\documentstyle[12pt,a4]{article}


\begin{document}

{\bf Here are my corrections to the 2000-11-23 version.}

\begin{itemize}

\item {\bf On page 1 :} The appropriate citations for the mSUGRA
are \cite{Hall,mSUGRArev}.


\item {\bf Subsection 2.1 rewritten }

The fact that supersymmetry is not observed 
at low energies requires the introduction of 
extra supersymmetry breaking interactions. 
This is achieved by adding to the
Lagrangian density, defined by the 
given $SU(3)_C\times SU(2)_L \times  U(1)_Y$ gauge symmetry
and and the superpotential {\cal W} , 
extra interaction terms that respect the gauge symmetry but break 
supersymmetry.
This breaking however should be such that no quadratic divergences appear
and the technical ``solution" to the hierarchy problem is not spoiled.
Such terms are generally called ``soft" SUSY breaking terms.
The most general supersymmetry breaking interaction Lagrangian
resulting from spontaneously broken Supergravity in the at flat 
limit ($M_P\rightarrow \infty$) contains
just four types of soft terms , 
i.e gaugino masses, $\Phi^*\Phi$-scalar masses, $\Phi\Phi\Phi$-scalar cubic
superpotential interactions and 
$\Phi\Phi$-scalar quadratic superpotential interactions.

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

As in the numerous \cite{REWSB} existing analyses 
of radiative symmetry breaking in the
MSSM, in the present article we employ 
the Renormalization Group. The Higgs boson
``running'' mass-squared matrix, although 
positive definite at large energy scales of the
order of Grand Unification scale $M_{GUT}$ ,
yields a negative eigenvalue at low energies causing the spontaneous
breakdown of the electroweak symmetry. The ``running" 
of mass parameters from large to
low scales is equivalent to computing 
leading logarithmic radiative corrections. Although
this scenario depends on the values of few (3 or 4) free parameters, 
one could interpret
it as leading to the prediction of $M_Z$ in terms of $M_{GUT}$ ,
or the Planck mass, and the
top quark Yukawa coupling. 
Another way to interpret the predictions of this model is to
consider MZ determined by the experiment and deriving the 
absolute value of the $\mu$-parameter as well the value
of the bilinear ``soft'' breaking parameter $B$ at a scale in the
vicinity of the EW scale,
from the minimization conditions of the effective potential,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\mu^2(Q) \ &=& \ \frac{\bar{m}_{H_1}^2-\bar{m}_{H_2}^2 \tan^2\beta}{\tan^2\beta
-1}-\frac{1}{2}M_Z^2(Q) \;, \label{min1} \\
B(Q) \ &=& \ -\frac{(\bar{m}_{1}^2+\bar{m}_{2}^2)\sin2\beta}{2\mu(Q)} \;,
\label{min2}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $Q=\sqrt{M_0^2+4 M_{1/2}^2}$   approximately coincides with 
the largest first and second generation  squark masses\footnote{We
consider this scale as the so called supersymmetry soft masses
breaking scale, $M_{SUSY}$,  since the first supersymmetry breaking evidence
becomes ``experimentally'' apparent at the scale $Q$ when we are approaching
the EW scale from a very high energy one. In addition, this coincides 
with the scale where the radiative corrections to the effective potential 
are rather small compared to any other scale.}. In 
Eqs.(\ref{min1},\ref{min2}) $\tan\beta$ is the ratio of the two vacuum 
expectation values of the Higgs fields  $H_2$ and $H_1$ responsible 
in giving masses to the up and down quarks, respectively. In 
Eqs.(\ref{min1},\ref{min2}), $\tan\beta$ is evaluated at the scale $Q$,
from the scale $M_Z$, where is considered as an input parameter\footnote{See
for example the discussion in the Appendix of \cite{Drees}.}. By 
$\bar{m}_{H_i}^2$ in (\ref{min1},\ref{min2}) we denote the radiative
corrected ``running '' Higgs soft breaking masses and 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\bar{m}_{i}^2 = {m}_{H_i}^2 + \mu^2 + \Sigma_{v_i} \;\; (i=1,2) \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Sigma_{v_i}, v_i=(v_1,v_2)$ are the one loop corrections
based on  the 1-loop Coleman-Weinberg effective potential $\Delta V$, 
$\Sigma_{v_i}=
\frac{1}{2v_i}\frac{\partial \Delta V}{\partial v_i}$,
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\Sigma_{v_i} = \frac{1}{64\pi^2} \sum_a (-)^{2J_a} (2J_a+1) C_a \Omega_a
\frac{M_a^2}{v_i} \frac{\partial M_a^2}{\partial v_i} \biggl [ \ln
\frac{M_a^2}{Q^2} - 1 \biggr ]\;,
\label{effpot}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
where $J_a$ is the spin of the $a$-particle, $C_a$ are the color degrees of
freedom, and $\Omega_a=1(2)$ for real scalar (complex scalar), 
$\Omega_a=1(2)$ for Majorana (Dirac) fermions. $Q$ is the energy scale 
and $M_a$'s are the field dependent mass matrices. Explicitly forms 
of $\Sigma$'s are given in the Appendices of Ref.\cite{barger,bagger} 
and in our 
analysis contain contributions from all the SUSY particles at 1-loop.
With $M_Z^2$ here we denote
the tree level `` running'' Z-boson mass, 
$M_Z^2(Q)=\frac{1}{2}(g^2+g_2^2)v^2$ extracted at the scale $Q$ from
its physical pole mass $M_Z=91.188$ GeV. The Radiative Electroweak
Symmetry Breaking (REWSB) is fullfiled, if and only if, there is
a solution to the Eqs.(\ref{min1},\ref{min2})\footnote{Sometimes in the
literature, the requirement of the REWSB is
described by the inequality $m_1^2(Q)m_2^2(Q)-|\mu(Q) B(Q)|^2 <0$. This
relation is automatically satisfied here from Eqs.(\ref{min1},\ref{min2})
and from the fact that the physical squared Higgs masses must
be positive.}.

