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\begin{flushright}
{\rm FERMILAB-TH-2002-xxx}\\
{\rm  July 2002} \\
\end{flushright}

\vspace*{\fill}

\begin{center}
{\Large \bf  Complementarity of the 
trilepton SUSY events and \\ [3mm]$B_s\rightarrow\mu^+\mu^-$ 
in the Minimal Supergravity Scenario \\[3mm]at Tevatron}\\[2.cm]

{{\Large A. Dedes$^1$}, {\Large H.~K. Dreiner$^1$},   
{\Large U. Nierste$^2$}, {\large and} {\Large P. Richardson$^3$}} \\[3mm]
{\it $^1$Physikalisches Institut der Universit\"at Bonn, 
 Nu\ss allee 12, 
 D-53115 Bonn, Germany
} \\[3mm]

{\it $^2$ Fermilab } 
\\[3mm]
{\it $^3$ Cambridge } 
\\[4mm]
\end{center}



\vspace*{1cm}

\begin{abstract}{\small\noindent
What Tevatron can see if supersymmetry breaking is mediated through
gravitational interactions with minimal kinetic terms? 
We present here results for two observables which have a 
fully supersymmetric origin : the trilepton final state events
and the rare $B_s$ meson decay to dimuons. Both observables are enhanced
when the supersymmetric particle spectrum is light of around 300 GeV,
and thus  within of the Tevatron reach.
However, the ratio of the vacuum expectation values, $\tan\beta$  
makes them leaving apart. If $\tan\beta$ is small (less than 30) then
Tevatron has a chance to see the trilepton events and not the rare
decay $B_s\to \mu^+\mu^-$, whereas if $\tan\beta$ is large the trileptons
become invisible (at least with a luminosity of 2$fb^{-1}$) and the
decay $B_s\to \mu^+\mu^-$ is observable even with 2$fb^{-1}$.  }
\end{abstract}

\vspace*{\fill}
\newpage
\setcounter{page}{1}

%\section{Introduction and motivation}
%\label{sec:intro}

%{\it 1. Minimal Supergravity Scenario  }

It is somehow embarasing that the MSSM even after 
the (by hand) imposition of the R-parity symmetry introduces
more than 100 parameters. Definitely a construction of a model
with few parameters is absolutely necessery not only for
aesthetic reasons but also for practical ones. The minimal
Supergravity scenario~\cite{mSUGRA} seems to be the most elegant one.
In the spontaneous breaking of local supersymmetry (Supergravity)
the spin 3/2 gravitino acquires a mass via the super-Higgs 
effect~\cite{Cremmer}. Then the minimum of the potential breaks
spontaneously Supergravity , but the quantum gravitational 
corrections are non-renormalizable. Formally speaking,
 taking the ``flat'' limit ($M=(8\pi G_N)^{-1/2} \rightarrow \infty$
we are left with an effective theory which is renormalizable. The
most general form of the low energy potential contains (apart from
the supersymmetry conserved F- and D-terms) mass terms
for the scalar particles, $m_{ij}$, trilinear  couplings $h_{ijk}$,
and the bilinear scalar mass $b_{ij}$ terms. These terms break
supersymmetry softly in the sense that no quadratic divergences 
appear in the theory. Furthermore, one
could impose a certain symmetry in the K\"ahler potential reducing
these parameters to just 3 : $m_{ij}=M_0 \delta_{ij}$, 
$h_{ijk}=A_0 Y_{ijk}$ and $B_{ij}=B_0\delta_{ij}$. 
In the 
MSSM and with  conserved R-parity symmetry there is only one
parameter $B_0$. This parameter, together with the 
$\mu$-parameter (but not its sign)
 of the superpotential are obtained at low energies
from  the requirement of succesful  electroweak symmetry breaking.
Finnaly, in mSUGRA one assumes common gaugino masses, $M_{1/2}$ at the
unification scale. Strictly speaking, one should impose these
boundary conditions at the Grand Unification scale only if the
MSSM is embedded in a unified scenario (i.e., superstring scenario)
at this scale. If above the GUT scale a unified group, such as
SO(10), exists then one must impose these conditions at the Planck
scale. Here we will impose three  universal boundary coditions,
 namely  squark masses $M_0$,  gaugino
masses $M_{1/2}$ and  trilinear couplings $A_0$ at the GUT scale,
and introduce the ratio $tan\beta$ of the Higgs vev's and the 
sign of the $\mu$-parameter at the electroweak scale. 
If an observable deviates substuntially from the SM in this
limited set of the parameter space then is certain that it
deviates in any variant of non-universality boundary conditions. 
However, in the latter, correlation of observables are usually
lost since there is more freedom to describe them individually. 

