%Paper: 
%From: ehret@ekpux7.physik.uni-karlsruhe.de (Ralf Ehret)
%Date: Fri, 20 May 1994 20:58:59 +0100 (WET DST)
%Date (revised): Mon, 4 Jul 1994 13:02:23 +0100 (WET DST)


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\newcommand{\unify}  {plot1}    % Unification of couplings
\newcommand{\runma}  {plot5}    % running of SUSY masses
\newcommand{\proton} {proton}   % 90% excluded region of proton decay
\newcommand{\mbvsmt} {plot3}    % mb vs. mtop for different tanb
\newcommand{\mhvsmt} {plot7}    % mh vs. mtop for diff. tanb,m0,m1/2
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\newcommand{\rG}   {{\rm GUT}}
\newcommand{\MG}   {{\ifmmode M_\rG         \else $M_\rG$          \fi}}
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\newcommand{\mze}  {{\ifmmode m_0           \else $m_0$            \fi}}
\newcommand{\mha}  {{\ifmmode m_{1/2}       \else $m_{1/2}$        \fi}}
\newcommand{\tb}   {{\ifmmode \tan\beta     \else $\tan\beta$      \fi}}
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\newcommand{\as}   {{\ifmmode \alpha_s      \else $\alpha_s$       \fi}}
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\begin{document}

%\input{titlepage}
%\input{section1}
%\input{section2}
%\input{section3}
%\input{section56}
%\input{appendix}
%\input{physlet}
\clearpage

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\begin{titlepage}
 \begin{flushright}
\vspace*{-2.2cm}
\noindent
           IEKP-KA/94-05    \\
          \\
    May, 1994        \\
 \end{flushright}
\vspace{1.5cm}
\begin{center} {\Large\bf Predictions of SUSY Masses \\
          in the Minimal Supersymmetric \\
          Grand Unified Theory  \\}
\vspace{0.6cm}
{\bf W.~de Boer\footnote{Bitnet: DEBOERW@CERNVM},
  R.~Ehret\footnote{E-mail: ehret@ekpux7.physik.uni-karlsruhe.de} and
  W.~Oberschulte\footnote{E-mail: wulf@ekpux5.physik.uni-karlsruhe.de}\\
}
{\it Inst.\ f\"ur Experimentelle Kernphysik, Univ.\ of Karlsruhe   \\}
{\it Postfach 6980, D-76128 Karlsruhe 1, FRG  \\} and \\
{\bf D.I. Kazakov\footnote{E-mail: kazakovd@thsun1.jinr.dubna.su} \\}
{\it Bogoliubov Lab. of Theor. Physics, Joint Inst. for Nucl. Research, \\}
{\it 141 980 Dubna, Moscow Region, RUSSIA \\}
\end{center}

\vspace{1cm}

\begin{center}
{\bf Abstract}
\end{center}

\vspace{0.3cm}

\begin{center}\parbox{13cm}{\small

The Minimal Supersymmetric Standard Model (MSSM) distinguishes
itself from other GUT's by a successful prediction of many
unrelated phenomena with a minimum number of parameters.

Among them: a)   Unification of the couplings constants;
b)  Unification of the masses;
c)  Proton decay;
d)  Electroweak symmetry breaking at a scale far below the unification
    scale
and the corresponding relation between
the gauge boson masses and the top quark mass.

A combined fit of the free parameters in the MSSM to these low energy
constraints  shows that the MSSM model can satisfy these constraints
simultaneously. From the fitted parameters the masses of the as yet
unobserved superpartners of the SM particles are predicted, the unknown
top mass is constrained to a range between 140 and 200 GeV, and the
second order QCD coupling constant is required  to be between 0.108 and
0.132. The complete second order renormalization group equations for
the gauge and Yukawa couplings are  used and analytical solutions for
the neutral gauge boson, the  Higgs masses and the sparticle  masses
are derived, taking into account the one-loop corrections to the
Higgs potential.

It is shown that a top mass  of $174\pm16$ GeV, as suggested recently
by the CDF Collaboration, constrains the mixing angle between the Higgs
doublets in the MSSM to: $1.2<\tb<5.5$ at the 90\% C.L.. The most
probable value corresponds to $\tb = 1.56$;
such a small value causes a large mixing in the stop sector, in which
case the lightest stop mass is likely to be
below the top mass.
In this case the stop production in $p\bar{p}$
collisions would contribute to the top signature.
This could be an explanation for the large effective
$t\bar{t}$ cross section observed by CDF.
               }
\end{center}
\vspace{0.5cm}
%\begin{center}

%(Submitted to
% Physics Letters B.)

%\end{center}


\end{titlepage}
\section{Introduction}

Grand Unified Theories (GUT) hold the promise of "explaining" the
difference between the electromagnetic, weak and strong nuclear forces:
 their different strenghts are simply due to radiative corrections.
Furthermore, they are candidates to explain several unrelated
observations about our universe, e.g. they almost automatically
lead to baryon number violation, thus providing a possible explanation
for the matter-antimatter asymmetry  in our universe~\cite{sak} and the
spontaneous symmetry breaking of the unified force into the known forces
at a sufficient high  energy can cause the inflationary scenario of the
universe, thus providing an explanation for the origin of matter and the
homogeneity of the universe on a large
scale~\cite{borner,kolb}.

The Grand Unification idea \cite{su5} has been recently subjected
to a new test using the new precise LEP data~\cite{ekn,abf,lalu}. The
result clearly indicates that the minimal Standard Model (SM), being
extrapolated to 15 orders of magnitude does not lead to
unification~\cite{abf}. Indeed, the coupling constants, as measured
precisely at LEP, do not become equal
at a single energy if extrapolated to high energies from their values at
$M_Z$.  On the contrary, within the supersymmetric extension of the
Standard Model, unification is achieved.

The Minimal Supersymmetric extension of the Standard Model (MSSM)
\cite{su5susy} has become
the leading candidate for a Grand Unified Theory (GUT).
Supersymmetry~\cite{susy,susyrev} presupposes a symmetry between fermions and
bosons, thus introducing spin 0 partners of the quarks and leptons
- - - - - - -- called squarks and sleptons -- and spin 1/2 partners of the
gauge
bosons and  Higgs particles -- called gauginos and Higgsinos.
Since these predicted particles have not been observed sofar,
these supersymmetric (SUSY) particles must be heavier than the known
particles, implying that supersymmetry must be broken.  However, from
the unification condition a first estimate of the SUSY breaking scale
could be made: it was found to be of the order of 1000 GeV, or more
precisely $10^{3\pm1}$  GeV~\cite{abf}. The uncertainty in this scale
is mainly caused by the uncertainty in the strong coupling constant.

Assuming soft symmetry breaking at the GUT scale, all SUSY masses can be
expressed in terms of 5 parameters and the masses at low energies are
then determined by the well known Renormalization Group (RG) equations.
So many parameters cannot be derived from the unification condition
alone. Further constraints can be considered:
\begin{itemize}
  \item $M_Z$ predicted from electroweak  symmetry
        breaking~\cite{Ino1} --
 \nocite{ewbr,ehnt83,grz90,Ibanez,rrb92,ir92,roskane}
        \cite{loopewbr}.
  \item Constraints from the unification of Yukawa
        couplings~\cite{rrb92,ir92},~\cite{Ross1} --
\nocite{copw,bbo,bbog,lanpol,bmaskln,cpr,op,copw,cpw}
        \cite{ara91}.
  \item Constraints from the lower limit on the proton
        lifetime~\cite{langac,arn,protkln}.
  \item Experimental lower limits on SUSY masses~\cite{higgslim,pdb}.
  \item Constraints from the top mass measured  by CDF~\cite{cdf}.
\end{itemize}

Of course, in many of the references given
above, several constraints are studied simultaneously, since
considering   one constraint at a time yields
only one relation between parameters.  Trying to
find complete solutions with only a few constraints requires then
additional assumptions, like naturalness, no-scale
models, fixed ratios for gaugino- and scalar masses or a fixed ratio
for the Higgs mixing parameter  and the scalar mass,  or
combinations of these assumptions.

Several ways to study the constraints simultaneously
have been pursued. One can either sample the whole
parameter space in a systematic or
random way and check the regions which are allowed
by the experimental constraints.

Alternatively, one can try a statistical
analysis, in which all constraints are
implemented in a $\chi^2$ definition and try to
find the most probable region of the parameter
space by minimizing the $\chi^2$ function.

In the first case one has to ask: which weight
should one give to the
various regions of parameter space and how large
is the parameter space? Some sample
the space logarithmically, thus emphasizing
the low energy regions~\cite{roskane}, others provide a linear
sampling~\cite{cpr,cpw,nanop,bor,ez}.
In the second case one is faced with the difficulty,
that the function to be  minimized is not monotonous,
because of the experimental  limits  on the particle
masses, proton lifetime and so on,
since at the transitions where these constraints become
effective, the derivative of the $\chi^2$ function
is not defined. Fortunately, good
minimizing programs in multidimensional parameter
space, which do not rely on the derivatives, exist~\cite{minuit}.
The advantage of such a statistical analysis
is that one obtains probabilities for the
allowed regions of the parameter space and
can calculate confidence levels.
The results of such an analysis will be
presented after a short description
of the experimental input values.

%Other analysis have obtained similar mass spectra
%for the predicted particles in the
%MSSM~\cite{rrb,arnnat,roskane,bor,ram1} or extended
%versions of the MSSM~\cite{nanopo}.

%It is the purpose of this paper to study all constraints simultaneously
%without any of the assumptions mentioned above in order to see if a
%solution within the {\it minimal} supersymmetric model (MSSM) exists at
%all, which is non-trivial as should be clear from the contradictions
%mentioned before.
%
%It turns out that a solution exists indeed and the 10 experimental
%constraints mentioned above are sufficient to obtain a unique solution
%in the multidimensional parameter space of the MSSM without any of
%the assumptions given above. The corresponding constraints on the
%SUSY mass spectrum, the strong coupling constant and the top mass
%are given.

The requirement of electroweak symmetry breaking leads to a
non-trivial minimum of the Higgs potential. Taking into account the
one-loop radiative corrections proportional to the top Yukawa coupling
we derive analytic expressions for the mass of the neutral gauge
boson, the $Z^0$, as function of the mass of the top quark. These
expressions can be used as strong constraints on the masses in the
Higgs potential.


The paper is organized as follows: In sect.~2 we describe the MSSM
together with the symmetry breaking part originated from supergravity
and introduce 5 soft breaking parameters mentioned above. Sect.~3
containes the one-loop renormalization group equations for all the
couplings and the parameters of the Higgs potential. One-loop analytic
solutions to them are given. In sect.~4 we consider the leading radiative
corrections to the Higgs potential and calculate their influence on
the masses.
For completeness we present   all the needed
formulas used in our analysis, including those that are well known
in the literature, since the notations usual are different and
difficult to combine.
Experimental constraints are discussed in sect.~5.
The results are summarized in sect.~6 and it is shown that the lightest stop
mass
is likely to be below the top mass.
\section{The Model}

\subsection{The Lagrangian}

Minimal supersymmetric extention of the Standard Model is described
by the Lagrangian containing the SUSY-symmetric part together with
SUSY breaking terms originated from supergravity~\cite{susy}.
The breaking terms of the Lagrangian are given by:
\begin{eqnarray} {\cal L}_{Breaking} & = &
- - - - - - -m_0^2\sum_{i}^{}|\varphi_i|^2-m_{1/2}\sum_{j}^{}\lambda_j
 \lambda_j \label{2} \\ & - &
   Am_0\left[h^u_{ab}Q_aU^c_bH_2+h^d_{ab}Q_aD^c_bH_1+
 h^e_{ab}L_aE^c_bH_1\right] - Bm_0\left[\mu H_1H_2\right],
 \nonumber \end{eqnarray}

\noindent
Here

\begin{tabular}{ll}
  $h^{u,d,e}_{ab}$ & are the Yukawa couplings, \ $a,b =1,2,3$ run
                     over the generations \\
  $Q_a$   & are the SU(2) doublet squark fields \\
  $U_a^c$ & are the SU(2) singlet charge-conjugated up-squark fields \\
  $D_b^c$ & are the SU(2) singlet charge-conjugated down-squark fields\\
  $L_a$ & are the SU(2) doublet slepton fields \\
  $E_a^c$ & are the SU(2) singlet charge-conjugated slepton fields \\
  $H_{1,2}$ & are the SU(2) doublet Higgs fields \\
  $\varphi_i$ & are all scalar fields \\
  $\lambda_j $ & are the gaugino fields, $j = 1,2,3$.\\
                    &
\end{tabular}

\noindent
 From the Lagrangian given above it follows that at the GUT scale   the
  squarks and sleptons  have a common mass $m_0$
and the gauginos a common mass $m_{1/2}$.
The full SUSY Lagrangian contains the following free parameters:
\begin{itemize}
\item 3 gauge couplings $\alpha_i$,
\item the Yukawa couplings $h_i$,
\item the Higgs fields mixing parameter  $\mu $.
\end{itemize}
\noindent
They are supplemented by the soft breaking ones:
\begin{itemize}
\item $m_0, \ m_{1/2}, \ A, \ B$,
where A and B are the coupling constants
for the Higgs fields.
\end{itemize}
These two sets of parameters determine completely the mass spectrum
of all quarks, leptons, Higgs bosons and their superpartners.

%\subsection{Neutralino and Chargino Masses}

The masses of the gauginos are denoted by $M_i$, where $i=1,2,3$
corresponds to the U(1), SU(2) and SU(3) groups, respectively.
For the gauginos of the
SU(2) $\otimes$ U(1) group one has to consider the mixing with the
Higgsinos, since they carry weak isospin, hypercharge and
spin $\frac{1}{2}$ too.
The mass terms in the full Lagrangian are~\cite{susy}:
\begin{equation}
   {\cal L}_{Gaugino-Higgsino}=
   -\frac{1}{2}M_3\bar{\lambda}_3^\alpha\lambda_3^\alpha
   -\frac{1}{2}\bar{\chi}M^{(0)}\chi -(\bar{\psi}M^{(c)}\psi + h.c.)
   \label{3}
\end{equation}
where $\lambda_3^\alpha$ are the 8 Majorana gluino fields and
$$ \chi = \left(\begin{array}{c}\tilde{B}^0 \\ \tilde{W}^3 \\
\tilde{H}^0_1 \\ \tilde{H}^0_2
\end{array}\right), \ \ \ \psi = \left( \begin{array}{c}
\tilde{W}^{+} \\ \tilde{H}^{+}
\end{array}\right),$$
are the Majorana neutralino and Dirac chargino fields.
The mass matrices are:
\begin{equation} M^{(0)}=\left(
\begin{array}{cccc}
M_1 & 0 & -M_Z\cos\beta \sin_W & M_Z\sin\beta \sin_W \\
0 & M_2 & M_Z\cos\beta \cos_W   & -M_Z\sin\beta \cos_W  \\
- - - - - - -M_Z\cos\beta \sin_W & M_Z\cos\beta \cos_W  & 0 & -\mu \\
M_Z\sin\beta \sin_W & -M_Z\sin\beta \cos_W  & -\mu & 0
\end{array} \right) \label{4} \end{equation}
\begin{equation} M^{(c)}=\left(
\begin{array}{cc}
M_2 & \sqrt{2}M_W\sin\beta \\ \sqrt{2}M_W\cos\beta & \mu
\end{array} \right) \label{5} \end{equation}
\subsection{The SUSY Mass Spectrum}

Owing to radiative corrections, all couplings and masses become scale
dependent (or running). Their scale dependence is described by the
renormalization group (RG) equations, which depend on
the particle content of the model in a given energy region.
In the minimal subtraction scheme~\cite{msbar}
one ignores the contribution of the particles
heavier than the energy scale at hand.  One can
distinguish the following energy scales: $M_{\rm GUT}, M_{\rm SUSY}$,
and $M_Z$, where   $M_{\rm SUSY}$ refers to  some averaged mass of  the
superpartners.

