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% \begin{flushright}
% {\rm CERN-TH-2001-xxx}\\
% {\rm  July 2001} \\
% \end{flushright}

%\vspace*{\fill}

%\twocolumn[\hsize\textwidth\columnwidth\hsize\csname@twocolumnfalse\endcsname
\title{\vspace{-1ex}\small
  BONN-???,~~~~~~~~~~CERN--TH/2001-???, \hfill hep-ph/yymmnnn\\[1mm]
\Large \bf \boldmath
 Correlation of $B_s \to \mu^+\mu^-$ and $(g-2)_\mu$ in Minimal Supergravity 
 \unboldmath
      }
%
\author{A.~Dedes$^1$, H.~K.~Dreiner$^1$ and 
 U.~Nierste$^2$}
%
\address{ 
 {\it $^1$Physikalisches Institut der Universit\"at Bonn, 
 Nu\ss allee 12, 
 D-53115 Bonn, Germany} \\[3mm]
 {\it $^2$ CERN, TH Division, CH-1211 Geneva 23, Switzerland
 } 
} 
\begin{abstract}
  We analyze the rare decay mode \bs\ in the minimal supergravity
  scenario (mSUGRA). We find a strong correlation with the muon
  anomalous magnetic moment $(g-2)_\mu$. An interpretation of the
  recently measured excess in $(g-2)_\mu$ in terms of mSUGRA
  corrections implies a substantial supersymmetric enhancement of the
  branching ratio $\cbs$: if $(g-2)_\mu$ exceeds the Standard Model
  prediction by $3\cdot 10^{-9}$, $\cbs$ is larger by a factor of {\bf
    XXX --- YYY} than in the Standard Model and well within reach of
  Run-II of the Tevatron.
\end{abstract}
\pacs{} 
\maketitle
\narrowtext

%{\it 1. Introduction  }
Supersymmetry (SUSY) is an attractive and widely studied extension of
the Standard Model. But, no superpartner has been directly observed to
date, which clearly reduces the available supersymmetric parameter
space. In addition, all precision observables (except for $(g-2)_\mu$
which we discuss here) are in agreement with the Standard Model
predictions. Supersymmetry can indirectly contribute to these
observables; the corresponding constraints on the parameter space are
equally strong to those from direct searches. The minimal supergravity
model (mSUGRA) \cite{mSUGRA} relates all supersymmetric parameters to
just 5 quantities: the universal scalar and gaugino masses $M_0$ and
$M_{1/2}$, the trilinear term $A_0$ the ratio $\tan \beta$ of the two
Higgs vacuum expectation values and $\mbox{sgn}\, \mu$, where $\mu$ is
the Higgsino mass parameter.\footnote{We do not consider new
  CP-violating phases which could occur in $\mu$ and $A_0$.}
\cite{history} These quantities are defined at a high, grand unified
energy scale and determine the physical parameters, which are then
evolved to low energies by the renormalization group equations.  The
small number of parameters makes mSUGRA sufficiently predictive to be
significantly tested by low energy precision observables.

Recently the Brookhaven National Laboratory (BNL) reported an excess
of the muon anomalous magnetic moment $a_{\mu}=(g-2)_{\mu}/2$ over its
Standard Model value \cite{BNL}. The difference $\delta a_{\mu} =
a_{\mu}^{exp} - a_{\mu}^{SM} = (43\pm 16) \cdot 10^{-10}$ corresponds
to a 2.6$\sigma$ deviation from the Standard Model. An mSUGRA
interpretation of this anomaly implies $\mu >0$ in the sign convention
with $M_{1/2}>0$ and equal signs of the diagonal elements of the
chargino mass matrix
\cite{Feng,Nath,Nanopoulos,Arnowitt,Choi,Martin,Gomez,Kim,Sven,Barger,
  Fauzi,Roszkowski,Lahanas}.  It further invites a large $\tan \beta$.
The discrepancy in the case of $a_\mu$ is by itself not significant
enough to justify the claim of new physics. Especially since the
calculation of $a_{\mu}^{SM}$ involves two hadronic quantities: the
hadronic contributions to the photon self-energy, which must be
obtained from other experiments, and the (smaller) light-by-light
scattering contribution, which can only be estimated with hadronic
models. A more conservative estimate of the latter would reduce the
BNL anomaly to a 2$\sigma$ effect. Hence in order to resolve the
possible ambiguity between the mSUGRA and alternative explanations of
$\delta a_{\mu}$ one ideally wishes to study other observables whose
sensitivity to supersymmetric loop corrections is correlated with
$\delta a_{\mu}$.  SUSY changes B physics observables if $\tan \beta$
is large, because the $b$ Yukawa coupling becomes sizable. Especially
sensitive are quantities which involve a chirality flip in the $b$
quark line like ${\cal B}(B \to X_s \gamma)$ and ${\cal B} (B \to
\ell^+ \ell^-)$. In mSUGRA the low energy value for the trilinear term
$A_t$ is dominated by $M_{1/2}$ with $A_t <0$ for $M_{1/2}>0$
\cite{Masiero}. Then $\mu>0$ implies that the charged-Higgs-top loop
and the chargino-stop loop tend to cancel in ${\cal B}(B \to X_s
\gamma)$, so that the sensitivity of ${\cal B}(B \to X_s \gamma)$ to
mSUGRA corrections is weakened.  A further disadvantage of this decay
mode is that experimentally it requires a cut on the photon energy,
which introduces some hadronic uncertainty. In \cite{Masiero} the
possible impact of flavor-blind SUSY on other B physics observables,
in particular those which enter the fit of the unitarity triangle, has
been studied and only small effects have been found. By contrast, in
SUSY ${\cal B} (B \to \ell^+ \ell^-)$ grows like $\tan^6 \beta$
\cite{Babu,Chankowski,Urban}, with possible order-of-magnitude
enhancements.
% This feature is caused by the fact that loop-induced
% flavor-changing neutral Higgs couplings do not vanish when the
% supersymmetric particles are decoupled with the non-standard Higgs
% masses kept fixed \cite{Maxim}. 
Since ${\cal B} (B \to \ell^+ \ell^-) \propto m_{\ell}^2$, the
branching ratio is largest for $\ell=\tau$.  Yet $\tau$'s are hard to
detect, so that the prime experimental focus at hadron colliders is on
the search for \bs. B factories running on the $\Upsilon(4S)$ resonance
produce no $B_s$ mesons.  Branching ratios of leptonic $B_d$ mesons
are smaller by a factor of $|V_{td}/V_{ts}|^2$. Since in B factories
the boost of the $B_d$ meson is known and the considered leptonic
decay rates are substantially enhanced over their SM values, we
encourage our colleagues at BaBar and BELLE to look for $B_d \to
\tau^+ \tau^-$ decays as well. From now on we specify to the decay
mode \bs for simplicity. In \cite{Babu,Chankowski,Urban} SUSY
corrections to $\cbs$ has been calculated at the one-loop level.  For
large $\tan \beta$, however, higher order corrections can be large,
eventually of order 1. In \cite{cgnw} $\tan \beta$-enhanced
supersymmetric QCD corrections have been summed to all orders in
perturbation theory. These dominant higher-order terms can be easily
incorporated by replacing the $b$ Yukawa coupling $h_b\propto m_b \tan
\beta$ with
\begin{eqnarray} 
h_b^{eff} &=& \frac{h_b}{1+\Delta m_b}, \label{heff}
\end{eqnarray} 
where $\Delta m_b \propto \mu \tan \beta$ depends on the gluino
and sbottom masses and can be found in \cite{cgnw}.  The dominant
contribution to $\cbs $ is proportional to $h_b^{eff\,4}$, so that the 
inclusion of $\Delta m_b$ modifies the large-$\tan \beta$ behaviour. 