In summary, in order to solve Eqs.(\ref{min1},\ref{min2})
we need to ``decide'' about the $sign(\mu)$, and input 
also the value of $\tan\beta(M_Z)$. 
The ``running'' soft SUSY breaking Higgs masses, $m_{H_1}$ and $m_{H_2}$,
are defined at the Electroweak scale after their evolution
from the GUT scale where we assume that they have a common mass, $M_0$.
In addition the radiative corrections, $\Sigma_{v_i}$ to the minimization
conditions are defined by knowing the low energy SUSY spectrum
and the masses of the SM particles, which in turn means knowledge 
of $M_0$, $M_{1/2}$ and $A_0$ at the GUT scale. Thus, apart from
the SM masses provided by the experimental data~\cite{pdg}, five parameters
are required to define the mSUGRA scenario:
%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}    
\{\; M_0\;,\;M_{1/2}\;,\;A_0\;,\;\tan\beta \;,\; sign(\mu)\; \} \;.
\label{mSUGRAparams}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
As for the numerical input values on (\ref{mSUGRAparams}),
we postpone here the discussion till 
the pragraph (\ref{subsec:numanalmsugra}). 




\item {\bf Regarding the subsection 2.5.3 and the REWSB}

REWSB: parameter sets that do not fulfill the REWSB condition
{\it i.e.,} do not satisfy the 1-loop minimization conditions
(\ref{min1},\ref{min2}). Within the mSUGRA scenario, 
the condition of REWSB leads e.g. to restrictions
on $\tan\beta$. For example, it is almost impossible
(or a huge fine tunning is required)
to find very large values of $\tan\beta >60$ which 
are allowed  in the mSUGRA scenario. This is because in that 
region both the RGEs of the Higgs soft breaking masses receive large
corrections not only from the top Yukawa coupling but also
from the bottom and the tau one. This results in 
the numerator of Eq.(\ref{min1}) in
such a way that the $\mu^2$ is negative and thus excluded from our
analysis. One  could also has a case where for very heavy spectrum
the $m_{H_2}^2$ never gets negative values and thus not triggering REWSB.  


\item {\bf Regarding the subsection 2.5.3 and the CCB constraints}

CCB: regions where charge or color breaking minima (CCB) occur, {\it i.e.,}
cases where the squared squark or slepton masses are becoming negative
at the scale $Q$. 
However, it could be possible for values of the scale Q greater than 
few  TeV,
that the lifetime of a color or charge
breaking minimum is longer than the present age of the universe and
surviving quantum tunneling and thus in practice not 
dangerous~\cite{riotto,kusenko,abel,subir}. As we will see below the scale
$Q$ in our analysis is taken to be $Q<2.4$ TeV and thus we consider 
those bounds. A detailed analysis in the mSUGRA parameter space
is rather complicated and certainly
beyond the aim of this article.   

 

\end{itemize}






\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Hall}
L.~Hall, J.~Lykken and S.~Weinberg,
%``Supergravity As The Messenger Of Supersymmetry Breaking,''
Phys.\ Rev.\  {\bf D27} (1983) 2359.