For a not too large mixing angle
$\tan \beta$ between the two Higgs doublets in the theory, as required
by the present data (see section \ref{ch6}), the bottom Yukawa coupling
is much smaller than the top Yukawa coupling. In this case the
contribution from the bottom Yukawa coupling can be neglected, which
allowed us to obtain analytical solutions for all superpartner masses.



In this section the first order RG equations and their solutions are
summarized in order to get a complete mass spectrum as function of the
parameters $\alpha_\rG, ~\MG, ~m_0, ~m_{1/2}, ~\mu, ~A$ and $\tan\beta$.
The RG equations for the masses and the other
SUSY breaking parameters  in the region
between \MG and $M_{\rm SUSY}$ are
 up to  one-loop    (we use the notation of the last
paper of ref.~\cite{Ibanez}):
\begin{eqnarray} \frac{d\tilde{m}^2_L}{dt} & = &
 \left( 3\tal_2M^2_2 + \frac{3}{5}\tal_1M^2_1\right)
\nonumber \\
%
\frac{d\tilde{m}^2_E}{dt} & = & ( \frac{12}{5}\tal_1M^2_1)
\nonumber \\
%
\frac{d\tilde{m}^2_Q}{dt} & = & (\frac{16}{3}\tal_3M^2_3 +
3\tal_2M^2_2 + \frac{1}{15}\tal_1M^2_1)
- - - - - - -\delta_{i3}Y_t(
\tilde{m}^2_Q+\tilde{m}^2_U+m^2_2+A^2_tm_0^2-\mu^2)
\nonumber \\
%
\frac{d\tilde{m}^2_U}{dt} & = & \left(\frac{16}{3}\tal_3M^2_3
+\frac{16}{15}\tal_1M^2_1\right)
- - - - - - -\delta_{i3}2Y_t(
\tilde{m}^2_Q+\tilde{m}^2_U+m^2_2+A^2_tm_0^2-\mu^2) \nonumber \\
\frac{d\tilde{m}^2_D}{dt} & =
 & \left(\frac{16}{3}\tal_3M^2_3
+ \frac{4}{15}\tal_1M^2_1\right) \label{11} \\
%
\frac{d\mu^2}{dt} & = & 3(
\tal_2 +\frac{1}{5}\tal_1 -Y_t)\mu^2 \nonumber \\
%
\frac{dm^2_1}{dt} & = &
3(\tal_2M^2_2 +\frac{1}{5}\tal_1M^2_1)+
3(\tal_2 +\frac{1}{5}\tal_1 -Y_t)\mu^2 \nonumber \\
%
\frac{dm^2_2}{dt} & = &
3(\tal_2M^2_2 +\frac{1}{5}\tal_1M^2_1)+
3(\tal_2 +\frac{1}{5}\tal_1)\mu^2-3Y_t(
\tilde{m}^2_Q+\tilde{m}^2_U+m^2_2+A^2_tm_0^2) \nonumber \\
%
\frac{dm^2_3}{dt} & = & \frac{3}{2}(
 \tal_2+\frac{1}{5}\tal_1-Y_t)m^2_3+3\mu m_0Y_tA_t-
3\mu(\tal_2M_2 +\frac{1}{5}\tal_1M_1)\nonumber \\
%
\frac{dA_t}{dt} & = & \left(\frac{16}{3}\tal_3\frac{M_3}{m_0}
+ 3\tal_2\frac{M_2}{m_0} +
\frac{13}{15}\tal_1\frac{M_1}{m_0}\right)-6Y_tA_t \nonumber \\
\frac{dB}{dt} & =
& 3\left(\tal_2\frac{M_2}{m_0} +
\frac{1}{5}\tal_1\frac{M_1}{m_0}\right)-3Y_tA_t, \nonumber \\
\frac{dM_i}{dt}& = & -b_i\tilde{\alpha_i}M_i. \nonumber
\end{eqnarray}
Here $m_U, m_D$ and $m_E$ refer to the masses of the superpartners
of the quark and lepton singlets, while $m_Q$ and $m_L$ refer to the
masses of the weak isospin doublet superpartners;  $m_1, m_2, m_3$ and
$\mu$ are the mass parameters of the Higgs   potential (see next section),
while A and B are the couplings   in
${\cal L}_{Breaking}$ as defined before;  $M_i$ are the gaugino
masses before any mixing.

The $\delta_{i3}$  term  in the  previous equations implies that
only the Yukawa coupling of the third
generation is taken into account.
The top Yukawa coupling obeys
\begin{eqnarray}
   \frac{dY_t}{dt} & = & Y_t\left(\frac{16}{3}\tal_3 +
                         3\tal_2 +
                         \frac{13}{15}\tal_1\right)-6Y_t^2 ,
\end{eqnarray}
while the RG equations for the gauge couplings in first order can
be written as:
\begin{eqnarray}
   \frac{d\tal_i}{dt}
                   & = & -b_i\tal_i^2 .
\end{eqnarray}
The higher order corrections to these RG equations will be discussed
in the next section.

Here
$$ \tal_i=\frac{\alpha_i}{4\pi}, \ Y_t=\frac{h_t^2}{(4\pi)^2},
\ t=\log(\frac{M_X^2}{Q^2}),$$ and the top Yukawa coupling $h_t$ is
related to the top mass by
$$ m_{t}=h_t(m_t)v\sin\beta.$$
Note that $m_t$ is the running mass.
Its relation to the physical pole mass
will be discussed in section \ref{ch6}.

The boundary conditions at $Q^2=M_{\rm GUT}^2$ or at $t=0$ are:
$$\tilde{m}^2_Q=\tilde{m}^2_U=\tilde{m}^2_D=\tilde{m}^2_L=\tilde{m}^2_E=
m_0^2;$$
$$\mu^2 = \mu_0^2;\ \ \  m_1^2=m^2_2=\mu_0^2+m_0^2; \ \ \ $$
$$M_i=m_{1/2};\ \ \ \tal_i(0)=\tal_{\rm GUT},\ \ \ i=1,2,3$$

Solutions to the one-loop RG equations can be obtained analytically. In
the MSSM they are~\cite{Ibanez}:
\begin{eqnarray}
   M_i(t)&=& \frac{\tal_i(t)}{\tal_i(0)}m_{1/2}
                                                                   \nn
   Y_t(t)&=& \frac{\displaystyle Y_0E(t)}{\displaystyle 1+6Y_0F(t)}
                                   \label{17}                      \\
   A(t)  &=& \frac{\displaystyle A_0}{\displaystyle 1+6Y_0F(t)}
            +\frac{m_{1/2}}{m_0}\left(H_2-
            \frac{\displaystyle 6Y_0H_3}{\displaystyle 1+6Y_0F(t)}
            \right)                                                \nn
   \mu^2(t) &=&  \frac{\displaystyle \mu_0^2}{\displaystyle
      (1+6Y_0F(t))^{1/2}}(1+\beta_2t)^{3/b_2}(1+\beta_1t)^{3/(5b_1)} \\
   m_1^2(t) &=& m_0^2+\mu^2(t) + m^2_{1/2}
                \tal_\rG(\frac{3}{2}f_2(t)+
                \frac{3}{10}f_1(t))                                \nn
   m^2_2(t) &=& \mu^2(t) + m^2_{1/2}e(t)+
                A_0m_0m_{1/2}f(t)+m_0^2(h(t)-k(t)A_0^2)            \nn
   m^2_3(t) &=& q(t)m^2_3(0) + r(t)\mu_0 m_{1/2} +
                s(t)A_0 m_0\mu_0                             \nonumber
\end{eqnarray}
with the notation defined in the appendix.

For the light squark  and slepton  masses one finds~\cite{Ibanez}:
\begin{eqnarray}
  \tilde{m}^2_{E_{L}}   & = & m^2_0+m^2_{1/2}\tal_\rG
          \left(\frac{3}{2} f_2(t)+\frac{3}{10}f_1(t)\right)
          -\cos(2\beta)M_Z^2(\frac{1}{2}-\sin^2\theta_W)        \nn
  \tilde{m}^2_{\nu_{L}} & = & m^2_0+m^2_{1/2}\tal_\rG
          \left(\frac{3}{2} f_2(t)+\frac{3}{10}f_1(t)\right)
          +\cos(2\beta)  \frac{1}{2}M_Z^2                       \nn
  \tilde{m}^2_{E_{R}}   & = & m^2_0+m^2_{1/2} \tal_\rG
          \left(\frac{6}{5}f_1(t)\right)
          -\cos(2\beta)M_Z^2\sin^2\theta_W                      \nn
  \tilde{m}^2_{U_{L}}   & = & m^2_0+m^2_{1/2}\tal_\rG
          \left(\frac{8}{3}f_3(t) +\frac{3}{2}f_2(t)
          +\frac{1}{30}f_1(t)\right) -\cos(2\beta)M_Z^2(-\frac{1}{2}
          +\frac{2}{3}\sin^2\theta_W)                           \nn
  \tilde{m}^2_{D_{L}}   & = & m^2_0+m^2_{1/2}\tal_\rG
          \left(\frac{8}{3}f_3(t) +\frac{3}{2}f_2(t)
          +\frac{1}{30}f_1(t)\right) -\cos(2\beta)M_Z^2(\frac{1}{2}
          -\frac{1}{3}\sin^2\theta_W)                           \nn
  \tilde{m}^2_{U_{R}}   & = & m^2_0+m^2_{1/2}\tal_\rG
          \left(\frac{8}{3}f_3(t) +\frac{8}{15}f_1(t)\right)+
          \cos(2\beta)M_Z^2(\frac{2}{3}\sin^2\theta_W)          \nn
  \tilde{m}^2_{D_{R}}   & = & m^2_0+m^2_{1/2}\tal_\rG
          \left(\frac{8}{3}f_3(t)  +\frac{2}{15}f_1(t)\right)-
          \cos(2\beta)M_Z^2(\frac{1}{3}\sin^2\theta_W)   \label{18}
\end{eqnarray}
where $\beta$ is the mixing angle between the two Higgs doublets.
For the third generation the effect of
the top  Yukawa coupling
is taken into account through the $\delta_{i3}$ terms in eqns \ref{11},
in which case the
solutions  given above are changed. We found:
\begin{eqnarray}
  \tilde{m}^2_{b_{R}} & = & \tilde{m}^2_{D_{R}}       \nn
  \tilde{m}^2_{b_{L}} & = & \tilde{m}^2_{D_{L}}+
          \left[\frac{1}{3}(m^2_2-\mu^2-m^2_0)
          -\frac{1}{2}\tilde{\alpha}_\rG
           \left(f_2(t)+\frac{1}{5}
           f_1(t)\right)m^2_{1/2}\right]              \nn
  \tilde{m}^2_{t_{R}} & = & \tilde{m}^2_{U_{R}}+2
           \left[\frac{1}{3}(m^2_2-\mu^2-m^2_0)
           -\frac{1}{2}\tilde{\alpha}_\rG\left(
           f_2(t)+\frac{1}{5}
           f_1(t)\right)m^2_{1/2}\right] +m_t^2       \nn
  \tilde{m}^2_{t_{L}} & = & \tilde{m}^2_{U_{L}}+
           \left[\frac{1}{3}(m^2_2-\mu^2-m^2_0)
           -\frac{1}{2}\tilde{\alpha}_\rG
           \left(f_2(t)+\frac{1}{5}
           f_1(t)\right)m^2_{1/2}\right]+m_t^2   \label{hsq}
\end{eqnarray}
Note that only   $b_L$  gets corrections from
the top-quark Yukawa coupling through a loop with a chargino
and a top-quark. The subscripts $L$ or $R$ do not indicate
the helicity, since the squarks and sleptons have no spin.
The labels just indicate in analogy to the non-SUSY particles, if they
are $SU(2)$ doublets or singlets.

A non-negligible Yukawa coupling
causes a mixing between the weak interaction eigenstates and the mass
eigenstates. Since the mixing is proportional to the Yukawa coupling,
we will only consider the mixing for the top quarks.
The mass matrix is~\cite{Ibanez}:
\begin{equation}
  \left(
  \begin{array}{cc}
     \tilde{m}^2_{t_{L}}        & -m_t(A_t m_0+\mu\cot\beta) \\
     -m_t(A_t m_0+\mu\cot\beta) & \tilde{m}^2_{t_{R}}
  \end{array}
  \right) \label{tlr}
\end{equation}
and the corresponding mass eigenstates are:
\begin{eqnarray}
  \tilde{m}^2_{t_{1,2}}&=&
  \frac{1}{2}\left[\tilde{m}^2_{t_{L}}+\tilde{m}^2_{t_{R}} \mp
  \sqrt{(\tilde{m}^2_{t_{L}}-\tilde{m}^2_{t_{R}}
  )^2+4m^2_t(A_t m_0 + \mu\cot\beta )^2}
  \right]\label{mt1t2}
\end{eqnarray}
$\tilde{m}_{t_1}$ is defined as the lightest stop.