...

 although broken at low energies,
will reveal her structure indirectly, through deviations from the 
-succesive after all- Standard Model (SM) predictions
and/or directly from decay signals of superparticles. Of course  
every deviation from the SM explained by SUSY will likely be accompanied 
with deviations and/or new predictions on other observables.
An example of such an observable's interplay is 

{\it i)} the recent ``hint'' from 
BNL~\cite{BNL} for  an approximate 2.6$\sigma$ diviation on
the muon anomalous magnetic moment, 

{\it ii)} a hint from LEP/SLAC on electroweak
precision measurements that the higgs boson should be light compared 
to TeV scale together with the fact that LEP in the end~\cite{LEPhiggs}
``prefers'' a 115.6 GeV higgs boson mass, and finally 

{\it iii)}
the SUSY prediction for the  decay \bs could be even three(!) orders
of magnitude different than its SM prediction. 

How and why these 
three observables are related in 
the Minimal Supergravity (mSUGRA) scenario~\cite{mSUGRA} is the question 
we want to address in this letter. Since the first and the second
``hint'' have been extensively discussed within the context
of the mSUGRA scenario in~\cite{Feng}-\cite{Lahanas}
%\cite{Feng,Nath,Nanopoulos,Arnowitt,Choi,Martin,
%Gomez,Kim,Sven,Barger,Fauzi,Roszkowski,Lahanas} 
and 
\cite{ASBS} respectively, we will discuss in (some) detail here 
only the point (iii).

The decays $\bar{B}_{s,d}\rightarrow l^+l^-$  are probably
the theoretically cleanest modes among the rare
B-decays~\cite{Burasreview}.
In the SM they are dominated from Z-boson penguins and box
diagrams involving top quark exchanges. Within the SM the 
Branching fraction, $\cbs$, is helicity suppressed i.e, proportional 
to the final state lepton mass,  and thus very small.
In particular, the SM prediction gives in LO approximation, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
2.4 \times 10^{-9} <  \cbs < 4.9 \times 10^{-9} \;\;\;\;\; {\rm (SM)} \;,
\label{smbmumu}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the error is dominated  from the variation
of $f_{B_s}=230\pm 30$ MeV~\cite{fbs} and the variation 
on $|V_{ts}|=0.04\pm 0.002$. In (\ref{smbmumu}) we  assume
fixed\footnote{
In the mSUGRA analysis below, $m_t(m_t)$ is not fixed due the 
variation of the strong coupling $\alpha_s$ with the SUSY
spectrum under the assumption of the unification of the gauge
couplings. However, the effect of this variation is small compared 
to the hadronic uncertainty.}   mass
$\bar{MS}$ value for the top quark, $m_t(m_t)=166.6$ GeV which corresponds 
to a pole mass $m_t=175$ GeV [there is an additional
-rather small- sencitivity of $\pm 0.3$
when scanning the top mass in its experimental value]. 
The above central values together with $m_b(m_b)=4.25$ GeV are
used throught this letter.  
>From the experimental point of view we have only an
upper bound by CDF~\cite{CDFbmumu} which at 95\% CL is
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\cbs < 2.6 \times 10^{-6} \;\;\;\;\; {\rm (expt)} \;,
\label{expbmumu}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
which is three orders of magnitude above the SM prediction in 
(\ref{smbmumu}).
CDF Run~II will be able to probe this value down to
$\sim 10^{-8}$ and only LHCb~\cite{LHCb} or BTeV will reach the
SM expectation of (\ref{smbmumu}). Thus any observation
of the rare decay mode \bs at CDF Run~II  will definitely mean Physics
Beyond the SM. 
Apart from the new physics motivation (see below), 
the decay mode \bs is an 
experimental challenge due to its small value 
predicted from the SM and also experimental
favourite thanks to its unique signature.  

