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Trilepton Events and $B_s\ra\mu^+\mu^-$: 
No-lose for mSUGRA at the Tevatron?
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hep-ph/yymmnnn\hfill
{\rm FERMILAB-Pub-02/129-T}\\
\mbox{~}\hfill
{\rm  June 2002} \\

%\vspace*{\fill}
\vspace*{1cm}

\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
%\boldmath
%  {\LARGE \bf Complementary mSUGRA signatures at the Tevatron : \\[3mm]
%   Trilepton Events and $B_s\ra\mu^+\mu^-$ }
%\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 $^c$ Cavendish Laboratory and DAMTP, University of Cambridge, UK} \\[4mm]
\end{center}



\vspace*{1cm}

\begin{abstract}{\small\noindent
    We study the Tevatron search potential for minimal supergravity
    (mSUGRA) and find two observables which 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 the universal gaugino mass, $M_{1/2}$ below 250 GeV
    and for common squark masses, $M_0$ other than 200-370 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.  For this study we perform an
    updated analysis of the trilepton signature which includes
    the full set  of the decay matrix elements.  This leads
    to a new, more promising search reach for the Tevatron.}


\end{abstract}

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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 {\it scalar
  potential} after supersymmetry breaking \cite{Soni:1983rm} contains
the following free parameters:\footnote{If R-parity is conserved there
  is only one bilinear 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 superpotential splits
into a hidden-sector and a visible sector part \cite{Nilles:1983ge}.
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 a further universal soft
supersymmetry breaking parameter: the common gaugino mass, $M_{1/2}$.
In the Higgs sector there is an additional parameter: $\tan\beta$, the
ratio of the vacuum expectation values of the two neutral, CP-even
Higgs fields. We thus have a total of 5 parameters in mSUGRA
\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 {\it 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 $M_X$.  In our numerical analysis, $\tan\beta$ and
$\sgn\mu$ are fixed at the weak scale.


The mSUGRA model with universal parameters is a simple, well motivated
model for the low-energy supersymmetric spectrum.  If an observable in
this model deviates substantially from the standard model (SM), 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) an indirect signature via a deviation from a SM prediction or 2) a
direct detection of a supersymmetric particle.  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{Dicus:1983cb,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.

In the following we present the actual status of leptonic $B_s$ meson
decays.  We then proceed and perform a brand new analysis of the
multilepton signatures in mSUGRA.  We confirm the results of Matchev
and Pierce \cite{Matchev:1999yn,Matchev:1999nb}.  We then go beyond
this work to include the full matrix elements for the supersymmetric
decays, which were not included in ISASUSY at the time.  This leads to
a more promising discovery reach for Run II of the Tevatron.
 




%{\bf (O$_1$) \underline{The $B$-meson decay \bs}}  
%{\bf \underline{1.~The $B$-meson decay \bs}}  
\section{The 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.  This 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 range. 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[t]\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Channel & Expt. & Bound (90\% CL) & SM prediction \cite{Buchalla}\\ \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{{\it The experimental status and the SM predictions for the 
  branching ratios ${\cal B}(B_{s,d}\to \ell^+\ell^{-}$). The error in
  the $B_s$ branching ratios mainly originates 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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 in 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~\ref{tab} 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~\ref{tab} as well.

From now on we restrict ourselves to the decay mode \bs.  In the SM,
the decay mode \bs is 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 at 95\% \cite{CDFbmumu}  
\begin{equation}
\cbs< 2.6 \times 10^{-6}\,,
\end{equation}
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 ${\cal B} (B \to
\ell^+ \ell^-)$.  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,Huang}. In the MSSM, however,
the amplitude involves three powers of $\tan \beta$, so that 
\begin{equation}
\cbs\propto \tan^6\beta. 
\end{equation}
This leads to an enhancement over the SM value by up to three orders
of magnitude.  This holds even in the flavour-universal MSSM, where
flavour changes occur in weak vertices only and come with the usual
CKM elements \cite{Babu,Huang,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 the
decay
\begin{equation}
B\ra (\pi,K) \ell^+\ell^-,
\end{equation}
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. In our
numerical analysis of $\cbs$ we use the formulae of \cite{Urban}
supplemented with an all-order resummation of $\tan\beta$-enhanced 
supersymmetric QCD corrections \cite{cgnw}. Other QCD corrections
(neglecting gluino contributions) were calculated in \cite{Buras}
and found to be small, if the renormalization scale is chosen close to
the masses of the SUSY particles propagating in the loop. 

