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\begin{document}

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\begin{flushright}
BNL--HET--00/BB\\
BONN--BBBB\\
CALT--CC--CCCC\\
CERN--TH/2000-348\\
hep-ph/yymmnnn \\
\end{flushright}


\vspace{1mm}

\begin{center}

{\large\sc {\bf Implications of different soft SUSY-breaking scenarios}}

\vspace{0.4cm}

{\large\sc {\bf on the Higgs boson sector}}
 
\vspace{1cm}


{\sc 
S.~Ambrosanio$^{1,2}$%
\footnote{email: ambros@mail.cern.ch}%
, A.~Dedes$^{3}$%
\footnote{email: dedes@th.physik.uni-bonn.de}%
, S.~Heinemeyer$^{4}$%
\footnote{email: Sven.Heinemeyer@bnl.gov}%
, S.~Su$^{5}$%
\footnote{email: shufang@theory.caltech.edu}%
~and G.~Weiglein$^{1}$%
\footnote{email: Georg.Weiglein@cern.ch}
}

\vspace*{1cm}

{\sl
$^1$CERN, TH Division, CH-1211 Geneva 23, Switzerland

\vspace*{0.2cm}

$^2$New Address, New City, New Country (from Jan 1, 2001)

\vspace*{0.2cm}

$^3$Physikalisches Institut der Universit\"at Bonn, 
 Nu\ss allee 12, 
 D-53115 Bonn, Germany

\vspace*{0.2cm}

$^4$HET, Physics Department, Brookhaven Natl.\ Lab., Upton, NY
11973, USA

\vspace*{0.2cm}

$^5$California Institute of Technology, Pasadena, CA 91125, USA

\vspace*{0.2cm}

}

\end{center}

\vspace*{0.2cm}

\begin{abstract}
We investigate the Higgs boson sector of the Minimal Supersymmetric
Standard Model (MSSM) in the three most
prominent soft SUSY-breaking scenarios mSUGRA, (m)GMSB and AMSB.
For each scenario we combine the evaluation of the 
parameters at the electroweak scale
obtained from the set of input variables at higher energy
scales (depending on the specific scenario) 
with the todays most accurate evaluation of Higgs
boson properties in the Feynman-diagrammatic approach. Results are
obtained for the maximum value of the mass of the lightest neutral
$\cp$-even MSSM Higgs
boson, $\mh$. Restrictions on $\tb$ and on the set of higher-energy scale
parameters are derived from the lower limit on $\mh$ obtained at LEP2,
as well as from the value of $\mh$ ``indicated'' by recent
measurements at LEP2, $\mh = 115^{+1.3}_{-0.9} \gev$.
The case that this mass corresponds to the heavier neutral $\cp$-even
Higgs boson is also considered.


\end{abstract}
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\section{Introduction}

The search for the light neutral Higgs boson is a crucial test of
Supersymmetry (SUSY) that can be performed with the present and the next
generation of high-energy colliders. The prediction of a relatively light
Higgs boson is common to all supersymmetric models whose couplings
remain in the perturbative regime up to a very high energy
scale~\cite{susylighthiggs}.  Finding the Higgs boson is thus one of
the main goals of todays high-energy physics.  
The running of LEP at $\sqrt{s} \gsim 206 \gev$ in the year 2000
has for the first time resulted in data
that could be interpreted as evidence of the Higgs
boson~\cite{LEPCOct2000}. The ``indicated'' value of the Higgs boson
mass is
\BE
\mh = 115^{+ 1.3}_{-0.9} \gev .
\label{mhiggs}
\EE
This mass value lies outside the theoretically allowed region for a
Standard Model (SM) Higgs boson, requiering perturbativity up to scales 
$\La \gsim 10^5 \gev$~\cite{riesselmann}. 
%
However, in the Minimal Supersymmetric Standard Model (MSSM) the mass of the
lightest Higgs boson, $\mh$, is only bounded 
from above by $\mh \simleq 135 \gev$~\cite{mhiggslong}, including
radiative corrections up to \twol\
order~\cite{mhiggslong,mhiggsRG1a,mhiggsRG1b,mhiggsRG2,mhiggsEP1,mhiggsEP2,mhiggsletter,maulpaul}. 
Therefore the evidence found hat LEP could be interpreted as a sign
of the lightest MSSM Higgs boson.


In the MSSM no restrictions are placed on the low energy parameters,
i.e.\ all parameters can be varied independently. 
However, the huge number of more than 100 free parameters
makes this model somewhat less attractive. On the other hand, models
that define all the low energy parameters in terms of a few parameters
at the GUT scale, employing a special soft SUSY-breaking scenario, are
much more predictive. The most prominent scenarios are minimal
Supergravity (mSUGRA)~\cite{Hall,mSUGRArev}, 
(minimal) Gauge Mediated SUSY Breaking ((m)GMSB)~\cite{GR-GMSB} 
and Anomaly Mediated SUSY Breaking (AMSB)~\cite{lr,giudice,wells}. 
Analyses of the Higgs sector in these scenarios have been
performed~\cite{higgsmsugra,higgsgmsb,higgsamsb}, mostly focusing only
on the maximum value of $\mh$. More recently also the LEP results have
been taken into account for special mSUGRA
realizations~\cite{higgs115msugra}. 


In this paper the three soft SUSY-breaking scenarios are analyzed in
view of their implications for Higgs physics in the following, 
similar way. From the GUT scale parameters, using renormalization
group equations (RGEs) at the \twol\ level, the low energy parameters
are derived. After transforming them into the corresponding on-shell
parameters~\cite{bse,FDRG1}, they are used as input for the most
precise evaluation of the neutral $\cp$-even Higgs boson sector, based
on the Feynman-diagrammatic (FD)
approach~\cite{mhiggslong,mhiggsletter,feynhiggs,mhiggsf1l}.
Further restrictions such as from precision observables and the
non-observation 
of SUSY particles are also taken into account.
Using this precise prediction of $\mh$ we obtain the maximum value of
$\mh$ for each soft SUSY-breaking scenario. Combining the
prediction of $\mh$ with the exclusion regions obtained at
LEP~\cite{tbexclexp} or with the evidence found at
LEP~\cite{LEPCOct2000} (see \refeq{mhiggs}), we derive limits on the
low energy parameter 
$\tb$ and on the GUT scale parameters of the corresponding scenario.


The rest of the paper is organized as follows. In Sect.~2 the three
soft SUSY-breaking scenarios as well as the evaluation of the
$\cp$-even Higgs boson sector of the MSSM 
are reviewed. Details about the combination of RGE and FD calculation
are given and the parameter restrictions used are listed. The
description of our data sets and the 
numerical analyses for the three scenarios is given in Sect.~3. The
conclusions can be found in Sect.~4.




