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\author{\sf Athanasios Dedes}
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\begin{document}
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\chead{\sf{\usebox{\Title}}}
\date{30/6/2000}
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\diad{\sf University of Bonn}
\lfoot{\diadvalue}
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%\epsffile{../ral/ppd.ps}}}
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\Trtitle{\sf $B_s \to \mu^+\mu^-$  in the Minimal
  Supergravity Scenario, ~~~~~~~~~~~~~~~~~ {\tt hep-ph/0108037}}
%
\Large
\sf
%
\begin{titlepage}


\setlength{\fboxrule}{2pt}

\vskip1cm

\begin{center}
%{\LARGE \Red{  Across the Present Energy Frontier : Probing the Origin
% of mass,
% \\ Corfu, Greece, 10 - 12 September 2001}}\\[0.4cm]
%                           Collider Physics          }} \\
%{\LARGE \Red{ 19 - 24 September 1999,
%St John's College, Durham }}\\[1cm]
\Green{Athanasios Dedes}\\
\end{center}
\vskip 0.2 cm
\begin{center}
{\large   \sl   University of Bonn, Germany  } \\
\end{center}

\vskip 0.2 cm

\begin{center}
XIV Workshop, Beyond the Standard Model, 
March 11 - 14, 2002, Bad Honnef, Germany
\end{center}
\vskip1.0cm

\begin{center}
\fcolorbox{blau}{groguenc}{\textcolor{blau}{\fontsize{10mm}{16}
\begin{tabular}{c}
\sf       \phantom{  ..... ............... .....}     \\
\sf        \Red{{\bf  $B_s \to \mu^+\mu^-$  }} \\
\sf \Red{{ \bf  in the Minimal
  Supergravity Scenario  }}\\
%\sf     \Red{{\bf  muon anomalous magnetic moment?  }} \\
% \\ \sf \Red{{\bf at hadron colliders in the MSSM}}    \\
\sf       \phantom{  ..... ............... .....}     \\
\end{tabular}}}
\end{center}

\vskip0.5cm
\begin{center}
\Green{In collaboration with:}\\[2mm]
\Red{H.~ Dreiner and U.~Nierste, 
Phys.\ Rev.\ Lett.\  {\bf 87}, 251804 (2001), {\tt hep-ph/0108037}} \\
\end{center}


\vskip0.5cm

%\LARGE{ \Blue{We show :}}
\Large{
\begin{enumerate}
\item \Blue{ mSUGRA scenario (definitions)}
\item \Blue{ The light Higgs boson mass}
\item \Blue{ Muon anomalous magnetic moment, $(g-2)_\mu$ }
\item \Blue{ The decay \bs\ }
\item \Blue{ Results}
\item \Blue{ Work in progress.....}
%\item \Blue{ Minimal SO(10) and \bs\ }
%\item \Blue{ Work to be done...}
\item \Blue{Conclusions}
\end{enumerate}
}


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%\epsffile{../ral/ppd.ps}}  \\
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%\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{titlepage}


%
% ----------------------------------------
%
% ----------------------------------------
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Theory , Channels ...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\include{susy}
\include{mssm}
\include{superpotential}
\include{intro_msugra}
\include{unification}
%\include{strong_CP}
%\include{edms_rparity}
%\include{bonn_conclusions}
%
\newpage
%
\CTran{The Light Higgs Boson in mSUGRA}

\spa

%\begin{figure}[t]
\centerline{\psfig{figure=P3-05fig1.eps,height=5.9in}}
%\end{figure}



\begin{center}
$50 ~{\rm GeV} \le M_0 \le 1 ~{\rm TeV}$

$50 ~{\rm GeV} \le M_{1/2} \le 1 ~{\rm TeV}$

$-3 ~{\rm TeV} \le A_0 \le 3 ~{\rm TeV} $

$1.5 \le \tan\beta \le 60$

$\mu=\pm 1$
\end{center}

{\small 
S.~Ambrosanio, A.~Dedes, S.~Heinemeyer, S.~Su and G.~Weiglein '01}

\newpage
%%%%%%%%%%%%%%%% PAGE 1 %%%%%%%%%%%%%%%%%
\CTran{$(g-2)_\mu$ : Before November 2001...}


