmetric parameters due to the dependence on
the CP-even mixing angle $\alpha$.
Since $\alpha$ and  $\Delta_b$  are governed by different
combinations of the supersymmetry breaking parameters, it is difficult
to exhibit in a simple way the behavior
of the radiatively corrected couplings of the CP-even Higgs bosons
to the bottom quarks as a function of the MSSM parameters.
% as can be done for  $g_{\ha b\bar b}$.


It is interesting to study different limits of the above couplings.
For $m_A \gg \mhmax$,
the lightest CP--even Higgs boson should behave like
the \SM\ Higgs boson.  To verify this assertion, one can use the result
for the CP-even mixing angle $\alpha$ in the decoupling limit
[\eq{cotalf}].  Plugging this result into \eq{barhb}, one indeed finds
that in the limit of large $\mha$,
% Indeed, in this limit
%$\cos\beta \simeq -\sin\alpha$ and $\sin\beta \simeq \cos\alpha$, and
$g_{\hl b\bar b}  = \sqrt{2}m_b/v$, which is precisely the coupling of
the \SM\ Higgs boson to $b\bar b$ \cite{loganetal}.
%(as expected from  the decoupling
%limit), and
Moreover, in the same limit of large $\mha$,
$g_{\hh b\bar b} \simeq h_b \sin \beta [1 + {\cal O}
(\Delta_b/\tan^2\beta)]$.  When $\mha$ approaches $\mhmax$,
due to the large $1/\cos\beta$ factor appearing in the definition
of the Yukawa coupling [\eq{barhb}],
a small departure from $\cos \beta \simeq - \sin \alpha$ can induce large
departures of the coupling $g_{\hl b\bar b}$
from the \SM\ value.
%similar to the ones found for ${\cal M}_{12}\to 0$ and
%$\Delta(m_b)=0$.
In contrast, for $\mha \ll\mhmax$, $\sin\beta \simeq -\sin\alpha \simeq 1$.
In this case, $g_{\hl b\bar b}\simeq
h_b \sin \beta [1 + {\cal O}(\Delta_b/\tan^2\beta)]$, while
the $\hh$ coupling to $b\bar b$ may deviate substantially from
the corresponding coupling of the \SM\ Higgs boson.
%as happens for vanishing values of the bottom mass corrections.


As discussed above, in the large $\tan\beta$
regime, the off--diagonal elements of the Higgs squared-mass
matrix can receive large radiative corrections
with respect to the tree--level value.
% $\left({{\cal M}^2_{12}}\right)^{tree}
%\simeq -(m_A^2 + M_Z^2)/\tan\beta$. When both $\Delta(m_b)$ and
%$\Delta(m_{\tau})$ are small, the coupling of the standard model--like
%Higgs boson to bottom quarks and $\tau$ leptons vanishes for
%vanishing $\sin 2 \alpha$ (see
%Eq. (\ref{sin2alpha})). The reason for this cancellation
%when $\sin2\alpha = 0$ is that the standard model--like Higgs boson
%becomes a pure $H_2^0$ state, which does not couple to bottom
%quarks and $\tau$--leptons at tree level.
If the bottom-quark and tau-lepton mass corrections
are large, then the bottom-quark and tau-lepton couplings to either
$\hl$ or $\hh$  do not vanish when $\sin 2\alpha=0$,
%for $\sin2\alpha = 0$,
but are given by $\Delta h_b$ and
$\Delta h_{\tau}$, respectively.  This result is exhibited explicitly in
\eqs{barhb}{tildehb}.  For example, in the limit of
$\sin\alpha = 0$, \eq{barhb} yields
\begin{equation}
g_{\hl b\bar b}={\sqrt{2} m_b\Delta_b \over
v\sinb(1+\Delta_b)}=\Delta h_b\,. \label{barhb2}
\end{equation}
The last step above is a consequence of \eq{yukbmass} and the
definition of $\Delta_b$ in terms of $\Delta h_b$.  Likewise, in
the limit of $\cosa=0$, \eq{tildehb} yields $g_{\hh b\bar
b}=\Delta h_b$.  In both cases, the Higgs couplings to $b\bar b$
are much smaller than the corresponding \SM\ coupling only if
$|\Delta_b| \ll 1$.\footnote{In a full one-loop computation, one must
take into account the momentum dependence of the two-point functions,
which leads to an effective momentum-dependent mixing angle.
Nevertheless, suppressions of Higgs branching ratios persist, depending
on the choice of the supersymmetric parameters, as shown in
\Ref{hffsusyprop}.}
Similar results apply to the Higgs couplings to $\tau^+\tau^-$.
%A similar expression to Eq.~(\ref{barhb2}) holds for the $\tau$ lepton
%coupling.

If $\Delta_b\simeq {\cal O}(1)$ (which is possible if $\tanb\gg
1$), then the Higgs couplings to $b\bar b$ are not particularly
small in the limiting cases considered above.  However, a strong
suppression of the couplings can still take place if the value of
the CP-even Higgs mixing angle $\alpha$ is slightly shifted away
from the limiting values considered above.  For example,
\eq{barhb} implies that $g_{\hl b\bar b}\simeq 0$ if
$\tan\alpha=\Delta_b/\tanb$. Inserting this result into the
corresponding expression for the $\hl\tau^+\tau^-$ coupling, it
follows that
%For values of
%$\Delta(m_b)$ of order 1, however, a strong suppression of the bottom
%coupling $\bar{h}_b$ can still occur for slightly different values of
%the Higgs mixing angle $\alpha$, namely
%\begin{equation}
%\tan\alpha \simeq \frac{\Delta (m_b)}{\tan\beta}.
%\end{equation}
%Under these conditions,
\begin{equation}
g_{\hl\tau^+\tau^-} = \frac{\sqrt{2}m_{\tau}\cosa}{v\sin\beta} \left(
\frac{ \Delta_{\tau} - \Delta_b}{ 1 + \Delta_{\tau}} \right)\,.
\end{equation}
Similarly, \eq{tildehb} implies
that $g_{\hh b\bar b}\simeq 0$ if $\tan\alpha=-\tanb/\Delta_b$.
Inserting this result into the corresponding expression for the
$\hh\tau^+\tau^-$ coupling, it follows that
\begin{equation}
g_{\hh\tau^+\tau^-} = \frac{\sqrt{2}m_{\tau}\sina}{v\sin\beta}\left(
\frac{ \Delta_{\tau} - \Delta_b}{ 1 + \Delta_{\tau}} \right)\,.
\end{equation}
In both cases, we see that although the Higgs coupling to $b\bar b$ can
be strongly suppressed for certain parameter choices, the corresponding
Higgs coupling to $\tau^+\tau^-$ may be  typically unsuppressed if
$\tan\beta$ is very large and $\Delta_b\simeq {\cal O}(1)$.  In such
cases, the $\tau^+\tau^-$ decay mode can be the dominant Higgs decay
channel for the CP-even Higgs boson with SM-like couplings to gauge
bosons.

%A similar expression is obtained for the coupling $\tilde{h}_{\tau}$
%in the case $\tilde{h}_b = 0$.
%Hence, if $\tan\beta$ is very large and $\Delta(m_b)$ is of order
%one, the $\tau$ Yukawa coupling may {\it not} be strongly suppressed
%with
%respect to the Standard Model case and can provide the {\it dominant}
%decay mode for a Standard Model--like Higgs boson.

To recapitulate, a cancellation in the off--diagonal element of
the CP-even Higgs squared-mass matrix (as a consequence of the
radiative corrections) can lead to a strong suppression of the
Higgs boson coupling to $b\bar b$ and $\tau^+\tau^-$. In general,
this implies a significant increase in the corresponding Higgs
branching ratios into gauge bosons and charm quark pairs. However,
for very large values of $\tan\beta$ and values of the
bottom-quark mass corrections $\Delta_b$ of order one, the Higgs
branching ratio into $\tau^+\tau^-$ may increase in the regions in
which the $b\bar b$ decays are strongly suppressed.

A similar analysis can be used to derive radiatively corrected couplings
of the charged Higgs boson to fermion pairs \cite{chhiggstotop2,eberl}.  
The tree-level couplings
of the charged Higgs boson to fermion pairs
[\eq{hpmqq}] are modified accordingly by replacing
$m_b \rightarrow m_b/(1 + \Delta_b)$ and
$m_{\tau} \rightarrow m_{\tau}/(1 + \Delta_{\tau})$, 
respectively.


%\bigskip
%Sola stuff.
%\bigskip
\subsubsection{Present Status of the MSSM Higgs Boson Searches}

Before turning to the relevant MSSM Higgs production processes
and decay modes at the
Tevatron, we shall survey the expected status of the MSSM Higgs
search at the end of the final LEP2 collider run.  At LEP2,
the MSSM Higgs boson production processes are $e^+e^- \rightarrow Z^*
\to Z\hl$ and $\hl \ha$.  In very tiny regions of MSSM parameter
space, the production of a $\hh$ instead of a $\hl$ is kinematically
possible and is also considered.
Since the $ZZ\hl$ coupling is proportional to $\sin(\beta-\alpha)$,
while the $Z\hl\ha$ coupling is proportional to $\cos(\beta-\alpha)$,
these two processes are complementary throughout the
$\mha$ {\it vs.} $\tan\beta$ plane.
%Given that the MSSM Higgs boson production cross
%sections at LEP2 depend on the couplings of the Higgs bosons to vector
%bosons, and, that these couplings depend importantly on $\tanb$ and
%$\mha$ (see \fig{cosgraph})), it follows that the above search
%channels are complementary in the search for a MSSM Higgs boson
%throughout the MSSM $\tanb$--$\mha$ plane.
The large $\tanb$ ($\tanb
>3$), low $\mha$ region [where $\cosbma$ is large, close to one] is
covered via $\hl \ha$ searches while the large $\mha$ region is
covered via $\hl Z$ searches.
The large $\mha$ region corresponds to the decoupling limit [where
$\sinbma\simeq 1$], and the $\hl Z$ production cross~section
approaches its SM value.  The discovery and
exclusion limits in the latter case can be determined from the
corresponding results of the Standard Model Higgs boson shown in
\fig{SM95}. In the low $\mha$
and low $\tanb$ region both the $Z\hl$ and $\hl\ha$ production channels
contribute.

