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Trilepton Events and $B_s\ra\mu^+\mu^-$: 
No-lose for mSUGRA at the Tevatron?
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\begin{flushright}
{\rm FERMILAB-Pub-02/xxx-T}\\
{\rm  July 2002} \\
\end{flushright}

\vspace*{\fill}

\begin{center}
%  {\Large \bf Trilepton Events and $B_s\ra\mu^+\mu^-$:
%    Complementary Signals for mSUGRA at the Tevatron}
\boldmath
  {\LARGE \bf Trilepton Events and $B_s\ra\mu^+\mu^-$:\\[3mm]
	No-lose for mSUGRA at the Tevatron?}
\unboldmath
\vspace{1cm}

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

{\it $^b$ Fermi National Accelerator Laboratory, Batavia, IL
60510-500, USA\footnote{Fermilab is operated by URA under DOE contract
No.~DE-AC02-76CH03000.}}  \\[3mm] 

{\it $^c$ 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 $\tan\this scenario beta$ 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$fb^{-1}$.  }
%% The possibility that supersymmetry breaking is mediated through
%% gravitational interactions with minimal kinetic terms is theoretically
%% appealing and widely studied. 
We study the potential of the Tevatron to find evidence for minimal
supergravity (mSUGRA) and find two observables to reveal complementary
information on the mSUGRA parameter space: the ``gold-plated'' decays
of charginos/neutralinos to trilepton final states and the rare decay
\bs. For supersymmetric particle masses below 300 GeV we face a
``no-lose'' situation for the Tevatron: If $\tan\beta \lsim 30$, the
Tevatron has a chance to see the trilepton events but not $B_s\to
\mu^+\mu^-$ during Run IIa, whereas for larger $\tan\beta$ the
trilepton events become invisible (at least with an integrated
luminosity of 2 fb$^{-1}$) while $B_s\to \mu^+\mu^-$ is enhanced to an
observable level.  }


\end{abstract}

\vspace*{\fill}
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\section{Introduction and Motivation}
\label{sec:intro}

%{\it 1. Minimal Supergravity Scenario  }

Even after imposing R-parity conservation by hand, the minimal
supersymmetric standard model (MSSM) has an embarrassingly large
number of parameters.  The construction of a model with only a few
parameters is desirable for {\ae}sthetic and for practical reasons.
The minimal supergravity (mSUGRA) scenario \cite{mSUGRA,Soni:1983rm}
is currently the most elegant solution to this problem.  In mSUGRA,
local supersymmetry (supergravity) is spontaneously broken in the
hidden-sector: the minimum of the scalar potential violates
supersymmetry.  The superpartner of the graviton, the spin 3/2
gravitino, acquires a mass, via this super-Higgs effect
\cite{Cremmer}.  Supersymmetry breaking is communicated to the
observable sector via gravity interactions.  The quantum gravitational
corrections are in principle non-renormalizable, but taking the
``flat'' limit where the Planck scale goes to infinity ($M_P=(8\pi
G_N)^{-1/2} \ra \infty)$ we are left with an effective theory
which is renormalizable.  The most general form of the low energy
scalar potential after supersymmetry breaking \cite{Soni:1983rm}
contains the following free parameters:\footnote{If R-parity is
  conserved there is only one parameter $b_{ij}=b$.}
\begin{enumerate}
\item mass terms for the scalar particles, $m_{ij}$, 
\item trilinear couplings, $h_{ijk}$,
\item the bilinear terms, $b_{ij}$.
\end{enumerate}
These terms break supersymmetry softly in the sense that no quadratic
divergences appear in the theory \cite{Girardello:1981wz}.  The number
of parameters is dramatically reduced if the K\"ahler potential is
minimal. Then these parameters are all universal
\begin{equation}
m_{ij}=M_0 \,\delta_{ij},\qquad h_{ijk}=A_0 \,Y_{ijk}, \qquad
B_{ij}=B_0\,\delta_{ij}.
\end{equation}  
$B_0$ and the magnitude of the superpotential Higgs mixing parameter
$\mu$ are fixed by electroweak symmetry breaking \cite{Ibanez:fr}.
Outside of the scalar sector there is also a universal soft
supersymmetry breaking parameter: the common gaugino mass, $M_{1/2}$.
In the Higgs sector we have one further parameter: $\tan\beta$, the
ratio of the vacuum expectation values of the two neutral Higgs
fields. We thus have a total of 5 parameters
\begin{equation}
M_0,\,A_0,
\,M_{1/2},\,sgn(\mu),\,\tan\beta.
\end{equation}

The parameters $M_0,\,A_0,\,M_{1/2},$ are expected to be universal at
the Planck scale, where supersymmetry breaking is mediated from the
hidden-sector to the observable sector.  In order to obtain weak-scale
predictions we must use the renormalization group equations (RGEs) to
evolve the parameters from $M_P$ down to $M_W$.  In addition, there
could be unification at the grand unified (GUT) scale $M_X={\cal O}
(10^ {16}\,GeV)$ in an unknown simple group such as $SU(5)$ or
$SO(10)$.  This would strongly modify the RGEs between the Planck scale
and the GUT scale.  In order to avoid this model dependence, we
consider $M_0,\,A_0,\,M_{1/2},$ to be universal at the GUT scale.  We
thus neglect any effects of the running of the RGEs between $M_P$ and
$M_X$.  $\tan\beta$ and $sgn(\mu)$ are fixed at the weak scale.