%
%------------- numerical values
%
The functions $f_i(t)$ depend on $\tal_\rG$ and
$t=\ln({\MG^2}/{Q^2})$ as shown explicitly in the appendix.
They were calculated for the parameters
from the typical fit shown in table \ref{t71}
($\alpha_\rG=1/24.3$, $\mgut=2.0\cdot10^{16}$ GeV, $\sws=0.2324$
and $A_t(0)=0$). For these values one finds the following numerical
formulae:
\begin{eqnarray}
  \tilde{m}^2_{E_{L}}(t=66)    & = &
        m^2_0+0.52m^2_{1/2}-0.27\cos(2\beta)M_Z^2\label{mslep}    \nn
  \tilde{m}^2_{\nu_{L}}(t=66)  & = &
        m^2_0+0.52m^2_{1/2}+0.5\cos(2\beta) M_Z^2                 \nn
  \tilde{m}^2_{E_{R}}(t=66)    & = &
        m^2_0+0.15 m^2_{1/2} -0.23\cos(2\beta)M_Z^2               \nn
  \tilde{m}^2_{U_{L}}(t=66)    & = &
        m^2_0+6.6m^2_{1/2}+0.35\cos(2\beta)M_Z^2 \label{msq}      \nn
  \tilde{m}^2_{D_{L}}(t=66)    & = &
        m^2_0+6.6m^2_{1/2}-0.42\cos(2\beta)M_Z^2                  \nn
  \tilde{m}^2_{U_{R}}(t=66)    & = &
        m^2_0+6.2m^2_{1/2}+0.15\cos(2\beta)M_Z^2                  \nn
  \tilde{m}^2_{D_{R}}(t=66)    & = &
        m^2_0+6.1m^2_{1/2}-0.07\cos(2\beta)M_Z^2                  \nn
  \tilde{m}^2_{b_{R}}(t=66) & = & \tilde{m}^2_{D_{R}}             \nn
  \tilde{m}^2_{b_{L}}(t=66) & = &
       \tilde{m}^2_{D_{L}}-0.48m^2_0-1.21m^2_{1/2}                \nn
  \tilde{m}^2_{t_{R}}(t=66) & = &
       \tilde{m}^2_{U_{R}} + m_t^2 -0.96m^2_0-2.42m^2_{1/2}       \nn
  \tilde{m}^2_{t_{L}}(t=66) & = &
       \tilde{m}^2_{U_{L}}+m_t^2 -0.48m^2_0-1.21m^2_{1/2}.
\label{sqsl}
\end{eqnarray}
After mixing the mass eigenstates of the stop matrix are
(using the same numerical input as for the light squarks):
\begin{eqnarray}
  \tilde{m}^2_{t_{1,2}}(t=66) & =      &
     \frac{1}{2}\left[\tilde{m}^2_{t_{L}}+\tilde{m}^2_{t_{R}} \mp
     \sqrt{(\tilde{m}^2_{t_{L}}-\tilde{m}^2_{t_{R}}
     )^2+4m^2_t(A_tm_0 + \mu/\tan\beta )^2}
     \right]                                              \nonumber \nn
                              &\approx &
     \frac{1}{2}\left[0.6m_0^2+9.2m_{1/2}^2+
     2m_t^2-0.19\cos(2\beta)M_Z^2 \right]                 \nonumber \nn
                             &         &
     \mp\frac{1}{2}\sqrt{\left[1.6m_{1/2}^2+
     0.5m_0^2-0.5\cos(2\beta)M_Z^2\right]^2+
     4m_t^2(A_tm_0 + \mu/\tan\beta )^2}.                  \nonumber \nn
     \label{t12}
\end{eqnarray}
 where   the values of $A_t$ and $\mu $ at the  weak
 scale can be calculated as:
 \bq A_t(M_Z)=4.6A_t(0)+1.7\frac{m_{1/2}}{m_0}
   \eq
\bq \mu(M_Z)=0.63\mu_0 \eq
%with $A_t(0)=0$.
Note that for large values of $A_t(0)$ or $\mu$ combined with a
small $\tan\beta$ the splitting becomes large and one of the stop
masses can become very small.


The   gauginos and Higgsinos
have similar quantum numbers (both spin 1/2), which causes a mixing between
the weak interaction eigenstates
and the mass eigenstates.
The masses are the eigenvalues of the mass matrices
given in  eqns. \ref{4} and \ref{5}.
The  two chargino eigenstates $\tilde{\chi}_{1,2}^{\pm}$
can be written easily in an analytical form:
\begin{eqnarray}
M^2_{1,2}&=&\frac{1}{2}\left[M^2_2+\mu^2+2M^2_W \right.\label{6}  \\
 & & \left. \mp \sqrt{(M^2_2-\mu^2)^2+4M^4_W\cos^22\beta
+4M^2_W(M^2_2+\mu^2+2M_2\mu \sin 2\beta )}\right], \nonumber
\end{eqnarray}
where at the GUT scale the masses of the gaugino fields of the
U(1), SU(2) and SU(3) groups are equal to $m_{1/2}$,
i.~e.~$M_1=M_2=M_3=m_{1/2}$.
The eigenvalues of the $4\times 4$ neutralino mass matrix can be solved
most easily by a numerical diagonalization.
If the  parameter $\mu$ is much larger
than $M_1$ and $M_2$, which is the case
as shown in the last section, the
mass eigenstates become
\bq \tilde{\chi}_i^0=[\tilde{B},\tilde{W}_3,
\frac{1}{\sqrt{2}}(\tilde{H}_1-\tilde{H}_2),
\frac{1}{\sqrt{2}}(\tilde{H}_1+\tilde{H}_2)] \eq
with eigenvalues $|M_1|,|M_2|, |\mu|,$ and $|\mu|$,
respectively.
In other words, the bino and neutral wino do not
mix with each other nor with the Higgsino eigenstates
in this limiting case.

\subsection{Radiative Corrections to the Higgs Potential}c
\label{cor_higgs_pot}

The MSSM needs two Higgs doublets:
$$ H_1(1,2,-\frac{1}{2})= \left(\begin{array}{c}H^0_1 \\
H^-_1\end{array}\right), \ \ \
 H_2(1,2,\frac{1}{2})= \left(\begin{array}{c}H^+_2 \\
H^0_2\end{array}\right).$$
If one keeps only the  radiative corrections proportional to the top
Yukawa coupling, the one-loop effective  Higgs potential for the
neutral components   can be written
as~\cite{cpr,erz,berz,drno,kz92,eqz}:
\begin{eqnarray}
  V(H_1,H_2) &=& m^2_1|H_1|^2+m^2_2|H_2|^2-m^2_3(H_1H_2+h.c.)+
\frac{g^2+g^{'2}}{8}(|H_1|^2-|H_2|^2)^2  \nonumber  \\
           &+&  \frac{3}{32\pi^2}\left[
\tilde{m}_{t_1}^4 ( \ln \frac{\tilde{m}_{t_1}^2}{Q^2}-\frac{3}{2})+
\tilde{m}_{t_2}^4 ( \ln \frac{\tilde{m}_{t_2}^2}{Q^2}-\frac{3}{2})
- - - - - - -   2  {m}_{t}^4 ( \ln \frac{
{m}_{t}^2}{Q^2}-\frac{3}{2})\right],
\label{21} \end{eqnarray}
where  $\tilde{m}_{t_i}$ and $m_t$ are field dependent masses:
\begin{equation}
   {m}_t^2=h_t^2\ |H_2^0|^2=h_t^2v^2\sin^2\beta, \label{22a}
\end{equation}
and
$\tilde{m}_{t_1}^2$ and $\tilde{m}_{t_2}^2$ are the eigenvalues
of the $2\times 2$ $\tilde{t}_L$ and $\tilde{t}_R$ mass-squared
matrix, given in eq.~(\ref{mt1t2}).

The mass parameters in the potential fulfil  the following boundary
conditions at the GUT scale:
\begin{eqnarray}
  m_1^2=m^2_2 & =& \mu^2_0+m_0^2 \mbox{\hspace{3mm} and}  \nn
        m^2_3 & = & -B\mu_0m_0,    \label{higgs_bound}
\end{eqnarray}
  where $\mu_0$ is the value of $\mu$ at the GUT scale.

After spontaneous symmetry breaking the minimimum of the potential
corresponds to non-zero vacuum expectation values of the Higgs fields:
\begin{equation}
  <H_1>\;\equiv
         \left(\begin{array}{c}v_1 \\
              0\end{array}\right)
   =      v           \cos\beta, \ \ <H_2>\;\equiv
  \left(\begin{array}{c}0 \\
  v_2\end{array}\right)
   =      v           \sin\beta,         \label{hv}
\end{equation}
$$ \tan\beta \equiv  \frac{v_2}{v_1}, \ \
v^2 =  v_1^2+v_2^2~~~~~(v \approx 174~{\rm GeV})$$




With $v_1, v_2$ defined in eq.~\ref{hv} the one-loop minimization
conditions are:
\begin{eqnarray}
  \frac{\partial V}{\partial H_1} & = & 2m_1^2v_1-2m_3^2v_2 +
        \frac{g^2+ g^{'2}}{2}(v_1^2-v_2^2)v_1 \nonumber  \\
        & & +\frac{3}{8\pi^2}h_t^2\mu(A_t m_0v_2+\mu v_1)
        \frac{b(\tilde{m}^2_{t1})-b(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
        \tilde{m}^2_{t2}}= 0 \label{22} \\
  \frac{\partial V}{\partial H_2} & = & 2m_2^2v_2-2m_3^2v_1 -
        \frac{g^2+ g^{'2}}{2}(v_1^2-v_2^2)v_2 \nonumber \\
        & & +\frac{3}{8\pi^2}\Bigg\{ h_t^2 A_t m_0(A_t m_0v_2+\mu v_1)
        \frac{b(\tilde{m}^2_{t1})-b(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
        \tilde{m}^2_{t2}} \nonumber  \\
        & & +[(b(\tilde{m}^2_{t1})+
        b(\tilde{m}^2_{t2})-2b(m^2_t)]h_t^2v_2\Bigg\}=0
  \label{23},
\end{eqnarray}
where
$$
  b(m^2)=m^2(\ln\frac{m^2}{m^2_t}-1).
$$
 From the minimization conditions one obtains:
\begin{eqnarray}
  v^2&=&\frac{\displaystyle 4
       }{\displaystyle (g^2+g^{'2})(\tan^2\beta -1)}\Bigg\{
       m_1^2-m_2^2\tan^2\beta \label{24} \\
  &-&\frac{3h_t^2}{16\pi^2}\left[
[b(\tilde{m}^2_{t1})+b(\tilde{m}^2_{t2})-2b(m^2_t)]\tan^2\beta+
(A_t^2m_0^2\tan^2\beta -\mu^2)
\frac{b(\tilde{m}^2_{t1})-b(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
\tilde{m}^2_{t2}}\right]\Bigg\} \nonumber  \\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
2m_3^2&=&(m_1^2+m_2^2)\sin 2\beta
  + \frac{3h_t^2 \sin 2\beta}{16\pi^2}\left\{
b(\tilde{m}^2_{t1})+b(\tilde{m}^2_{t2})-2b(m^2_t)\right. \nonumber \\
 & & \left. +(A_t m_0 +\mu\tan\beta)
(A_t m_0 +\mu\cot\beta)
\frac{b(\tilde{m}^2_{t1})-b(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
\tilde{m}^2_{t2}}
\right\}  \label{25}
\end{eqnarray}
 From the above equations one  can derive easily:
\begin{eqnarray}
M^2_Z&\equiv&  \frac{\displaystyle (g^2+g^{'2})}{2}v^2=
\frac{\displaystyle m^2_1-m^2_2 \tan^2\beta -
\Delta^2_Z}{\tan^2\beta -1}, \label{26} \\
\Delta^2_Z&=&\frac{3g^2}{32\pi^2}\frac{m^2_t}{M^2_W\cos^2\beta}
\left[
b(\tilde{m}^2_{t1})+b(\tilde{m}^2_{t2})+2m^2_t +
(A^2_t m_0^2 -\mu^2\cot^2\beta )
\frac{b(\tilde{m}^2_{t1})-b(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
\tilde{m}^2_{t2}}\right] \nonumber
\end{eqnarray}
The Higgs masses corresponding to this one loop potential
are~\cite{erz,berz,drno,kz92,eqz,cpr}:
\begin{eqnarray} m^2_A&=&m_1^2+m_2^2+\Delta^2_A,   \label{27}     \\
\Delta^2_A&=& \frac{3g^2}{32\pi^2}\frac{m^2_t}{M^2_W\sin^2\beta}\left[
b(\tilde{m}^2_{t1})+b(\tilde{m}^2_{t2})+2m^2_t +(A^2_t m_0^2+\mu^2 )
\frac{b(\tilde{m}^2_{t1})-b(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
\tilde{m}^2_{t2}}\right] \nonumber \\
m^2_{H^{\pm}}&=&m^2_A+M^2_W+\Delta^2_H, \label{28} \\
\Delta^2_H&=&-\frac{3g^2}{32\pi^2}\frac{m^4_t\mu^2}{\sin^4\beta
M^2_W}\frac{c(\tilde{m}^2_{t1})-c(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}-
\tilde{m}^2_{t2}} \nonumber \\
m^2_{h,H}&= &
\frac{1}{2}\left[m^2_A+M^2_Z +\Delta_{11}+\Delta_{22} \right. \label{29}
\\
&&\left. { ^-_+} \sqrt{\begin{array}{ll}
(m^2_A+M_Z^2+\Delta_{11}+\Delta_{22})^2 & -4m^2_AM_Z^2\cos^22\beta
- - - - - - -4(\Delta_{11}\Delta_{22}-\Delta_{12}^2) \\
- - - - - - -4(\cos^2\beta
M^2_Z+\sin^2\beta M^2_A)\Delta_{22} & -4(\sin^2\beta M^2_Z+\cos^2\beta
M^2_A)\Delta_{11}\\
- - - - - - -4\sin2\beta (M^2_Z+M^2_A)\Delta_{12}&
\end{array} }\right] \nonumber \\
\Delta_{11}&=&\frac{3g^2}{16\pi^2}\frac{m^4_t}{\sin^2\beta
M^2_W} \left[\frac{\mu(A_t m_0+ \mu\cot\beta
)}{\tilde{m}^2_{t1}-\tilde{m}^2_{t2}}
\right]^2d(\tilde{m}^2_{t1},\tilde{m}^2_{t2}), \nonumber \\
\Delta_{22}&=&\frac{3g^2}{16\pi^2}\frac{m^4_t}{\sin^2\beta M^2_W}\left[
\ln (\frac{\tilde{m}^2_{t1}\tilde{m}^2_{t2}}{m^4_t})+
\frac{2A_t m_0 (A_t m_0+
\mu\cot\beta )}{\tilde{m}^2_{t1}-\tilde{m}^2_{t2}} \ln
(\frac{\tilde{m}^2_{t1}}{\tilde{m}^2_{t2}}) \right.\nonumber \\
 & & \left.+\left[
 \frac{A_t m_0(A_t m_0 +\mu\cot\beta
)}{\tilde{m}^2_{t1}-\tilde{m}^2_{t2}}\right]^2
d(\tilde{m}^2_{t1},\tilde{m}^2_{t2})\right], \nonumber \\
\Delta_{12}&=&\frac{3g^2}{16\pi^2}\frac{m^4_t}{\sin^2\beta M^2_W}
\frac{\mu(A_t m_0 +
\mu\cot\beta )}{\tilde{m}^2_{t1}-\tilde{m}^2_{t2}} \left[
\ln (\frac{\tilde{m}^2_{t1}}{\tilde{m}^2_{t2}}) +
\frac{A_t m_0 (A_t m_0+\mu\cot\beta
)}{\tilde{m}^2_{t1}-\tilde{m}^2_{t2}}
d(\tilde{m}^2_{t1},\tilde{m}^2_{t2})\right] \nonumber
\end{eqnarray}
where
$$
  c(m^2)=\frac{m^2}{m^2-\tilde{m}^2_{q}}\ln \frac{m^2}{\tilde{m}^2_{q}},
$$
$$
  d(m^2_1,m^2_2)=2-\frac{m^2_1+m^2_2}{m^2_1-m^2_2}\ln
  \frac{m^2_1}{m^2_2}
$$
and $\tilde{m}^2_q$ is the mass of a light squark.
 From eqns.~(\ref{higgs_bound}) and (\ref{25}) it follows that
$ \beta$ is related to $B$, and it is customary to use
$\tb  $  as free parameter  instead of B.
%
%
\begin{table}[tbh]
\begin{center}
\begin{minipage}{13cm}
\begin{minipage}[b]{5cm}
  %
  %---------------- table of 1.order beta-function-updates
  %
  \begin{center}
  $$
  \begin{array}{|c|c|c|c|}  \hline
  \mbox{Particle}  & b_1 & b_2 & b_3 \\
  \hline
  \hline
  \rule{0cm}{0.5cm}
  \tilde{g} & 0 & 0 & 2  \\
  \rule{0cm}{0.5cm}
  \tilde{l}_l & \frac{3}{10}  & \frac{1}{2} & 0  \\
  \rule{0cm}{0.5cm}
  \tilde{l}_r & \frac{3}{5}  &            0& 0  \\
  \rule{0cm}{0.5cm}
  \tilde{w}   &            0 &  \frac{4}{3}& 0  \\
  \rule{0cm}{0.5cm}
  \tilde{q}-\tilde{t} & \frac{49}{60}  &            1&\frac{5}{3}  \\
  \rule{0cm}{0.5cm}
  \tilde{t}_l         & \frac{1}{60}  & \frac{1}{2} &\frac{1}{6}  \\
  \rule{0cm}{0.5cm}
  \tilde{t}_r         & \frac{4}{15}  &           0 &\frac{1}{6}  \\
  \rule{0cm}{0.5cm}
  \tilde{h}           & \frac{2}{5}  & \frac{2}{3} &          0  \\
  \rule{0cm}{0.5cm}
             H        & \frac{1}{10}  & \frac{1}{6} &         0   \\
  \rule{0cm}{0.5cm}
         t            & \frac{17}{30}  &  1        &\frac{2}{3}  \\
                     & & & \\
  \hline
                     & & & \\
  \rule{0cm}{0.5cm}
   \mbox{SM}          & \frac{41}{10}  & -\frac{19}{6} & -7    \\
  \rule{0cm}{0.5cm}
   \mbox{MSSM}        & \frac{33}{5}  &            1  & -3    \\
                     & & & \\
  \hline
   \end{array}
$$
  \end{center}
   \caption[Contributions to first order RGE coefficients.]
           {\label{tbi} Contributions to first order RGE coefficients.}
\end{minipage}\hfill
\begin{minipage}[b]{7cm}
 \begin{center}
 \renewcommand{\arraystretch}{1.545}
$$
 \begin{array}{|c|c|}
 \hline
  \mbox{Particle} & b_{ij}          \\
 \hline
 \hline
  \tilde{g} &
  \left(\begin{array}{rrr}
   \rule{0cm}{0.5cm}
               0&            0&    0        \\
   \rule{0cm}{0.5cm}
               0&            0&    0        \\
   \rule{0cm}{0.5cm}
              0      &      0      &    48
  \end{array}\right)   \\
 \hline
  \tilde{w} &
  \left(\begin{array}{rrr}
   \rule{0cm}{0.5cm}
               0      &      0      &     0 \\
   \rule{0cm}{0.5cm}
               0      & \frac{64}{3}&    0        \\
   \rule{0cm}{0.5cm}
               0      &      0      &     0
  \end{array}\right) \\
 \hline
  \tilde{q},\tilde{l},\mbox{gauginos} &
  \left(\begin{array}{rrr}
   \rule{0cm}{0.5cm}
   \frac{19}{15} & \frac{3}{5} & \frac{44}{15} \\
   \rule{0cm}{0.5cm}
   \frac{ 1}{5} &-\frac{7}{3} &       4       \\
   \rule{0cm}{0.5cm}
   \frac{11}{30} & \frac{3}{2} &-\frac{8}{3}
  \end{array}\right)   \\
 \hline
  \begin{array}{l} \mbox{Heavy Higgses} \\ \mbox{and Higgsinos}
  \end{array} &

