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\def\gappeq{\mathrel{\rlap {\raise.5ex\hbox{$>$}}
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{\lower.5ex\hbox{$\sim$}}}}

\def\Toprel#1\over#2{\mathrel{\mathop{#2}\limits^{#1}}}
\def\FF{\Toprel{\hbox{$\scriptscriptstyle(-)$}}\over{$\nu$}}


\title{Decline and Fall of the Standard Model?}

\author{John Ellis}

\address{Theoretical Physics Division, CERN,
        CH-1211 Geneva 23\\E-mail: John.Ellis@cern.ch}

%\author{Elsie Tan,  Jessie Tan and R. Sankaran}

%\address{World Scienfitic Publishing Co Ltd, 
%57 Shelton Street, Covent Garden\\ London WC2H 9HE, England\\
%E-mail: wspc@wspc.ox.uk}  

\twocolumn[\maketitle\abstract{
Motivations for physics beyond the Standard Model are reviewed, with
particular emphasis on supersymmetry at the TeV scale. Constraints on the
minimal supersymmetric extension of the Standard Model with universal soft
supersymmetry-breaking terms (CMSSM) are discussed. These are also
combined with the supersymmetric interpretation of the anomalous magnetic
moment of the muon. The prospects for observing supersymmetry at
accelerators are reviewed using benchmark scenarios to focus the
discussion. Prospects for other experiments are discussed, including the
detection of cold dark matter, $\mu \to e \gamma$ and related processes,
as well as proton decay. \\
~~\\
\begin{center}
CERN-TH/2001-275~~~~~~
\end{center}
}]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The empire of the Standard Model has resisted all attacks by accelerator
data. Nevertheless, we theorists are driven to overcome our ignorance of
the barbarian territory beyond its frontiers. In the gauge sector, the
Standard Model has three independent gauge couplings and (potentially) a
CP-violating phase in QCD. In the Yukawa sector, it has six random-seeming
quark masses, three charged-lepton masses, three weak mixing angles and
the Kobayashi-Maskawa phase. Finally, the symmetry-breaking sector has at
least two free parameters. Moreover, this list of 19 parameters in the
Standard Model begs the more fundamental questions of the origins of the
particle quantum numbers. As if this were not enough, non-accelerator
neutrino experiments~\cite{nus} now convince us that we need three
neutrino mass parameters, three neutrino mixing angles and three
CP-violating phases in the neutrino sector: one observable in oscillation
experiments and two that affect $\beta\beta_{0\nu}$ experiments, without
even talking about the mechanism of neutrino mass generation. Moreover, we
should not forget about gravity, with at least two parameters to
understand: Newton's constant $G_N \equiv m_P^{-2} \sim (10^{19}$
GeV)$^{-2}$ and the cosmological `constant', which recent data suggest is
non-zero~\cite{lambda}, and may not even be constant. Talking of 
cosmology, we would need at least one
extra parameter to produce an inflationary potential, and at least one
other to generate the baryon asymmetry, which cannot be explained within
the Standard Model.

Confronted by our ignorance of so much barbarian territory, we legions
of theorists organize our explorations on three main fronts: {\it
unification} -- the quest for a single framework for all gauge
interactions, {\it flavour} -- the quest for explanations of the
proliferation of quark and lepton types, their mixings and CP violating
phases, and {\it mass} -- the quest for the origin of particle masses and
an explanation why they are so much smaller than the Planck mass $m_P \sim
10^{19}$ GeV. Beyond all these beyonds, other scouting parties of
theorists seek a {\it Theory of Everything} that includes gravity,
reconciles it with quantum mechanics, explains the origin of space-time
and why we live in four dimensions (if we do so).

Physics beyond the Standard Model is therefore a very broad subject.
However, many aspects are discussed here by other speakers: electroweak
flavour physics~\cite{RB}, CP violation~\cite{MN}, the Higgs
sector~\cite{FZ}, $g_\mu - 2$~\cite{g-2}, searches for new
particles~\cite{GH}, neutrinos~\cite{HM}, dark matter~\cite{KKG}, strings
and extra dimensions~\cite{LR}. Therefore, in this talk I seek a
complementary approach.

For reasons that I describe in Section 2, many theorists believe that
supersymmetry is the inescapable framework for discussing physics at the
TeV scale and beyond. In the rest of this talk, I first discuss the
constraints imposed on (the simplest) supersymmetric models by the
available experimental and cosmological constraints, then address the
prospects for understanding $g_\mu - 2$ in supersymmetric models, the
prospects for detecting sparticles directly at present and future
colliders, and the prospects for non-collider experiments, including the
searches for dark matter, $\mu\rightarrow e\gamma$ and proton decay.

\section{The Electroweak Vacuum}

The generation of particle masses requires the breaking of gauge symmetry in
the vacuum:
\beq
m_{W,Z} \not= 0 \Leftrightarrow <0\vert X_{I,I_3}\vert 0> \not= 0
\label{one}
\eeq
for some field $X$ with isospin $I$ and third component $I_3$. The measured
ratio
\beq
\rho \equiv {m^2_W\over m^2_Z ~\cos^2\theta_W} \simeq 1
\label{two}
\eeq
tells us that $X$ mainly has $I = 1/2$~\cite{RV}, which is also what is 
needed to
generate fermion masses. The key question is the nature of the field $X$:
is it elementary or composite? A fermion-antifermion condensate $v \equiv
<0\vert X\vert 0> = <0\vert\bar FF\vert 0> \not= 0$ would be analogous to
what we know from QCD, where $<0\vert\bar qq\vert 0> \not= 0$, and
conventional superconductivity, where $<0\vert e^-e^-\vert 0> \not= 0$.
However, analogous `technicolour' models of electroweak symmetry 
breaking~\cite{TC}
fail to fit the values of the radiative corrections $\epsilon_i$ to $\rho$
and other quantities extracted from the precision electroweak data
provided by LEP and other experiments, as seen in 
Fig.~\ref{fig:TC}~\cite{Aetal}. One 
cannot
exclude the possibility that some calculable variant of technicolour might
emerge that is consistent with the data, but for now we focus on
elementary Higgs models.

\begin{figure}%1
\epsfxsize180pt
\figurebox{120pt}{160pt}{generationTC.eps} 
\caption[]{Predictions for the radiative corrections $\epsilon_i$
in the Standard Model and a minimal one-generation model~\cite{EFL} are 
compared with the precision electroweak data~\cite{Aetal}. 
\label{fig:TC}}
\end{figure}


Within this framework, the data favour a relatively light Higgs boson,
with $m_H \simeq$ 115 GeV, just above the exclusion unit
provided by direct searches at LEP, being the
`most-probable'~\cite{precision}. This is one reason why many theorists
were excited by the possible sighting during the last days of LEP of a
Higgs boson, with a preferred mass of 115.6 GeV~\cite{GH}. If this were to
be confirmed, it would suggest that the Standard Model breaks down at some
relatively low energy $\lappeq 10^3$ TeV~\cite{ER}. As seen in
Fig.~\ref{fig:HR}, above this scale the effective Higgs potential of the
Standard Model becomes unstable as the quartic Higgs self-coupling is
driven negative by radiative corrections due to the relatively heavy top
quark~\cite{tdown}. This is not necessarily a disaster, and it is possible
that the present electroweak vacuum might be metastable, provided that its
lifetime is longer than the age of the Universe~\cite{Setal}. However, we
would surely feel more secure if such instability could be avoided.

\begin{figure}%1
\epsfxsize180pt
\figurebox{120pt}{160pt}{smallHR.eps}
\caption[]{The range allowed for the mass of the Higgs boson if the 
Standard Model is to remain valid up to a given scale $\Lambda$. In the 
upper part of the plane, the effective potential blows up, whereas in the 
lower part the present electroweak vacuum is unstable~\cite{tdown}.} 
\label{fig:HR} 
\end{figure}

