%Paper: 
%From: Alberto Casas <casas@iem.csic.es>
%Date: Tue, 19 Dec 1995 20:47:43 UTC+0100

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\begin{document}
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\begin{flushright}
SCIPP 95/48\\
IEM--FT--113/95 \\
% \\
\end{flushright}
\vskip 0.3in
\begin{center}{\large\bf
New and Strong Constraints on the Parameter Space of the MSSM from
Charge and Color Breaking Directions
\footnote{Work supported in part by
the European Union (contract CHRX-CT92-0004) and
CICYT of Spain
(contract AEN94-0928).}  } \\
\vspace*{6.0ex}
{\large J.A. Casas}
\footnote{On leave of absence from Instituto
de Estructura de la Materia, CSIC, Serrano 123, 28006-Madrid,
Spain.}\\
%
\vspace*{1.5ex}
{\large\it Santa Cruz Institute for Particle Physics, Univ. of California,
Santa Cruz, CA 95064}\\
\vspace{2cm}
Talk given at the SUSY-95 International Workshop\\
``Supersymmetry and Unification of Fundamental Interactions''\\
Palaiseau, Paris, May 1995
\vspace{2cm}
\end{center}
\vskip.5cm
\begin{center}
{\bf Abstract}
\end{center}
%\begin{quote}
%\vbox{ \baselineskip 14pt

Although the possible existence of dangerous charge and color breaking
(CCB) directions in the MSSM has been known since the early 80's, only
particular directions in the field-space have been considered, thus obtaining
necessary but not sufficient conditions to avoid dangerous CCB minima.
Furthermore, the radiative corrections to the potential were not normally
included in a proper way, often leading to an overestimation of the
restrictive power of the bounds. It turns out that when correctly evaluated,
the ``traditional" CCB bounds are very weak. I give here a brief survey of
recent results on this subject, which represent a complete analysis, showing
that the new CCB bounds are very strong and, in fact, there are extensive
regions in the parameter space that become forbidden. This produces important
bounds, not only on the value of $A$, but also on the values of $B$ and
$M_{1/2}$. The form of strongest one of the new bounds, the so called UFB-3
bound, is explicitely given.
%\end{quote}
%}
%\vskip1.cm

\begin{flushleft}
%CERN-TH/95-xxx\\
November 1995 \\
\end{flushleft}

\end{titlepage}
%
\newpage
It is well known\cite{Frere,Claudson}, \cite{Drees,Gunion,Komatsu,Gamberini}
that the presence of scalar fields with
color and electric charge in supersymmetric (SUSY) theories induces the
possible existence of dangerous charge and color breaking (CCB)
minima.
However,
a complete study of this crucial issue is still lacking. This is
mainly due to two reasons. First, the enormous complexity of the
scalar potential, $V$, in a SUSY theory, which has motivated that
only analyses examining particular directions in the field--space
have been performed. Second, the radiative
corrections to $V$ have not been normally included in a proper way.
Concerning the first point, the tree-level scalar potential, $V_o$,
in the minimal supersymmetric standard model (MSSM) is given by
$V_o=V_F + V_D + V_{\rm soft}$, where $V_F$ and $V_D$ are the F-- and
D--terms respectively and $V_{\rm soft}$ are the soft breaking terms, i.e.
\bea
\label{Vsoft}
V_{\rm soft}&=&\sum_\alpha m_{\phi_\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)\;\; ,
\eea
in a standard notation.
$V_o$ is extremely
involved since it  has a large number of independent fields and parameters.
Even assuming universality of the soft breaking terms at
the unification scale, there are five undetermined
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.
(Notice that
$M$ does not appear explicitely in $V_o$, but it does through the
renormalization group equations (RGEs) of all the remaining parameters.)


As mentioned above, the complexity of $V$ has made that only
particular directions in the field-space have been explored, thus
obtaining necessary but not sufficient CCB conditions to avoid
dangerous CCB minima. By far the most extensively used CCB condition is
the ``traditional" bound, first studied in ref.\cite{Frere,Claudson}.
Namely, given a particular trilinear scalar coupling,
e.g. $\lambda_u A_u Q_u H_2 u$, assuming
equal vacuum expectation values (VEVs) for the three fields involved
in it, i.e. $|Q_u| = |H_2| = |u|$, it turns out that
a very deep CCB minimum appears {\em unless} the famous constraint
\be
\label{frerebound}
|A_u|^2 \leq 3\left( m_{Q_u}^2 + m_{u}^2 + m_2^2\right)
\ee
is satisfied. In the previous equation $m_{Q_u}^2, m_{u}^2, m_2^2$ are the
mass parameters of $Q_u$, $u$, $H_2$, where
$m_2^2$ is the sum of the $H_2$ squared soft mass, $m_{H_2}^2$, plus
$\mu^2$. Further analogous constraints have been derived in the
existing literature \cite{Drees,Gunion,Komatsu,Gamberini}.

