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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
 } 
} 
\maketitle
\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. 
  {\bf a remark on Higgs masses or SO(10)?}
\end{abstract}
\pacs{} 
\narrowtext

%{\it 1. Introduction  }
Supersymmetry (SUSY) is an attractive and widely studied extension of
the Standard Model (SM). 
% 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 SM
% 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. Precision measurements, which are
affected by SUSY corrections through loop effects, play an important
role to constrain the supersymmetric parameter space.  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
SM 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 SM. 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 modifies 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 $ h_b^{{\rm eff}} = h_b/(1+\Delta m_b)$,
where $\Delta m_b \propto \mu \tan \beta$ depends on the gluino and
sbottom masses and can be found in \cite{cgnw}. $\Delta m_b$ is positive for
$\mu>0$. The dominant contribution to $\cbs $ is proportional to
$h_b^{{\rm eff}\,4}$, so that the inclusion of $\Delta m_b$ tempers the
large-$\tan \beta$ behaviour.

The considered branching ratio can be expressed as
\begin{eqnarray}
\lefteqn{\!\! \cbs \; =} \nn
&& 6.0 \cdot 10^{-7} 
\lt(\frac{|V_{ts}|}{0.040}\rt)^2
\lt(\frac{f_{B_s}}{230\,\mev}\rt)^2
\frac{m_{\mu}^2}{m_{B_s}^2}
\sqrt{1-\frac{4 m_\mu^2}{m_{B_s}^2}} \nonumber \\
 && \Biggl \{
 \biggl ( 1-\frac{4 m_\mu^2}{M_{B_s}^2}\biggr ) 
  \lt| \frac{m_{B_s}^2 C_S}{m_{\mu}} \rt|^2 + 
 \lt| \frac{m_{B_s}^2 C_P}{m_{\mu}} - 2 C_A \rt|^2 \Biggr \} .
 \label{brancing}
\end{eqnarray}
Here $|V_{ts}|=0.040\pm 0.002$ is the relevant element of the
Cabibbo-Kobayashi-Maskawa matrix and $f_{B_s}=(230 \pm 30)\; \mev$
\cite{fbs} is the decay constant of the $B_s$ meson. In \eq{brancing}
we have kept the dependence on the lepton mass $m_{\mu}$, so that the
generalization to $B_d \to \tau^+ \tau^-$ is straightforward. The
Wilson coefficients $C_S$, $C_P$ and $C_A$, which contain the
short-distance physics, are normalized as in \cite{Nierste}.  The
coefficients $c_S$, $c_P$ and $c_{10}$ defined in \cite{Urban} are
related to ours by $C_S=-2 c_S \sin^2 \theta_W$, $C_A = -2 c_{10}
\sin^2 \theta_W$ and $C_P = 2 c_P \sin^2 \theta_W$. Within the SM
$C_S$ and $C_P$ are negligibly small and the next-to-leading order
result for $C_A$ can be well approximated by $C_A=1.99
(\ov{m}_t/166.6\; \gev)^1.55$ \cite{Buchalla}. Here $\ov{m}_t\equiv
\ov{m}_t(m_t) $ is the top quark mass in the $\ov{\rm MS}$ scheme.
$\ov{m}_t = 166.6$ GeV corresponds to a pole mass of $m_t=175\; \gev$.
$f_{B_s}$ dominates the uncertainty of the SM prediction, which reads
$\cbs = (3.7 \pm 1.1) \cdot 10^{-9}$.  During Run-I of the Tevatron CDF
found the upper bound \cite{CDFbmumu}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\cbs < 2.6 \times 10^{-6}
\qquad \mbox{ at 95\% C.L.} 
\label{expbmumu}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}.  That is, 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 will imply
250-500 events in this example.


The Wilson coefficients for a type-II two-Higgs doublet model can be
found in \cite{Nierste}: contribution from charged Higgs loops to
$C_S$ and $C_P$ grow at most as $\tan^2 \beta$ and, in most of the
parameter space, tend to diminish ${\cal B}$. SUSY corrections to
these coefficients, however, grow like $\tan^3 \beta$ and can enhance
${\cal B}$ by several orders of magnitude
\cite{Babu,Chankowski,Urban}.  Hence the signatures of the extended
Higgs sector and of the superparticles are very different. The
resummation of the $\tan\beta$-enhanced SUSY-QCD corrections comprised
in $h_b^{\rm eff}$ can be easily implemented into the formulae of
\cite{Babu,Chankowski,Urban} by replacing the leading $\tan^3\beta$
term with $\tan\beta^3/(1+\Delta m_b)^2$.  The dominant dependence of
$\delta a_{\mu}$ and $\cbs$ on the SUSY parameters is given by
\begin{eqnarray}
\delta a_{\mu} & \propto & \frac{\tan \beta}{\mu M_2}, \qquad 
\cbs \; \propto \;  \frac{m_{\tilde{q}}^2\tan^6 \beta}{M_{A^0}^4},
\label{damu}
\end{eqnarray}
where $m_{\tilde{q}}$ represents the mass of stop and scharm, $M_2$ is
the wino mass parameter and $M_{A^0}\simeq M_{H^0}$ is the mass of the
pseudoscalar Higgs. Hence \eq{damu} reveals that the correlation of
the two quantities in a general MSSM is weak and restricted to the
fact that both grow with powers of $\tan \beta$. The much stronger 
correlation within mSUGRA scenarios only comes in after relating the
low energy parameters in \eq{damu} to the GUT parameters: 
{\bf Sakis and Herbi, please correct and continue, old text:}
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 $\cbs$. {\bf end of old text}

% 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.   

