\def\E{\,\rlap/\!E_T}
\def\srts{\sigma_{\!\!\!\sqrt s}^{\vphantom y}}
\def\mumu{$\mu^+\mu^-$}
%\def\etal{{\it et al.}}

%% to help placement of floats:   (normally helps, but not this time)
\renewcommand{\topfraction}{1.0}   
\renewcommand{\bottomfraction}{1.0}
\renewcommand{\textfraction}{0.0}  

\section{THE PHYSICS POTENTIAL OF MUON COLLIDERS}

\subsection{Brief overview}

The physics agenda at a muon collider falls into three categories: 
First Muon Collider (FMC) 
physics at a machine with center-of-mass energies of 100 to 500~GeV;
Next Muon Collider (NMC) physics at 3--4~TeV 
center-of-mass energies; and
front-end physics with a high-intensity muon source.

The FMC will be a unique facility for neutral Higgs boson (or  
techni-resonance) studies through $s$-channel resonance production.  
Measurements can also be made of the threshold cross sections for 
production of $W^+W^-$,  
$t\bar t$, $Zh$, and pairs of supersymmetry particles ---
$\chi_1^+\chi_1^-$, $\chi_2^0\chi_1^0$,  
$\tilde\ell^+\tilde\ell^-$ and $\tilde\nu\bar{\tilde\nu}$ --- that will  
determine the corresponding masses to high precision.
A $\mu^+\mu^-\to Z^0$ factory, utilizing the partial  
polarization of the muons, could allow significant improvements in  
$\sin^2\theta_{\rm w}$ precision and in $B$-mixing and CP-violating studies.
In Fig.~\ref{fmc_han}, we show the cross sections 
for standard model (SM) processes versus the CoM energy at the FMC. For the 
unique $s$-channel Higgs boson production, where $\sqrt s_{\mu\mu} =m_H$,
results  for three different beam energy resolutions are presented.
%
\begin{figure*}[tbh!]
\centering\leavevmode
\epsfxsize=5in\epsffile{fmc_sm_han1.eps}
\caption[Cross sections for SM processes versus the 
CoM energy at the FMC]{Cross sections for SM processes versus the 
CoM energy at the FMC. 
$\sigma_{pt}\equiv \sigma(\mu^+\mu^-\to \gamma^*\to e^+e^-)$.
For the $s$-channel Higgs boson production, 
three different beam energy
resolutions of 0.003\%, 0.01\% and 0.1\% are presented.
}
\label{fmc_han}
\end{figure*}
%

The NMC will be particularly valuable for reconstructing supersymmetric  
particles of high mass from 
their complex cascade decay chains. Also, any $Z'$  
resonances within the kinematic reach of the machine would give enormous  
event rates. The effects of virtual $Z'$ states would be detectable to high  
mass. If no Higgs bosons exist below $\sim$1~TeV, then the NMC would be the  
ideal machine for the study of strong $WW$ scattering at TeV energies.


At the front end, a high-intensity muon source will permit searches for rare  
muon processes sensitive to branching ratios 
that are orders of magnitude below  
present upper limits. Also, a high-energy muon-proton collider can be  
constructed to probe high$-Q^2$ phenomena beyond the reach of the HERA $ep$  
collider. In addition, 
the decaying muons will provide high-intensity neutrino  
beams for precision neutrino cross-section measurements and for long-baseline  
experiments \cite{sgeer,sgeerjhf,sgeerphyrevd,moha1,harris1,bjknu1,bjknu2,ref8b,yu1,cline80}.
Plus, there are numerous other new physics possibilities 
for muon facilities \cite{ref6a,mup} that we will not
discuss in detail in this document.

\subsection{Higgs boson physics}

The expectation that there will be a light (mass below $2M_W$) 
SM-like Higgs boson
provides a major motivation for the FMC, since such a Higgs boson
can be produced with a very high rate directly in the $s$-channel.
Theoretically, the lightest Higgs boson $h^0$ of the most
general supersymmetric model is predicted to have mass below $150$~GeV
and to be very SM-like in the usual decoupling limit. Indeed,
in the minimal supersymmetric model, which contains
the five Higgs bosons $h^0,\, H^0,\, A^0,\, H^\pm$, one finds
$m_{h^0}\alt 130$~GeV and the $h^0$ is SM-like if $m_{A^0}\agt 130$~GeV. 
 Experimentally, 
global analyses of precision electroweak data now indicate a strong 
preference for a light SM-like Higgs boson.
 The goals of the FMC for studying the SUSY
Higgs sector via $s$-channel resonance production are: to measure
the light Higgs mass, width, and branching fractions with
high precision, in particular sufficient to differentiate the minimal supersymmetric standard model (MSSM) $h^0$ from the SM $h_{\rm SM}$; and, to find and study 
the heavier neutral Higgs bosons $H^0$ and $A^0$.