  \left(\begin{array}{rrr}
   \rule{0cm}{0.5cm}
   \frac{9}{50} & \frac{9}{10} &    0 \\
   \rule{0cm}{0.5cm}
   \frac{3}{10}    & \frac{29}{6}&    0        \\
   \rule{0cm}{0.5cm}
               0      &      0      &     0
  \end{array}\right)   \\
 \hline
 \end{array}
$$
 \renewcommand{\arraystretch}{1.0}
 \end{center}
  \caption{\label{tbij} Contributions to second order RGE
           coefficients.}
%  \vspace{17.5pt}
\end{minipage}
\end{minipage}
\end{center}
\end{table}
%
%-------------------- table of c_ij from Y_top
%
\begin{table}[htb]
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|c|c|c|}
%\begin{tabular}{|c|c|cccc|}
\hline
                &            & \multicolumn{4}{|c|}{ $\bf c_{ij}$}  \\
                              \cline{3-6}
  \rb{Region I} & \rb{$\bf c_i$} & $j=1$ & $j=2$ & $j=3$ & $j=4$    \\
\hline
\hline
  $i=1$         & $ \frac{13}{15}$
                & $(\frac{13}{15}b_1+\frac{169}{450})$
                & 1 & $\frac{136}{45}$ & $\frac{6}{5}$              \\
  $i=2$         & $ 3$  &   0          & $(3b_2+\frac{9}{2})$
                & 8   & 6                                           \\
  $i=3$         & $ \frac{16}{3}$        & 0 & 0
                & $(\frac{16}{3}b_3+\frac{128}{9})$ & 16            \\
  $i=4$         & $-6$            &   0   & 0     &  0   & $-22$    \\
\hline
\hline
  Region II     &            &       &       &       &              \\
\hline
\hline
  $i=1$         & $\frac{17}{20}$ & $\frac{1187}{600}$ & $-\frac{9}{20}$
                & $\frac{19}{15}$ & $\frac{393}{80}$                \\
  $i=2$         & $\frac{9}{4}$   &        0           & $-\frac{23}{4}$
                & 9               & $ \frac{225}{16}$               \\
  $i=3$         & 8               &     0              & 0
                & $-108$          & $ 36$                           \\
  $i=4$         & $-\frac{9}{2}$  &     0              & 0
                & 0               & $-12$                           \\
\hline
%\hline
%  Region III    &            &       &       &       &              \\
%\hline
%\hline
%  $i=1$         & 0 & 0 & 0 & 0 & 0 \\
%  $i=2$         & 0 &   & 0 & 0 & 0 \\
%  $i=3$         & 0 &   &   & 0 & 0 \\
%  $i=4$         & 0 &   &   &   & 0 \\
%\hline
  \tabs{2cm}    & \tabs{1cm} & \tabs{2.1cm} & \tabs{2.1cm}
                & \tabs{2.1cm} & \tabs{2.1cm}
\end{tabular}
\renewcommand{\arraystretch}{1.}
\caption[One and two loop coefficients $c_i$ and
  $c_{ij}$ of the RG equation for the top Yukawa coupling $Y_t$.]
  {\label{tcij} One and two loop coefficients $c_i$ and
  $c_{ij}$ of the RG equation for the top Yukawa coupling $Y_t$,
  eq.~(\ref{ytop}). The region of integration is divided into two parts:
 $M_{\rm SUSY} < m < M_{\rm GUT}, \ \
\ m_t < m < M_{\rm SUSY}. $
 Below    the
  top mass  all coefficients  $c_i$ and $c_{ij}$ are zero.}
\end{center}
\end{table}

%
%-------------------- table of d_ij from Rbtau
%
\begin{table}[t]
\vspace*{-2cm}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
                &            & \multicolumn{4}{|c|}{ $\bf d_{ij}$}  \\
                              \cline{3-6}
  \rb{Region I} & \rb{$\bf d_i$} & $j=1$ & $j=2$ & $j=3$ & $j=4$    \\
\hline
\hline
  $i=1$         & $-\frac{2}{3}$  & $-(\frac{2}{3}b_1+\frac{34}{45})$
                & $-\frac{2}{5}$  & $\frac{4}{9}$ & $ \frac{2}{5}$  \\
  $i=2$         & 0               & 0             & 0
                & 4               & 0                               \\
  $i=3$         & $\frac{8}{3}$   & 0             & 0
                & $(\frac{8}{3}b_3+\frac{64}{9})$ & 0               \\
  $i=4$         & $-\frac{1}{2}$  & 0             & 0
                & 0               & $-\frac{5}{2}$                  \\
\hline
\hline
  Region II     &            &       &       &       &              \\
\hline
\hline
  $i=1$         & $-1$            & $-\frac{53}{15}$
                & $-\frac{27}{20}$
                & $\frac{31}{30}$ & $-\frac{79}{160}$               \\
  $i=2$         & $0$             & 0                  &  $0$
                & $\frac{9}{2}$   & $\frac{9}{32}$                  \\
  $i=3$         & $4$             & 0                  &   0
                & $-54$           & $-8$                            \\
  $i=4$         & $\frac{3}{4}$   & 0                  &   0
                &  0              & $\frac{13}{4}$                  \\
\hline
\hline
  Region III    &            &       &       &       &              \\
\hline
\hline
  $i=1$         & $-1$            & $-\frac{53}{15}$
                & $-\frac{27}{20}$
                & $\frac{31}{30}$ & $0$                             \\
  $i=2$         & $0$             &  0                 &  $0$
                & $\frac{9}{2}$   & $0$                             \\
  $i=3$         & $4$             &  0                 &   0
                & $-54$           & $0$                             \\
  $i=4$         & $0$             &  0                 &   0
                &  0              & $0$                             \\
\hline
\hline
  Region IV    &            &       &       &       &              \\
\hline
\hline
  $i=1$         & $-1$            & $-\frac{53}{15}$
                &
                & $\frac{31}{30}$ & $0$                             \\
  $i=2$         & $0$             &                    &  $0$
                &                 & $0$                             \\
  $i=3$         & $4$             &                    &
                & $-54$           & $0$                             \\
  $i=4$         & $0$             &                    &
                &                 & $0$                             \\
\hline
  \tabs{2cm}    & \tabs{1cm} & \tabs{2.1cm} & \tabs{2.1cm}
                & \tabs{2.1cm} & \tabs{2.1cm}
\end{tabular}
\renewcommand{\arraystretch}{1.}
\caption[One and two loop coefficients $d_i$ and
  $d_{ij}$ of the RG equation for the ratio $R_{b\tau}$.]
  {\label{tdij} One and two loop coefficients $d_i$ and
  $d_{ij}$ of the RG equation for the ratio $R_{b\tau}$,
  eq.~(\ref{Rbtau}).
The region of integration is divided into four parts:
$M_{\rm SUSY} < m < M_{\rm GUT}, \ \
\ m_t < m < M_{\rm SUSY}, \ \ \
M_Z < m < m_t ,  \ \ \
m_b < m < m_Z.
 $
The running of $m_\tau $ below $m_b$ is ignored.
In these four regions the superpartners are
decoupled in regions II, III and IV,
the top in regions III and IV, and the Z and W in region IV.
With the superpartners one has to be careful, since some are
light and still survive in region II.}
\end{center}
\end{table}




\subsection{Second order RG Equations for the Couplings}

After calculating the complete mass spectrum, one is able to check
the unification conditions. Considering the wide range from \MG to
$M_Z$, one needs the second order RG equations for the three couplings
$\alpha_i$, the top Yukawa coupling $Y_t$ and
the ratio $R_{b\tau} = {m_b}/{m_\tau}= \sqrt{ {Y_b}/{Y_\tau}}$.

SUSY particles influence the evolution only through their
appearance in the loops, so they enter only in higher order.
Therefore it is sufficient to consider the loop corrections
to the masses in first order, in which case simple analytical
solutions can be found,  as shown in the previous section.
There is one exception: the corrections to the bottom and tau mass
are compared directly with data, which implies that second order
solutions have to be used for the RGE predicting the ratio of the
bottom and tau mass. Since this ratio involves
the top Yukawa coupling $Y_t$, the RGE for $Y_t$
has to be considered in second order too. These
second order corrections are  important for the
bottom mass, since the strong coupling constant becomes
large at the small scale of the bottom mass:
$\alpha_s(m_b)\approx 0.2$.

Defining a vector
\boldmath$\vec{\alpha}\; \mbox{\unboldmath$ =
\tal_{i=1,..,4} = (\tal_1,\tal_2,\tal_3,Y_t)$}\;$\unboldmath
the RG equations in second order can be written in a compact form:

\begin{eqnarray}
   \frac{d\tal_i}{dt} & = &
            -b_i \tal_i^2 -\tal_i^2\left( \sum_{j=1}^3
            b_{ij}\tal_j-a_i Y_t\right), \;\; i=1,2,3 \label{coupl}   \\
   \frac{dY_t}{dt} & = & Y_t
           \sum_{i=1}^4 \left( c_i \tal_i -
           \sum_{j\geq i}^4 c_{ij} \tal_i \tal_j \right)
                                                \label{ytop} \\
   \frac{dR_{b\tau}}{dt}& = & R_{b\tau}
           \sum_{i=1}^4 \left( d_i \tal_i -
           \sum_{j\geq i}^4 d_{ij} \tal_i \tal_j \right)
                                               \label{Rbtau}
\end{eqnarray}
Since there are no analytic solutions for the two-loop RG equations,
we solve this system of 5 coupled differential equations numerically.


All coefficients depend on the particle content  of the theory.
Running the RG equations from \MG down to \mz, the
coefficients   will be updated every time a particle or
sparticle threshold is passed. So we start with the MSSM values
and the contribution of $m_i$ to the coefficients
is subtracted, as soon as the scale $Q$ is below $m_i$.
The contributions of the various particles to $b_i$ and $b_{ij}$ have
been summarized in tables \ref{tbi} and \ref{tbij}
(from \cite{bbo,ekn2,Iba85,einjon,bjo,fis}).


