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\begin{document}

\title{Updated constraints on the minimal supergravity model}

\author{
H. Baer, C. Bal\'azs\footnote{Talk given by C. Bal\'azs at SUSY02.}, A. Belyaev
{\it (Department of Physics, Florida State University)} \\
%Tallahassee, FL, USA 32306\\
%E-mail: \email{baer@hep.fsu.edu}, \email{balazs@hep.fsu.edu},
%        \email{belyaev@hep.fsu.edu}}
J.~K. Mizukoshi
{\it (Instituto de F\'{\i}sica, Universidade de S\~ao Paulo)} \\
%C.P.\ 66318, 05315--970, S\~ao Paulo, Brazil. \\
%E-mail: \email{mizuka@fma.if.usp.br}} 
X. Tata, Y. Wang 
{\it (Department of Physics and Astronomy, University of Hawaii)} \\
%Honolulu, HI 96822, USA    \\
%E-mail: \email{tata@phys.hawaii.edu}, \email{yili@phys.hawaii.edu}
}
\date{}
\maketitle
%\thispagestyle{empty}

\begin{abstract}
Recently, refinements have been made on both the
theoretical and experimental determinations of 
{\it i}.) the mass of the lightest Higgs scalar,
{\it ii}.) the relic density of cold dark matter in the universe,
{\it iii}.) the branching fraction for the radiative $b \to s \gamma$ decay,
{\it iv}.)  the muon anomalous magnetic moment, and
{\it v}.) the flavor violating decay $B_s\to \mu^+\mu^-$. 
%Each of these quantities can be predicted in the MSSM, and each depends 
%in a non-trivial way on the spectra of SUSY particles.
In this work, we present constraints from each of these quantities
on the minimal supergravity model as embedded in the updated version of the
computer program ISAJET v7.64. 
Improvements and updates since our published work are especially emphasized.
The combination of constraints
points to certain favored regions of model parameter space where
collider and non-accelerator SUSY searches may be more focused.
\end{abstract}

\section{Introduction}
\label{sec:intro}

%The search for weak scale supersymmetric matter is one of the prime objectives
%of present and future collider experiments. 
Particle physics models including
supersymmetry solve a host of problems occurring in non-supersymmetric
theories, and predict a variety of new matter states--- the sparticles---
at or around the TeV scale%
%\cite{Martin:1997ns,Drees:1996ca,Bagger:1996ka,Tata:1997uf,Dawson:1997tz}.
\cite{reviews}. 
%Supersymmetric models can be classified
%by the mechanism for communicating SUSY breaking from the hidden
%sector to the observable sector. Possibilities include gravity
%mediated SUSY breaking (SUGRA)\cite{sugra}, 
%gauge mediated SUSY breaking (GMSB)\cite{gmsb},
%anomaly mediated SUSY breaking (AMSB)\cite{amsb} and gaugino mediated SUSY
%breaking (inoMSB)\cite{inomsb}. Of these, the SUGRA models may be perceived as
%the most conservative, since they do not require the introduction
%of either extra dimensions or new messenger fields, and because gravity
%exists. 
%
The so-called {\it minimal} supergravity (mSUGRA) model (sometimes also
referred to as the CMSSM) has traditionally been the most popular
choice for phenomenological SUSY analyses. In mSUGRA, it is assumed that
the Minimal Supersymmetric Standard Model (MSSM) is valid from the weak
scale all the way up to the GUT scale $M_{GUT}\simeq 2\times 10^{16}$
GeV, where the gauge couplings $g_1$ and $g_2$ unify. In many of the
early SUGRA models\cite{sugra}, a simple choice
of K\"ahler metric $G_i^j$ and gauge kinetic function $f_{AB}$ led to
{\it universal} soft SUSY breaking scalar masses ($m_0$), gaugino masses
($m_{1/2}$) and $A$-terms ($A_0$) at $M_{GUT}$.  This assumption of
universality in the scalar sector
leads to the phenomenologically required suppression of
flavor violating processes that are supersymmetric in origin.  
%However,
%there is no known physical principle which gives rise to the desired
%form of $G_i^j$ and $f_{AB}$; indeed, for general forms of $G_i^j$ and
%$f_{AB}$, non-universal masses are expected\cite{sw}. In addition, even
%if nature did select a SUGRA model leading to tree level universality,
%quantum corrections would (without further assumptions) lead to large
%deviations from universality\cite{munoz}.  Hence, the universality
%assumption nowadays is regarded as being ad hoc--- entirely motivated by
%the phenomenological need for suppression of flavor violating processes
%in the MSSM.
%
In the mSUGRA model, we thus assume universal scalar masses, gaugino
masses (as a consequence of assuming grand unification) and $A$-terms. 
%
We will also require that electroweak symmetry is
broken radiatively (REWSB), allowing us to fix the magnitude,
but not the sign, of the superpotential Higgs mass term $\mu$ so as to
obtain the correct value of $M_Z$. Finally, we
trade the bilinear soft
supersymmetry breaking (SSB) parameter $B$ for $\tan\beta$ (the ratio of
Higgs field vacuum expectation values).  Thus,
the parameter set
%
\begin{equation}
m_0,\ m_{1/2},\ A_0,\ \tan\beta ,\ \ {\rm and}\ \ sign(\mu )
\end{equation}
completely determines the spectrum of supersymmetric matter and physical 
Higgs fields.