Concerning the radiative corrections it should be noted that the usual
CCB bounds (e.g. eq.(\ref{frerebound})) come from the tree-level potential,
$V_o$. However, $V_o$ is strongly dependent on the renormalization scale,
$Q$ and the one-loop radiative corrections to it, namely
$\Delta V_1={\displaystyle\sum_{\alpha}}{\displaystyle\frac{n_\alpha}{64\pi^2}}
M_\alpha^4\left[\log{\displaystyle\frac{M_\alpha^2}{Q^2}}
-\frac{3}{2}\right]$
are crucial to make it stable against variations of $Q$
\cite{Gamberini,CC}. In the previous expression $M_\alpha$ are
the tree-level mass eigenstates, which in general are field-dependent
quantities, so
$\Delta V_1$ is a complicated function of all the scalar fields.
However, a good approximation is to still work just
with $V_o$, but at an appropriate choice for the value of $Q$,
so that $\Delta V_1$ is
small and the predictions of $V_o$ and $V_o+\Delta V_1$ essentially coincide.
This occurs for a value of $Q$ of the order of the most significant $M_\alpha$
mass appearing in $\Delta V_1$, which in turn depends on what is the
direction in the field-space that is
being analyzed. In the usual calculations, however, the CCB bounds
are imposed at any scale between $M_X$ and $M_Z$ and, therefore, their
restrictive power has been overestimated.

In this talk I give a brief survey of the results of our article, ref.
\cite{CCB}, where we have tried to completely classify all the possible
dangerous directions in the MSSM, extracting the corresponding improved
(and hopefully complete) bounds and analyzing numerically their restrictive
power.

It is important to keep in mind that the Higgs part of the potential
must be in such a way that it developes a {\em realistic} minimum at
$|H_1|=v_1$, $|H_2|=v_2$, with
$v_1^2+v_2^2=2M_W^2 / g_2^2$, which
corresponds to the standard vacuum.
This requirement fixes the value of $\mu$ in terms of the
other independent parameters, i.e. $m,M,A,B,\mu$.
Furthermore one has to demand that all the physical
particles have masses compatible with their observed values
(or upper bounds).

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
realistic minimum. A complete classification of the UFB and CCB
constraints can be obtained from ref. \cite{CCB}. Since there
is no room here to list the precise form of these bounds,
let us mention here their most important characteristics and
what is the most important bound.

Concerning the UFB directions (and corresponding constraints),
there are three of them, labelled as UFB-1, UFB-2, UFB-3
in \cite{CCB}. The relevant scalar fields involved are
$\{H_1,H_2\}$, $\{H_1,H_2, L\}$, $\{H_2, L,{e_L}_j,{e_R}_j\}$ respectively,
where $L$ is a slepton taking the VEV along the $\nu_L$ direction and
${e_L}_j, {e_R}_j$ are selectrons of the $j-$generation. The
UFB-3 bound turns out to be the {\it strongest} one of {\it all}
the UFB and CCB constraints in the parameter space of the MSSM,
so it deserves to be exposed in greater detail.

\begin{description}
\item[UFB-3]

${}^{}$\\
It is possible, by simple analytical minimization, 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 $|H_2|<M_X$ satisfying
\be
\label{SU6}
|H_2| > \sqrt{ \frac{\mu^2}{4\lambda_{e_j}^2}
+ \frac{4m_{L}^2}{g'^2+g_2^2}}-\frac{|\mu|}{2\lambda_{e_j}} \ ,
\ee
the value of the potential along the UFB-3 direction is simply given
by
\be
\label{SU8}
V_{\rm UFB-3}=(m_2^2 -\mu^2+ m_{L}^2 )|H_2|^2
+ \frac{|\mu|}{\lambda_{e_j}} ( m_{L_j}^2+m_{e_j}^2+m_{L}^2 ) |H_2|
-\frac{2m_{L}^4}{g'^2+g_2^2} \ ,
\ee
otherwise
\be
\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| + \frac{1}{8}
(g'^2+g_2^2)\left[ |H_2|^2+\frac{|\mu|}{\lambda_{e_j}}|H_2|\right]^2 \ .
\ee
In eqs.(\ref{SU8},\ref{SU9}) $\lambda_{e_j}$ is the leptonic Yukawa
coupling of the $j-$generation (see eq.(\ref{Vsoft}). Then, the
UFB-3 condition reads
\be
\label{SU7}
V_{\rm UFB-3}(Q=\hat Q) > V_{\rm real \; min} \ ,
\ee
where $V_{\rm real \; min}=-\frac{1}{8}\left(g^2 + g'^2\right)
\left(v_2^2-v_1^2\right)^2$ is the realistic minimum
and the $\hat Q$ scale is given by
$\hat Q\sim {\rm Max}(g_2 |e|, \lambda_{top} |H_2|,
g_2 |H_2|, g_2 |L|, M_S)$
with
$|e|$=$\sqrt{\frac{|\mu|}{\lambda_{e_j}}|H_2|}$ and
$|L|^2$=$-\frac{4m_{L}^2}{g'^2+g_2^2}$+($|H_2|^2$+$|e|^2$).
Finally $M_S$ is the typical scale of SUSY masses (normally a good
choice for $M_S$ is an average of the stop masses, for more details
see refs.\cite{Gamberini,CC,CCB})
{}From (\ref{SU8}-\ref{SU7}), it is clear that the larger
$\lambda_{e_j}$ the more restrictive
the constraint becomes. Consequently, the optimum choice of
the $e$--type slepton should be the third generation one, i.e.
${e_j}=$ stau.

