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\begin{document}

\def\simlt{\stackrel{<}{{}_\sim}}
\def\simgt{\stackrel{>}{{}_\sim}}

\vspace{-4 in}
\rightline{
KAIST-TH 97/17}
\rightline{FTUAM 97/20}

\title{Charge and Color Breaking in 
Supersymmetry and Superstring\footnote{Based on talks given at
`Beyond the desert: accelerator and non--accelerator approaches',
Castle Ringberg, Tegernsee (Germany), June 1997;
`8th Miniworkshop on Particle and Astroparticle Physics',
Pusan (South Korea), May 1997.}}

\author{Carlos Mu\~noz\dag\footnote{Permanent address:
Departamento de F\'{\i}sica Te\'orica, Universidad Aut\'onoma de
Madrid, Cantoblanco, 28049 Madrid, Spain.}} 
%Al Troyano\dag\ and J A Revill\ddag\footnote{E-mail:
%jim.revill@ioppublishing.co.uk.}}

\affil{\dag\ Department of Physics, Korea Advanced Institute of Science
and Technology, Taejon 305-701, Korea}

%\affil{\ddag\ Editorial Department, Institute of Physics Publishing, 
%Dirac House, Temple Back, Bristol, 
%BS1~6BE, UK}

\beginabstract
Charge and color 
breaking minima in SUSY theories might 
make the standard vacuum unstable.
In this talk a brief review of this issue is performed.
When a complete analysis of all the potentially dangerous directions in the
field space of the theory is carried out, 
imposing that the standard vacuum should be the global minimum, 
the corresponding constraints
turn out to be very strong and, in fact, there are extensive regions in the
parameter space of soft SUSY--breaking terms that become forbidden.
For instance, in the context of the MSSM
with universal soft terms,
this produces important bounds, not only on the value of $A$, but also
on the values of $B$, $M$ and $m$. 
In specific SUSY scenarios, as fixed point models, 
no--scale supergravity, 
gauge--mediated SUSY breaking and superstrings, the
charge and color breaking constraints are also very important.
%These constraints can also be applied
%to specific supersymmetric scenarios. 
For example, 
if the dilaton is the source of SUSY breaking
in four--dimensional superstrings,
%the results 
%indicate that the 
the whole parameter space ($m_{3/2}$,$B$) 
is excluded on these grounds. Cosmological analyses are also
briefly reviewed.
\endabstract

\section{Introduction and summary}
As is well known, 
the presence of scalar fields with color and electric charge in
supersymmetric (SUSY) theories induces the possible existence of
dangerous charge and color breaking minima, which would make
the standard vacuum unstable \cite{reciente}. 
This is not necessarily a shortcoming since many SUSY models can
be discarded on these grounds, thus improving the predictive power
of the theory. 
A complete
analysis of all the potentially dangerous directions in the field
space of the minimal supersymmetric standard model (MSSM) was carried
out in \cite{CCB1}. It was shown there that,
imposing that the SUSY standard vacuum should be {\it deeper} than
the charge and color breaking minima, the corresponding constraints
on the soft parameter space 
%($m$, $M$, $A$, $B$) 
are very strong (see also \cite{olive}). 
For instance, in the universal case and assuming
radiative symmetry breaking with nothing but the MSSM in between the
weak scale and the grand unification scale $M_{GUT}$,
there are extensive regions of this space that
become {\it forbidden} producing
important bounds, not only on the value of the trilinear scalar parameter 
($A$), but also on the values of the bilinear scalar parameter ($B$) and the
scalar and gaugino masses ($m, M$ respectively). 
The above mentioned constraints were used 
in \cite{baer} finding a
preferred region of SUSY particle masses after imposing in addition
dark matter and naturalness constraints. 
Very strong bounds can also be obtained applying the above mentioned 
constraints to particular
SUSY scenarios. This is the case of the infrared fixed 
point model \cite{CCB2} and a SO(10) GUT \cite{strumia}.
No--scale supergravity models where the limit $m=0$ is obtained would be
excluded on these grounds \cite{CCB1,CCB2}.
Charge and color breaking constraints were also studied in the context
of gauge--mediated SUSY--breaking models \cite{randall}. In most of them
the global vacuum does not preserve QCD. 
On the other hand, 
the stability of the corresponding
constraints with respect to variations of the initial scale for the
running of the soft breaking parameters
was analyzed in \cite{CCB2}, finding that the larger the scale
is, the stronger the bounds become. In particular, by taking the Planck
scale rather than $M_{GUT}$ for the initial scale, substantially stronger
constraints are found.
These issues are
reviewed in section 2. Let us finally remark that the stability of the
standard vacuum also imposes constraints on flavor--mixing trilinear
soft terms which are stronger than the laboratory bounds coming from
the absence of FCNC \cite{dimo}.

The low--energy limit of four--dimensional superstrings is a 
SUSY field theory. This allows us to apply the above mentioned general
constraints to SUSY/string scenarios. The analysis can be in principle
more
predictive since in four--dimensional superstrings 
it is possible to obtain
information about the structure of soft SUSY--breaking terms
\cite{BIM3}. 
%The basic idea is to identify some chiral fields whose auxiliary 
%components could break SUSY by acquiring a vacuum expectation value 
%(VEV). This is the case of the dilaton and the moduli fields.  
In particular,
in the dilaton--dominated SUSY--breaking scenario, 
%\cite{Kaplunovsky}, 
%where only the 
%dilaton $S$ contributes to SUSY breaking
%is specially interesting.
%Not only the soft terms, at string tree level, 
%are independent of the four--dimensional
%string considered but also their expressions are quite simple. 
the soft terms 
are universal and depend on only two parameters, the gravitino
mass $m_{3/2}$ and $B$. 
It was shown in \cite{CCB3} that 
charge and color breaking
constraints are so important that
the {\it whole} parameter space is forbidden and, as a consequence, the
dilaton--dominated SUSY breaking is excluded on these grounds.
In section 3 this analysis is reviewed. The possibility
of assuming that the moduli fields contribute to SUSY breaking is
also discussed.
%The dilaton field, whose VEV determines the gauge coupling, is
%present in any four--dimensional string and couples at string tree--level in
%a universal manner to all particles.
%Therefore, this limit is quite {\it model independent} and, as a consequence,
%the soft terms are independent of the four--dimensional string
%considered.
%Besides, their expressions are quite simple since they are universal and
%essentially depend on only two parameters, the gravitino mass $m_{3/2}$ and 
%$B$.
%From all the above reasons it is clearly of the utmost importance
%to study the consistency of the dilaton--dominated SUSY--breaking
%scenario
%with the possible existence of dangerous charge and color breaking minima.
%This analysis was carried out in \cite{CCB3}. It was shown that
%charge and color breaking
%constraints are so important that
%the whole parameter space is {\it forbidden} and, as a consequence, the
%dilaton--dominated SUSY breaking is excluded on these grounds.
%In section 3 this analysis is reviewed. The possibility
%of assuming that the moduli fields contribute to SUSY breaking is
%also discussed.