If an observable deviates substantially from the standard model (SM)
in mSUGRA with this limited set of universal parameters then we would
expect to find a similar deviation in a more general non-universal
case, modulo accidental cancellations. Thus we expect to be able to
obtain more general results already in this simplified model. However,
correlations between observables in mSUGRA such as \bs\ versus
$(g-2)_\mu$ \cite{sakis} are typically lost in the more general case
due to the additional parameter freedom.


After more than 25 years there is still no experimental evidence for
supersymmetry (SUSY).  In principle we expect two kinds of signatures:
1) direct detection of a supersymmetric particle or 2) an indirect
signature via a deviation from a SM prediction.  It is the purpose of
this letter to demonstrate the complementarity of these two approaches
in two specific examples.  For the direct search we consider the
trilepton signature, which was first proposed in the context of
supersymmetry in \cite{Baer:1985at,Baer:1986dv,Baer:1986vf}, and has
since become a standard search mode for supersymmetry production. We
wish to investigate the correlation of this signature in mSUGRA with
the rare $B_s$ meson decay \bs. We expect these two signatures to be
observable at Run~II of the Tevatron.
 





 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}  
	          & $<5.4\times 10^{-5}$ 
		  & $(8.9 \pm 2.3 )\times 10^{-14}$ \\ \hline
$B_s \to \mu^+ \mu^-$ & CDF~\cite{CDFbmumu} 
		  & $<2.0\times 10^{-6}$ 
		  & $(3.8\pm1.0) \times 10^{-9}$ \\ \hline
$B_s \to \tau^+ \tau^-$  & LEP~\cite{Nardi} 
	         & $<0.05$  
		 & $(8.2\pm 2.1) \times 10^{-7}$ \\ \hline
$B_d \to e^+ e^-$ & CLEO~\cite{CLEO} 
		  & $<8.3\times 10^{-7}$ 
		  & $(2.4\pm 0.7 \pm 0.7) \times 10^{-15}$\\ \hline
$B_d \to \mu^+ \mu^-$ & CLEO~\cite{CLEO} 
		  & $<6.1\times 10^{-7}$ 
		  & $(1.0\pm 0.3\pm 0.3) \times 10^{-10}$  \\ \hline
$B_d \to \tau^+ \tau^-$  & LEP~\cite{Nardi} 
			 & $<0.015$ 
			 & $(2.1 \pm 0.6\pm 0.6)\times 10^{-8}$ \\ \hline
\end{tabular}\end{center}
\caption{The experimental status and the
SM predictions for the branching ratio ${\cal B}(B_{s,d}\to 
\ell^+\ell^{-}$). The error in the $B_s$ branching ratios stems from
the uncertainty in $f_{B_s}$, the two errors in the $B_d$ branching
ratios correspond to the uncertainties in $f_{B_d}$ and $|V_{td}|$.  
}\label{tab}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%{\bf (O$_1$) \underline{The $B$-meson decay \bs}}  
%{\bf \underline{1.~The $B$-meson decay \bs}}  
\section{The $B$-meson decay \boldmath{\bs}}
% Among the rare $B$ decays the leptonic decays $\bar{B}_{s,d}\ra 
% \ell^+\ell^{-}$ are special:
In the SM the leptonic decays $B_{s,d}\ra \ell^+\ell^{-}$ are
dominated by Z-boson penguin and box diagrams involving top quark
exchange. The decay amplitude is helicity suppressed and thereby
proportional to the lepton mass.  The resulting small branching ratios
are well below the upper bounds set by past and present experiments;
the current situation is summarized in Table~\ref{tab}.  The
uncertainty in the SM predictions \cite{Buchalla} of Table~\ref{tab}
mainly stems from the decay constants $f_{B_d}=(200\pm 30)$ MeV and
$f_{B_s}=(230\pm 30)$ MeV~\cite{fbs}. While the element
$|V_{ts}|=0.040\pm 0.002$ of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix entering the $B_s$ decays is well-known from CKM unitarity, the
$B_s$ decays suffer from an additional uncertainty due to
$|V_{td}|$. From a global fit to the unitarity triangle one finds
$0.15 \leq |V_{td}/V_{ts}| \leq 0.23$ \cite{ckmfit}.  We use the
$\overline{\rm MS}$ b-quark mass $m_b(m_b)=4.25$ GeV and assume a
fixed\footnote{ In the mSUGRA analysis below, $m_t(m_t)$ is not fixed
due to the variation of the strong coupling constant $\alpha_s$ with
the SUSY spectrum under the assumption of gauge coupling unification.
However, the effect of this variation is small compared to the
hadronic uncertainty.}  $\overline{\rm MS}$ value for the top quark
mass, $m_t(m_t)=167$ GeV which corresponds to a pole mass of $m_t=175$
GeV.  There is an additional small uncertainty of $\pm 0.3\times 10^{-9}$ in
${\cal B}($\bs$\!)$ when scanning the top mass in its experimental
value. Since we are interested in order-of-magnitude effects from
supersymmetry, we can safely neglect the hadronic uncertainty and work
with the central values of the decay constants and CKM elements.