The production of Higgs bosons in the $s$-channel with interesting rates is  
a unique feature of a muon collider \cite{Barger,bbgh}. The resonance cross  
section is
%
\begin{equation}
\sigma_h(\sqrt s) = {4\pi \Gamma(h\to\mu\bar\mu) \, \Gamma(h\to X)\over
\left( s - m_h^2\right)^2 + m_h^2 \left(\Gamma_{\rm tot}^h \right)^2}\,.
\end{equation}
%
Gaussian beams with root-mean-square (rms) 
energy resolution  down to $R=0.003\%$ are  
realizable. The corresponding rms spread $\srts$ in CoM energy is
%
\begin{equation}
\srts = (2{\rm~MeV}) \left( R\over 0.003\%\right) \left(\sqrt s\over  
100\rm~GeV\right) \,.
\end{equation}
%
The effective $s$-channel Higgs cross section convolved with a Gaussian spread,
%
\begin{equation}
\bar\sigma_h(\sqrt s) = {1\over \sqrt{2\pi}\,\srts} \; \int \sigma_h  
(\sqrt{\hat s}) \; \exp\left[ -\left( \sqrt{\hat s} - \sqrt s\right)^2 \over  
2\sigma_{\sqrt s}^2 \right] d \sqrt{\hat s},
\end{equation}
%
is illustrated in Fig.~\ref{s-chan-higgs} for 
$m_h = 110$~GeV, $\Gamma_h = 2.5$~MeV, and  
resolutions $R=0.01\%$, 0.06\% 
and 0.1\%. 
A resolution $\srts \sim \Gamma_h$ is needed to be 
sensitive to the Higgs width. 
The light Higgs  width is predicted to be
\begin{equation}
\begin{array}{lll}
\Gamma \approx 2\mbox{ to 3 MeV}& \rm if& \tan\beta\sim1.8,\\
\Gamma \approx 2\mbox{ to 800 MeV}& \rm if& \tan\beta\sim20,
\end{array}
\end{equation}
%
for $80{\rm~GeV}\alt m_h\alt120$~GeV, where the smaller values
apply in the decoupling limit of large $m_{A^0}$. We note
that, in the MSSM, $m_{A^0}$ is required to be in the decoupling regime 
in the context of minimal super gravity (mSUGRA) boundary conditions in order that
correct electroweak symmetry breaking arises after evolution of parameters
from the unification scale. In particular, decoupling applies
in mSUGRA at $\tan\beta\sim1.8$, corresponding to the 
infrared fixed point of the top quark Yukawa coupling.

\begin{figure*}[htb!]
\centering\leavevmode  % we may want to uncomment this line
%\epsfxsize=5in\epsffile{fermi-fig3.eps}
\includegraphics[width=5.75in]{fermi-fig3.eps}

\caption[Effective $s$-channel Higgs cross section $\bar\sigma_h$]{Effective $s$-channel Higgs cross section $\bar\sigma_h$ obtained  
by convoluting the Breit-Wigner resonance formula with a Gaussian 
distribution  
for resolution $R$. From Ref.~\cite{Barger}.\label{s-chan-higgs}}
\end{figure*}

At $\sqrt s = m_h$, the effective $s$-channel Higgs cross section is
%
\begin{equation}
\bar\sigma_h \simeq {4\pi\over m_h^2} \; {{\rm BF}(h\to\mu\bar\mu) \,
{\rm BF}(h\to X) \over \left[ 1 + {8\over\pi} \left(\srts\over\Gamma_{\rm  
tot}^h\right)^2 \right]^{1/2}} \,.
\end{equation}
%
BF denotes the branching fraction for $h$ decay; also,
note that $\bar\sigma_h\propto 1/\srts$ for $\srts>\Gamma_{\rm tot}^h$. At  
$\sqrt s = m_h \approx 110$~GeV, the $b\bar b$ rates are
%
\begin{eqnarray}
\rm signal &\approx& 10^4\rm\ events\times L(fb^{-1})\,,\\
\rm background &\approx& 10^4\rm\ events\times L(fb^{-1})\,,
\end{eqnarray}
%
assuming a $b$-tagging efficiency $\epsilon \sim 0.5$ and an energy resolution of $0.003\%.$ The effective  
on-resonance cross sections for other $m_h$ values and other channels ($ZZ^*,  
WW^*$) are shown in Fig.~\ref{sm-higgs} for the SM Higgs. 
The rates for the MSSM Higgs are nearly the same as the SM rates 
in the decoupling regime of large $m_{A^0}$.


\begin{figure*}[hbt!]
\centering\leavevmode
%\epsfxsize=5.5in\epsffile{fermi-fig4.eps}
\includegraphics[width=5.5in]{fermi-fig4.eps}