%\vspace*{0.5cm}

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

{\bf the review here , Herbi ??}

\subsection{Numerical Analysis}
In general our strategy for the analysis of the multilepton signatures
closely follows that of Matchev and Pierce
\cite{Matchev:1999yn,Matchev:1999nb}. The major differences between
these studies are:
\begin{itemize}
\item While Matchev and Pierce \cite{Matchev:1999yn,Matchev:1999nb}
  used leading-order (in QCD) cross sections for both the signal and
  background processes we made use of next-to-leading order
  calculations for the background and most important signal processes.
    
\item In Refs.~\cite{Matchev:1999yn,Matchev:1999nb} the authors used
  ISAJET 7.42 for the simulation of the signal. For the background
  they used PYTHIA 6.115 \cite{pythia}. In Ref.\cite{Matchev:1999nb}
  they used a combination of PYTHIA and COMHEP \cite{Pukhov:1999gg}
  for the $WZ$ background.\footnote{This brief summary is based on a
    private communication with Konstantin Matchev and differs slightly
    from the presentation in the papers.}  Instead, 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 momentum and spin 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}

% Konstantin's change here
% While the analysis of \cite{Matchev:1999yn,Matchev:1999nb} used the SHW detector
%  simulation 

\noindent In our simulation we make use of the PGS \cite{pgs} detector
simulation.  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 is less than 2~GeV.
\item The missing transverse energy, \met, which is calculated in PGS using the
      calorimeter cells, iss 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
  Ref.~\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 Ref.~\cite{Matchev:1999nb}.

\subsection{Background and Signal}
The major backgrounds to multilepton production are:
\begin{description}
\item[\underline{WW}:] Here the background is from the production of
  leptons in the cascade 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[\underline{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[\underline{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[\underline{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
    Ref.~\cite{Matchev:1999nb} in order to estimate this rate from the
    Run~I 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 Ref.~\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[\underline{Z+jets}:] In this case the background comes from
    either the leptons produced in the cascade 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[\underline{\boldmath{${\rm t}{\rm \bar{t}}$}}:] The background
    comes from 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
    Ref.~\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 produced
  $b$-flavoured hadrons. However, this should be negligible after the
  isolation and cuts on the transverse momentum $p_T$ of 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 give the dominant contribution to
the SUSY cross section, were normalized to the next-to-leading order
cross section using the results of \cite{Beenakker:1999xh}.  The
leading-order cross section from HERWIG was used for the remaining
processes.



\subsection{Cuts}

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: 
  channel-independent cuts  and those which are specific to a given channel.


  The channel independent cuts we used were:
\begin{enumerate}
  
\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 pair 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{enumerate}

  The channel dependent cuts were:
\begin{enumerate}
  
\item For the trilepton and like-sign lepton signature we 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 we require that the tau decay
  only has 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 is greater than 80\%, as suggested in
  \cite{Guchait:2001gk}.
  
\item A cut on the transverse momenta of the leptons is also applied
  depending on the channel, the cuts for the various different
  channels are given in Table~\ref{tab:ptcut}.

\end{enumerate}
These cuts are the same as used in
Ref.~\cite{Matchev:1999yn,Matchev:1999nb} {\it apart} from one extra missing
transverse energy cut and the cut on the momentum of the track in the
tau jets.


\begin{table}[tbh]
\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{{\it 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 in 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 indicate the $5\sigma$
reach for an integrated luminosity of 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
contours are iso-mass contours of the lightest neutral Higgs
boson. The solid contour (only for $\tan\beta=50$) indicates the
prediction for the branching ratio $\cbs$.  In the left
($\tan\beta=5$) plot the solid line indicates where
$m_{\tilde{\chi}^\pm_1}=m_{\tilde{\tau}_1}$ and the dot-dashed line
$m_{\tilde{\chi}^\pm_1}=m_{\tilde{\nu}_\tau}$.  The large red regions
are excluded by theory and experiment \cite{sakis}.}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Trilepton (3L) Channel}