As ${\cal B} (B \to \ell^+ \ell^-) \propto m_{\ell}^2$, the branching
ratio is largest for $\ell=\tau$.  Yet $\tau$-lepton searches are very
difficult at hadron colliders. Therefore the best mode to search for
new physics is \bs.  Still the decay to $\tau^+\tau^-$ has a very weak
bound and we believe that an analysis of the existing LEP data could
result to a stronger bound.  Leptonic branching ratios of $B_d$ mesons
are smaller by a factor of $|V_{td} /V_{ts}|^2\lsim 0.05$ than the
leptonic decays of $B_s$ mesons.  From Table~I we see that
CLEO~\cite{CLEO} has provided strong bounds for the electron and muon
final states. Again, a dedicated search for the final state $\tau$'s
remains to be performed. The BaBar and Belle experiments could improve
on Table~I, as well.  From now on we restrict ourselves to the decay
mode \bs.  In the SM the decay mode \bs experimentally challenging due
to its small branching ratio.  However it has a quite distinctive and
unique signature.  During Run I of the Tevatron the CDF experiment has
set an upper bound of $2.6 \times 10^{-6}$ for $\cbs$, based on an
analysis using very tight cuts to achieve an essentially
background-free framework. When extrapolated to Run IIa, which
corresponds to an integrated luminosity of 2 fb$^{-1}$, this method
yields a single event sensitivity of $\cbs =1.0 \cdot 10^{-8}$
\cite{fnalrep}.  A recent analysis~\cite{Tanaka} found that CDF can
discover \bs in Run~IIb with an integrated luminosity of 15 fb$^{-1}$,
if $\cbs > 1.2 \times 10^{-8}$.  Currently further sophisticated
studies are underway, and better cuts may well improve the explorable
range of $\cbs$. The SM range will certainly be accessible to the LHC
experiments~\cite{LHCb}.

The helicity suppression gives the leptonic $B$ decays a very special
r\^ole in the search for new physics, because they are sensitive to
new chirality-flipping interactions like extended Higgs sectors with
enhanced Yukawa couplings. This scenario occurs in the
two-Higgs-doublet model (2HDM) and the MSSM, if $\tan \beta$ is large.
In particular no other observable in $B$ physics is more sensitive to
effects from non-standard neutral Higgs bosons than $\cbs$.  In the
type-II 2HDM the extra contribution to the decay amplitude grows like
$\tan^2\beta$ and is roughly of the same size as the SM amplitude
\cite{Nierste}. In the MSSM, however, the amplitude involves three
powers of $\tan \beta$, so that $\cbs \propto \tan^6\beta$. This leads
to an enhancement over the SM value by up to three orders of
magnitude, even in the flavour-universal MSSM, where flavour changes
occur in weak vertices only and come with the usual CKM elements
\cite{Babu,Urban}.  Moreover, in the limit of infinite SUSY mass
parameters the MSSM contribution does not vanish, because the MSSM
approaches a general 2HDM with flavour-changing neutral Higgs
couplings in this limit. This feature and the resulting importance of
$B$ physics to probe the large $\tan\beta$ regime of the MSSM was
first emphasized in \cite{Maxim}. The supersymmetric contributions
enhancing $\cbs$ will also affect the branching ratio and
forward-backward asymmetry in $B\ra (\pi,K) \ell^+\ell^-$, but
these effects are small and strongly correlated with $\cbs$ and will
only be measurable, if $\cbs$ is close to its experimental upper bound
\cite{Urban,Olive}.

The first analysis of \bs in the minimal supergravity scenario
(mSUGRA) revealed an interesting correlation of $\cbs$ with the muon
anomalous magnetic moment $(g-2)_\mu$ and the mass of the lightest
Higgs boson \cite{sakis}: If $(g-2)_\mu$ exceeds the SM prediction by
$40\times 10^{-10}$ (which is the maximum allowed from
BNL \cite{BNL,Narison}), $\cbs$ is larger by a factor of 10--100 than
in the SM and within reach of Run II of the Tevatron.  If the decay
\bs is observed at Run II of the Tevatron, then in mSUGRA the mass of
the lightest neutral CP-even Higgs boson should be less than 120 GeV.
The results of Ref.~\cite{sakis} were recently confirmed in
Ref.~\cite{Tanaka}, where also constraints from dark matter were
included.  A nice observation was made in Ref.~\cite{Baek}: alternative
SUSY breaking scenarios imprint signatures on $(g-2)_\mu$ and $\cbs$
which are very different from those found for mSUGRA in
Ref.~\cite{sakis}.  A substantial enhancement of $\cbs$ in
anomaly-mediated SUSY breaking scenarios is impossible and also hard
to accommodate with gauge-mediated SUSY-breaking.  Thus an observation
of \bs\ at Run II of the Tevatron can be viewed as a signature of
supersymmetry with gravity mediated supersymmetry breaking.



%\vspace*{0.5cm}

%{\it O2)}
%\underline{Trilepton SUSY events, }
\section{Multilepton SUSY events}