In our calculations, we use ISAJET v7.64 \cite{isajet} since this version 
includes a number of improvements in calculating the SUSY particle mass spectrum
compared to v7.58 used in Ref.\cite{Baer:2002gm}.
%Working in the $\overline{DR}$ regularization 
%scheme, the weak scale values of gauge and third generation Yukawa couplings
%are evolved via 2-loop RGEs to $M_{GUT}$. At $M_{GUT}$, universal
%SSB boundary conditions are imposed, and all SSB masses along with gauge 
%and Yukawa couplings are evolved to the weak scale $M_{weak}$. 
%Using an optimized scale choice $Q_{SUSY}=\sqrt{m_{\tst_L}m_{\tst_R}}$, 
%the RG-improved one-loop effective potential is minimized
%and the entire spectrum of SUSY and Higgs particles is calculated.
%Our values of $m_h$ are in close accord with those generated by the
%FeynHiggsFast program\cite{FHF}.
%Yukawa couplings are updated\cite{bagger} 
%to account for SUSY threshold corrections,
%and the entire parameter set is iteratively run between $M_{weak}$ and
%$M_{GUT}$ until a stable solution (within tolerances) is obtained.
Once the SUSY and Higgs masses and mixings are known, then a host of
observables may be calculated, and compared against experimental
measurements. The most important of these include: 
\begin{itemize}
\vspace{-2.6mm}
\item lower limits on sparticle and Higgs boson masses from new particle
searches at LEP2,
\vspace{-2.6mm}
\item the relic density of neutralinos originating from the Big Bang,
\vspace{-2.6mm}
\item the branching fraction of the flavor changing decay $b\to
s\gamma$,
\vspace{-2.6mm}
\item the value of muon anomalous magnetic moment $a_\mu = (g-2)_\mu/2$ and
\vspace{-2.6mm}
\item the lower bound on the rate for the rare decay $B_s\to\mu^+\mu^-$.
\end{itemize}
\vspace{-2.6mm}
%
Our goal is to delineate the mSUGRA parameter space region consistent
with all these constraints.  
%Similar studies have recently been
%presented in Refs. \cite{Gomez:2002tj,ellis,leszek,drees,deboer}.  
%We use ISAJET v7.64, 
%which includes several improvements over previous versions. 
%The most important of these improvements are the evaluation of
%the bottom Yukawa coupling and the value of $m_A$, especially for large
%values of $\tan\beta$. Complete 1-loop self energy corrections as given
%in Ref. \cite{bagger} have been incorporated.
%\footnote{On the technical
%side, the numerical precision in ISAJET has been improved to facilitate
%a better analysis near the boundary of the region excluded by
%electroweak symmetry breaking constraints.}  
%Comparisons of ISAJET v7.64 with other similar codes are available 
%in Ref. \cite{kraml}.
%
%Returning to 
% cut <
% >>> cut
In our analysis,
we incorporate a new calculation of the neutralino relic density
%$\Omega_{\tz_1}h^2$ 
that has recently become available\cite{bbb}.  
%In Ref. \cite{bbb}, all relevant neutralino annihilation and
%co-annihilation processes are calculated, and the neutralino relic
%density is evaluated using {\it relativistic} thermal averaging (see
%also Refs. \cite{bb} and recently Belanger {\it et al.},
%Ref. \cite{belanger}). The latter is especially important in evaluating
%the relic density when $s$-channel annihilation resonances occur, as in
%the mSUGRA model at large $\tan\beta$, when $\tz_1\tz_1\to A,\ H\to
%f\bar{f}$, where the $f$s are SM fermions\cite{bb,dn,an,ellis_ltb,nano}. 
We also present improved $b\to s\gamma$ branching fraction predictions in
accord with the current ISAJET release. We discuss constraints
imposed by the measurement of the muon anomalous magnetic moment.
%, using updated calculations from two somewhat different analyses. 
Finally, we delineate
the region of mSUGRA parameter space excluded by the CDF lower
limit\cite{cdf} on the branching fraction of $B_s\to \mu^+\mu^-$.  This
constraint is important 
%only 
for very large 
%values of
$\tan\beta$'s\cite{bmm}.

Within the mSUGRA framework, the parameters $m_0$ and $m_{1/2}$ are the
most important for fixing the scale of sparticle masses. The
$m_0$-$m_{1/2}$ plane (for fixed values of other parameters) is convenient
for
a simultaneous display of these constraints, and hence, of parameter
%By imposing all the above constraints on a single plot, it is
%possible to find favored regions of model parameter space.
regions in accord with all experimental data.
%Physicists interested in the mSUGRA model may 
%wish to focus their attention on these regions.
%We also present five mSUGRA model cases illustrating distinctive 
%characteristics of the mSUGRA particle spectrum for parameter choices which 
%are consistent with all experimental constraints.
  
%The remainder of this paper is organized as follows. 
%In Sec. \ref{sec:constraints}, we discuss the various constraints
%on the mSUGRA model, and present some details of our calculations. 
%In Sec. \ref{sec:results}, we show our main results as regions of the
%$m_0\ vs.\ m_{1/2}$ parameter space plane for different values
%of $\tan\beta$ and sign of $\mu$. Our conclusions and sample
%points are presented in Sec. \ref{sec:conclude}.  
 

\section{Constraints and calculations in the mSUGRA model}
\label{sec:constraints}

{\bf ~~~ Constraints from LEP2 searches}

%The LEP2 collaborations have finished taking data, and
%significant numbers of events were recorded at 
%$e^+e^-$ CM energies ranging up to $\sqrt{s}\simeq 208$ GeV.
Based on negative searches for superpartners at LEP2,
we require 
\begin{itemize}
\vspace{-2.6mm}
\item $m_{\tw_1}>103.5$ GeV ~~~ and ~~~ $m_{\te_{L,R}}>99$ GeV 
provided $m_{\tell}-m_{\tz_1}>10$ GeV, %\cite{lep2_w1}, 
\end{itemize}
\vspace{-2.6mm}
which is the most stringent of the slepton mass limits.
%
The LEP2 experiments also set a limit on the SM Higgs boson mass:
$m_{H_{SM}}>114.1$ GeV\cite{lep2_h}. In our mSUGRA parameter space scans, the
lightest SUSY Higgs boson $h$ is almost always SM-like. The exception occurs
when the value of $m_A$ becomes low at very large values 
of $\tan\beta$. For clarity, we show contours where
\bi
\vspace{-2.6mm}
\item $m_h>114.1$ GeV,
\vspace{-2.6mm}
\ei
and will direct the reader's attention to any regions where this bound might
fail. \\

{\bf Neutralino relic density}

Measurements of galactic rotation curves, binding of galactic clusters, 
and the large scale structure of the universe all point to the need for
significant amounts of cold dark matter (CDM) in the universe. In addition, 
recent measurements of the power structure of the cosmic
microwave background, and measurements of distant supernovae, point
to a cold dark matter density\cite{cdm}
\bi
\vspace{-2.6mm}
\item $0.1 <\Omega_{CDM}h^2<0.3$.
\vspace{-2.6mm}
\ei
The lightest neutralino of mSUGRA is an excellent candidate for
relic CDM particles in the universe.
The upper limit above represents a true constraint, while the 
corresponding lower
limit is flexible, since there may be additional sources of CDM such
as axions, or states associated with the hidden sector and/or extra
dimensions.