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\section{The Higgs sector in soft SUSY-breaking scenarios}
\label{sec:higgs}

The fact that supersymmetry is not observed 
at low energies requires the introduction of 
extra supersymmetry breaking interactions. 
This is achieved by adding to the
Lagrangian density (defined by the 
given $SU(3)_C\times SU(2)_L \times  U(1)_Y$ gauge symmetry
and the superpotential $W$)
some extra interaction terms that respect the gauge symmetry but break 
supersymmetry.
This breaking however should be such that no quadratic divergences appear
and the technical ``solution'' to the hierarchy problem is not spoiled.
Such terms are generally called ``soft SUSY-breaking'' terms.
The most general supersymmetry breaking interaction Lagrangian
resulting from spontaneously broken Supergravity in the at flat 
limit ($M_P\rightarrow \infty$) contains
just four types of soft SUSY-breaking terms~\cite{softSUSYbreaking}, 
i.e.\ gaugino masses, $\Phi^*\Phi$-scalar masses, 
$\Phi\Phi\Phi$-scalar cubic superpotential interactions and 
$\Phi\Phi$-scalar quadratic superpotential interactions.
Assuming that $R$-parity is conserved, which we do in this paper
for all SUSY breaking scenarios, reduces the amount of 
new soft terms allowed in the Lagrangian. 
Choosing a particular soft SUSY-breaking pattern allows further
reduction of the number of free parameters and the construction
of predictive models. 


In this section, we first explain how we employ the principle of
radiative electroweak symmetry breaking (REWSB). Then we
present the three most attractive soft 
SUSY-breaking scenarios and describe the general method used to derive
the low-energy Higgs sector, which applies to all scenarios. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Radiative electroweak symmetry breaking}
\label{subsec:rewsb}

As in the numerous existing analyses 
of radiative electroweak symmetry breaking in the
MSSM~\cite{REWSB}, in the present article we employ 
the RGEs. The Higgs boson
``running'' mass-squared matrix, although 
positive definite at large energy scales of the
order of Grand Unification scale $M_{GUT}$,
yields a negative eigenvalue at low energies causing the spontaneous
breakdown of the electroweak (EW) symmetry. The running 
of mass parameters from high to
low scales is equivalent to computing the 
leading logarithmic radiative corrections to the soft Higgs mass
parameters. Although 
this scenario depends on the values of few (3 or 4) free parameters, 
one could interpret
it as leading to the prediction of $\MZ$ in terms of $M_{\rm GUT}$
(or the Planck mass) and the top quark Yukawa coupling. 
Another way to interpret the predictions of this model is to
consider $\MZ$ determined by the experiment and deriving the 
absolute value of the $\mu$-parameter as well the value
of the bilinear soft-SUSY breaking parameter $B$ at a scale in the
vicinity of the EW scale,
from the minimization conditions of the effective potential,
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\label{min1} 
\mu^2(Q) \ &=& \ \frac{\bar{m}_{H_1}^2-\bar{m}_{H_2}^2 \tan^2\beta}
                      {\tan^2\beta -1}
                -\frac{1}{2} \MZ^2(Q) \;, \\
B(Q) \ &=& \ -\frac{(\bar{m}_{1}^2+\bar{m}_{2}^2)\sin2\beta}
                   {2\mu(Q)} \;,
\label{min2}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
where 
%$Q=\sqrt{M_0^2+4 M_{1/2}^2}$   approximately coincides with 
%the largest first and second generation  squark masses%
%\footnote{
%We consider this scale as the so called supersymmetry soft masses
%breaking scale, $\msusy$, since the first supersymmetry breaking evidence
%becomes ``experimentally'' apparent at the scale $Q$ when we are approaching
%the EW scale from a very high energy one. In addition, this coincides 
%with the scale where the radiative corrections to the effective potential 
%are rather small compared to any other scale.
%}%
%. 
$Q$ is derived from the scalar fermion sector. It is usually chosen
such that radiative corrections to the effective potential are rather
small compared to other scales.
In \refeqs{min1},(\ref{min2}) $\tb$ is the ratio of the two vacuum 
expectation values of the Higgs fields  $H_2$ and $H_1$ responsible 
in giving masses to the up-type and down-type quarks, respectively. In 
\refeqs{min1},(\ref{min2}), $\tb$ is evaluated at the scale $Q$,
from the scale $\MZ$, where it is considered as an input parameter%
\footnote{
See
for example the discussion in the Appendix of \citere{Drees}.
}%
. By 
$\bar{m}_{H_i}^2$ in \refeqs{min1},(\ref{min2}) we denote the radiatively
corrected ``running '' Higgs soft-SUSY breaking masses and 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\bar{m}_{i}^2 = {m}_{H_i}^2 + \mu^2 + \Sigma_{v_i} \;\; (i=1,2) \;,
\label{mbar}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
where $\Sigma_{v_i}, v_i=(v_1,v_2)$ are the one loop corrections
based on  the 1-loop Coleman-Weinberg effective potential $\Delta V$, 
$\Sigma_{v_i}=
\frac{1}{2v_i}\frac{\partial \Delta V}{\partial v_i}$,
%
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\Sigma_{v_i} = \frac{1}{64\pi^2} \sum_a (-)^{2J_a} (2J_a+1) 
               C_a \Omega_a
               \frac{M_a^2}{v_i} 
               \frac{\partial M_a^2}{\partial v_i} 
               \KKL \ln \frac{M_a^2}{Q^2} - 1 \KKR \;.
\label{effpot}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
%
Here $J_a$ is the spin of the $a$-particle, $C_a$ are the color degrees of
freedom, and $\Omega_a=1(2)$ for real scalar (complex scalar), 
$\Omega_a=1(2)$ for Majorana (Dirac) fermions. $Q$ is the energy scale 
and the $M_a$ are the field dependent mass matrices. Explicitly formulas
of the $\Sigma$'s are given in the Appendices of \citeres{barger,bagger}.
In our analysis contributions from all SUSY particles at \onel\ level
are incorporated.
With $\MZ^2$ here we denote
the tree level ``running'' $Z$~boson mass, 
$\MZ^2(Q)=\frac{1}{2}(g^2+g_2^2)v^2$ extracted at the scale $Q$ from
its physical pole mass $\MZ = 91.187 \gev$. The Radiative Electroweak
Symmetry Breaking (REWSB) is fulfilled, see \refse{subsubsec:phenorest}, 
if and only if, there is 
a solution to the \refeqs{min1},(\ref{min2})%
\footnote{
Sometimes in the
literature, the requirement of the REWSB is
described by the inequality $m_1^2(Q)m_2^2(Q)-|\mu(Q) B(Q)|^2 <0$. This
relation is automatically satisfied here from \refeqs{min1},(\ref{min2})
and from the fact that the physical squared Higgs masses must
be positive.
}%
%
.