Finally, section 4 is left for some final comments including 
cosmological considerations. 


\section{Charge and color breaking in supersymmetry}

A complete study of this crucial issue is in principle very involved. This
is mainly due to two reasons. First, the enormous complexity of the
scalar potential, $V$, in a SUSY theory. Second, the radiative corrections
to $V$ must be included in a proper way. Concerning the first point,
the tree--level scalar potential, using a standard notation, is given by
$V_0 = V_F + V_D + V_{\rm soft}$, with
%
%\begin{eqnarray}
%\label{V0}
%V_0 = V_F + V_D + V_{\rm soft}\;\; ,
%\end{eqnarray}
%
%with
%\subequations{
\begin{eqnarray}
\label{VF}
V_F = \sum_\alpha \left| \frac{\partial W}{\partial \phi_\alpha}
\right| ^2\;\;,\; \;
V_D = \frac{1}{2}\sum_a g_a^2\left(\sum_\alpha\phi_\alpha^\dagger
T^a \phi_\alpha\right)^2\;\; ,
\end{eqnarray}
%\begin{eqnarray}
%\label{VD}
%V_D = \frac{1}{2}\sum_a g_a^2\left(\sum_\alpha\phi_\alpha^\dagger
%T^a \phi_\alpha\right)^2\;\; ,
%\end{eqnarray}
\begin{eqnarray}
\label{Vsoft}
V_{\rm soft}&=&\sum_\alpha m_{\alpha}^2
|\phi_\alpha|^2\ +\ \sum_{i\equiv generations}\left\{
A_{u_i}\lambda_{u_i}Q_i H_2 u_i + A_{d_i}\lambda_{d_i} Q_i H_1 d_i
\right.
\nonumber \\
&+& \left. A_{e_i}\lambda_{e_i}L_i H_1 {e_i} + {\rm h.c.} \right\}
+ \left( B\mu H_1 H_2 + {\rm h.c.}\right)\;\; ,
\end{eqnarray}
%}
%\endsubequations
where $W$ is the MSSM superpotential
\begin{eqnarray}
\label{W}
W=\sum_{i\equiv generations}\left\{
\lambda_{u_i}Q_i H_2 u_i + \lambda_{d_i}Q_i H_1 d_i
+ \lambda_{e_i} L_i H_1 e_i \right\} +  \mu H_1 H_2\;\; ,
\end{eqnarray}
and $\alpha$ runs over all the canonically normalized 
scalar components of the chiral
superfields. 
The first observation is that the 
previous potential is extremely
involved since it  has a large number of independent fields.
Furthermore, even assuming universality of the soft breaking terms at
$M_{GUT}$, it contains a large number of
independent parameters: $m$, $M$, $A$, $B$, $\mu$.
%, i.e. the universal
%scalar and gaugino masses, the universal
%coefficients of the trilinear and bilinear scalar terms, and
%the Higgs mixing mass, respectively. 
In addition, there are the
gauge ($g$)
and Yukawa ($\lambda$)
couplings which are constrained by the experimental data. Notice that
$M$ does not appear explicitely in $V_0$, but it does through the
renormalization group equations (RGEs) of all the remaining parameters.

Concerning the radiative corrections it should be noted that
the tree--level scalar potential $V_0$ is strongly dependent on
the renormalization scale $Q$, and the one--loop radiative corrections
to it, namely
$\Delta V_1=\sum_{\alpha}\frac{n_\alpha}{64\pi^2}
M_\alpha^4\left[\log\frac{M_\alpha^2}{Q^2}-\frac{3}{2}\right]$,
%  
%\begin{eqnarray}
%\label{DeltaV1p}
%\Delta V_1={\displaystyle\sum_{\alpha}}{\displaystyle\frac{n_\alpha}{64\pi^2}}
%M_\alpha^4\left[\log{\displaystyle\frac{M_\alpha^2}{Q^2}}
%-\frac{3}{2}\right]\;\;,
%\end{eqnarray}
%
are crucial to make the potential stable against variations of the $Q$ scale.
In the previous expression 
$M_\alpha^2(Q)$ are the improved tree--level
squared mass eigenstates and $n_\alpha=(-1)^{2s_\alpha} (2s_\alpha+1)$, 
where $s_\alpha$ is the spin of the corresponding particle. 
Clearly the complete one--loop potential $V_1=V_0+\Delta V_1$
has a structure that is even far more involved than $V_0$. Notice 
that $M_\alpha^2(Q)$ are in general
field--dependent quantities since they are the eigenvalues of
the $(\partial^2 V_0/\partial \phi_i\partial\phi_j)$ matrix. Hence, the 
values of $M_\alpha^2(Q)$ depend on the values of the fields and thus
on which direction in the field space is being analyzed.
This makes in practice the minimization of the complete $V_1$ an
impossible task. However, in the region of $Q$ where $\Delta V_1$ 
is small, the predictions of $V_0$ and $V_1$ essentially coincide.
This occurs for a value of $Q$ of the order of the most significant 
$M_{\alpha}$ mass appearing in 
%(\ref{DeltaV1p})
$\Delta V_1$, which in turns
depends on what is the direction in the field space that is being
analyzed. Therefore one can still work just with $V_0$, but with the
approximate choice of $Q$. 