The total contributions from the SM particles can be summarized as:
\begin{equation}
a_i=\left( \begin{array}{r} a_1 \\ a_2 \\a_3 \end{array} \right)
   =
\left( \begin{array}{r}           17/10     \\
                             3/2  \\
                                2   \end{array} \right)
\end{equation}
\begin{equation}
b_i=\left( \begin{array}{r} b_1 \\ b_2 \\b_3 \end{array} \right)
   =
\left( \begin{array}{r}           0    \\
                             - 22 / 3  \\
                                -11    \end{array} \right) +N_{Fam}
\left( \begin{array}{r}         4 / 3  \\
                                4 / 3  \\
                                4 / 3  \end{array} \right) + N_{Higgs}
\left( \begin{array}{r}         1 / 10 \\
                                1 / 6  \\
                                  0    \end{array} \right) ,
\label{15}
\end{equation}
while for the MSSM one finds:
\begin{equation}
a_i=\left( \begin{array}{r} a_1 \\ a_2 \\a_3 \end{array} \right)
   =
\left( \begin{array}{r}           26/5     \\
                               6  \\
                                4   \end{array} \right)
\end{equation}
\begin{equation}
b_i=\left( \begin{array}{r} b_1 \\ b_2 \\b_3 \end{array} \right)
   =
\left( \begin{array}{r}          0     \\
                                -6     \\
                                -9      \end{array} \right) +N_{Fam}
\left( \begin{array}{r}          2     \\
                                 2     \\
                                 2      \end{array} \right) +N_{Higgs}
\left( \begin{array}{r}          3/10  \\
                                 1/2   \\
                                 0      \end{array} \right)     ,
\label{16}
\end{equation}
Here  $N_{Fam}$ is the number of families of matter supermultiplets
and $N_{Higgs}$ is the number of Higgs doublets. We use $N_{Fam}=3$
and $N_{Higgs}=1$ or 2, which corresponds to the minimal SM or minimal
SUSY model, respectively. Below the top mass $a_i = 0$.

%
%------------------- Contributions to b_i
%
\noindent
The $b_{ij}$'s for the standard $SU(3)\otimes SU(2)\otimes U(1)$ are:
\begin{equation}      b_{ij}=
\left(\begin{array}{rrr}
\rule{0cm}{0.5cm}
0&            0&            0\\
\rule{0cm}{0.5cm}
                                   0&-\frac{136}{3}&            0\\
\rule{0cm}{0.5cm}
                                   0&            0&         -102
\end{array}\right)   + N_{Fam}
\left(\begin{array}{rrr}
\rule{0cm}{0.5cm}
\frac{19}{15}&\frac{3}{5}  &\frac{44}{15}\\
\rule{0cm}{0.5cm}
                        \frac{1}{5}  &\frac{49}{3} &    4        \\
\rule{0cm}{0.5cm}
                        \frac{11}{30}&\frac{3}{2}  &\frac{76}{3}
\end{array}\right)   + N_{Higgs}
\left(\begin{array}{rrr}
\rule{0cm}{0.5cm}
\frac{ 9}{50}&\frac{9}{10} &    0        \\
\rule{0cm}{0.5cm}
                        \frac{3}{10} &\frac{13}{6} &    0        \\
\rule{0cm}{0.5cm}
                              0      &      0      &    0
\end{array}\right).
\end{equation}
For the         SUSY    model they become:
\begin{equation}   b_{ij}=
\left(\begin{array}{rrr}
\rule{0cm}{0.5cm}
           0&            0&            0\\
\rule{0cm}{0.5cm}
                                   0&          -24&            0\\
\rule{0cm}{0.5cm}
                                   0&            0&          -54
\end{array}\right)   + N_{Fam}
\left(\begin{array}{rrr}
\rule{0cm}{0.5cm}
\frac{38}{15}&\frac{6}{5}  &\frac{88}{15}\\
\rule{0cm}{0.5cm}
                        \frac{2 }{5 }&     14      &    8        \\
\rule{0cm}{0.5cm}
                        \frac{11}{15}&      3      &\frac{68}{3}
\end{array}\right)   + N_{Higgs}
\left(\begin{array}{rrr}
\rule{0cm}{0.5cm}
\frac{ 9}{50}&\frac{9}{10} &    0        \\
\rule{0cm}{0.5cm}
                        \frac{3}{10 }&\frac{7}{2}  &    0        \\
\rule{0cm}{0.5cm}
                              0      &      0      &    0
\end{array}\right)   .
\label{susy2}
\end{equation}
Note that the $\alpha_i^2(\mu)$ coefficients are changed when
taking the second order contributions into account
and the running of each $\alpha_i$ depends
on the values of the two other     coupling constants.
The second order effects are small, because the
           $b_i$'s               are multiplied by
$     {\alpha_i}/{4\pi}\leq 0.01$. Higher orders are presumably
even smaller.



For the coefficients $c_i, c_{ij}$ of eq.~(\ref{ytop}) and $d_i, d_{ij}$
of eq.~(\ref{Rbtau}), we consider four regions; region (I) from \MG to
$M_{\rm SUSY}$, region (II) from $M_{\rm SUSY}$ to $m_{top}$, region
(III) from $m_{top}$ to $M_Z$ and region (IV) from $M_Z$  to $m_b$;
$M_{\rm SUSY}$ is some averaged mass of superpartners. The coefficients
in these different regions are shown in tables \ref{tcij} and
\ref{tdij}.
\section{Comparison of the MSSM   with  Data}\label{ch6}
In this chapter the various low energy GUT predictions are
compared with data. The most  restrictive constraints are
the coupling constant unification and the requirement that the
unification scale has to be above $10^{15}$ GeV from the proton
lifetime limits, assuming decay via s-channel exchange of heavy
gauge bosons. They exclude the SM~\cite{ekn2,abf,lalu} as well
as many other models~\cite{abf,abfI,yana} with
either a more complicated Higgs sector
or models, in which one searches for the minimum
number of new particles  needed for unification.
 From the many models tried, only a few yielded
unification at the required energies, but these
models have particles introduced ad-hoc without
the appealing properties of Supersymmetry.
Therefore we will concentrate here on the
supersymmetric models
and ask if the predictions of the simplest, i.e. {\it minimal}
models~\cite{su5susy} are consistent with
all constraints from data at low energy.

\subsection{Unification of the Couplings}
In the SM based on the group $\rm SU(3)\times
SU(2)\times U(1)$   the
 couplings are defined as:
%
\bq\label{SMcoup}{\matrix{
\alpha_1&=&(5/3)g^{\prime2}/(4\pi)&=&5\alpha/(3\cos^2\theta_W)\cr
\alpha_2&=&\hfill g^2/(4\pi)&=&\alpha/\sin^2\theta_W\hfill\cr
\alpha_3&=&\hfill g_s^2/(4\pi)\cr}}
\eq%
where $g'~,g$ and $g_s$ are the $U(1)$, $SU(2)$ and $SU(3)$ coupling
constants; the first two coupling constants are related to the fine
structure constant by:
\bq
  \label{SW}
  e = \sqrt{4\pi\alpha}=g\sin\theta_W=g'\cos \theta_W.
\eq
The factor of $5/3$ in the
definition of $\alpha_1$ has been included for
the proper normalization at
the unification point.
 The couplings, when defined as
effective values including loop corrections in
the gauge boson propagators,
become energy dependent (``running'').  A running coupling requires
the specification of a renormalization prescription, for which one
usually uses the modified minimal subtraction ($\overline{MS}$)
scheme~\cite{msbar}.

In this scheme the world averaged values of the couplings at the
Z$^0$ energy are
%
\begin{eqnarray}
  \label{worave}
  \alpha^{-1}(M_Z)             & = & 127.9\pm0.1\\
  \sin^2\theta_{\overline{MS}} & = & 0.2324\pm0.0005\\
  \alpha_3                     & = & 0.123\pm0.006.
\end{eqnarray}
The value of $\alpha^{-1}$ is given in ref.~\cite{dfs} and the value
of $\sin^2\theta_{\overline{MS}}$ has been  been taken from a detailed
analysis of all available data by
Langacker and Polonsky~\cite{sinms2}, which agrees
with the latest analysis of the LEP data~\cite{lep}.
The error includes the uncertainty
from the top quark. We have not used the smaller
error of 0.0003 for a given value of $\mt$, since
the fit was only done within the SM, not the MSSM,
so we prefer to use the more conservative error including
the uncertainty from $\mt$.


The $\alpha_3$ value
corresponds to the value at \mz\ as determined from quantities
calculated in the ``Next to Leading Log Approximation''~\cite{asresum}.
These quantities are less sensitive to the renormalization scale,
which is an indicator of the unknown higher order corrections;
they are the dominant uncertainties in quantities relying on second
order QCD calculations~\cite{asrev}. This $\alpha_s$ value is in
excellent agreement with a preliminary value of $0.120\pm 0.006$
from a fit to the $Z^0$ cross sections and
asymmetries measured at LEP~\cite{lep}, for which   the
third order QCD corrections have been calculated too;
the renormalization scale uncertainty is correspondingly small.

The top quark mass was simultaneously fitted to all electroweak data
and found to be~\cite{lep}:
\bq M_{top}=166^{+m17~+19}_{-19~-22}~{\rm GeV},\label{mtop}\eq
where the first error is statistical and the
second error corresponds to a variation
of the Higgs mass between 60 and 1000 GeV.
The central value  corresponds to a Higgs mass of 300 GeV.
This value is in good agreement with recent results  quoted by
the CDF Collaboration~\cite{cdf}:
\bq M_{top}=174^{+10~+13}_{-10~-12}~{\rm GeV},\label{mtop1}\eq
where the first error is statistical and the
second error systematic.
%the world average from the values shown in Fig.~4.
%They originate from a wide range of different process:
%the ratio  of hadronic- and leptonic width of $\tau$ decays $R_\tau$,
%taken from Ref.~\pic, the scaling violation in deep inelastic
%lepton-nucleon scattering\dis,  the quarkonium decays\kob,
%the beauty production in $p\overline{p}$ collisions\uaone,
%the W+jet  production in $p\overline{p}$ collisions\uatwo,
%studies of event shape variables and ratios of hadronic - and leptonic
%cross sections $R$
%in $e^+e_-$ collisions\rev, including  the variables for which
%the Next to Leading Logarithms (NLLA) have been resummed\resum and the
%recently published scaling violations in the fragmentation functions
%of hadronic final states                in
%$e^+e_-$   collisions\svi, which is the analogon from the
%scaling violations in DIS in the timelike scattering region.


%                  Although the world average has only a total error of
%$0.002$, we have conservatively
%taken $0.005$ as a $68\%$ C.L.~error, which
%is the smallest systematic
%error of a single $\alpha_3$ measurement.
%The $\chi^2$   of the        average is 0.5 per degree of freedom
%for an error of 0.005, so from this point of view the error
%is already overestimated. However, since the
%errors originate mainly from    the unknown higher order corrections,
%this error is hard to estimate reliably and has been debated
%very much\rev. We therefore have repeated
%the analysis for a range of $\alpha_3$ values.
%We find that the general properties of unification are not
%sensitive to the
%precise value of $\alpha_3$, although the intermediate scales for
%possible new physics do depend on it.

For SUSY models, the dimensional reduction $\overline{DR}$
scheme is a more appropriate renormalization scheme~\cite{akt}.
This scheme also has the advantage
that all thresholds can be treated by simple step
approximations.  Thus
unification occurs in the $\overline{DR}$ scheme if all
three $\aii(\mu)$ meet
exactly at one point.
                     This crossing point then gives
the mass of the heavy
gauge bosons.  The $\overline{MS}$ and $\overline{DR}$ couplings
differ by a small offset
%
\bq\label{MSDR}{{1\over\alpha_i^{\overline{DR}}}=
{1\over\alpha_i^{\overline{MS}}}-{C_i\over\strut12\pi}
}\eq
%
where the $C_i$ are the quadratic Casimir coefficients
of the group ($C_i=N$
for SU($N$) and 0 for U(1) so $\alpha_1$ stays the same).
 Throughout the
following, we use the $\overline{DR}$ scheme for the MSSM.

\subsection{$M_Z$    from Electroweak Symmetry Breaking}

In the MSSM at least two Higgs doublets have   to be introduced.
Radiative corrections from the heavy top and stop quarks can drive
one of the Higgs masses negative, thus causing spontaneous symmetry
breaking in the electroweak sector. In this case the Higgs potential
does not have its minimum for all fields equal zero, but the
minimum is obtained for non-zero vacuum expectation  values of the
fields.  The scale, where symmetry breaking occurs depends on the
 starting values of the mass parameters at the GUT scale,
the top mass and the evolution of the couplings and masses.
This gives strong constraints between the known $Z^0$ mass and the
SUSY mass parameters, as demonstrated e.g. in ref.~\cite{rrb92}.

After including  the one-loop corrections to the
potential~\cite{erz,berz,drno,kz92,eqz,cpr},
the $M_Z$ mass becomes dependent on the top- and stop quark masses too
(see eq. \ref{26}).
Note that the corrections $\Delta_Z$ are zero if the top- and stop
quark masses are identical, i.e. if supersymmetry would be exact.
They grow with the difference $\tilde{m}^2_t-m_t^2$, so these
corrections become unnaturally large for large values of the
stop masses, as will be discussed later.

\subsection{Top Mass Constraints}
\label{s65}
The top mass can be expressed as:
\bq
  \mt^2=(4\pi)^2\ Y_t(t)\ v^2\ \sin^2(\beta), \label{mt}
\eq
where the running of the Yukawa coupling as function of
$t=\log(\frac{M_\rG^2}{Q^2})$ in first order\footnote{Throughout the
analysis we have used the second order RG equations, for which
no analytical solution exists, but this will not
change the following arguments dramatically.}
is given by~\cite{Ibanez,ytfix}:
\bq
  Y_t(t)=\frac{\displaystyle Y_t(0)E(t)}{\displaystyle 1+6Y_t(0)F(t)},
  \label{ytt}
\eq
where $E$ and $F$ are functions of the couplings only~(see appendix).
One observes that $Y_t(t)$ becomes independent
of $Y_t(0)$ for large values of $Y_t(0)$, implying
an upper limit on the top mass~\cite{ir92,bbo,bbog}.
 Requiring electroweak symmetry breaking
implies a minimal value of the top Yukawa
coupling, typically $Y_t(0)\ge {\cal O}(10^{-2})$.
In this case the term
   $6Y_t(0)F(t)$ in the denominator of eq.~(\ref{ytt})
is much larger than one, since $F(t)\approx 290$ at
the weak scale, where  $t\approx 66$.
 In this case $Y_t(t)=E(t)/6F(t)$, so from eq.~(\ref{mt}) it follows:
\bq m_t^{2}=\frac{(4\pi)^2\ E(t)}{6F(t)}\ v^2\ \sin^2(\beta)\approx
(190~{\rm GeV})^2\sin^2(\beta),\eq
The physical (pole) mass is about 6\% larger
than the running mass~\cite{runmas}:
\bq M_{t}^{pole}=m_t\left(1+\frac{4}{3}\frac{\as}
{\pi}\right)\approx (200~{\rm GeV})\sin\beta,\label{topm}.\eq

The electroweak breaking conditions
require $\pi/4<\beta<\pi/2$ ; hence
the equation above implies for the MSSM approximately:
\bq 145 < M_{t}^{pole} < 200 ~{\rm GeV},\label{mtlim} \eq
which is consistent with
the experimental values given in eqns.~(\ref{mtop}) and (\ref{mtop1}).
%Although  the latter  value was determined
%from a fit using the SM, one does not expect shifts
%outside the errors, if the fit would be made  for the MSSM.