\Green{$\bullet$} \Blue{Based on Davier-H\"ocker calculation for the
  Hadronic
contributions to the muon 
$g-2$, BNL reports a 2.6$\sigma$ deviation from the SM}

\spa


There are four possible interpretations
of this discrepancy :
\begin{itemize}

\item a statistical fluctuation with 0.9\% probability.


\item something is ``wrong'' with the experiment


\item something is ``wrong'' with the theoretical analysis of the
hadronic  and/or the light by light contribution to
the muon anomalous magnetic moment.



\item the by far most exciting possibility: 
\end{itemize}


%\Green{$\bullet$} 
\Blue{
    there is \Red{``New Physics''}~ BSM}
\begin{eqnarray}
21.5 \times 10^{-10} \lsim \delta a_\mu \lsim 63.7\times 10^{-10}
\;\;\;\;
{\rm @ 90 \% ~CL}\nonumber
\end{eqnarray}
\begin{itemize}

\item{\Blue{$\delta a_\mu$ is positive}}

\item{\Blue{$\delta a_\mu$ is of the same order of the EW
      contributions, 
$\delta a_\mu^{NP} \propto \frac{G_F m_\mu^2}{4\pi^2\sqrt{2}}$}}
\spa

\end{itemize}
\bc
\Green{$\bullet$} \Blue{mSUGRA}
\ec

%\spa

\bc
\huge\Red{$(\delta a_\mu)_{SUSY}\propto~\tan\beta~\frac{
f(M_0)}{M_{1/2} ^2}$}
\ec


\Blue{~~~~~~~~invites large $\tan\beta \gsim 10$ and $M_{1/2}\lsim
  750$ GeV}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\CTran{$(g-2)_\mu$ : After November 2001...}

{\small M.~Knecht, A.~Nyffeler, M.~Perrottet and E.~De Rafael Nov. '01\\
M.~Hayakawa and T.~Kinoshita, Dec. '01}

\Green{$\bullet$} : \Blue{A mistake on the light 
by light scattering calculation :}

\spa

\begin{eqnarray}
-4.2 \times 10^{-10} \lsim \delta a_\mu \lsim 41.3\times 10^{-10}
\;\;\;\;
{\rm @ 90 \% ~CL}\nonumber
\end{eqnarray}

\spa 

\Green{$\rightarrow$} only just above 1$\sigma$ significance...

\spa

\Red{$\surd$} \Blue{A more definite claim is waiting for
a more precise $a_\mu$ measurement...}

\Green{$\rightarrow$} \Blue{one ideally wishes to study other
observables whose sensitivity to supersymmetric loop corrections is
correlated with $\delta a_{\mu}$.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\CTran{$B_s\to \mu^+ \mu^-$ vs. $B\rightarrow X_s \gamma$}

\Green{$\bullet$} \Blue{SUSY modifies B-meson observables if
  $\tan\beta$
is large [$\Rightarrow Y_b$ large ]}

\Green{$\bullet$} \Blue{Especially sensitive observables : $ B\rightarrow
  X_s \gamma , B\rightarrow l^+ l^- $}


\begin{itemize}

\item $B\rightarrow X_s \gamma$ : relies on a cancelation between
$H^\pm-t$ and $\tilde{\chi}-\tilde{t}$ loops.

\item  In
mSUGRA the low energy value for the trilinear term $A_t$ is dominated
by $M_{1/2}$,  \Red{$A_t\simeq -2 M_{1/2}$}. 
Then
\Red{$\mu>0$} implies that the charged-Higgs-top loop and the chargino-stop
loop tend to cancel in \Red{${\cal B}(B \to X_s \gamma)$}, so that the
sensitivity to mSUGRA corrections \Red{is weakened}. 