\end{description}


Let us briefly turn to the CCB constraints in the strict sense, i.e. those
coming from the possible existence of charge and color
breaking (CCB) minima in the potential deeper than the
realistic minimum. We have already mentioned the ``traditional" CCB
constraint \cite{Frere} of eq.(\ref{frerebound}).
Other particular CCB constraints have been explored in the
literature \cite{Drees,Gunion,Komatsu,Langacker}.
In ref.\cite{CCB} it has been performed a complete analysis of
the CCB minima, obtaining a set of ``improved" analytic constraints
that represent the
necessary and sufficient conditions to avoid the dangerous ones.
For certain regions of values of the initial parameters, the CCB constraints
``degenerate" into the above-mentioned UFB constraints since the
minima become unbounded from below directions. In this sense, the
CCB constraints comprise the UFB bounds, so the latter can be
considered as special (but extremely important) limits of the former.

It is not possible to give here an account of the general CCB constraints
obtained in ref.\cite{CCB}, so let us mention their most outstanding
characteristics. First, the most dangerous, i.e. the
deepest, CCB directions in the MSSM potential involve only one particular
trilinear soft term of one generation. Then, for each trilinear soft term
there are three possible (optimized) types of constraints, which in
\cite{CCB} were named CCB-1,2,3. Of course these constraints, which
have an analytical form not very different from the ``traditional"
ones (see eq.(\ref{frerebound})), include the latter and are much
more stronger than them. It is important to recall here that the
CCB bounds must be evaluated at a correct renormalization scale,
$\hat Q$, in order to avoid an overestimation of their restrictive power.
That scale is always of order $\frac{A}{4\lambda}$, where $A$ and
$\lambda$ are, respectively, the coefficient of the trilinear scalar
term and the Yukawa coupling constant associated to the Yukawa coupling
under consideration (a more precise recipe for the value of $\hat Q$
can be found in \cite{CCB}). The numerical analysis shows that the
the ``traditional" CCB bounds when correctly evaluated
turn out to be very weak (see Fig.1a). On the contrary,
the new improved CCB-1,2,3 constraints obtained in ref.\cite{CCB}
(and not explicitely written here) are much more restrictive (see Fig.1b).


Anyway, as mentioned above, the most important restrictions come from
the UFB constraints, in particular from the UFB-3 one, explicitely shown
in eqs.(\ref{SU6}-\ref{SU7}). This is clearly exhibited in Fig.2,
where we summarize all the constraints
plotting also the excluded region due to (conservative)
experimental bounds on SUSY
particle masses. The allowed region left at the end of the day
(white) is quite small.

When the whole MSSM parameter space is scanned it is observed that, as
a general trend, the smaller the value of $m$, the more restrictive the
constraints become. In the limiting case $m=0$ essentially the
{\it whole} parameter space turns out to be excluded. This has obvious
implications, e.g. for no-scale models. Let us also mention that,
contrary to a common believe, the UFB and CCB constraints are
very strong and
put  important bounds not only on the
value of $A$ (soft trilinear parameter), but also on the values of $B$
(soft bilinear parameter) and $M$ (gaugino masses). This is a new and
interesting feature.

\newpage
%\hspace{0.1cm}
%
%\vspace{-0.1cm}

%\newpage
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\begin{figure}[htb]
%\psdraft
\centerline{
\psfig{figure=fig1ab.ps,height=11.5cm,angle=180
%,bbllx=9.5cm,bblly=1.cm,bburx=19.cm,bbury=14cm
}}
\caption{{\bf Fig.1:} Excluded regions in the parameter space of the Minimal
Supersymmetric Standard Model, with $B=A-m$, $m=100$ GeV and
$M^{\rm phys}_{\rm top}=174$ GeV. The darked region is excluded because
there is no solution for $\mu$ capable of producing the correct electroweak
breaking. a) The circles and diamonds indicate regions excluded by the
``traditional"
CCB constraints associated with
the $e$ and $d$-type trilinear terms respectively.
b) The same as (a) but using our ``improved" CCB
constraints. The triangles correspond to the $u$-type trilinear terms.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace{-3cm}

%\end{document}
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\begin{figure}[htb]
%\psdraft
\centerline{
\psfig{figure=fig2a.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 Minimal
Supersymmetric Standard Model, with $B=A-m$ and
$M^{\rm phys}_{\rm top}=174$ GeV.
The small filled squares indicate regions excluded by our
Unbounded From Below constraints, mainly the UFB-3 one.
The circles indicate regions excluded
by our ``improved" CCB constraints. The filled diamonds
indicate regions excluded by the experimental lower bounds on supersymmetric
particle masses.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\section*{Acknowledgements}

I thank my collaborators A. Lleyda and C. Mu\~noz for an enjoyable
work in this project