\caption[The SM Higgs cross sections and backgrounds in $b\bar b,\ WW^*$  
and $ZZ^*$. ]{The SM Higgs cross sections and backgrounds in $b\bar b,\ WW^*$  
and $ZZ^*$. Also shown is the luminosity needed for a 5~standard deviation  
detection in $b\bar b$. From Ref.~\cite{Barger}.\label{sm-higgs}}
\end{figure*}


The important factors that make $s$-channel Higgs physics studies possible at  
a muon collider are energy resolutions $\srts$ of order a few MeV, little  
bremsstrahlung and no beamstrahlung smearing, and precise tuning of the beam  
energy to an accuracy $\Delta E\sim10^{-6}E$ through continuous spin-rotation  
measurements \cite{ref7}. As a case study, we consider a SM-like
Higgs boson with $m_h \approx 110$~GeV.  
Prior Higgs discovery is assumed at the Tevatron 
(in $Wh, t\bar th$ production  
with $h\to b\bar b$ decay) or at the LHC (in $gg\to h$ production with $h\to  
\gamma\gamma, 4\ell$ decays with a mass measurement of $\Delta m_h \sim  
100$~MeV for an integrated luminosity of $L=300\rm~fb^{-1}$) or possibly at a  
NLC (in $Z^*\to Zh, h\to b\bar b$ giving $\Delta m_h \sim 50$~MeV for  
$L=200\rm~fb^{-1}$). A muon collider ring design would be optimized to run at  
energy $\sqrt s= m_h$. For an initial Higgs-mass uncertainty of $\Delta  
m_h\sim 100$~MeV, the maximum number of scan points required to locate the  
$s$-channel resonance peak at the muon collider is
%
\begin{equation}
n = {2\Delta m_h\over \srts} \approx 100
\end{equation}
%
for a $R=0.003\%$ resolution of $\srts \approx 2$~MeV. 
The necessary luminosity per scan  
point ($L_{\rm s.p.}$) to observe or eliminate the $h$-resonance at a  
significance level of $S/\sqrt B = 3$ is $L_{\rm s.p.} \sim
1.5\times10^{-3}\,\rm   fb^{-1}$. (The scan luminosity requirements increase
for $m_h$ closer to   $M_Z$; at $m_h\sim M_Z$ the $L_{\rm s.p.}$ needed is a
factor of 50 higher.)   The total luminosity then needed to tune to a Higgs
boson with $m_h = 110$~GeV   is $L_{\rm tot} = 0.15\rm~fb^{-1}$. If the machine
delivers    $1.5\times10^{31}\rm\, cm^{-2}\, s^{-1}$ (0.15~fb$^{-1}$/year),
then one year   of running  would suffice to complete the scan and measure the
Higgs mass to   an accuracy  $\Delta m_h \sim 1$~MeV.  Figure~\ref{bbar-events}
illustrates a simulation of such a scan.


Once the $h$-mass is determined to $\sim1$~MeV, a 3-point fine  
scan \cite{Barger} can be made across the peak with higher luminosity,  
distributed with $L_1$ at the observed peak position in $\sqrt s$ and $2.5L_1$  
at the wings ($\sqrt s = {\rm peak} \pm 2\srts$). Then, with $L_{\rm tot}=  
0.4\rm~fb^{-1}$ the following accuracies would be achievable: 16\% for  
$\Gamma_{\rm tot}^h$, 1\% for $\sigma\rm BF(b \bar b)$ and 5\% 
for $\sigma\rm  
BF(WW^*)$. The ratio $r = {\rm BF}(WW^*)/ {\rm BF} (b\bar b)$ is sensitive to  
$m_{A^0}$ for $m_{A^0}$ values below 500~GeV.
For example, $r_{\rm MSSM}/r_{\rm SM} =  
0.3, 0.5, 0.8$ for $m_{A^0} = 200, 250, 400$~GeV \cite{Barger}. Thus, 
using $s$-channel measurements of the $h$, 
it may be possible not only to distinguish the $h^0$ from the
SM $h_{SM}$ but also to infer $m_{A^0}$. 

The study of the other neutral MSSM Higgs bosons at a muon collider via the  
$s$-channel is also of major interest. Finding the $H^0$ and $A^0$ may not be  
easy at other colliders. At the LHC the region $m_{A^0}>200$~GeV is deemed to be  
inaccessible for $3\alt\tan\beta\alt5$--10 \cite{froid}. At an NLC the  
$e^+e^-\to H^0 A^0$ production process may be kinematically inaccessible if  
$H^0$ and $A^0$ are heavy
(mass $>230$~GeV for $\sqrt s=500$~GeV). 
At a $\gamma\gamma$ collider, very high luminosity  
(${\sim}200\rm\ fb^{-1}$) would be needed for $\gamma\gamma\to H^0, A^0$  
studies.