\subsection{$m_b$ from the $m_b/m_\tau$ Mass Ratio  }

Unification of the Yukawa couplings for a given generation at the
GUT scale predicts relations for quark and lepton masses within a given
family. Unfortunately, for the light quarks are the masses uncertain,
but the ratio of b-quark and $\tau$-lepton masses can be correctly
predicted by the radiative  mass
corrections~\cite{bmas,Ibanez,lanpol,bmaskln,bbo,bbog}.

Assuming the simplest possible GUT model based on SU(5) gauge group, one
has at the GUT scale: $m_b = m_\tau $. To calculate the
experimentally observed mass ratio the RG equations for the
running masses have to be used. By a physical
mass we understand the value of the running mass at the energy scale
equal to the mass itself. This definition of the mass is used
throughout this paper.

 From the RG equations for the Yukawa couplings one can easily
obtain the RGE  for the ratio
$$R_{b\tau } \equiv \frac{m_b}{m_\tau } = \sqrt{\frac{Y_b}{Y_\tau }},$$
see eq.  \ref{Rbtau} and  table \ref{tdij}.

For the running mass of the b-quark we used~\cite{runmas}:
\bq m_b=4.25\pm0.3~ {\rm GeV}.\label{bmas}\eq
This mass depends on the choice of scale
  and the value of $\as(m_b)$.
Consequently, we have assigned a rather
conservative error of 0.3 GeV instead of
the proposed value of 0.1 GeV~\cite{runmas}.
Note that the running mass (in the
$\overline{MS}$ scheme) is related to the
physical (pole) mass $M_b^{pole}$ by~\cite{runmas}:
\bq m_b=M_b^{pole}\left(1-\frac{4}{3}\frac{\as}
{\pi}-12.4(\frac{\as}{\pi})^2\right)\approx 0.825 \;M_b^{pole}, \eq
so $m_b=4.25$ corresponds to $M_b^{pole}\approx 5$ GeV.
We ignore the running of $m_\tau $ below
$m_b$ and use for the $\tau$ mass:
$M_\tau=1.7771\pm 0.0005$ GeV~\cite{taumas}.

For   large top masses, the b-quark
mass becomes a sensitive function of $\mt$
and of the starting values of the gauge
couplings at $M_\rG$, as can be checked  from the
first order solution of the RGE for the ratio $R_{b\tau}$:
\begin{eqnarray}
R_{b\tau}(m_b) &=&\left[
\frac{\tilde{\alpha}_3(m_b)}{\tilde{\alpha}_3(m_Z)}\right]^{12/23}
\left[
\frac{\tilde{\alpha}_1(m_b)}{\tilde{\alpha}_1(m_Z)}\right]^{30/103}
\\
 &\times &
\left[
\frac{\tilde{\alpha}_3(m_Z)}{\tilde{\alpha}_3(m_t)}\right]^{12/23}
\left[
\frac{\tilde{\alpha}_1(m_Z)}{\tilde{\alpha}_1(m_t)}\right]^{15/53}
\\
 &\times &
\left[
\frac{\tilde{\alpha}_3(m_t)}{\tilde{\alpha}_3(m_S)}\right]^{4/7}
\left[
\frac{\tilde{\alpha}_1(m_t)}{\tilde{\alpha}_1(m_S)}\right]^{10/41}
[1+9/2Y_t(t_S)F(t_t)]^{1/6}
\\
 &\times &
\left[
\frac{\tilde{\alpha}_3(m_S)}{\tilde{\alpha}_3(m_G)}\right]^{8/9}
\left[
\frac{\tilde{\alpha}_1(m_S)}{\tilde{\alpha}_1(m_G)}\right]^{10/99}
[1+6Y_t(t_G)F(t_S)]^{-1/12},
\label{rbtausol}
\end{eqnarray}
where the values of the  Yukawa coupling at the GUT and SUSY scales,
indicated by $m_G$ and $m_S$ or $t_G$ and $t_S$, respectively, are
related to the Yukawa coupling at $m_t$:
$$ Y_t(t_S)=\frac{Y_t(t_t)}{E(t_t)-6Y_t(t_t)F(t_t)}, \ \ \
 Y_t(t_G)=\frac{Y_t(t_S)}{E(t_S)-9/2Y_t(t_S)F(t_S)}.$$
The correlation between $m_t$ and $m_b$ originates from the
$Y_t$ terms, which are large as will be shown later.

\subsection{Experimental Lower Limits on SUSY Masses}
SUSY particles have not been found so far
and from the searches
at LEP one knows that the lower limit on the
charged leptons and charginos is
about half the $Z^0$ ~ mass (45 GeV)~\cite{pdb}
and the Higgs mass has to be above
62 GeV~\cite{higgslim}. The lower limit on
the lightest neutralino is 18.4 GeV~\cite{pdb},
while the sneutrinos have to
be above 41 GeV~\cite{pdb}.
  These limits require  minimal values for the
SUSY mass parameters.

There exist also limits on squark and gluino
masses from the hadron colliders~\cite{pdb}, but these
limits depend on the assumed decay modes.
Furthermore, if one takes the limits given above
into account, the  constraints from the limits of all other
particles are usually fulfilled, so they
do not provide additional reductions of  the
parameter space in case of the {\it minimal} SUSY model.



\subsection{Proton Lifetime Limits}

GUT's predict proton decay \cite{pdecay}
 and the present lower limits
on the proton lifetime yield quite strong  constraints
on the GUT scale and the SUSY parameters.
The direct decay $p\rightarrow e^+\pi^0$
via s-channel exchange requires
the GUT scale to be above $10^{15}$ GeV. This is not fulfilled
in the SM, but always fulfilled in the MSSM. Therefore we do not
consider this constraint.
However, the decay via box diagrams with winos and Higgsinos
predict much shorter lifetimes, especially in the preferred mode
 $p\rightarrow \overline{\nu} K^+$.
 From the present experimental lower limit of  $10^{32}$ yr for
this decay mode Arnowitt and Nath~\cite{arn}  deduce an upper limit
on the parameter B:
\bq
   B <\; (293\pm 42)\;M_{H_3}/3M_\rG\ GeV^{-1}
\eq
Here $M_{H_3}$ is the Higgs triplet mass, which is expected to be
of the order of $M_\rG$. To obtain a conservative upper limit
on $B$, we allow  $M_{H_3}$ to become an order of magnitude heavier
than $M_\rG$, so we require
\bq
  B< 977\pm 140\ GeV^{-1}.
\eq
The uncertainties from the unknown heavy Higgs mass  are large
compared with the contributions from the first and third generation,
which contribute through the mixing in the CKM matrix.
Therefore we only consider the second order generation
contribution, which can be written as~\cite{arn}:
\bq
  B=\frac{-2 \alpha_2}{ \alpha_3  \sin(2\beta)}
  \frac{ m_{\tilde{g}}}{ m^2_{\tilde{q}}} ~10^6
  \label{prot}
\eq
where $\alpha_2$ and $\alpha_3$ are the coupling
constant of the $SU(2)$ and $SU(3)$ groups at the SUSY scale,
respectively.
One observes that the upper limit on $B$ favours small gluino masses
$m_{\tilde{g}}$, large squark masses $ m_{\tilde{q}} $, and small
values of $\tb $.
 To fulfill this constraint requires
\bq
   \tan \beta < 10
\label{tanb}
\eq
for the whole parameter space. Arnowitt and Nath note that requiring
the gluino mass to be below 500 GeV  implies the mass of the scalar
particles ($m_0$) at the GUT scale to be above 600 GeV. We will not
impose this requirement on the gluino mass.  Furthermore, they require
$M_{H_3}<3~\MG$, so they obtained tighter limits on $\tan \beta $,
since we allow $M_{H_3}<10~\MG$.

\subsection{Fit Strategy}\label{s61}

As mentioned before, given the five parameters in the MSSM plus $\agut$
and $\mgut$, all other SUSY masses, the b-quark mass, and $\mz $ can be
calculated by performing the complete evolution of the couplings
including all thresholds.

The proton lifetime limits prefer small values of $\tb$, as
discussed in the previous section,
while all SUSY masses are expected to be below 1 TeV from the
fine-tuning argument.

Therefore   the following strategy was adopted:
 \mze\ and \mha\ were varied between 0 and 1000 GeV and
$\tb$ between 1 and 10.
The trilinear coupling $A_t(0)$   at $\mgut$
was kept mostly at zero, but the large radiative
corrections to it were taken into account,
so at lower  energies it is unequal zero.
Varying $A_t(0)$ between $+3\mze$ and $-3\mze$
did not change the
results significantly, so   the following
results are quoted for $A_t(0)=0$.


The remaining four parameters  - $\agut,\ \mgut,\   \mu,\  $
and $Y_t(0)$ - were fitted for each choice of $m_0, m_{1/2}$
and $\tan\beta$. Alternatively, fits were performed
in which $\mha$ or $\tb$ were left free too.

For unification in the $\overline{DR}$ scheme, all three couplings
$\aii(\mu)$
must cross at a single unification point. Thus in these models
one can fit the couplings at $\mz$ by extrapolating from a single
starting point at \MG back to $\mz$ for each of the $\alpha_i$'s and
taking into account all light thresholds. The fitting
program  will then adjust the starting values of the four
high energy  parameters ($\mgut\ ,\agut,\   \mu $ and $Y_t(0)$) until
the five low energy values (three coupling constants, $\mz$ and $\mb$)
are ``hit''. The fit is repeated for all values of $\mze$ and $\mha$
between 100 and 1000 GeV and $\tb$  between 1 and 10.

The light thresholds are taken into account by
changing the coefficients of the RGE at the
value $Q=m_i$, where
the threshold masses $m_i$ are obtained from
the analytical solutions of the corresponding RGE.
These solutions depend on the integration range,
which was chosen between $m_i$ and $\mgut$.
However, since one does not know $m_i$ at the
beginning, an iterative procedure has to be used:
one  first uses $\mz$ as a lower integration limit,
calculates $m_i$,
and uses this as lower limit in the next iteration.
Of course, since the  coupling constants are running,
the latter have to be iterated too, so the values
of $\alpha_i(m_i)$ have to be used for calculating
the mass at the scale $m_i$~\cite{rrb92,acpz}.
Usually three to five iterations are enough to find a stable solution.

Following Ellis, Kelley and Nanopoulos~\cite{ekn2} the
possible effects from heavy thresholds are set to zero, since
the proton lifetime limits  forbid the Higgs triplet masses to be below \MG.
These heavy thresholds have been considered
by other authors   for different assumptions~\cite{baha,sinms2,acz}.



The most probable parameter values were obtained  by minimizing
the following $\chi^2$ function\footnote{We use the MINUIT
program from F. James and M. Roos, {\em MINUIT Function Minimization
and Error Analysis\/}, CERN Program Library Long Writeup D506;
Release 92.1, from March 1992. Our $\chi^2$ has discontinuities
due to the  experimental bounds on various quanitities, which
become ``active''only for specific regions of the
parameter space. Consequently the derivatives are
not everywhere defined. The option SIMPLEX, which does
not rely on derivatives,  can be used to find
the monotonous region and the option MIGRAD to
optimize inside this region.}:
\begin{eqnarray} \chi^2 & = &
{\sum_{i=1}^3\frac{(\aii(\mz)-\alpha^{-1}_{MSSM_i}(\mz))^2}
{\sigma_i^2}}                             \nn
 & &+\frac{(\mz-91.18)^2}{\sigma_Z^2}                           \nn
 & &+\frac{(\mb-4.25)^2}{\sigma_b^2}                            \nn
 & &+{\frac{(B  - 997)^2}{\sigma_B^2}} {(for ~B > 997)}         \nn
 & &+{\frac{(D(m1m2m3))^2}{\sigma_D^2}} {(for~ D > 0)}          \nn
&&+{\frac{(\tilde{M}-\tilde{M}_{exp})^2}{\sigma_{\tilde{M}}^2}}
 {(for~\tilde{M} > \tilde{M}_{exp})}.                  \label{chi2}
\end{eqnarray}
The first term is the contribution of the
difference between the three
calculated and measured gauge coupling
constants at \mz~and  the following
two terms are the contributions from the
\mz-mass ~and \mb-mass  constraints.
The last three terms impose constraints from the proton
lifetime limits, from
electroweak symmetry
breaking, i.e. $D=V_H(v_1,v_2)-V_H(0,0) < 0$,
and from experimental lower limits on the SUSY masses.
The top mass, or equivalently, the top
Yukawa coupling enters sensitively into the
calculation of $\mb$ and $\mz$.
Instead of the top Yukawa coupling
one could have taken the top mass as a parameter.
However, if the couplings are evolved from $\mgut$
downwards, it is more convenient to run also the
Yukawa coupling downward, since the RG equations of the
gauge and Yukawa couplings form a set of coupled
differential equations in second order,
see eqs.~(\ref{coupl})~-~(\ref{Rbtau}).
Once the Yukawa coupling is known at $\mgut$,
the top mass can be calculated at any scale.
 In principle the experimental value of the top mass
can be taken as a constraint too, as will be discussed
in the next section.


The following errors were attributed:
$\sigma_i$ are the experimental errors in the
coupling constants,
as given above, $\sigma_b$=0.3 GeV,
$\sigma_B=140~ \mbox{GeV}^{-1}$, while  $\sigma_D$ and
$\sigma_{\tilde{M}}$ were set to 10 GeV.
 The values of the  latter errors are not   critical,
 since the corresponding terms in the numerator
 are  zero
 in case of a good fit and even for the 90\% C.L.  limits
 these constraints could be fulfilled and the $\chi^2$ was determined
 by  the other terms, for which one knows the errors.


In total one has to solve a system of 18 coupled differential
equations: 5 second order ones (for the 3 gauge couplings,
eq.~(\ref{coupl}), $Y_t$, eq.~(\ref{ytop}), and $Y_b/Y_\tau$,
eq.~(\ref{Rbtau})), and 13 first order ones (for the masses and
parameters in the Higgs sector, see sect.~3).
As mentioned before, the SUSY particles influence the evolution only through
their
appearance in loops, so it is sufficient
to consider the RGE for the masses only in first order.
The  second order RG equations  are solved numerically\footnote{The program
DDEQMR from the CERN library was used for the solution of these
coupled second order differential equations.} taking
into account the thresholds of the light particles using
the iteration procedure discussed above. Note that from the starting
values of the parameters at $\mgut$ one can calculate all light
thresholds from the simple first order equations before one starts the
numerical integration of the five second order equations. Consequently,
the program is fast in finding the optimum solution, even if before
each iteration the light thresholds have to be recalculated.