%%%%%%%%%%%%%%%%%%--- References
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 }
\def\ZPC #1 #2 #3 {{\em Z.~Phys.}~{\bf C#1}\ (#2) #3 }
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%
\bibitem{Frere}J.M. Frere, D.R.T. Jones and S. Raby, \NPB 222 1983 11;
L. Alvarez-Gaum\'e, J. Polchinski and M. Wise, \NPB 221 1983 495;
J.P. Derendinger and C.A. Savoy, \NPB 237 1984 307;
C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quir\'os, \NPB 236 1984 438.
%
\bibitem{Claudson}M. Claudson, L.J. Hall and I.Hinchliffe, \NPB 228 1983 501.
%
\bibitem{Drees}M. Drees, M. Gl\"uck and K. Grassie, \PLB 157 1985 164.
%
\bibitem{Gunion}J.F. Gunion, H.E. Haber and M. Sher, \NPB 306 1988 1.
%
\bibitem{Komatsu}H. Komatsu, \PLB 215 1988 323.
%
\bibitem{Gamberini}G. Gamberini, G. Ridolfi and F. Zwirner, \NPB 331 1990 331.
%
\bibitem{CC}B. de Carlos and J.A. Casas, \PLB 309 1993 320.
%
\bibitem{CCB} J.A. Casas, A. Lleyda and C. Mu\~noz,
``Strong Constraints on the Parameter Space of the MSSM from
Charge and Color Breaking Minima", preprint FTUAM 95/11, IEM--FT--100/95,
sent to {\em Nucl.~Phys.}~{\bf B}.
%
%
\bibitem{Langacker} P. Langacker and N. Polonsky, {\it UPR-0594T},
.

\end{thebibliography}



\end{document}


[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[


\section{Constraints on the parameter space}

In the previous sections, a complete analysis of all the potentially
dangerous unbounded from below (UFB)
and charge and color breaking (CCB)
directions has been carried out. In particular, the analytical form of the
constraints obtained on the parameter space of the MSSM
has been summarized in sect.5.
Now, we will analyze numerically those constraints. We will see that they
are very important and, in fact,
there are {\it extensive regions} in the parameter space which are forbidden.
%increasing
%the predictive power of the theory.


Our analysis will be quite general in the sense that we will consider the
whole parameter space of the MSSM, $m$, $M$, $A$, $B$, $\mu$, with the only
assumption of universality\footnote{Let us
remark, however, that the constraints
found in previous sections are general and they could also be applied
for the non-universal case.}. Actually, universality
of the soft SUSY-breaking terms at $M_X$ is a
desirable property not only to reduce the number of
independent parameters, but also for phenomenological reasons, particularly
to avoid flavour-changing neutral currents (see, e.g. ref.\cite{Ross}).
As discussed in sect.2, the requirement of correct electroweak breaking
fixes one
of the five independent parameters of the MSSM, say $\mu$, so we are left
with only four parameters ($m$, $M$, $A$, $B$). Although we will perform
the numerical analysis on this space, it is worth noticing that
particularly interesting values of $B$ can be obtained from
Supergravity (SUGRA). In this sense we will first consider two values of
$B$ as guiding examples to get an idea of
how strong the different constraints are
and then we will vary $B$ in order to obtain the most
general results. Hence, let us first justify, theoretically and
phenomenologically, the two specific values of $B$.



%Even assuming universality of the soft SUSY-breaking terms at $M_X$, the
%theory still contains a large number of (in principle) independent
%parameters: $m$, $M$, $A$, $B$, $\mu$. To carry out the numerical analysis
%it will be interesting to fix some of them. In fact we will work with
%two (theoretically well-motivated) particular values of $B$ as guiding
%%examples
%and, we will comment below on what happens as we vary this parameter.

The particular values of the soft terms depend on the type of Supergravity
theory from which the MSSM derives and, in general, on the mechanism of
SUSY-breaking. But, in fact, is still possible to learn things about soft
terms without knowing the details of
SUSY-breaking \cite{Yo}. Let us consider the simple
case\footnote{We will assume from now on a vanishing cosmological constant.}
of canonical kinetic terms for hidden and observable matter fields (i.e. a
K\"ahler potential $K=\sum_\alpha |\phi_\alpha|^2$). Then,
irrespective of the SUSY-breaking mechanism, the scalar masses are
automatically universal. Furthermore, if the observable part of the
superpotential $W$ is
assumed to be as in eq.(\ref{W}), $\mu$ being an initial parameter, then
the $B$ term and the universal
$A$ terms are automatically
generated and they are related to each other (assuming
that Yukawa couplings and $\mu$ are hidden field independent \cite{Yo}) by the
well known relation \cite{Weinberg}
%\be
%\footnote{
%In fact, the universality of the $A$ terms and relation (\ref{BAm}) can only
%be obtained if the Yukawa couplings ($\lambda$) and $\mu$ are hidden field
%independent \cite{Yo}.}.
\be
\label{BAm}
B = A - m \ .
\ee
Finally, if the gauge kinetic function is the same for the different gauge
groups of the theory $f_a=f$ (where $a$ is associated with $SU(3)$, $SU(2)_L$
and $U(1)_Y$), the gaugino masses are also universal.
This SUGRA theory is attractive for its simplicity and for the natural
explanation that it offers to the universality of the soft terms.
However, this scenario has a serious drawback. It is well known that, in
order to get appropriate $SU(2)_L\times U(1)_Y$ breaking, the $\mu$ parameter
has to be of the same order of magnitude ($M_W$) as the soft SUSY-breaking
terms discussed above. This is in general unexpected since the $\mu$ term
is a SUSY term whereas the soft terms are originated after SUSY-breaking.
In principle, the natural scale of $\mu$ would be the Planck mass.
The unnatural smalleness of the $\mu$ parameter is the so-called
$\mu$ problem. We will briefly explain here three interesting
scenarios considered in SUGRA in order to solve the problem,
illustrating them in the case
of canonical kinetic terms:

{\em (a)} In ref.\cite{Casas} was pointed out that the presence of a
non-renormalizable term in the
superpotential, $\lambda W H_1 H_2$, characterized by the coupling $\lambda$,
yields dynamically a
$\mu$ parameter when the hidden sector part of $W$ acquires a VEV, namely
$\mu = m_{3/2} \lambda $, where $m_{3/2}$ is the gravitino mass.
The fact that $\mu$ is of the electroweak scale order is a consequence
of our assumption of a correct SUSY-breaking scale
$ m_{3/2} = O(M_W) $. Now, with this solution to the $\mu$ problem, the
$B$ parameter can be straightforwardly evaluated.
%Considering again the simple
%case of canonical kinetic terms for matter fields, in order to guarantee
%universal soft breaking terms,
The simple result (in the case of
$\lambda$ independent of the hidden fields \cite{Yo}) is
\be
\label{B2m}
B = 2 m \ .
\ee
For this mechanism to work, the $\mu H_1 H_2$ term in eq.(\ref{W})
must be initially absent (otherwise the natural scale for $\mu$ would be the
Planck
mass), a fact that remarkably enough, is automatically guaranteed in the
framework of Superstring theory as we will see below.

{\em (b)} In refs.\cite{Masiero, Casas}
it was shown that if a term, $Z H_1 H_2 + h.c.$,
characterized by the coupling $Z$
is present in the K\"ahler potential, an effective low-energy $B$ term is
naturally generated.
In the case of $Z$ independent of the hidden fields, this mechanism
for solving the $\mu$ problem is equivalent \cite{Casas} to the previous
one {\em (a)} and therefore
the value of $B$ is again given by eq.(\ref{B2m}). Now, the size
of $\mu$ is $\mu = m_{3/2} Z$.

{\em (c)} In ref.\cite{Giudice} the observation was made that in the framework
of
any SUSY-GUT, starting again with $\mu = 0$, an effective $\mu$ term is
generated by the integration of the heavy degrees of freedom.
The prediction for $B$ is once more given by eq.(\ref{B2m}).


The solutions discussed here in order to solve the $\mu$ problem
are {\it naturally present} in Superstring theory. In ref.\cite{Casas}
was first remarked that the $\mu H_1 H_2$ term is naturally absent
as already mentioned above. The reason is that in SUGRA theories coming
from Superstring theory mass terms for light fields are forbidden in the
superpotential.
Then a realistic example where non-perturbative
SUSY-breaking mechanisms like gaugino-squark condensation induce
superpotentials of the type {\em (a)} was given. In ref.\cite{Narain}
the same kind of superpotential was obtained using pure gaugino condensation
in the context of orbifold models.
The alternative mechanism {\em (b)} in which there is an extra term in the
K\"ahler
potential originating a $\mu$-term
is also naturally present in some large classes of four-dimensional
Superstrings \cite{Kaplu,Lust,Narain}. In Superstring theory, neither the
kinetic terms are in general canonical nor the couplings
(Yukawas, $\lambda$, $Z$) and the mass term ($\mu$)
are independent of hidden fields.
However, it is still possible to obtain (the phenomenologically
desirable) universal soft terms in the so-called
dilaton-dominated limit \cite{Kaplu,Brignole}. This limit is not only
interesting because of that, but also because
it is quite {\it model independent}
(i.e. for any compactification scheme the results for the soft terms are the
same).
It is also remarkable, that in this
limit once again the value of $B$ for the two mechanisms {\em (a), (b)}
coincides \cite{Yo} with that of eq.(\ref{B2m}).
If, alternatively, we just assume that a small ($\sim M_W$) dilaton-independent
mass $\mu$
is present in the superpotential, then the result for $B$ is
now given \cite{Brignole} by eq.(\ref{BAm}) as in the case of canonical
kinetic terms.

{}From the above analysis, it is clear that eqs.(\ref{BAm},\ref{B2m})
give us two values of $B$ very interesting from the theoretical and
phenomenological point of view. Thus, we will consider, for the moment, in
our numerical study of the UFB and CCB constraints both possibilities.
In fact, the value
of $\mu$ is also fixed once we choose a particular mechanism for
solving the $\mu$ problem, e.g. mechanisms {\em (a), (b)} (see above).
However, this value still depends on the couplings
$\lambda$ and $Z$
which are in general model dependent\footnote{
For an
analysis of the MSSM from Superstring theory taking into account a particular
value of $Z$ coming from
orbifold compactifications, and therefore a fixed value of
$\mu$, see ref.\cite{BIMS}.}, so we prefer to eliminate $\mu$ in terms of
the other parameters by imposing appropriate symmetry-breaking
at the weak scale as mentioned above.
%Then we are left with only three independent parameters ($m$,$M$,$A$).
Let us now turn to the numerical results.