At a muon collider the resolution requirements for $s$-channel $H^0$ and  
$A^0$ studies are not as demanding as for the $h^0$ because the $H^0, A^0$  
widths are broader; typically $\Gamma\sim30$~MeV for $m_{A^0}<2m_t$ and  
$\Gamma\sim3$~GeV for $m_{A^0}>2m_t$. Consequently $R\sim0.1\%$ ($\srts \sim  
70$~MeV) is adequate for a scan. This is important, since higher
instantaneous luminosities (corresponding to $L\sim 2-10~{\rm fb}^{-1}
/{\rm yr}$)
are possible for $R\sim 0.1\%$ (as contrasted with the $L\sim 0.15~{\rm
fb}^{-1}/{\rm yr}$ for the much smaller $R\sim 0.003\%$ preferred for studies
of the $h^0$). A luminosity per scan point $L_{\rm s.p.}\sim  
0.1\rm~fb^{-1}$ probes the parameter space with $\tan\beta>2$.
The $\sqrt s$-range over which the scan should be made depends on other  
information available to indicate 
the $A^0$ and $H^0$ mass range of interest. A wide
scan would not be necessary if $r$ is measured
with the above-described precision 
to obtain an approximate value of $m_{A^0}$.



\begin{figure*}[tbh!]
\centering\leavevmode
%\epsfxsize=4in\epsffile{fermi-fig5.eps}
\includegraphics[width=4in]{fermi-fig5.eps}
\caption[Number of events and statistical errors in the $b\bar b$ final  
states ]{Number of events and statistical errors in the $b\bar b$ final  
states as a function of $\sqrt s$ in the vicinity of $m_{h_{\rm SM}}=110$~GeV,  
assuming $R=0.003\%$. From Ref.~\cite{Barger}.\label{bbar-events}}
\end{figure*}

In the MSSM, $m_{A^0}\approx m_{H^0}\approx m_{H^\pm}$ at large $m_{A^0}$
(as expected for mSUGRA boundary conditions), with a  
very close degeneracy in these masses for large $\tan\beta$. In such a  
circumstance, only an $s$-channel scan 
with the good resolution possible at a muon collider may allow  
separation of the $A^0$ and $H^0$ states; see Fig.~\ref{H0-A0-sep}.

\begin{figure*}[hbt!]
\centering\leavevmode
\epsfxsize=4in\epsffile{fermi-fig6.eps}

\caption[Separation of $A^0$ and $H^0$ signals for $\tan\beta=10$]{Separation of $A^0$ and $H^0$ signals for $\tan\beta=10$. From  
Ref.~\cite{Barger}. \label{H0-A0-sep}}
\end{figure*}


\subsection{Light particles in technicolor models}

In most technicolor models, there will be light neutral and colorless
technipion resonances, $\pi_T^0$ and $\pi_T^{0\prime}$,
with masses below 500~GeV. Sample models include the recent 
top-assisted technicolor models \cite{techni}, in which
the technipion masses are typically above 100~GeV,
and models \cite{dominici} in which the masses of the 
neutral colorless resonances come
primarily from the one-loop effective potential
and the lightest state typically has mass as low as 10 to 100~GeV.
The widths of these light neutral and colorless
states in the top-assisted models
will be of order 0.1 to 50~GeV \cite{bhat}. In the one-loop models,
the width of the lightest technipion is  typically 
in the range from 3 to 50~MeV. Neutral technirho and techniomega
resonances are also a typical feature of technicolor models.
In all models, these resonances 
are predicted to have substantial Yukawa-like couplings to muons
and would be produced in the $s$-channel at a muon collider,
%
\begin{equation}
\mu^+\mu^-\to\pi^0_T, \, \pi^{0\prime}_T,\, \rho^0_T,\, \omega^0_T,
\end{equation}
%
with high event rates.
The peak cross sections for these processes are estimated to be  
$\approx 10^4$--$10^7$~fb \cite{bhat}. The dominant decay modes 
depend on eigenstate composition and other details but
typically are \cite{bhat}
%
\begin{eqnarray}
\pi^0_T &\to& gg,\, b\bar b, \, \tau\bar\tau,\, c\bar c,\, t\bar t\,,\\ 
\pi_T^{0\prime} &\to& gg,\, b\bar b,\, c\bar c,\, t\bar t,\, \tau^+\tau^-\,,\\
\rho^0_T &\to& \pi_T\pi_T,\, W\pi_T,\, WW \,,\\
\omega^0_T &\to& c\bar c,\, b\bar b,\, \tau\bar\tau,\, t\bar t,\,  
\gamma\pi^0_T,\, Z\pi^0_T \,.
\end{eqnarray}
%
Such resonances would be easy to find and study at a muon collider.