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\subsection{mSUGRA}
\label{subsec:msugra}

A dramatic simplification of the structure of the 
SUSY breaking interactions
is provided either by Grand Unification assumptions or by Superstrings.
For example, SU(5) unification implies at tree level equality relations
between the  scalar 
soft-SUSY breaking masses $m_{\tilde{Q}}=m_{\tilde{U}^c}=m_{\tilde{E}^c}$,
and $m_{\tilde{L}}=m_{\tilde{D}^c}$, equality between the soft
breaking gaugino masses $M_1=M_2=M_3$ and for the two of the
trilinear soft breaking couplings, $A_d=A_e$.
Furthermore, SO(10) unification implies  equality 
of all scalar particle masses, equality of
Higgs masses and equality of the three types of cubic couplings.
The simplest possible
choice at tree level is to take all scalar particle and Higgs 
masses equal to a common mass
parameter $M_0$, 
all gaugino masses are chosen to be equal to the parameter $M_{1/2}$ 
and all cubic couplings flavor blind and equal to $A_0$.
This situation is common in the effective Supergravity
theories resulting from Superstrings 
but there exist more complicated alternatives.
Interestingly, the contribution of the family--anomalous $U(1)$
universal D-term 
to the scalar quark masses may be intra-family non-universal, 
and may differ from the usual (universal) boundary conditions assumed
in the MSSM~\cite{faraggi}. Another alternative are for example
 Superstrings with massless 
string modes of different modular weights that lead to
different scalar particle masses at tree level~\cite{Nilles}.
Thus, it seems that in almost all the ``realistic''  models 
motivated by Grand Unified Theories (GUTs)
 or Superstrings the universality assumption
is broken and certainly one should address each model separately 
in order to study its phenomenology at low energies. On the other
hand, one should note 
that such non-minimal alternatives like flavor dependent
scalar particle masses are constrained by limits on Flavor Changing
Neutral Currents (FCNC) processes. In what follows,
we shall therefore consider the simplest case 
(which is considered as a rather
simplification assumption here) of the three  parameters 
at the GUT scale, namely $M_0$, $M_{1/2}$ and $A_0$, which is
usually called the mSUGRA scenario.


In order to solve \refeqs{min1},(\ref{min2}), i.e.\ in order to impose
the constraint of REWSB, 
one needs furthermore to ``decide'' about the $sign(\mu)$ and the
input value of $\tb(\MZ)$. 
The running soft SUSY-breaking Higgs mass parameters, 
$m_{H_1}$ and $m_{H_2}$,
are defined at the EW scale after their evolution
from the GUT scale where we assume that they have a common value, $M_0$.
In addition the radiative corrections $\Sigma_{v_i}$
to the minimization conditions~\ref{effpot} 
are defined from the low energy SUSY spectrum
and the masses of the SM particles, which in turn means knowledge 
of $M_0$, $M_{1/2}$ and $A_0$ at the GUT scale. Thus, apart from
the SM masses provided by the experimental data~\cite{pdg}, 
$4\edz$~parameters are required to define the mSUGRA scenario:
%
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}    
\{\; M_0\;,\;M_{1/2}\;,\;A_0\;,\;\tb \;,\; sign(\mu)\; \} \;.
\label{mSUGRAparams}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%




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\subsection{(m)GMSB}
\label{subsec:gmsb}

A very promising alternative to (m)SUGRA is based on the hypothesis
that the SUSY breaking (SSB) occurs at relatively low energy scales and it 
is mediated mainly by gauge interactions 
(GMSB)~\cite{oldGMSB,newGMSB,GR-GMSB}. 
This scheme provides a natural, automatic suppression of the SUSY 
contributions to flavor-changing neutral currents and $\cp$-violating processes. 
Furthermore, in the simplest versions of GMSB (denoted hereafter with mGMSB), 
the MSSM spectrum and other (not all) observables depend on just 
$4\frac{1}{2}$ parameters,
\begin{equation}
\{\; M_{\rm mess}, \; N_{\rm mess}, \; \Lambda, \; 
     \tb, \; {\rm sign}(\mu) \; \} \;,
\label{eq:pars}
\end{equation}
where $M_{\rm mess}$ is the overall messenger scale; $N_{\rm mess}$ is the 
so-called messenger index, parameterizing the structure of the messenger
sector; $\Lambda$ is the universal soft SUSY breaking scale felt by the
low-energy sector; $\tb$ is the ratio of the vacuum expectation 
values of the two Higgs doublets; sign($\mu$) is the ambiguity
left for the SUSY higgsino mass after imposing a correct REWSB (see
\refse{subsec:rewsb} and e.g.\ 
\citeres{GMSBmodels1,GMSBmodels2,AKM-LEP2,AB-LC,AMPPR}).

The phenomenology of GMSB (and more in general of any theory with low-energy
SSB) is characterized by the presence of a very light gravitino 
$\tilde{G}$ with mass~\cite{Fayet} given by  
$m_{3/2} = m_{\tilde{G}} = \frac{F}{\sqrt{3}M'_P} \simeq 
\left(\frac{\sqrt{F}}{100 \tev}\right)^2 2.37 \; {\rm eV}$,  
where $\sqrt{F}$ is the fundamental scale of SSB and 
$M'_P = 2.44 \times 10^{18}$~GeV is the reduced Planck mass.
Since $\sqrt{F}$ is typically of order 100 TeV, the $\tilde{G}$ is always the 
lightest SUSY particle (LSP) in these theories. Hence, if $R$-parity is 
conserved, any MSSM particle will decay into the gravitino. 
Depending on $\sqrt{F}$, the interactions 
of the gravitino, although much weaker than gauge and Yukawa interactions,
can still be strong enough to be of relevance for collider physics. 
In most cases, the last step of any SUSY decay chain is 
the decay of the next-to-lightest SUSY particle (NLSP), which can  
occur either outside or inside a typical detector, possibly 
close to the interaction point. 
The nature of the NLSP -- or, better, of the scalar particle(s) having a large 
branching ratio for decaying into the gravitino and the relevant SM 
partner -- determines four main scenarios giving rise to qualitatively 
different phenomenology \cite{AKM-LEP2}. 