In Fig.1 we have presented in detail the case $B=A-m$ with $m=100$ GeV, to get
an idea of how strong the different constraints are, plotting the excluded
regions 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 already mentioned in sect.2,
not for all the parameter space
it is possible to choose the boundary condition of $\lambda_{top}$
so that the experimental mass of the top is reproduced, since the RG infrared
fixed point of $\lambda_{top}$ puts an upper bound on $M_{\rm top}$,
namely $M_{\rm top}\simlt 197\sin\beta$ GeV \cite{infrared},
where $\tan\beta=v_2/v_1$.
In this way, the upper and lower darked regions are forbidden because
$M^{\rm phys}_{\rm top}=174$ GeV cannot be reached.
Furthermore, the small central darked region is also forbidden because
there is no value of $\mu$ capable of producing the correct electroweak
breaking.

Fig.1a shows the region excluded by  the ``traditional" CCB
bounds of the type of
eq.(\ref{frerebound}), evaluated at an {\em appropriate} scale
(see subsect.4.5). For a point in the parameter
space to be excluded we have also demanded that the corresponding
CCB minimum is deeper than the realistic one (this is especially
relevant for the bounds coming from the top trilinear term).
Clearly, the ``traditional" bounds, when correctly evaluated, turn out to
be very weak. In fact, only the leptonic (circles) and the $d$--type
(diamonds) terms do restrict, very modestly, the parameter space.
Let us recall here that it has been a common (incorrect) practice
in the literature to evaluate these traditional bounds at all the scales
between $M_X$ and $M_W$, thus obtaining very important (and of course
overestimated) restrictions in the parameter space.
Fig.1b shows the region excluded by our ``improved" CCB constraints
obtained in sect.4 and summarized in sect.5. Comparing Figs.1a
and 1b it is clear that the excluded region becomes dramatically increased.
Notice also that all the trilinear couplings (except the top one in this
case) give restrictions, producing areas constrained by different
types of bounds simultaneously. The restrictions coming from the UFB
constraints, obtained in sect.3 and summarized in sect.5, are shown
in Fig.1c. By far, the most restrictive bound is the UFB--3 one (small
filled squares). Indeed, the UFB--3 constraint is the {\em strongest}
one of {\em all} the UFB and CCB constraints, excluding extensive
areas of the parameter space, as is illustrated in the figure. In our
opinion, this is a most remarkable result.
Finally, in
Fig.1d we summarize all the constraints
plotting also the excluded region due to the (conservative)
experimental bounds on SUSY
particle masses (filled diamonds) of eq.(\ref{Expb}).  More precisely,
this forbidden area comes from too
small masses for the gluino, lightest chargino, lightest
neutralino, left sbottom, and left and right $u,c$ squarks.
The allowed region left at the end of the day
(white) is quite small.
Figs.2a, 2b, 2c give, in a summarized way, the same analysis as that
of Fig.1, but for three different values of $m$ ($m=100$ GeV,
$m=300$ GeV, $m=500$ GeV).
For the plots with $m$ bigger
than 100 GeV the gluino, lightest stop, lightest chargino and
lightest neutralino are responsible for
the excluded region due to experimental
bounds on masses.
The ants indicate regions
which are excluded by negative squared
mass eigenvalues, in this case
the lightest stop.
The figures show a clear trend in the sense that
the smaller the value of $m$, the more restrictive the
constraints become.
This is mainly due to
the effect of the UFB-3 constraint (note the almost exact $m$--invariance
of the CCB bounds).
In the limiting case $m=0$ (not represented in the
figures) essentially the
{\it whole} parameter space turns out to be excluded. This has obvious
implications, e.g. for no-scale models\footnote{We thank J. L\'opez
for a comment stressing us the possible implications of the CCB and UFB
bounds for no-scale models.} \cite{noscale}.
Anyway,
extensive areas in the parameter space are forbidden in all cases.

The same conclusions are obtained for the other (theoretically and
phenomenologically well-motivated) value of $B$, $B=2m$. The
results in this scenario are shown
in Fig.3, where the whole darked region is forbidden because
$M^{\rm phys}_{\rm top}=174$ GeV cannot be reached. Unlike the Fig.2,
now in some cases the left sbottom
may also get a negative squared mass eigenvalue.