We first discuss the trilepton (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 is thus only relevant in regions
with small $M_{1/2}$, {\it i.e.} $M_{1/2}\lsim 300$ GeV. We present
our results in Fig.~1 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 190~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=10 fb^{-1} \;, \\
M_0 \gsim 360~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 220~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=30 fb^{-1} \;, \end{array} \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
Region~I is larger than in Ref.~\cite{Matchev:1999yn,Matchev:1999nb}
due to the extra missing transverse energy cut we have made.
Region~II is larger than in \cite{Matchev:1999yn,Matchev:1999nb} due
to use of the matrix elements for the decay distributions, which are
included in HERWIG.  This is important since at the boundary the
neutralino mass difference is getting closer to $M_Z$.  Overall, we
thus obtain a more promising trilepton search reach at Run~II of the
Tevatron than in previous studies.  This is one of the main
conclusions of this paper. Although we present our analysis in terms
of the fundamental parameters of mSUGRA $M_0,M_{1/2},\tan\beta$, 
there exist mass sum rules~\cite{Martin}
and approximately we can  translate these parameters 
onto the physical ones (up to a 10\% accuracy),
%%%%%%%%%%%%
\begin{eqnarray}
2 m_{\tilde{\chi}_1^0} & \simeq & m_{\tilde{\chi}_2^0} 
\simeq m_{\tilde{\chi}_1^\pm} \simeq \frac{1}{3} m_{\tilde{g}} \simeq
0.8 M_{1/2} \;, \nonumber \\
m_{\tilde{q}} & \simeq & \sqrt{M_0^2+4 M_{1/2}^2} \;,
\end{eqnarray}
%%%%%%%%%%%%%%%
where $m_{\tilde{\chi}_i^0}$ are the neutralinos with the 
lightest being the LSP, $ m_{\tilde{\chi}_i^\pm}$ the charginos,
$m_{\tilde{g}}$ the gluino and $m_{\tilde{q}}$ the heavy first 
and second generation squark masses. Thus the Tevatron reach is 
relevant for a light chargino  and next to lightest neutralino
masses  below 200 GeV and (indirectly)
for a gluino mass below 600 GeV.

The calculation of the light Higgs boson mass strictly follows
the procedure described in   Ref.\cite{FeynHiggs}.
In Region~I the light Higgs boson mass varies from
$(106 \div 112)\pm 3$ GeV where the error denotes a theoretical error
in the calculation~\cite{Sven}.  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 considered as the minimum $\tan\beta$ value
where trilepton events can appear at the Tevatron since lower values
will be in conflict with the LEP Higgs bound. The Higgs mass increases
with $\tan\beta$.

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
will imply an at least moderate enhancement of $a_\mu$ by 
\begin{equation}
\dam \gsim 7\times 10^{-10}\;,
\end{equation}
due to the light charginos and sneutrinos.  This is not the case in
Region~II, where $\,\dam$ approaches zero.  

There is a ``null'' $M_0-M_{1/2}$ region at the Tevatron where no (3L)
can be observed.  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 of the gauginos destructively interfere with
the gauge boson mediated decays.  Since the decay $B_s\ra\mu^+\mu^-$
cannot be seen for such small values of $\tan\beta$ the ``null'' area
has to be covered by other Tevatron searches 
(possibly Higgs searches~\cite{Carena}).
  In Region~II, the 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, 
\begin{equation}
\dam \lsim 5\times 10^{-10}.
\end{equation}  