Taking into account all the above points one should carry out a complete
analysis of all the possible dangerous directions in the field space
along which the potential develops a charge and color breaking
minimum deeper than the realistic one.
The latter, given by
$V_{\rm real\;min}
=- \frac{1}{8} (g'^2+g_2^2) (v_2^2-v_1^2)^2$, 
%
%\begin{eqnarray}
%\label{Vreal}
%\hspace{-0.1cm}V_{\rm real\;min}
%=- \frac{1}{8} (g'^2+g_2^2) (v_2^2-v_1^2)^2\;\; , 
%= - \frac{ \left \{ \left[ ( m_1^2+m_2^2 )^2-4 |m_3|^4  \right]
%^{1/2}  - m_1^2+m_2^2  \right \} ^2  } {2 \  (g'^2+g_2^2) }
%\end{eqnarray} 
%
where $|H_1^0|=v_1$, $|H_2^0|=v_2$ with 
$v_1^2+v_2^2=2M_W^2 / g_2^2$, corresponds to the standard vacuum.
Several comments with respect to this minimum are in order. 
First, note that result $V_{\rm real\;min}$ is obtained by minimizing
just the tree-level part of the Higgs potential.
As explained above this procedure is correct if the minimization
is performed at some sensible scale $Q\equiv M_S$, which should be of the order
of the most relevant mass entering $\Delta V_1$.
%(\ref{DeltaV1p}). 
Since we are dealing here with the 
Higgs--dependent part of the potential, that mass is necessarily
of the order of the largest Higgs--dependent mass, namely the
largest stop mass. A more precise estimate of $M_S$, using a certain average
of typical SUSY masses, can be found in 
\cite{CCB1}. 
Second,
the requirement of correct electroweak breaking fixes one
of the five independent parameters of the MSSM, say $\mu$, in terms
of the others ($m$,$M$,$A$,$B$). Third, we must be sure that the realistic
minimum is really a minimum in the whole field space. This simply implies
that all the scalar squared mass eigenvalues (charged Higgses, squarks
and sleptons) must be positive. This is guaranteed for the charged Higgs
fields since in the MSSM the minimum of the Higgs potential always lies at
$H_2^+=H_1^-=0$, but not for the rest of the sparticles. Finally, we must
go further and demand that all the not yet observed particles
%, i.e.
%gluino (g), charginos ($\chi^{\pm}$), neutralinos ($\chi^0$), Higgses,
%squarks (q) and sleptons (l), 
have masses compatible with the
experimental bounds. 
%Conservatively enough 
%one can take, in an obvious notation
%\begin{eqnarray}
%\label{Expb}
%& &M_{\tilde g} \geq 120\ {\rm GeV}        \;,\;\;
%   M_{\tilde \chi^{\pm}}\geq 45\ {\rm GeV} \;,\;\;
%   M_{\tilde \chi^o} \geq 18\ {\rm GeV}    \;,\nonumber \\
%& &M_{\tilde q}\geq 100\ {\rm GeV}  \;,\;\;\;
%   M_{\tilde t} \; \geq 45\ {\rm GeV}\;,\;\;\; \;
%   M_{\tilde l} \; \geq 45\ {\rm GeV}\; .
%\end{eqnarray}
%

There are two types of charge and color breaking constraints:
the ones arising from directions in the field--space along
which the (tree--level) potential can become unbounded from below (UFB),
and those arising from the existence of charge and color
breaking (CCB) minima in the potential deeper than the
standard minimum. Since it is not possible to give here an account of
the explicit form of the constraints we refer the interested reader
to 
\cite{CCB1}. Here we will mention only their most 
important characteristics.


Concerning the CCB constraints, let us mention that the 
``traditional'' 
bound, first studied by Frere et al. and subsequently by
others \cite{reciente}, when correctly evaluated (i.e. including the
radiative corrections in a proper way) turns out to be extremely weak.
However, the 
``improved'' 
set of analytic constraints obtained in
\cite{CCB1}, which represent the necessary and sufficient conditions
to avoid dangerous CCB minima, is much stronger.

Concerning the UFB directions (and corresponding constraints),
there are three of them, labelled as UFB--1, UFB--2, UFB--3
in \cite{CCB1}. It is worth mentioning here that in general the
unboundedness is only true
at tree level since radiative corrections eventually raise the potential for
large enough values of the fields, but still these minima can be deeper than
the realistic one and thus dangerous.
The UFB--3 direction, which involves
the fields
$\{H_2,\nu_{L_i},e_{L_j},e_{R_j}\}$ with $i \neq j$
and thus leads also to electric charge
breaking, yields the {\it strongest} bound among {\it all}
the UFB and CCB constraints so it deserves to be exposed in greater detail. 
The explicit form of this bound
is as follows.
By simple analytical minimization it is possible to write the
value of all the relevant fields along the UFB--3 direction in
terms of the $H_2$ one. Then, for any value of $|H_2|<M_{GUT}$ satisfying
\begin{eqnarray}
\label{SU6}
|H_2| > \sqrt{ \frac{\mu^2}{4\lambda_{e_j}^2}
+ \frac{4m_{L_i}^2}{g'^2+g_2^2}}-\frac{|\mu|}{2\lambda_{e_j}} \ ,
\end{eqnarray}
the value of the potential along the UFB-3 direction is simply given
by
\begin{eqnarray}
\label{SU8}
V_{\rm UFB-3}&=&(m_2^2 -\mu^2+ m_{L_i}^2 )|H_2|^2
+ \frac{|\mu|}{\lambda_{e_j}} ( m_{L_j}^2+m_{e_j}^2+m_{L_i}^2 ) |H_2|
\nonumber\\ &&
-\ \frac{2m_{L_i}^4}{g'^2+g_2^2} \ .
\end{eqnarray}
Otherwise
\begin{eqnarray}
\label{SU9}
V_{\rm UFB-3}&=& (m_2^2 -\mu^2 ) |H_2|^2
+ \frac{|\mu|} {\lambda_{e_j}} ( m_{L_j}^2+m_{e_j}^2 ) |H_2| 
\nonumber\\ &&
+\ \frac{1}{8}
(g'^2+g_2^2)\left[ |H_2|^2+\frac{|\mu|}{\lambda_{e_j}}|H_2|\right]^2 \ .
\end{eqnarray}
In (\ref{SU8}) and (\ref{SU9}) $\lambda_{e_j}$ is the leptonic Yukawa
coupling of the $j-$generation and $m_2^2$ is the sum of the $H_2$ squared
soft mass, $m_{H_2}^2$, plus $\mu^2$. Then, the
UFB--3 condition reads
$V_{\rm UFB-3}(Q=\hat Q) > V_{\rm real \; min}$,
%
%\begin{eqnarray}
%\label{SU7}
%V_{\rm UFB-3}(Q=\hat Q) > V_{\rm real \; min} \ ,
%\end{eqnarray} 
%
where $V_{\rm real \; min}$
was given above 
%is given in (\ref{Vreal})
%=-\frac{1}{8}\left(g'^2 + g_2^2\right)
%\left(v_2^2-v_1^2\right)^2$, with $v_{1,2}$ the VEVs of the Higgses $H_{1,2}$,
%is the realistic minimum evaluated at $M_S$ 
%and the $\hat Q$ scale is given by \linebreak
%$\hat Q\sim {\rm Max}(g_2 |e|, \lambda_{top} |H_2|,
%g_2 |H_2|, g_2 |L_i|, M_S)$
%with
%$|e|$=$\sqrt{\frac{|\mu|}{\lambda_{e_j}}|H_2|}$ and
%$|L_i|^2$=$-\frac{4m_{L_i}^2}{g'^2+g_2^2}$ \linebreak +($|H_2|^2$+$|e|^2$).
and the $\hat Q$ scale is given by 
$\hat Q\sim {\rm Max}(g_2 |e|, \lambda_{top} |H_2|,
g_2 |H_2|, g_2 |L_i|, M_S)$
with
$|e|$=$\sqrt{\frac{|\mu|}{\lambda_{e_j}}|H_2|}$ and
$|L_i|^2$=$-\frac{4m_{L_i}^2}{g'^2+g_2^2}$ 
+($|H_2|^2$+$|e|^2$).
Notice from (\ref{SU8}) and (\ref{SU9}) 
that the negative contribution to $V_{UFB-3}$
is essentially given by the $m_2^2-\mu^2$ term, which can be very sizeable in 
many instances. On the other hand, the positive contribution is dominated by 
the term $\propto 1/\lambda_{e_j}$, thus the larger
$\lambda_{e_j}$ the more restrictive
the constraint becomes. Consequently, the optimum choice of
the $e$--type slepton is the third generation one, i.e.
${e_j}=$ stau.