To estimate the relic density of neutralinos in the mSUGRA model,
we use the recent calculation in Ref. \cite{bbb}. 
%In Ref. \cite{bbb},
In that work, 
all relevant neutralino annihilation and co-annihilation
reactions are evaluated at tree level using the CompHEP\cite{comphep}
program.
% for automatic evaluation of the 
%associated $7618$ Feynman diagrams. 
The annihilation cross section times velocity is relativistically thermally
averaged\cite{graciela}, 
which is important for obtaining the correct neutralino relic 
density in the vicinity of annihilations through $s$-channel resonances. \\

{\bf The $b\to s\gamma$ branching fraction}

The branching fraction $BF(b\to s\gamma )$ has recently been measured by
the BELLE\cite{belle}, CLEO\cite{cleo} and ALEPH\cite{aleph}
collaborations.  Combining statistical and systematic errors in
quadrature, these measurements give $(3.36\pm 0.67)\times 10^{-4}$
(BELLE), $(3.21\pm 0.51)\times 10^{-4}$ (CLEO) and $(3.11\pm 1.07)\times
10^{-4}$ (ALEPH). A weighted averaging of these results yields $BF(b\to
s\gamma )=(3.25\pm 0.37) \times 10^{-4}$. The 95\% CL range corresponds
to $\pm 2\sigma$ away from the mean. To this we should add uncertainty
in the theoretical evaluation, which within the SM dominantly comes from
the scale uncertainty, and is about 10\%.
%\footnote{We caution the reader
%that the SUSY contribution may have a larger theoretical uncertainty,
%particularly if $\tan\beta$ is large. An additional theoretical
%uncertainty that may increase the branching ratio in the SM is pointed
%out in Ref. \cite{Gambino:2001ew}.} 
Together, these imply the bounds,
\bi
\vspace{-2.6mm}
\item $2.16\times 10^{-4}< BF(b\to s\gamma )< 4.34 \times 10^{-4}$.
\vspace{-2.6mm}
\ei
%Other computations of the range of $BF(b\to s\gamma )$ include for instance
%Ellis {\it et al.}\cite{ellis}: 
%$2.33\times 10^{-4}<BF(b\to s\gamma )<4.15\times 10^{-4}$,
%and Djouadi {\it et al.}\cite{drees}: 
%$2.0\times 10^{-4}<BF(b\to s\gamma )<5.0 \times 10^{-4}$.
In our study, we show contours of $BF(b\to s\gamma )$ of
2, 3, 4 and $5\times 10^{-4}$.
%, allowing the reader the flexibility
%of their own interpretation.

The calculation of $BF(b\to s\gamma )$ used here is based upon the
program of Ref. \cite{bsg}. 
%That calculation uses an effective field 
%theory approach to evaluating radiative corrections to the 
%$b\to s\gamma$ decay rate. In running from $M_{GUT}$ to $M_{weak}$,
%when any sparticle threshold is crossed, the corresponding sparticle is 
%integrated out of the theory, and a new basis of decay-mediating operators
%multiplied by Wilson coefficients (WCs) is induced. 
%The evolution of the WCs can be calculated by RG methods. We adopt the
%Anlauf procedure\cite{anlauf} in our calculation, which implements a tower 
%of effective field theories, corresponding to each sparticle threshold
%which is crossed. 
%This procedure sums large logarithms that can occur 
%from a disparity between different scales involved in the loop 
%calculations. 
In our calculations, we also implement the running $b$-quark mass
including SUSY threshold corrections as calculated in ISAJET;
these effects can be important at large values of the 
parameter $\tan\beta$\cite{degrassi}.
%Once the relevant operators and Wilson coefficients are 
%known at $Q=M_W$, then the SM WCs are evolved down to $Q=m_b$ via
%NLO RG running. At $m_b$, the $BF(b\to s\gamma )$ is evaluated at NLO,
%including bremsstrahlung effects. 
Our value of the SM $b\to s\gamma$
branching fraction yields $3.4\times 10^{-4}$, with a scale uncertainty
of 10\%. \\

{\bf Muon anomalous magnetic moment}

The muon anomalous magnetic moment $a_\mu =(g-2)_\mu/2$ has been
recently measured to high precision by the E821 experiment\cite{Bennett:2002jb}:
$ a_\mu=11659204(7)(5)\times 10^{-10}$.
%In addition, additional data analyses should soon be finished, and we
%may anticipate a further reduction in the experimental error by a factor
%of 2.  The initially reported\cite{e821} $2.6\sigma$ deviation from the
%SM value of Ref.\cite{marciano} has since been tempered somewhat by
%correcting the sign of the SM light-by-light contribution to
%$a_\mu$\cite{lbl}. A correction of the sign of the hadronic light by
%light contribution reduces the significance of the deviation from the SM
%to $1.6\sigma$, {\it i.e.}  at $2\sigma$:  
%
%\bi
%\vspace{-2.6mm}
%\item $-6<\delta a_\mu\times 10^{10}<58$ (CM). 
%\vspace{-2.6mm}
%\ei
%
%An alternative evaluation of theory uncertainties in the SM $a_\mu$
%calculation by Melnikov\cite{melnikov} leads to (including the LBL
%correction):
%
%\bi
%\vspace{-2.6mm}
%\item $-29.9< \delta a_\mu\times 10^{10}<62.3$ (Melnikov).
%\vspace{-2.6mm}
%\footnote{Melnikov--- who uses the analysis of Ref.\cite{jag}
%which does not use tau decay data for the evaluation of the hadronic
%vacuum polarization and has a more conservative error on the light by
%light contribution--- finds $\delta a_{\mu}\times 10^{10}= 16.2 \pm
%14.0|_{stat} \pm 6.0|_{sys} \pm 15.6|_{theory}$. The conservative
%``$2\sigma$'' range reported here can be obtained by linearly combining
%the theory error with the $2\sigma$ experimental error.} 
%\ei
%
The most challenging parts of the SM calculation are the hadronic 
light-by-light\cite{lbl} and vacuum polarization (HVP)\cite{HVPee} contributions 
and their uncertainties. Presently these results are in dispute. In the case of 
the HVP the use of tau decay data can reduce the error, but the interpretation 
of these data is somewhat controversial\cite{HVPtau}.
%
Thus, the deviation of the measurement from the SM depends on which prediction 
is taken into account. According to the recent analysis by Hagiwara et 
al.\cite{HVPee}:
\bi
\vspace{-2.6mm}
\item $11.5<\delta a_\mu\times 10^{10}<60.7$. 
% this is 36.1[below their Eq.(17)]+-(7.4+2*8.6) [their Eq.(17) and sqrt(7^2+5^2)]
\vspace{-2.6mm}
\ei
%An alternative evaluation of theory uncertainties in the SM $a_\mu$
%calculation by Melnikov\cite{HVPee} leads to:
A different assessment of the theoretical uncertainties\cite{HVPee}
using the procedure described in ref.\cite{Baer:2002gm} gives,
\bi
\vspace{-2.6mm}
\item $-16.7< \delta a_\mu\times 10^{10}<49.1$.
% this is the numbers in our JHEP paper +-2*(15.2-8.6)
\vspace{-2.6mm}
\ei
%
In view of the
theoretical uncertainty, 
%and the impending new experimental analysis, 
we only
present contours of $\delta a_\mu$, as calculated using the program
developed in \cite{bbft}, and leave it to the reader to decide the
extent of the parameter region allowed by the data. \\