The CLEO/Belle/ALEPH allowed band for the branching ratio ${\cal
  B}(b\to s \gamma)$ 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 no.~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 of the large $\tan\beta=50$ plot in
Fig.~1.  
The new Region~I (low $M_0$)  is not there any more (see below).
The new Region~II is given by 
\begin{eqnarray}
{\rm Region~II}~(3L, \tan\beta=50) &:&\Biggl \{\begin{array}{c} 
M_0 \gsim 480~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 185~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=10 fb^{-1} \;, \\
M_0 \gsim 360~{\rm GeV} \;\;\;,\;\;\; M_{1/2} \lsim 230~{\rm GeV} 
\;\;\;,\;\;\; {\cal L}=30 fb^{-1} \;. \end{array} \nonumber
\end{eqnarray}
As before, this is also somewhat larger than in
Ref.~\cite{Matchev:1999yn,Matchev:1999nb} due to the added decay
matrix elements.  All the same, the tri-lepton search only covers a
small range of the $M_0-M_{1/2}$-plane.  There is a further strong
constraint.  The light Higgs boson mass varies in the range $(110\div
119)\pm 3$ GeV from bottom to top where the contour $m_h=115$ GeV is
indicated in Fig.~1 as a dotted line.  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.  Using the
results in Ref.\cite{Drees} we find that the excluded region due to
the $b\to s \gamma$ constraint (accidentally) co\"{\i}ncides almost
exactly with the area below the $m_h=115$ GeV Higgs mass contour.
We thus do not redraw it. 
This leaves only a small area for the 3L events to appear at the
Tevatron
\begin{eqnarray}
{\rm Region~II'}~(3L, \tan\beta=50) &:& 
M_0 \gsim 800~{\rm GeV}\,; \;\;M_{1/2} \sim 190\div230~{\rm GeV},
\;\; {\cal L}=30 fb^{-1} \;.  \nonumber
\end{eqnarray}
Notice also that this area is bounded for large $M_0$ due to the
radiative electroweak symmetry breaking requirement.



By contrast, the decay mode \bs is very promising for $\tan\beta=50$.
The branching ratio varies from a handful of events for large $M_0$ up
to 100 events for low $M_0$ at the Tevatron already with ${\cal L}=2
fb^{-1}$.  This very nicely complements the trilepton search and is
the second main result of our paper: the tri-lepton search covers most
of the low $\tan\beta$ region, the branching ratio $\cbs$ covers most
of the high $\tan\beta$ region.

Fig.1 also verifies what had been seen in Ref.\cite{sakis}: the
branching $\cbs$ and the anomalous magnetic moment of the muon,
$\dam$, go hand-in-hand, {\it i.e.} there is a strong correlation.
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 $-4.2\times 10^{-10} \lsim \dam \lsim 41.3 \times
10^{-10}$, at 90\% C.L ~\cite{BNL,Narison} with the central value
$\dam = 18 \times 10^{-10}$~\cite{Narison}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\begin{figure}[t]
\centerline{\hbox{\psfig{figure=m150lll.ps,height=3in,angle=90}}
{\psfig{figure=m300lll.ps,height=3in,angle=90}}}
\caption{{\it Tevatron 5$\sigma$ reach in the trilepton (3L) channel
in the $M_{1/2}-\tan\beta$ 
plane for the mSUGRA scenario with  $M_0=150$ GeV 
(left)
and  $M_0=300$ GeV (right)
 and $A_0=0$, $\mu>0$ and $m_t=175$ GeV. The shaded
areas indicate integrated luminosities of 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 contours the Higgs boson mass. The
solid contours indicate the prediction
for  $\cbs$.  }}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our next step is to simulate the 3L events on $M_{1/2}-\tan\beta$ 
plane for different values of $M_0$. Our results are depicted in
 Fig.2. In the case of $M_0=150$ GeV, the Tevatron with ${\cal L}=30 fb^{-1}$
can reach 5$\sigma$ for the 3L events, only in the region with
$\tan\beta \lsim 30$ and $M_{1/2}\lsim 270$ GeV with always 
a Higgs mass below 114 GeV. Tevatron with 
${\cal L}=2 fb^{-1}$ can observe the 3L events in a region with 
small $\tan\beta \le 5$ and $M_{1/2}\le 210$ GeV. In this region 
the maximum of the Higgs mass obtained is 109 GeV. As $\tan\beta$ increases,
the stau mass dicreases and  the gaugino  decays 
are mediated mostly by staus rather than sleptons. Gradually the  3L
event rate is decreasing as we increase $\tan\beta$.  
When 3L events are no longer possible the Branching ratio $\cbs$ opens up 
with an observable rate of up to 30 events with ${\cal L}=2 fb^{-1}$
 at Tevatron Run II.   
Moving to higher values of $M_0=300$ GeV (see Fig.2), the 3L
events dissapear for the reason   we have already explain in discussing
 the Fig.1 . However,
a large portion of the parameter space with $\tan\beta \gsim 30$ 
 is still accesible  to  \bs ~ searches. The parameter space with
$\tan\beta$ values below 30 has to be covered by other searches, most
probably Higgs searches (squarks and gluinos are heavy to be directly 
discovered at Tevatron in the mSUGRA scenario~\cite{mSUGRATEVREP}.)
The two cases $M_0=150$ and $M_0=300$ has to be considered respectively as
the ``optimistic'' and the ``nightmare'' benchamrk scenarios
 for the Tevatron Run II.
For a complete analysis for the \bs ~mode and other observables
on $M_{1/2}-\tan\beta$ plane  the reader 
is refered to Refs.~\cite{sakis,Tanaka}. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 
as in Fig.1.}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 