The SM Z-boson penguins and W-box diagrams were calculated 
first in~\cite{Inami}. SM Higgs penguins found~\cite{Krawczyk} to
be suppressed by $m_b^2/M_W^2$. Next to Leading Order (NLO)
corrections to $\cbs$ can be found in \cite{Buchalla}. 
New physics enters the Branching ratio \bs through 
the Higgs-boson exchange diagrams~\cite{He}-\cite{Chankowski}.
In the two Higgs doublet model although there is  an enhancement
with $\tan^2\beta$ from boxes, penguins and self energies 
involving charged Higgs
boson masses in the loop, there is no significant effect on $\cbs$
for $40<tan\beta<60$~\cite{Nierste,Urban}. 
The importance of the Higgs boson mediated Flavour Changing Neutral 
Currents in even ``flavour blind'' 
SUSY models for large $\tan\beta$ has been noted for the
first time in \cite{Maxim}. In the case of \bs here,
SUSY contributions with
chargino and squark loop from self energy diagrams would give 
a $\tan^3\beta$ enhansement~\cite{Babu} which for large $\tan\beta$ leads
the $\cbs$. Chargino and squark boxes and penguins behave like
$\tan^2\beta$ and give a subleading effect.

Before we describe the corresponding formulae, we should
remark that although the above contributions give sizable
contributions to $\bar{B}\rightarrow K l^+l^-$ mode, this is 
already ruled out from the \bs mode~\cite{Urban}. 
In addition, one of course
expects similar enhancement to ${\cal B}(\bar{B}_d\rightarrow \mu^+ \mu^-)$
modes but this decay mode is ussually one order of magnitude
smaller than the $\cbs$ which is  experimentaly first evident. 
For $\bar{B}_s \rightarrow \tau^+\tau^-$ and  
$\bar{B}_s \rightarrow e^+e^-$ one expects branching ratios of
order ${\cal O}(10^{-6})$ and  ${\cal O}(10^{-13})$ respectively in
the SM~\cite{Burasreview}. 
Compination of all these modes together with the observed
$b\rightarrow s\gamma$ within the mSUGRA scenario
is of course a very interesting subject and we postpone it for
the future. For recent works on  B-meson observables and the muon
the anomalous magnetic moment we refer to \cite{Masiero,Ali}.  

After integrating out the electroweak and SUSY scale particles
the process \bs can be discribed at low energies of order 
${\cal O}$(5 GeV) with the effective Hamiltonian\footnote{We neglect
the operators $O_S^\prime = m_s(\bar{s}_\alpha P_L b_\alpha)(\bar{\mu}\mu)$
and  $O_P^\prime = m_s(\bar{s}_\alpha P_L b_\alpha)(\bar{\mu}\gamma_5\mu)$
since they are 50 times smaller then the leading $m_b$ ones. For large
values of $\tan\beta$ the $m_s\tan^3\beta$ Wilson coefficients, 
$c_{S,P}^\prime$ could be only comparable with the $m_b\tan^2\beta$ 
coefficients $c_{S,P}$ but not competitive to the leading
$m_b\tan^3\beta$ self energy $c_{P,S}$'s.}
for the process $b\rightarrow s \mu^+ \mu^-$, 
%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\cal H}_{eff} &=& -\frac{G_F}{\sqrt{2}} \frac{\alpha(M_Z)}{\pi}
V_{tb} V_{ts}^* \left\{ c_S(\mu) O_S(\mu) +c_P(\mu)O_P(\mu)\right. 
\nonumber \\
&& \left.+c_A(\mu) O_A(\mu) \right\} \;,
\label{hamiltonian}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
with the  Scalar, Pseudoscalar and Axial local operators 
%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
O_S &=& m_b(\bar{s}_\alpha P_R b_\alpha)(\bar{\mu}\mu),\\
O_P &=& m_b(\bar{s}_\alpha P_R b_\alpha)(\bar{\mu}\gamma_5\mu),\\
O_S &=& (\bar{s}_\alpha \gamma^\mu P_L b_\alpha)(\bar{\mu}\gamma_\mu
\gamma_5 \mu)\;\;,
\label{operators}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
and $c_{S,P,A}$ being the corresponding Wilson coefficients.
We assume CP-violation {\it only} through the CKM phase and we do not
take into account additional SUSY CP-violating phases.

{\bf  analytic formula of the leading Wilson here : Sakis?}

Following \cite{Urban} the branching ratio for the process 
\bs is given by
%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\cbs &=& \frac{G_F^2 \alpha^2 M_{B_s} \tau_{B_s}}{16\pi^3}
|V_{tb}V^*_{ts}|^2\sqrt{1-\frac{4 m_\mu^2}{M_{B_s}^2}} \nonumber \\
&& \hspace{-2cm}\Biggl \{
\biggl ( 1-\frac{4 m_\mu^2}{M_{B_s}^2}\biggr ) |F_S|^2+
|F_P+2 m_\mu F_A|^2 \Biggr \} \;, 
\label{brancing}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%
with form factors 
%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_S&=&-\frac{i}{2}M_{B_s}^2 f_{B_s} \Biggl (\frac{c_S~ m_b}{m_b+m_s}
\Biggr ), \\ 
F_P&=&-\frac{i}{2}M_{B_s}^2 f_{B_s} \Biggl (\frac{c_P~ m_b}{m_b+m_s}
\Biggr ), \\
F_A&=&-\frac{i}{2} f_{B_s} c_{10} \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%