The parameter sets used in this model have been created by using the
program 
{\em SUSYFIRE}~\cite{SUSYFIRE} and adopting the phenomenological 
approach of \citeres{AKM-LEP2,AB-LC,AMPPR}, see also \citere{higgsgmsb}.
The origin of $\mu$ is not specified, nor the assumption $B\mu = 0$ is
made at the messenger scale. Instead, correct REWSB is imposed to trade $\mu$
and $B\mu$ for $\MZ$ and $\tb$, leaving the sign of $\mu$
undetermined, see \refse{subsec:rewsb}. 
However, note that to build a satisfactory GMSB model, one should also 
find a more fundamental solution to the latter problem, perhaps providing a 
dynamical mechanism to generate $\mu$ and $B\mu$, possibly with values
of the same order of magnitude. This might be accomplished radiatively 
through some new interaction. However, in this case, the other
soft terms in the Higgs potential, namely $m^2_{H_{1,2}}$, will be also  
affected and this will in turn change the values of $|\mu|$ and $B\mu$ 
coming from REWSB conditions. 
\mpar{non-minimal models?}

To determine the MSSM spectrum and low-energy parameters, the RGE evolution 
is solved with boundary conditions at the $M_{\rm mess}$ scale, where
\begin{eqnarray}
M_a & = & N_{\rm mess} \Lambda 
g\left(\frac{\Lambda}{M_{\rm mess}}\right) \alpha_a, 
\; \; \; (a=1, 2, 3) \nonumber \\
\tilde{m}^2 & =  & 2 N_{\rm mess} \Lambda^2 
f\left(\frac{\Lambda}{M_{\rm mess}}\right) 
\sum_a \left(\frac{\alpha_a}{4\pi}\right)^2 C_a,
\label{eq:bound}
\end{eqnarray} 
respectively for the gaugino and the scalar masses. 
The exact expressions for $g$ and $f$  at the one and two-loop level
can be found, e.g., in Ref.~\cite{AKM-LEP2},
and $C_a$ are the quadratic Casimir invariants for the scalar fields.
As usual, the scalar trilinear couplings $A_f$ are assumed to vanish
at the messenger scale, as suggested by the fact that they (and not
their square) are generated via gauge interactions with the messenger 
fields at the two loop-level only. 

The interesting region of the GMSB parameter space is selected as follows.
Barring the case where a neutralino is the NLSP and decays outside
the detector (large $\sqrt{F}$), the GMSB signatures are very spectacular
and the SM background is generally negligible or easily subtractable. 
%With this in mind and being interested in GMSB phenomenology at the LHC, 
Therefore, also in accordance with negative LEP2
searches~\cite{lepsusySep2000},  
\mpar{Ok, Sandro?}
only models where the NLSP mass is larger than 100~GeV are considered, 
assuming that searches at LEP and Tevatron, if unsuccessful, will at the 
end exclude a softer spectrum in most cases. 
Other requirements are: $M_{\rm mess} > 1.01 \Lambda$, to prevent 
an excess of fine-tuning of the messenger masses; the mass of the 
lightest messenger scalar be at least 10 TeV; 
$M_{\rm mess} > M_{\rm GUT} \times \exp(-125/N_{\rm mess})$,
to ensure the perturbativity of gauge interactions up to the 
GUT scale; $M_{\rm mess} \simleq 10^{5} \Lambda$, for simplicity. 
As a result of this and other constraints, the messenger index 
$N_{\rm mess}$, which is assumed to be an integer independent of the gauge 
group, cannot be larger than 8. To prevent the top Yukawa coupling from 
blowing up below the GUT scale, $\tb > 1.2$ is required.
Models with $\tb \gsim 55$ (with a mild dependence on 
$\Lambda$) are forbidden by the REWSB requirement, see
\refse{subsubsec:phenorest}, and typically fail 
to give $m^2_A > 0$.

The model samples are generated using {\em SUSYFIRE} with logarithmic
steps for $\Lambda$ (between about 45 TeV/$N_{\rm mess}$ and about 220 
TeV/$\sqrt{N_{\rm mess}}$, which corresponds to excluding models with
scalar particle masses above $\sim 4$ TeV), $M_{\rm mess}/\Lambda$ 
(between about 
1.01 and $10^5$) and $\tb$ (between 1.2 and about 60), subject
to the constraints described above. 
{\em SUSYFIRE} starts from the values of particle masses and gauge
couplings at the weak scale and then evolves them up to the messenger
scale through RGEs. At the messenger scale, it imposes the boundary 
conditions (\ref{eq:bound}) for the soft scalar particle masses and then 
evolves the RGEs back to the weak scale. The decoupling of each 
scalar particle at the proper threshold is taken into account. 
Two-loop RGEs are used for gauge couplings, third generation Yukawa
couplings and gaugino soft masses. The other RGEs are taken at the
one-loop level. At the scale 
%$\sqrt{\mste\mstz}$, where $\msti, i=1,2$
%denote the two scalar top quark masses (see \refse{subsec:mssmhiggs}),
$Q$, derived from the scalar fermion sector,
REWSB conditions are imposed by means of the one-loop effective potential 
approach. For the $\Si_{v_i}$ in \refeq{effpot} all dominant
corrections from the stop, sbottom and stau sector are included.
The program then evolves up again to $M_{\rm mess}$ and so on.
Three or four iterations are usually enough to get a good approximation
for the MSSM spectrum. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{AMSB}
\label{subsec:amsb}

The newly proposed Anomaly Mediated SUSY Breaking (AMSB) 
scenario \cite{lr,giudice} provides an alternate way to give mass to 
all the SUSY particles. In this model, SUSY breaking happens on a
separate brane and is  
communicated to the visible world via the super-Weyl anomaly. 
The overall scale of scalar particle masses are set by $m_{\rm aux}$, 
which is the VEV of the auxiliary field in the supergravity multiplet.
In the AMSB scenario, the low energy soft supersymmetry 
breaking parameters $M_i$ (gaugino masses, $i$=1$-$3),  
$m_{\rm scalar}^2$  and $A_{y}$ at the GUT scale are given by \cite{lr,wells}

\begin{eqnarray}
M_{\rm i}        &=& \frac{\beta_{g_i}}{g_{\rm i}}m_{\rm aux},
 \label{m1} \\
m_{\rm scalar}^2 &=& -\frac{1}{4}
 \left(\frac{\partial\gamma}{\partial{g}}\beta_{g} + 
       \frac{\partial\gamma}{\partial{y}}\beta_{y}\right)m_{\rm aux}^2
 + m_{0}^2,
 \label{m2} \\
A_{y}            &=& -\frac{\beta_{y}}{y}m_{\rm aux}.
 \label{m3}
\end{eqnarray}

Notice that the slepton squared-masses would be negative if $m_{0}$ is absent. 
There have been several proposals to solve this tachyonic slepton 
problem: bulk contributions \cite{lr}, non-decoupling effects 
of ultra-heavy vectorlike matter fields \cite{negative}, 
coupling extra Higgs doublets to the leptons \cite{clm}, contributions
from the $R$-parity violating couplings 
in the \refeq{m2} (with $m_0=0$) \cite{ben}, and the heavy mass 
threshold contribution at higher orders \cite{kss}. 
Here we have adopted a phenomenological approach and have introduced
an additional  
mass scale $m_{0}$ at the GUT scale in order to keep the 
slepton masses positive \cite{wells}.  
For simplification, we choose $m_{0}$ to be the 
same for all the super scalar particles.  Therefore, 
in the minimal case, the particle spectrum can be determined by 
$3 \edz$  parameters:  
\begin{equation}
\{m_{\rm aux},\ m_{0},\ \tb,\ {\rm sign}(\mu) \} .
\end{equation}

Eq.~(\ref{m1}), (\ref{m2}) and (\ref{m3}) would be true at all 
scales if  $m_{0}$ is absent.  However, once $m_{0}$ is introduced
at the GUT scale,  the above definitions of $M_{i}$, $m_{\rm
scalar}^2$  and $A_{y}$ set the boundary conditions and
the entire SUSY spectrum can be obtained via the running of
supersymmetric RGEs down to a lower
scale. 