This may be done by introducing new bosons $\phi$ coupled to the Higgs
field~\cite{ER}:
\beq
\lambda_{22}\vert H\vert^2~\vert\phi\vert^2~~:~~M^2_0 \equiv \lambda_{22}
v^2
\label{three}
\eeq
As seen in Fig.~\ref{fig:ER}a, the effective potential is very sensitive 
to the
coupling parameter $M_0$: for $M_0 \leq$ 70.9 GeV in this example, the
potential still collapses, whereas for $M_0 \geq$ 71.0 GeV the potential
blows up instead. Thus the bosonic coupling (\ref{three}) must be finely
tuned~\cite{ER}. This occurs naturally in supersymmetry, in which the 
Higgs bosons
are accompanied by fermionic partners $\tilde H$. As seen in 
Fig.~\ref{fig:ER}b,
again the Higgs coupling blows up in the absence of the $\tilde H$,
whereas it is well behaved in the minimal supersymmetric extension of the
Standard Model (MSSM).

\begin{figure}%1
\epsfxsize180pt
\figurebox{120pt}{160pt}{smallER1.2.eps}   
%\caption{(a)}
%\label{fig:HR}
\epsfxsize180pt
%\figurebox{120pt}{160pt}{ER2.eps}   
\caption[]{(a) If the quartic coupling $M_0$ (\ref{three}) is too large, the 
effective potential blows up (solid line), whereas it is unstable if $M_0$
is too 
small (dotted line), indicating a need for fine tuning. (b) This occurs
naturally in a 
supersymmetric model (solid line) but not if the $\tilde H$ are omitted 
(dotted line)~\cite{ER}.} 
\label{fig:ER} 
\end{figure}


The avoidance of fine tuning has long been the primary motivation for
supersymmetry at the TeV scale~\cite{hierarchy}. This issue is normally 
formulated in
connection with the hierarchy problem: why/how is $m_W \ll m_P$, or
equivalently why is $G_F \sim 1/m^2_W \gg G_N = 1/m^2_P$, or equivalently
why does the Coulomb potential in an atom dominate over the Newton
potential, $e^2 \gg G_N m_p m_e \sim (m/m_P)^2$, where $m_{p,e}$ are the
proton and electron masses? One might think naively that it would be
sufficient to set $m_W \ll m_P$ by hand. However, radiative corrections
tend to destroy this hierarchy. For example, one-loop diagrams generate
\beq
\delta m^2_W = {\cal O}\left({\alpha\over\pi}\right)~\Lambda^2 \gg m^2_W
\label{four}
\eeq
where $\Lambda$ is a cut-off representing the appearance of new physics, and
the
inequality in (\ref{four}) applies if $\Lambda\sim 10^3$ TeV, and even 
more so if  $\Lambda \sim m_{GUT} \sim
10^{16}$
GeV or $ \sim m_P \sim 10^{19}$ GeV. If the radiative corrections to a 
physical
quantity
are much larger than its measured values, obtaining the latter requires
strong
cancellations, which in general require fine tuning of the bare input
parameters.
However, the necessary cancellations are natural in supersymmetry, where 
one has
equal
numbers of bosons $B$ and fermions $F$ with equal couplings, so that
(\ref{four})
is replaced by
\beq
\delta m^2_W = {\cal O}\left({\alpha\over\pi}\right)~\vert m^2_B - 
m^2_F\vert~.
\label{five}
\eeq
The residual radiative correction is naturally small if
\beq
\vert m^2_B - m^2_F\vert \lappeq 1~{\rm TeV}^2
\label{six}
\eeq
Note that this argument is logically distinct from that in the previous
paragraph. There supersymmetry was motivated by the control of logarithmic
divergences, and here by the absence of quadratic divergences.

\section{The MSSM}

The MSSM has the same gauge interactions as the Standard Model, and
similar Yukawa couplings. A key difference is the necessity of two Higgs
doublets, in order to give masses to all the quarks and leptons, and to
cancel triangle anomalies. This duplication is important for
phenomenology: it means that there are five physical Higgs bosons, two
charged $H^\pm$ and three neutral $h, H, A$. Their quartic
self-interactions are determined by the gauge interactions, solving the
vacuum instability problem mentioned above and limiting the possible mass
of the lightest neutral Higgs boson. However, the doubling of the Higgs
multiplets introduces two new parameters: $\tan\beta$, the ratio of Higgs
vacuum expectation values and $\mu$, a parameter mixing the two Higgs
doublets.

There are two key experimental hints in favour of supersymmetry. One is
provided by the LEP measurements of the gauge couplings, that are in very
good agreement with supersymmetric GUTs~\cite{susyGUTs} if sparticles 
weigh $\sim 1$~TeV. This agreement
appears completely fortuitous in composite Higgs models~\cite{TC}, and is 
difficult
(though not impossible~\cite{DDG}) to reproduce accurately in models with 
large extra
dimensions~\cite{GR}. The other experimental hint is provided by the
preference of the precision electroweak data for a relatively light Higgs
boson~\cite{precision}. In the MSSM, one predicts $m_h \lappeq$ 130
GeV~\cite{Hmass,FZ}, right in the preferred range, whereas composite Higgs
model generally predict heavier effective Higgs masses.

The gauge symmetries of the MSSM would permit the inclusion of interactions
that violate baryon number and/or lepton number~\cite{Rviol}:
\beq
\lambda LLE^c + \lambda^\prime QD^cL + \lambda^" U^cD^cD^c
\label{seven}
\eeq
where the $L(Q)$ are left-handed lepton (quark) doublets and the
$E^c(D^c,U^d)$ are conjugates of the right-handed lepton (quark) singlets.
Their possible appearance is ignored in this talk, in which case the
lightest supersymmetric particle is stable, and hence a candidate for dark
matter~\cite{EHNOS}. In the following this is assumed to be a neutralino, 
i.e., a
mixture of the $\tilde\gamma, \tilde H$ and $\tilde Z$.

The final ingredient in the MSSM is the soft supersymmetry breaking, in
the form of scalar masses $m_0$, gaugino masses $m_{1/2}$ and trilinear
couplings $A$~\cite{DG}. These are presumed to be inputs from physics at 
some
high-energy scale, e.g., from some supergravity or superstring theory,
which then evolve down to lower energy scale according to well-known
renormalization-group equations. In the case of the Higgs multiplets, this
renormalization can drive the effective mass-squared negative, triggering
electroweak symmetry weaking~\cite{renn}. In this talk, it is assumed that 
the $m_0$
are universal at the input scale~\footnote{Universality between the
squarks and sleptons of different generations is motivated by upper limits
on flavour-changing neutral interactions~\cite{FCNC}, but universality 
between the
soft masses of the $L, E^c, Q^c, D^c$ and $U^c$ is not so well
motivated.}, as are the $m_{1/2}$ and $A$ parameters. In this case the
free parameters are
\beq
m_0, m_{1/2}, A \quad {\rm and}\quad \tan\beta~,
\label{eight}
\eeq
with $\mu$ being determined by the electroweak vacuum conditions, up to a 
sign.

This constrained MSSM (CMSSM) serves \\
as the basis for the subsequent
discussion. It has the merit of being sufficiently specific that the
different phenomenological constraints can be combined meaningfully. On
the other hand, it is just one of the phenomenological possibilities
offered by supersymmetry~\cite{othersusy}.