\subsection{Results}\label{s62}

We first consider fits without proton lifetime constraints, since they
basically only influence only the lower limits on the SUSY particles
and require $\tb$ to be below 10.

The upper part of fig.~\ref{\unify}  shows the evolution of the
coupling constants in the MSSM for two cases: one for the minimum
value of the $\chi^2$ function given in eq.~\ref{chi2} (solid lines)
and one corresponding to the 90\% C.L. upper limit of the thresholds
of the light SUSY particles (dashed lines). The  position of the light
thresholds is shown in the bottom part as jumps in the first order
$\beta$ coefficients, which are increased according to the entries in
table {\ref{tbi} as soon as a new threshold is passed.
Also the second order coefficients
are changed correspondingly (see table \ref{tbij}),
but their effect on the evolution
is not visible in the top figure in contrast
to the first order effects, which change the
slope of the lines considerably in the top figure.
One observes that the changes  in the coupling constants
occur  in a rather
narrow energy regime, so qualitatively
this picture is very similar to the case,
in which all sparticles were assumed to be
degenerate at an  effective SUSY mass scale $M_{\rm SUSY}$~\cite{abf}.
Since the running of the couplings depends only
logarithmically on the sparticle masses, the
90\% C.L. upper limits are as large as several TeV, as
shown by the dashed lines in fig.~\ref{\unify} and more
quantitatively in table \ref{t71}.
With the fitted SUSY parameters given at the top of the table,
the corresponding masses of the SUSY particles
can be calculated. Their values are given in the
lower part of the table and their running is
displayed in fig. \ref{\runma}. The upper and lower limits in table \ref{t71}
will be discussed below.



The parameters  $\mze, ~\mha $ and $\mu$ are
correlated, as shown in fig.~\ref{\mufit},
where the value of $\mu$ is shown for all
combinations of $\mze$ and $\mha$  between
100 and 1000 GeV.
 One observes that $\mu$ increases
with increasing \mze~ and $\mha$.

 The $\chi^2$ slowly increases    with increasing  values of  $\mu$ and $\mha$,
as
 shown    in fig.~\ref{\mucormh}.
The steep walls  originate from  the experimental lower
limits on the SUSY masses and the requirement
of radiative symmetry breaking.
In the minimum the $\chi^2$ value is zero, but one notices
a long valley, where the $\chi^2$ is only slowly increasing.
Consequently, the upper limits on the sparticle masses,
which grow with increasing values of $\mu$ and $\mha$,
become several TeV, as shown in table \ref{t71}.
The 90\% C.L. upper limits were obtained by
requiring an increase in $\chi^2$ of 1.64.
%Of course, the upper limits  depend  on the
%errors, so  the rather
%conservative  errors  defined above were used.


The correlation between $\mha$ and $\mu$ originates
mainly from the electroweak symmetry breaking
condition, but also from the fact that  the thresholds in
the running of the gauge couplings all have
to occur at a similar scale. For example, from
fig.~\ref{\unify} it is obvious that the  dashed
lines for $1/\alpha_1$ and $1/\alpha_2$ from the 90\% C.L.~
will not meet with the solid line of $1/\alpha_3$,
simply because the thresholds are too different;
the thresholds in $1/\alpha_3$ are mainly
determined by $\mha$, while the thresholds
for the upper two lines include the winos
and higgsinos too, so one obtains automatically a positive
correlation between $\mu$ and $\mha$.

The 90\% C.L. limits on the SUSY masses are obtained by scanning $\mze$
and $\mha$ till
the $\chi^2$ value increases by 1.64, while optimizing all other
parameters ($\tb,~ \mu,~,\agut,~Y_t(0)$ and $\mgut$) each time.
The upper limits  are a sensitive function of the
central value of $\as$: decreasing the central  value
of $\as$ by two standard deviations (i.e.
0.012) can increase   the thresholds of  sparticles
several TeV. Acceptable fits are only obtained for
input $\as$ values between 0.108 and   0.132, if the error
is kept at 0.006. Outside this range all requirements cannot
be met simultaneously any more, so the MSSM predicts $\as$
in this range.


As discussed previously, sparticle masses in the TeV range
spoil the cancellation of the quadratic divergences.
This can be seen explicitly in the corrections to $\mz$ (eq. \ref{26}):
$\Delta_Z$ is exactly zero if the masses of stop-- and top quarks
are identical, but the corrections grow quickly if the degeneracy
is removed, as shown in fig.~\ref{\dmz}.
For the  SUSY masses at the minimum value of $\chi^2$ the
corrections to $\mz$ are small. If one requires that only
solutions are allowed for which the corrections to $\mz$
are not large compared with $\mz$ itself, one has to limit
the mass of the heaviest stop quark to about one TeV. The
corresponding 90\% C.L. upper limits of the individual
sparticles masses are given in the right hand column of table
\ref{t71}. The correction to $\mz$ is 6 times \mz~ in this case.

 The lower
limits on the SUSY parameters are shown in the left column of
table \ref{t71}. The lowest values of $\mze$
and $\mha$ are required to have simultaneously a
sneutrino mass above 42 GeV and a wino  mass above 45 GeV.
If the proton lifetime limit is included, either $\mze$ or $\mha$ have
to be above a certain limit as  is apparent
from eq. \ref{prot}.
Since the squarks and gauginos are much more sensitive to
$\mha$ than $\mze$, one obtains the lower limits by increasing $\mze$.
The minimum value for $\mze$ is about 400 GeV in this case.
But in both cases the $\chi^2$ increase for the
lower limits is  due to the b-mass,
which is predicted to be 4.6 GeV from the
parameters determining the lower limits, so it gives a contributions to
the $\chi^2$ function,
which requires $m_b$=4.25 GeV (see eq. \ref{bmas}).

The b-quark mass is a strong function of both, $\tb$ and $\mt$,
as  shown in fig.~\ref{\mbvsmt}; this dependence originates
from the $W-t$ loop to the bottom quark. The horizontal band
corresponds to the running mass of the b-quark (eq. \ref{bmas}):
$\mb=4.25\pm 0.3$ GeV.
The top mass depends on $\tb$   (see eq. \ref{mt}); consequently the
correlation between $m_b$ and $m_t$ depends
 on $\tb$, as shown by the two curves in
fig.~\ref{\mbvsmt}.
Since also $\mz$ is a strong function
of the same parameters, the requirement of gauge and Yukawa
coupling unification together with electroweak symmetry breaking
strongly constrains  the SUSY particle spectrum. A typical fit
with a $\chi^2$ equal zero is given in the central column of
table \ref{t71}, but is should be noted that the values in the
other columns provide acceptable fits too at the 90\% C.L.~.


The proton lifetime constraint (eq.~\ref{prot}) requires $m_{1/2}$
and $m_0$ to be above a certain minimum value, as can be seen
from fig. \ref{\chisqp}: the preferred
region, i.e. the lowest $\chi^2$ is
obtained for values away from the origin.
The increase in the corner is completely due
to the constraint from the proton lifetime.
This plot was made for $\tb=2$. For larger values the region excluded
by proton decay quickly increases; for $\tb=10$ practically the whole
region is excluded, as shown in fig.~\ref{\proton}.

The mass  of the lightest Higgs particle, called $h$ in table \ref{t71},
is a rather strong function of \mt, as shown in fig.~\ref{\mhvsmt}
for various choices of $\tb$, $\mze$ and $\mha$. All other parameters
were optimized for these inputs and after the fit the values of the
Higgs and top mass were calculated and plotted. One observes that the
mass of the lightest Higgs particle varies between 60 and 150 GeV and
the top mass between 134 and 190  GeV. Furthermore,
it is evident
that $\tb$ almost uniquely determines
the value of $\mt$ (through eq. \ref{mt}), since even
if $\mha$ and $\mze$ are varied between 100
and 1000 GeV, one finds
practically the same $\mt$ for a given
$\tb$.  The value of $\mt$
varies between 134 and 190 GeV, if $\tb$
is varied between 1.2 and 10.
This range is in excellent agreement with
the estimates given in
eq.~\ref{mtlim}, if one takes into account that
$M_t^{pole}\approx 1.06 \mt$ (see eq.~\ref{topm}).

The shaded area in   fig.~\ref{\mhvsmt}  ~indicates
the results on the top mass quoted by the CDF Collaboration\cite{cdf}.
It clearly favours low values
of $\tb$.
Adding to the $\chi^2$ a term \mbox{$(M_t-174)^2/16^2$} yields
after minimization:
\bq 1.2<\tb<5.5~{\rm at ~the~ 90\% ~C.L.} \eq
The most probable value corresponds to $\tb=1.56$ as indicated by the
star in the figure.
Such a low value leads to a large mixing in the stop sector, in which
case the value of the lightest stop
is below the top mass (see the typical fit in table
\ref{t71}), although stop masses above the top mass
are not excluded, as shown by the upper limits in table
 \ref{t71}. Also a change in sign of  the Higgs mixing
parameter $\mu$ leads to stop masses above the top mass,
but   the $\chi^2$ value is hardly worse in that case,
so this cannot be excluded either.
Varying $A_t(0)$ between
$+3\mze$ and $-3\mze$
does not influence the results very much, since
it is usually compensated by a change in  $\mu$, so
 $A_t(0)$  was kept  zero,
but its non-zero value at lower energies      was taken
into account.

If the  stop mass is below the top mass, it
cannot decay into the top, but can decay as follows:
$$
  \tilde{t}_1\rightarrow \tilde{\chi}^\pm_1+b
  \rightarrow\tilde{\chi}^0_1+W+b \rightarrow
  \tilde{\chi}^0_1+lepton+\nu+b,
$$
which is experimentally very similar to the normal top decay
signature \cite{baer}.
Additional stop production could be an explanation for the excess of
events seen by the CDF Collaboration: they observe an effective cross
section for top pair production  of $13.9^{+6.1}_{-4.8}$ pb, while the
calculated $t\bar{t}$ cross section is only $5.8^{+0.8}_{-0.4}$
pb~\cite{cdf}.



\section{Summary}

The MSSM model has many predictions, which can be compared
with experiment, even in the energy range where the predicted
SUSY particles are out of reach. Among these predictions:

\begin{itemize}
\item $M_Z$.
\item $m_b/m_\tau$.
\item Proton decay.
\item Limits on $m_t$.
\end{itemize}
It is surprising, that the {\it minimal} supersymmetric model
can fulfil  all experimental constraints for these predictions.
As far as we know, supersymmetric   models are the only ones, which
are  consistent
 with all these observations simultaneously. Other models
can yield unification too~\cite{abfI}, but they do not exhibit
the elegant symmetry properties of supersymmetry, they offer no
explanation for dark matter and no explanation for the electroweak
symmetry breaking. Furthermore the quadratic divergencies do not cancel.

As mentioned in the introduction, this is the first analysis using
a $\chi^2$ definition to determine the probability of each point
in the SUSY parameter space, which allows us to determine 90\% C.L.~
limits and the most probable values ( see table \ref{t71} ).
The fit to all data simultaneously yields at the 90\% C.L.~the
following parameter ranges (if extreme finetuning
is to be avoided, i.e.~$\tilde{m}_{t2}< 1$ TeV and excluding
the proton lifetime constraint):
\begin{eqnarray*}
   65~  <& ~m_0~        &<~ 1000~\mbox{GeV}    \\
   37~  <& ~m_{1/2}~    &<~  475~\mbox{GeV}    \\
  117~  <& ~|\mu|~      &<~ 1100~\mbox{GeV}    \\
  1.2~  <& ~\tan \beta~ &<~ 5.5                 \\
  145~  <& ~M_t^{pole}~ &<~ 200 ~\mbox{GeV~(from fig.~\ref{\mhvsmt}.)}\\
0.108~  <& ~\alpha_s~   &<~0.132
\end{eqnarray*}
The upper limit on $m_0$ originates from the fine-tuning
constraint, the upper limit on $\tan \beta $ from the top mass
estimates. If the proton lifetime limit is considered too, the lower
limit on $m_0$ would be 250 GeV (see fig. \ref{proton}).
The fact that $\tan \beta $ is so much smaller than the ratio
of the top-- and b-quark masses implies that the Yukawa coupling
of the b-quark is negligibly small, so one does not have
to consider its contributions in the RG equations.

Good fits are only obtained for $\alpha_s$ between 0.108 and 0.132,
if the error on $\alpha_s$ is taken to be 0.006. The bottom mass
constraint  together with the given couplings
require the top mass to be between 134 and 190 GeV,
in perfect agreement with the experimental values.

 From the allowed ranges for the parameters one finds the
  corresponding constraints on the
SUSY masses   (see table \ref{t71} for details):
\begin{eqnarray*}
 18~<  &  \chi^0_1(\tilde{\gamma})  &<~ 202 ~\mbox{GeV}  \\
 39~<  &  \chi^0_2(\tilde{Z}) ,
          \chi^{\pm}_1(\tilde{W})   &<~  386 ~\mbox{GeV} \\
 109~ <& \tilde{g}                  &<~ 1104 ~\mbox{GeV} \\
 115~ <& \tilde{q}                  &<~ 1070 ~\mbox{GeV} \\
 140~ <& \tilde{t}_1                &<~  725 ~\mbox{GeV} \\
 218~ <& \tilde{t}_2                &<~ 1000 ~\mbox{GeV} \\
  82~ <& \tilde{e}_L                &<~  521 ~\mbox{GeV} \\
  67~ <& \tilde{e}_R                &<~  440 ~\mbox{GeV} \\
  41~ <& \tilde{\nu}_L              &<~  516 ~\mbox{GeV} \\
 109~ <& \chi^0_3(\tilde{H}_1)      &<~  799 ~\mbox{GeV} \\
 120~ <& \chi^0_4(\tilde{H}_2)      &<~  812 ~\mbox{GeV} \\
 129~ <& \chi^\pm_2(\tilde{H}^\pm)  &<~  831 ~\mbox{GeV} \\
 121~ <& H^\pm                      &<~ 1034 ~\mbox{GeV} \\
 118~ <&       {H}                  &<~ 1033 ~\mbox{GeV} \\
  92~ <&       {A}                  &<~ 1031 ~\mbox{GeV} \\
  60~ <&       {h}                  &<~  142
                   ~\mbox{GeV ~(from fig.~\ref{\mhvsmt}.)}
\end{eqnarray*}

The lower limits will all increase as soon as
the LEP limits on sneutrinos, winos and the lightest Higgs increase.
The lightest Higgs particle  is certainly within reach of
experiments at present or future accelerators~\cite{lep200,higgshunter}.
Its observation in the predicted mass range of 60 to 140 GeV would
be a strong case   in support of this minimal version of a
supersymmetric grand unified theory.