{\bf the review here , Herbi ??}

  In general our strategy for the analysis of the multilepton signatures closely
  follows that of \cite{Matchev:1999yn,Matchev:1999nb}. The major differences
  between these studies are:
\begin{itemize}
\item While \cite{Matchev:1999yn,Matchev:1999nb} used the leading-order cross
      sections for both the signal and background processes we have made use of
      next-to-leading order calculations for the background and
      most important signal processes.
\item In \cite{Matchev:1999yn,Matchev:1999nb} PYTHIA6.115\cite{pythia}
      was used for the signal
      simulation and a combination of PYTHIA and COMPHEP \cite{Pukhov:1999gg}
      for the background simulation.
      We used HERWIG6.4 \cite{HERWIG64}
      to simulate the signal and a combination of HERWIG and
      COMPHEP to simulate the background.
      This has a number of advantages due to recent improvements in HERWIG.
      While PYTHIA performs the production and decay of the SUSY
      particles independently, and
      often does not use a matrix element to give the distributions and the decay  
      products, HERWIG6.4 makes use of the method described in 
      \cite{Richardson:2001df} to perform the SUSY decays including all the
      correlations. One of the major advantages for our studies
      is that the spin correlation algorithm used in HERWIG can
      use the TAUOLA decay package \cite{Jadach:1993hs} to give the correct helicity
      of the tau on an event-by-event basis, rather than
      averaging as was done in \cite{Matchev:1999yn,Matchev:1999nb}.
\end{itemize}
  While the analysis of \cite{Matchev:1999yn,Matchev:1999nb} used the SHW detector
  simulation we made use of the PGS \cite{pgs} detector simulation. However
  this is simply a more recent version of the SHW package.
  The default PGS parameters were used together with the following additions
\begin{enumerate}
\item A requirement that the total transverse energy in a cone
      of size $\Delta R=\sqrt{\Delta\phi^2+\Delta\eta^2}=0.4$ about the
      direction of a muon was less than 2~GeV.
\item The missing transverse energy, \met, which is calculated in PGS using the
      calorimeter cells, was corrected to include any muons.
\item A cut on the ratio of the electromagnetic, $E_{\rm em}$,
      to hadronic energy, $E_{\rm had}$, in a jet $ E_{\rm em}/E_{\rm had}>10$
      to avoid electrons being considered as jets.
\end{enumerate}
  These modifications are virtually identical\footnote{In \cite{Matchev:1999nb} the
  tracking coverage was extended from $|\eta|\leq1.5$ to $|\eta|\leq2.0$ however this
  is now the default in PGS.} to those made in \cite{Matchev:1999nb}

  As discussed in \cite{Matchev:1999yn,Matchev:1999nb} the major backgrounds to
  multilepton production are:
\begin{description}
\item[WW] Here the background is from the production of leptons in the decay of the
	  W bosons. These events were simulated using HERWIG6.4 and the
          results normalized to the next-to-leading order result from
          MCFM \cite{Campbell:1999ah}.
\item[WZ] In this case the leptons come from the decay of the gauge bosons.
          This is the main physics background in most of the
          channels. As this channel is so important rather than use
          HERWIG, which does not include all the diagrams, we used
	  COMPHEP to generate all the Feynman diagrams for the four-fermion
          final state. HERWIG was then used to generate the initial-state
          radiation and hadronization of the partonic events generated with
          COMPHEP. The results were normalized to the next-to-leading order
          cross section calculated using MCFM \cite{Campbell:1999ah}.
\item[ZZ] The background comes from the production of leptons in the decay of 
          the Z bosons. These events were simulated using HERWIG6.4 with
          a next-to-leading order normalization from MCFM \cite{Campbell:1999ah}.
\item[W+jets] In these events the background comes from one real lepton
              produced in the decay of the W and additional fake leptons
              from the mis-identification of isolated hadrons as leptons.
              The simulation of isolated hadrons producing fake electrons and muons
              cannot
              be performed reliably using Monte Carlo event generators
              and therefore we adopted the same procedure as in 
	      \cite{Matchev:1999nb} in order to estimate this rate from the
              RunI CDF data.
	      The SHW simulation, which was the predecessor to PGS,
              gives a reasonable simulation of fake tau production
              \cite{pgstau,Matchev:1999nb}, and we therefore relied on the
              Monte Carlo simulation for fake tau production.
	      HERWIG6.4, which includes the matrix-element correction to 
		the parton shower as described in \cite{Corcella:1999gs},
	      was used to simulate this background.
	      The results of the Monte Carlo simulation were
	      re-scaled to the next-to-leading-order cross section
              calculated using MCFM \cite{Campbell:2002tg}.
\item[Z+jets] In this case the background comes from either the
              leptons produced in the decay of the Z, with additional
              fake leptons as described above, charge mis identification,
              or leptons from the Z itself depending on the channel.
	      Again HERWIG6.4 which includes the matrix-element correction
	      to the initial-state radiation was used to simulate this process.
              As with the W+jets background the results of the simulation
	      were normalized using the  next-to-leading-order cross section
              calculated with MCFM \cite{Campbell:2002tg}. The fake lepton rate
              was calculated as for the W+jets events.
\item[\boldmath{${\rm t}{\rm \bar{t}}$}]
	      The ${\rm t}{\rm \bar{t}}$ background comes the semi-leptonic
	      decay of the top quarks. We simulated this background
              using HERWIG6.4 and used the next-to-leading-order
              with next-to-leading-log resummation calculation
              of \cite{Bonciani:1998vc} to normalize the results.
\end{description}

  There is also a potential background from the production of 
  ${\rm b}{\rm \bar{b}}$ followed by the semi-leptonic decay of the 
  B hadrons produced however this should be negligible after the isolation and
  transverse momentum, $p_T$, cuts on the leptons.

  In all cases we used HERWIG6.4 to simulate the signal.
  The cross sections for $\tilde{\chi}^0_2\tilde{\chi}^\pm_1$
  and $\tilde{\chi}^+_1\tilde{\chi}^-_1$, which are give the
  dominant contribution to the SUSY cross section,
  were normalized to the 
  next-to-leading cross section using the results of \cite{Beenakker:1999xh}.
  The leading-order cross section from HERWIG was used for the remaining
  processes.