\section{Constraints on the CMSSM}

Important constraints on the CMSSM parameter space are provided by direct
searches at LEP and the Tevatron collider~\cite{GH}, as seen in
Fig.~\ref{fig:CMSSM}. One of these is the limit $m_{\chi^\pm} \gappeq$ 103
GeV provided by chargino searches at LEP, where the third significant
figure depends on other CMSSM parameters. LEP has also provided lower
limits on slepton masses, of which the strongest is $m_{\tilde e}\gappeq$
99 GeV, again depending only sightly on the other CMSSM parameters, as
long as $m_{\tilde e} - m_\chi \gappeq$ 10 GeV. The most important
constraints on the $u, d, s, c, b$ squarks and gluinos are provided by the
Tevatron collider: for equal masses $m_{\tilde q} = m_{\tilde g} \gappeq$
300 GeV. In the case of the $\tilde t$, LEP provides the most stringent
limit when $m_{\tilde t} - m_\chi$ is small, and the Tevatron for larger
$m_{\tilde t} - m_\chi$. Their effect is almost to exclude the range of
parameter space where electroweak baryogenesis is possible~\cite{leptog}.

\begin{figure}%1
\epsfxsize200pt
\figurebox{120pt}{160pt}{CMSSM1.eps}
%\caption{(a)}
%\label{fig:HR}
\epsfxsize200pt
\figurebox{120pt}{160pt}{CMSSM2.eps}
\caption[]{Compilations of phenomenological constraints on the CMSSM for 
$\tan \beta = 10$ and (a) $\mu > 0$, (b) $\mu < 0$. Representative 
contours of the selectron, chargino and Higgs masses are indicated, as 
is the likely physics reach of Run II of the Tevatron Collider in (a). The
dark 
shaded regions are excluded because the LSP is charged, whereas a 
neutralino LSP has acceptable relic density (\ref{ten}) in the 
light-shaded regions~\cite{EGNO}. The medium-shaded region in (b) is
excluded by $b \to s \gamma$~\cite{bsg}.} 
\label{fig:CMSSM} 
\end{figure}

Another important constraint is provided by the LEP limit on the Higgs mass:
$m_H
> $ 114.1 GeV. This holds in the Standard Model, for the lightest Higgs 
boson $h$
in the general MSSM for $\tan\beta \lappeq 5$, and in the CMSSM for all
$\tan\beta$, at least as long as CP is conserved~\footnote{The lower 
bound on the lightest MSSM Higgs boson may be relaxed significantly if 
CP violation feeds into the MSSM Higgs sector~\cite{CEPW}.}. Since $m_h$ 
is sensitive 
to sparticle masses, particularly $m_{\tilde t}$, via loop corrections:
\beq
\delta m^2_h \propto {m^4_t\over m^2_W}~\ln\left({m^2_{\tilde t}\over
m^2_t}\right)~ + \ldots
\label{nine}
\eeq
the Higgs limit also imposes important constraints on the CMSSM parameters,
principally $m_{1/2}$ as seen in Fig.~\ref{fig:CMSSM}.

Also shown in Fig.~\ref{fig:CMSSM} is the constraint imposed by
measurements of $b\rightarrow s\gamma$~\cite{bsg}. These agree with the
Standard Model, and therefore provide bounds on chargino and charged Higgs
masses, for example. For moderate $\tan\beta$, the $b\rightarrow s\gamma$
constraint is more important for $\mu < 0$, as seen in
Fig.~\ref{fig:CMSSM}b, but it is also significant for $\mu > 0$ when
$\tan\beta$ is large.

Fig.~\ref{fig:CMSSM} also displays the regions where the supersymmetric 
relic density $\rho_\chi = \Omega_\chi \rho_{critical}$ falls within the 
preferred range
\beq
0.1 < \Omega_\chi h^2 < 0.3
\label{ten}
\eeq
The upper limit is rigorous, since astrophysics and cosmology tell us that
the
total matter density $\Omega_m \lappeq 0.4$, and the Hubble expansion rate
$h
\sim 1/\sqrt{2}$ to within about 10 \% (in units of km/s/Mpc). On the 
other hand, the lower limit
in
(\ref{ten}) is optional, since there could be other important contributions
to
the overall matter density.

As is seen in Fig.~\ref{fig:CMSSM}, there are generic regions of the CMSSM 
parameter space
where the relic density falls within the preferred range (\ref{ten}). What
goes
into the calculation of the relic density? It is controlled by the
annihilation
rate~\cite{EHNOS}:
\beq
\rho_\chi = m_\chi n_\chi : n_\chi \sim {1\over
\sigma_{ann}(\chi\chi\rightarrow\ldots)}
\label{eleven}
\eeq
and the typical annihilation rate $\sigma_{ann} \sim 1/m_\chi^2$. For this
reason, the relic density typically increases with the relic mass, and
this combined with the upper bound in (\ref{ten}) then leads to the common
expectation that $m_\chi \lappeq$ 1 TeV. However, there are various ways
in which the generic upper bound on $m_\chi$ can be increased along
filaments in the $(m_{1/2},m_0)$ plane. For example, if the
next-to-lightest sparticle (NLSP) is not much heavier than $\chi$: $\Delta
m/m_\chi \lappeq 0.1$, the relic density may be suppressed by
coannihilation: $\sigma (\chi + $NLSP$ \rightarrow \ldots )$~\cite{coann}. 
In this way,
the allowed CMSSM region may acquire a `tail' extending to large $m_\chi$,
as in the case where the NLSP is the lighter stau: $\tilde\tau_1$ and
$m_{\tilde\tau_1} \sim m_\chi$ as seen in 
Fig.~\ref{fig:coann}~\cite{ourcoann}. Another mechanism for
extending the allowed CMSSM region to large $m_\chi$ is rapid annihilation
via a direct-channel pole when $m_\chi \sim {1\over 2} m_{Higgs, 
Z}$~\cite{funnel,EFGOSi}. This
may yield a `funnel' extending to large $m_{1/2}$ and $m_0$ at large
$\tan\beta$, as seen in Fig.~\ref{fig:funnel}~\cite{EFGOSi}. Another 
allowed region at 
large $m_{1/2}$
and $m_0$ is the `focus-point' region~\cite{focus}, which is adjacent to 
the boundary
of the region where electroweak symmetry breaking is possible, as seen in
Fig.~\ref{fig:EO}. However, in this region $m_\chi$ is not particularly 
large.

\begin{figure}%1
\epsfxsize200pt
\figurebox{120pt}{160pt}{coann1.eps}
\caption[]{The large-$m_{1/2}$ `tail' of the $\chi - {\tilde \tau_1}$ 
coannihilation region 
for $\tan \beta = 10$ and $\mu < 0$~\cite{ourcoann}.}
\label{fig:coann}
\end{figure}

\begin{figure}%1
\epsfxsize200pt 
\figurebox{120pt}{160pt}{funnel.eps}
\caption[]{The region where the cosmological relic density is in the 
preferred range (\ref{ten}) for $\tan \beta = 50$ and $\mu > 0$. Note 
the rapid-annihilation `funnel' at intermediate $m_0 / 
m_{1/2}$~\cite{EFGOSi}.} 
\label{fig:funnel} 
\end{figure}  

\begin{figure}%1
\epsfxsize180pt 
\figurebox{120pt}{160pt}{EO.eps}
\caption[]{The $m_{1/2}, m_0$ plane for $\tan \beta = 10$ and $\mu > 0$, 
including the `focus-point' region~\cite{focus} at large $m_0$, close to
the boundary of the shaded region where electroweak symmetry breaking
occurs, and exhibiting 
contours of the cosmological sensitivity (\ref{twelve})~\cite{EO}.} 
\label{fig:EO}
\end{figure}  

These filaments extending the preferred CMSSM parameter space are clearly
exceptional, in some sense, so it is important to understand the sensitivity
of the relic density to input parameters, unknown higher-order effects, 
etc. One proposal is the relic-density fine-tuning measure~\cite{EO}
\beq
\Delta^\Omega \equiv \sqrt{\sum_i ~~{\partial\ln (\Omega_\chi h^2)\over
\partial
\ln a_i}}
\label{twelve}
\eeq
where the sum runs over the input parameters, which might include
(relatively) poorly-known Standard Model quantities such as $m_t$ and
$m_b$, as well as the CMSSM parameters $m_0, m_{1/2}$, etc. As seen in
Fig.~\ref{fig:EO}, the sensitivity $\Delta^\Omega$ (\ref{twelve}) is 
relatively small
in the `bulk' region at low $m_{1/2}$, $m_0$, and $\tan\beta$. However, it
is somewhat higher in the $\chi - \tilde\tau_1$ coannihilation `tail', and
at large $\tan\beta$ in general. The sensitivity measure $\Delta^\Omega$
(\ref{twelve}) is particularly high in the rapid-annihilation `funnel' and
in the `focus-point' region. This explains why published relic-density
calculations may differ in these regions~\cite{otherOmega}, whereas they 
agree well when
$\Delta^\Omega$ is small: differences may arise because of small
differences in the treatments of the inputs.


