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\begin{center}
{\Large Note written by A. Dedes$^1$} \\[3mm]
%, H.~K.~Dreiner$^1$ and U. Nierste$^2$} \\[3mm]
{\it$^1$ Physikalisches Institut, Universit\"at Bonn,
Nu{\ss}allee 12, D-53115 Bonn, Germany} 
\\
% {\it $^2$CERN, TH Division, CH-1211 Geneva 23, Switzerland} 
%  \\[10mm]
\end{center}

{\it [Based on 
``Correlation of $B_s \to \mu^+ \mu^-$ and $(g-2)_\mu$
 in minimal supergravity,'' A.~Dedes, H.~K.~Dreiner and U.~Nierste,
 arXiv:hep-ph/0108037.]}
\\[3mm]

Another interesting probe of supersymmetry is
the decay \bs . The SM prediction
is given by $\cbs = (3.7 \pm 1.2) \cdot 10^ {-9}$, with the
uncertainty ($\pm 25\%$) dominated by $f_{B_s}$.
The current experimental bound on the Branching ratio, (${\cal B}$),
has been set during  Run-I of the Tevatron, where CDF~\cite{CDFbmumu}
determined $\cbs < 2.6 \times 10^{-6}$ at 90\% C.L. 
In addition to the experimental challenge, the almost three 
orders of magnitude gap between the current experimental bound and
the SM prediction makes this mode an excellent laboratory for
new physics. In contrast to observables which enter the unitarity 
triangle~\cite{Masiero}, in the MSSM, the branching 
ratio $\cbs$ grows like $\tan^6\beta$
\cite{Babu,Chankowski,Urban,Isidori}, with a possible several orders of
magnitude enhancement. More interestingly, it has been very recently 
shown~\cite{dedes} that in the minimal supergravity
scenario (mSUGRA) there is 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 $(\delta a_\mu)_{SUSY}=4\cdot 10^{-9}$, 
  $\cbs$ is larger by a factor of { 10--100}
  than in the Standard Model and within reach of Run-II of the
  Tevatron. The single event sensitivity 
of CDF at Run-IIa is estimated to be $1.0
\cdot 10^{-8}$, for an integrated luminosity of 2 fb$^{-1}$
\cite{fnalbrep}. Thus if mSUGRA  corrections enhance $\cbs$ to
e.g.\ $5\cdot 10^{-7}$, one will see 50 events in Run-IIa. Run-IIb may
collect 10-20 fb$^{-1}$ of integrated luminosity, which implies 
250-500 events in this example.

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[b]
\centerline{\psfig{figure=btomumu_m12mo.ps, angle=90}}
\caption{Contour plots  of the $\cbs$ (solid) and on $(\delta a_
  \mu)_{SUSY}$ (dashed)  in the ($M_{0},M_{1/2}$)-plane for
  mSUGRA parameter values  as shown. Contours of the light Higgs boson mass
  (dot-dash line) are also shown. From Ref.~\cite{dedes}. }
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Following Ref.~\cite{dedes}, 
in Fig.\ref{fig1} we show the contours of $\cbs$ (solid) and
$(\delta a_\mu)_{SUSY}$ (dashed) in this plane, for $\tan\beta=50$,
$A_0=0$, $\mu>0$ and $m_t=175 \, {\rm GeV}$.   The  shaded
region is excluded through the various theoretical and experimental
contstraints.  A sensitivity of $\cbs\OOrd
2\cdot10^{-7}$ at CDF now corresponds to a sensitivity of
$M_{1/2}\Ord 280 \, {\rm GeV}$ and $M_0\Ord 400 { \,{\rm GeV}}$, 
respectively.
While CDF is not able to see squark masses directly up to 0.7 TeV
(corresponding to $M_{1/2}=M_0\simeq 300$ GeV,), it will nevertheless
be able to prepare the ground for LHC by observing the \bs\ mode.
Even better, after 10$\,{\rm fb}^{-1}$ CDF will probe $M_{1/2} \Ord
450$ GeV and $M_0 \Ord 600\,{\rm GeV}$ (for $\tan \beta=50$) which in
mSUGRA corresponds to masses for the heaviest superpartners of 1 TeV.
We conclude the discussion of Fig.\ref{fig1} with the prediction of
the light Higgs boson mass $M_h$ (dot-dashed line) for $\tan\beta=50$
in the mSUGRA scenario~\cite{ASBS}.  Any measurement of $\cbs$ by
itself implies a useful \emph{upper}\ bound on $M_h$. The simultaneous
information of $\cbs$ and $\delta a_{ \mu}$ fixes $M_h$ in most
regions of the $(M_{1/2},M_0)$-plane.  A Higgs mass around 115.6 GeV
results in $ 10^{-8} \Ord \cbs \Ord 3\cdot 10^{-7}$ which would most
likely be measured before the Higgs boson is discovered.




\begin{thebibliography}{99}

\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{Masiero}
A.~Bartl, T.~Gajdosik, E.~Lunghi, A.~Masiero, 
W.~Porod, H.~Stremnitzer and O.~Vives,
%``General flavor blind MSSM and CP violation,''
Phys.\ Rev.\ D {\bf 64}, 076009 (2001)
[arXiv:hep-ph/0103324].


\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;%%
C.~Huang,  W.~Liao and Q.~Yan,
%``The promising process to distinguish supersymmetric models with large
%tan(beta) from the standard model: B $\to$ X/s mu+ mu-,''
Phys.\ Rev.\ D {\bf 59} (1999) 011701
[hep-ph/9803460].
%%CITATION = HEP-PH 9803460;%%

\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{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-,''
Phys.\ Rev.\ D {\bf 64}, 074014 (2001)
[arXiv:hep-ph/0104284].
%%CITATION = HEP-PH 0104284;%%

\bibitem{Isidori}
G.~Isidori and A.~Retico,
%``Scalar flavour-changing neutral currents in the large-tan(beta) limit,''
JHEP {\bf 0111}, 001 (2001)
[arXiv:hep-ph/0110121].
%%CITATION = HEP-PH 0110121;%%

\bibitem{dedes}
A.~Dedes, H.~K.~Dreiner and U.~Nierste,
%``Correlation of B/s $\to$ mu+ mu- and (g-2)(mu) in minimal supergravity,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 251804 (2001)
arXiv:hep-ph/0108037.
%%CITATION = HEP-PH 0108037;%%


\bibitem{fnalbrep}
D.~Atwood et al., 
\emph{B Physics at the Tevatron: Run-II and Beyond},
FERMILAB-Pub-01/197 (preliminary).




\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,''
arXiv:hep-ph/0106255.
%%CITATION = HEP-PH 0106255;%%

\end{thebibliography}

\end{document}