For more details see, S.~Weinberg,
``The quantum theory of fields.  Vol. 3: Supersymmetry,''
{\it  Cambridge, UK: Univ. Pr. (2000) 419 p}.


\bibitem{mSUGRArev}
For reviews  see also, 

H.~P.~Nilles,
%``Supersymmetry, Supergravity And Particle Physics,''
Phys.\ Rept.\  {\bf 110} (1984) 1.

A.~B.~Lahanas and D.~V.~Nanopoulos,
%``The Road To No Scale Supergravity,''
Phys.\ Rept.\  {\bf 145} (1987) 1.

H.~E.~Haber and G.~L.~Kane,
%``The Search For Supersymmetry: Probing Physics Beyond The Standard Model,''
Phys.\ Rept.\  {\bf 117} (1985) 75.

\bibitem{faraggi}
The sparticle spectrum and phenomenology of this model has been studied in
A.~Dedes and A.~E.~Faraggi,
%``D-term spectroscopy in realistic heterotic-string models,''
Phys.\ Rev.\  {\bf D62} (2000) 016010 
[hep-ph/9907331].


\bibitem{Nilles}
D.~Matalliotakis and H.~-P.~Nilles, Nucl. \ Phys. {\bf B435} (1995) 115;

A.~Lleyda and C.~Munoz, Phys. \ Lett. \ {\bf B317} (1993) 82;

N.~Polonsky and A.~Pomarol, Phys. \ Rev. \ {\bf D51} (1995)6532 .


\bibitem{REWSB}
L.E.~Iba\~nez  and G.G.~Ross, {\em Phys. Lett.}
{\bf 110} (1982) 215; \\
K.~Inoue, A.~Kakuto, H.~Komatsu and S.~Takeshita,
{\em Progr. Theor. Phys.} {\bf 68} (1982) 927; {\it ibidem}, {\bf 71} (1984)
96;\\
J.~Ellis, D.V.~Nanopoulos and K.~Tamvakis,
{\em Phys. Lett.} {\bf B 121} (1983) 123;\\
L.E.~Iba\~nez , {\em Nucl. Phys.} {\bf B 218} (1983) 514;\\
L.~Alvarez-Gaum\'e, J.~Polchinski and M.~Wise, {\em Nucl. Phys.} {\bf B 221}
(1983) 495;\\
J.~Ellis, J.S.~Hagelin, D.V.~Nanopoulos and K.~Tamvakis, {\em Phys. Lett.}
{\bf B 125} (1983) 275;\\
L.~Alvarez-Gaum\'e, M.~Claudson and M.~Wise, {\em Nucl. Phys.} {\bf B 207}
(1982) 96.


\bibitem{Drees}
M.~Drees and M.~M.~Nojiri,
%``One loop corrections to the Higgs sector in minimal supergravity models,''
Phys.\ Rev.\  {\bf D45} (1992) 2482.

\bibitem{barger}
V.~Barger, M.S.~Berger and P.~Ohmann,
{\em Phys.\ Rev.} {\bf D 49}, 4908 (1994)
hep-ph/9311269.


\bibitem{bagger}
D.~M.~Pierce, J.~A.~Bagger, K.~Matchev and R.~Zhang,
%``Precision corrections in the minimal supersymmetric standard model,''
Nucl.\ Phys.\  {\bf B491} (1997) 3
[hep-ph/9606211].

\bibitem{pdg} Part. Data Group,
              {\em Eur.\ Phys.\ Jour.\ }{\bf C15} (2000) 1.

\bibitem{riotto}
A.~Riotto and E.~Roulet,
%``Vacuum decay along supersymmetric flat directions,''
Phys.\ Lett.\  {\bf B377} (1996) 60
[hep-ph/9512401].

\bibitem{kusenko}
A.~Kusenko, P.~Langacker and G.~Segre,
%``Phase Transitions and Vacuum Tunneling Into Charge and Color Breaking Minima in the MSSM,''
Phys.\ Rev.\  {\bf D54} (1996) 5824
[hep-ph/9602414].

\bibitem{abel}
S.A. Abel, C.A. Savoy, Phys. Lett. B444 (1998) 119,
hep-ph/9809498; S. Abel, T. Falk, Phys. Lett. B444 (1998) 427,
hep-ph/9810297 ; S.~A.~Abel and B.~C.~Allanach,
JHEP {\bf 0007} (2000) 037
[hep-ph/9909448].

\bibitem{subir}
For an overview on bounds see for example
Subir Sarkar, Rept. Prog. Phys. 59 (1996) 1493, hep-ph/9602260.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}



\end{document}