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[$M_{1/2}=450\; \gev, M_0=350\; \gev, A_0=0, \mu>0$ 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 bound from (\ref{expbmumu}) for $\cbs$ are also
shown in the figure.  The solid (dashed) curve represents the $\cbs$
result with (without) resummation of the $\tan \beta$-enhanced
SUSY-QCD corrections.  In this example the resummation suppresses
$\cbs$ by 20\% for $\tan\beta=50$. {\bf This looks a bit smallish to
  me.  What is the value of $\mu$?}  Hence 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 $\cbs \gsim
10^{-7}$, which will be oberved by CDF at Run~IIa.  Another
interesting 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 cancellation of the pure higgs boson
contributions (2HDM contributions) and the SUSY box sneutrino-squark
diagrams (which are similar in this case) with the SM contribution to
$cbs$. This effect has been described in detail within the context of
the 2HDM in \cite{Nierste}.  We finally that varying $f_{B_s}$ in
200-260 MeV changes $\cbs$ by roughly $\mp$ 25\% with respect to the
values shown in Fig.(\ref{fig1}).

While we fixed $M_{1/2}$ and $M_0$ in Fig.(\ref{fig1}) we next display
the impact of $\cbs$ on the $M_{1/2}-M_0$ plane for fixed
$\tan\beta=50$, $A_0=0$ GeV and $\mu>0$ in Fig.(\ref{fig2}).  The
solid contours represent $\cbs$ with resummed SUSY-QCD corrections.
First we find 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}). Second, CDF RUN~IIa will probe $M_{1/2}$ and
$M_0$ up to 500 GeV and 1.2 TeV respectively.  Third, the
shaded-dotted region representing the 2$\sigma$ range ($11 < \delta
a_\mu < 75$) of the BNL experiment nicely reveals the correlation with
\bs: if the BNL anomaly persists and is due to mSUGRA corrections, CDF
will certainly confirm this evidence through the observation of \bs.
Hence while 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], it will
nevertheless be able to prepare the ground for LHC by observing 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 (for $\tan \beta=50$) which in turn
means masses for the heaviest superpartners of 2.4 TeV.  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 find that Higgs masses less than 119 GeV are correlated
with an enhancement of $\cbs$ over the SM value by two orders of
magnitude.  The recent ``hint'' by LEP~\cite{LEPhiggs} of a mass
around 115.6 GeV results in $\cbs \gsim 5 \times 10^{-7}$ which will
be measured before Higgs boson discovery.


{\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??}

A strong theoretical motivation for a large $\tan\beta$ is SO(10)
Yukawa unification. It has recently been found~\cite{Stuart,Baer} that
a narrow parameter region is consistent with the current $g-2$
constraint and $b\rightarrow s\gamma$. In this region $\mu$ and
$M_{1/2}$ are light. This implies also a light CP-odd Higgs boson mass
probably less than 300 GeV, again enhancing the \bs\ decay.  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.}
 
In conclusion we have found a striking correlation between the muon
anomalous magnetic moment $a_{\mu}$ and the decay \bs\ in mSUGRA
scenarios. If the reported excess of ${\cal O} (4\cdot 10^{-9})$ in the
measured value of $a_{\mu}$ \cite{BNL} is caused by mSUGRA
corrections, one faces a huge, order-of-magnitude enhancement of
$\cbs$ over its SM value. The enhanced branching ratio is within reach
of Run-II of the Tevatron. A measurement of $\cbs$ will further
constrain the mass of the lightest Higgs bosons. SO(10) SUSY
explanations of the measured $a_{\mu}$ are hardly compatible with the 
present upper bound on $\cbs$. 

\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,
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366;
%%CITATION = PHLTA,B115,193;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
A.~H.~Chamseddine, R.~Arnowitt and P.~Nath,
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Phys.\ Rev.\ Lett.\  {\bf 49} (1982) 970;
%%CITATION = PRLTA,49,970;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
R.~Barbieri, S.~Ferrara and C.~A.~Savoy,
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L.~Hall, J.~Lykken and S.~Weinberg,
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%%%%%%%%%%History of Hidden Sector Supergravity Breaking %%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%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{fnalbrep}
D.~Atwood et al., Report of the workshop 
\emph{B Physics at the Tevatron: Run-II and Beyond},
FERMILAB-Pub-01/197 (preliminary).


\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}