At present, the published LEP bounds for the MSSM CP-even Higgs bosons are:
$\mhl> 88.3$~GeV and $\mha> 88.4$~GeV at 95\% CL \cite{LEPHiggs}.
These results correspond to the large $\tan\beta$ region in which $Z\hl$
production is suppressed.  The Higgs mass limits then arise from the
non-observation of $\hl\ha$ production.  These limits are expected to
improve slightly when the LEP2 data from 2000 are fully analyzed.  
Projected coverage of the MSSM parameter space via
Higgs searches after the final run of LEP2 are shown in
\fig{mh200} \cite{pjanot}.
%Suppose that no Higgs signal is seen at LEP.  This would then rule
%out a portion of the MSSM parameter space.
The projected 95\% CL exclusion contours 
in the $(a) \tanb$ {\it vs.} $\mhl$ and $(b) \tanb$ {\it vs.} $\mha$
planes are exhibited.
To make such plots, the value of $\MSUSY$ and
the squark mixing, which affect the radiatively corrected Higgs
masses and couplings, must be specified.
In general, the most conservative limits
in the $\tan\beta$ {\it vs.} $\mhl$ [or $\mha$] plane are
obtained in the case of maximal mixing chosen for these
plots.\footnote{The contours shown in \fig{mh200} do not employ the most
recent set of two-loop Higgs mass radiative corrections (discussed in
Section~I.C.2).  When these corrections are included, the theoretical
upper bound on $\mhl$ is slightly increased.  This implies that the
region of MSSM parameter space {\it not} ruled out by the projected 
LEP Higgs searches is slightly larger than the one shown in \fig{mh200}.} 
However, there are some
regions of MSSM parameter space, in which the Higgs couplings to
vector boson pairs or fermion pairs may be suppressed and small holes
may appear for other choices of the supersymmetric parameters 
(see {\it e.g.} refs.~\cite{benchmark} and \cite{tomjunk}) in areas
that are otherwise covered in the maximal mixing case shown in
\fig{mh200}.  
%In practice, the latter will only occur for very
%extreme values of the mixing squark sector parameters which are
%theoretically disfavored.

%In particular, for special values of the MSSM parameters, it may occur
%that $\sinbma$ is too small suppressing the $\hl Z$ production cross
%section and at the same time that the production of $\hl \ha$ is not
%yet kinematically allowed. The other possibility is that the coupling
%of the Higgs bosons to $b$ quarks is strongly suppressed. Given the
%mass reach of LEP it follows that the latter will only occur for very
%extreme values of the mixing squark sector parameters which are
%theoretically disfavored.

\begin{figure}[!ht]
\centering
\centerline{\psfig{file=theory/mhtbmax2000.eps,width=8cm}
\hfill
\psfig{file=theory/matbmax2000.eps,width=8cm}}
\vskip1pc
\caption[0]{\label{mh200} Projected LEP2 contours of the 95\% C.L. exclusion
limits for MSSM Higgs sector parameters as a function of $\tan\beta$ and
(a) $\mhl$ and (b) $\mha$ (in GeV), taken from \Ref{pjanot}.
In (a), the dark shaded region is
theoretically forbidden.  The light shaded regions in (a) and (b) are
expected to be ruled out after the final year of LEP2 running.  The
solid [dashed] line boundary between the light shaded and unshaded
region corresponds to an assumed value of $M_t=174.3$ [180]~GeV, which
governs the size of the radiative corrections to the Higgs masses.
The contours shown have been obtained assuming an average top-squark
squared-mass of $(1~{\rm TeV})^2$ and maximal top-squark mixing.}
% assuming a collider energy of 200 GeV
%and 100 or 200 pb$^{-1}$ of data per experiment.}
\end{figure}

From \fig{mh200} $(a)$ and $(b)$,
it follows that, independent of the value
of $\tanb$, LEP2 will be sensitive to a light Higgs boson
with a mass up to about 93 GeV \cite{pjanot},
depending slightly on the final luminosity collected.
For low values of $\tanb$, the coverage is better and the search
will cover up to the theoretically largest allowed value of the
lightest Higgs mass, which is obtained for large values of the CP-odd
Higgs mass. In this decoupling limit,
the lightest Higgs has SM-like properties.  The maximal LEP2
reach for a light Higgs with SM-like couplings to the $Z$ boson is the same
as the SM one, namely up to a mass of about 112 GeV for a $5\sigma$ discovery.

The anticipated final LEP lower bounds on the SM-like Higgs mass
significantly constrain the MSSM parameter space.
In particular, it is possible to derive a lower bound on $\tanb$ as a
function of the stop masses and mixing angles.
As an example, \fig{ccpw}
%from \Ref{CCPW}
shows the bounds on $\tanb$ that will be obtained in the case that no Higgs
boson is found at LEP with a mass below 115~GeV in the $Z \hl$ channel.
%\footnote{In the
%numerical computations, the experimental Higgs mass
%bounds have been lowered by 2 GeV with
%  respect to this limit, to partially account for theoretical
%  uncertainties in the computations.}
This figure illustrates that the lower bound on $\tan\beta$ will range
between about 2.3 and 5 depending on the value of the stop masses and
mixing.  It is interesting to note that even for very large values
of the stop masses and mixing angles, the bound on $\tanb$ resulting
from the direct bound on the Higgs mass becomes stronger than the
bound on $\tanb$ that one can get by the requirement of perturbative
consistency of the theory up to scales of order $M_{\rm PL}$ (associated
with the infrared fixed point solution of the top Yukawa coupling)
\cite{CCPW}.
The constraints on the
MSSM parameters that will be available from Higgs searches
after the final run of LEP2 will be useful for guiding supersymmetric
and Higgs particle searches at the Tevatron and the LHC.


\begin{figure}[!ht]
\begin{center}
\centerline{\psfig{file=theory/htanb3r_p2.ps,width=8cm}}
\end{center}
\vspace*{-1pc}
\caption[0]{\label{ccpw} Bounds on $\tanb$ obtained for
$M_t$ = 175 GeV and for a lower bound on the Higgs mass
of 115 GeV, as a function of the heavier stop mass $M_{\tilde{t}_2}$,
for different values of the stop mass splitting $\Delta M_{\tilde t}\equiv
M_{\tilde{t}_2}-M_{\tilde{t}_1}$ ranging from 0 to 500 GeV
(subject to the constraint that the lighter top squark mass is greater
than 65~GeV).
%The absence of a contour for $\Delta M_{\tilde t}=0$
%indicates that LEP will be able to rule out all values of
%$\tan\beta$ below 5.2 in the case of minimal mixing
%if no Higgs boson is seen.
As $\Delta M_{\tilde t}$  increases, the contours move to the left
until the 400~GeV contour is reached.  (The contours between 300
GeV and 450 GeV are all nearly on top of the 400~GeV contour.)  The
dashed line [the 500 GeV contour] shows that the contours begin to move
back to the right as $\Delta M_{\tilde t}$ increases further.
This plot is an updated version of one that was first shown in
\Ref{CCPW}, and was produced using \Ref{feynhiggs}.
  }
\end{figure}


\medskip

One of the goals of the Tevatron Higgs Working Group is to
examine the potential for the upgraded Tevatron to extend the LEP2
MSSM Higgs search.  We begin by examining the most promising channels
for MSSM Higgs discovery at the upgraded Tevatron.

\subsubsection{MSSM Higgs Boson Decay Modes}

In the MSSM, we must consider the decay properties of three neutral
Higgs bosons ($\hl$, $\hh$ and $\ha$) and one charged Higgs pair
($\hpm$).  In the region of parameter space where $\mha\gg m_Z$
and the masses of supersymmetric particles are large, the
decoupling limit applies, and we find that the properties of $\hl$
are indistinguishable from the Standard Model Higgs
boson.\footnote{If supersymmetric particles are light, then
the decoupling limit does not strictly apply even in the limit of
$\mha\gg m_Z$.  In particular, the $\hl$ branching ratios are
modified, if the decays of $\hl$ into supersymmetric particles are
kinematically allowed.  In addition, 
if light superpartners exist that can couple
to photons and/or gluons, then the one-loop $gg$ and $\gamma\gamma$
decay rates would also deviate from the corresponding Standard Model
Higgs decay rates due to the extra contribution of the
light superpartners appearing in the loops.}
In this
case, the discussion of Section~I.A.2 applies, and the decay
properties of $\hl$ are precisely those of the Standard Model Higgs
boson.  In this case, the heavier Higgs states, $\hh$, $\ha$ and
$\hpm$, are roughly mass degenerate and cannot be observed at the
upgraded Tevatron.