\subsection{Exotic narrow resonance possibilities}

There are important types of exotic physics that would be best
probed in $s$-channel production of a narrow resonance at a muon
collider. Many extended Higgs sector models
contain a doubly-charged Higgs boson $\Delta^{--}$ (and its $\Delta^{++}$
partner) that couples to $\mu^-\mu^-$ via a Majorana coupling.
The $s$-channel process
$\mu^-\mu^- \rightarrow \Delta^{--}$ has been shown \cite{gunmummum}
to probe extremely small values of this Majorana coupling, in particular
values naturally expected in models where such couplings
are responsible for neutrino mass generation. 
In supersymmetry, it is possible that there is R-parity violation.
If R-parity violation is of
the purely leptonic type, the coupling $\lambda_{\mu\tau\mu}$ for 
$\mu^-\mu^+\to\tilde\nu_\tau$ is very possibly the largest
such coupling and could be related to neutrino mass
generation.  This coupling can be probed down to
quite small values via $s$-channel
$\tilde\nu_\tau$ production at the muon collider \cite{fgh}.
\subsection{$Z$-factory}

A muon collider operating at the $Z$-boson resonance energy is an interesting  
option for measurement of polarization asymmetries, $B_s^0$--$\bar B_s^0$  
mixing, and of CP violation in the $B$-meson system \cite{demarteau}. 
The muon collider advantages are the partial muon beam polarization, 
%the separation of  $b$ and $\bar b$ in $Z\to b\bar b$ events,
and the long $B$-decay length for  
$B$-mesons produced at this $\sqrt s$. The left-right asymmetry $A_{LR}$ is  
the most accurate measure of $\sin^2\theta_{\rm w}$, since the uncertainty is  
statistics dominated. The present LEP and SLD polarization measurements show  
deviations from the Standard Model prediction by $2.4\sigma $ in $A_{LR}^0$, $1.9\sigma $ in $A_{FB}^{0,b}$ and $1.7\sigma $ in  
$A_{FB}^{0,\tau}$ \cite{e-l}. The CP angle $\beta$
%(see Fig.~\ref{uni-triangle})
could be  measured from $B^0\to K_s J/\psi$ decays. 
To achieve significant improvements over existing measurements and 
those at future $B$-facilities, a data sample of $10^8\,Z$-boson events/year  would be needed. 
This corresponds to a luminosity $>0.15\rm~fb^{-1}$ /year, 
which is well within the domain of muon collider expectations;
 $R\sim 0.1\%$ would be more than adequate, given the
substantial $\sim 2.4$~GeV width of the $Z$.

\subsection{Threshold measurements at a muon collider}

With 10~fb$^{-1}$ integrated luminosity devoted to a measurement of a  
threshold cross-section, the following precisions on particle masses may be  
achievable \cite{bbgh2}:
\begin{equation}
\begin{array}{ll}
\mu^+\mu^-\to W^+W^-& \Delta M_W = 20\rm\ MeV\,,\\
\mu^+\mu^-\to t\bar t & \Delta m_t = 0.2\rm\ GeV\,,\\
\mu^+\mu^-\to Zh& \Delta m_h = 140 \rm\ MeV\ \ 
\end{array}
\end{equation}
(if\  $m_h = 100\rm\ GeV)\,.$ 
%
Precision $M_W$ and $m_t$ measurements allow important tests of electroweak  
radiative corrections through the relation
%
\begin{equation}
M_W = M_Z \left[ 1 - {\pi\alpha\over \sqrt 2 \, G_\mu \, M_W^2 (1-\delta r)}  
\right]^{1/2} \,,
\end{equation}
%
where $\delta r$ represents loop corrections. In the SM, $\delta r$ depends  
on $m_t^2$ and $\log m_h$. The optimal precision for tests of this relation is  
$\Delta M_W \approx {1\over 140}\Delta m_t$, so the uncertainty on $M_W$ is  
the most critical. With $\Delta M_W=20$~MeV the SM Higgs mass could be  
inferred to an accuracy
%
\begin{equation}
\Delta m_{h_{\rm SM}} =  30{\rm\ GeV} \left(m_h\over 100\rm\ GeV\right)\,.
\end{equation}
%
Alternatively, once $m_h$ is known from direct measurements, SUSY loop  
contributions can be tested.

In top-quark production at a muon collider above the threshold region, modest  
muon polarization would allow sensitive tests of anomalous top quark  
\mbox{couplings \cite{parke}}.

One of the important physics opportunities for the First Muon Collider is the  
production of the lighter chargino, $\tilde\chi_1^+$ \cite{carprot}. Fine-tuning arguments in mSUGRA suggest that it should be lighter than  
200~GeV. A search at the upgraded Tevatron for the process $q\bar  
q\to\tilde\chi_1^+\tilde\chi_2^0$ with $\tilde\chi_1^+\to  
\tilde\chi_1^0\ell^+\nu$ and $\tilde\chi_2^0\to\tilde\chi_1^0\ell^+\ell^-$  
decays can potentially reach masses $m_{\tilde\chi_1^+}\simeq  
m_{\tilde\chi_2^0}\sim 170$~GeV with 2~fb$^{-1}$ luminosity and $\sim230$~GeV  
with 10~fb$^{-1}$ \cite{teva2000}. The mass difference $M(\tilde\chi_2^0) -  
M(\tilde\chi_1^0)$ can be determined from the $\ell^+\ell^-$ mass  
distribution.
\begin{figure*}[tbh!]
\centering\leavevmode
\epsfxsize=4in\epsffile{fermi-fig7.eps}
\caption[Diagrams for production of the lighter chargino]{Diagrams for production of the lighter chargino.\label{light-chargino}}
\end{figure*}