{\bf $B_s\to\mu^+\mu^-$ decay}

%While all SUSY models contain two doublets of Higgs superfields, there
%are no tree level flavor changing neutral currents because one doublet
%${\hat H}_u$ couples only to $T_3=1/2$ fermions, while the other doublet
%$\hat{H}_d$ couples just to $T_3= -1/2$ fermions. At one loop, however,
%couplings of ${\hat H}_u$ to down type fermions are induced. These
%induced couplings grow with $\tan\beta$. As a result, down quark Yukawa
%interactions and down type quark mass matrices are no longer
%diagonalized by the same transformation, and flavor violating couplings
%of neutral Higgs scalars $h$, $H$ and $A$ emerge. Of course, in the limit of
%large $m_A$, the Higgs sector becomes equivalent to the SM
%Higgs sector with the light Higgs boson $h=H_{SM}$, and the flavor
%violation decouples. The interesting thing is that while
%this decoupling occurs as $m_A \to \infty$, {\it there is no decoupling
%for sparticle masses becoming large.}

%An important consequence of this coupling is the possibility of the
%decay $B_s \to \mu^+\mu^-$, whose branching fraction has been
The branching fraction of $B_s$ to a pair of muons has been
experimentally
bounded by CDF\cite{cdf}:
%
\bi
\vspace{-2.6mm}
\item $BF(B_s\to\mu^+\mu^- )< 2.6\times 10^{-6}$.
\vspace{-2.6mm}
\ei
%
%This decay is mediated by 
A potentially important contribution to this decay is mediated by
the neutral states in the Higgs sector of supersymmetric
models. While this branching fraction is very small within the SM
($BF_{SM}(B_s \to \mu^+\mu^-)\simeq 3.4 \times 10^{-9}$), the amplitude
for the Higgs-mediated decay of $B_s$ grows as $\tan^3\beta$ within the
SUSY framework, and hence can completely dominate the SM contribution if
$\tan\beta$ is large.  
%Several groups\cite{bmm} have analyzed the
%implications of this decay within the mSUGRA framework. 
%A subset of us have recently performed an independent analysis of this
%decay. In the following, we use the results of this analysis to
%delineate
In our analysis we use the results from the last paper in Ref.\cite{bmm} 
to delineate 
the region of mSUGRA parameters excluded by the CDF upper limit on its 
branching fraction. 

%Tevatron experiments should be able to probe this decay in the near
%future.  With an integrated sample of 2~fb$^{-1}$ they should be
%sensitive to a branching fraction for $B_s \to \mu^+\mu^-$ down to $\sim
%10^{-7}$. With a still bigger data sample (that is expected to accumulate
%before the Large Hadron Collider begins operation) the sensitivity
%should be even greater.


\section{Results}
\label{sec:results}

To generate numerical results, in this work we use ISAJET v7.64 that includes 
several improvements over v7.58 which was used in Ref.\cite{Baer:2002gm}. These 
changes lead to important differences in the figures when compared
with Ref.\cite{Baer:2002gm}. Notably, 
%due to ..., 
the boundary of the region excluded by the lack of REWSB moved to higher 
$m_{0}$ values and the allowed relic density region along this boundary changed,
especially for the lower $\tan\beta$ values.
Furthermore, 
%due to the reduced widths of the heavy neutral Higgses,  
for high $\tan\beta$ and $\mu < 0$ the diagonal corridors allowed by the 
relic density are considerably shifted and narrowed.
Finally, the area allowed by relic density near the boundary of the stau LSP 
region shrank at low $\tan\beta$'s.

\begin{figure}
\epsfig{file=sug10m.eps,width=8cm} 
\epsfig{file=sug10p.eps,width=8cm} 
\caption{Plot of constraints for the mSUGRA model in the 
$m_0\ vs.\ m_{1/2}$ plane for $\tan\beta =10$ and $A_0=0$.
We plot contours of the CDM relic density, $m_h=114.1$ GeV, 
the muon anomalous magnetic moment $a_\mu$ ($\times 10^{10}$) and
contours of $b\to s\gamma$ branching fraction ($\times 10^{4}$).}
\label{fig:sug10}
\end{figure}

Our first results are plotted in Fig.\ref{fig:sug10}. Here, we
show the $m_0\ vs.\ m_{1/2}$ plane for $A_0=0$, $\tan\beta =10$ and
both signs of $\mu$.
The red shaded regions are excluded either due to a
lack of REWSB (right-hand side), or a stau LSP (left-hand side).
The magenta region is excluded by searches  for charginos and
sleptons at LEP2. The region below the red contour is excluded by LEP2
Higgs searches, since here $m_h<114.1$ GeV. In addition, we show
regions of neutralino relic density with green contours marking
$\Omega_{\tz_1} h^2 =0.1$ (dotted), 0.3 (dashed) and 1.0 (solid). 
The region right to the solid green contour has 
$\Omega_{\tz_1} h^2>1$, 
and would thus be excluded since the age of the universe would be less
than 10 billion years. 
%
There is no constraint arising from $B_s\to \mu^+\mu^-$ decay at 
$\tan\beta =10$.