For values of $\mha\sim {\cal O}(\mz)$, all Higgs boson states lie
below 200~GeV in mass, and could in principle be accessible at an
upgraded Tevatron (given sufficient luminosity).  In this parameter
regime, there is a significant area of the parameter space in which
none of the neutral Higgs boson decay properties approximates that of
the Standard Model Higgs boson.  For example, when
$\tan\beta$ is large, supersymmetry-breaking effects can significantly
modify the $b\bar b$ and/or the $\tau^+\tau^-$ decay rates with
respect to those of the Standard Model Higgs boson.
Additionally, the Higgs bosons can decay into new channels, either
containing lighter Higgs bosons or supersymmetric particles.
In the following,
the decays of the neutral Higgs bosons $\hl$, $\hh$ and
$\ha$ (sometimes denoted collectively by $\phi$) and 
the decays of charged Higgs
bosons are discussed with particular emphasis on differences
from Standard Model expectations.

As discussed earlier (see \fig{mh200}), if the
Higgs boson has not yet been discovered by the time the upgraded
Tevatron begins its run, then the anticipated LEP limits will rule out
values of $\tan\beta$ below about 2.5.\footnote{LEP will not be able to
rule out small values of $\tan\beta\simeq 0.5$ in a narrow range of
MSSM parameter space.  However, theoretical arguments favor values of
$\tan\beta\gsim 1$, so we do not consider further the possibility of
$\tan\beta\lsim 1$.}
Thus, in the following discussion, two cases are considered to illustrate
the difference between ``low'' and ``high'' $\tan\beta$:
$\tan\beta=6$ and 30.
%The results shown below include the effects of the dominant
%radiative corrections, which affect
%the masses and the couplings of the Higgs sector as described in the
%previous section.

%\break
\vskip6pt\noindent
\centerline{a.~Neutral Higgs Boson Decays}
\vskip6pt

\vskip6pt\noindent
\underline{$\phi\to f\bar f$} \\[0.2cm]
%              =================
In the MSSM, the decay modes $\hl,\hh,\ha \to b\bar b$, $\tau^+\tau^-$
dominate
the neutral Higgs decay modes for large $\tanb$, while for small $\tanb$
they are important for neutral Higgs masses $\lsim 150$ GeV as can be
seen from \fig{fg:2}a--c.
%The dominant decay
%modes of the charged Higgs particles are $H^+\to \tau^+\nu_\tau, t\bar
%b$ [see \fig{fg:2}d].
As in the \SM\ case, the QCD corrections~\cite{hffsmqcd}
reduce the partial
decay widths into $b,c$ quarks by about 50--75\% as a result of the
running quark masses, while they are moderate for decays into top
quarks \cite{DSZ}.
The dominant Higgs propagator corrections of ${\cal O}(G_F m_t^4/m_W^2)$
and ${\cal O}(G_F \alpha_s m_t^4/m_W^2)$ can to a good approximation be
absorbed into the effective mixing angle $\alpha$~\cite{hffsusyprop}. As
explained above, as a consequence of this universal correction the
coupling of $\hl$ to $b\bar b$ and $\tau^+\tau^-$
can be strongly suppressed for
small $\mha$ and large $\tan\beta$. For the decays into $b\bar b$,
the supersymmetric-QCD
corrections~\cite{hffsusy1l,hffsusyqcd,bfpt,CMW1,eberl,hffsusyprop,loganetal}
can be very significant for large values of $\mu$ and $\tan\beta$.
Their dominant effect manifests
itself in the relation between $m_b$ and the Yukawa couplings
[resulting in \eqns{barhb}{tildehb}].
The dominant electroweak corrections from
higgsinos to $\hl \to f \bar f$ enter in the same way.
The remaining
process-specific electroweak one-loop corrections are typically at the
level of only a few percent~\cite{hffsusy1l}.



\begin{figure}
\centering
\centerline{\psfig{file=theory/brlh6.ps,width=8cm,angle=-90}
\hfill
\psfig{file=theory/brlh30.ps,width=8cm,angle=-90}}
\vskip2pc
\centerline{\psfig{file=theory/brha.ps,width=8cm,angle=-90}
\hfill
\psfig{file=theory/brc.ps,width=8cm,angle=-90}}
\vskip1pc
\caption[0]{\label{fg:2} Branching ratios of the MSSM Higgs bosons
(a) $\hl$ and $\hh$, with $\tan\beta=6$ (b)  $\hl$ and $\hh$, with
$\tan\beta=30$ (c) $\ha$, with $\tan\beta=6$ and $30$ and
(d) $\hpm$ with $\tan\beta=6$ and $30$.  The Higgs masses and branching ratios
are sensitive to the third generation squark spectrum through
radiative corrections.  The above plots were made under the assumption
that the average top and bottom squark masses are 1~TeV and top-squark
mixing is maximal.  In addition, other supersymmetric parameters are
chosen such that there are no supersymmetric particle decay modes in
the Higgs mass ranges shown above.}
\end{figure}



\vskip6pt\noindent
\underline{$\phi\to WW$, $ZZ$} \\[0.2cm]
%          ============
In the MSSM, the decays $\hl, \hh\to WW,ZZ$ are suppressed by
kinematics and by angle factors [\eq{vvcoup}].  Hence, these decays 
are less important than in the
SM. Their branching ratios turn out to be sizeable only for small and
moderate $\tanb$
or in the decoupling regime, where the light scalar Higgs particle $\hl$
reaches the upper bound of its mass.

\break
\vskip6pt\noindent
\underline{$\phi\to gg$, $\gamma\gamma$} \\[0.2cm]
%          ============
The gluonic (photonic) decays of the Higgs bosons $\hl,\hh,\ha\to gg
(\gamma\gamma)$ reach branching ratios of $\sim 10\%$ ($\sim 10^{-2}$) in the
MSSM and are expected to be unimportant at the Tevatron.
A light top squark can enhance the $\gamma\gamma$ branching ratio
when its coupling has the opposite sign to the top quark contribution,
and it will reduce the $gg$ branching ratio in this case.

In addition to the above decay modes, which also arise in the Standard
Model, there exist new Higgs decay channels that involve scalars of
the extended Higgs sector and supersymmetric final states.
The unambiguous observation of these modes would clearly constitute
direct evidence of new physics beyond the Standard Model.

\vskip6pt\noindent
\underline{$\hh\to \hl\hl, \ha\ha$; $\hl\to \ha\ha$} \\[0.2cm]
%          =============
The decay $\hl\to\ha\ha$ is kinematically 
possible only for small values of $\mha$,
which would seem to have been ruled out by the present LEP 
Higgs mass bounds.  However,
there may be small regions in the MSSM parameter space, in which this
decay is still kinematically allowed, consistent with the LEP Higgs
search. The one-loop radiatively corrected rate for $\hl\to\ha\ha$ has
been computed in \Ref{htoaa}.

For $\mhh\lsim 200$~GeV, the decay modes $\hh\to \hl\hl$, $\ha\ha$ are
significant in the MSSM only for a very small range of $\hh$ masses
(as shown in \fig{fg:2}).  However, the process $\hh\to\hl\hl$ is
not kinematically possible given the present LEP bounds on $\mhl$.
The present LEP limits on $\mhl$ and $\mha$ also suggest that
$\hh\to\ha\ha$ is ruled out, although there may be 
small regions of the MSSM parameter space (as noted above) 
in which this latter decay mode is still allowed.
%Thus, the decay of $\hh$ into Higgs boson pairs is not relevant for
%the Tevatron Higgs mass search.  
For completeness, we note that for
values of $\tanb\lsim 5$, the branching ratio of $\hh\to \hl\hl$
is dominant for a Higgs mass range of $200~{\rm GeV}\lsim\mhh\lsim 2m_t$.
The dominant radiative corrections to this decay arise from
the corrections to the self-interaction $\lambda_{\hh\hl\hl}$ in the
MSSM and are large \cite{7}.
%The decay mode $\hh\to \ha\ha$ is only
%important at the lower end of the heavy scalar Higgs mass range.


\vskip6pt\noindent
\underline{$\hh\to\ha Z$, $\hpm W^\mp$; $\ha\to\hl Z$} \\[0.2cm]
%          =====================
The decay modes $\hh\to \ha Z$, $\hpm W^\mp$ and
$\ha\to \hl Z$ are important for
small $\tanb$ below the $t\bar t$ threshold.  Below the
corresponding thresholds, decays into off-shell Higgs and gauge bosons turn out
to be important especially for small $\tanb$ \cite{13}.  However, the
MSSM parameter space where these effects are important for the
Tevatron Higgs search are close to being ruled out by the anticipated
LEP limits.  In particular, present LEP limits already imply that
the two-body decays $\hh\to \ha Z$, $\hpm W^\mp$ are kinematically
forbidden.  Similarly, the decay $\ha\to\hl Z$ can be relevant at the
Tevatron only for $\mha\gsim 200$~GeV and $\tan\beta$ near its
experimental lower limit.


%\newpage

\vskip6pt\noindent
\underline{\it $\phi\to$~supersymmetric particles} \\[0.2cm]
%              ==========================
Higgs decays into
charginos, neutralinos and third-generation squarks and sleptons can become
important, once they are kinematically allowed \cite{13a}.
For Higgs masses below 130~GeV, the range of supersymmetric parameter space
in which supersymmetric decays are dominant is rather narrow
(when the current bounds on supersymmetric particle masses is taken
into account).  One interesting possibility is a significant branching
ratio of $\hl\to\widetilde\chi^0 \widetilde\chi^0$, which could arise
for values of $\mhl$ near its upper theoretical limit (assuming the
decay is kinematically possible).  Such an invisible decay mode would
clearly complicate the Higgs search at the Tevatron.
Note that if supersymmetric particles are directly produced and observed
at the Tevatron, then new sources of Higgs boson production (either in
association with a supersymmetric particle or in the decay chains of
supersymmetric particle decays) may exist \cite{scottkon}.