Once we cross the squark  threshold, the squarks decouple and we are
left with an effective field  theory with two Higgses and all the
standard model particles%
\footnote{
Gluinos  are also decoupled since their
masses are close to the squark masses.  The contributions of Bino and Winos to
the Higgs sector can be neglected   since the 
${\rm U(1)}_{Y}$ and SU(2) gauge couplings are small.%
}.  
The two unknown parameters $|\mu|$
and $B$ are determined by the minimization of the Higgs effective
potential as explained in \refse{subsec:rewsb}, where details used for
the AMSB scenario can be found in \citere{minimize}.
Therefore, the low-energy
spectrum  is fixed once the values of $m_{\rm aux}$, $m_{0}$ 
$\tb$ and the sign of $\mu$ are known. 
%The couplings of Higgs particles to the other 
%particles can be obtained by the diagonalization of the Higgs 
%mass matrices.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Evaluating the Higgs boson sector in the MSSM}
\label{subsec:mssmhiggs}

As described in \refses{subsec:msugra} -- \ref{subsec:amsb} the
low-energy parameters are derived in each of the soft SUSY-breaking
scenarios. The most relevant parameters for Higgs boson phenomenology
are the mass of the $\cp$-odd Higgs boson, $\MA$, the ratio of the two
vacuum expectation values, $\tb$, the scalar top masses and mixing
angle, $\mste, \mstz, \tst$ (and for large $\tb$ also the scalar
bottom masses and mixing angle, $\msbe, \msbz, \tsb$), the Higgs
mixing parameter, $\mu$, the 
gluino mass, $\mgl$, and the soft SUSY-breaking parameters of the
gaugino sector, $M_1$ and $M_2$. Since these parameters are derived
from RGEs, they are \msbar\ parameters. 


The MSSM Higgs sector is evaluated in the Feynman-diagrammatic
approach, see below. Therefore the \msbar\ parameters obtained from
the soft-SUSY-breaking scenarios have to be transformed into on-shell
parameters~\cite{bse}. It has been shown that this is most relevant
for the parameters of the $\Stop$ sector. Therefore we transform the
$\Stop$ masses and mixing angle to their on-shell value, taking into
account the full \onel\ contribution presented in~\citeres{bse,FDRG1}. 


In the FD approach the masses of the two $\cp$-even
Higgs bosons, $\mh$ and $\mH$, are derived beyond tree level 
by determining the poles of the $h-H$-propagator
matrix whose inverse is given by
\BE
\left(\Delta_{\rm Higgs}\right)^{-1}
= - i \ML q^2 -  m_{H,{\rm tree}}^2 + \hSi_{H}(q^2) &  \hSi_{hH}(q^2) \\
     \hSi_{hH}(q^2) & q^2 -  m_{h,{\rm tree}}^2 + \hSi_{h}(q^2) \MR,
\label{higgspropagatormatrixnondiag}
\EE
where the $\hSi$ denote the renormalized Higgs boson
self-energies. Determining the
poles of the matrix $\Delta_{\rm Higgs}$ in
\refeq{higgspropagatormatrixnondiag} is equivalent to solving
the equation
\BE
\left[q^2 - m_{h,{\rm tree}}^2 + \hat\Sigma_{hh}(q^2) \right]
\left[q^2 - m_{H,{\rm tree}}^2 + \hat\Sigma_{HH}(q^2) \right] - 
\left[\hat\Sigma_{hH}(q^2)\right]^2 = 0 . 
\EE

In our calculation the complete \onel\ result for the
Higgs boson self-energies in the on-shell scheme~\cite{mhiggsf1l} 
is combined with 
the dominant \twol\ contributions of
$\oaas$~\cite{mhiggslong,mhiggsletter}
and further subdominant corrections~\cite{mhiggsRG1b,mhiggsRG2}, see
\citere{mhiggslong} for details.
The matrix \refeq{higgspropagatormatrixnondiag} therefore
contains the renormalized Higgs boson self-energies
\BE
\hSi_s(q^2) = \hSie_s(q^2) + \hSiz_s(0), \quad s = h, H, hH,
\label{besttwoloopse}
\EE
where the momentum dependence is neglected only in the \twol\
contribution.

Neglecting the momentum dependence also in the \onel\ part of the
renormalized 
self-energies~\refeq{besttwoloopse}, the Higgs boson masses $\mh$ and
$\mH$ are
obtained by diagonalizing the higher order corrected mass matrix with
the mixing angle $\aeff$ (see e.g.~\citere{mhiggslle}):
\BE
 \aeff = {\rm arctan}\KKL 
  \frac{-(\MA^2 + \MZ^2) \Sb \Cb - \hSi_{\PePz}}
       {\MZ^2 \CQb + \MA^2 \SQb - \hSi_{\Pe} - \mh^2} \KKR~,~~
  -\frac{\pi}{2} < \aeff < \frac{\pi}{2}~.
\label{aeff}
\EE
Inserting $\aeff$ in the tree-level formulas for Higgs production and
decay, the higher-order dominant contributions are
automatically included~\cite{hff,eehZhA}.

The complete \onel\ calculation together with the leading \twol\
corrections and the other corrections beyond $\oaas$
have been implemented into
the Fortran code \fh~\cite{feynhiggs}.
The transition from \msbar\ to on-shell parameters is also included in
the program. 
\fh\ is available via its WWW page {\tt www.feynhiggs.de}.