{\bf some words on the tanb ressumation here: Uli??}


We turn now the discussion into the correlation of the three
observables. The muon anomalous magnetic moment in the MSSM
is  set by the scale of the slepton masses. As we have seen before
the $\cbs$ value is set by the CP-Odd Higgs mass which also sets 
the light Higgs mass in the MSSM.
All these three observables have one common : they are affected
substantially by large\footnote{In the case
of the Higgs mass the word ``large" means $\tan\beta \gsim 10$, in the
case of the muon $g-2$, $\tan\beta\gsim 20$ and as we will see
in the case of \bs, $\tan\beta \gsim 40$.} 
values of $\tan\beta$ . In addition, in the
mSUGRA scenario, these scales are related : they are 
equal at the GUT scale, with a common
mass $M_0$ and driven at the Electroweak (EW) through their RGEs
which are mostly affected by the gaugino masses, assumed to be
equal to $M_{1/2}$ at the GUT scale\footnote{The value for the
trilinear soft SUSY breaking couplings,  $A_0$ is set to zero 
throughout this analyses. We have explicitly check that its variation
within $\pm$500 GeV modifies our result for \bs within the $f_{B_s}$
error. We leave it out here in order to make the analysis simpler.}.
Furthermore, by demanding succesful Radiative Electroweak symmetry 
breaking (REWSB)~\cite{REWSB} in the mSUGRA scenario the CP-Odd Higgs
mass gets reduced~\cite{ASBS} with large values of $\tan\beta$. The latter
enhances the ratio $\tan^3\beta/M_A^2$ and eventually  the braching
fraction $\cbs$.

 

In Fig.\ref{fig1} we plot the SUSY contribution to the anomalous 
magnetic moment of the muon against $\cbs$ for
$10<\tan\beta<58$~\footnote{The
upper limit comes from the REWSB requirement.}
shown on the top of the figure. We have chosen one specific
point[$M_{1/2}=450, M_0=350, A_0=0, \mu>0$ and $m_t=175$
 GeV]\footnote{Actually 
this point coincides with the point L quoted in \cite{Ellis}.}
of the mSUGRA parameter space. The SM prediction and the
CDF~\cite{CDFbmumu}  for $\cbs$ 
(see eq.(\ref{smbmumu}) and eq.(\ref{expbmumu})) are also
shown in the figure. 
The dashed curve represents the $\cbs$ result before the ressumation
and the solid one after. We obtain a suppresion on the $\cbs$ 
around 20\% for $\tan\beta$ values around 50. The effect diminishes
for small $\tan\beta$ values around 10.  
The conclusion is straightforward :
In order for SUSY to account for a new physics contribution $\delta
a_\mu$ within 1$\sigma$ of the current BNL measurement~\cite{BNL} 
we need a large value of $\tan\beta \gsim 50$.
At the same time we obtain  $\cbs \gsim 10^{-7}$ will be oberved at
CDF Run~II with 2 ${\rm fb}^-1$. With 10 ${\rm fb}^-1$ we expect
the sencitivity to reach a branching fraction of $1.3\times 10^{-8}$
which in turn means that $\delta a_\mu \gsim 22\times 10^{-10}$ 
{\it i.e, BNL will persist deviating from the SM},
CDF will probe $\tan\beta$ 
values above 40.   Another stricking feature from Fig.\ref{fig1}
is the destructive interference in $\cbs$ for moderate values
of $20\lsim \tan\beta \lsim 30$. This is a result of a conspirancy
between the pure higgs boson contributions (2HDM contributions) plus
the SUSY box sneutrino+squark diagram [which are acually more or less
equal in this case] against the SM contribution
to $cbs$. This effect has been described in detail within 
the context of the 2HDM
in \cite{Nierste} and is analogous in here. 
The $\cbs$ becomes just a half of the SM expectation and only
LHCb can probe this effect. Finnaly we note that for $\tan\beta
\lsim 10$ the branching fraction $\cbs$ reaches its SM expectation.
We should mention also that the effect on $\cbs$ in varying $f_{B_s}$ in
200-260 MeV is about $\mp$ 20\% with respect to the values shown in 
Fig.(\ref{fig1}). As a general fact, for large values of $\tan\beta
\gsim 35$ the self energy Wilson coefficients are 25 to 50 times
bigger
than the SUSY penguins and boxes as well as the corresponding 
pure Higgs contributions. 