  The basic idea of the analysis is to take several different cuts, which 
  should differentiate between the signal and the background, consider
  all possible combinations of the cuts at each point in parameter space
  and optimize the choice of cuts in order to maximize $S\sqrt{B}$,
  where $S$ and $B$ are the number of signal and background events respectively.

  The cuts can be split into two types: those which were applied regardless of 
  the channel and those which are specific to a given channel.

  The channel independent cuts we used were:
\begin{itemize}
\item Five possible cuts on the missing transverse energy, 
      $\not\!\!E_T> \{15,20,25,30\}$ or no cut;
\item An optional veto on the presence of QCD jets with
      $E_T>\{10,15,20,25,30\}$ in the event.
\item A cut on the minimum invariant mass of any opposite sign same flavour electron or
      muon (OSSF)  pairs 
      in the event $m_{\ell\ell}>\{5,10,15,20,25,30,35,40,45,50,55,60\}$ or no cut.
\item A cut on the maximum invariant mass of any OSSF pairs
      in the event \mbox{$m_{\ell\ell}<\{50,60,70,80\}$}, or
      a cut on the minimum difference between the mass of any OSSF pairs and
      the Z mass $|M_{\rm Z}-m_{\ell\ell}|>\{10,15\}$ or no cut.
\item A cut removing any events which have an electron or muon with transverse
      mass $65\,\rm{GeV} \leq m_T \leq 85\,\rm{GeV}$. The transverse mass is given by
      \mbox{$m^2_T=2|p_{T\ell}||p_{T\rm{miss}}|(1-\cos\Delta\phi)$},
      where $p_{T\ell}$ is the transverse momentum of the lepton, $p_{T\rm{miss}}$
      is the missing transverse momentum and $\Delta\phi$ is the azimuthal
      angle between
      the lepton and the missing transverse momentum.
\end{itemize}

  The channel dependent cuts were
\begin{itemize}
\item For the trilepton and like-sign lepton require a central electron or muon
      with $|\eta|\leq1$ and $p_T>{11, 15, 20}$, or no requirement for a central lepton.
\item For the lepton+ tau jet channel a requirement that the tau only
      had one prong\footnote{\ie\ the tau jet only had one charged track.} and
      that the fraction of the tau jet's momentum carried by the charged track 
      was greater than 80\%, as suggested in \cite{Guchait:2001gk}.
\item A cut on the transverse momenta of the leptons 
      was also applied depending on the channel,
      the cuts for the various different channels are given in Table~\ref{tab:ptcut}.
\end{itemize}
  These cuts are the same as used in \cite{Matchev:1999yn,Matchev:1999nb} apart
  from one extra missing transverse energy cut and the cut on the
  momentum of the track in the tau jets.


\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{4}{|c|}{Trilepton channel}\\
\hline
Cut Set        & $p_T(\ell_1)$ & $p_T(\ell_2)$ & $p_T(\ell_3)$ \\
\hline
1       & 11 & 5  & 5 \\
2       & 11 & 7  & 5 \\
3       & 11 & 7  & 7 \\
4       & 11 & 11 & 11\\
5       & 20 & 15 & 10\\
\hline
\multicolumn{4}{|c|}{Like-sign dilepton channel}\\
\hline
 Cut Set        & $p_T(\ell_1)$ & $p_T(\ell_2)$&  \\
\hline
1      & 11 &  9 &\\
2      & 11 & 11 &\\
3      & 13 & 13 & \\
4      & 15 & 15 &\\
5      & 20 & 20 & \\
\hline
\multicolumn{4}{|c|}{Dilepton+hadronic tau channel}\\
\hline
Cut Set  & $p_T(\ell_1)$ & $p_T(\ell_2)$ & $p_T(\tau)$ \\
\hline
1     &  8  &  5  &10\\
2     &  8  &  5  &15\\
3     &  11 &  5  &10\\
4     &  11 &  5  &15\\
5     &  15 & 10  &15\\ 
\hline
\end{tabular}
\end{center}
\caption{Lepton $p_T$ cuts.}
\label{tab:ptcut}
\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\begin{figure}[t]
\centerline{\hbox{\psfig{figure=tan5lll.ps,height=3in,angle=90}},
{\psfig{figure=tan50lll.ps,height=3in,angle=90}}}
\caption{{\it Tevatron 5$\sigma$ reach in the trilepton (3L) channel
in the $M_0-M_{1/2}$ plane for the mSUGRA scenario for small $\tan\beta=5$ 
(left)
and large $\tan\beta=50$ (right)
 and $A_0=0$, $\mu>0$ and $m_t=175$ GeV. The shaded
areas indicated are shown 30$fb^{-1}$ (magenta), 
10$fb^{-1}$ (blue) and 2$fb^{-1}$ (green) and from 
top to bottom respectively. Dashed contours represent the SUSY 
contribution to the muon anomalous magnetic moment (in units
of $10^{-10}$) and  the dotted contour the Higgs boson mass. The
solid contour (only for $\tan\beta=50$) indicates the prediction
for  the $\cbs$.  On the low ($\tan\beta=5$) plot
the solid line indicates where $m_{\tilde{\chi}^\pm_1}=m_{\tilde{\tau}_1}$
and the dot-dashed line $m_{\tilde{\chi}^\pm_1}=m_{\tilde{\nu}_\tau}$. }}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