Now, we will analyze numerically the above constraints. We will see that
they are very important and, in fact, there are {\it extensive regions}
in the parameter space which {\it are forbidden}. 
Our analysis will be quite general in the sense that we will consider
the whole parameter space of the MSSM
with the only assumption of universality,
i.e. $m$, $M$, $A$, $B$. Let us remark, however, that the constraints
reviewed above are general and they could also be applied for the 
non--universal case. In Fig.1 we have presented in detail, as a guiding
example, the 
(well--known minimal supergravity) case $B=A-m$ with $m$=100 GeV to get an idea
of how strong the different constraints are and then we will vary $B$ and $m$
freely
in order to obtain the most general results. 
The excluded regions are plotted in the remaining parameter space
($A/m$, $M/m$). 
It is worth 
noticing here that even before imposing CCB and UFB constraints, the parameter
space is strongly restricted by the experiment
as explains in the Figure caption.
%Actually, it is not always possible to choose the boundary condition of the
%top Yukawa coupling $\lambda_{top}$ so that the physical (pole) mass is
%reproduced because RG infrared fixed point of
%$\lambda_{top}$ puts an
%upper bound on the running top mass $M_{top}$, namely
%$M_{top}\simlt 197 sin\beta$
%GeV, where $tan \beta$=$ v_2/v_1$.
%%$tan \beta$=$\langle H_2\rangle/\langle H_1\rangle$.
%In this way, the upper and lower black regions in Fig.1 
%are forbidden. Furthermore, the small central black region is also
%forbidden because there is no value of $\mu$ capable of producing the
%correct electroweak breaking. 
%The region excluded by the CCB bounds is denoted in the figure by circles.
The restrictions coming from the UFB constraints 
%(small filled squares)
are very strong. {\it Most} of the parameter space is in fact excluded by 
the UFB--3 constraint. 
%Finally, we have also plotted in Fig.1, using
%filled diamonds, the region
%excluded by the experimental bounds on SUSY particles masses (\ref{Expb}).
Notice from Fig.1 that there are areas that are simultaneously constrained
by different types of bounds. 
Besides, the values of $A$ and $M$ are both bounded from below and above
in a correlated way.
At the end of the day, the allowed region left
(white) is quite small.
%%%%%%%%%%%%%%%%%%%%%%%%%%



%\vspace{-3cm}

%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
%\psdraft
\centerline{
\epsfig{file=fig1.ps,height=11.5cm,angle=180
%\epsfig{figure=fig1.ps,height=11.5cm,angle=180
%,bbllx=9.5cm,bblly=1.cm,bburx=19.cm,bbury=14cm
}}
\vspace{-3.5cm}
\caption{
%{\bf Fig.2:} 
Excluded regions in the parameter space of the MSSM,
%with $B=A-m$ and
with $M^{\rm phys}_{\rm top}=174$ GeV.
The central black region is excluded because there is no solution
for $\mu$ capable of producing the correct electroweak breaking.
The upper and lower black regions are excluded because it is not possible
to reproduce the experimental mass of the top due to the infrared fixed
point of $\lambda_{top}$.
The filled diamonds
indicate regions excluded by the experimental lower bounds on SUSY particle
masses.
% (\ref{Expb}).
The small filled squares indicate regions excluded by 
UFB constraints,
mainly the UFB-3 one.
The circles indicate regions excluded
by CCB constraints.} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage





%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to show now that the CCB and UFB constraints put important
bounds not only on the value of $A$ and $M$, but also on the values
of $B$ and $m$, we generalize
the previous analysis by varying first the value of $B$. For
a particular value of $m$, the larger the value of $B$ the smaller the
allowed region becomes. In general, for      
$m\simlt 500$ GeV 
(larger values of $m$ would conflict absence--of--fine--tuning
requirements for electroweak breaking),
$B$ has to satisfy the bound
$|B|\simlt 3.5\ m$.
%
%\begin{eqnarray}
%\label{bound1}
%|B|\simlt 3.5\ m\ .  
%\end{eqnarray} 
%
%This behaviour comes mainly from the enhancement of the forbidden
%areas by the UFB--3 constraint and the requirement of obtaining the
%experimental mass of the top. Both facts are due to the decreasing of
%$tan \beta$ as the low--energy value of $B$ grows. Then higher top Yukawa
%couplings are needed in order to reproduce the experimental mass of the top.
%On the one hand, this cannot be always accomplished due to the infrared
%fixed point limit on the top mass. On the other hand, the larger the
%top Yukawa coupling is, the stronger the UFB--3 becomes.  
The results also indicate
that the smaller the value of $m$, the more restrictive the constraints
become. 
%This is also mainly due to the effect of the UFB--3 constraint. 
In fact, it is possible to find a value of $m$ for which the whole
parameter space turns out to be excluded. This interesting lower bound
on $m$ is 
$m\geq 50\ GeV$.  
%
%\begin{eqnarray}
%\label{bound2}
%m\geq 50\ GeV\ .  
%\end{eqnarray} 
%
{}From this discussion it is evident that the limiting case $m=0$
is also excluded. Of course, this has obvious implications for no--scale
supergravity models since that limit is usually obtained. 
Figures illustrating these numerical results, as well as a discussion about 
the physical reasons underlying them, can be found in \cite{CCB1}.