It is important to note that the relic-density fine-tuning measure
(\ref{twelve}) is distinct from the traditional measure of the fine-tuning
of the electroweak scale~\cite{EENZ}:
\beq
\Delta_i \equiv {\partial \ln m_W\over \partial \ln a_i}
\label{thirteen}
\eeq
This electroweak fine-tuning is a completely different issue, and values
of the $\Delta_i$ are not necessarily related to values of
$\Delta^\Omega$.  Electroweak fine-tuning is sometimes used as a criterion
for restricting the CMSSM parameters. However, the interpretation of the
$\Delta_i$ (\ref{thirteen}) is unclear. How large a value of $\Delta_i$ is
tolerable? Different physicists may well have different pain thresholds.
Moreover, correlations between input parameters may reduce its value in
specific models.

\section{Muon Anomalous Magnetic Moment}

As reported at this meeting~\cite{g-2}, the BNL E821 experiment has 
recently reported a
2.6-$\sigma$ deviation of $a_\mu\equiv {1\over 2} (g_\mu -2)$ from the
Standard Model prediction~\cite{BNL}:
\beq
a_\mu^{exp} - a_\mu^{th} = (43 \pm 16) \times 10^{-10}
\label{forteen}
\eeq
The largest contribution to the error in (\ref{forteen}) is the
statistical error of the experiment, which will soon be significantly
reduced, as many more data have already been recorded. The next-largest
error is that due to strong-interaction uncertainties in the Standard
Model prediction. Recent estimates converge on an estimate of about
7$\times 10^{-10}$ for the error in the hadronic vacuum polarization
constribution to (\ref{forteen})~\cite{DHSNTY}, and the error in the 
hadron
light-by-light scattering contribution is generally thought to be 
smaller~\cite{JB}.
Therefore, if the central value in (\ref{forteen}) does not change
substantially with the new data, this would be strong evidence for new
physics at the TeV scale.

As many authors have 
pointed 
out~\cite{susygmu}, the 
discrepancy 
(\ref{forteen}) could
well be explained by supersymmetry if $\mu > 0$ and $\tan\beta$ is not too
small, as exemplified in Fig.~\ref{fig:g-2}. Good consistency with all 
the experimental and cosmological
constraints on the CMSSM is found for $\tan\beta\lappeq 10$ and $m_\chi
\simeq$ 150 to 350 GeV. Already before the measurement (\ref{forteen}),
the LHC was thought to have a good chance of discovering 
supersymmetry~\cite{susyLHC}. If
the result (\ref{forteen}) were to be confirmed, this would be almost
guaranteed, as we now discuss.

\begin{figure}%1
\epsfxsize180pt
\figurebox{120pt}{160pt}{g-2.eps}
\caption[]{The medium-shaded region is that compatible with the BNL E821 
measurement of $g_\mu - 2$ at the 2-$\sigma$ level~\cite{g-2,BNL}, the
light-shaded region has a relic density in the preferred range
(\ref{ten}), and the dark-shaded region does not have a neutralino LSP.
Good 
compatibility is found between $g_\mu - 2$ and the other phenomenological
constraints for $\tan \beta \sim 5$ or more~\cite{susygmu}.} 
\label{fig:g-2}
\end{figure}

\section{Prospects for Observing Supersymmetry at Accelerators}

As an aid to the assessment of the prospects for detecting sparticles at
different accelerators, benchmark sets of supersymmetric parameters have
often been found useful~\cite{benchmarks}, since they provide a focus for
concentrated discussion. A set of post-LEP benchmark scenarios in the
CMSSM has recently been proposed~\cite{benchmark}, and are illustrated
schematically in Fig.~\ref{fig:Bench}. They take into account
the direct searches for sparticles and Higgs bosons, $b\rightarrow
s\gamma$ and the preferred cosmological density range (\ref{ten}). About a
half of the proposed benchmark points are consistent with $g_\mu -2$
(\ref{forteen}) at the $2-\sigma$ level, but this was not imposed as an
absolute requirement.

\begin{figure}%1
\epsfxsize180pt
\figurebox{120pt}{160pt}{Bench.eps}
\caption[]{
Schematic overview of the benchmark points proposed in~\cite{benchmark}.
They were chosen to be compatible with the indicated experimental
constraints, as well as have a relic density in the preferred range
(\ref{ten}). The points are intended to illustrate the range of available
possibilities.} 
\label{fig:Bench} 
\end{figure}

The proposed points were chosen not to provide an `unbiased' statistical
sampling of the CMSSM parameter space, whatever that means in the absence
of a plausible {\it a priori} measure, but rather are intended to
illustrate the different possibilities that are still allowed by the
present constraints~\cite{benchmark}. Five of the chosen points are in the
`bulk' region at small $m_{1/2}$ and $m_0$, four are spread along the
coannihilation `tail' at larger $m_{1/2}$ for various values of
$\tan\beta$, two are in the `focus-point' region at large $m_0$, and two
are in rapid-annihilation `funnels' at large $m_{1/2}$ and $m_0$. The
proposed points range over the allowed values of $\tan\beta$ between 5 and
50.  Most of them have $\mu > 0$, as favoured by $g_\mu - 2$, but there
are two points with $\mu < 0$.

Various derived quantities in these
supersymmetric benchmark scenarios, including the relic density, $g_\mu -
2, b \rightarrow s\gamma$, electroweak fine-tuning $\Delta$ and the
relic-density sensitivity $\Delta^\Omega$, are given in~\cite{benchmark}.
These enable
the reader to see at a glance which models would be excluded by which
refinement of the experimental value of $g_\mu - 2$. Likewise, if you find
some amount of fine-tuning uncomfortably large, then you are free to
discard the corresponding models.

The LHC collaborations have analyzed their reach for sparticle detection
in both generic studies and specific benchmark scenarios proposed
previously~\cite{susyLHC}. Based on these studies,
Fig.~\ref{fig:Manhattan} displays estimates how many different sparticles
may be seen at the LHC in each of the newly-proposed benchmark
scenarios~\cite{benchmark}. The lightest Higgs boson is always found, and
squarks and gluinos are usually found, though there are some scenarios
where no sparticles are found at the LHC. The LHC often misses heavier
weakly-interacting sparticles such as charginos, neutralinos, sleptons and
the other Higgs bosons.

\begin{figure}%1
\epsfxsize200pt 
\figurebox{120pt}{160pt}{Manhattan.eps}
\caption[]{Estimates of the numbers of different types of CMSSM particles 
that may be detectable~\cite{benchmark} at (a) the LHC, (b) a 1-TeV linear 
$e^+ e^-$ collider~\cite{LC}, and (c,d) a 3(5)-TeV $e^+ e^-$~\cite{CLIC}
or $\mu^+ 
\mu^-$ collider~\cite{MC}. Note the complementarity between the sparticles 
detectable at the LHC and at a 1-TeV linear $e^+ e^-$ collider.} 
\label{fig:Manhattan} 
\end{figure}  

The physics capabilities of linear $e^+e^-$ colliders are amply documented
in various design studies~\cite{LC}. Not only is the lightest MSSM Higgs
boson observed, but its major decay modes can be measured with high
accuracy, as seen in Fig.~\ref{fig:Battaglia}. Moreover, if sparticles are
light enough to be produced, their masses and other properties can be
measured very precisely, enabling models of supersymmetry breaking to be
tested~\cite{Zerwas}.

\begin{figure}%1
\epsfxsize200pt 
\figurebox{120pt}{160pt}{Battaglia.eps}
\caption[]{Analysis of the accuracy with which Higgs decay branching 
ratios may be measured with a linear $e^+ e^-$ collider~\cite{BD}.}
\label{fig:Battaglia}
\end{figure}  

As seen in Fig.~\ref{fig:Manhattan}, the sparticles visible at an $e^+e^-$
collider largely complement those visible at the LHC~\cite{benchmark}. In
most of benchmark scenarios proposed, a 1-TeV linear collider would be
able to discover and measure precisely several weakly-interacting
sparticles that are invisible or difficult to detect at the LHC. However,
there are some benchmark scenarios where the linear collider (as well as
the LHC) fails to discover supersymmetry. Only a linear collider with a
higher centre-of-mass energy appears sure to cover all the allowed CMSSM
parameter space, as seen in the lower panels of Fig. 12, which illustrate
the physics reach of a higher-energy lepton collider, such as
CLIC~\cite{CLIC} or a multi-TeV muon collider~\cite{MC}.