%\begin{figure}
%\centering
%\centerline{\psfig{file=theory/brsusy.ps,width=9cm,angle=-90}}
%\vskip1pc
%\caption[0]{\label{fg:3} Branching ratios of the light scalar MSSM Higgs
%boson $h$ decays into charginos/neutralinos as a function of its mass for
%$\tanb=30$. The mixing parameters have been chosen as $\mu=150$ GeV,
%$A_t=2.45$ TeV, $A_b=0$ and the squark masses of the first two generations as
%$M_{\widetilde{Q}}=1000$ GeV. The gaugino mass parameter has been set to
%$M_2=136$ GeV. The masses of the lightest gauginos are $m_{\tilde
%\chi_1^0} = 56.5$ GeV and $m_{\tilde \chi_1^\pm} = 94.1$ GeV.
%{\bf THIS PLOT SHOULD BE REMADE}}
%\end{figure}

\vskip6pt
\centerline{b.~Charged Higgs Boson Decays}
\vskip6pt

The anticipated constraints on the MSSM Higgs parameter space from LEP
(see \fig{mh200}) restrict the possible values of $\mhpm$ and
$\tan\beta$.  Using the tree-level relation of \eq{susymhpm}, it
follows that $\mhpm\gsim 120$~GeV.\footnote{Using the
radiatively-corrected expression for the charged Higgs mass does not
significantly change this estimate.}  Moreover, if no Higgs signal is
found at LEP, $\tan\beta$ will be constrained to be
larger than roughly 2.5.   
%For values of $\tan\beta$ close to this
%limit, the lower bound on $\mha$ (and therefore on
%$\mhpm$) would be significantly larger.
As will be seen in the next section, the prospects for detecting
charged Higgs bosons at the upgraded Tevatron are poor unless they
are light enough to appear among the decay products of the top quark.
In this case, $t\bar t$ production followed by $t\to bH^+$ (or
$\bar t\to \bar b H^-$) provides the most promising mechanism for
discovering a charged Higgs boson at the Tevatron.  These decays are
kinematically allowed when $\mhpm<m_t-m_b$, implying that
$\mha\lsim 150$~GeV.  Thus, there is a relevant range of MSSM
parameter space ($95~{\rm GeV}\lsim \mha\lsim 150$~GeV and
$\tan\beta\gsim 5$) in which the charged Higgs boson of the MSSM
is accessible to the Tevatron Higgs search.  

However, one can take a broader view.  In non-supersymmetric two-Higgs
doublet models, the phenomenology of the Higgs sector depends on the
pattern of Higgs couplings to fermion-pairs.  Type-II two-Higgs
doublet models possess the same Higgs-fermion couplings as in the
MSSM.  Models with a different Higgs-fermion coupling pattern can also
be constructed.
In non-supersymmetric two-Higgs-doublet models, all the
Higgs masses and the angles $\alpha$ and
$\beta$ are independent parameters.  Thus, the
restrictions on the Higgs parameters derived above 
in the case of the MSSM Higgs
sector do not apply in the more general non-supersymmetric model.
%Moreover, the charged Higgs sector is particularly simple, in that
%it is parameterized by only two parameters---$\mhpm$ and $\tan\beta$.
%Thus,
and it is quite simple to extend the charged Higgs phenomenology of
the MSSM to the more general case.  Additional complications arise
only when the neutral Higgs bosons of the model affect the production
and/or decay modes of the charged Higgs boson.
Furthermore, using present experimental and theoretical
knowledge, the indirect lower bound on the charged
Higgs mass coming from measurement of the inclusive $b \rightarrow s
\gamma$ decay is about 165 GeV \cite{bsga}
in the non-supersymmetric Type-II two-Higgs-doublets model. 
In the MSSM, no such limit exists for $\mhpm$, since 
supersymmetric particles can be exchanged in loops
contributing to $b \rightarrow s \gamma$ decay; these effects can
approximately cancel out the corresponding charged-Higgs 
exchange contribution in some regions of MSSM parameter space.

\vskip6pt\noindent
\underline{\it $\hpm\to ff'$} \\[0.2cm]
%          =====================
%We focus here on charged Higgs masses such that
%$\mhpm<m_t-m_b$\footnote{In
%this case, $q\bar q$, $gg\to t\bar t$ followed by $t\to bH^+$
%[$\bar t\to \bar bH^-$] provides the dominant mechanism for charged
%Higgs production at the Tevatron.}.
%In the MSSM, $\mhpm\gg m_W$ at tree level [\eq{largema}].  One-loop
%radiative effects can only significantly decrease this lower bound at
%small values of $\mha$ and $\tan\beta<1$, a region not preferred by
%theoretical models and current experimental mass bounds.  Using the
%present experimental bounds on neutral MSSM Higgs bosons, it follows
%that the possible decay mode $\hpm \rightarrow \phi^0 W^\pm$
%($\phi^0 = \hl$, $\hh$, or $\ha$) is kinematically
%forbidden\footnote{We are assuming that supersymmetric particle
%decays of $\hpm$ are also kinematically forbidden.}.
%
%This leaves $\hpm$ decays into quarks and leptons as the only open
%two-body tree-level decay channels.
%
Because charged Higgs couplings are
proportional to fermion masses [see \eq{hpmqq}], the decays to third
generation quarks and leptons are dominant.  In particular,
for $\mhpm<m_t+m_b$ (so that the channel $H^+\to t\bar b$ is closed),
$H^+\to\tau^+\nu_\tau$ is favored if $\tan\beta\gsim 1$, while
$H^+\to c\bar s$ is favored only if $\tan\beta$ is small.
Indeed, ${\rm BR}(H^+\to\tau^+\nu_\tau) \simeq 1$ if $\tan\beta\gsim 5$.
The  branching fractions of the two-body decay modes of
the charged Higgs boson are shown in \fig{fg:2}d as a function of  $\mhpm$,
and   also in \fig{fg:br_h}
as a function of $\tan\beta$ for several values of $\mhpm$.
These results apply generally to Type-II two-Higgs doublet models.
For $\mhpm\gsim 180$---$200$~GeV, the decay
$H^+\to t\bar b\to W^+b \bar b$ is the dominant decay mode.

The results of \fig{fg:2}d and
\fig{fg:br_h} include QCD quantum effects in all decays when relevant,
besides in the three body decay mode $H^{\pm}\to W^{\pm} b\bar b$.
The QCD radiative corrections have been computed
in \Ref{chhiggsdecay}.
Their effect on the charged Higgs branching
ratios may be significant only in the region of $\tan\beta$ where the
$c\bar s$ and $\tau^+\nu_\tau$ decay modes are competitive or for
sufficiently large values of $\tan \beta$ for the decay mode $H^+\to t\bar b$.
In principle, depending on the supersymmetric parameter space,
virtual supersymmetric (SUSY) particle exchange can also influence the
branching ratios.  The computation of radiative 
SUSY-QCD and SUSY-electroweak loop effects to the width of $H^+\to t\bar b$
can be found in \Refs{chhiggstotop}{chhiggstotop2}. In particular,
the latter includes the
resummation of the large QCD quantum effects and the dominant
contributions for 
SUSY quantum effects, which are those associated to
the redefinition of the Yukawa couplings due to bottom mass
corrections (as discussed in section~I.B.3).



\vskip6pt\noindent
\underline{\it $H^\pm\to W^\pm b\bar b$} \\[0.2cm]
%          =====================
The decay $H^+\to t^\ast b$ (where $t^\ast$ is an off-shell
$t$-quark) can become competitive (and even dominant) with respect to
the two body fermion decays.  
%For example,
%if $\tan\beta\lsim 2$ and $130~{\rm GeV}\lsim M_{H^+} \lsim
%180$~GeV, then
%${\rm BR}(H^+ \rightarrow t^\ast \bar b \rightarrow W^+b \bar b)$
%exceeds ${\rm BR}(H^+\to c\bar s)$~\cite{hwbb} and is the dominant
%charged Higgs decay channel, as shown in \fig{fg:br_h}.
Only the tree-level prediction has been displayed
in \fig{fg:br_h}.  QCD radiative corrections have been computed
in \Ref{chhiggsdecay3}.  It is found that the pure QCD corrections
raise the decay width of $H^+\to W^+b\bar b$ by about $12\%$, and the
SUSY-QCD  corrections (due to gluino and third generation
squark exchanges) can be comparable or even larger than the pure QCD
corrections in some regions of the MSSM parameter space.

\vskip6pt\noindent
\underline{$H^\pm\to W^\pm\hl$} \\[0.2cm]
%          =====================
The decay $H^+\to W^+\hl$ is kinematically allowed only for charged
Higgs bosons that are heavier than the top quark.
Hence, this mode is
not relevant for the mass region which is most likely to be accessible
at the Tevatron.  The related
decays $H^+\to W^*\ha$, $W^*\hl$, where the $W$  is off-shell, can be
sizeable for small $\tanb$ and $M_{H^+}< m_t+m_b$ \cite{bordjo}.  However, as
discussed above, this region of the MSSM parameter space is expected
to be ruled out by the end of the LEP run.  In more general
non-supersymmetric two Higgs doublet models, these modes could be
present, although their predicted rates are more model dependent
(depending on unconstrained parameters from the neutral Higgs sector).


\begin{figure}[!ht]
\centerline{\psfig{figure=theory/brctgb.ps,height=9.0cm,width=9.0cm}}
\caption{Branching fractions of 
charged Higgs bosons as functions of $\tan\beta$:
BR$(H^+\to\tau^+\nu_\tau)$ (solid), BR$(H^+\to c\bar s)$ (dashed),
and BR$(H^+\to W^+b\bar b)$ (dotted), for four
values of $\mhpm$:
(a) 100 GeV, (b) 120 GeV, (c) 140 GeV, and (d) 160 GeV.}
\label{fg:br_h}
\end{figure}

\vskip6pt\noindent
\underline{$H^\pm\to$~supersymmetric particles} \\[0.2cm]
%          =====================
Charged Higgs decay into charginos/neutralinos, and third generation
squarks and sleptons can be important once they are kinematically
allowed.  However, for $\mhpm\lsim m_t-m_b$ (most relevant for
Tevatron phenomenology), the area of MSSM parameter space in which the
two-body supersymmetric decay modes are kinematically allowed is not
large \cite{baeretal,bordjo}.
Still, this scenario cannot be ruled out at present (the most
likely supersymmetric decay mode is $\hpm\to\widetilde\chi^\pm
\widetilde\chi^0$).