We first discuss the 3L channel. The production cross section for 
the chargino-neutralino pair, $\tilde{\chi}_1^\pm \tilde{\chi_2}^0$,
scales like $M_{1/2}^{-11/2}$ and thus relevant only in regions
with small $M_{1/2}$, less than 300 GeV. We present our results 
in Fig.~1 and on the $M_0-M_{1/2}$ plane, for two different
values of $\tan\beta$: 5 and 50. Starting with the (3L) events
in the small $\tan\beta$ plot, we observe that Tevatron could 
reach a 5$\sigma$ discovery depending on its luminosity in the regions 
%%%%%%%%%%%%%
\begin{eqnarray}
{\rm Region~I}~(3L, \tan\beta=5) &:& \Biggl \{\begin{array}{c} 
M_0 \lsim 175~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 230~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=2 fb^{-1} \;, \\
M_0 \lsim 190~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 270~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=10 fb^{-1} \;, \\
M_0 \lsim 200~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 290~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=30 fb^{-1} \;, \end{array} \nonumber \\[3mm]
{\rm Region~II}~(3L, \tan\beta=5) &:&\Biggl \{\begin{array}{c} 
M_0 \gsim 480~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 290~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=10 fb^{-1} \;, \\
M_0 \gsim 370~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 320~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=30 fb^{-1} \;, \end{array} \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
In the Region~I the light Higgs boson mass varies~\cite{FeynSSG} from 
$(106 \div 112)\pm 3$ GeV where the error denotes a theoretical error
in the calculation from unknown higher order corrections. 
The current status for the Higgs boson mass bound
from  LEP is~\cite{LEPHiggs} : $M_h \gsim 113.5~~{\rm GeV}$ for
$\sin^2(\beta-\alpha)\simeq 1$, 
where $\alpha$ is the mixing angle in the CP-even Higgs sector.
Thus $\tan\beta=5$ has to be consider as the minimum $\tan\beta$ value
where trilepton events can appear at Tevatron since lower values will
be in conflict with the LEP higgs bound. The Higgs mass
increases as $\tan\beta$ is increasing. For small values of
$\tan\beta$ the supersymmetric contribution to the muon anomalous
magnetic moment ($\dam$) is in general small~\cite{Moroi}. 
However, observing a trilepton
signal in Region I shows a rather amplified value for 
$\dam \gsim 7\times 10^{-10}$
due to the light charginos and sneutrinos. This is not the case
in  Region~II $\dam$ approaches zero. 
 There is a ``null'' $M_0-M_{1/2}$
region at Tevatron where no (3L) can be observe. This ``null'' region
spans the parameter space where $200\lsim M_0 \lsim 370$ GeV and is due
to the fact that off-shell slepton mediated decays  
destructively interfere with 
the gauge boson mediated decays of the chargino-neutralino pair. Since 
$\cbs$ cannot be seen for such small values of $\tan\beta$ the ``null''
area has to be covered by other Tevatron searches 
(squark or slepton searches). In the Region~II, Tevatron 5$\sigma$
reach can be achieved with ${\cal L}=10,30 fb^{-1}$. The light Higgs
boson mass varies between $(106\div 111)\pm 3$ GeV and the supersymmetric
contributions to the muon anomalous magnetic moment are small, 
$\dam \lsim 5\times 10^{-10}$. The CLEO/Belle/ALEPH band for the 
${\cal B}(b\to s \gamma)$ does not set
 any constraint in the low $\tan\beta=5$ region.
In what follows, we adopt the  conservative value\footnote{We follow here 
 the discussion in the footnote 18 of Ref.\cite{Drees}.} 
 $2\times 10^{-4}\lsim {\cal B}(b\to s \gamma)\lsim 5\times 10^{-4}$.

 
We now turn to the discussion on large $\tan\beta=50$ plot in Fig.~1.
The decay mode \bs appears now with a Branching ratio  varying from a
handful of events up to 100 events at Tevatron with ${\cal L}=2 fb^{-1}$.
Fig.1 also verifies  what is seen in Ref.\cite{sakis} : $\cbs$ and 
$\dam$ go hand by hand. 
New experimental results for the anomalous magnetic moment of the muon
are expected to appear as this manuscript is written.
 The up to date BNL
excess is read at 90\% C.L ~\cite{BNL,Narison}
$-4.2\times 10^{-10} \lsim \dam \lsim 41.3 \times 10^{-10}$
with the central value being at $\dam = 18 \times 10^{-10}$~\cite{Narison}.
The Tevatron 5$\sigma$ contours have now moved onto Region~II and 
{\it no} trilepton signature is left in the region with $M_0\lsim 370$ GeV.
However, CDF/D0, even  with  ${\cal L}=2 fb^{-1}$ (Run~IIa)
 will be able to detect  the  decay, \bs . If this will  be true in the
mSUGRA scenario then
$\dam$ has to stay in disagreement with the SM and in positive direction.
The light Higgs boson mass varies on $(110\div 119)\pm 3$ GeV from bottom
to top where   the contour  $m_h=115$ GeV is indicated in Fig.~1. The LEP
preliminary bound, $m_h \gsim 113$ GeV, is evaded (almost) 
everywhere in this plot.
However, this is not true for the $b\to s \gamma$ constraint. Following
the results of Ref.\cite{Drees}, (and accidentally) the area which is below
the 115 Higgs contour is excluded, from the $b\to s \gamma$ constraint. 
This leaves a small piece for the 3L events to appear at Tevatron with
${\cal L}=30 fb^{-1}$ with $M_0\gsim 800$ GeV and 
$M_{1/2} \lsim 230$ GeV. Notice also that this area is banded for large $M_0$
due to the radiative electroweak symmetry breaking requirement. In conclusion,
{\it for large $\tan\beta$ values although 3L events are difficult to discover at
Tevatron, they can be complement by  the rare B-decay \bs .}