For $\mu <0$ the magenta contours denote values of
$BF(b\to s\gamma )= 4$ and $5\times 10^{-4}$ and the blue
contours denote values of $\delta a_\mu =-30, -10, -5, -2$ and 
$-1\times 10^{-10}$, moving from lower left to upper right.
%
An intriguing feature of the plot is that the region with the allowed 
relic density in the lower left part, where neutralinos mainly annihilate
via $t$-channel slepton exchange 
to lepton-anti-lepton pairs
is essentially excluded by the $m_h$, $b\to s\gamma$ and
$\delta a_\mu$ constraints. That leaves two
allowed regions
with a preferred relic density: one that runs near the stau LSP region, where
$\ttau_1-\tz_1$ co-annihilation effects reduce an otherwise
large relic density 
(as pointed out by Ellis {\it et al.}\cite{ellis_co}).
This region has a highly fine-tuned relic density, since a slight change in
$m_0$ leads to either too light or too heavy of a $\ttau_1$ mass
to give $0.1<\Omega h^2<0.3$\cite{ellis_ft,bbb}. 
The other runs parallel
to the REWSB excluded region for $m_{1/2} > 400$ GeV in the ``focus point'' SUSY 
region. It occurs when the $\tz_1$ has a sufficiently
large higgsino component that annihilation into $WW$, $ZZ$ and $Zh$
pairs reduces the relic density\cite{feng_relic,bbb}. 
%This region corresponds to what is known
%as ``focus point'' SUSY, and since $m_0$ is large, SUSY scalar masses
%are also large, leading to some degree of suppression of FCNC and CP violating 
%processes\cite{feng}. The narrowness of the region indicates again
%that some fine-tuning of parameters is needed to achieve the 
%right relic density, although the amount of fine-tuning is less than
%in the $\ttau_1$ co-annihilation case.

%A similar plot is shown in Fig. \ref{fig:sug10}, but in this case f
For $\mu >0$ 
%Much of the labeling is similar to Fig. \ref{fig:sug10},
%although now the $b\to s\gamma$ contour denotes a branching fraction of
%$3\times 10^{-4}$. In this case 
almost the entire plane shown is in
accord with the measured branching fraction of $b\to s\gamma$. % In addition,
The blue contours denote values of $\delta a_\mu =60$, 40, 20, 10, 5, 2 and
$1\times 10^{-10}$. Constraints from $\delta a_{\mu}$ as
well as from $B_s \to \mu^+\mu^-$ are not relevant for this case.
%
%For $\tan\beta =10$ and $\mu >0$, 
In this case 
the slepton annihilation region
of relic density has a small surviving region just beyond 
the Higgs mass contour. 
%In this case, $BF(b\to s\gamma )$ and $\delta a_\mu$ are in accord
%with experiment over almost the entire plane, with the exception being
%very small values of both $m_0$ and $m_{1/2}$. 
For the most part, to attain a 
preferred value of neutralino relic density, one must again live in the 
stau co-annihilation
% ? , or the focus point 
region. A final 
possibility is to be in the slepton annihilation region, but then the 
value of $m_h$ should be slightly beyond the LEP2 limit; 
in this case,
%the LEP2 Higgs candidate events might turn out to be signal. 
a Higgs boson signal may be detected in Run 2 of the Fermilab
Tevatron\cite{run2}.

\begin{figure}
\epsfig{file=sug30m.eps,width=8cm} 
\epsfig{file=sug30p.eps,width=8cm} 
\caption{Same as Fig. \ref{fig:sug10}, but for $\tan\beta =30$. 
The light blue contour labeled 0.1 denotes where $B(B_s \to
\mu^+\mu^-)= 0.1 \times 10^{-7}$. In subsequent figures these branching
fractions contours are all labeled in units of $10^{-7}$. }
\label{fig:sug30}
\end{figure}

We next turn to our results for $\tan\beta=30$ shown in Fig.\ref{fig:sug30}.
%We next turn to the $m_0\ vs.\ m_{1/2}$ plane for $\tan\beta =30$. 
The gray region in the bottom left corner of the plot is
excluded because $m_{\ttau_1}^2 < 0$. In this case, the allowed
region of the relic density in the lower-left has expanded considerably
owing to enhanced neutralino annihilation to $b\bar{b}$ and
$\tau\bar{\tau}$ at large $\tan\beta$.  Both lighter values of
$m_{\ttau_1}$ and $m_{\tb_1}$ and also large $\tau$ and $b$ Yukawa
couplings at large $\tan\beta$ enhance these $t$-channel annihilation
rates through virtual staus and sbottoms. Unfortunately, for $\mu < 0$ 
the region excluded by $BF(b\to s\gamma )$ and by $\delta a_\mu$ 
%(even with the conservative constraint of Melnikov) 
also expands, and most of the cosmologically preferred region is
again ruled out.
As before, we are left with the corridors of stau
co-annihilation and an enlarged focus point scenario\cite{feng_relic,bbb} 
as the only surviving regions.
 
%The corresponding plot is shown f
For 
%$\tan\beta =30$ but 
$\mu >0$
%in Figure \ref{fig:sug30}. In this case, 
the magenta contours of
$BF(b\to s\gamma )$ correspond to $2$ and $3\times 10^{-4}$. Thus, the lower
left region is excluded since it leads to too {\it low} a value
of $BF(b\to s\gamma )$. The $\delta a_\mu $ contours begin from lower left 
with $60\times 10^{-10}$, then proceed to 40, 20, 10, 5, 2 and $1\times 10^{-10}$.
A fraction of the slepton annihilation region of relic density is
excluded also by too large a value of $\delta a_\mu$. Of course,
a reasonable relic density may also be achieved in the stau co-annihilation 
and focus point regions of parameter space.

Next, we turn to Fig.\ref{fig:sug4552} where we examine 
%Next, we examine 
the mSUGRA parameter plane for very large values of
$\tan\beta =45$ and $\mu <0$. 
%If we take $\tan\beta$ much bigger than 45
%for this sign of $\mu$, the entire parameter space is excluded due to
%lack of REWSB. 
The gray and red regions are as in previous figures. The
blue region is excluded because $m_A^2<0$, denoting again a lack of
appropriate REWSB. The inner and outer red dashed lines are contours of
$m_A=100$ and $m_A= 200$~GeV, respectively. The former is roughly the
lower bound on $m_A$ from LEP experiments. In between these contours,
$h$ is not quite SM-like, and the mass bound from LEP may be somewhat lower
than $m_h=114.1$~GeV shown by the solid red contour, but outside the
200~GeV contour this bound should be valid.
%
%The tiny black region in the figure is excluded by 
%constraints\cite{zprop} on the
%width of the $Z$ boson (specifically, the decay $Z \to hA+HA$ 
%leads to too large a value for $\Gamma_Z$). 
%We also see that m
Much of the lower-left region is excluded by too high a
value of $BF(b\to s \gamma )$ and too low a value of $\delta a_\mu$.  In
addition, in this plane, the experimental limit on $B_s\to\mu^+\mu^-$
enters the lower-left, where values exceeding $26\times 10^{-7}$ are
obtained.
It seems that in the upper region which is  favored by the $b \to s\gamma$
constraint, detection of $B_s \to \mu^+\mu^-$ at the Tevatron will be
quite challenging.