\subsubsection{MSSM Higgs Boson Production at the Tevatron}



\vskip6pt\noindent
\centerline{a.~Neutral Higgs Boson Production Processes}


\vskip6pt\noindent
\underline{$q\bar q\to V^*\to V\phi$ $[V=W,Z]$, [$\phi=\hl,\hh$]}
\\[0.2cm]
%          ================================
%According to \eq{vvcoup}, both the $\hl VV$ and $\hh VV$ couplings are
%reduced relative to that of the Standard Model, subject to the sum rule:
%$g_{\hl VV}^2+g_{\hh VV}^2=g_{\hsm VV}^2$.
Over most of the MSSM Higgs
parameter space, one of the two CP-even Higgs bosons has very
suppressed couplings to $VV$, while the other one couples to $VV$ with
Standard Model strength.  For the latter scalar state,
the process $q\bar q\to V^*\to V + \phi$ [$V=W^\pm$ or $Z$]
can be important.  Note that the CP-even scalar
$\phi$ has SM like-couplings to the vector bosons in two cases:
(i)~in the decoupling regime for the lightest Higgs boson, where
$\phi=\hl$ and (ii)~for large $\tanb$ and low $\mha$, where 
$\phi=\hh$.  In either case, the corresponding scalar 
$\phi$ has a mass less than about 130~GeV.
This behavior is evident from the total cross-section curves shown in
\fig{fg:5}a,b.  These cross-sections include first-order QCD
corrections; the corresponding $K$ factors are approximately 30\%
\cite{23}.  The SUSY-QCD corrections due to the
exchange of virtual squarks and gluinos are not included; these are
known to be small \cite{djospi}.

\begin{figure}
\centering
\centerline{\psfig{file=theory/cxnh6.ps,width=8cm,angle=-90}
\hfill
\psfig{file=theory/cxnh30.ps,width=8cm,angle=-90}}
\vskip2pc
\centerline{\psfig{file=theory/cxna6.ps,width=8cm,angle=-90}
\hfill
\psfig{file=theory/cxna30.ps,width=8cm,angle=-90}}
\vskip1pc
\caption[0]{\label{fg:5} Neutral MSSM Higgs production cross-sections 
at the Tevatron [$\sqrt{s}=2$ TeV] for gluon fusion $gg\to \phi$,
vector-boson fusion $qq\to qqV^*V^* \to qqh$,
$qqH$, vector-boson bremsstrahlung $q\bar q\to V^* \to \hl V/\hh V$ and
the associated
production $gg,q\bar q \to \phi b\bar b/ \phi t\bar t$ including all known
QCD corrections, where $\phi=\hl$, $\hh$ or $\ha$.   
As in \fig{fg:4}, in the vector boson fusion process,
$qq$ refers to both $ud$ and $q\bar q$ scattering.
The four panes exhibited above show 
(a) $h,H$ production for $\tanb=6$, (b) $\hl$, $\hh$
production for
$\tanb=30$, (c) $\ha$ production for $\tanb=6$, (d) $A$ production for
$\tanb=30$.}
\end{figure}

\vskip6pt\noindent
\underline{$gg\to \phi$ [$\phi=\hl$, $\hh$, $\ha$]} \\[0.2cm]
%          ============
The gluon fusion processes are
mediated by heavy top and bottom quark triangle loops and the
corresponding supersymmetric partners \cite{29a,spiramssm,21}. Gluon fusion is
the dominant neutral Higgs
production mechanism at the Tevatron, even for large $\tanb$, with
cross-sections ranging from $0.03$--$30$~pb
(for $100~{\rm GeV}\lsim m_\phi\lsim 200$~GeV), 
as shown in \fig{fg:5}. However,
as in the case of the \SM\ Higgs boson, the
dominant $b\bar b$ final states are
overwhelmed by the huge QCD background of $b\bar b$ production.
Hence, it is not possible to detect $gg\to\phi\to b\bar b$
at the Tevatron.  Only the
$\tau^+\tau^-$ decay modes may be promising for large $\tanb$,
especially if the Higgs bosons are produced in association with a jet.

As in the case of the $gg\to\hsm$, the two-loop QCD corrections are
dominated by soft and collinear gluon radiation for small $\tanb$ in the MSSM
\cite{29}. The $K$ factor
remains nearly the same after including squark loops, since the dominant
soft and collinear gluon effects are universal, thus suppressing the (s)quark
mass dependence \cite{21}.

\break
\vskip6pt\noindent
\underline{$V^*V^*\to \phi$ [$V=W^\pm$ or $Z$; $\phi=\hl$, $\hh$]} \\[0.2cm]
%          ============
As noted previously, one typically finds that one of
the two CP-even Higgs bosons has very
suppressed couplings to $VV$, while the other one couples to $VV$ with
Standard Model strength.
The process $qq\to qqV^*V^* \to qq\phi$
(CP-even Higgs production via vector boson fusion, $V=W^\pm$ or $Z$)
can be significant only for the SM-like scalar state.
In either case the corresponding scalar $\phi$ has a mass less than
about 130~GeV.  In this mass range, observation of the Higgs boson
via this channel at the Tevatron will be extremely difficult.
The corresponding cross-sections are shown in
\fig{fg:5}a,b.  These cross-sections include first-order QCD
corrections; the corresponding $K$ factors are small, enhancing the
cross-section by roughly 10\%
\cite{32}.  The corresponding SUSY-QCD corrections are
known to be small \cite{djospi}, and are not included.



\vskip6pt\noindent
\underline{$q\bar q, gg\to \phi t\bar t,
\phi b\bar b$ [$\phi=\hl$, $\hh$, $\ha$]} \\[0.2cm]
%          ==============================================
In the MSSM, Higgs boson
radiation off bottom quarks becomes important for large $\tanb$ with
cross-sections of order 1 pb for $\hl$ and $\ha$ with masses of
order
100 GeV and 0.1 pb for  $\hh$ and $\ha$ with masses of
order  200 GeV,
%exceeding 10 $pb$ for the light scalar ($\hl$) and the
%pseudoscalar ($\ha$) Higgs particles,
see \fig{fg:5}. Thus, the
theoretical predictions, including full NLO computations, are
crucial to do realistic simulations of the MSSM Higgs signals in these
channels.\footnote{As mentioned in the corresponding discussion for
$q\bar q$, $gg\to b\bar b\hsm$ [see Section~I.B.3], one also needs to
evaluate $gb \to b\phi$ and $b\bar b\to \phi$ with suitable subtraction
of the logarithms due to quasi-on-shell quark exchange (to avoid double
counting) in order to obtain the total inclusive cross-section for
$\phi$ production.  For example, the $\tan\beta$ enhancement
can lead to copious $s$-channel production of Higgs bosons via $b$-quark
fusion \cite{bhy}.  A complete calculation of this type, fully
consistent up to ${\cal O}(\alpha_s^2)$ does not yet exist in the
literature.}
Moreover, as discussed in Section~I.B.3, SUSY-QCD and SUSY-electroweak
corrections to the $b \bar b \phi$ coupling as those shown
in eqs.~(\ref{bhCP}), (\ref{barhb}) and (\ref{tildehb})
play a very important role in enhancing or suppressing (depending on
the exact supersymmetric spectrum) these production
cross-sections at large $\tanb$ \cite{CMW1,CMW2,bdhty}.


\vskip6pt\noindent
\underline{\it Higgs Boson Pair Production} \\[0.2cm]
%              ===========
Light scalar Higgs pair production $gg\to \hl \hl$ yields a
cross-section at the Tevatron with $\sigma \gsim 10$ fb \cite{33,34}.
The cross-section for $q\bar q, gg\to \hl\ha$ is of similar size in
some
regions of the MSSM parameter space [see \fig{fg:hpair}]. Since
the process $gg\to \hh\to \hl\hl$ is sensitive to the trilinear
coupling $\lambda_{\hh\hl\hl}$ it is important for a partial
reconstruction
of the Higgs potential. One may hope that the dominant $b\bar bb\bar b,
b\bar b\tau^+\tau^-$ final states can be extracted from the QCD
backgrounds due to the different event topologies. The two-loop QCD
corrections have recently been calculated [for $gg$ initial states in
the limit of heavy top quarks, thus leading to a reliable result for
small $\tanb$]. They enhance the $gg\to hh, hA$ production cross-sections 
by about 70--90\% and the Drell--Yan-like $q\bar q \to
\hl\ha$ cross-section by about 30\% \cite{34}.  SUSY-QCD corrections,
considered in \Ref{djospi}, are small and not included here.

\begin{figure}

  \begin{center}
\centerline{\psfig{file=theory/cxnpair6.ps,width=9cm,angle=-90}}
  \end{center}
  \caption[0]{\label{fg:hpair}  QCD corrected production cross-sections of
$hh, hA$ pairs at the Tevatron [$\sqrt{s}=2$ TeV] as a function of the
pseudoscalar Higgs mass for $\tanb = 6$. The labeled axis 
at the top of the figure exhibits the
corresponding values of the light and heavy scalar Higgs masses $\mhl$,
$\mhh$.}
%The bump in the $gg\to hh$ cross-section originates from resonance
%$gg\to H\to hh$ production.}
\end{figure}