After making the $\tan\beta$ analysis in Fig.(\ref{fig1}) for
a specific mSUGRA point we now turn to discuss the branching
fraction $\cbs$ and its correlation with the muon $g-2$ and the
higgs mass on a plane $M_{1/2}-M_0$ for fixed $\tan\beta=50$, $A_0=0$
GeV and $\mu>0$~\footnote{Needed not to comply with $g-2$ and
$b\rightarrow s\gamma$ constraints.} in Fig.(\ref{fig2}). 
The solid contours represent
the ressumed in $\tan\beta$ result for the branching fraction
$\cbs$. First of all we obtain that the area bounded by
 $M_{1/2}\lsim 200$ GeV and
$M_0\lsim 350$ GeV is excluded by the CDF limit in
eq.(\ref{expbmumu}). Secondly, CDF RUN~II with 2${\rm fb}^-1$ will
probe $M_{1/2}$ and $M_0$ up to 500 GeV and 1.2 TeV respectively.
Thirdly, if BNL persists deviating from the SM within 2$\sigma$
($11 < \delta a_\mu < 75$) the CDF will certainly confirm 
the evidence for the decay \bs. This is the main result of this
letter. We must notice here that although CDF is not able to
see squark masses up to 1.5 TeV [corresponding to $M_{1/2}=500$ GeV
and $M_0=1.2$ TeV] will be able  prepare the ground 
for the LHC after the indirect observation of the \bs mode. 
Even better after 10${\rm fb}^-1$ the CDF Run~II will probe 
$M_{1/2}=750$ GeV and 1.8 TeV which in turn means masses for the
heaviest superpartners of 2.4 TeV. As it happens with every
observable, the decay \bs will be like the SM one if the 
superpartner masses are heavy. The SM limit of eq.(\ref{smbmumu}) 
is reached for $M_{1/2} \gsim 1.3$ TeV. This region will only
be explored at the LHCb. We conclude the discussion of 
Fig.(\ref{fig2}) with the light Higgs boson prediction (dot-dashed line)
for $\tan\beta=50$ in the mSUGRA scenario. We obtain that 
a possible Higgs mass of less than 119 GeV is also prefered by
a two orders of magnitude deviation on the $\cbs$. The recent
``hint'' by LEP~\cite{LEPhiggs} of a mass qround 115.6 GeV results 
in $\cbs \gsim 5 \times 10^{-7}$ which might be discovered at CDF
with before the Higgs boson.


{\bf Other contraints here : dark matter b to s gamma e.t.c
I think the paper by Roszckowski et.al~\cite{Roszkowski} is enough
here
Some words also on the recent paper \cite{Shadmi}. Herbi could you
please write this here??}

The real motivation for large $\tan\beta$ values comes from the
SO(10) Yukawa unification. It has recently been 
found~\cite{Stuart,Baer}
that a narrow region consistent with the current $g-2$ constraint and
$b\rightarrow s\gamma$ survives. This region must have a light $\mu$
and $M_{1/2}$ parameters. This implies also light CP-Odd Higgs boson mass
probably less than 300 GeV. But this is what exactly the \bs decay
mode needs to be significant. As an application, we find the $\cbs$ 
for the best fitted points found in~\cite{Stuart}: $M_A=110$ GeV,
$m_{\tilde{\chi}_1} \lsim 250$ GeV, $A_t \lsim 5$ TeV,
$m_{\tilde{t}} \lsim 1$ TeV and
$\tan\beta \simeq 50$. We find that within the hadronic uncertainties
$\cbs \gsim 10^{-5}$ which already is excluded by
 CDF~\cite{CDFbmumu}. The only way out from the \bs  contraint in the 
SO(10) Models is to have  $M_A \gsim 200$ GeV 
[or smaller values of $\tan\beta < 45$]. That in turn means that one has to
relax the Yukawa unification assumption by relying on large GUT
threshold corrections. In few words:
 {\it If an SO(10) GUT model supposed to
describe the nature then the decay \bs is just around the corner.}   
 


{\bf Conclusions here: Herbi, Uli??}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{thebibliography}{99}

\bibitem{BNL}
H.~N.~Brown {\it et al.}  [Muon g-2 Collaboration],
%``Precise measurement of the positive muon anomalous magnetic moment,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 2227 (2001)
[hep-ex/0102017].
%%CITATION = HEP-EX 0102017;%%
For a theory review see, A.~Czarnecki and W.~J.~Marciano,
%``The muon anomalous magnetic moment: A harbinger for 'new physics',''
Phys.\ Rev.\ D {\bf 64}, 013014 (2001)
[hep-ph/0102122].
%%CITATION = HEP-PH 0102122;%%

\bibitem{LEPhiggs}
The ALEPH, DELPHI, L3 and OPAL Collaborations,
%``Searches for the neutral Higgs bosons of the MSSM: Preliminary combined  results using LEP data collected at energies up to 209-GeV,''
hep-ex/0107030.
%%CITATION = HEP-EX 0107030;%%



\bibitem{mSUGRA} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H.~P.~Nilles,
%``Dynamically Broken Supergravity And The Hierarchy Problem,''
Phys.\ Lett.\ B {\bf 115} (1982) 193; Nucl.\ Phys.\ B {\bf 217} (1983)
366;
%%CITATION = PHLTA,B115,193;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
A.~H.~Chamseddine, R.~Arnowitt and P.~Nath,
%``Locally Supersymmetric Grand Unification,''
Phys.\ Rev.\ Lett.\  {\bf 49} (1982) 970;
%%CITATION = PRLTA,49,970;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
R.~Barbieri, S.~Ferrara and C.~A.~Savoy,
%``Gauge Models With Spontaneously Broken Local Supersymmetry,''
Phys.\ Lett.\ B {\bf 119} (1982) 343;
%%CITATION = PHLTA,B119,343;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
L.~Hall, J.~Lykken and S.~Weinberg,
%``Supergravity As The Messenger Of Supersymmetry Breaking,''
Phys.\ Rev.\ D {\bf 27} (1983) 2359;
%%CITATION = PHRVA,D27,2359;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
S.~K.~Soni and H.~A.~Weldon,
%``Analysis Of The Supersymmetry Breaking Induced By N=1 Supergravity
%Theories,''
Phys.\ Lett.\ B {\bf 126} (1983) 215.
%%CITATION = PHLTA,B126,215;%%
%%%%%%%%%%History of Hidden Sector Supergravity Breaking %%%%%%%%%%%%%%%%%%%%%
\bibitem{history} To our knowledge the first hidden sector, gravity
  mediated supersymmetry breaking supergravity model was constructed
  in the first reference in \cite{mSUGRA}. Here the ``$A$''-terms were
  also first introduced.