The evaluation of the Higgs boson sector implemented in \fh\ has been
combined with the program {\em SUITY}~\cite{sakis2}, 
used for the evaluation of the low-energy spectrum of the mSUGRA scenario.
This new combined code, {\em FeynSSG}~\cite{FeynSSG}, will be
available at {\tt www-itp.physik.uni-karlsruhe.de/feynssg/}.
The results presented for the mSUGRA scenario below are derived from
the {\em FeynSSG} output.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Low-energy constraints}
\label{subsec:constraints}

\subsubsection{Precision observables}
\label{subsubsec:precobs}

The whole set of low-energy SUSY particles contributes to electroweak
precision observables. The most prominent role plays the
$\rho$-parameter~\cite{rhoparameter}, which gives the main
contribution to other electroweak precision observables, such as the
$W$~boson mass, $\MW$, and the effective leptonic mixing angle,
$\sweff$. Here we take into account the corrections 
arising from $\Stop/\Sbot$ loops up to $\oaas$~\cite{delrhosusyqcd}.
A value of $\De\rho$ outside the experimentally preferred region of 
$\dr^{\SU} \simleq 2\times 10^{-3}$~\cite{pdg} indicates experimentally
disfavored $\Stop$  and $\Sbot$ masses.
The evaluation of $\dr^{\SU}$ is implemented into \fh.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Experimental bounds on SUSY particle masses}
\label{subsubsec:susymassbounds}

The search for SUSY particles has been one of the main tasks pursued
at the Tevatron and at LEP. The searches all turned out to be
negative, thus lower limits on the SUSY particle masses have been
set. In order to restrict the allowed parameter space in the three
soft SUSY-breaking scenarios we employed the following constraints on
their low-energy mass 
spectrum~\cite{pdg,lepsusySep2000,lepFestOct2000,CDF,Iashvili,deVivie}:
%
\BEA
m_{\tilde e} &>& 95 \gev \non \\ %{\rm ~(mSUGRA, GMSB)}  \non \\
m_{\tilde \mu} &>& 85 \gev \non \\ %{\rm ~(mSUGRA, GMSB)} \non \\
m_{\tilde \tau} &>& 71 \gev \non \\
m_{\tilde \nu} &>& 43 \gev \non \\
\mst &>& 95 \gev \non \\
\msb &>& 85 \gev \non \\
\mgl &>& 190 \gev \non \\
m_{\tilde{\chi}^\pm} &>& 103 \gev ~({\rm mSUGRA}, 
                                      m_{\tilde\nu} > 300 \gev) \non \\
m_{\tilde{\chi}^\pm} &\gsim& 84.6 \gev ~({\rm mSUGRA}, 
                                      m_{\tilde\nu} < 300 \gev) \non \\
m_{\tilde{\chi}^\pm} &>& ??? {\rm ~(GMSB)} \non \\
m_{\tilde{\chi}^\pm} &\gsim& 50 \gev {\rm ~(AMSB)} \non \\
m_{\tilde{\chi}^0_1} &\gsim& 36 \gev {\rm ~(mSUGRA, AMSB(??))} \non \\
m_{\rm NLSP} &>& 100 \gev {\rm ~(GMSB)} \non
\EEA
%
Although it has recently been shown that the case for a light $\Sbot$
with $\msb \approx \mb$ is not ruled out~\cite{lightsbottheo} and that
such a light $\Sbot$ is hard to detect~\cite{lightsbotexp}, we do not
take this possibility into account.
We furthermore demanded that the lightest SUSY particle is uncolored
and uncharged, see also the next section.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Other phenomenological restrictions}
\label{subsubsec:phenorest}

Besides constraints from precision observables and from unsuccessful
direct search for SUSY particles, we also take into account the
following restrictions (if not indicated otherwise):

\begin{itemize}
\item
For the top-quark mass most part of the analysis is based on the 
value $\mt = 175 \gev$. However, implications of deviations by 
$\pm 5 \gev$, i.e.\ $\mt = 170, 180 \gev$ are also investigated.
As a rule of thumb, a change in $\mt$ by $1 \gev$ also results in a
change in $\mh$ of about $1 \gev$.

\item 
The GUT scale parameters are taken to be real, no SUSY $\cp$-violating
phases are assumed. 

\item
In all models under consideration the $R$-parity
symmetry is taken to be conserved.

\item
Parameter sets that do not fulfill the condition of 
radiative electroweak symmetry breaking (REWSB), 
i.e.\ they do the \onel\ minimization conditions of
\refeqs{min1},(\ref{min2}), are discarded. Although these conditions
have been specified for the mSUGRA scenario, similar conditions apply
in all soft SUSY-breaking scenarios considered here.

Within all soft SUSY-breaking scenarios considered here,
the condition of REWSB leads to restrictions
on $\tb$. For example, it is almost impossible
(or a huge fine tuning is required)
to find very large values of $\tb$, $\tb \gsim 60$ which 
pass this constraint. This is because in that 
region both the RGEs of the Higgs soft breaking masses receive large
corrections not only from the top Yukawa coupling but also
from the bottom and the tau Yukawa couplings. This drives the $\mu^2$
in the numerator of \refeq{min1} negative, thus excluding this
parameter set from our analysis. 
Another possibility of not fulfilling the REWSB condition is a very
heavy soft SUSY-breaking spectrum, so that $m_{H_2}^2$ does not reach
negative values and thus not triggering REWSB.

\item
Parameter sets that do not fulfill the CCB constraints are
discarded, i.e.\ models for which charge or color breaking minima occur.

This corresponds to cases where the squared scalar quark or lepton masses 
are becoming negative at the scale $Q$, where $Q$ is the energy scale
at which the low-energy parameters are decoupled.
\mpar{How about that?}
However, it could be possible for values of the scale Q greater than 
few TeV that the lifetime of a color or charge
breaking minimum is longer than the present age of the universe,
% and surviving quantum tunneling and 
and thus are in practice not 
dangerous~\cite{riotto,kusenko,abel,subir}. 
Within the mSUGRA analysis e.g.\ the scale
$Q$ is taken to be smaller than a few TeV. In this way these
CCB bounds are taken into account in the present analysis.
A detailed analysis in the whole mSUGRA parameter space
is rather complicated and beyond the aim of this article.   

\item
Within the mSUGRA and AMSB scenario, the lightest SUSY particle (LSP)
is required to be the neutralino. Parameter sets that result in a
different LSP are excluded.
\mpar{LSP correct, Shufang?}

\item 
We do not take into account the constraints coming from 
$b \to s \gamma$, since the uncertainty
is still relatively large and the parameter space where the obtained
results are valid are still quite restricted~\cite{btosgamma}.


\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Numerical analyses}
\label{sec:numanal}

\subsection{Experimental bounds from the MSSM Higgs sector}
\label{subsec:higgsscenarios}

The running of LEP in the year 2000 has excluded a large part of the
MSSM parameter space. On the other hand a few events have been found,
which can be interpreted as evidence of the Higgs
boson~\cite{LEPCOct2000}. For our numerical analysis we will focus on
three different cases implying different restrictions on the MSSM
parameter space:

\begin{itemize}
\item[(I)]
The MSSM Higgs boson have been searched in three different benchmark
scenarios~\cite{benchmark}. From the non-observation of a Higgs boson
lower bounds of $\mh \gsim 90 \gev$ and $\MA \gsim 90 \gev$ have been
obtained~\cite{tbexclexp}, depending for both searches on $\tb$ and
for $\mh$ also on the assumed $\MA$ value. These results are used to
infer bounds on the parameters of the three soft SUSY-breaking
scenarios. 
More precisely, we made use of the analysis presented in
\citere{Barate:2000zr}. 
In this reference the excluded Higgs boson mass is presented with respect
to the reduced $ZZh$ coupling squared (i.e.\ $\sin^2(\be-\aeff)$, \
see \refeq{aeff}) at 95\% CL for $m_t=175$ GeV.
It automatically puts bounds also on the $\cp$-odd Higgs boson mass since
the channel $e^+e^- \to Ah$ is included in the analysis.
The theoretical prediction obtained at the \twol\ level 
of $\mh$ and $\sqbaeff$ for each parameter set is then compared with
the excluded region given in \citere{Barate:2000zr}.
This allows to set stricter bounds on the Higgs boson masses and on
the $\tb$, compared to those derived by directly applying the
$\mh$--$\tb$ exclusion plots for the unconstrained MSSM.
%(see for example Fig.7 in~\cite{Barate:2000zr}). 