\subsection{Other than 3L Channels}
Apart from the trilepton (3L) analysis, we also study 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 channels
alternative to 3L, only the (2L) and (2L1T) survive and only in the
small $\tan\beta \simeq 5$ region.  The results are shown in Fig.3.
Depending on the luminosity, we observe a 5$\sigma$ reach at the
Tevatron, in the mSUGRA region :
%%%%%%%%%%%%%%%%%%%%%%

 %%%%%%%%%%%%%
\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 $M_0-M_{1/2}$. In the $M_{1/2}-\tan\beta$ plane and
for the optimistic benchamark $M_0=150$ GeV
2L events appear only with ${\cal L}=10 (30) fb^{-1}$ in 
$M_{1/2} \lsim 190, \tan\beta \lsim 3$ 
($M_{1/2} \lsim 220, \tan\beta \lsim 6$).



\section{Conclusions}
We have presented two complementary observables to study the parameter
space of minimal supergravity at Run-II of the Fermilab Tevatron: the 
production of charginos or neutralinos decaying into trilepton final
states and the branching ratio of the rare decay \bs. The simulation 
of the trilepton events has been improved over previous results in
various respects. In particular next-to-leading QCD corrections have
been incorporated where available and the full momentum and spin
correlations between production and decay of the supersymmetric
particles have been taken into account. We have also presented modified cuts.
With these enhancements we find
the Tevatron reach for trilepton events increased. The two studied
observables reveal very useful complementary information on the mSUGRA
parameter space, covering the regions of small and large $\tan\beta$,
respectively. For example, for $M_0=150\,\gev$, $A_0=0$ and $\mu>0$ we
find that a Run-IIb with 30 $fb^{-1}$ will probe values of
$M_{1/2}$ in the range 
\begin{eqnarray}
M_{1/2} & \lsim & 250\,\gev \; - \; 70 \, \gev \times \frac{\tan \beta}{23}
\no ,  
\end{eqnarray}
which decreases with $\tan \beta$. On the contrary, if the Tevatron 
can measure $\cbs$ down to $10^{-8}$ (which requires less
than 30 $fb^{-1}$ of integrated luminosity), this will probe the 
region with $\tan \beta > 32$ and the range 
\begin{eqnarray}
M_{1/2} & \lsim &  19 \, \gev \times \tan \beta \; 
    - \; 260\,\gev 
\no   
\end{eqnarray}
for the chosen values of $M_0$, $A_0$ and $\mu$. 

This work should be considered as a first attempt of 
relating ``direct'' SUSY searches such as the 3L events
with an ``indirect'' rare B-decay, such as \bs
at the Tevatron. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\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. We would like to thank F. Kr\"uger, J. Urban 
  for illuminating discussions on the \bs calculation. We also like
  to thank K. Matchev for his valuable comments on the trilepton analysis.
  A.D would like to thank T. Kamon for discussions on the Tevatron 3L events,
  P.~Slavich and S.~Heinemeyer for useful discussions on the updates of the
  Higgs boson mass calculations.  Preliminary results
  of this analysis have been presented by (U.N) and  (A.D) at 
  the Fermilab Users' meeting  and SUSY02 conference respectively.
  A.D. would like to acknowledge financial support from the Network
  RTN European Program HPRN-CT-2000-00148 ``Physics Across the Present
  Energy Frontier: Probing the Origin of Mass''.
\begin{thebibliography}{99}