%%%%%%%%%%g-2 and other constraints papers%%%%%%%%%%%%%%%%%%%%%
\bibitem{Feng}
J.~L.~Feng and K.~T.~Matchev,
%``Supersymmetry and the anomalous anomalous magnetic moment of the muon,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 3480 (2001)
[hep-ph/0102146].
%%CITATION = HEP-PH 0102146;%%

\bibitem{Nath}
U.~Chattopadhyay and P.~Nath,
%``Upper limits on sparticle masses from g-2 and the possibility for  discovery of SUSY at colliders and in dark matter searches,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 5854 (2001)
[hep-ph/0102157].
%%CITATION = HEP-PH 0102157;%%

\bibitem{Nanopoulos}
J.~R.~Ellis, D.~V.~Nanopoulos and K.~A.~Olive,
%``Combining the muon anomalous magnetic moment with other constraints on  the CMSSM,''
Phys.\ Lett.\ B {\bf 508}, 65 (2001)
[hep-ph/0102331].
%%CITATION = HEP-PH 0102331;%%


\bibitem{Arnowitt}
R.~Arnowitt, B.~Dutta, B.~Hu and Y.~Santoso,
%``Muon g-2, dark matter detection and accelerator physics,''
Phys.\ Lett.\ B {\bf 505}, 177 (2001)
[hep-ph/0102344].
%%CITATION = HEP-PH 0102344;%%

\bibitem{Choi}
K.~Choi, K.~Hwang, S.~K.~Kang, K.~Y.~Lee and W.~Y.~Song,
%``Probing the messenger of supersymmetry breaking by the muon anomalous  magnetic moment,''
Phys.\ Rev.\ D {\bf 64}, 055001 (2001)
[hep-ph/0103048].
%%CITATION = HEP-PH 0103048;%%

\bibitem{Martin}
S.~P.~Martin and J.~D.~Wells,
%``Muon anomalous magnetic dipole moment in supersymmetric theories,''
Phys.\ Rev.\ D {\bf 64}, 035003 (2001)
[hep-ph/0103067].
%%CITATION = HEP-PH 0103067;%%


\bibitem{Gomez}
D.~F.~Carvalho, J.~R.~Ellis, M.~E.~Gomez and S.~Lola,
%``Charged-lepton-flavour violation in the CMSSM in view of the muon  anomalous magnetic moment,''
hep-ph/0103256.
%%CITATION = HEP-PH 0103256;%%



\bibitem{Kim}
Y.~G.~Kim and M.~M.~Nojiri,
%``Implications of muon anomalous magnetic moment for direct detection of  neutralino dark matter,''
hep-ph/0104258.
%%CITATION = HEP-PH 0104258;%%


\bibitem{Sven}
J.~R.~Ellis, S.~Heinemeyer, K.~A.~Olive and G.~Weiglein,
%``Observability of the lightest CMSSM Higgs boson at hadron colliders,''
hep-ph/0105061.
%%CITATION = HEP-PH 0105061;%%



\bibitem{Barger}
V.~Barger and C.~Kao,
%``Implications of new CMB data for neutralino dark matter,''
hep-ph/0106189.
%%CITATION = HEP-PH 0106189;%%


\bibitem{Fauzi}
G.~Belanger, F.~Boudjema, A.~Cottrant, R.~M.~Godbole and A.~Semenov,
%``The MSSM invisible Higgs in the light of dark matter and g-2,''
hep-ph/0106275.
%%CITATION = HEP-PH 0106275;%%

\bibitem{Roszkowski}
L.~Roszkowski, R.~Ruiz de Austri and T.~Nihei,
%``New cosmological and experimental constraints on the CMSSM,''
hep-ph/0106334.
%%CITATION = HEP-PH 0106334;%%


\bibitem{Lahanas}
A.~B.~Lahanas and V.~C.~Spanos,
%``Implications of the pseudo-scalar Higgs boson in determining the  neutralino dark matter,''
hep-ph/0106345;
%%CITATION = HEP-PH 0106345;%%
A.~B.~Lahanas, D.~V.~Nanopoulos and V.~C.~Spanos,
%``Dark matter direct searches and the anomalous magnetic moment of muon,''
hep-ph/0107151.
%%CITATION = HEP-PH 0107151;%%

\bibitem{cgnw}
M.~Carena, D.~Garcia, U.~Nierste and C.~E.~Wagner,
%``Effective Lagrangian for the anti-t b H+ interaction in the MSSM 
%and  charged Higgs phenomenology,''
Nucl.\ Phys.\ B {\bf 577}, 88 (2000).
%%CITATION = HEP-PH 9912516;%%

\bibitem{ASBS}
S.~Ambrosanio, A.~Dedes, S.~Heinemeyer, S.~Su and G.~Weiglein,
%``Implications of the Higgs boson searches on different soft  SUSY-breaking scenarios,''
hep-ph/0106255 and references therein.
%%CITATION = HEP-PH 0106255;%%


\bibitem{Burasreview}
For an excelent  review  on CP-violation and Rare decays see, A.~J.~Buras,
%``Weak Hamiltonian, CP violation and rare decays,''
hep-ph/9806471.
%%CITATION = HEP-PH 9806471;%%

\bibitem{fbs}
For the most updated lattice value on $f_{B_s}$ see, C.~Bernard,
%``Heavy quark physics on the lattice,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 94}, 159 (2001)
[hep-lat/0011064].
%%CITATION = HEP-LAT 0011064;%%