In this figure, the relic density regions are qualitatively
different
from the lower $\tan\beta$ plots. A long diagonal strip running from
lower-left to upper-right occurs because in this region, neutralinos
annihilate very efficiently through $s$-channel $A$ and $H$ Higgs
graphs, where the total Higgs widths are very large due to the large $b$
and $\tau$ Yukawa couplings for the high value of $\tan\beta$ in this
plot. Adjacent to this region allowed regions where
neutralino annihilation is still dominated by the $s$-channel Higgs
graphs, but in this case the annihilation is somewhat off-resonance. The
$A$ and $H$ widths are so large
%, typically $10-30$ GeV, 
that even if
$|2m_{\tz_1}-m_{A(H)}|$ is relatively large, efficient annihilation can
still take place. (An improvement of the Higgs widths is adopted for these
plots compared to Ref.\cite{Baer:2002gm}.)
% The relic density changes so slowly on the flanks of the
%annihilation corridor that little fine tuning of parameters is needed
%to achieve a favored value of $\Omega_{\tz_1}h^2$.

\begin{figure}
\epsfig{file=sug45m.eps,width=8cm} 
\epsfig{file=sug52p.eps,width=8cm} 
\caption{Same as Fig. \ref{fig:sug10}, but for $\tan\beta =45$, $\mu <0$
and for $\tan\beta =52$, $\mu >0$.
The inner and outer red dashed lines are contours of $m_A=100$ and 
$m_A= 200$~GeV, respectively.}
\label{fig:sug4552}
\end{figure}

For the case of $\mu >0$, 
%$\tan\beta$
%values ranging up to $60$ can be allowed, although the mSUGRA parameter
%space becomes very limited for $\tan\beta \agt 55$. 
%Hence, 
we show %in Fig. \ref{fig:sug4552} 
the mSUGRA parameter space plane for $\tan\beta =52$. 
In this plane, the relic density annihilation corridor 
occurs near the boundary of the excluded $\ttau_1$ LSP region. 
The width of the $A$ and $H$ Higgs scalars is very wide, 
%ranging from 15 GeV
%for $m_{1/2}\sim 400$ GeV, to 55-65 GeV for $m_{1/2}\sim 2000$ GeV.
so efficient $s$-channel annihilation through the Higgs poles can occur
throughout much of the allowed parameter space. But the 
annihilation is not overly efficient due to the large breadth of the 
Higgs resonances.
%In fact, {\it none} of the entire parameter plane is excluded 
%by $\Omega_{\tz_1}h^2>1$!
%The change in $\Omega_{\tz_1}h^2$ is so slow over almost the entire
%parameter plane that little fine-tuning occurs. 
%
In much of the region with $m_{1/2}<400$ GeV, the value of
$BF(b\to s\gamma )$ is below
$2\times 10^{-4}$, so that some of the lower allowed relic density region
where annihilation occurs through $t$-channel stau exchange is excluded.
In contrast, the value of $\delta a_\mu$ is 
in the range of $10-40\times 10^{-10}$, which is in accord with
the E821 measurement. The value of $m_h$ is almost always
above 114.1 GeV, and the $BF(B_s\to\mu^+\mu^- )$ is always
below $10^{-7}$, and could (if at all) be detected with several years of
main injector operation. 
%
%Aside from the somewhat low value of
%$BF(b\to s\gamma )$ in the lower left, large part of this plane 
%represents a very attractive area
%of mSUGRA model parameter space. If the model parameters are indeed in
%this range, the Tevatron signal for $B_s \to \mu^+\mu^-$ will be
%small, and $BF(b\to s\gamma )$ will turn out somewhat below the SM prediction,
%while $\delta a_\mu$ will be somewhat above the SM value. 

%\section{Conclusions}
%\label{sec:conclude}

In conclusion, we have presented updated constraints on the mSUGRA model
from {\it i.}) the LEP2 constraints on sparticle and Higgs boson masses, 
{\it ii.}) the neutralino relic density $\Omega_{\tz_1}h^2$, 
{\it iii.}) the branching fraction $BF(b\to s\gamma )$, 
{\it iv.}) the muon anomalous magnetic moment $a_\mu$ and
{\it v.}) the leptonic decay $B_s\to\mu^+\mu^-$. Putting all five
constraints together, we find favored regions of parameter space which
may be categorized by the mechanism for annihilating relic neutralinos
in the early universe:
%
\bi
\vspace{-2.6mm}
\item {\bf 1.} annihilation through $t$-channel slepton exchange
(low $m_0$ and $m_{1/2}$),
\vspace{-2.6mm}
\item {\bf 2.} the stau co-annihilation region 
(very low $m_0$ but large $m_{1/2}$),
\vspace{-2.6mm}
\item {\bf 3.} the focus point region (large $m_0$ but low to 
intermediate $m_{1/2}$) and
\vspace{-2.6mm}
\item {\bf 4.} the flanks of the neutralino $s$-channel annihilation via
$A$ and $H$ corridor at large $\tan\beta$ when $\Gamma_A$ and $\Gamma_H$
are very large.
\vspace{-2.6mm}
\ei
%

%In previous years, there may have been a preference for region {\bf 1.}
%as offering the most natural channel for obtaining a reasonable value
%of relic density. However, recently much of this region is ruled out by
%a combination of LEP2 limits on $m_h$ at low $\tan\beta$, too high 
%(for $\mu <0$) or too low (for $\mu >0$ and large $\tan\beta$) a value of
%$BF(b\to s\gamma )$, and too low a value of $a_\mu$ (for $\mu <0$ and
%intermediate to large $\tan\beta$). In addition, the CDF
%$BF(B_s\to \mu^+\mu^-)$ constraint is starting to become important
%for $\mu <0$ and large $\tan\beta$. Nevertheless, some 