Apart from the trilepton (3L) analysis,
we also studied the modes with like sign dileptons (2L), dilepton 
plus tau-jet (2L1T),  lepton plus two tau-jets (1L2T), three tau-jets (3T),
like sign lepton plus tau-jet (1L1T), and two like sign tau-jets (2T) in the
final state. The idea is to identify alternative 
regions of the parameter space 
where the (3L) signal is suppressed. For all these alternative to 3L  channels,
only  the (2L) and (2L1T) survive and only in  the small 
$\tan\beta \simeq 5$ region.
The results are shown in Fig.2. Depending on the luminosity, we observe
a 5$\sigma$ reach at Tevatron, in the mSUGRA region :
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\hbox{\psfig{figure=tan5ll.ps,height=3in,angle=90}},
{\psfig{figure=tan5llt.ps,height=3in,angle=90}}}
\caption{{\it Tevatron reach in the like sign dilepton channel (2L) (left)
and and dilepton plus tau jet channel (2L1T) (right) for low $\tan\beta=5$
and all the other parameters as in Fig.1. We do not observe these two
modes at large $\tan\beta=50$. Contours and shaded areas are given 
like in Fig.1.}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%
\begin{eqnarray}
(2L, \tan\beta=5) &:&\Biggl \{\begin{array}{c} 
M_0 \lsim 120~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 175~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=2 fb^{-1} \;, \\
M_0 \lsim 150~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 200~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=10 fb^{-1} \;, \\
M_0 \lsim 160~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 230~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=30 fb^{-1} \;, \end{array} \nonumber \\[3mm]
(2L1T, \tan\beta=5)  &:& \Biggl \{\begin{array}{c} 
40 \lsim M_0 \lsim 80~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 180~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=2 fb^{-1} \;, \\
50 \lsim M_0 \lsim 100~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 260~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=10 fb^{-1} \;, \\
M_0 \lsim 120 {\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 230~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=30 fb^{-1} \;, \end{array} \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
Comparing with the Region~I of the 3L signal region, we see
that an observation of 2L or/and 2L1T will pin down the 
parameters $M_0-M_{1/2}$.









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



\section*{Acknowledgments}

  We would like to thank M. Kr\"{a}mer and T. Plehn for providing the
  code used in \cite{Beenakker:1999xh} for next-to-leading order electroweak
  gaugino production.
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\bibitem{Baer:1986dv}
H.~Baer, K.~Hagiwara and X.~Tata,
%``Can The Cern P Anti-P Collider Limit Gaugino Masses?,''
Phys.\ Rev.\ Lett.\  {\bf 57} (1986) 294.
%%CITATION = PRLTA,57,294;%%

%\cite{Baer:1986vf}
\bibitem{Baer:1986vf}
H.~Baer, K.~Hagiwara and X.~Tata,
%``Gauginos As A Signal For Supersymmetry At P Anti-P Colliders,''
Phys.\ Rev.\ D {\bf 35} (1987) 1598.
%%CITATION = PHRVA,D35,1598;%%


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

%%%%%%%%%%%%% SAKIS %%%%%%%%%%%%%%%%%%
\bibitem{FeynSSG}
For SUSY and Higgs spectra we use the code {\tt FeynSSG v1.0}.
Details for the code are given in 
S.~Ambrosanio, A.~Dedes, S.~Heinemeyer, S.~Su and G.~Weiglein,
%``Implications of the Higgs boson searches on different soft  SUSY-breaking scenarios,''
Nucl.\ Phys.\ B {\bf 624}, 3 (2002)
[arXiv:hep-ph/0106255], 
%%CITATION = HEP-PH 0106255;%% 
and
A.~Dedes, S.~Heinemeyer,  G.~Weiglein, in G.~Azuelos {\it et al.},
%``The beyond the standard model working group: Summary report,''
arXiv:hep-ph/0204031.
%%CITATION = HEP-PH 0204031;%%



\bibitem{LEPHiggs}
 [LEP Higgs Working Group Collaboration],
``Searches for the neutral Higgs bosons of the MSSM: Preliminary combined  results using LEP data collected at energies up to 209-GeV,''
arXiv:hep-ex/0107030. The result quoted here is preliminary.
%%CITATION = HEP-EX 0107030;%%


\bibitem{Moroi}
T.~Moroi,
Phys.\ Rev.\ D {\bf 53}, 6565 (1996)
[Erratum-ibid.\ D {\bf 56}, 4424 (1996)]
[arXiv:hep-ph/9512396].
%%CITATION = HEP-PH 9512396;%%



\bibitem{Drees}
A.~Djouadi, M.~Drees and J.~L.~Kneur,
%``Constraints on the minimal supergravity model and prospects for SUSY  particle production at future linear e+ e- colliders,''
JHEP {\bf 0108}, 055 (2001)
[arXiv:hep-ph/0107316].
%%CITATION = HEP-PH 0107316;%%