Finally, let us remark that the previous analyses were performed assuming
universality of the soft terms at $M_{GUT}$. 
As mentioned in the introduction, 
the larger the initial scale for the running of the soft terms
is, 
the stronger the bounds become.
This can be understood from our discussion about the UFB--3
direction above: the larger the initial scale for the running is, the
more important the negative contribution $m_2^2-{\mu}^2$ to the potential
(see (\ref{SU8}) and (\ref{SU9})) becomes.
In particular,
in the standard supergravity framework, where SUSY is broken in 
a hidden sector, the natural initial scale to implement the boundary
conditions for the soft terms is $M_P\equiv M_{Planck}/\sqrt{8\pi}$
rather than $M_{GUT}$. Using the scale $M_P$ the constraints are substantially
increased. For instance, regions of large $M$ which were previously allowed
for $m>100$ GeV
become now completely excluded, also 
%bounds (\ref{bound1}) and (\ref{bound2})
the above bounds on $B$ and $m$ 
become $|B|\simlt 3m$ and $m\geq 55$ GeV respectively.
Figures illustrating these results can be found in \cite{CCB2}. 

The CCB and UFB constraints can be applied to particular SUSY scenarios
as mentioned in the introduction.
For instance, 
%very strong bounds can be obtained in a 
%in a SO(10) GUT 
%\cite{strumia}. 
in the case of the infrared fixed point model, 
the parameter space turns out to be severely constrained, including
the bound $|M/m|\simlt 1$. Figures can
be found again in \cite{CCB2}. 
SUSY/string scenarios are reviewed in the next section.


 


\section{Charge and color breaking in superstrings}

Let us briefly review the basic ingredients required for this analysis.
First we will concentrate on the form of soft SUSY--breaking terms.
The
general form of the soft SUSY--breaking Lagrangian in the context of the
MSSM for instance is given by
${\cal L}_{soft}=\frac{1}{2}(\sum_a M_a 
{\lambda}_a {\lambda}_a + h.c.)
- V_{\rm soft}$,
%
%\begin{eqnarray}
%{\cal L}_{soft} &=& \frac{1}{2}(\sum_a M_a 
%{\lambda}_a {\lambda}_a + h.c.)
%- V_{\rm soft}
%- \sum_{\alpha} m_{\alpha}^2 |\phi_{\alpha}|^2
%\nonumber\\ &&
%-\
%\left(\frac{1}{6} \sum_{\alpha\beta\gamma} 
%A_{\alpha \beta \gamma} {\lambda}_{\alpha \beta \gamma}
%	    {\phi}_{\alpha} {\phi}_{\beta} {\phi}_{\gamma}
%  + B {\mu} {H}_1 {H}_2+h.c.\right)\ ,
%\label{F6}
%\end{eqnarray}
%
where 
%$ C^\alpha=Q_L, u_L^c, d_L^c, L_L,e_L^c, H_1, H_2$ and
%$\phi_{\alpha}$ and 
$\lambda_a$ are gaugino canonically normalized fields
%respectively,
and $V_{\rm soft}$ is given in (\ref{Vsoft}).
%we have taken for simplicity diagonal Yukawa couplings
%$ Y_{\alpha \beta \gamma}=Y_u,Y_d,Y_e $, in a
%self-explanatory notation.
%Although the above soft parameters must be added by hand 
%in a generic SUSY theory,
The above soft parameters are free in the context of the pure
MSSM but 
can be obtained dynamically in a supergravity theory through
the spontaneous breaking of local SUSY in a hidden sector \cite{BIM3}.
%. Indeed, if 
%one considers the SUSY standard model and couples it to supergravity,
%the spontaneous breaking of local SUSY in a hidden sector generates
%explicit soft SUSY--breaking terms \cite{BIM3}. 
%Their particular values depend on the
%type of supergravity theory from which the MSSM derives. 
%In particular
%once the gauge kinetic functions $f_a$, the K\"ahler potential $K$ and 
%the superpotential $W$ are known,
%the supergravity theory is determined and the soft terms can be computed.
%However, one can think of many possible supergravity models (with
%different $K$, $W$ and $f_a$) leading to different results for the
%soft terms. This arbitrariness can be ameliorated in supergravity models
%deriving from superstring theory, where $K$, $f_a$, and the hidden sector
%are more constrained.
In supergravity models 
obtained from superstring compactifications
there is a natural hidden sector built--in: the complex dilaton field
$S$ and the complex moduli fields $T_i$. 
%(where we are denoting the 
%$T$-- and $U$--type moduli collectively by $T_i$). 
%These gauge singlet fields
%are generically present in four--dimensional models: the dilaton arises from
%the gravitational sector of the theory and the moduli parametrize
%the size and shape of the compactified variety. 
Assuming that the auxiliary fields of those multiplets are the seed
of SUSY breaking, interesting predictions about soft terms are obtained.
%For any four--dimensional superstring
%the gauge kinetic function, at string tree level, is 
%independent of the moduli sector
%and is simply given by
%
%\begin{eqnarray}
%{f_a} &=& k_a S\ , 
%\label{kahler3}
%\end{eqnarray}
%
%where $k_a$ is the Kac--Moody level of the gauge factor.
%Usually (level one case) one takes $k_3=k_2=\frac{3}{5}k_1=1$, but this is
%irrelevant for the string tree-level computation since 
%$k_a$ will not contribute to
%the soft parameters.
%Likewise, at string tree level, 
%the associated effective N=1 supergravity 
%K\"ahler potentials, to lowest order in the matter fields, 
%are of the type:
%\begin{eqnarray}
%K &=& 
%-\log(S+S^*)\ +\ {\hat K}(T_i,T_i^*)\ 
%+\
%{\tilde K}_{{\overline{\alpha }}{ \beta }}(T_i,T_i^*){C^*}^{\overline {\alpha}}
%C^{\beta } 
%\nonumber\\ &&
% +\ \left[\frac{1}{2} Z_{{\alpha }{ \beta }}(T_i,T_i^*){C}^{\alpha}
%C^{\beta } + h.c. \right]\ ,
%\label{kahler}
%\end{eqnarray}
%%
%where we remark that we are denoting the $T$- and $U$-type 
%(K\"ahler class and complex structure in the Calabi-Yau language)
%moduli
%collectively by $T_i$, and the hat in the matter fields means that they
%are un--normalized. 
%The first piece in (\ref{kahler}) is the usual term corresponding 
%to the complex dilaton $S$ that is present for any compactification.
%The second piece is the K\"ahler potential of the moduli fields,
%which in general depends on the compactification scheme and can
%be a complicated function. For the moment we leave it generic.
%The same comment applies to 
%${\tilde K}_{{\overline{\alpha }}{ \beta }}(T_i,T_i^*)$ and 
%$Z_{{\alpha }{ \beta }}(T_i,T_i^*)$.
%With the above information it is 
%now a straightforward
%exercise,
%plugging (\ref{kahler3}) and (\ref{kahler}) into the standard supergravity
%formulas for soft terms \cite{BIM3} to compute these as functions
%of the VEVs of the dilaton and moduli auxiliary fields, 
%$F^S$ and $F^i$ respectively. 
%On the one hand, since the string tree--level gauge kinetic functions are
%given for any four--dimensional superstring by (\ref{kahler3}), the
%string tree--level gaugino masses are universal, independent of the
%moduli sector, and simply given by:
%
%\begin{eqnarray}
%M_a     &=&   \sqrt{3 m_{3/2}^2 + \frac{V_0}{M_P^2}} \;\; 
%\sin\theta e^{-i \gamma_S} \;\;,
%\label{gauginos}
%\end{eqnarray}
%
%where $M_P\simeq 2.4\times 10^{18}$ is the reduced Planck mass, 
%$V_0$ is the 
%the VEV of the scalar potential 
%at supergravity tree--level, the angle $\theta$ just parametrize the
%direction of the goldstino in the $S$, $T_i$, field space
%and $\gamma_S$ is a possible phase of the
%dilaton F-term $F^S$.
%%
%The second basic ingredient of our analysis concerns the constraints
%%associated with the existence of dangerous directions in the field
%space. These were explained in the previous section for a generic
%SUSY theory and therefore can be applied for any four--dimensional
%superstring model.
%In order to show this, two cases can be considered: dilaton--dominated
%SUSY breaking and dilaton/moduli--dominated SUSY breaking scenarios. 