The two contributing diagrams in the chargino pair production process are  
shown in Fig.~\ref{light-chargino}; 
the two amplitudes interfere destructively. The  
$\tilde\chi_1^+$  and $\tilde\nu_\mu$ 
masses can be inferred from the shape of  
the cross section in the threshold region \cite{bbh-new}. The chargino decay  
is $\tilde\chi_1^+\to f\bar f' \tilde\chi_1^0$. Selective cuts suppress the  
background from $W^+W^-$ production and leave $\sim 5\%$ signal efficiency for  
4\,jets${}+\E$ events. Measurements at two energies in the threshold region  
with total luminosity $L=50\rm~fb$ and resolution $R=0.1\%$ can give the  
accuracies listed in table~\ref{chargino-table} on the chargino mass for the specified values of  
$m_{\tilde\chi_1^+}$ and $m_{\tilde\nu_\mu}$.

\begin{table*}[bht!]
\caption[Achievable uncertainties $\Delta m$ with 50~fb$^{-1}$ luminosity]{Achievable uncertainties with 50~fb$^{-1}$ luminosity on the mass  
of the lighter chargino for representative $m_{\tilde\chi_1^+}$ and  
$m_{\tilde\nu_\mu}$ masses. From Ref.~\cite{bbh-new}. \label{chargino-table}}
\centering\leavevmode
%\begin{tabular}{c}
\begin{tabular}{ccc}
%\epsfxsize=3in\epsffile{fermi-tab1.eps}
$\Delta m_{\tilde\chi_1^+}$ (MeV) & $m_{\tilde\chi_1^+}$ (GeV) & $m_{\tilde\nu_\mu}$ (GeV)\\
35 & 100 & 500\\
45 & 100 & 300\\
150 & 200 & 500\\
300 & 200 & 300\\
\end{tabular}
\end{table*}


\subsection{Heavy particles of supersymmetry}

The requirements of gauge coupling unification can be used to predict the  
mean SUSY mass scale, given the value of the strong coupling constant 
at the $Z$-mass  
scale. Figure~\ref{alpha_s} shows the SUSY GUT 
predictions versus $\alpha_s(M_Z)$. For the  
value $\alpha_s(M_Z) = 0.1214\pm0.0031$ from a new global fit to precision  
electroweak data \cite{e-l}, a mean SUSY mass of order 1~TeV is expected. 
Thus, it is likely that some SUSY particles will have masses at the TeV scale.
Large masses for the squarks of the first family are perhaps the
most likely in that this would provide a simple cure for possible flavor
changing neutral current difficulties. 

\begin{figure}[bht!]
\centering\leavevmode
\epsfxsize=3.25in\epsffile{barbi-gut.eps}
\bigskip
\caption[$\alpha_s$ prediction in supersymmetric GUT]{$\alpha_s$ prediction in supersymmetric GUT with minimal particle content in the Dimensional Regularization scheme. \label{alpha_s}}
\end{figure}

At the LHC, mainly squarks and gluinos will be produced; 
these decay to lighter  
SUSY particles. The LHC will be  a great SUSY machine, but some sparticle  
measurements will be very difficult or impossible there \cite{hinch,paige},  
namely: (i)~the determination of the LSP mass (LHC measurements give SUSY mass  
differences); (ii)~study of sleptons of mass $\agt200$~GeV because Drell-Yan  
production becomes too small at these masses; (iii)~study of heavy gauginos  
$\tilde\chi_2^\pm$ and $\tilde\chi_{3,4}^0$, which are mainly Higgsino and  
have small direct production rates and small branching fractions to channels  
usable for detection; (iv)~study of heavy Higgs bosons $H^\pm,\ H^0,\ A^0$  
when the MSSM $\tan\beta$ parameter is not large and their
masses are larger than $2m_t$, so that cross sections
are small and decays to $t\bar t$ are likely to be
dominant (their detection is deemed impossible if SUSY decays dominate).