\vskip6pt\noindent
\centerline{b.~Charged Higgs Boson Production Processes}


\vskip6pt\noindent
\underline{\it $t\to bH^+$} \\[0.2cm]
%              ========================
If $\mhpm<m_t-m_b$, then the charged Higgs boson $H^\pm$ can be
produced in the decay of the top quark via $t\to bH^+$ (and $\bar t\to
\bar b H^-$) \cite{Roy1,Roy2,TDECAY,bordjo}.  The $t \to bH^+$ decay
mode can be competitive with the dominant \SM\ decay mode, $t\to
bW^+$, depending on the value of $\tan\beta$.  Here, the case of a
general Type-II two-Higgs doublet model [based on the Higgs-fermion
couplings of \eq{qqcouplings}] is considered.  Since the Higgs sector
of the MSSM is a Type-II two-Higgs-doublet model, the following
considerations also apply to the MSSM under the assumption that there
are no open charged Higgs decay channels into supersymmetric final
states, {\it i.e.}  ${\rm BR}(t\to bH^+)+{\rm BR}(t\to bW^+) = 1$.
\Fig{fg:br_t} shows
the $t\to bH^+$ branching fraction as a function of
$\tan\beta$ for various choices of $\mhpm$.
Note that ${\rm BR}(t\to b H^+)$ is significant for very small or
very large values of $\tan\beta$, while it is suppressed for
intermediate values of $\tan\beta$.

\begin{figure}
\centering
\centerline{\psfig{file=theory/qcdbr_logtb.eps,width=8cm,angle=0}
\hfill
\psfig{file=theory/mssmbr_logtb.eps,width=8cm,angle=0}}
\vskip1pc
\caption[0]{\label{fg:br_t} (a)
BR($t\to H^+ b$) as a function of $\tan\beta$ for
four values of $\mhpm$: 100 GeV (solid), 120 GeV (dotted),
140 GeV (dashed), and 160 GeV (dot--dashed), with $M_t = 175$~GeV and
$m_b=5$~GeV in the general two-Higgs-doublet model (without supersymmetry).
One-loop QCD corrections are included.
(b) BR($t\to H^+ b$) in
the MSSM for $\mhpm=120$~GeV.  The tree-level prediction is
depicted by the dotted curve.  The one-loop QCD corrected result
[from (a)] is shown by the solid curve, and the two dashed curves
incorporate additional one-loop MSSM radiative corrections for two
different choices of MSSM parameters as described in the text.}
\end{figure}

The results of \fig{fg:br_t}(a) incorporate
one-loop QCD corrections (first computed in \Ref{toptochhiggsdecay})
and are thus applicable to the general Type-II two-Higgs doublet model.
The QCD corrections to the $t\to bH^+$ branching
ratio are sizeable for values of $\tan \beta \geq 20$.
In the MSSM, one-loop virtual supersymmetric particle
exchange can also significantly influence the top quark branching
ratios, depending on the choice of supersymmetric parameters.
As an example, in \fig{fg:br_t}(b), the effects of one-loop MSSM
radiative corrections is exhibited by the dashed lines for the case of
$\mhpm=120$~GeV and the following supersymmetric parameters:
$m_{\tilde g}=M_2=m_{\tilde
{b}_{1}}= m_{\tilde{t}_{1}}=|A_{t}|=|A_{b}|=1$~TeV and
$|\mu|=300$~GeV.  The
upper [lower] dashed curve corresponds to a negative [positive] value
of $\mu$ and positive [negative] values of $A_t$ and $A_b$.
A full one-loop calculation of $\Gamma (t\rightarrow
H^{+}\,b)$  in the MSSM including
all sources of large Yukawa couplings was presented in
Ref \cite{susytoptochhiggsdecay}.
As explained in section~I.B.3, the dominant
SUSY-QCD corrections (due to gluino exchange)
and SUSY-electroweak corrections are those
associated with the redefinition of the Yukawa couplings due to
$b$-quark mass renormalization effects.
A treatment including
resummation of the leading QCD quantum effects and the dominant
contributions from loop effects arising from supersymmetric particle
exchange can be found in ref.~\cite{chhiggstotop2}.

The total cross-section for charged Higgs production in the
mass region under consideration is then given by:

\begin{equation} \label{sighplus}
\sigma(p\bar p\to \hpm+X)=\left(1-[{\rm BR}(t\to bW^+)]^2\right)
\sigma(p\bar p\to t\bar t+X)\,.
\end{equation}
With $\sigma(p \bar p \rightarrow t \bar t) \simeq 7$ pb at $\sqrt{s} = 2.0$
TeV \cite{ttbarxsec},
roughly 1400 $t\bar t$ events per detector will be produced in
Run 2 of the Tevatron.  Folding in the top quark branching ratio given
in \fig{fg:br_t}, it is a simple matter to compute the inclusive charged
Higgs cross-section for $\mhpm<m_t-m_b$.  For larger values of $\mhpm$,
the top quark that produces the charged Higgs boson must be off-shell,
and one must compute the full $2\to 3$ processes $p\bar p\to
H^+\bar t b+X$ and $p\bar p \to H^-t \bar b+X$.  This computation will be
described in the next section.  Note that if one evaluates
$\sigma(p\bar p\to H^+\bar t b+X)$ in the region of $\mhpm<m_t-m_b$,
one obtains the {\it single} charged Higgs
inclusive cross-section,
$\sigma(p\bar p\to H^+ +X)={\rm BR}(t\to bH^+)\sigma(p\bar p
\to t\bar t+X)$ (which is plotted in \fig{TeVsum}\footnote{In 
this computation, the renormalization and
factorization scales have been chosen to be $\mu=2m_t$.}), 
rather than full charged Higgs inclusive cross-section
of \eq{sighplus}.\footnote{The latter is not quite a factor of two
larger than the former since $X$ can contain a charged Higgs boson; one
must subtract off $[{\rm BR}(t\to bH^+)]^2\sigma(p\bar p\to t\bar t+X)$
to avoid double-counting.}  
As expected, in the
region of $\mhpm<m_t-m_b$, the single inclusive charged Higgs
cross-section is completely dominated by on-shell $t\bar t$ production
followed by the decay of one of the top quarks into a charged Higgs
boson.

\begin{figure}
\centering
\centerline{\psfig{file=theory/sigmatbh_chic_band.eps,width=8cm,angle=0}
\hfill
\psfig{file=theory/sigmatbh_chic_tb_band.eps,width=8cm,angle=0}}
\vskip1pc
\caption[0]{\label{TeVsum} (a) The leading-order production cross-section
$\sigma(p\bar{p}\to \bar b t H^- +X)$, for $\sqrt{s} = 2$~TeV and
$M_t=175$~GeV, as a function of the charged Higgs mass, is shown for
$\tan\beta = 50$.
The cross-section is obtained by adding the
contribution of the $2 \to 2$ processes, $g b \to t H^-$, to those of
the $2 \to 3$ processes, $ g g \to t \bar{b} H^-$ and
$ q \bar{q} \to t \bar{b} H^-$ (suitably subtracted to avoid double
counting as explained in the text).  By definition, no final state
$\bar b$ is detected in the $2 \to 2$ process (in this case, the initial
$b$ was made via $g\to b\bar b$ which implies that the attendant $\bar
b$ is produced in the forward direction), while the final state
$\bar b$ is detected in the $2 \to 3$ process.
The solid lines correspond to
the prediction of a general two-Higgs doublet with no supersymmetry.
The shaded band corresponds to
a variation of the Higgs--fermion Yukawa coupling corresponding to
a $b$-quark mass which is varied in the interval 2.5~GeV~$\leq m_b\leq
5$~GeV, which corresponds to the difference of the running $b$-quark
mass evaluated at the Higgs mass and the $b$-quark pole mass.   This
uncertainty is meant to represent the uncertainty of the unknown QCD
corrections.  The leading-order parton distribution functions
CTEQ4M \cite{CTEQ} are used.  The dot-dashed line corresponds to
the tree-level value of $\sigma(p\bar p\to t\bar t+X)\times {\rm
BR}(t\to bH^+)$.  Finally, the dashed lines exhibit the effect of
the leading MSSM radiative corrections, for a set of MSSM parameters
as described in the text.
(b) The production cross-section for
$\sigma(p\bar{p}\to \bar b t H^- +X)$ is shown as a function of $\tan\beta$
for two values of the charged Higgs mass.  For the case of
$\mhpm=140$~GeV, the top quark that decays into the observed charged
Higgs boson is on-shell, while for $\mhpm=200$~GeV, the corresponding
top quark is off-shell.  The interpretation of the solid and dashed
lines and the shaded area are the same as in (a).
These results are based on the calculations of
\Refs{chhiggstotop}{francesca}.}
\end{figure}




\vskip6pt\noindent
\underline{\it $gb\to tH^-$; $gg$, $q\bar q\to t\bar b H^-$} \\[0.2cm]
%          =====================
If $\mhpm>m_t-m_b$, then charged Higgs boson production occurs
mainly through radiation off a third generation quark.
Single charged Higgs associated
production proceeds via the $2\to 3$ partonic processes
$gg,\, q\bar q\to t\bar b H^-$ (and the charge conjugate final state).
As in the case of $b\bar b\hsm$ production,
large logarithms $\ln(\mhpm^2/m^2_b)$ arise
for $\mhpm\gg m_b$ due to quasi-on-shell
$t$-channel quark exchanges, which can be resummed by absorbing them
into the $b$-quark parton densities.  Thus, the proper procedure for
computing the charged Higgs production cross-section is to add the
cross-sections for $gb\to tH^-$ and $gg\to t\bar b H^-$
and subtract out the large logarithms accordingly
from the calculation of the $2\to 3$ process \cite{soper,OT}.
This procedure avoids double-counting of the large logarithms at
${\cal O}(\alpha_s)$, and correctly resums the leading logs to all
orders.  In particular,
the contribution to the total cross-section coming from the kinematical
region of the gluon-initiated $2\to 3$ process in which one of the
two gluons splits into a pair of $b$-quarks (one of which is
collinear with the initial proton or antiproton), is incorporated into
the $b$-quark parton density.  A cruder calculation would
omit the contribution of the $2\to 2$ process and simply include
the results of the unsubtracted $2\to 3$ process.  The latter procedure
would miss the resummed leading logs that are incorporated into the
$b$-quark density.  However, the numerical difference between the two
procedures is significant only for $\mhpm\gg m_t$.