\bibitem{CDFbmumu}
F.~Abe {\it et al.}  [CDF Collaboration],
%``Search for the decays B/d0 $\to$ mu+ mu- and B/s0 $\to$ mu+ mu- in  p anti-p collisions at s**(1/2) = 1.8-TeV,''
Phys.\ Rev.\ D {\bf 57}, 3811 (1998).
%%CITATION = PHRVA,D57,3811;%%


\bibitem{LHCb} For a review on the B decays at the LHC see for
  instance,
P.~Ball {\it et al.},
%``B decays at the LHC,''
hep-ph/0003238.
%%CITATION = HEP-PH 0003238;%%

%%%%%%%%%%%%%%%%%%%%%%B->mu mu papers%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Inami}
T.~Inami and C.~S.~Lim,
%``Effects Of Superheavy Quarks And Leptons In Low-Energy Weak Processes K(L) $\to$ Mu Anti-Mu, K+ $\to$ Pi+ Neutrino Anti-Neutrino And K0 <--->
Anti-K0,''
Prog.\ Theor.\ Phys.\  {\bf 65}, 297 (1981)
[Erratum-ibid.\  {\bf 65}, 1772 (1981)].

\bibitem{Krawczyk}
B.~Grzadkowski and P.~Krawczyk,
%``Higgs Particle Effects In Flavor Changing Transitions,''
Z.\ Phys.\ C {\bf 18}, 43 (1983);
%%CITATION = ZEPYA,C18,43;%%
P.~Krawczyk,
%``Flavor Changing Yukawa Coupling Of The Standard Higgs Boson: Effects Of The External Quark Masses,''
Z.\ Phys.\ C {\bf 44}, 509 (1989).
%%CITATION = ZEPYA,C44,509;%%

\bibitem{Buchalla}
G.~Buchalla and A.~J.~Buras,
%``The rare decays K+ $\to$ pi+ neutrino anti-neutrino and K(L) $\to$ mu+ mu- beyond leading logarithms,''
Nucl.\ Phys.\ B {\bf 412}, 106 (1994)
[hep-ph/9308272];
%%CITATION = HEP-PH 9308272;%%
M.~Misiak and J.~Urban,
%``{QCD} corrections to FCNC decays mediated by Z-penguins and W-boxes,''
Phys.\ Lett.\ B {\bf 451}, 161 (1999)
[hep-ph/9901278].
%%CITATION = HEP-PH 9901278;%%

%%%2HDM%%%
\bibitem{He}
X.~G.~He, T.~D.~Nguyen and R.~R.~Volkas,
%``B Meson Rare Decays In Two Higgs Doublets Models,''
Phys.\ Rev.\ D {\bf 38}, 814 (1988);
%%CITATION = PHRVA,D38,814;%%
J.~L.~Hewett, S.~Nandi and T.~G.~Rizzo,
%``B $\to$ Mu+ Mu- In The Two Higgs Doublet Model,''
Phys.\ Rev.\ D {\bf 39}, 250 (1989);
%%CITATION = PHRVA,D39,250;%%
M.~J.~Savage,
%``Constraining flavor changing neutral currents with B $\to$ mu+ mu-,''
Phys.\ Lett.\ B {\bf 266}, 135 (1991);
%%CITATION = PHLTA,B266,135;%%
Y.~Grossman,
%``Phenomenology of models with more than two Higgs doublets,''
Nucl.\ Phys.\ B {\bf 426}, 355 (1994)
[hep-ph/9401311].
%%CITATION = HEP-PH 9401311;%%
For $B_s\rightarrow \tau^+ \tau^-$ see,
W.~Skiba and J.~Kalinowski,
%``B(s) $\to$ tau+ tau- decay in a two Higgs doublet model,''
Nucl.\ Phys.\ B {\bf 404}, 3 (1993).
%%CITATION = NUPHA,B404,3;%%


\bibitem{Nierste}
H.~E.~Logan and U.~Nierste,
%``B/s,d $\to$ l+ l- in a two-Higgs-doublet model,''
Nucl.\ Phys.\ B {\bf 586}, 39 (2000)
[hep-ph/0004139].
%%CITATION = HEP-PH 0004139;%%

\bibitem{Urban}
C.~Bobeth, T.~Ewerth, F.~Kruger and J.~Urban,
%``Analysis of neutral Higgs-boson contributions to the decays anti-B/s  $\to$ l+ l- and anti-B $\to$ K l+ l-,''
hep-ph/0104284. For the numerical results on $\cbs$ presented here
we follow the formulae of this article.
%%CITATION = HEP-PH 0104284;%%

\bibitem{Choudhury}
S.~R.~Choudhury and N.~Gaur,
%``Dileptonic decay of B/s meson in SUSY models with large tan(beta),''
Phys.\ Lett.\ B {\bf 451}, 86 (1999)
[hep-ph/9810307].
%%CITATION = HEP-PH 9810307;%%

\bibitem{Babu}
K.~S.~Babu and C.~Kolda,
%``Higgs-mediated B0 $\to$ mu+ mu- in minimal supersymmetry,''
Phys.\ Rev.\ Lett.\  {\bf 84}, 228 (2000)
[hep-ph/9909476].
%%CITATION = HEP-PH 9909476;%%

\bibitem{Chankowski}
P.~H.~Chankowski and L.~Slawianowska,
%``B0/d,s $\to$ mu- mu+ decay in the MSSM,''
Phys.\ Rev.\ D {\bf 63}, 054012 (2001)
[hep-ph/0008046].
%%CITATION = HEP-PH 0008046;%%