\bibitem{mSUGRA} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H.~P.~Nilles,
%``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;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Soni:1983rm}
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;%%

%%%%%%%%%%%%%%%%%%%%%%%%%5
\bibitem{Cremmer}
E.~Cremmer, B.~Julia, J.~Scherk, P.~van Nieuwenhuizen, S.~Ferrara and L.~Girardello,
%``Superhiggs Effect In Supergravity With General Scalar Interactions,''
Phys.\ Lett.\ B {\bf 79}, 231 (1978);
%%CITATION = PHLTA,B79,231;%%
E.~Cremmer, B.~Julia, J.~Scherk, S.~Ferrara, L.~Girardello and P.~van Nieuwenhuizen,
%``Spontaneous Symmetry Breaking And Higgs Effect In Supergravity Without 
%Cosmological Constant,''
Nucl.\ Phys.\ B {\bf 147}, 105 (1979).
%%CITATION = NUPHA,B147,105;%%

\bibitem{Girardello:1981wz}
L.~Girardello and M.~T.~Grisaru,
%``Soft Breaking Of Supersymmetry,''
Nucl.\ Phys.\ B {\bf 194} (1982) 65.
%%CITATION = NUPHA,B194,65;%%

%\cite{Nilles:1983ge}
\bibitem{Nilles:1983ge}
H.~P.~Nilles,
%``Supersymmetry, Supergravity And Particle Physics,''
Phys.\ Rept.\  {\bf 110} (1984) 1.
%%CITATION = PRPLC,110,1;%%



%\cite{Ibanez:fr}
\bibitem{Ibanez:fr}
L.~E.~Ibanez and G.~G.~Ross,
%``SU(2)-L X U(1) Symmetry Breaking As A Radiative Effect Of 
%Supersymmetry Breaking In Guts,''
Phys.\ Lett.\ B {\bf 110} (1982) 215.
%%CITATION = PHLTA,B110,215;%%

\bibitem{sakis}
A.~Dedes, H.~K.~Dreiner and U.~Nierste,
%``Correlation of B/s $\to$ mu+ mu- and (g-2)(mu) in minimal supergravity,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 251804 (2001)
[arXiv:hep-ph/0108037].
%%CITATION = HEP-PH 0108037;%%

%\cite{Dicus:1983cb}
\bibitem{Dicus:1983cb}
D.~A.~Dicus, S.~Nandi and X.~Tata,
%``W Decay In Supergravity Guts,''
Phys.\ Lett.\ B {\bf 129} (1983) 451
[Erratum-ibid.\ B {\bf 145} (1984) 448].
%%CITATION = PHLTA,B129,451;%%

%\cite{Baer:1985at}
\bibitem{Baer:1985at}
H.~Baer and X.~Tata,
Phys.\ Lett.\ B {\bf 155} (1985) 278.
%%CITATION = PHLTA,B155,278;%%


%\cite{Baer:1986dv}
\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;%%


\bibitem{L3}
W.~Adam {\it et al.}  [DELPHI Collaboration],
%``Study of rare b decays with the DELPHI detector at LEP,''
Z.\ Phys.\ C {\bf 72}, 207 (1996).
%%CITATION = ZEPYA,C72,207;%%

\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{Nardi} 
The bound on ${\cal B} (B\to \tau^+\tau^-)$ from LEP~I 
is a byproduct of the search for $B^-\to \tau^- \bar{\nu}$ pointed out in: 
Y.~Grossman, Z.~Ligeti and E.~Nardi,
%``B $\to$ tau+ tau- (X) decays: First constraints and
% phenomenological implications,''
Phys.\ Rev.\ D {\bf 55}, 2768 (1997) [arXiv:hep-ph/9607473].
%%CITATION = HEP-PH 9607473;%%
This bound can be improved with the existing LEP data. We would like
to thank M. Kobel and N. Wermes for discussions on this matter.