%\subsection{Dilaton--dominated SUSY breaking}
%Here we will focus on the very interesting 
Let us first focus on the very interesting 
case where the dilaton field is
the source of all the SUSY breaking 
%(i.e. $F^i=0$) 
\cite{BIM3}. 
Since, at string tree--level, 
the dilaton couples in a universal manner to all particles, 
this limit is quite model {\it independent}. 
%Indeed, the expressions
%for the soft parameters
%obtained using (\ref{kahler3}) and the first
%piece of (\ref{kahler}), 
%are quite simple and 
%independent of the four--dimensional superstring considered. 
The soft parameters
%assuming vanishing cosmological constant and phases in order
%to fulfil experimental restrictions, 
are:
%
%\begin{eqnarray}
%\label{softtermsV}
%m_{\alpha}^2   &=& m_{3/2}^2 + \frac{V_0}{M_P^2}                             \;\;,\nonumber\\
%M_a     &=&   \sqrt{3 m_{3/2}^2 + \frac{V_0}{M_P^2}} \;\; 
%e^{-i \gamma_S} \;\;,\nonumber\\
%A_{\alpha\beta\gamma} &=& - M_a \;\;,
%\sqrt{3 m_{3/2}^2 + 
%\frac{V_0}{M_P^2} \;\; e^{-i \gamma_S} \;\;.
%\label{dilaton}
%\end{eqnarray}
%
%The previous expressions for the soft terms can be simplified taking
%into account several experimental restrictions.  From the limits on
%the electric dipole moment of the neutron it seems reasonable to
%impose in what follows $\gamma_S=0$ mod $\pi$.  On the other hand,
%experimental constraints in present cosmology allow us to assume
%vanishing cosmological constant $V_0=0$ (we will see later on that our
%conclusions will not be modified if we give up this assumption). Then
%
%\begin{eqnarray} 
%\label{softterms}
%m_{\alpha}^2 = m_{3/2}^2 \;, \;\; M_a = \pm \sqrt{3} \; m_{3/2}
%\;, \;\; A_{\alpha\beta\gamma} = - M_a \;\;,
%\end{eqnarray} 
%
$m_{\alpha}^2 = m_{3/2}^2$, $M_a = \pm \sqrt{3} \; m_{3/2}$,
$A_{\alpha\beta\gamma} = - M_a$,
where $A_{\alpha\beta\gamma}=A_u, A_d, A_e$ in a self--explanatory notation.
This dilaton--dominated scenario 
is attractive for its simplicity and for
the natural explanation that it offers to the {\it universality} of the
soft terms. 
%Actually, universality is a desirable property not
%only to reduce the number of independent parameters in the MSSM, but also
%for phenomenological reasons, particularly to avoid FCNC. 
%Because of the simplicity of this scenario, the corresponding 
%low--energy predictions 
%are quite precise.
% \cite{BLM,BIM,LNZ,KMV}. 
%For example, the first and second generation squarks are almost degenerate
%with the gluino and are much heavier than sleptons.
Since the value of $B$ 
%the bilinear parameter $B$ 
is more model dependent, 
%and deserves
%some additional comments. Indeed, $B$ depends not only on the 
%dilaton--dominance assumption but 
%also on the particular mechanism which could
%generate the associated (electroweak size) $\mu$ term \cite{BIM3}. 
%For example, the interesting possibilities of generating it through the
%superpotential and/or the
%quadratic term associated with $Z_{\alpha\beta}$ in the 
%K\"ahler potential (\ref{kahler})
%\cite{Giudice, Casas, Kaplunovsky, Lopes, Antoniadis} 
%\cite{Casas, Antoniadis} give rise, 
%in the dilaton--dominated scenario, to the following value 
%of $B$\ :
%\cite{Brignole, Kaplunovsky, Munoz2, Munoz1, Brignole2}
%$B = 2 m_{3/2} + \frac{V_0}{m_{3/2}M_P^2}$.
%It is worth noticing here that the value of $\mu$
%is compactification dependent even in this dilaton--dominated scenario.
%For example a simple result $\mu=m_{3/2}$ can be obtained   
%in orbifold models where the source of $\mu$ is a $Z$ term in the
%K\"ahler potential. 
%Notice that, when such a property holds, the whole SUSY spectrum
%with the above experimental restrictions depends only on 
%one parameter: $m_{3/2}$. Since this parameter can be fixed from the
%phenomenological requirement of correct electroweak breaking,
%at the end of the day we are left essentially with no free parameters.
%In \cite{BIM2} the consistency of the above 
%boundary conditions with the
%appropriate radiative electroweak symmetry breaking was explored. 
%Unfortunately, it was 
%found that they are not consistent with the measured value of the top-quark
%mass, namely the mass obtained in this scheme turns out to be too small.
%In this sense
%in order to analyze charge and color breaking constraints
%in the dilaton--dominated scenario, 
%it is better to take the
%expressions of the soft terms given by (\ref{softterms}),
%consider $B$ as a free parameter and eliminate $\mu$ in terms of the
%other parameters by imposing appropriate electroweak breaking.
%The reason being that this provides the most general analysis.
it is better to take it as a free parameter in order to carry out the most
general analysis.

  
The second basic ingredient of our analysis concerns the constraints
associated with the existence of dangerous directions in the field
space. These were explained in section 2 for a generic
SUSY theory and therefore can be applied for any four--dimensional
superstring model.
In the particular case of the dilaton--dominated scenario,
the restrictions coming from the UFB constraints are very strong and
the
whole parameter space ($m_{3/2}$,$B$) turns out to be excluded. Most of it is in fact 
excluded by the UFB--3 constraint. Figures illustrating this result
can be found in \cite{CCB3}.