Detection and study of the many scalar particles predicted in
supersymmetric models could be a particularly valuable contribution
of a high energy lepton collider. However, since pair production of  
scalar particles at a lepton collider is $P$-wave suppressed,
energies well above threshold are needed for sufficient production rates; see  
Fig.~\ref{pair-product}. For scalar particle masses of order 1 TeV a
collider energy of 3 to 4 TeV is needed to get past the threshold
suppression. A muon collider operating in this energy range
with high luminosity ($L\sim10^2$ to  
$10^3\rm~fb^{-1}/year$) would provide sufficient event rates to reconstruct  
heavy sparticles from their complex cascade decay chains \cite{paige,lykken}.
\begin{figure*}[tbh!]
\centering\leavevmode
\epsfxsize=5.5in\epsffile{fermi-fig9.eps}
\caption[Cross sections for pair production of Higgs bosons and scalar  
particles]{Cross sections for pair production of Higgs bosons and scalar  
particles at a high-energy muon collider. 
From Ref.~\cite{sanfran95}. \label{pair-product}}
\end{figure*}

In string models, it is very natural to have
extra $Z$ bosons in addition to low-energy supersymmetry.
The $s$-channel production of a $Z'$ boson at the  
resonance energy would give enormous event rates at the NMC.  Moreover, the  
$s$-channel contributions of $Z'$ bosons with mass far above the kinematic  
reach of the collider could be revealed as contact interactions \cite{god}.
\subsection{Strong scattering of weak bosons}

The scattering of weak bosons can be studied at a high-energy muon collider  
through the process in Fig.~\ref{strong-ww}. 
The amplitude for the scattering of  
longitudinally polarized $W$-bosons behaves like
%
\begin{equation}
A(W_LW_L\to W_LW_L) \sim m_H^2/v^2
\end{equation}
%
if there is a light Higgs boson, and like 
%
\begin{equation}
A(W_LW_L\to W_LW_L) \sim s_{WW}^{\vphantom y} /v^2
\end{equation}
%
if no light Higgs boson exists; here $s_{WW}^{\vphantom y}$ is the square of  
the $WW$ CoM energy and $v=246$~GeV. In the latter scenario, partial-wave  
unitarity of $W_LW_L\to W_LW_L$ requires that the scattering of 
weak bosons  becomes strong at energy scales of order 1 to 2 TeV. 
Thus, subprocess energies  
$\sqrt{\smash{s_{WW}^{\vphantom y}}}\agt 1.5$~TeV are needed to probe strong  
$WW$ scattering effects.

\begin{figure}[bth!]
\centering\leavevmode
\epsfxsize=3in\epsffile{fig-strongww.eps}
\caption{Symbolic diagram for strong $WW$ scattering. \label{strong-ww}}
\end{figure}

The nature of the dynamics in the $WW$ sector is unknown. Models for this
scattering assume heavy resonant particles (isospin scalar and vector) or a  
non-resonant amplitude based on
a unitarized extrapolation of the low-energy theorem behavior  
$A\sim s_{WW}^{\vphantom y}/v^2$. In all models, impressive signals of strong  
$WW$ scattering are obtained at the NMC, with cross sections typically of  
order 50~fb \cite{W-mass}. Event rates are such that the
various weak-isospin channels ($I=0,1,2$) could be studied in detail
as a function of $s_{WW}$. After several years of operation,
it would even be possible to perform such a study after projecting
out the different final polarization states ($W_LW_L$, $W_LW_T$
and $W_TW_T$), thereby enabling one to verify that it is the
$W_LW_L$ channel in which the strong scattering is taking place.

\subsection{Front end physics}

New physics is likely to have important lepton flavor dependence and
may be most apparent for heavier flavors. The intense muon source
available at the front end of the muon collider will provide many
opportunities for uncovering such physics.

\subsubsection{Rare muon decays}

The planned muon flux of ${\sim}10^{14}$ muons/sec for a muon collider  
dramatically eclipses the flux, ${\sim}10^8$ muons/sec, of present sources. 
With an intense source, 
the rare muon processes $\mu\to e\gamma$ (for which the current branching  
fraction limit is $0.49\times10^{-12}$), 
$\mu N\to eN$ conversion, and the muon  
electric dipole moment can be probed at very interesting levels. A generic  
prediction of supersymmetric grand unified theories is that these lepton  
flavor violating or CP-violating processes should occur via loops at  
significant rates, e.g.\ BF$(\mu\to e\gamma)\sim 10^{-13}$. 
Lepton-flavor violation can also occur via $Z'$ bosons, leptoquarks, 
and heavy neutrinos \cite{marciano}.

\subsubsection{Neutrino flux}

The decay of a muon beam leads to neutrino beams of 
well defined flavors. 
A muon collider would yield a neutrino flux 1000  
times that presently available \cite{chuckandsteve}. This would result
in ${\sim}10^6$ $\nu N$  
and $\bar\nu N$ events per year, which could be used
to measure charm production  
($\sim$6\% of the total cross section) and measure $\sin^2\theta_{\rm w}$  
(and infer the $W$-mass to an accuracy $\Delta M_W \simeq 30$--50~MeV in one  
year) \cite{sgeer,sgeerjhf,sgeerphyrevd,moha1,harris1,bjknu1,bjknu2,ref8b,yu1}.