\bibitem{CLEO}
T.~Bergfeld {\it et al.}  [CLEO Collaboration],
%``Search for decays of B0 mesons into pairs of leptons: B0 $\to$ e+ e-, B0  
%$\to$ mu+ mu- and B0 $\to$ e+- mu-+,''
Phys.\ Rev.\ D {\bf 62}, 091102 (2000)
[arXiv:hep-ex/0007042].
%%CITATION = HEP-EX 0007042;%%

\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{fbs}
C.~Bernard,
%``Heavy quark physics on the lattice,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 94}, 159 (2001)
[hep-lat/0011064].
%%CITATION = HEP-LAT 0011064;%%


\bibitem{ckmfit}
\emph{CKM fitter}\ group,\\ 
{\tt http://www.slac.stanford.edu/\~{}laplace/ckmfitter/ckm\_wellcome.html}



\bibitem{fnalrep}
K.~Anikeev {\it et al.},
%``B physics at the Tevatron: Run II and beyond,''
arXiv:hep-ph/0201071.
%%CITATION = HEP-PH 0201071;%%


\bibitem{Tanaka}
R.~Arnowitt, B.~Dutta, T.~Kamon and M.~Tanaka,
%``Detection of B/s $\to$ mu+ mu- at the Tevatron run II 
%and constraints on  the SUSY parameter space,''
arXiv:hep-ph/0203069.
%%CITATION = HEP-PH 0203069;%%



\bibitem{LHCb} For a review on B physics at the LHC see for
  instance,
P.~Ball {\it et al.},
%``B decays at the LHC,''
hep-ph/0003238.
%%CITATION = HEP-PH 0003238;%%

%%%%%%%%%%%%%%%%%%%%%%B->mu mu papers%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibitem{Inami}
% T.~Inami and C.~S.~Lim,
% %``Effects Of Superheavy Quarks And Leptons In Low-Energy 
%Weak Processes K(L) $\to$ Mu Anti-Mu, K+ $\to$ Pi+ Neutrino Anti-Neutrino And K0 <--->
% %Anti-K0,''
% Prog.\ Theor.\ Phys.\  {\bf 65}, 297 (1981)
% [Erratum-ibid.\  {\bf 65}, 1772 (1981)].

% \bibitem{Krawczyk}
% B.~Grzadkowski and P.~Krawczyk,
% %``Higgs Particle Effects In Flavor Changing Transitions,''
% Z.\ Phys.\ C {\bf 18}, 43 (1983);
% %%CITATION = ZEPYA,C18,43;%%
% P.~Krawczyk,
% %``Flavor Changing Yukawa Coupling Of The Standard Higgs Boson: 
%Effects Of The External Quark Masses,''
% Z.\ Phys.\ C {\bf 44}, 509 (1989).
% %%CITATION = ZEPYA,C44,509;%%





\bibitem{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{Huang}
C.~S.~Huang, W.~Liao, Q.~S.~Yan and S.~H.~Zhu,
%``B/s $\to$ l+ l- in a general 2HDM and MSSM,''
Phys.\ Rev.\ D {\bf 63}, 114021 (2001)
[Erratum-ibid.\ D {\bf 64}, 059902 (2001)]
[arXiv:hep-ph/0006250];
%%CITATION = HEP-PH 0006250;%%
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{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]; 
G.~Isidori and A.~Retico,
%``Scalar flavour-changing neutral currents in the large-tan(beta) limit,''
JHEP {\bf 0111} (2001) 001
[arXiv:hep-ph/0110121].
%%CITATION = HEP-PH 9909476;%%


\bibitem{Urban}
C.~Bobeth, T.~Ewerth, F.~Kr\"uger 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-,''
Phys.\ Rev.\ D {\bf 64}, 074014 (2001)
[arXiv:hep-ph/0104284].
%%CITATION = HEP-PH 0104284;%%


\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{Olive}
D.~A.~Demir, K.~A.~Olive and M.~B.~Voloshin,
%``The forward-backward asymmetry of B $\to$ (pi, K) l+ l-: Supersymmetry at  work,''
arXiv:hep-ph/0204119.
%%CITATION = HEP-PH 0204119;%%