\section{Prospects for Other Experiments}

\noindent
\underline{Detection of cold dark matter}\\

Fig.~\ref{fig:DM} shows rates for the elastic spin-independent scattering
of supersymmetric relics~\cite{EFFMO}, including upper limits from the
UKDMC, CDMS and Heidelberg experiments~\cite{KKG}, as well as the range
suggested by the DAMA collaboration~\cite{DAMA}. Also shown are the rates
calculated in the proposed benchmark scenarios discussed in the previous
section, which are considerably below the DAMA range, but may be within
reach of future projects. Indirect searches for supersymmetric dark matter
via the products of annihilations in the galactic halo or inside the Sun
also have prospects in some of the benchmark scenarios~\cite{EFFMO}.

\begin{figure}%1
\epsfxsize180pt 
\figurebox{120pt}{160pt}{smallEFFMO102.eps}
%\caption{Caption for the figure will come here.}
%\label{fig:coann}
%\epsfxsize180pt 
%\figurebox{120pt}{160pt}{EFFMO2.eps}
\caption[]{Rates for the elastic spin-independent scattering  
of supersymmetric relics on (a) protons and (b) neutrons calculated in 
benchmark scenarios~\cite{EFFMO}, compared with upper limits from the 
UKDMC, CDMS and Heidelberg experiments~\cite{KKG}, as well as the range 
suggested by the DAMA collaboration~\cite{DAMA}.}
\label{fig:DM}
\end{figure}  


\noindent
\underline{$\mu\rightarrow e\gamma$ and related processes}\\

The BNL E821 report of a possible deviation from the Standard Model
suggests that a non-trivial $\mu-\mu-\gamma$ vertex is generated at a
scale $\lappeq$ 1 TeV. Neutrino oscillations indicate that there are
$\Delta L_\mu \not= 0$ processes~\cite{HM}, so it is natural to expect
that there might also be a non-trivial $\mu-e-\gamma$ vertex. This is
indeed the case in a generic supersymmetric GUT, where neutrino mixing
induces slepton mixing~\cite{mueg}. Within this framework, the measurement
of $g_\mu -2$ fixes the sparticle scale, and $\Gamma (\mu\rightarrow
e\gamma)$ may then be calculated within any given flavour texture. Very
approximately, if $g_\mu-2$ is within one or two $\sigma$ of the present
central value, one may expect $B(\mu\rightarrow e\gamma)$ with one or two
orders of magnitude of the present experimental upper limit, as 
illustrated in Fig.~\ref{fig:mueg}~\cite{CEGL}.

\begin{figure}%1
\epsfxsize190pt
\figurebox{120pt}{160pt}{mueg.eps}
\caption[]{Illustration in one particular lepton-flavour texture model, 
for $\tan \beta = 30$ and $\mu > 0$,
indicating that $\mu \to e \gamma$ decay may occur at a rate close to 
the present experimental upper limit, in the CMSSM with parameters 
chosen~\cite{CEGL} to match the measured value of 
$g_\mu-2$~\cite{g-2,BNL}.} \label{fig:mueg} \end{figure}

The decay $\mu\rightarrow 3e$ and $\mu\rightarrow e$ conversion on nuclei
are expected to occur with branching ratios within two or three orders of
magnitude of $B(\mu\rightarrow e\gamma)$, and it is in principle possible
to measure CP violation in $\mu\rightarrow 3e$ decay. This may provide
another interesting interface with neutrino physics and
cosmology~\cite{CPmu3e}. The minimal supersymmetric seesaw model has six
CP-violating phases: the MNS phase $\delta$, two light-neutrino Majorana
phases, and three phases arising from neutrino Dirac Yukawa couplings,
which may be responsible for our existence via leptogenesis in the early
Universe~\cite{HM}.  The CP-violating neutrino phases induce phases in
slepton mass matrices, which may show up in $\mu\rightarrow 3e$ decay,
$\tau\rightarrow 3e/\mu$ decays and leptonic electric dipole moments. In
principle, the leptogenesis phases might be obtainable by comparing
CP-violating measurements in the charged-lepton and neutrino
sectors~\cite{CPmu3e}.

\noindent
\underline{Proton decay}\\

This could be within reach, with $\tau (p\rightarrow e^+\pi^0)$ via a
dimension-six operator possibly $\sim 10^{35}y$ if $m_{GUT} \sim 10^{16}$
GeV as expected in a minimal supersymmetric GUT. Such a model also
suggests that $\tau (p\rightarrow \bar\nu K^+) < 10^{32} y$ via
dimension-five operators~\cite{dim5}, unless measures are taken to
suppress them~\cite{fsu5}. This provides motivation for a next-generation
megaton experiment that could detect proton decay as well as explore new
horizons in neutrino physics~\cite{UNO}.

\section{Conclusions}

As we have seen, future colliders such as the LHC and a TeV-scale linear
$e^+e^-$ collider have good prospects of discovering supersymmetry and
making detailed measurements. In parallel, $B$ and $\nu$ factories have
good prospects of making inroads on the flavour and unification problems.
Searches for dark matter, stopped-muon experiments and searches for proton
decay also have interesting prospects.

Looking further beyond the Standard Model, how can one hope to test a
Theory of Everything, including quantum gravity?  This should be our
long-term ambition, our analogue of the `faint blue dot' towards which
exoplanetary science is directed, and which motivates much of their
funding. Testing a quantum theory of gravity will be relatively easy if
there are large extra dimensions~\cite{LR,AB}. Much more challenging would 
be
the search for observable effects if the gravitational scale turns out,
after all, to be of the same order as the Planck mass $\sim 10^{19}$ GeV.
Perhaps the only way to reconcile relativity with quantum mechanics is to
modify one or the other, or both~\cite{EMN}? Testing the Theory of
Everything may require thinking beyond the standard `Beyond the Standard
Model' box.

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J.~Ellis, K.~Enqvist, D.~V.~Nanopoulos and F.~Zwirner,
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H.~N.~Brown {\it et al.}  [Muon g-2 Collaboration],
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R.~Alemany, M.~Davier and A.~Hocker,
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Phys.\ Lett.\ B {\bf 435} (1998) 427
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S.~Narison,
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Phys.\ Lett.\ B {\bf 513} (2001) 53
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J.~F.~De Troconiz and F.~J.~Yndurain,
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M.~Hayakawa and T.~Kinoshita,
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L.~L.~Everett, G.~L.~Kane, S.~Rigolin and L.~Wang,
Phys.\ Rev.\ Lett.\  {\bf 86}, 3484 (2001)
;
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J.~L.~Feng and K.~T.~Matchev,
Phys.\ Rev.\ Lett.\  {\bf 86}, 3480 (2001)
;
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%\bibitem{Baltz:2001ts}
E.~A.~Baltz and P.~Gondolo,   
Phys.\ Rev.\ Lett.\  {\bf 86}, 5004 (2001)
;
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%\bibitem{Chattopadhyay:2001vx}
U.~Chattopadhyay and P.~Nath,
%``Upper limits on sparticle masses from g-2 and the possibility for
% discovery of SUSY at colliders and in dark matter searches,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 5854 (2001)
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S.~Komine, T.~Moroi and M.~Yamaguchi,
Phys.\ Lett.\ B {\bf 506}, 93 (2001)
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S.~P.~Martin and J.~D.~Wells,
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Phys.\ Rev.\ D {\bf 64}, 035003 (2001)
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%\bibitem{Baer:2001kn}
H.~Baer, C.~Balazs, J.~Ferrandis and X.~Tata,
%``Impact of muon anomalous magnetic moment on supersymmetric models,''
Phys.\ Rev.\ D {\bf 64}, 035004 (2001)
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J.~Ellis, D.~V.~Nanopoulos and K.~A.~Olive,
Phys.\ Lett.\ B {\bf 508} (2001) 65
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R.~Arnowitt, B.~Dutta, B.~Hu and Y.~Santoso,
Phys.\ Lett.\ B {\bf 505} (2001) 177  
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ATLAS Collaboration, {\it ATLAS detector and physics
performance
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S.~Abdullin {\it et al.}  [CMS Collaboration],
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S.~Abdullin and F.~Charles,
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I.~Hinchliffe, F.~E.~Paige, M.~D.~Shapiro, J.~Soderqvist and W.~Yao,
Phys.\ Rev.\ D {\bf 55} (1997) 5520;
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TESLA Technical Design Report, DESY-01-011,
Part III, {\it Physics at an $e^+e^-$ Linear Collider} (March 2001).