The single inclusive charged Higgs cross-section at the Tevatron is
shown in \fig{TeVsum}(a) as a function of the charged Higgs mass, for
$\tan\beta=50$.
These results are based on the calculations of
\Refs{chhiggstotop}{francesca} and include the contributions of 
the $2\to 2$ process
and suitably subtracted $2\to 3$ process as described above.
\Fig{TeVsum}(b) depicts the single inclusive charged Higgs cross-section 
as a function of $\tan\beta$, for
$\mhpm=140$ and $200$~GeV.
In the computations that produced these two figures, the solid lines
correspond to the prediction of a general two-Higgs doublet model
(without supersymmetry).  Only tree-level diagrams are evaluated, with
the Higgs--fermion Yukawa coupling determined by a $b$-quark mass
which is varied in the interval 2.5~GeV~$\leq m_b\leq 5$~GeV
(corresponding to the shaded band).  The renormalization
and factorization scales were fixed at the
threshold value $m_t + m_{H^\pm}$.  The variation of these scales
would change the values of the cross-section presented here.
For example, a variation in the interval between $(m_t +\mhpm)/2$ and
$2(m_t + \mhpm)$, can produce deviations up to $\pm 30\%$ with respect
to the values shown in the figures.  In low-energy supersymmetric models,
the effects of the leading SUSY-electroweak corrections can
significantly modify the tree-level $H^+ t\bar b$ vertex depending on
the values of the MSSM parameters.  This is illustrated by the dashed
lines in \fig{TeVsum} which are displaced from the corresponding solid
lines due to the virtual supersymmetric effects.\footnote{The leading
MSSM effects shown in \fig{TeVsum} have been implemented only in the
computation of the $2\to 3$ process, $gg$, $q\bar q\to t\bar b H^-$.}
The effects
displayed correspond to the following supersymmetric parameters:
$m_{\tilde{g}}=300$~GeV, $m_{\tilde
{b}_{1}}=250$~GeV, $m_{\tilde{t}_{1}}=200$~GeV,
$A_{t}=A_{b}=300$~GeV, and $\left|\mu\right|=150$~GeV.  The upper
[lower] dashed curve corresponds to a positive [negative] value of
$\mu$.


%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%After adding the processes $q\bar q,gg\to b\bar b
%\hsm$, $gb \to b\hsm$ and $b\bar b\to \hsm$ and subtracting the
%logarithms accordingly
%and through the
%$2 \to 2$ process $qg \to q^{\prime}H^{\pm}$ (see
%Figs.~1, 2, and 3). The relative cross-sections are clearly
%maximized when $q$, $q^\prime$ are third-generation quarks.


%Charged-Higgs production from $t$-quark decays plays the most
%important role when kinematically allowed, that is for
%$m_{H^{\pm}} <m_{t}-m_{b}$; it was extensively studied by various
%authors~\cite{TDECAY}.  The Higgs-strahlung in the $2\to 3$ channels
%$gg,\, qq \to t H^{-} \bar{b} $ encompasses the resonant
%production of a $t\bar{t}$ pair followed by the decay
%$t \to H^+ b$, in the same kinematical region
%$m_{H^{\pm}}< m_{t}-m_{b}$, and provides, in addition,
%the intrinsically off-shell
%associated production of $H^\pm$ beyond this kinematical limit,
%throughout all possible ranges of $m_{H^\pm}$.
%
%Higgs radiation off third generation-quarks is also realized in the
%$2\to 2$ channel $g b \to t H^-$. Since the $b$-quark is contained
%in the proton via a gluon, both types of partonic processes, the
%$2\to 3$ and the $2\to 2$, give rise to inclusive processes that are
%formally of the same order in an $\alpha_s$ expansion.  Note that away
%from the resonance region, $H^+$ ($H^-$) decays dominantly into
%$t\bar{b}$ ($\bar{t} b$), and
%$t$ into $W^+b$.  Thus, the $2\to 3$ processes
%$gg,\, qq \to t H^{-}\bar{b} $ give rise to four $b$-quarks in the final
%state, whereas only three $b$'s are produced in the $2 \to 2 $
%processes $g b \to t H^-$.  Both elementary processes contribute to
%the inclusive production of a single $H^\pm$ at hadron colliders, when
%at most three $b$-quarks are tagged and used to identify the final
%particle configuration.  The contributions from the two elementary
%processes have to be properly combined, using the same subtraction
%procedure as discussed in the case of associated production of a neutral
%Higgs. This simply corrects for the fact that the
%contribution to the total cross-section coming from the kinematical
%region of the gluon-initiated $2\to 3$ process, in which one of the
%two gluons splits into a pair of $b$-quarks, one of them being
%collinear to the initial proton (antiproton), is also contained in the
%cross section coming from the $2\to 2$ process.


%The $2\to 3$ production cross section for the Tevatron,
%$\sigma(p \bar{p} \to t H^- \bar{b} X)$, is shown by the solid lines
%in Fig.~\ref{TeVmassdep} as a function of the Higgs mass, for three
%different values of $\tan\beta$, $\tan \beta =2$, $10$, and $50$, and
%in Fig.~\ref{TeVtanbdep} as a function of $\tan\beta$, for
%$m_{H^{+}} = 100$, $140$, $200\,$GeV.  All calculations are done at
%the leading order in QCD.  The leading-order parton distribution
%functions CTEQ4L~\cite{CTEQ} are used and the renormalization
%($\mu_R$) and factorization ($\mu_f$) scales are always fixed to the
%threshold value $m_t + m_{H^\pm}$.  The variation of these scales may
%result in changes in the cross section presented here: a
%variation in the interval between $(m_t + m_{H^\pm})/2$ and
%$2(m_t + m_{H^\pm})$,
%can produce deviations up to $\pm 30\%$ with respect to the values
%shown in the figures.
%
%Higher-order corrections, therefore, may be important, as was shown
%in the case of associated production of the neutral
%Higgs.  Their inclusion is called upon, but this is left for
%future work.
%
%(As a check of our cross section calculation and phase space
%integration procedure, we have reproduced the well-known
%$gg, q \bar q \to t \bar t h$ cross section~\cite{Kunstzetc} for
%the production of the neutral Higgs $h$ in association with top quarks.)
%


%Note that, as expected, the production cross section
%$\sigma(p \bar{p} \to t H^- \bar{b} X)$ is enhanced in the resonance
%region $m_{H^{\pm}} < m_{t}-m_{b}$, when $H^\pm$ is obtained as a
%decay product of one of two $t$-quarks produced on shell. The
%resonance $t$-quark propagator is regularized by the width of the top
%quark, calculated from the Standard Model
%decay $t\to b W^+$ and from the decay
%$t \to b H^+$, for each value of $m_{H^\pm}$ and $\tan \!\beta$.
%%
%In this region the full process cross section is not distinguishable
%from the convolution of $\sigma (p \bar{p} \to t \bar{t}X)$ with the
%branching fraction $Br(t \to bH^+)$.  The $\tan\beta$ dependence,
%shown explicitly in Fig.~\ref{TeVtanbdep}, has the same
%pattern as that for the branching ratio $Br(t \to bH^+)$, i.e. large
%enhancements for very small and very large values of $\tan \! \beta$
%and a minimum for $\tan\beta \simeq \sqrt{m_t/m_b}$.
%
%Away from the resonance region, the cross section diminishes rather
%drastically and the Tevatron reach is cut off at
%$m_{H^{\pm}} \lesssim 250$ GeV.


%The leading-order cross section that
%originated from the $2 \to 2 $ process
%only, $\sigma(p \bar{p} \to t H^- \bar{b} X)$, is shown in dashed lines
%in Figs.~\ref{TeVmassdep} and~\ref{TeVtanbdep}.  Notice that, away
%from the resonance region $m_{H^\pm} < m_t - m_b$, and for fixed
%$\sqrt{s}$, the relative size of the two classes of cross sections
%depends on the value of $m_{H^\pm}$.  At the Tevatron energy, the
%inclusive cross section from
%$2\to 3$ processes is still larger than that from the
%$2\to 2$ one, with a cross-over at about $m_{H^\pm} \sim 265\,$GeV,
%for the particular choice of scales $\mu_R$ and $\mu_f$
%made here.



%The appropriately summed inclusive cross section is
%shown in Fig.~\ref{TeVsum}.  As in the case of the cross section
%from $2\to 3$
%channels, a reach of $\sim 250$ GeV seem naively possible
%(with 10--30 fb$^{-1}$ luminosity), although background and other
%identification considerations will somewhat decrease this
%reach. These issues should be the subject of further study.