\item $B\rightarrow l^+ l^- $ : large for large $\tan\beta$, almost 
independent from the sign of $\mu$. No accidental cancelation required!

\item Since ${\cal B} (B \to \ell^+ \ell^-) \propto
m_{\ell}^2$, the branching ratio is largest for $\ell=\tau$: hard to 
detect!

\item branching ratios of $B_d$ mesons are smaller by a factor of $|V_{td}
/V_{ts}|^2\lsim 0.06$.

\item BaBar and BELLE could look for \Red{$B_d \to \tau^+ \tau^-$} ?


\end{itemize}

{\small K.~S.~Babu and C.~Kolda, '99,\\
 C.~Huang,  W.~Liao and Q.~Yan, '99\\
P.~H.~Chankowski and L.~Slawianowska, '01,\\
 C.~Bobeth, T.~Ewerth, F.~Kruger and J.~Urban, '01 \\
G. Isidori, A. Retico, '01. }

%\spa
%
%\centerline{\epsfig{figure=self.eps}}
%
%\spa

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\CTran{$B_s \to \mu^+ \mu^-$}


\spa

\begin{eqnarray}
\Blue{{\cal H}_{eff} = -\frac{4 G_F}{\sqrt{2}} V_{tb} V_{ts}^* \biggl [
c_{10} {\cal O}_{10}+ \Red{c_S {\cal O}_S + c_P {\cal O}_P} \biggr ]}
\nonumber
\end{eqnarray}

\begin{eqnarray}\Blue{
{\cal O}_{10} &=& \frac{e^2}{16\pi^2} (\bar{s}\gamma^\mu P_L b)
(\bar{l}\gamma_\mu \gamma_5 l)} \nonumber \\ \Red{
{\cal O}_S &=& \frac{e^2}{16\pi^2} m_b (\bar{s}P_R b)(\bar{l}l) \nonumber \\
{\cal O}_P &=& \frac{e^2}{16\pi^2} m_b (\bar{s}P_R b) (\bar{l}\gamma_5 l)}
\nonumber 
\end{eqnarray}


\Green{$\bullet$} \Blue{SM $\rightarrow {\cal O}_{10}$}

\spa 

\centerline{\epsfig{figure=bmumu_SM.eps}}

\newpage

\spa

\Green{$\bullet$} \Red{MSSM $\rightarrow {\cal O}_{S}$}

\spa\spa


\centerline{\epsfig{figure=bmumu_MSSM_CS.eps,height=7.5in}}

\newpage

\spa

\Green{$\bullet$} \Red{MSSM $\rightarrow {\cal O}_{P}$}

\spa\spa


\centerline{\epsfig{figure=bmumu_MSSM_CP.eps,height=7.5in}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\Green{$\bullet$} \Blue{Branching Ratio :}


\begin{eqnarray}
& &{\cal B}(B_s\to \mu^+ \mu^-) =
\frac{G_F^2 \alpha^2 M_{B_s}^3 \tau_{B_s} f^2_{B_s}}{64\pi^3} 
|V_{tb}^*V_{ts}|^2 \sqrt{1-\frac{4 m_{\mu}^2}{M_{B_s}^2}}
\nonumber \\[5mm]
&\times& \Biggl \{ \biggl (1-\frac{4 m_{\mu}^2}{M_{B_s}^2}\biggr )
\Biggl |\frac{M_{B_s}}{m_b+m_s} \Red{C_S} \Biggr |^2 %\nonumber \\[2mm] &+&
+\Biggl |\frac{2 m_\mu}{M_{B_s}} \Blue{C_{10}}-\frac{M_{B_s}}{m_b+m_s}\Red{C_P}
\Biggr |^2 \Biggr \} \nonumber
\end{eqnarray}

