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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
  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
     }
\title{Correlation of $B_s \to \mu^+\mu^-$ and $(g-2)_\mu$ in Minimal Supergravity}
%
\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} 
 {\it $^2$ CERN, TH Division, CH-1211 Geneva 23, Switzerland
 } 
} 
\maketitle
\begin{abstract}
  We analyse 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 100
  than in the Standard Model and well within reach of Run-II of the
  Tevatron. Thus an experimental search for \bs is a stringent test of
  the mSUGRA GUT scale boundary conditions. If the decay \bs is not
  observed at Run-II of the Tevatron we predict the mass of lightest
  supersymmetric Higgs boson to be larger than {\bf XXX GeV}. Going
  beyond mSUGRA, the decay \bs can also significantly probe the
  favoured parameter range in SO(10) GUT models, possibly excluding
  them.
\end{abstract}
\pacs{} 
]


%\narrowtext

%{\it 1. Introduction  }
Supersymmetry (SUSY) is an attractive and widely studied extension of
the Standard Model (SM).  The minimal supergravity model (mSUGRA)
\cite{mSUGRA} relates all supersymmetric parameters to just 5 real
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. 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
renormalisation group equations. Precision observables, which are
affected by SUSY corrections through loop effects, play an important
role in constraining the supersymmetric parameter space. The small
number of parameters makes mSUGRA sufficiently predictive to be
significantly tested by low energy precision measurements. In this
letter we would like to show that the decay \bs is a stringent test of
the mSUGRA scenario, in particular when correlated with $(g-2)_\mu$.

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{\nrefs,Sven,Barger,Fauzi,Roszkowski,Lahanas}. It further invites
a large $\tan\beta\gsim10$. 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 \cite{BNL}.
Hence in order to resolve the possible ambiguity between mSUGRA and
alternative explanations of $\delta a_{\mu}$ one ideally wishes to
study other observables whose sensitivity to supersymmetric loop
corrections is correlated with $\delta a_{\mu}$. It is our purpose
here to show the strong correlation between $\cbs$ and $\delta a_{
\mu}$ in the mSUGRA model.

SUSY modifies B meson 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
the branching ratio ${\cal B}(B \to X_s \gamma)$ 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 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 {\it flavour-blind} SUSY on
other B physics observables, in particular those which enter the fit
of the unitarity triangle, were studied and only small effects were
found. This did not include the decay \bs.  In contrast to the
observables in \cite{Masiero}, the branching ratio $\cbs$ grows like
$\tan^6\beta$ \cite{Babu,Chankowski,Urban}, with a possible several
orders of magnitude enhancement. We here go beyond this work to study
the effect of \bs in the mSUGRA model, which has parametrized boundary
conditions at the GUT scale. 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.  Leptonic
branching ratios of $B_d$ 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 in SUSY, we encourage our
colleagues at BaBar and BELLE to look for $B_d \to \tau^+ \tau^-$
decays as well. From now on we restrict ourselves to the decay mode
\bs.

In \cite{Babu,Chankowski,Urban} the SUSY corrections to $\cbs$ were
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
generalisation 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 normalised 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 $m_t=175\; \gev$.
The SM prediction is then given by $\cbs = (3.7 \pm 1.2) \cdot 10^
{-9}$, with the uncertainty dominated $f_{B_s}$.


Within mSUGRA the dominant coefficients are $C_{S,P}$ since they are
proportional to $\tan^3\beta$. An approximate formula, which is good
to within $50\%$ for $\tan\beta>45$ and $M_{1/2}<600$ GeV is given by
$|C_{S,P}|\approx({m_\mu F\tan^3\beta})/( {4M_W M_{1/2}})$. $F$ is a
loop factor which can be approximated by 1. Here we have made use of
the fact that within mSUGRA the low-energy values of both $\mu$ and
the squark masses are dominated by $M_{1/2}$ through the
renormalization group equations. For large values of $\tan\beta$ the
supersymmetric contribution will dominate in $\cbs$, and we expect it
to scale as $|C_{S,P}|^2\propto \tan^6\beta/M_{1/2}^2$. A similar
estimate of the supersymmetric contribution to $a_\mu$ yields $\delta
a_\mu \propto \tan\beta/M_{1/2}^2$. Thus within mSUGRA we expect a
strong correlation between $\cbs$ and $\delta a_\mu$. Furthermore, if
an experimental discrepancy in $a_\mu$ requires a supersymmetric
explanantion with large $\tan\beta$ then we would expect $\cbs$ to be
strongly enhanced. If however the supersymmetric explanation of
$\delta a_\mu$ requires small $M_{1/2}$ and moderate to small values
of $\tan\beta$ then we would expect the relative enhancement of $\cbs$
to be much smaller.









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}. 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 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.




%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 at the GUT scale
%  throughout this analyses. We have explicitly checked that its
%  variation within $\pm$500 GeV modifies our result for \bs within the
%  $f_{B_s}$ error.}. Furthermore, by demanding successful 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 branching 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.   