\subsubsection{Neutrino oscillations}

A special purpose muon ring has been proposed \cite{sgeerjhf} to store  
${\sim}10^{21}$ $\mu^+$ or $\mu^-$ per year and obtain ${\sim}10^{20}$  
neutrinos per year from muon decays along $\sim$75-m straight sections of 
the  
ring, which would be pointed towards a distant neutrino detector. 
The  neutrino fluxes from $\mu^-\to\nu_\mu\bar\nu_e e^-$ or from  
$\mu^+\to\bar\nu_\mu\nu_e e^+$ decays can be calculated with little systematic
error. Then, for example, from the decays of  
stored $\mu^-$'s, the following neutrino oscillation channels could be studied  
by detection of the charged leptons from the interactions of neutrinos in the  
detector:
$$ \begin{array}{cc}
\underline{\rm\ oscillation\ }& \underline{\rm\ detect\ }\\
\nu_\mu\to\nu_e& e^-\\
\nu_\mu\to\nu_\tau& \tau^-\\
\bar\nu_e\to\bar\nu_\mu& \mu^+\\
\bar\nu_e\to\bar\nu_\tau& \tau^+
\end{array}
$$
The detected $e^-$ or $\mu^+$ have the ``wrong sign" from the leptons  
produced by the interactions of the $\bar\nu_e$ and $\nu_\mu$ flux. The known  
neutrino fluxes from muon decays could be used for long-baseline oscillation  
experiments at any detector on Earth. The probabilities for vacuum  
oscillations between two neutrino flavors are given by
%
\begin{equation}
P(\nu_a\to\nu_b) = \sin^2 2\theta \, \sin^2(1.27\delta m^2 L/E)
\end{equation}
%
with $\delta m^2$ in eV$^2$ and $L/E$ in km/GeV.  In a very long baseline  
experiment from Fermilab to the Gran Sasso laboratory 
or the Kamioka mine
($L={\cal O}(10^4)$~km) with  
$\nu$-energies $E_\nu=20$ to 50~GeV ($L/E = 500$--200~km/GeV),  neutrino  
charged-current interaction rates of ${\sim}10^3$/year would result. 
In a long baseline experiment from Fermilab to the Soudan mine 
(L=732 km), the corresponding interaction rate is ${\sim}10^4$/year.
Such an  experiment would have sensitivity to oscillations down to $\delta  
m^2\sim 10^{-4}\,-\,10^{-5}\rm\,eV^2$ for $\sin^22\theta=1$ \cite{sgeerjhf}.

\subsubsection{$\mu p$ collider}

The possibility of colliding 200-GeV muons with 1000-GeV protons at Fermilab  
is under study. This collider would reach a maximum  
$Q^2\sim 8\times 10^5$~GeV$^2$, which is $\sim$8~times the reach of the HERA  
$ep$ collider, and deliver a luminosity ${\sim}10^{33}\rm\,cm^{-2}\,s^{-1}$,  
which is $\sim$300 times the HERA luminosity. The $\mu  p $ collider would  
produce ${\sim}10^6$ neutral-current deep-inelastic-scattering events per year  
at $Q^2>5000\rm~GeV^2$, which is more than a factor of
$10^3$ higher than at HERA.  In the new physics realm,
leptoquark couplings and contact interactions,
if present, are likely to be larger for muons than for electrons.
This $\mu p$ collider would have sufficient sensitivity 
to probe leptoquarks up to a  
mass $M_{LQ} \sim 800$~GeV and contact interactions to a scale  
$\Lambda\sim6$--9~TeV \cite{cheung}. 

\subsection{Summary of the physics potential}

The First Muon Collider offers unique probes of supersymmetry (particularly  
$s$-channel Higgs boson resonances) and technicolor models (via
$s$-channel production of techni-resonances),
high-precision threshold measurements of  
$W,\ t$ and SUSY particle masses, tests of SUSY radiative corrections that  
indirectly probe the existence of high-mass squarks, and a possible $Z^0$  
factory for improved precision in polarization measurements and for  
$B$-physics studies of CP violation and mixing.

The Next Muon Collider guarantees access to heavy SUSY scalar particles and  
$Z'$ states or to strong $WW$ scattering if there are no Higgs bosons and no  
supersymmetry.


The Front End of a muon collider offers dramatic improvements in sensitivity  
for flavor-violating transitions (e.g., \ $\mu\to e\gamma$), 
access to high-$Q^2$  
phenomena in deep-inelastic muon-proton and neutrino-proton interactions, and  
the ability to probe very small $\delta m^2$
via neutrino-oscillation studies in long-baseline experiments.

The muon collider would be crucial to unraveling the flavor dependence
of any type of new physics that is found at the next generation
of colliders.

Thus, muon colliders are robust options for probing new  
physics that may not be accessible at other colliders.