It should be noted that for the typical fit in table \ref{t71} the
mass of the lightest stop is below  the top mass,
mainly due to the low value of $\tb$ and relatively
large value of $\mu$, see eq.~(\ref{t12}). In this case the stop
cannot decay into the top, but can decay into
a chargino and b-quark,
which is experimentally very similar to the normal top decay
signature \cite{baer}.
Additional stop production could be an explanation for the excess of
events seen by the CDF Collaboration: they observe an effective cross
section for top pair production  of $13.9^{+6.1}_{-4.8}$ pb, while the
calculated $t\bar{t}$ cross section is only $5.8^{+0.8}_{-0.4}$
pb~\cite{cdf}.
\medskip\\
% \clearpage
\bigskip
\bigskip
\centerline{\bf ACKNOWLEDGMENTS}
\medskip
The research described in this publication was made possible in part by
 support from
  the Human Capital and Mobility Program
 (Contract ERBCHRXCT 930345)  from the European Communities, and by
support from the German Bundesministerium f\"ur Forschung und Technologie
(BMFT)
(Contract 05-6KA16P).
\newpage
\begin{table}[h]
\vspace*{-1.7cm}
\renewcommand{\arraystretch}{1.30}
\renewcommand{\rb}[1]{\raisebox{1.75ex}[-1.75ex]{#1}}
\begin{center}
\begin{tabular}{|c|r|r||r||r|r|}
\hline
Symbol& \multicolumn{2}{|c||}{ \makebox[4.6cm]{Lower limits }}&
      \makebox[2.3cm]{{\bf Typical fit}} &
      \multicolumn{2}{|c|}{ \makebox[4.6cm]{90\% C.L.~Upper limits }} \\
\hline
\hline
Constraints & \makebox[2.3cm]{ GEY } &\makebox[2.3cm]{ GEY+P } &
 \makebox[2.3cm]{ {\bf  GEY+(PF) }}&
 \makebox[2.3cm]{ GEY+ (P)  }      &
 \makebox[2.3cm]{ GEY+(P)+F  }                                        \\
\hline
\hline
 \multicolumn{6}{|c|}{ Fitted SUSY parameters }                       \\
\hline
 $m_0$   &  65 &400&{\bf 400} &  400 & 400                            \\
\hline
 $m_{1/2}$  &37 &80 &{\bf 111} & 1600 & 475                           \\
\hline
 $\mu$  &-117&330&{\bf 870} & 1842 & 1101                             \\
\hline
 $\tan\beta$  &3.0 &3.0&{\bf 1.56} &  8.5& 2.9                        \\
\hline
 $Y_t(0)$   &0.0158 &0.0035&{\bf 0.0150} & 0.0023 & 0.0084            \\
\hline
%$M_t^{pole}$   &195 & 182&{\bf 175} &  178 & 189                     \\
$M_t^{pole}$   & -- & -- &{\bf 175} &  178 & 189                      \\
\hline
$m_t$   & -- & -- &{\bf 165} &  168 & 178                             \\
\hline
 $1/\alpha_\rG$  &23.8&24.3&{\bf 24.5} & 25.9 & 25.2                  \\
\hline
 $\MG$  &$2.3\;10^{16}$&$2.0\;10^{16}$& $
 {\bf 2.0\;10^{16}}$ & $0.8\;10^{16}$ & $1.3\;10^{16}$                \\
\hline
\hline
 \multicolumn{6}{|c|}{SUSY masses in [GeV]}                           \\
\hline
\hline
  $\chi^0_1(\tilde{\gamma})$   & 18 &25 &{\bf  41}& 720  &  202       \\
\hline
  $\chi^0_2(\tilde{Z})$  &39&52&{\bf  80}  & 1346  &  386             \\
\hline
  $\chi^{\pm}_1(\tilde{W})$   &  46&48&{\bf  79}  & 1347 &  386       \\
\hline
  $\tilde{g}$    &  109 &217&{\bf 293}& 3377 & 1105                   \\
\hline  \hline
  $\tilde{e}_L$      &  82 &406&{\bf 409}& 1160  &  521               \\
\hline
  $\tilde{e}_R$    &  67&401&{\bf 402} & 729  &  440                  \\
\hline
  $\tilde{\nu}_L$     &  41&400&{\bf406}  & 1157  &  516              \\
\hline  \hline
  $\tilde{q}_L$   &  120&443&{\bf 477} & 3030  & 1071                 \\
\hline
  $\tilde{q}_R$   &  115&440&{\bf 471} & 2872  & 1030                 \\
\hline
 $\tilde{b}_L$   &  112&352&{\bf 369} & 2610  & 903                   \\
\hline
  $\tilde{b}_R$   &  119&440&{\bf 471} & 2862  & 1027                 \\
\hline
% $\tilde{t}_1$    &181&240&{\bf 144} & 2333  &  725                  \\
  $\tilde{t}_1$    & --& --&{\bf 144} & 2333  &  725                  \\
\hline
% $\tilde{t}_2$  & 311 & 414&{\bf 467} & 2817  & 1008                 \\
  $\tilde{t}_2$  &  -- &  --&{\bf 467} & 2817  & 1008                 \\
\hline        \hline
  $ \chi^0_3(\tilde{H}_1)$  &  109  & 292&{\bf  540}& 1771  &  799    \\
\hline
  $\chi^0_4(\tilde{H}_2)$   &  120&313&{\bf 556}& 1780  &  812        \\
\hline
  $\chi^{\pm}_2(\tilde{H}^{\pm})$& 129&315&{\bf 566}& 1816  &  831    \\
\hline   \hline
% $       h $    &   105&96&{\bf  87}&  146 &  127                    \\
  $       h $    &  --  &--&{\bf  87}&  146 &  127                    \\
\hline
  $       H $   & 118&523&{\bf 812}& 2218  &  1033                    \\
\hline
  $       A $   & 92 &521&{\bf 810}& 2217  &  1031                    \\
\hline
  $       H ^{\pm}$    &121&527&{\bf 813 }& 2219  &  1034             \\
\hline
\end{tabular} \end{center}
\caption[]{\label{t71} Values of SUSY masses and parameters for
  various constraints: G=gauge coupling unification;
  E=electroweak symmetry breaking;
  Y=Yukawa coupling unification;
  P=Proton lifetime constraint;
  F=finetuning constraint. Constraints in brackets indicate that they
  are fulfilled but not required. The minimum values of the lightest
  Higgs mass, the stop mass and the top mass can't be reached
  for the parameters minimizing the squarks and slepton masses.
  One needs smaller values of $\tb$ in that case.}
\end{table}
\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1cm}
{\bf APPENDIX A}

\vspace{1cm}

We present here the notation used above:
\begin{eqnarray*}
E(t)&=&(1+\beta_3t)^{16/(3b_3)}(1+\beta_2t)^{3/b_2}(1+\beta_1t)^{13/(15b_1)}\\
F(t)&=&\int\limits_{0}^{t}E(t')dt' \\
f_i(t)&=&\frac{1}{\beta_i}\left(1-\frac{1}{(1+\beta_it)^2}\right) \\
h_i(t)&=&\frac{t}{(1+\beta_it)},
\ \ \ \beta_i=b_i~\tilde{\alpha}_{\rm GUT} \\
e(t)&=&\frac{3}{2}\left[\frac{G_1(t)+Y_0G_2(t)}{D(t)}+
\frac{(H_2(t)+6Y_0H_4(t))^2}{3D^2(t)}+H_8(t)\right] \\
f(t)&=&-\frac{6Y_0H_3(t)}{D^2(t)} \\
h(t)&=&\frac{1}{2}(\frac{3}{D(t)}-1) \\
k(t)&=&\frac{3Y_0F(t)}{D^2(t)} \\
q(t)&=&\frac{\displaystyle 1}{\displaystyle
(1+6Y_0F(t))^{1/4}}(1+\beta_2t)^{3/(2b_2)}(1+\beta_1t)^{3/(10b_1)}\\
r(t)&=&\left(\frac{3Y_0H_3(t)}{D(t)}-H_7(t)\right)q(t)\\
s(t)&=&\frac{3Y_0F(t)}{D(t)}q(t)\\
D(t)&=&1+6Y_0F(t) \\
H_2(t)&=&\tilde{\alpha}_{\rm GUT}(\frac{16}{3}h_3(t)
+3h_2(t)+\frac{13}{15}h_1(t)) \\
H_3(t)&=& tE(t)-F(t) \\
H_4(t)&=&F(t)H_2(t)-H_3(t) \\
H_5(t)&=&\tilde{\alpha}_{\rm GUT}(-\frac{16}{3}f_3(t)+6f_2(t)
- - - - - - -\frac{22}{15}f_1(t)) \\
H_6(t)&=&\int\limits_{0}^{t}H_2^2(t')E(t')dt' \\
H_7(t)&=&\tilde{\alpha}_{\rm GUT}(3h_2(t)+\frac{3}{5}h_1(t)) \\
H_8(t)&=&\tilde{\alpha}_{\rm GUT}(-\frac{8}{3}f_3(t)+f_2(t)
- - - - - - -\frac{1}{3}f_1(t)) \\
G_1(t)&=&F_2(t)-\frac{1}{3}H^2_2(t) \\
G_2(t)&=&6F_3(t)-F_4(t)-4H_2(t)H_4(t)+2F(t)H^2_2(t)-2H_6(t) \\
F_2(t)&=&\tilde{\alpha}_{\rm GUT}
(\frac{8}{3}f_3(t)+\frac{8}{15}f_1(t))\\
F_3(t)&=&F(t)F_2(t)-\int\limits_{0}^{t}E(t')F_2(t')dt' \\
F_4(t)&=&\int\limits_{0}^{t}E(t')H_5(t')dt' \\
\end{eqnarray*}
\clearpage
%
%----------------- Evolution of couplings
%
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=15cm
  \epsfysize=18cm
  \epsffile{\unify.eps}
 \end{center}
 \caption{\label{\unify}
 Evolution of the inverse of the three couplings in the
 MSSM.                   The line  above $\MG $ follows the prediction
 from the supersymmetric SU(5) model.
The SUSY thresholds have been indicated in the lower part  of the curve:
they are treated as step functions in the
     first order $\beta$ coefficients in the  renormalization group
equations, which correspond to a change in slope in the evolution
of the couplings in the top figure.
The dashed lines correspond to the 90\% C.L. upper limit for the
SUSY thresholds.
}
\end{figure}
%
%----------------- running of masses
%
\begin{figure}[bht]
\begin{center}
%  HNS   A
\mbox{\epsfysize=12.cm\epsfxsize=10.cm\epsfbox{\runma.eps}}
%  HNS   E
\caption{Typical running of the squark ($\tilde{q}$),
slepton ($\tilde{e}_L$), and  gaugino ($M_1,~M_2,~ M_3$)
masses (solid lines). The dashed lines indicate the
running of the four neutralinos and two charginos.}
\label{\runma}
\end{center}
\end{figure}
%
%---------------- mu vs. m_0,m_1/2
%
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=15cm
  \epsfysize=18cm
  \epsffile{\mufit.eps}
 \end{center}
\caption{\label{\mufit}
  The fitted MSSM parameter $\mu$ as function of  $\mze$ and
  $\mha$ for $\tb=2$.}
\end{figure}
%
%---------------- chi^2 vs. mu,m_1/2
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=15cm
  \epsfysize=18cm
  \epsffile{\mucormh.eps}
 \end{center}
 %\vspace{18cm}
  \caption{\label{\mucormh}
   The correlation between
   $\mha$ and $\mu$ for $\mze$=500 GeV.     }
\end{figure}
%
%--------corrections to Z-mass vs. m0,m_1/2
%
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=15cm
  \epsfysize=18cm
  \epsffile{\dmz.eps}
 \end{center}
\caption{\label{\dmz}
  The one-loop correction factor to $\mz$
  as function of  $\mze$ and $\mha$.}
\end{figure}
%
%---------------- m_b vs. m_top
%
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=15cm
  \epsfysize=18cm
  \epsffile{\mbvsmt.eps}
 \end{center}
 \caption{\label{\mbvsmt}
  The correlation between the physical pole masses $M_b^{pole}$ and
  $M_t^{pole}$ for $\mze$=400 GeV and two values of $\tb$.
  The hatched area indicates the experimental value for $M_b^{pole}$.}
\end{figure}
%
%---------------- chi^2 vs. m_0,m_1/2
%
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=15cm
  \epsfysize=18cm
  \epsffile{\chisqp.eps}
 \end{center}
\caption{\label{\chisqp}
   The  $\chi^2$ of the fit as function of $\mze$ and $\mha$ for
   $\tb=2$. The sharp increase in $\chi^2$ in the corner is caused
   by the lower limit on the proton lifetime.
  }
\end{figure}
%
%---------------- proton decay
%
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=14cm
  \epsfysize=14cm
  \epsffile{\proton.eps}
 \end{center}
 \caption{\label{\proton}
The 90\% C.L. contours from proton lifetime  limits for different values of
$\tb$.
 For $\tan\beta = 1.2 $ only the
  small corner on the lower left part is excluded (small $\tilde{q}$
  masses), whereas for $\tan\beta = 10$ nearly the whole region
  is excluded. Only the lower right corner (small $m_{1/2}$ and large
  $m_0$) is then still allowed. Consequently the proton decay requires
  $\tan\beta < 10$ for practically the whole parameter space.}
\end{figure}
%
%--------------- lightest Higgs vs. m_top
%
\begin{figure}
 \vspace{-1cm}
 \begin{center}
  \leavevmode
  \epsfxsize=13cm
  \epsfysize=16cm
  \epsffile{\mhvsmt.eps}
 \end{center}
 \caption[ ]
  {The mass of the lightest Higgs particle as function of the
   top quark mass for values of $\tb$ between 1.2 and 10 and values
   of $\mze$ and $\mha$ between 100 and 1000 GeV.
   The parameters  of $\mu, ~\mgut,~ \agut$ and $Y_t(0)$~are optimized
   for each choice of these parameters; the corresponding values of
   the top and lightest Higgs mass  are shown as symbols.
   For small values of $\mha$ the Higgs mass increases with $\mze$,
   as shown for a ``string'' of points, each representing a step of
   100 GeV in $\mze$ for a given value of $\mha$, which is increasing
   in steps of 100 GeV, starting with the low values for the lowest
   strings. At high values of $\mha$ the value of $\mze$ becomes
   irrelevant and the ``string'' shrinks to a point. Note the strong
   positive correlation between $m_{higgs}$ and all other parameters:
   the highest value of the Higgs mass corresponds to the maximum
   values of the input parameters, i.e. $\tb=10$, $\mze=\mha=1000$ GeV;
   this value does not correspond to the minimum $\chi^2$. More likely
   values are: $m_{higgs}\approx 87$ GeV for $\mha=100$ GeV,
   $\mze=400$ GeV, $\mu=822$ GeV and $\tb=1.56$, as indicated by the
   star. The hatched area corresponds to the top mass range measured
   by~\cite{cdf}.
  }
\label{\mhvsmt}

\end{figure}
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\end{thebibliography}

\end{document}