After more than 20 years of both theoretical and experimental
effort, Supersymmetry (SUSY)  still remains a ``theoretical'' 
symmetry. Supersymmetry, although broken at low energies,
will reveal its structure indirectly, through deviations from the 
-succesful after all- Standard Model (SM) predictions,
and/or directly from decay signals of superparticles. 
 The scope of this letter is  to  search for 
a complementarity [or no lose] between an ``indirect'' observable,
such as the rare $B_s$ meson decay \bs and the ``direct'' search
for charginos and neutralinos when they decay into the final state
leptons and missing energy. The collider these observables could show
up is Tevatron Run~II. 
In more detail the two observables are :

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Channel & Expt. & Bound (90\% CL) & SM prediction \\ \hline 
$B_s \to e^+ e^-$ & L3~\cite{L3} & $<5.4\times 10^{-5}$ & $9 \times 10^{-14}$ \\ \hline
$B_s \to \mu^+ \mu^-$ & CDF~\cite{CDFbmumu} 
& $<2.0\times 10^{-6}$ & $4 \times
10^{-9}$ \\ \hline
$B_s \to \tau^+ \tau^-$  & LEP~\cite{Nardi} & $<0.05$  &
$1 \times 10^{-6}$ \\ \hline
$B_d \to e^+ e^-$ & CLEO~\cite{CLEO} & $<8.3\times 10^{-7}$ & $6\times 10^{-15}$\\ \hline
$B_d \to \mu^+ \mu^-$ & CLEO~\cite{CLEO} & $<6.1\times 10^{-7}$ & $2\times
10^{-10}$  \\ \hline
$B_d \to \tau^+ \tau^-$  & LEP~\cite{Nardi} & $<0.015$ 
& $6\times 10^{-8}$ \\ \hline
\end{tabular}\end{center}\caption{The experimental status and the
SM predictions for the ${\cal B}(B_{s,d}\to l^+l^{-}$).}\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




{\it O1)}
\underline{The  $B$-meson decay,  \bs.}
The decays $\bar{B}_{s,d}\rightarrow l^+l^{-}$  are probably
the theoretically cleanest (after $B\to X_{s,d} \nu\bar{\nu}$)
 modes among the rare B-decays~\cite{Burasreview}.
The current experimental status and the
SM expectation is summarized in Table~I.
In the SM the Lepton Flavor conserved 
B-meson decays, $\bar{B}_{s,d}\rightarrow l^+l^{-}$,
are dominated from Z-boson penguins and box
diagrams involving top quark exchanges. Within the SM their 
Branching ratio,  is helicity suppressed i.e, proportional 
to the final state lepton mass,  and thus very small.
SM predictions given in Table~1 are dominated by
$f_{B_d}=200\pm 30$ MeV, 
$f_{B_s}=230\pm 30$ MeV~\cite{fbs} and the value of  
$|V_{ts}|=0.04\pm 0.002$. 
For the SM model predictions on Table~1  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.}   
$\bar{MS}$ mass 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.  
Since ${\cal B} (B \to \ell^+ \ell^-) \propto
m_{\ell}^2$, the branching ratio is largest for $\ell=\tau$.
Difficulties to search for $\tau$'s at hadron colliders 
turns the experimental focus on the search for \bs. Still 
the decay to $\tau^+\tau^-$ has a very weak bound and we
believe that a analysis of the existed LEP data could result to 
a much stronger bound. 
Leptonic
branching ratios of $B_d$ mesons are smaller by a factor of $|V_{td}
/V_{ts}|^2\lsim 0.06$.  
From Table~I  we see that
CLEO~\cite{CLEO}   has provided  strong bounds for the 
electron and muon final states. Again, a search for the final state
tau's remain to be seen. BaBar and Bell could add to this Table~I  too.
From now on we restrict ourselves to the decay mode \bs.

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 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{fnalrep}. 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. 
A recent analysis~\cite{Tanaka} showed that Run~II/CDF  with 15$fb^{-1}$
could be sensitive to    $\cbs < 1.2 \times 10^{-8}$.
Only ATLAS, CMS or LHCb~\cite{LHCb} 
are expected to reach the SM value for $\cbs$ displayed in Table 1.
Thus any observation
of the rare decay mode \bs at Tevatron Run~II  will definitely signals
Physics Beyond the SM.  

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. 
We should
emphasize here that although the above contributions give sizable
contributions to $\bar{B}\rightarrow K l^+l^-$ mode~\cite{Babu,Urban}
 or to the FB asymmetry in  $\bar{B}\rightarrow (\pi,K) l^+l^-$ mode
\cite{Olive}, this happens 
when $\cbs$ reaches its upper experimental bound. 