%We found that a small part 
%of region {\bf 1.} remains 
%viable, especially for $\mu >0$, where we expect a lightest Higgs boson 
%just beyond the bounds from LEP2.
%%
%The stau co-annihilation region\cite{ellis_co} {\bf 2.} 
%is intriguing because one can always take
%$m_{1/2}$ large enough for any $\tan\beta$ value to evade constraints
%on deviations from SM predictions. However, 
%there is 
%%this region is exceptionally 
%%narrow in the parameter $m_0$, and slight deviations cause either too high
%%or too low a value of relic density. This indicates 
%a high degree of 
%fine-tuning in the determination of the relic density in this region.
%%
%The focus point region %\cite{feng} 
%{\bf 3.} also leads to a reasonable relic density.
%%,
%%this time because the higgsino component of $\tz_1$ is large enough
%%that efficient annihilation can occur to $WW$, $ZZ$ and $ZH$ states.
%It may also be preferable based on possibly low electroweak fine-tuning, and
%because matter scalar masses are high enough to offer some degree of 
%suppression of SUSY induced FC and CP violating processes. %\cite{feng}.
%This region also suffers some degree of fine-tuning of the relic density.
%%,
%%since too high or too low a higgsino component of $\tz_1$ can result in
%%too low or too high a value of relic density. For $\tan\beta\sim 10$ and
%% ? $\mu >0$, the focus point region is in accord with all constraints. 
%For $\mu <0$, the focus point region usually gives too high a value of
%$BF(b\to s\gamma )$; for $\mu >0$, we would expect ultimately
%experiment to measure a somewhat lower value of $BF(b\to s\gamma )$ 
%than the SM prediction, and a somewhat higher value of $a_\mu$.
%%
%Finally, at very large $\tan\beta\sim 45-55$ there can exist wide regions of
%parameter space where $\tz_1\tz_1$ annihilation can occur in the early 
%universe through very broad $s$-channel $A$ and $H$ resonances,
%giving rise to a reasonable value of relic density: %\cite{dn,bb}: 
%region {\bf 4.}. 
%Unfortunately, much of 
%this region is excluded for $\mu <0$ by too large a value of 
%$BF(b\to s\gamma )$. Here also a rather low a value of $a_\mu$ less than the
%SM prediction is generated. In this case, very large values of $m_0$ and 
%$m_{1/2}$ are needed to escape experimental constraints, perhaps
%placing the SUSY spectrum in conflict with naturalness considerations\cite{diego}.
%%
%For very large $\tan\beta$ and $\mu >0$, however, broad regions of 
%parameter space can be found with a reasonable relic density, and also 
%which are in accord with all other low energy constraints. 
%In this region, we expect $BF(b\to s\gamma )$ somewhat below the SM 
%prediction, and $a_\mu$ somewhat above the SM prediction.
%The current $BF(b\to s\gamma )$ measurement suggest 
%$m_{1/2}$ values $\agt 500$ GeV, giving rise to sparticles 
%typically at the TeV scale or beyond.

%To summarize these various regions, we present in Table \ref{table} 
%five parameter space points indicating the SUSY spectrum
%that might occur in each region. 
%We also list the relic density, $BF(b\to s\gamma )$, $a_\mu$ and
%$BF(B_s\to\mu^+\mu^- )$ values. 
%
%Point 1 occurs in region {\bf 1.}, and is characterized by light
%sparticle masses, especially tau sleptons.  The light Higgs scalar $h$
%is slightly beyond the LEP2 bound.  The search channel at the Fermilab
%Tevatron would be $p\bar{p}\to\tw_1\tz_2 X$, with $\tz_2\to\tau\ttau_1$
%and $\tw_1\to\ttau\nu_\tau$ though detection appears to be
%difficult\cite{tevatron}.  The CERN LHC would be awash in
%signals\cite{lhc}, and many new SUSY states would be accessible to a
%linear collider (LC) with $\sqrt{s}\simeq 500-800$ GeV\cite{nlc}.
%
%Point 2 is in the stau co-annihilation region {\bf 2.}, but at low
%enough $m_{1/2}$ that $t$-channel stau exchange is still important in
%the $\tz_1\tz_1$ annihilations. It is characterized by a rather small
%mass gap between $\ttau_1$ and $\tz_1$. Only the Higgs boson $h$
%would be observable at the Tevatron. At the LHC, a variety of 
%leptonic signatures would occur in gluino and squark cascade decays, 
%including a high rate of $\tau$ production. The LC would have to operate
%slightly above $\sqrt{s}\sim 500$ GeV to access even the first SUSY states.
%
%Point 3 in the focus point region {\bf 3.} has TeV scale scalars, but
%light $\tw_1$ and $\tz_2$. The LSP is a mixture of gaugino-higgsino.
%The $\tw_1$ decays into three bodies dominated by $W^*$ exchange, and the
%$\tz_2$ decay is dominated by $Z^*$ exchange. Because this point is in
%the region with small $|\mu|$ (but with a not too small
%$m_{\tw_1}-m_{\tz_1}$), a low rate of 
%trilepton events may be accessible to Fermilab Tevatron experiments.
%At the CERN LHC, $\tg\tg$ and $\tw_1\tz_2$ production would be dominant.
%A LC would be able to find most of the charginos and neutralinos, since
%these are the lowest lying states.
%
%Point 4a lies in region {\bf 4.} on the edge of the Higgs annihilation 
%corridor for large $\tan\beta$
%and $\mu <0$. To evade constraints, the SUSY spectrum is very heavy:
%only the Higgs $h$ would be seen at the Tevatron or a LC, 
%and in fact it would even be challenging to discover SUSY at the LHC.
%This point is disfavored by naturalness arguments.
%
%Point 4b also in region {\bf 4.} lies in the Higgs annihilation 
%corridor for large
%$\tan\beta$ and $\mu>0$. A significantly lighter spectrum can be tolerated
%than in the 4a case, although SUSY scalars are still in the TeV range.
%Only the $h$ would be accessible at the Tevatron. SUSY signals
%corresponding to point 4b
%should be readily visible at the LHC, although a LC with $\sqrt{s}>1$ TeV
%would be required to access even the lowest lying SUSY states.