% \bibitem{Huang}
% C.~Huang, W.~Liao, Q.~Yan and S.~Zhu,
% %``B/s $\to$ l+ l- in a general 2HDM and MSSM,''
% Phys.\ Rev.\ D {\bf 63}, 114021 (2001)
% [hep-ph/0006250].
% %%CITATION = HEP-PH 0006250;%%


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

\bibitem{Maxim}
C.~Hamzaoui, M.~Pospelov and M.~Toharia,
%``Higgs-mediated FCNC in supersymmetric models with large tan(beta),''
Phys.\ Rev.\ D {\bf 59}, 095005 (1999)
[hep-ph/9807350].
%%CITATION = HEP-PH 9807350;%%

%%%%%%%%%%%%%g-2 and CP-violation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Masiero}
A.~Bartl, T.~Gajdosik, E.~Lunghi, A.~Masiero, W.~Porod, 
H.~Stremnitzer and O.~Vives,
%``General flavor blind MSSM and CP violation,''
hep-ph/0103324.
%%CITATION = HEP-PH 0103324;%%

\bibitem{Ali}
A.~Ali and E.~Lunghi,
%``Extended minimal flavor violating MSSM and implications for B physics,''
hep-ph/0105200.
%%CITATION = HEP-PH 0105200;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\bibitem{REWSB} L.E.~Iba\~nez  and G.G.~Ross, 
                {\em Phys. Lett.} {\bf 110} (1982) 215; \\
                %%CITATION = PHLTA,B110,215;%%
                K.~Inoue, A.~Kakuto, H.~Komatsu and S.~Takeshita,
                {\em Progr. Theor. Phys.} {\bf 68} (1982) 927;
                %%CITATION = PTPKA,68,927;%%
                {\em Progr. Theor. Phys.} {\bf 71} (1984) 413;\\
                %%CITATION = PTPKA,71,413;%%
                J.~Ellis, D.V.~Nanopoulos and K.~Tamvakis,
                {\em Phys. Lett.} {\bf B 121} (1983) 123;\\
                %%CITATION = PHLTA,B121,123;%%
                L.~Alvarez-Gaum\'e, J.~Polchinski and M.~Wise, 
                {\em Nucl. Phys.} {\bf B 221} (1983) 495;\\
                %%CITATION = NUPHA,B221,495;%%
                J.~Ellis, J.S.~Hagelin, D.V.~Nanopoulos and K.~Tamvakis, 
                {\em Phys. Lett.} {\bf B 125} (1983) 275;
                %%CITATION = PHLTA,B125,275;%%




\bibitem{Ellis}
M.~Battaglia {\it et al.},
%``Proposed post-LEP benchmarks for supersymmetry,''
hep-ph/0106204.
%%CITATION = HEP-PH 0106204;%%

%%%%%%%%%%%%%%%%%%%%%%OTHER CONSTRAINTS%%%%%%%%%%%%%%%%
\bibitem{Shadmi}
J.~L.~Feng, K.~T.~Matchev and Y.~Shadmi,
%``Theoretical expectations for the muon's electric dipole moment,''
hep-ph/0107182.
%%CITATION = HEP-PH 0107182;%%


%%%%%%%%%%%%%%%%%%%%%%SO(10)%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{Stuart}
T.~Blazek, R.~Dermisek and S.~Raby,
%``Predictions for Higgs and SUSY spectra from SO(10) Yukawa unification  with mu > 0,''
hep-ph/0107097.
%%CITATION = HEP-PH 0107097;%%

\bibitem{Baer}
H.~Baer and J.~Ferrandis,
%``Supersymmetric SO(10) GUT models with Yukawa unification and a positive  mu term,''
hep-ph/0106352.
%%CITATION = HEP-PH 0106352;%%

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURES%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%555
\onecolumn
\begin{figure}[t]
\centerline{\psfig{figure=btomumu-g-2.ps, angle=90,height=9cm}}
\caption{The predicted SUSY contribution to the muon anomalous magnetic 
  $\delta a_\mu$ (bottom), versus the predicted branching ratio \bs~
  (left) for the input values of $\tan\beta$ (top) in an indicative
  mSUGRA point [$M_{1/2}=450, M_0=350, A_0=0, \mu>0, m_t=175$ GeV]. We
  plot the both LO and NLO corrections to \bs~ together with the SM
  prediction and the present bound by CDF~\protect\cite{CDFbmumu}, \bs$ <
  2.6\times 10^{-6}$ at 95\% CL.  The regions where the SUSY
  contribution to the muon anomalous magnetic moment is such that the
  present deviation with the SM is recovered at $1\sigma$ and
  $2\sigma$ is also shown. $f_{B_s}$ has been chosen to be 230 MeV.}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\psfig{figure=btomumu_m12mo.ps, angle=90,height=9cm}}
\caption{Contour plots (solid lines) of the predicted Branching Ratio \bs on
  the plane $M_{0},M_{1/2}$ in the mSUGRA scenario and for
  $\tan\beta=50, A_0=0, \mu>0, m_t=175$ GeV.  The fully shaded region
  is excluded from both theoretical [REWSB(left-upper corner),not
  neutralino LSP(down)] and experimental [susy and higgs searches
  (mainly left strip till 1 TeV $M_0$ approximately)] facts.  The
  shaded-dotted region shows the prediction of this mSUGRA point for
  the muon anomalous magnetic moment [in units $10^{-10}$] being
  within $2 \sigma $ of the value reported in \protect\cite{BNL}.
  Contours with the light Higgs boson mass equal to 115,120,121 are
  also shown (dotted-dashed lines).  Here $f_{B_s}=230$ MeV. }
\label{fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\end{document}