%%%%%%%%%%%%%%%%%%%%%%%%%555
%\onecolumn
%\widetext
\begin{figure}[t]
\centerline{\psfig{figure=btomumu-g-2.ps, angle=90,height=8cm}}
\caption{
  The SUSY 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].
  Shown also,  the SM
  prediction, the present bound by CDF~\protect\cite{CDFbmumu}, 
  on $\cbs$ aswell as the present 1$\sigma$ and 2$\sigma$ bound on
   $\delta a_\mu$ from BNL~\protect\cite{BNL}.
  $f_{B_s}$ has been chosen to be 230 MeV.}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%555
%\onecolumn
%\widetext
\begin{figure}[t]
\centerline{\psfig{figure=btomumu-g-2_light.ps, angle=90,height=8cm}}
\caption{Same as in Fig.\ref{fig1} but with $M_{1/2}=250, M_0=200$ GeV.}
\label{fig1b}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Fig.\ref{fig1} we plot the mSUGRA 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{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
50\% for $\tan\beta=50$. Hence in order for mSUGRA to account for
$\delta a_\mu$ within 1$\sigma$ of the current BNL
measurement~\cite{BNL} we need a large value of $\tan\beta \gsim 50$.
Due to the strong correlation we have shown within mSUGRA we then
predict $\cbs \gsim 10^{-7}$, which should be observed by CDF at
Run~IIa.  

%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\psfig{figure=btomumu_m12mo.ps, angle=90,height=7cm}}
\caption{Contour plots  of the $\cbs$ (solid) on
  the plane $M_{0},M_{1/2}$   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.  
  Contours of the light Higgs boson mass (red dot-dash line) 
  and of the $\delta a_\mu$ [in units $10^{-10}$] (dash) are also depicted.}
%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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Another interesting feature in Fig.\ref{fig1} is the dip in
the curve of $\cbs$ for moderate values of $20\lsim \tan\beta \lsim
30$. This is the result of a destructive interference of the pure
higgs boson contributions (2HDM contributions) and the SUSY box
sneutrino-squark diagrams (which are similar in this case) on the one
side with the SM contribution. This effect has been described in
detail within the context of the 2HDM in \cite{Nierste}. The
uncertainty in Fig.\ref{fig1} is mainly given by $f_{B_s}$. Changing
it in the range 200-260 MeV changes $\cbs$ by roughly $\mp$ 25\% with
respect to the values shown in Fig.\ref{fig1}.





In Fig.\ref{fig2} we show the dependence of $\cbs$ on the mSUGRA
parameters $M_{1/2}$ and $M_0$ for fixed $\tan\beta=50$, $A_0=0$ GeV,
and $\mu>0$. The solid contours represent $\cbs$ with resummed
SUSY-QCD corrections.  The lower left-hand corner in the figure is
excluded by the CDF limit given in Eq.(\ref{expbmumu}) and indicated
in the figure. If as expected, CDF can probe down to $\cbs\gsim2\cdot
10^{-7}$ at RUN~IIa, we see from Fig.\ref{fig2} that this corresponds
to a sensitivity of $M_{1/2} \lsim 500\,\gev$ and $M_0\lsim\,1.2$ TeV,
respectively. The shaded-dotted region represents the 2$\sigma$ range
($11 < \delta a_\mu < 75$) of the BNL experiment. It nicely reveals
the correlation with $\cbs$: 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 directly 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${\rm fb}^{-1}$ CDF will probe
$M_{1/2}=750$ GeV and 1.8 TeV (for $\tan \beta=50$) which in mSUGRA
corresponds to 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 would
be measured before the Higgs boson is discovered.


{\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 large values of $\tan\beta$ we have been considering are
theoretically very well motivated within supersymmetric SO(10) Yukawa
unification. There it has recently been found \cite{Baer,Stuart} that
a narrow parameter region can explain the observed $\delta a_\mu$
while still being consistent with the constraint from $b\rightarrow
s\gamma$. This is {\it not} within the context of mSUGRA. However, in
this parameter region both $\mu$ and $M_{1/2}$ are light, while the
CP-odd Higgs boson mass is less than 300 GeV, and $\tan\beta\approx
50$. Therefore the \bs\ decay is strongly enhanced and the bound from
CDF in Eq.(\ref{expbmumu}) further restricts the allowed parameter
space. As an example we determine $\cbs$ 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 is already excluded by CDF~\cite{CDFbmumu}. Thus the
SO(10) models should be reconsidered in the light of the experimental
bound on $\cbs$.  The only possible escape we can see for these 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. Or, turning
it around, if an SO(10) GUT model is the correct description of nature
then the decay \bs must be 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{mSUGRA} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%``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;%%

\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;%%
For the supersymmetric calculation see for example
%\cite{Moroi:1996yh}
T.~Moroi,
%``The Muon Anomalous Magnetic Dipole Moment in the Minimal Supersymmetric Standard Model,''
Phys.\ Rev.\ D {\bf 53} (1996) 6565
[Erratum-ibid.\ D {\bf 56} (1996) 4424]
[hep-ph/9512396].
%%CITATION = HEP-PH 9512396;%%


%%%%%%%%%%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;%%

%%%%%%%%%%%%%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{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{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{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{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{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{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;%%


\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).

%\cite{Ibanez:1992rk}
\bibitem{REWSB} 
For a review see, {\it e.g.} L.~E.~Ibanez and G.~G.~Ross,
%``Electroweak breaking in supersymmetric models,''
hep-ph/9204201.
%%CITATION = HEP-PH 9204201;%%

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

\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;%%



%%%%%%%%%%%%%%%%%%%%%%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{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;%%

\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;%%








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

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURES%%%%%%%%%%%%%%%%%%%%%%%%%%%





\end{document}




\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{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;%%



%%%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{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{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;%%



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