\spa\spa



\Green{$\bullet$} \Blue{Standard Model :}

{\small G.~Buchalla and A.~J.~Buras, '94\\
M.~Misiak and J.~Urban, '99}


\begin{eqnarray}
\cbs  =(3.7 \pm 1.2) \times 10^{-9}
\nonumber
\end{eqnarray}

\spa 

\Green{$\bullet$} \Blue{MSSM : Large $\tan\beta \gsim 40$}

\begin{eqnarray}
C_{S,P} \simeq \pm \frac{m_\mu~}{4 s_W^2 M_W^2}\times 
 \tan^3\beta\times \frac{\mu A_t}{ M_A^2}\times
C_0(\mu,m_{\tilde{t}_1},m_{\tilde{t}_2})\nonumber
\end{eqnarray}


\Green{$\bullet$} \Blue{MSUGRA : Large $\tan\beta \gsim 40$}

\begin{eqnarray}
\cbs \propto
\tan^6\beta/M_{1/2}^4  ~~~{\rm for} ~~~M_{1/2} \gg M_0
\nonumber
\end{eqnarray}

\Green{$\bullet$} \Blue{For large $\tan \beta$, higher order
corrections can be large}

\Green{$\bullet$} \Blue{ $\tan\beta$-enhanced 
supersymmetric QCD corrections have been summed to all
orders in perturbation theory. }

{\small M.~Carena, D.~Garcia, U.~Nierste and C.~E.~Wagner, '00}

\Green{$\bullet$} \Blue{We have incorporated 
these dominant higher order corrections
 by simply replacing the $b$ Yukawa coupling
\Red{$h_b\propto m_b \tan \beta$} with \Red{$ h_b^{{\rm eff}} = h_b/(1+\Delta
m_b)$}}

\Green{$\bullet$} \Blue{QCD corrections to $B_s\to \mu^+\mu^-$
 in the MSSM have been recently calculated.}

{\small C.~Bobeth, A.~J.~Buras, F.~Kruger and J.~Urban, Dec. '01}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage

\spa


\centerline{\Green{$\bullet$} \underline{Experiment}}

\Green{$\bullet$} \Blue{Only an experimental bound from CDF Run~I}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\cbs < 2.6 \times 10^{-6},
\qquad \mbox{ at 95\% C.L.} 
\nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\spa 

\Green{$\bullet$} \Blue{Near Future experiment: CDF Run IIa and IIb}

\Red{$\rightarrow$} \Blue{Single event sensitivity (${\cal S}$) :
Branching ratio for which we would expect to observe one event 
in 2 $fb^{-1}$ }

\begin{eqnarray}
{\cal S}(B_s \to \mu^+ \mu^-) = 1.0 \times 10^{-8} \times 
\frac{2~{\rm fb}^{-1}}{\int{ {\cal L}}} \nonumber 
\end{eqnarray}

\Red{$\rightarrow$} \Blue{ $5 \times 10^{-7}$ for 50  events  with luminosity
  2 $fb^{-1}$}

\Red{$\rightarrow$} \Blue{ $5 \times 10^{-7}$ for 250-500 events  
with luminosity
  10-20 $fb^{-1}$}

{\small K.~Anikeev {\it et al.},
``B physics at the Tevatron: Run II and beyond,''
arXiv:hep-ph/0201071.}