Finally, in Figs.4a, 4b we generalize the previous analyses by
varying the value
of $B$ for different values of $m$, namely $m=100$ GeV,
$m=300$ GeV.
The final allowed regions from all types of bounds
in the parameter space of the
MSSM are shown.
Both figures exhibit a similar trend. For a particular value of $m$,
the larger the value of
$B$ the smaller the allowed region becomes. More precisely, the maximum
allowed value of $B$ is $B=2.5m$ for $m=100$ GeV and $B=3.5m$
for $m=300$ GeV. This fact comes mainly
from the enhancement of the forbidden areas by the UFB-3 constraint and the
requirement of $M^{\rm phys}_{\rm top}=174$ GeV.
Both facts are due to the decreasing of $\tan \beta$ as $B$ grows.
Then higher top Yukawa couplings are needed in order to reproduce
the experimental top mass. 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, the
stronger the UFB-3 bound becomes.
For negative values of $B$
the corresponding figures can easily be deduced from the previous ones,
taking into account that they are invariant
under  the transformation
$B,A,M \rightarrow -B,-A,-M$.
%(the sign of $\mu$, which is not plotted in the figures, is simply
%the appropriate one to yield the electroweak breaking).

{}From the various figures it is clear that the CCB and UFB constraints
{\em put
important bounds} not only on the value of $A$, but also on the values of
$B$ and $M$, which is an interesting novel fact.

\section{Conclusions}

Although the possible existence of dangerous charge and color breaking
minima in the supersymmetric standard model has been known since the
early 80's, a complete study of this crucial issue was still lacking. This
was due to two reasons: First, the complexity of the SUSY scalar potential,
$V$,
caused that only particular directions in the field-space were considered, thus
obtaining necessary but not sufficient conditions to avoid dangerous
charge and color breaking minima. Second, the radiative corrections to $V$
were not normally included in a proper way.
%, often leading to an overestimation
%of the restrictive power of the bounds.

In the present paper we have carried out a
complete analysis of all the potentially
dangerous directions in the field-space of the MSSM, obtaining the
corresponding constraints on the parameter space. These are completely general
and can be applied to the non-universal case. The
constraints turn out to be very important and, in fact, there are
{\it extensive regions} in the parameter space which are forbidden, increasing
the predictive power of the theory.

The constraints can be clasified in two types. First, the ones
associated with the existence of charge and color breaking (CCB) minima in
the potential deeper than the realistic minimum.
%E.g. this is
%the origin of the ``traditional" bounds (see eq.(\ref{frerebound})).
Second, the constraints associated with
directions in the field-space along which the potential becomes
unbounded from below (UFB). It is worth mentioning here that 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.
%This is the origin, for example, of the well
%known condition (\ref{ufbhiggs}).

We have performed a complete analysis of both types of directions
obtaining {\it new and very restrictive bounds}, expressed in an analytic
way, that represent a set of necessary and sufficient constraints.
They are summarized in sect.5.
For certain values of the initial parameters the CCB constraints
``degenerate" into the UFB constraints since the minima become unbounded from
below directions. In this sense, the CCB constraints comprise the UFB bounds,
which can be considered as special (but extremely important) limits of the
former.

We have also taken into
account the radiative corrections to $V$ in a proper way. To this respect,
let us remember that, usually, the scalar potential
is considered at tree-level, improved by
one-loop RGEs, so that all the parameters appearing in it are running
with the renormalization scale, $Q$. Then it is often demanded that the
CCB and UFB constraints are satisfied at any scale between
$M_X$ and $M_Z$. However, this is not correct since the tree-level scalar
potential is strongly $Q$-dependent and the one-loop
radiative corrections to it are crucial to make the potential stable against
variations of the scale. Using the scale independence of $V$, instead
of minimizing the complete one-loop potential, which would be an impossible
task, we have demanded
that the previous (tree-level-like) bounds are satisfied at the renormalization
scale, $Q$, at which the one-loop correction to the
potential is essentially negligible. This simplifies enormously
the analysis, producing equivalent results. We have also given explicit
expressions of the appropriate scale to evaluate the different types of bounds.

The usual lack in the literature
of an optimum scale to evaluate the constraints
implies that their restrictive power has normally been overestimated.
E.g., the ``traditional" CCB bounds (see eq.(\ref{frerebound}))
when (incorrectly) analyzed
at $M_X$ are very strong. However, we have seen that
when correctly evaluated, they turn out to
be very weak (see Fig.1a).
The new CCB constraints obtained here are much more restrictive and, in fact,
the excluded region becomes dramatically increased (see Fig.1b). On the
other hand, the restrictions coming from the new UFB constraints are by
far the most important ones, excluding extensive areas of the parameter
space (see e.g. Fig.1c).

We have performed a numerical analysis of how our UFB and CCB constraints
put restrictions on the whole parameter space of the MSSM. As already
mentioned they are very strong producing important bounds not only on the
value of $A$ (soft trilinear parameter), but also on the values of $B$
(soft bilinear parameter) and $M$ (gaugino masses). This is a new and
interesting feature. This analysis is summarized in Figs.2--4.
As a general trend, the smaller the value of $m$, the more restrictive the
constraints become. In the limiting case $m=0$ essentially the
{\it whole} parameter space turns out to be excluded. This has obvious
implications, e.g. for no-scale models.


Finally, let us mention that all the constraints that has been obtained here
come from the requirement that the standard vacuum is the global minimum of
the theory.
Although the possibility of living in a metastable
vacuum with a lifetime larger than the present age of the Universe
\cite{Claudson} does not seem specially attractive, it cannot be excluded.
Since the constraints on the parameter space found in this paper are very
strong, this dynamical question deserves further analysis
\cite{Nosotros}.