The rare decay mode \bs was analyzed first 
in the minimal supergravity scenario (mSUGRA) in Ref.\cite{sakis}. 
 A strong correlation with the muon
 anomalous magnetic moment $(g-2)_\mu$ was found.
Thus  if $(g-2)_\mu$ exceeds the Standard Model prediction by
$40*10^{-10}$ (which is the maximum upper bound now allowed 
from BNL~\cite{BNL,Narison}), $\cbs$ is larger 
by a factor of 10-100 than in the Standard Model 
and within reach of Run-II of the Tevatron. 
If the decay \bs  is
observed at Run-II of the Tevatron, 
 the mass of the lightest supersymmetric Higgs boson 
should be less than 120 GeV and again within reach at Tevatron.
Similar results with Ref.\cite{sakis} reported   recently in 
Ref.\cite{Tanaka}.
A nice observation was made very recently in Ref.\cite{Baek} :
the only scenario which can accomodate large values
of $\cbs$ is the mSUGRA scenario or related gravity mediated SUSY
breaking scenarios. Thus 
if the mode \bs will be found at Tevatron Run II then that means
only those type of scenarios survive.
But what Tevatron ``directly'' can see ??

\vspace*{0.5cm}

{\it O2)}
\underline{Trilepton SUSY events, }


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




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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%5
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This bound can be improved with the existed LEP data. We would like to
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%%CITATION = HEP-EX 0007042;%%

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%%CITATION = HEP-PH 9806471;%%


\bibitem{fbs}
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%%CITATION = HEP-LAT 0011064;%%


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%%CITATION = HEP-PH 0201071;%%


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%%CITATION = HEP-PH 0203069;%%



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%``B decays at the LHC,''
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%%CITATION = HEP-PH 0003238;%%

%%%%%%%%%%%%%%%%%%%%%%B->mu mu papers%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%2HDM%%%
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%%CITATION = NUPHA,B404,3;%%

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%%CITATION = HEP-PH 0008046;%%



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%%CITATION = HEP-PH 0004139;%%



\bibitem{Maxim}
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%%CITATION = HEP-PH 9807350;%%



\bibitem{Babu}
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[arXiv:hep-ph/9803460];
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[hep-ph/9810307].
%%CITATION = HEP-PH 9810307;%%
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%%CITATION = HEP-PH 9909476;%%
\bibitem{Huang:2000sm}
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%%CITATION = HEP-PH 0006250;%%


\bibitem{Urban}
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%%CITATION = HEP-PH 0104284;%%
For the numerical results on $\cbs$ presented here
we follow the formulae of this article.
%%CITATION = HEP-PH 0104284;%%


\bibitem{Olive}
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arXiv:hep-ph/0204119.
%%CITATION = HEP-PH 0204119;%%


\bibitem{sakis}
A.~Dedes, H.~K.~Dreiner and U.~Nierste,
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[arXiv:hep-ph/0108037].
%%CITATION = HEP-PH 0108037;%%



\bibitem{BNL}
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Phys.\ Rev.\ Lett.\  {\bf 86}, 2227 (2001)
[hep-ex/0102017];
%%CITATION = HEP-EX 0102017%%


\bibitem{Narison}
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%%CITATION = HEP-PH 0103199;%%




\bibitem{Baek}
S.~w.~Baek, P.~Ko and W.~Y.~Song,
%``Implications on SUSY breaking mediation mechanisms from observing  B/s $\to$ mu+ mu- and the muon (g-2),''
arXiv:hep-ph/0205259.
%%CITATION = HEP-PH 0205259;%%





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




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

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURES%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%555
%\begin{figure}
%\centerline{\psfig{figure=btomumu_higgs.ps, angle=90,height=5in}}
%\caption{Contour plots of the predicted \bs in the unconstraint 
%MSSM on the $M_A-\tan\beta$ plane 
%and for the input point indicated on the top [this is actually
%the point considered by LEP~\cite{LEPhiggs} for the higgs
%bosons searches-the so called
%maximal mixing scenario, $X_t=A_t-\mu\cot\beta=2 TeV$]. Contours
%of the light Higgs boson masses are also shown for $m_h=91,110,127$
%GeV. Values of $M_A<91$ are excluded by LEP~\cite{LEPhiggs}
% at 95\% CL. The upper left shaded corner is  excluded by
%by CDF Run I~\cite{CDFhiggs} at 95\% CL. $f_{B_s}$ has 
%been chosen to be 230 MeV. Departures from this value are discribed
%in the text.}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%555
%\begin{figure}[t]
%\centerline{\psfig{figure=btomumu-g-2.ps, angle=90,height=5in}}
%\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~\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=5in}}
%\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 \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}