Another important constraint on $\mh$ comes from the searches in 
$p\bar p$ collisions at CDF~\cite{cdfhiggs}.
No evidence of a signal of a type 
$p \bar p \to b\bar b h \to b\bar b b\bar b$ has been found. This sets
lower mass limits for the light Higgs boson in the region of
large $\tb$, $\tb \gsim 35$. This bound is taken into account in our
analysis (see Fig.~2 in \citere{cdfhiggs}).

\item[(II)]
The data collected at $\sqrt{s} \gsim 206 \gev$ resulted in a $3\si$
excess~\cite{LEPCOct2000} for a SM-like Higgs boson with a
``measured'' mass of 
\BE
%\mh = 115^{+ 1.3}_{-0.9} \gev .
\mh = 115 \pm 2 \gev .
\label{mhiggs2}
\EE
This ``measurement'' interpreted as the mass of the lightest MSSM
Higgs boson is used to derive constraints on the parameter space of
the three soft SUSY-breaking scenarios.

\item[(III)]
Finally it is also possible to interpret the evidence of the Higgs as
a measurement of the heavy $\cp$-even Higgs boson mass. 
Therefore it is necessary that $H$ has SM-like couplings,
i.e.\ $\cos^2(\be-\aeff) \gsim 0.8$. 
Two interpretations are possible:

\begin{itemize}
\item[(IIIA)]
There exists also a $3\si$ excess in the $\mh$-$\MA$ plane at
$\mh \approx \MA \approx 95 \gev$~\cite{tbexclexp}. 
Although it is likely that at least
at one place in this plane a $3\si$ excess occurs, we interpret this
as an indication of the two Higgs boson masses (with errors of 
$\pm 2 \gev$)

\item[(IIIB)]
For $\cos^2(\be-\aeff) \gsim 0.8$ the $h$ production via Higgs
strahlung, $e^+e^- \to Zh$, is highly suppressed, whereas the
associated production $e^+e^- \to Ah$ could be beyond the kinematic
reach of LEP. In this scenario a bound of $\mh + \MA > 206 \gev$ is
assumed. 
\end{itemize}


\end{itemize}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{mSUGRA}
\label{subsec:numanalmsugra}


%%%%%%%%%%%%%%%%%%%%%%SAKIS NEW HERE 5/11/2000%%%%%%%%%%%%%%%%%
In our numerical procedure we follow a two loop 
renormalization group analysis for
all parameters involved, i.e couplings and dimensionful parameters, 
v.e.v's included. We
start with the $\overline{MS}$
values for the gauge couplings at $M_Z$ giving a trial input value for
the strong coupling constant $\alpha_s$ in the vicinity of 0.120, 
which are then converted to their $\overline {DR}$ values. 
These are run down to $M_b=4.9$ GeV, $M_\tau=1.777$ GeV
with the $SU_c(3)\times U_{em}(1)$
RGE's~\cite{arason} to know
the running bottom and tau masses extracted from their pole masses
including SUSY at 1-loop and QCD at 2-loop  corrections 
as given in \cite{bagger}.
From these and by making use of the RGE's
for the running masses $m_b$, $m_\tau$ at 2-loops 
we run upwards to know their $\overline{MS}$ values at $M_Z$ which
are subsequently converted to their corresponding $\overline{DR}$ values. 
This provides us with the
bottom and tau Yukawa couplings at the scale $M_Z$. 
The top Yukawa coupling is known
from the input  top quark pole mass $M_t=175$ GeV  which is
subsequently converted to the $\overline{DR}$, $m_t(M_t)$ value
where the top Yukawa coupling is defined.
The evolution of all
couplings from $M_Z$ running upwards to high energies 
determines the unification scale
$M_{GUT}$ and the value of the unification coupling $\alpha_{GUT}$ by
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\alpha_1(M_{GUT})|_{\overline{DR}}=\alpha_2(M_{GUT})|_{\overline{DR}}
=\alpha_{GUT} \;.
\label{unification}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
At the GUT scale we set the boundary conditions for the 
soft SUSY breaking parameters in the region,
%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
50~{\rm GeV} \le &M_0& \le 1~{\rm TeV} \;, \nonumber \\
50~{\rm GeV} \le  &M_{1/2}& \le 1~{\rm TeV} \;, \nonumber \\
-3~{\rm TeV} \le &A_0& \le 3~{\rm TeV} \;.
\label{msugraparam}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
Running down from $M_{GUT}$ 
to $M_Z$ the trial input value for $\alpha_s$ has now changed.
We also vary the $\tan\beta$ in the region,
%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
1.4 \le &\tan\beta & \le 60 \;,
\label{mSUGRAtanb}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%
As we have already mentioned earlier the parameters $|\mu|$   and $B$
are calculated from the minimization conditions (\ref{min1}) and
(\ref{min2}), respectively. The sign of the $\mu$-parameter is input
and for every data point in (\ref{msugraparam},\ref{mSUGRAtanb}) we scann
for both the signs of $\mu$. For the calculation of the soft breaking
masses at the Electroweak scale we use the so called step function
approximation~\cite{sakis}. Thus, if
the equation at hand is the Renormalization Group equation 
for a particular running
mass $m(Q)$, then $Q_0$ is the corresponding physical mass 
determined by the condition $m(Q_0) =Q_0$. 
This procedure is iterated several times until convergence is reached.