\bibitem{benchmark}
M.~Battaglia {\it et al.}, \\
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LBL-46299, Jul 2000, ;
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R.~W.~Assmann {\it et al.} [CLIC Study Team], {\it A 3-TeV
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G.~A.~Blair, W.~Porod and P.~M.~Zerwas,
Phys.\ Rev.\ {\bf D63} (2001) 017703
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M.~Battaglia and K.~Desch,
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J.~Ellis, J.~L.~Feng, A.~Ferstl, K.~T.~Matchev and K.~A.~Olive, \\
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J.~Hisano, T.~Moroi, K.~Tobe and M.~Yamaguchi,
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D.~F.~Carvalho, J.~R.~Ellis, M.~E.~Gomez and S.~Lola,
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J.~R.~Ellis, J.~Hisano, S.~Lola and M.~Raidal,
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H.~Murayama and A.~Pierce,
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\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Guidelines}%1

\subsection{Producing the Hard Copy}\label{subsec:prod}%1.1

The hard copy may be produced using the instructions given in 
the file {\it stlread.1st}, which are repeated in this section. 
You should have three files in total.\footnote{
You can obtain these files from our WWW pages at:\\
\noindent {\sf www.wspc.com/others/style\_files/authorsf\_new.html}\\
\noindent {\sf www.wspc.com.sg/others/style\_files/authorsf\_new.html}\\
\noindent or by ftp from\\
\noindent {\sf ftp.wspc.com.sg,\\ cd /pub/style\_files/proceedings/stwocol/.}
}

\noindent {\sf readme.txt} ---  the preliminary guide.

\noindent {\sf  ws-p10x7.cls}   ---  the style file that provides the 
		     higher level latex2e commands for the
		     proceedings. Don't change 
		     these parameters.

\noindent {\sf  ws-p10x7.tex}  ---  the main text.  

You can delete our  sample text and replace it with your
  own contribution to the volume, however we  
recommend keeping an initial version of the 
file for reference.  Strip off any mail headers   
and then latex the tex file.  The command for latexing is {\sf latex  ws-p10x7}, do this twice to sort out the cross-referencing. 

We have provided the running page number, jobname and submission informations
at the bottom of each page with overfull box for identification purpose only.
 The same will be removed at the final
stage and the new folio will be set by us according to the lineup of the volume.


If you wish to use some other form of 
word-processor, some guidelines are given in Sec.~\ref{subsec:wpp} below. 
These files will work with standard Latex2e. If there is an 
abbreviation defined in the new definitions at the top of the 
file {\it  ws-p10x7.tex} that conflicts with one of your own macros, 
then delete the appropriate command and revert to longhand. 
Failing that, please consult your local texpert to check for 
other conflicting macros that may be unique to your computer 
system.  Page numbers are included at the bottom of the page for 
your guidance. Do not worry about the final pagination of the 
volume which will be done after you submit the paper.


\begin{figure}%1
\epsfxsize120pt
\figurebox{120pt}{160pt}{}
\caption[]{Caption for the figure will come here.}
\label{fig:radk}
\end{figure}

\begin{table*}[t]
\begin{equation}%1
U = D(\delta_1 , \delta_2 , \delta_3 , \delta_4)
R_{12}(a,\delta_5)R_{13}(b,\delta_6)R_{14}(c,\delta_7)R_{23}(d,\delta_8)
R_{24}(e,\delta_9)R_{34}(f,\delta_{10})
\label{eq:murnf}
\end{equation}
\end{table*}

\subsection{Using Other Word-Processing Packages}\label{subsec:wpp}%1.1

If you want to use some other form of word-processor to 
construct your output, and are using the final hard copy version 
of these files as guidelines; then please follow the style given 
here for headings, table and figure captions, and the footnote 
and citation marks.  For this size of volume, the final page 
dimensions will be 10 by 7 inches, however you 
should submit the copy on standard A4 paper. The text area 
(which includes the page numbers) should be 8.25 by 5.6 inches 
and the separation between the columns is 0.20in.  The text should be in 10pt roman for the title, section 
heads and the body of the text, using capitals for the title and 
authors, bold face for the title and headings, and italics for 
the subheadings.  The abstract, footnotes, figure and table 
captions should be in 8pt.

It's also important to reproduce the spacing of the text and 
headings as shown here. Text should be slightly more than 
single-spaced; use a baselineskip (which is the average distance 
from the base of one line of text to the base of an adjacent 
line) of 13 pts and 10 pts for footnotes. All headings should be 
separated from the text preceding it by a baselineskip of about 
26 pts and use a baselineskip of about 18 pts for the following 
text.

Paragraphs should have a first line indented by about 0.25in 
(6mm), except where the paragraph is preceded by a heading, and 
the abstract should be indented on both sides by 0.25in (6mm)  
from the main body of the text.


\subsection{Headings and Text}

Please preserve the style of the headings, text fonts and line 
spacing to provide a uniform style for the proceedings volume.  
In a two column format there are more difficulties when finding 
suitable line and page breaks. We recommend that you leave such 
problems until preparing the final draft and after you have 
checked the placing of the two-column wide tables etc.  (See 
Sec.~\ref{sec:plac} below).


\subsection{Equations}

Equations should be confined to one column wherever possible, as 
in Eq.~(\ref{eq:it}), and the {\it eqnarray} environment may be 
used to split equations into several lines, for example in 
Eq.~(\ref{eq:sp}), or to align several equations.  An alternative 
method is given in Eq.~(\ref{eq:spa}) for long sets of equations 
where only one referencing equation number is wanted.


If it's essential to have a two-column wide equation then use 
the method
of Eq.~(\ref{eq:murnf}) above. The surrounding environment is 
important here.
In the text file {\it  ws-p10x7.tex} make sure  that you keep the 
declarations
{\it $\backslash$begin\{table$\ast$\}} and
 {\it $\backslash$end\{table$\ast$\}}
and only change the  equation and its label within the inner equation
environment.


\begin{table}%1
\caption[]{This is a Small Table.}\label{tab:smtab}
\begin{tabular}{|c|c|c|c|} 
 
\hline 
 
\raisebox{0pt}[12pt][6pt]{Title} & 
 
\raisebox{0pt}[12pt][6pt]{$\epsilon^{\prime}$} & 
 
\raisebox{0pt}[12pt][6pt]{$\lambda$} &
 
\raisebox{0pt}[12pt][6pt]{$\gamma$} \\
 
\hline
 
\raisebox{0pt}[12pt][6pt]{3.5687} & 
 
\raisebox{0pt}[12pt][6pt]{3.4567} & 
 
\raisebox{0pt}[12pt][6pt]{3.8746} & 
 
\raisebox{0pt}[12pt][6pt]{2.8934} \\
\cline{1-2}
\multicolumn{2}{|c|}{\raisebox{0pt}[12pt][6pt]{Trans Process
 for Decay}} & &\\\cline{1-2}
\raisebox{0pt}[12pt][6pt]{6.8977} & 
 
\raisebox{0pt}[12pt][6pt]{8.9087} & 
 
\raisebox{0pt}[12pt][6pt]{2.8989} & 
 
\raisebox{0pt}[12pt][6pt]{4.2928} \\\hline
\end{tabular}
\end{table}

 
For problems of placement of a wide equation, see 
Sec.~\ref{sec:plac} below.
Please note, do not use square brackets in two-column 
wide figures, tables and
equations. This is a bug due to the declaration
{\it $\backslash$twocolumn$[$...$]$} which is hidden in 
the definition of
{\it table$\ast$} and {\it figure$\ast$}.