To summarize, we find the five constraints considered in this work
to be highly restrictive. Together, they rule out large regions of parameter
space of the mSUGRA model, including much of the region where
$t$-channel slepton annihilation of neutralinos occurs in the early universe.
The surviving regions {\bf 1.}-{\bf 4.} have distinct characteristics of their
SUSY spectrum, and should lead to distinctive SUSY signatures at colliders.

%\begin{table}
%\begin{center}
%\caption{Representative weak scale sparticle masses 
%(in GeV units) and parameters for five selected mSUGRA models.
% We use $A_0=0$ and $m_t=175$ GeV. The value of $\mu$ is also
%shown since it is sometimes regarded as a measure of fine tuning. }
%\bigskip
%\begin{tabular}{lrrrrr}
%\hline
%parameter & \multicolumn{5}{c}{value}  \\
%\hline
% $point$           &   1   &    2   &    3   &    4a  &    4b  \\
% $m_0$             &   100 &    165 &   1200 &   2750 &    800 \\
% $m_{1/2}$         &   300 &    550 &    250 &   1800 &    800 \\
% $\tan(\beta)$     &    10 &     10 &     10 &     45 &     52 \\
% $sgn(\mu)$        &     1 &      1 &      1 &   $-$1 &      1 \\
%\hline
% $m_{\tg}$         & 701.4 & 1225.6 &  658.0 & 3810.8 & 1757.6 \\
% $m_{\tu_L}$       & 630.7 & 1099.1 & 1271.1 & 4185.0 & 1715.3 \\
% $m_{\tu_R}$       & 611.1 & 1060.0 & 1269.1 & 4080.7 & 1662.4 \\
% $m_{\td_L}$       & 635.6 & 1102.0 & 1273.6 & 4185.8 & 1717.1 \\
% $m_{\td_R}$       & 610.0 & 1055.7 & 1269.6 & 4067.4 & 1656.5 \\
% $m_{\tb_1}$       & 584.9 & 1020.8 & 1072.4 & 3501.7 & 1484.4 \\
% $m_{\tb_2}$       & 610.7 & 1053.4 & 1260.8 & 3537.6 & 1539.4 \\
% $m_{\tst_1}$      & 471.7 &  858.5 &  825.2 & 3213.2 & 1328.2 \\
% $m_{\tst_2}$      & 648.1 & 1064.0 & 1084.3 & 3529.6 & 1533.6 \\
% $m_{\tnu_{e}}$    & 216.4 &  396.7 & 1203.1 & 2972.3 &  952.4 \\
% $m_{\te_L}$       & 230.4 &  404.5 & 1205.7 & 2973.4 &  955.8 \\
% $m_{\te_R}$       & 155.5 &  264.8 & 1201.7 & 2822.1 &  851.7 \\
% $m_{\tnu_{\tau}}$ & 215.6 &  395.4 & 1198.0 & 2750.8 &  834.0 \\
% $m_{\ttau_1}$     & 147.5 &  257.6 & 1191.0 & 2320.8 &  524.2 \\
% $m_{\ttau_2}$     & 233.4 &  405.2 & 1200.8 & 2752.6 &  847.9 \\
% $m_{\tz_1}$       & 117.5 &  225.1 &   88.6 &  785.0 &  336.2 \\
% $m_{\tz_2}$       & 215.1 &  416.9 &  144.1 & 1235.1 &  620.2 \\
% $m_{\tz_3}$       & 398.5 &  668.1 &  198.2 & 1249.2 &  848.7 \\
% $m_{\tz_4}$       & 417.8 &  682.8 &  260.9 & 1461.7 &  862.3 \\
% $m_{\tw_1}$       & 214.7 &  416.9 &  136.5 & 1234.5 &  620.3 \\
% $m_{\tw_2}$       & 418.0 &  682.6 &  260.3 & 1461.7 &  862.5 \\
% $m_h      $       & 114.7 &  119.0 &  114.4 &  123.9 &  121.5 \\
% $m_H      $       & 443.9 &  766.5 & 1204.9 & 1495.2 &  810.1 \\
% $m_A      $       & 443.3 &  765.7 & 1203.9 & 1494.2 &  809.5 \\
% $m_{H^+}  $       & 450.7 &  770.4 & 1207.4 & 1498.4 &  816.5 \\
% $\mu$             & 392.0 &  664.9 & 188.7  & -1247.2&  845.8 \\
%%
% $\Omega h^2$      & 0.232 &  0.218 &  0.262 &  0.210 &  0.181 \\
%
% $BF(b\to s\gamma)\times 10^4$       
%                   &  3.12 &   3.46 &   3.20 &   3.92 &   2.85 \\
% $a_\mu^{SUSY}\times 10^{10}$         
%                   &  22.6 &   7.13 &   2.65 &$-$1.48 &   10.2 \\
% $BF(B_s\to\mu^+\mu^-)\times 10^7$   
%                   & 0.0399& 0.0389 & 0.0384 & 0.0306 & 0.0870 \\
%
%\hline
%\label{table}
%\end{tabular}  
%\end{center}   
%\end{table}    


\section*{Acknowledgments}
 
This research was supported in part by the U.S. Department of Energy
under contracts number DE-FG02-97ER41022 and DE-FG03-94ER40833, and by 
Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo
(FAPESP).
	
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%
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%%
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%
%\bibitem{lep2_w1} Joint HEP2 Supersymmetry Working Group, {\it Combined 
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%%CITATION = ;%%

%\bibitem{marciano} A.~Czarnecki and W.~Marciano,
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M.~Hayakawa and T.~Kinoshita, \hepph{0112102};
I.~Blokland, A.~Czarnecki and K.~Melnikov,
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%\cite{Nyffeler:2002cf}
%\bibitem{Nyffeler:2002cf}
A.~Nyffeler,
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%Energy Physics Seminar, University of Hawaii, March 2002.

\bibitem{HVPee}
K.~Melnikov, \ijmpa{16}{2001}{4591}
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F.~Jegerlehner, \hepph{0104304};
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%
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%
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%K.~Matchev and D.~Pierce, \prd{60}{1999}{075004} and
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%H.~Baer, M.~Drees, F.~Paige, P.~Quintana and X.~Tata, \prd{61}{2000}{095007};
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%%
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%S.~Abdullin {\it et al.}  (CMS Collaboration), \hepph{9806366}.
%%
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%Collaboration), \hepph{0106315}.

\end{thebibliography}

\end{document}