%Single production of $H^\pm$ was previously studied
%for the LHC center-of-mass energy~\cite{OTHERS}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\vskip6pt\noindent
\underline{\it $q\bar q$, $gg\to H^+ H^-$} \\[0.2cm]
%          =====================
Pair production of charged Higgs bosons occurs via Drell-Yan $q\bar q$
annihilation.  The dominant contribution, which  arises from the
annihilation of $u$ and $d$ quarks into a virtual photon or $Z$, is
independent of $\tan\beta$.  Some $\tan\beta$ dependence enters through
$b\bar b$ annihilation via $t$-channel top-quark exchange, although this
effect is more than one order of magnitude suppressed relative to the
dominant contribution.  The tree-level result for $\sigma(p\bar p\to H^+
H^- +X)$ for $\sqrt{s}=2$~TeV, taken from \Ref{kniehl1}, is shown in
\fig{kniehl}.  These results are obtained with
the Higgs--fermion Yukawa coupling based on
a fixed $b$-quark pole mass of $M_b=4.7$~GeV.   Note that the inclusive
$H^+H^-$ cross-section lies below the cross-section for single charged
Higgs associated production
(see \fig{TeVsum}) over the entire Higgs mass range shown.


\vskip6pt\noindent
\underline{\it $b\bar b$, $gg\to \hpm W^\mp+X$} \\[0.2cm]
%          =====================
Associated production of a charged Higgs boson and a $W^\pm$ can occur
via $b\bar b$ annihilation and $gg$-fusion \cite{kniehl2}.  The
contribution of
$b\bar b$ annihilation to $\sigma(p\bar p\to \hpm W^\mp+X)$ (both charge
state pairs are included)
is shown as function of the charged Higgs mass for $\tan\beta =
6$ and 30 in \fig{kniehl}.
The $b\bar b$ annihilation always
dominates over $gg$-fusion
except in the parameter regime where $\tan\beta$ is close to one
and the charged Higgs mass is greater than about 200 GeV.
The $gg$ fusion contribution is greatly suppressed
if $\tan\beta\gsim 6$, independent of the value of the charged Higgs
mass.
%For all values of $\tan\beta$, the latter exhibits a dip located
%about $\mhpm = m_t$, which arises from resonating top-quark propagators in
%the box amplitude.

Given the small number of events predicted (before cuts, efficiencies
and backgrounds are taken into account), it seems unlikely that this
process would provide a charged Higgs discovery signature at the
Tevatron.

\begin{figure}

  \begin{center}
\centerline{\psfig{file=theory/kniehl_new.ps,width=9cm}}
  \end{center}
  \caption[0]{\label{kniehl}
Total cross-section (in fb) for (i) $p\bar p\to H^+ H^-+X$
\cite{kniehl1} (solid line) and
(ii) $p\bar p\to \hpm W^\mp +X$ \cite{kniehl2} (dashed lines)
as a function of $\mhpm$ for $\tan\beta=6$ and 30.  The
dependence of process (i) on $\tan\beta$ is negligible.
The contribution of $b\bar b$ annihilation to process (ii) dominates
over the $gg$ fusion scattering mechanism.
$M_t=174.3$~GeV and a fixed $b$-quark pole mass
of $M_b=4.7$~GeV are used to fix the Higgs--fermion Yukawa coupling.
The leading-order parton
distribution functions CTEQ5L \cite{CTEQ} are used.}
\end{figure}

\newpage

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

\vspace{0.1in}


%{\bf {Effective Lagrangians for Higgs Interactions}}

In the previous section, the particular case of the Minimal
Supersymmetric Standard Model was discussed in great detail.
A more agnostic approach to the study of physics beyond the
Standard Model is to construct an effective Lagrangian.
The effective Lagrangian is a 
%(fairly)\footnote{It is implicitly
%assumed that the new physics admits an expansion in powers of
%momentum over the typical high--energy scale.  It is not obvious
%that this will be true for a generic, strongly--coupled theory.}
model--independent approach that
describes the effects of new physics associated with an
energy scale $\Lambda$ which lies beyond the scale where
experiments are performed.  The effective Lagrangian can be
constructed out of higher dimensional (hence non--renormalizable)
operators and depends only
on the particle content of the low energy theory.
The discussion of this section is limited to the
case of having a light Higgs boson
contained in these operators. Hence, a linearly realized
\cite{linear,hisz} $SU_L(2) \times U_Y(1)$ invariant effective
Lagrangian is assumed to describe the bosonic sector of the SM,
keeping the fermionic sector unchanged.

There are eleven dimension--6 operators involving the gauge
bosons and the Higgs scalar field which respect local $SU_L(2)
\times U_Y(1)$, $C$ and $P$ symmetries \cite{linear}.  Six of
these operators either affect only the Higgs self--interactions
or contribute to the gauge boson two--point functions at tree
level and are severely constrained from low energy physics below
the present sensitivity of high energy experiments \cite{hisz}.
From the remaining five ``blind'' operators, four affect the
Higgs couplings and can be written as
\cite{linear,hisz},
\begin{equation}
{\cal L}_{\mbox{eff}} 
= \frac{1}{\Lambda^2} \Bigl[
 f_W (D_{\mu} \Phi)^{\dagger} \hat{W}^{\mu \nu} (D_{\nu} \Phi)
 + f_B (D_{\mu} \Phi)^{\dagger} \hat{B}^{\mu \nu} (D_{\nu} \Phi)
+ f_{WW} \Phi^{\dagger} \hat{W}_{\mu \nu} \hat{W}^{\mu \nu} \Phi
+ f_{BB} \Phi^{\dagger} \hat{B}_{\mu \nu} \hat{B}^{\mu \nu} \Phi
  \Bigr]\,,
\label{lagrangian}
\end{equation}
where $\Phi$ is the Higgs field doublet,   $\hat{B}_{\mu\nu} = i
(g'/2) B_{\mu \nu}$, and $\hat{W}_{\mu \nu} = i (g/2) \sigma^a
W^a_{\mu \nu}$ with $B_{\mu \nu}$ and $ W^a_{\mu \nu}$ being the
field strength tensors of the $U(1)$ and $SU(2)$ gauge fields
respectively.
Anomalous $H\gamma\gamma$, $HZ\gamma$, $HZZ$ and $HWW$
couplings are generated by Eq.~(\ref{lagrangian}), which modify the
Higgs boson production and decay \cite{hagiwara2}. In the unitary
gauge they are given by
\beqa
{\cal L}_{\mbox{eff}}^{\mbox{H}} &=
g_{H \gamma \gamma} H A_{\mu \nu} A^{\mu \nu} +
g^{(1)}_{H Z \gamma} A_{\mu \nu} Z^{\mu} \partial^{\nu} H
+ g^{(2)}_{H Z \gamma} H A_{\mu \nu} Z^{\mu \nu}
+ g^{(1)}_{H Z Z} Z_{\mu \nu} Z^{\mu} \partial^{\nu} H \nonumber \\
&+ g^{(2)}_{H Z Z} H Z_{\mu \nu} Z^{\mu \nu} +
g^{(2)}_{H W W} H W^+_{\mu \nu} W^{- \, \mu \nu}
+ g^{(1)}_{H W W} \left (W^+_{\mu \nu} W^{- \, \mu} \partial^{\nu} H
+ \mbox{h.c.} \right)\,,
\label{H}
\eeqa
where $A(Z)_{\mu \nu} = \partial_\mu A(Z)_\nu - \partial_\nu
A(Z)_\mu$. The effective couplings $g_{H \gamma \gamma}$,
$g^{(1,2)}_{H Z \gamma}$, and $g^{(1,2)}_{H Z Z}$  and
$g^{(1,2)}_{H WW}$ are related to the coefficients of the
operators appearing in Eq.~(\ref{lagrangian}) and can be found  elsewhere
\cite{hagiwara2}. Of special interest is the
Higgs couplings to two photons, which is given by
\begin{equation}
g_{H \gamma \gamma} =
- \left( \frac{g \sin^2\theta_W M_W}{2 \Lambda^2} \right)
                      (f_{BB} + f_{WW})  \,.
\label{g}
\end{equation}

Eq.~(\ref{lagrangian}) also generates  new contributions to
the triple gauge boson vertex \cite{linear,hisz}. The operators
proportional to ${f}_{W}$  ${f}_{B}$ give rise to both anomalous
Higgs--gauge boson couplings and to new triple and quartic
gauge boson self--couplings. On the other hand,
the operators proportional to
${f}_{WW}$ and ${f}_{BB}$ only affect $HVV$ couplings
and cannot be constrained by the study of anomalous trilinear
gauge boson couplings.

A summary of the present bounds on the couplings introduced in
\eqs{lagrangian}{H} from searches at the Tevatron and LEP,
and an analysis of the possible bounds attainable at the upgraded Tevatron
will be discussed in Section III.C.

New physics can also modify the couplings of Higgs bosons to fermions.
For example, in the topcolor model \cite{chill} the coupling of a
composite charged Higgs boson to charm and bottom quarks can be so
large that the signal of the charged Higgs boson via the $s$-channel
process is spectacular \cite{bhy}.  A similar conclusion applies to
the generic type-III two-Higgs-doublet model, in which both Higgs
doublets couple to up-type and down-type
fermions.\footnote{Constraints on Type-III two-Higgs doublet models
have been recently addressed in \Ref{typethree}.}

One can develop a more model independent approach 
for the effects of new physics on 
fermion--Higgs boson couplings by following the
chiral Lagrangian methods as in the case of the vector boson--Higgs boson
couplings discussed above.  Such a generic analysis has yet to be
applied to the Tevatron search for Higgs bosons.