\subsection{Tables}

The tables are designed to have a uniform style throughout the 
proceedings volume. It doesn't matter how you choose to place 
the inner lines of the table, but we would prefer the border 
lines to be of the style shown in Tables~\ref{tab:smtab} 
and~\ref{tab:exp}.  For either a single or a double column 
table, the top and bottom horizontal lines should be single 
(using {\it $\backslash$hline}), and there should be single 
vertical lines on the perimeter, (using 
{\it  $\backslash$begin\{tabular\}\{$|...|$\}}) as in 
Tables~\ref{tab:smtab} and~\ref{tab:exp}.  For the inner lines 
of the table, it looks better if they are kept to a minimum. 
We've chosen more complicated examples purely as an illustration  
of what is possible.


We recommend the use of single column-wide tables wherever possible. For the
page wide tables, use the environment given in the example of
Table~\ref{tab:exp}. Do not change the latex commands from
{\it $\backslash$begin\{table$\ast$\}} to 
{\it $\backslash$begin\{tabular\}},
or from
{\it $\backslash$end\{tabular\}} to {\it $\backslash$end\{table$\ast$\}}, 
apart
from inserting your own caption heading and table label.
The caption heading for a table should be placed at the top 
of the table.


\subsection{Figures}

The same arguments apply as are given above for tables, i.e.~it 
is preferable to have figures that fit into one column of the text. 
If this is not possible, then use the example of  
figure~\ref{fig:radish} and  use the commands 
{\it $\backslash$begin\{figure$\ast$\}} and  
{\it $\backslash$end\{figure$\ast$\}}.  The {\it $\backslash$figurebox\{\}\{\}\{\}} command is defined with three arguments. 
First argument is for the figure height, second argument is for figure 
width, and the third will be for the actual figure/image file 
name, (i.e. eps/ps file name).
 
 If you wish to `embed' a {\it eps/postscript} figure in the file, then 
include the figure/image {\it (eps/ps)} file name in the third argument with
an extension, (i.e. {\it figure.eps or figure.ps}), etc. While embedding 
the figure it is 
not necessary to fill the first and second arguments 
and if the thrid argument is empty, 
the system will read the first two arguments and generate the box based on the measurements provided.

If you like to
reduce or enlarge (scaling) the eps/ps figure,   please remove 
the \% mark
from the declaration of the {\it $\backslash$epsfxsize} and provide the exact width of the
figure.  Latex2e will scale the 
figure based on the width provided and adjust the height accordingly.
You can use any form of the units of measurement described 
in the \LaTeX\  Book. 


\begin{table*}[t]
\caption{Experimental Data bearing on $\Gamma(K \rightarrow \pi \pi \gamma)$
for the $K^0_S$, $K^0_L$ and $K^-$ mesons.\label{tab:exp}}
\begin{tabular}{|c|c|c|p{187pt}|}  
\hline
\raisebox{0pt}[16pt][6pt]{Meson} &
$\Gamma(\pi^+\pi^-)\; s^{-1}$ &
$\Gamma(\pi^+\pi^-\gamma)\; s^{-1}$ &{} \\[3pt]
\hline 
\raisebox{0pt}[16pt][6pt]{$K^0_S$} 	& $0.769 \times 10^{10}$ 	& $5.46 \times 10^7$ & 
No DE observed, not even (IB)-E1 
interference, despite large statistics, for $E^{\ast}_{\gamma} > 
20$~MeV.\\[6pt] 
\hline
\raisebox{0pt}[16pt][6pt]{$K^0_L$} &
$3.93 \times 10^4$ &
\raisebox{-6pt}[16pt][6pt]{$\displaystyle{0.90 \times 10^3\atop (\rm{DE} = 0.62 \times 10^3)}$} &
DE prominent, exceeding IB over the 
range of measurement $20 < E^{\ast}_{\gamma} < 160$~MeV.\\[6pt]
\hline\hline
\raisebox{0pt}[16pt][6pt]{}& $\Gamma(\pi^- \pi^0)\; s^{-1}$ &
$\Gamma(\pi^- \pi^0 \gamma)\; s^{-1}$ &\\[6pt] \hline
\raisebox{0pt}[16pt][6pt]{$K^-$} &  $1.711 \times 10^7$ 
&\raisebox{-6pt}[16pt][6pt]{$\displaystyle {2.22 \times 10^4\atop (\rm {DE} = 1.46 \times 10^3)}$} &
No (IB)-E1 interference seen but data shows 
excess events relative to IB over $E^{\ast}_{\gamma} = 80$ to $100$~MeV\\[6pt]
\hline
\end{tabular}
\end{table*}

If instead you wish to use some other method, then it's most  
important to leave the right amount of vertical space in the 
figure declaration to accommodate your figure (i.e.~remove the 
lines and change the space in the example).  Send the original hard copy 
of the figure on a separate page, clearly identifying where it should 
be placed on the final hard copy. The original figure you send 
should be correctly scaled as this ensures that the details will 
be visible in the final version.  The figure caption should be placed 
below the figure.

\subsection{Limitations on the Placement of Equations, Tables 
and Figures}\label{sec:plac}

In the final stages of preparing the document,
try to declare the two-column wide figures,
tables or equations at a point in
{\it  ws-p10x7.tex} that is prior to the top of the column of 
hard-copy text
where you would like the item to appear.\footnote{By `declaring'
we refer to placing the chunk of material
describing the table or whatever at a particular point in the text file.}
Very large figures and tables should be placed on a page by 
themselves. One
can use the instruction {\it $\backslash$begin\{figure$\ast$\}$[$p$]$} or
{\it $\backslash$begin\{table$\ast$\}$[$p$]$}
to position these, and they will appear on a separate page 
devoted to
figures and tables. Again, we would recommend making any necessary
adjustments to the layout of the figures and tables
only in the final draft. It is also simplest to sort out line and
page breaks in the last stages.
 
 
\subsection{Footnotes, the Bibliography, Appendices and Acknowledgments}
Acknowledgments to funding bodies etc. may be placed in a separate
section at the end of the text, before the Appendices. 
This should not
be numbered so use {\it $\backslash$section$\ast$\{Acknowledgements\}}.

It's preferable to have no appendices in a brief article, but 
if more
than one is necessary then simply copy the
{\it $\backslash$section$\ast$\{Appendix\}}
heading and type in Appendix A, Appendix B etc. between the brackets.

Footnotes are denoted by a letter superscript in the text, and references
are denoted by a number superscript.
We have used {\it $\backslash$bibitem} to produce the bibliography.
Citations in the text use the labels defined in the bibitem 
declaration,
for example, the first paper by Jarlskog\cite{ja} is cited using the command
{\it $\backslash$cite\{ja\}}.

If you more commonly use the method of square brackets in 
the line of text
for citation than the superscript method,
please note that you need  to adjust the punctuation
so that the citation command appears after the punctuation mark.

 
\subsection{Final Manuscript}
 
The final hard copy that you send must be absolutely clean and unfolded.
It will be printed directly without any further editing. Use a printer
that has a good resolution (300 dots per inch or higher). There should
not be any corrections made on the printed pages, nor should adhesive
tape cover any lettering. Photocopies are not acceptable.

The manuscript will not be reduced or enlarged when filmed so please ensure
that indices and other small pieces of text are legible.