%\subsection{Dilaton/Moduli--dominated SUSY breaking}

Given the above dramatic conclusions about the dilaton--dominated
scenario, let us briefly discuss a possible way--out. The most
straightforward possibility is to assume that also the moduli
fields $T_i$ contribute to SUSY breaking, which is in fact a more general
situation. Then
%i.e. $F^i\neq 0$.
the soft terms are modified, new free parameters beyond 
$m_{3/2}$ and $B$ appear, and possibly some regions in the
parameter space will be allowed. 
%Of course, this amounts to a 
%departure of the pure dilaton--dominated scenario. 
%On the other hand,
%It is interesting to note that explicit possible scenarios of
%SUSY breaking by gaugino condensation in strings, when analyzed at the
%string one--loop level, lead to the mandatory inclusion of the moduli
%in the game (in fact the moduli are the main source of SUSY breaking in
%these cases).
This situation is more model dependent since different compactification
schemes have different numbers and types of moduli and different couplings
of them to matter, therefore giving rise to different soft terms.
%The moduli sector of the K\"ahler potential (\ref{kahler}) depends
%on the compactification scheme  and therefore the computation of the
%bosonic soft parameters will be model dependent. We will concentrate
%here on the interesting case of (0,2) symmetric Abelian orbifolds
%with diagonal moduli and matter metrics  
In the simple case of (0,2) symmetric Abelian orbifolds with diagonal
moduli and matter metrics the soft terms have been computed.
To assume that SUSY breaking is equally shared among $T_{i}$'s, i.e. 
the ``overall modulus'' $T$ scenario is a good starting point in the
analysis of charge and color breaking since essentially only one more
free parameter must be added \cite{prepa}.

%To illustrate the main features of mixed dilaton/moduli SUSY breaking,
%we will concentrate mainly on the case of diagonal moduli and matter 
%metrics. For instance, under this assumption the parametrization (\ref{auxi})
%is simplified to
%
%\begin{eqnarray}
%F^S &=& \sqrt{3}Cm_{3/2}{\hat K}_{{\overline S}S}^{-1/2}\sin\theta 
%e^{-i\gamma _S}\ \ , 
%\nonumber \\
%F^i &=& \sqrt{3}Cm_{3/2}{\hat K}_{{\overline i}i}^{-1/2}\cos\theta \Theta_{i}
%e^{-i\gamma_i}
%\ , 
%\label{auxili}
%\end{eqnarray}
%
%where $\sum_i \Theta_i^2=1$.

%Since the moduli part of the K\"ahler potential (\ref{kahler})
%has been computed for $(0,2)$ symmetric Abelian orbifolds, 
%we will concentrate here on these models.
%They contain  generically
%three $T$-type moduli 
%(the exceptions are the orbifolds $Z_3$, 
%$Z_4$ and $Z'_6$, which have 9, 5 and 5 respectively, and
%are precisely the ones with off-diagonal
%metrics) 
%and, at most, three $U$-type moduli.
%We will denote them collectively by $T_i$, 
%where e.g. $T_i=U_{i-3}$; $i=4,5,6$.
%For this  class of models the K\"ahler potential has the 
%form 
%\begin{eqnarray}
%K &=& -\log(S+S^*) - \sum _i \log(T_i+T_i^*) 
%%+ \sum _{\alpha } |C^{\alpha }|^2 \Pi_i(T_i+T_i^*)^{n_{\alpha }^i} \ .
%\label{orbi}
%\end{eqnarray}
%Here $n_{\alpha }^i$ are (zero or negative) fractional numbers usually 
%called ``modular weights" of the matter fields $C^{\alpha }$. 
%For each given Abelian orbifold,
%independently of the gauge group or particle content, the possible
%values of the modular weights are very restricted. For a classification of
%modular weights for all Abelian orbifolds see \cite{IL}.
%%As a matter of fact, the K\"ahler potentials which appear in the large-$T$
%%limit of Calabi-Yau compactifications \cite{calabi} and 
%4-D fermionic Strings \cite{fermionic}
%are quite close to the above one. Thus the results that we will
%obtain below will probably be more general than just for orbifold 
%compactifications.
%The piece proportional to $Z_{\alpha\beta}$ in (\ref{kahler})
%has been shown to be present in Calabi--Yau compactifications and 
%orbifolds. In particular, in the case of orbifolds, such a term
%arises when the untwisted sector has at least one complex--structure 
%field $U$ and has been explicitly computed.
%We will analyze separately this case below, as well as the associated 
%$\mu$ and $B$ parameters, whereas we will concentrate 
%here on the other bosonic soft parameters. Plugging 
%the particular form (\ref{orbi}) of the K\"ahler potential and
%the parametrization (\ref{auxili}) 
%in (\ref{mmmatrix}) and (\ref{mmmatrix2}) 
%we obtain
%the following results
%\footnote{This analysis was also carried out for the
%particular case of the three diagonal moduli $T_i$
%in ref.\cite{japoneses} and \cite{BC}
%in order to obtain unification of gauge coupling constants
%and to analyze  
%FCNC constraints respectively.
%Some particular multimoduli examples were also considered in
%ref.\cite{FKZ}.} 
%for the scalar masses and trilinear parameters \cite{BIMS,KSYY,BC}:
%
%\begin{eqnarray}
%m_{\alpha }^2 &=& m_{3/2}^2\left(1 + 3C^2\cos^2\theta\ {\vec {n_{\alpha }}}.
%{\vec {\Theta ^2}}\right)\ +\ V_0\ ,
%\label{masorbi2}\\
%&  M = \  \sqrt{3}m_{3/2}\sin\theta e^{-i{\gamma }_S} \ , &
%\nonumber\\
%A_{\alpha \beta \gamma } &=& -\sqrt{3} Cm_{3/2} \left( \sin\theta e^{-i{\gamma
%}_S}
%\right.
%\nonumber\\ &&
%+ \cos\theta \sum _{i=1}^6 e^{-i\gamma _i}    {\Theta }_i 
%\left[1\right.\right.
%\nonumber\\ &&
%\left.\left. +\
%n^i_{\alpha }+n^i_{\beta
%}+n^i_{\gamma
%}-
%(T_i+T_i^*) \partial_i \log Y_{\alpha \beta \gamma}\! \right]\right)\ . 
%\label{masorbi}
%\end{eqnarray}
%
%It is easy to check that the results (\ref{uno}) and (\ref{tres}) 
%are recovered in the limit where $\cos\theta \rightarrow 0$.
%Notice that neither the scalar (\ref{masorbi2}) nor the gaugino masses 
%(\ref{gaugin}) have any explicit dependence on $S$ or $T_i$: they 
%only depend on the gravitino mass and the goldstino angles.
%This is one of the advantages of a parametrization in terms of such angles.
%Although in the case of the $A$-parameter 
%an explicit $T_i$-dependence may appear in
%the term proportional to $\partial_i \log Y_{\alpha \beta \gamma }$, 
%it disappears in 
%several interesting 
%cases \cite{BIMS}. 
%Using the above information, we can now analyze the structure of
%soft parameters available in Abelian orbifolds. 