All the information relevant to the light Higgs boson
mass in the mSUGRA scenario 
has been collected in our Figures.(\ref{fig1mSUGRA},
\ref{fig2mSUGRA},\ref{fig3mSUGRA}). This fugures show 
the variation of the light Higgs boson mass with respect to 
$\tan\beta$ for the various cases of interest in the paragraph 
(\ref{subsec:higgsscenarios}) for up to 50000 models scanned
in the region of (\ref{msugraparam},\ref{mSUGRAtanb}). 
From the fig.(\ref{fig1mSUGRA}) we obtain that the general behaviour
of $m_h^{max}$ upper bound 
is to be increased as $\tan\beta\le 12 $ is increasing. For larger
$\tan\beta$ values $m_h^{max}$ takes on rather constant values  
until $\tan\beta \simeq 50$ where it starts declining. Larger 
than $\tan\beta >60$ values are not allowed die to REWSB constraint.
The upper bound on the light Higgs boson mass in the mSUGRA scenario
is\footnote{This bound is 5 GeV lower than that  obtained in
Ref.\cite{higgsmsugra}. This is mainly because of two reasons :
a) The bound of $\Delta\rho$ we apply here and b) the sort of ``naturalness''
bound on the squark masses to be less than 2.5 TeV which is also applied
here but not in~\cite{higgsmsugra}. We also have to 
admit that several upgrades in both programs {\tt SUITY} and {\tt FeynHiggs}
had been made since then which might (in principle) couse such a discrepancy.}
%%%%%%%%%%%%%
\begin{equation}
m_h^{max}  \simleq 123~{\rm GeV} \;\;\; {\rm (mSUGRA)} \;.
\end{equation}
%%%%%%%%%%%%%%%%%
This upper limit on $m_h$ takes place on a large region
of moderate and large values of $\tan\beta$, {\it i.e.,}
$20<\tan\beta<50$. LEP Higgs searches seem to
exclude values of $\tan\beta$ with
%%%%%%%%%%%%%%%%%%%
\begin{equation}
\tan\beta \simleq 50 \;\;,\;\; m_h \simleq 107.7~{\rm GeV} 
\;\;\; {\rm (mSUGRA)~ EXCLUDED} \;.
\end{equation}
%%%%%%%%%%%%%%%%%%%%
This is a new result in the mSUGRA scenario which differs
from the case of the uncontraint MSSM where $\tan\beta$ is
excluded up to 2 (4) for the maximal (minimal) squark mixing 
scenario. $sin^2(\beta-\alpha)$ becomes smaller  than the unity only
for large $\tan\beta \simgeq 50$ values. Thus, we obtain\footnote{Very 
preliminary LEPC~\cite{LEPCOct2000} 
results have pushed this limit upwards for 3-4 GeV. Although, we follow here
the published results of~\cite{Barate:2000zr}, one can straitforwordly
extract the relevant info from our Figures.}
%%%%%%%%%%%%%%%%%%%
\begin{equation}
 m_h \simleq 91.2~{\rm GeV}\;\;\;,\;\;\; {\rm for~all~\tan\beta~} 
\;\;\; {\rm (mSUGRA)~ EXCLUDED} \;.
\end{equation}
%%%%%%%%%%%%%%%%%%%% 
We have also investigated the possibility raised very resently 
by LEP~\cite{Acciarri:2000ke,Barate:2000ts} where there are 
evidences of a Higgs with a mass of around 115 GeV. We explore 
the parameter space where the Higgs departs with $\pm$2 GeV 
from its value (this estimate comes from various theoretical 
uncertainties). From Fig.(\ref{fig2mSUGRA}) we obtain
%%%%%%%%%%%%%%%%%%%
\begin{equation}
 m_h = 113~(117)~{\rm GeV}\Rightarrow \tan\beta \simgeq 3.6~(5.1) 
\;\;\; {\rm (mSUGRA)} \;.
\end{equation}
%%%%%%%%%%%%%%%%%%%%
As far the constraints to the other parameters of the
model $M_0$, $M_{1/2}$ and $A_0$ it is rather impossible
to extract some results. In fact, in almost all the variations
of these three parameters one can find a Higgs which weighs 115 GeV.
Further restriction of the parameter space is possible 
when one includes other constraints, i.e., $b\rightarrow s\gamma$,
proton decay~\cite{matchev}, FCNC, dark matter constraints~\cite{olive} 
e.t.c.

Cases IIIA and IIIB indeed take place in the mSUGRA 
scenario as one can see from Fig.(\ref{fig1mSUGRA}) for
only large values of $\tan\beta$
%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\begin{array}{c}
{\rm IIIA~:}~m_h\simeq M_A \simeq 95~{\rm GeV} \\
{\rm IIIB~:}~{\rm h~invisible}\Rightarrow M_H=115\pm2~{\rm GeV}
\end{array}\biggr \}\Rightarrow 50\simleq \tan\beta \simleq 55 \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
In the case IIIB the invisible light Higgs takes
on values in the region 103-113 GeV for the ``observable''
heavy Higgs to be in the LEP favour value $115\pm 2$ GeV.
We have to say here that the case IIIA  is  under threat 
of being excluded by the CDF bounds~\cite{cdfhiggs}.
However, it is not clear to us whether this bound is applied
in our case since bounds from CDF are dispayed
for the maximal and minimal squark-mixing scenario in the unconstraint
MSSM {\bf Agree?}. This point certainly needs further investigation.






 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\epsfig{figure=mSUGRA1.eps,height=5in}
\caption{The light Higgs boson mass $m_h$ as a function of 
$\tb$ in the mSUGRA scenario.The various cases discussed in the 
paragraph (\ref{subsec:higgsscenarios}) are displayed. All the black
small dots are excluded either by direct Higgs searches or by SUSY 
experimental ones. The mass of the top quark has been taken to be fixed
at 175 GeV. The CDF~\cite{cdfhiggs} bound on the Higgs mass has not been applied here
(see the discussion on the text). }
\label{fig1mSUGRA}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\epsfig{figure=mSUGRA2.eps,height=5in}
\caption{As in Fig.(\ref{fig1mSUGRA}) but focusing in the excluded
region of the $\tan\beta$.}
\label{fig2mSUGRA}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\epsfig{figure=mSUGRA3.eps,height=5in}
\caption{As in Fig.(\ref{fig1mSUGRA}) but focusing 
 on the upper limit of the light 
Higgs boson mass.}
\label{fig3mSUGRA}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%
 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{GMSB}
\label{subsec:numanalgmsb}

\mda
For this paper, about XXXXX GMSB models were generated under
well defined hypotheses described in \refse{subsec:gmsb}, using the program
{\em SUSYFIRE}~\cite{SUSYFIRE} and adopting the phenomenological 
approach of \citeres{AKM-LEP2,AB-LC,AMPPR}, see also \citere{higgsgmsb}.
\mua


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{AMSB}
\label{subsec:numanalamsb}

\mda
According to the description presented in \refse{subsec:amsb} 
about $\sim YYYYY$ models have been created. The
GUT scale parameters have been varied in the ranges
\BEA
20 \tev \le &m_{\rm aux}& \le 100 \tev \non \\
0 \le &M_0& \le 2 \tev \non \\
1.2 \le &\tb& \le 60 ,
\label{gmsbparams}
\EEA
and both signs of $\mu$.
\mua


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

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\end{thebibliography} 


\end{document}