 
\section{Sample Text}
 
The following may be (and has been) described as `dangerously irrelevant'
physics. The Lorentz-invariant phase space integral for
a general n-body decay from a particle with momentum $P$
and mass $M$ is given by:


\begin{equation}
I((P - k_i)^2, m^2_i, M) = \frac{1}{(2 \pi)^5}\!
\int\!\frac{d^3 k_i}{2 \omega_i} \! \delta^4(P - k_i).
\label{eq:it}
\end{equation}


The only experiment on $K^{\pm} \rightarrow \pi^{\pm} \pi^0 \gamma$ since 1976
is that of Bolotov {\it et al}.\cite{bu}
The photon spectrum observed certainly exceeds the IB spectrum for 
$E^{\ast}_{\gamma} \geq 70$~MeV. These authors report definite 
evidence for DE, however they ``conclude that [IB-DE] 
interference is not observed'' in their experiment but  such 
interference is expected to 
be seen. 


Although PDG has recorded the Serphukov experiment as $K^+$ 
decay, it is in fact a $K^-$ experiment. This is not a trivial 
difference. Assuming CPT invariance to be satisfied, CP 
violation is equivalent with T violation. The latter has to do 
with phases in the DE processes.  Part of the calculated phase 
will arise from the final-state interactions, but there will 
also be a non-zero phase in the initial Lagrangian which may 
feed through to the phases in the final amplitudes. These 
Lagrangian phases will be different for the $K^+$ 
and $K^-$ processes, and can give rise to differences between 
spectra and rates for $K^+$ and $K^-$ mesons.

\begin{figure*}
\epsfxsize30pc
\figurebox{16pc}{32pc}{}
\caption{Dynamics of two limit cycle oscillators. The repulsive 
case $(K<0)$ and attractive case $(K>0)$ are shown. The rate of 
phase separation, $f$, is plotted versus the phase separation 
between the oscillators. \label{fig:radish}}
\end{figure*}

 
\subsection{Parametrizations of the CKM Matrix}

It is emphasised that there are two necessary conditions 
required for any acceptable parametrization of the quark mixing 
matrix. The first is that the matrix must be unitary, and the 
second is that it should contain a CP violating phase $\delta$. 
In Sec.~\ref{sec:four} the connection between invariants (of 
form similar to J) and unitarity relations will be examined 
further for the more general $ n \times n $ case.  For the 
present, it's sufficient to note that J is equal to just twice 
the area of any one of these triangles. This does not mean that 
a non-zero J follows from unitarity alone; if J equalled zero, 
the unitarity constraint would still hold, but the triangle 
would collapse to a straight line and the measurement of J would 
be of no use in determining the existence of CP violation if the 
quark mixing matrix was in fact represented by such a matrix. 
The reason is that such a matrix is not a faithful 
representation of the group, i.e.~it does not cover all of the  
parameter space available.

 
\begin{eqnarray}
T & = & Im[V_{11} {V_{12}}^* {V_{21}}^* V_{22}]\nonumber \\[4pt]
&&{} + Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}] \nonumber \\[4pt]
&&{} - Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}] \nonumber \\[4pt]
&&{} - Im[V_{33} {V_{31}}^* {V_{13}}^* V_{11}].\label{eq:sp}
\end{eqnarray}

\noindent 
There are only 162 quark mixing matrices using these parameters 
which are to first order in the phase variable $e^{i \delta}$ as 
is the case for the Jarlskog parametrizations, and for which $J$ 
is not identically zero.\footnote{An example of a matrix which has elements
containing the phase variable $e^{i \delta}$ to second order, i.e.
elements with a phase variable $e^{2i \delta}$ is given at the 
end of this section.} It should be emphasised that 
these are physically identical and form just one true parametrization.

 
\subsection{Four and N-Generation Mixing Matrix} \label{sec:four}

Murnaghan\cite{ma} provided the most general representation of 
a $4 \times 4$ unitary matrix given in Eq.~(\ref{eq:murnf}). We 
have calculated the possible combinations and have found that 
there are eight distinct parametrizations. 


The unitary nature of the matrix imposes eight conditions on the connections
between adjacent rows and columns, analagous to the six unitarity triangles
for the three family case, but for four generations the unitarity
condition forms a quadrilateral in the imaginary plane.


We have found only one set of invariants that are independent of  
their positions in the matrix, i.e.~for which one can choose any  
element to be the `starting point' element $V_{j,\alpha}$ in the  
definitions of {\bf K}, {\bf L} \rm and {\bf M}  given below (where
the invariants are the sums or differences of the imaginary 
parts of four plaquettes). These are not however independent of  
the choice of parametrization but are invariant for each of the 
individual choices.

\begin{eqnarray}
\bf{K} & = & \Im[V_{j, \alpha} {V^*_{j,\alpha + 1}}
{V^*_{j + 1,\alpha}} V_{j + 1, \alpha + 1}] \nonumber \\[4pt]
 &&{} + \Im[V_{k, \alpha + 2} {V^*_{k,\alpha + 3}}
{V^*_{k + 1,\alpha + 2}} V_{k + 1, \alpha + 3}] \nonumber \\[4pt]
 &&{} + \Im[V_{j + 2, \beta} {V^*_{j + 2,\beta + 1}}
{V^*_{j + 3,\beta}} V_{j + 3, \beta + 1}] \nonumber \\[4pt]
 &&{} + \Im[V_{k + 2, \beta + 2} {V^*_{k + 2,\beta + 3}}
{V^*_{k + 3,\beta + 2}}] \nonumber\\[12pt]
\bf{L} & = & \Im[V_{j + 1,\alpha} {V^*_{j + 1, \alpha + 1}}
{V^*_{k, \alpha + 2}} V_{k,\alpha + 3}]  \nonumber\\[4pt]
 &&{} - \Im[V_{j, \alpha} {V^*_{j,\alpha + 1}}
{V^*_{k + 1,\alpha + 2}} V_{k + 1, \alpha + 3}]  \nonumber\\[4pt]
 &&{} + \Im[V_{j + 3,\beta} {V^*_{j + 3, \beta + 1}}
{V^*_{k + 2, \beta + 2}} V_{k + 2,\beta + 3}]  \nonumber\\[4pt]
 &&{} - \Im[V_{j + 2, \beta} {V^*_{j + 2,\beta + 1}}
{V^*_{k + 3,\beta + 2}} V_{k + 3, \beta + 3}] \nonumber\\[12pt]
 \bf{M} & = & \Im[V_{j,\alpha + 1} {V^*_{j, \alpha}}
{V^*_{j + 1, \alpha + 1}} V_{j + 1,\alpha}]  \nonumber\\[4pt]
 &&{} + \Im[V_{k, \alpha + 2} {V^*_{k,\alpha + 3}}
{V^*_{k + 1,\alpha + 2}} V_{k + 1, \alpha + 3}]  \nonumber \\[4pt]
 &&{} + \Im[V_{j + 2,\beta + 1} {V^*_{j + 2, \beta}}
{V^*_{j + 3, \beta + 1}} V_{j + 3,\beta}] \nonumber\\[4pt]
 &&{} + \Im[V_{k + 2, \beta + 2} {V^*_{k + 2,\beta + 3}}
{V^*_{k + 3,\beta + 2}}],
\label{eq:spa}
\end{eqnarray}

\noindent 
where $ k = j$ or $j+1$ and $\beta = \alpha$ or $\alpha+1$, but if
$k = j + 1$, then $\beta \neq \alpha + 1$ and similarly, if
$\beta = \alpha + 1$ then $ k \neq j + 1$.
 
\section*{Acknowledgments}
This is where one places acknowledgments for funding bodies etc.
Note that there are no section numbers for the Acknowledgments and
Appendix. The style file will automatically generate the heading 
for references.
 
\begin{appendix}
%\section*{Appendix}
We can insert an appendix here and place equations so that they are
given numbers such as Eq.~(\ref{eq:app}). 
\begin{equation}
x = y.
\label{eq:app}
\end{equation}
\end{appendix}
 
\begin{thebibliography}{99}

\bibitem{ja}
C Jarlskog in {\it CP Violation}, ed. C Jarlskog
(World Scientific, Singapore, 1988).

\bibitem{ma}
L. Maiani, {\it Phys. Lett.} B {\bf 62}, 183 (1976).

\bibitem{bu}
J.D. Bjorken and I. Dunietz, {\it Phys. Rev.} D {\bf 36}, 2109 (1987).

\end{thebibliography}
 
\end{document}


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