\section{Final comments and outlook}

We have shown in this review that charge and color breaking constraints
on the parameter space of generic SUSY theories are very strong. This is
particularly true in the case of SUSY theories deriving from weakly 
coupled heterotic
superstring where information about the structure of soft terms can be
obtained. Since the dilaton--dominated SUSY--breaking scenario is excluded
on these grounds, it would be very interesting
to study possible way--outs to the previous dramatic conclusion.
As mentioned in section 3 one possibility is to asume that also the moduli
fields contribute to SUSY breaking \cite{prepa}. 
%Of course, this amounts to a departure
%of the pure dilaton--dominated scenario. 
Another possibility, 
is 
%still in the
%context of this scenario, is to consider possible string
%non--perturbative contributions whose size could be large
%modifying in a sensible way the formulas for soft terms
%\cite{dilatonio}. 
to think that the perturbative and non-perturbative corrections
to the ``standard'' string tree--level dilaton--dominated scenario 
are important
and can modify the previous conclusions. 
%Due to analyticity and
%non-renormalization theorems these contributions are likely to affect in
%a substantial way only the K\"ahler potential 
%\cite{Banksdine}
%(in fact, the gauge kinetic function does also receive perturbative
%corrections at one loop level, but not beyond). 
Actually, some
one--loop string corrections 
%to the K\"ahler potential (and the gauge 
%kinetic function) 
have been
calculated for orbifold models 
%\cite{Kaplder} 
and they are rather small
for sensible values of the moduli. 
%Thus, it is reasonable to expect
%that further perturbative corrections will be even smaller. 
However,
this is not the case for the string non-perturbative corrections, whose
size could be much larger.  
%(see e.g. ref.\cite{Shenker}). 
modifying in a sensible way the formulas for soft terms
\cite{dilatonio}. 
Finally, recently some information has been obtained 
in the sense that all superstring theories
seem to correspond to some points in the parameter space of a unique
strongly coupled eleven--dimensional underlying theory, M--theory.
%Although the structure of this theory is largely unknown, some
%preliminary attempts have been made to extract some information
%of phenomenological interest. 
Once the structure of soft terms is known
charge and color breaking constraints should be applied to determine
their phenomenological viability. 



All the strongs constraints on the soft parameter space of SUSY
theories that have been reviewed here come from the requirement that
the standard vacuum is the global minimum of the theory.
In this sense, one possibility to avoid some of the above constraints
is to accept that we live in a metastable vacuum, provided that its
lifetime is longer than the present age of the universe \cite{claudson}, thus
rescuing points in the parameter space. 
In order to carry out this study one might consider two possibilities:
quantum tunneling at zero temperature from the standard vacuum to the
charge and color breaking one and thermal effects in the hot early universe.
Regarding the latter, although 
there is
a thermal energy to cross the barrier, due to the high temperature of the early
universe, the barrier is also higher \cite{riotto}.
In the case of quantum tunneling at zero temperature,
for instance the CCB minima
associated with the top--quark Yukawa coupling are the only ones to which
the standard vacuum might decay within the lifetime of the 
universe \cite{claudson,riotto,langacker,strumia}.
The CCB minima associated with other Yukawas are deeper 
but the
height of the barrier ($h\sim 1/\lambda^2$) 
is too large to allow an efficient tunnelling
probability ($\sim e^{-ch}$). 
In this sense the bounds that we reviewed here are basically 
the most conservative ones (in the sense of safe ones). Needless to say
that in any case, the identification of the dangerous CCB and UFB minima
is the first necessary step for the cosmological analysis.
In the context of gauge--mediated SUSY--breaking models the
standard vacuum seems to be stable cosmologically, but only if 
certain couplings are sufficiently small \cite{randall}.

Let us remark however that the possibility of living in a metastable vacuum
poses several problems. First of all, as was first suggested in \cite{CCB3},
it is hard to understand 
how  the cosmological constant is vanishing precisely
in such local ``interim'' vacuum.  
Even if a solution to that problem is found we would still
have to face the rather bizarre (but mathematically possible) 
situation of a future cosmological catastrophe, which does not seem
very attractive (at least for our descendants!).
Finally, from a more scientific (and less philosophical) point of view 
one needs to explain (without invoking an
anthropic principle) how does the universe manage to reach the 
standard minimum in the first place in spite of being local and metastable.
This requires the analysis of all possible cosmological scenarios.
In particular one can consider scenarios where the initial conditions
are dictated by thermal effects or inflationary scenarios. In the former
the standard vacuum is the closest one to the origin and therefore it
is the thermal equilibrium state at large temperatures \cite{claudson}. 
The inflationary
scenario may be much more dangerous and involved due to large fluctuations
of all the scalar fields, that could be driven in this way to the
dangerous minima. Whether this is the case or not is a complex issue
that is hotly discussed \cite{vilja}.




\section*{Acknowledgments}

Research supported in part by KOSEF, under the Brainpool Program;
the CICYT, under contract
AEN93-0673; the European Union,
under contracts CHRX-CT93-0132 and
SC1-CT92-0792.



%%%%%%%%%%%%%%%%%   References  %%%%%%%%%%%%%%%%%%%%%%
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\def\MPL #1 #2 #3 {{\em Mod.~Phys.~Lett.}~{\bf#1}\ (#2) #3 }
\def\NPB #1 #2 #3 {{\em Nucl.~Phys.}~{\bf B#1}\ (#2) #3 }
\def\PLB #1 #2 #3 {{\em Phys.~Lett.}~{\bf B#1}\ (#2) #3 }
\def\PR  #1 #2 #3 {{\em Phys.~Rep.}~{\bf#1}\ (#2) #3 }
\def\PRD #1 #2 #3 {{\em Phys.~Rev.}~{\bf D#1}\ (#2) #3 }
\def\PRL #1 #2 #3 {{\em Phys.~Rev.~Lett.}~{\bf#1}\ (#2) #3 }
\def\PTP #1 #2 #3 {{\em Prog.~Theor.~Phys.}~{\bf#1}\ (#2) #3 }
\def\RMP #1 #2 #3 {{\em Rev.~Mod.~Phys.}~{\bf#1}\ (#2) #3 }
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