\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%%


\bibitem{Narison}
M.~Knecht and A.~Nyffeler,
%``Hadronic light-by-light corrections to the muon g-2: 
%The pion-pole  contribution,''
arXiv:hep-ph/0111058.
%%CITATION = HEP-PH 0111058;%%
M.~Knecht, A.~Nyffeler, M.~Perrottet and E.~De Rafael,
Phys.\ Rev.\ Lett.\  {\bf 88}, 071802 (2002)
[arXiv:hep-ph/0111059].
%%CITATION = HEP-PH 0111059;%%
The bound quoted in the text is taken from
S.~Narison,
%``Muon and tau anomalies updated,''
Phys.\ Lett.\ B {\bf 513}, 53 (2001)
[Erratum-ibid.\ B {\bf 526}, 414 (2002)]
[arXiv:hep-ph/0103199].
%%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;%%

\bibitem{cgnw}
L.~J.~Hall, R.~Rattazzi and U.~Sarid,
%``The Top quark mass in supersymmetric SO(10) unification,''
Phys.\ Rev.\ D {\bf 50} (1994) 7048
[arXiv:hep-ph/9306309];
%%CITATION = HEP-PH 9306309;%%
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} (2000) 88 [arXiv:hep-ph/9912516].
%%CITATION = HEP-PH 9912516;%%

\bibitem{Buras}
C.~Bobeth, A.~J.~Buras, F.~Kruger and J.~Urban,
%``QCD corrections to anti-B $\to$ X/d,s nu anti-nu, anti-B/d,s $\to$ l+ l-,  K $\to$ pi nu anti-nu and K(L) $\to$ mu+ mu- in the MSSM,''
Nucl.\ Phys.\ B {\bf 630}, 87 (2002)
[arXiv:hep-ph/0112305].
%%CITATION = HEP-PH 0112305;%%


%\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{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{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{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.~K\"uhn,
%``The tau decay library TAUOLA: Version 2.4,''
Comput.\ Phys.\ Commun.\  {\bf 76} (1993) 361.
%%CITATION = CPHCB,76,361;%%


\bibitem{pgs}
PGS detector simulation package {\bf 
http://www.physics.rutgers.edu/~jconway/soft/pgs/pgs.html.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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{pgstau}
Talk given by K. Matchev at the  conference \emph{Higgs and Supersymmetry},
Gainesville, USA, 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: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;%%

%\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;%%





\bibitem{Martin}
S.~P.~Martin and P.~Ramond,
%``Sparticle spectrum constraints,''
Phys.\ Rev.\ D {\bf 48}, 5365 (1993)
[arXiv:hep-ph/9306314].
%%CITATION = HEP-PH 9306314;%%


\bibitem{mSUGRATEVREP}
S.~Abel {\it et al.}  [SUGRA Working Group Collaboration],
%``Report of the SUGRA working group for run II of the Tevatron,''
arXiv:hep-ph/0003154.
%%CITATION = HEP-PH 0003154;%%


%%%%%%%%%%%%% SAKIS %%%%%%%%%%%%%%%%%%
\bibitem{FeynHiggs}
For the Higgs spectrum we use the code {\tt FeynHiggs v1.2} in
M.~Frank, S.~Heinemeyer, W.~Hollik and G.~Weiglein,
%``FeynHiggs1.2: Hybrid MS-bar / on-shell renormalization for the CP-even  
%Higgs boson sector in the MSSM,''
arXiv:hep-ph/0202166 and references therein. 
 The two loop bottom Yukawa corrections to the Higgs mass have been 
recently calculated in  
 A.~Brignole, G.~Degrassi, P.~Slavich and F.~Zwirner,
%``On the two-loop sbottom corrections to the neutral Higgs boson masses  in the MSSM,''
arXiv:hep-ph/0206101 .
These corrections are  negligible for our choice of the sign of the Higgsino
mixing  parameter. 
%%CITATION = HEP-PH 0206101;%%
%%CITATION = HEP-PH 0202166;%%

%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{Sven}
S.~Heinemeyer and G.~Weiglein,
%``Higgs production and decay in the MSSM: Status and perspectives,''
arXiv:hep-ph/0102117.
%%CITATION = HEP-PH 0102117;%%


\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{Carena}
M.~Carena {\it et al.},
%``Report of the Tevatron Higgs working group,''
arXiv:hep-ph/0010338.
%%CITATION = HEP-PH 0010338;%%


\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;%%




\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}