%\cite{Matchev:1999yn}
\bibitem{Matchev:1999yn}
K.~T.~Matchev and D.~M.~Pierce,
%``New backgrounds in trilepton, dilepton and dilepton plus tau jet SUSY  signals at the Tevatron,''
Phys.\ Lett.\ B {\bf 467} (1999) 225
[arXiv:hep-ph/9907505].
%%CITATION = HEP-PH 9907505;%%


%\cite{Matchev:1999nb}
\bibitem{Matchev:1999nb}
K.~T.~Matchev and D.~M.~Pierce,
%``Supersymmetry reach of the Tevatron via trilepton, like-sign dilepton  and dilepton plus tau jet signatures,''
Phys.\ Rev.\ D {\bf 60} (1999) 075004
[arXiv:hep-ph/9904282].
%%CITATION = HEP-PH 9904282;%%

\bibitem{HERWIG64}
G.~Corcella {\it et al.},
%``HERWIG 6: An event generator for hadron emission reactions with  interfering gluons (including supersymmetric processes),''
JHEP {\bf 0101} (2001) 010
[arXiv:hep-ph/0011363];
%%CITATION = HEP-PH 0011363;%%
G.~Corcella {\it et al.},
%``HERWIG 6.4 release note,''
arXiv:hep-ph/0201201;
%%CITATION = HEP-PH 0201201;%%
S.~Moretti, K.~Odagiri, P.~Richardson, M.~H.~Seymour and B.~R.~Webber,
%``Implementation of supersymmetric processes in the HERWIG event  generator,''
JHEP {\bf 0204} (2002) 028
[arXiv:hep-ph/0204123].
%%CITATION = HEP-PH 0204123;%%

%\cite{Richardson:2001df}
\bibitem{Richardson:2001df}
P.~Richardson,
%``Spin correlations in Monte Carlo simulations,''
JHEP {\bf 0111} (2001) 029
[arXiv:hep-ph/0110108].
%%CITATION = HEP-PH 0110108;%%


\bibitem{Jadach:1993hs}
S.~Jadach, Z.~Was, R.~Decker and J.~H.~Kuhn,
%``The tau decay library TAUOLA: Version 2.4,''
Comput.\ Phys.\ Commun.\  {\bf 76} (1993) 361.
%%CITATION = CPHCB,76,361;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{pgstau}
Talk given by K. Mathev at the Higgs and Supersymmetry Conference,
Gainesville FL, March 7-11,1999.


\bibitem{Corcella:1999gs}
G.~Corcella and M.~H.~Seymour,
%``Initial state radiation in simulations of vector boson production at  hadron colliders,''
Nucl.\ Phys.\ B {\bf 565} (2000) 227
[arXiv:hep-ph/9908388].
%%CITATION = HEP-PH 9908388;%%

\bibitem{Campbell:1999ah}
J.~M.~Campbell and R.~K.~Ellis,
%``An update on vector boson pair production at hadron colliders,''
Phys.\ Rev.\ D {\bf 60} (1999) 113006
[arXiv:hep-ph/9905386].
%%CITATION = HEP-PH 9905386;%%
\bibitem{Campbell:2002tg}
J.~Campbell and R.~K.~Ellis,
%``Next-to-leading order corrections to W + 2jet and Z + 2jet production  at hadron colliders,''
arXiv:hep-ph/0202176.
%%CITATION = HEP-PH 0202176;%%
\bibitem{Bonciani:1998vc}
R.~Bonciani, S.~Catani, M.~L.~Mangano and P.~Nason,
%``NLL resummation of the heavy-quark hadroproduction cross-section,''
Nucl.\ Phys.\ B {\bf 529} (1998) 424
[arXiv:hep-ph/9801375].
%%CITATION = HEP-PH 9801375;%%

\bibitem{Pukhov:1999gg}
A.~Pukhov {\it et al.},
%``CompHEP: A package for evaluation of Feynman diagrams and integration  over multi-particle phase space. User's manual for version 33,''
arXiv:hep-ph/9908288.
%%CITATION = HEP-PH 9908288;%%



\bibitem{pythia}
T.~Sjostrand,
%``High-energy physics event generation with PYTHIA 5.7 and JETSET 7.4,''
Comput.\ Phys.\ Commun.\  {\bf 82} (1994) 74;
%%CITATION = CPHCB,82,74;%%
S.~Mrenna,
%``SPYTHIA, a supersymmetric extension of PYTHIA 5.7,''
Comput.\ Phys.\ Commun.\  {\bf 101} (1997) 232
[arXiv:hep-ph/9609360].
%%CITATION = HEP-PH 9609360;%%
\bibitem{pgs}
PGS detector simulation package {\bf 
http://www.physics.rutgers.edu/~jconway/soft/pgs/pgs.html.}


%\cite{Beenakker:1999xh}
\bibitem{Beenakker:1999xh}
W.~Beenakker, M.~Klasen, M.~Kr\"{a}mer, T.~Plehn, M.~Spira and P.~M.~Zerwas,
%``The production of charginos/neutralinos and sleptons at hadron  colliders,''
Phys.\ Rev.\ Lett.\  {\bf 83} (1999) 3780
[arXiv:hep-ph/9906298].
%%CITATION = HEP-PH 9906298;%%
\bibitem{Guchait:2001gk}
M.~Guchait and D.~P.~Roy,
%``Using tau polarization to sharpen up the SUGRA signal at Tevatron,''
arXiv:hep-ph/0109096.
%%CITATION = HEP-PH 0109096;%%


\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}