\section*{Acknowledgements}

I thank my collaborators A. Lleyda and C. Mu\~noz for an enjoyable
work in this project


%%%%%%%%%%%%%%%%%%--- References
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 }
\def\ZPC #1 #2 #3 {{\em Z.~Phys.}~{\bf C#1}\ (#2) #3 }
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C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quir\'os, \NPB 236 1984 438.
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\bibitem{CC}B. de Carlos and J.A. Casas, \PLB 309 1993 320.
%
\bibitem{CCB} J.A. Casas, A. Lleyda and C. Mu\~noz,
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Charge and Color Breaking Minima", preprint FTUAM 95/11, IEM--FT--100/95,
sent to \NPB.
{}.
%

\end{thebibliography}


\newpage

\section*{Figure Captions}

\begin{description}
\item[Fig.~1] Excluded regions in the parameter space of the Minimal
Supersymmetric Standard Model, with $B=A-m$, $m=100$ GeV and
$M^{\rm phys}_{\rm top}=174$ GeV. The central darked region is excluded because
there is no solution for $\mu$ capable of producing the correct electroweak
breaking. The upper and lower darked regions are
excluded because it is not possible to reproduce the experimental mass of the
top. a) The circles and diamonds indicate regions excluded by the
``traditional"
Charge and Color Breaking constraints associated with
the $e$ and $d$-type trilinear terms respectively.
b) The same as (a) but using our ``improved" Charge and Color Breaking
constraints. The triangles correspond to the $u$-type trilinear terms.
c) The crosses, squares and small filled squares indicate
regions excluded by the Unbounded From Below-1,2,3 constraints respectively.
d) The previous excluded regions together with the one
arising from the experimental lower bounds on
supersymmetric particle masses (filled diamonds).

\item[Fig.~2] Excluded regions in the parameter space of the Minimal
Supersymmetric Standard Model, with $B=A-m$ and
$M^{\rm phys}_{\rm top}=174$ GeV, for
different values of $m$.
The central darked region is excluded because
there is no solution for $\mu$ capable of producing the correct electroweak
breaking. The upper and lower darked regions are
excluded because it is not possible to reproduce the experimental mass of the
top. The small filled squares indicate regions excluded by our
Unbounded From Below constraints. The circles indicate regions excluded
by our ``improved" Charge and Color Breaking constraints. The filled diamonds
indicate regions excluded by the experimental lower bounds on supersymmetric
particle masses. The ants indicate regions excluded by
negative scalar squared mass eigenvalues.

\item[Fig.~3] The same as Fig.~2 but with $B=2m$. Now, the whole
darked region is excluded because it is not
possible to reproduce the experimental mass of the top.


\item[Fig.~4] Contours of allowed regions in the parameter space of the Minimal
Supersymmetric Standard Model, with
$M^{\rm phys}_{\rm top}=174$ GeV  and different
values of $B$ and $m$, by the whole set of constraints.



\end{description}






\end{document}
















]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]


((((((((((((((((((
For example, for the trilinear coupling
$\lambda_u Q_u H_2u_R$, the CCB-1 bound reads as follows.
If
$3m_{L_i}^2-(m_{Q_u}^2+m_{u}^2)+2(m_2^2-\mu^2) > 0$,
then the optimized CCB-1 bound is
\bea
\label{CCCB1a}
|A_u|^2 \leq  3 \left[ m_2^2-\mu^2+m_{Q_u}^2+m_{u}^2\right]\;.
\eea
Otherwise
\bea
\label{CCCB1b}
|A_u|^2 \leq \left(1+\frac{2}{\alpha^2}\right)
\left[ m_2^2-\mu^2+(m_{Q_u}^2+m_{u}^2)\alpha^2
+ m_{L_i}^2(1-\alpha^2) \right]
\eea
with $\alpha^2=\sqrt{\frac{2(m_{L_i}^2+m_2^2-\mu^2)}
{m_{Q_u}^2+m_{u}^2-m_{L_i}^2}}.$

If $ m_2^2-\mu^2+m_{L_i}^2<0$, then the CCB-1 bound is automatically
violated. In fact the minimization of the
potential in this case gives $\alpha^2 \rightarrow 0$, and we are exactly
led to the UFB-3 direction shown above, which represents the correct
analysis in this instance.


\vspace{0.3cm}
\noindent
\vspace{0.3cm}
\noindent

)))))))))))))))))))))))))))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
%\psdraft
\centerline{
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\psfig{figure=susy1.ps,height=9.5cm,bbllx=9.5cm,bblly=1.cm,bburx=19.cm,bbury=14cm}}
\caption{{\bf Left panel:} Plot of $\phi_{\rm min}(t)/\xi(t)$ as a function of
$\mu(t)$ in the
LL (dashed line) and NTLL (solid line) approximation. The dotted line shows the
would-be scale independent result.  {\bf Right panel:} Plot of $m_{H,{\rm
der}}(t)/m_H(t)$
as a function of $\mu(t)$. In both cases $m_t=160$ GeV and the supersymmetric
parameters
are $M_S=1$ TeV, $X_t=0$ and $\tan\beta\gg 1$.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}