\newpage
\CTran{Results}

\Green{$\bullet$} \Blue{$M_{1/2}=450, M_0=350, A_0=0, \mu>0, m_t=175$ GeV}

\Green{$\surd$} \Blue{Fine with $b\to s \gamma$ and direct searches...}

%{\small A.~Djouadi, M.~Drees and J.~L.~Kneur, '01}




\vspace*{2cm}

\centerline{\epsfig{figure=btomumu-g-2.munich.ps,angle=90}}

{\small A.~Dedes, H.~K.~Dreiner and U.~Nierste, '01}

\newpage

\vspace*{3cm}

\centerline{\epsfig{figure=btomumu_m12tanb.ps,angle=90}}

{\small A.~Dedes, H.~K.~Dreiner and U.~Nierste, '01}


\newpage

\vspace*{3cm}

\centerline{\epsfig{figure=btomumu_m12mo.ps,angle=90}}

{\small A.~Dedes, H.~K.~Dreiner and U.~Nierste, '01}


\newpage
\CTran{Trileptons and \bs  at Tevatron} 

\spa

\Green{$\bullet$} \Blue{The Mode} 
%$\rightarrow$ 
%\Red{Trilepton final states, $lll$, $l^\pm l^\pm$}

%\Green{$\bullet$} \Blue{ $q \bar{q}' \stackrel{W}\rightarrow \chi^\pm \chi_2^0%}
%
%\Red{$\bullet$} \Blue{ $\chi_1^\pm \rightarrow
%  (\tilde{l},\tilde{\tau}) \nu
%\rightarrow (l,\tau) \chi_1^0 \nu$
%
%\Red{$\bullet$} \Blue{ $\chi_1^\pm \rightarrow
%  (\tilde{l},\tilde{\tau}) \tilde{\nu}
%\rightarrow (l,\tau) \chi_1^0 \nu$
%\spa

\centerline{\epsfig{figure=tril.eps,height=3in}}

\spa 

\centerline{\epsfig{figure=chardecays.ps,height=4.5in}}


\Green{$\bullet$} \Blue{Final State :} \Red{ $lll$, $l^\pm l^\pm$, $l^\pm l^\pm \tau$,
$l^\pm \tau\tau$, $\tau\tau\tau$, $\tau^\pm\tau^\pm$}

\newpage

\CTran{Work in progress..........}

\spa 

{\small A. Dedes, H. Dreiner, U. Nierste, P. Richardson}

\spa\spa

\centerline{\epsfig{figure=btomumu_m12mo_tev.ps,angle=90}}


\newpage

\CTran{ SO(10) Unification }

\spa

\centerline{\epsfig{figure=raby.eps,height=7.5in}}

{\small T. Blazek, R. Dermisek, S. Raby  '02}


$\rightarrow$ 
\Red{$\cbs \gsim 10^{-5}~~~~{\rm already~~ excluded~~ by~~ CDF}$}

\newpage


\centerline{\epsfig{figure=raby2.eps,height=7.8in}}

{\small T. Blazek, R. Dermisek, S. Raby  '02}

\Red{!!!!} \Blue{Detailed analysis for $B_s \to \mu^+ \mu^-$ is not done..}

\Red{!!!!} \Blue{SO(10) predicts \bs~ just around the corner...}






\newpage

\spa

\CTran{Conclusions}

\spa

\Green{$\surd$} \Blue{Striking correlation between the muon
anomalous magnetic moment $a_{\mu}$ and the branching ratio $\cbs$ in
mSUGRA scenarios.}

\spa

\Green{$\surd$} \Blue{If a reported excess on $a_{\mu}$  is
caused by mSUGRA corrections with large $\tan\beta$, one faces more
than an order of magnitude enhancement of $\cbs$ over its SM value.}

\spa

\Green{$\surd$} \Blue{Well within reach of Run-II of the Tevatron.}


\spa

\Green{$\surd$} \Blue{The two combined measurements 
significantly constrain the mSUGRA parameters,
allowing a
determination of $\tan\beta$ and $M_{1/2}$. }

\spa

\Green{$\surd$} \Blue{A measurement of $\cbs$
will further constrain the mass of the lightest Higgs boson.}

%\spa
%
%\Green{$\surd$} \Blue{SO(10)
%SUSY explanations of the measured $a_{\mu}$ are barely compatible with
%the present upper bound on $\cbs$.}







\end{document}





