\section{INTRODUCTION}
The Standard Model of electroweak and strong interactions
has passed precision experimental tests at the highest 
energy scale accessible today.
Theoretical arguments indicate that new physics 
\textit{beyond  the Standard Model} associated with the 
electroweak gauge symmetry breaking and fermion mass
generation will emerge in 
% particle-antiparticle
% quark-antiquark and lepton-antilepton    glue-glue dominates in p-pbar!
parton collisions at or
approaching the TeV energy scale. It is likely that both hadron-hadron  and lepton-antilepton 
colliders will be required to discover and make precision measurements of the
new phenomena.  The next big step  forward in advancing the hadron-hadron
collider energy frontier will be  provided by the CERN Large Hadron Collider
(LHC), a proton-proton collider with a center-of-mass (CoM) energy of 14~TeV 
 which is due to come into operation in 2005. Note that in a high energy hadron beam, valence quarks carry momenta which are, approximately, between ${1\over 6}$ to ${1\over 9}$ of the hadron momentum. The LHC will therefore provide hard parton-parton collisions with typical center of mass energies of $2.3 - 1.5 $~TeV.

The route towards TeV-scale lepton-antilepton colliders is less clear. The lepton-antilepton colliders built so far have been $e^+e^-$ colliders, such as the
Large Electron Positron collider (LEP) at CERN and the Stanford Linear  Collider (SLC) at SLAC. In a circular ring such as LEP the  energy lost per revolution in keV is
$88.5\times E^4 / \rho,$ where the electron energy $E$ is in GeV, and the 
radius of
the orbit $\rho$ is in meters. Hence, the energy loss grows rapidly as $E$
increases. This limits the center-of-mass energy that  would be achievable in a
LEP-like collider. The problem can be  avoided by building a linear machine
(the SLC is partially linear), but with current technologies, such a        
machine must be very long (30-40~km) to attain the TeV energy scale.
Even so, radiation during the beam-beam interaction (beamstrahlung) limits
the precision of the CoM energy \cite{Tigner92}.

\begin{figure*}[bth!]
\centerline{\epsfig{file=machine_comparison.ps,height=4.75in,width=6.in}}
\vspace{0.5cm}
\caption[Comparative sizes of various proposed high energy colliders compared with the FNAL and BNL sites.]
{Comparative sizes of various proposed high energy colliders compared with the FNAL and BNL sites. The energies in parentheses give for lepton colliders their CoM energies and for hadron colliders the approximate range of CoM energies attainable for hard parton-parton collisions.}
\label{compare}
\end{figure*}

For a lepton with mass $m$ the radiative energy losses 
are inversely proportional to $m^4$. Hence, the energy-loss  problem can be
solved by using heavy leptons. In practice this means using  muons, which have
a mass $\approx 207$ times that of an electron. The resulting % significant 
reduction in
radiative losses enables higher energies to be reached and  smaller collider
rings to be used \cite{ref01,ref01a}. Parameters for 10 to 100~TeV collider have been discussed\cite{skrinskiultimate,king_highe}. 
Estimated sizes of the accelerator  complexes required for
0.1-TeV, 0.5-TeV and 4-TeV muon colliders are compared with  the sizes of other
possible future colliders,  and with the FNAL and BNL sites in 
Fig.~\ref{compare}. Note that muon colliders with CoM energies up to $\approx 4$~TeV would fit on these existing
laboratory sites. The cost  of building a muon collider is not yet known. However, since muon colliders are relatively  small, they may be significantly less expensive than alternative machines. 

Since muons decay quickly, large numbers of them must be produced to operate
a muon collider at high luminosity.  Collection of muons from the decay of
pions produced in proton-nucleus interactions results in a large initial
phase volume for the muons, which must be reduced (cooled) by a factor of $10^6$ for a practical collider. This may be compared with the antiproton stochastic cooling achieved in the Tevatron. In this case the 6-dimensional (6-D) phase space is reduced by approximately a factor of $10^6,$ while with stacking the phase space density \cite{tevcooling,reviewtev} is increased by a factor of $10^{10}.$  The technique of ionization cooling
is proposed for the $\mu^+\mu^-$ collider \cite{Kolomensky,Budker67,Ado,Balbekov96}. This technique is uniquely applicable to muons because of
their minimal interaction with matter.

Muon  colliders also offer some significant physics advantages. The small
radiative losses  permit very small beam-energy spreads to be achieved.  For
example, momentum spreads as low as $\Delta P/P = 0.003\%$ are believed  to be
possible for a low-energy collider. By measuring the time-dependent decay
asymmetry resulting from the naturally polarized  muons, it has been
shown \cite{ref7} that the beam energy could be determined with a precision of 
$\Delta E/E = 10^{-6}$. The small beam-energy spread, together with the precise
energy determination, would facilitate measurements of the masses
and widths of any new resonant states scanned by the collider. In addition,
since the cross-section for producing a Higgs-like scalar particle in the
s-channel (direct lepton-antilepton annihilation) is proportional to $m^2$,
this extremely important process could be studied only at a muon collider and not at an $e^+e^-$ collider \cite{ref2b}. Finally, 
the decaying muons will produce copious quantities of neutrinos.  Even 
short straight sections in a muon-collider ring will result in neutrino
beams several orders of magnitude higher in intensity than presently
available, permitting greatly extended studies of neutrino
oscillations,  nucleon structure functions, the CKM matrix, and
 precise indirect measurements of the $W$-boson mass\cite{stevealw} (see section~\textbf{II.I}).

The concept of muon colliders was introduced by 
G.~I.~Budker \cite{ref01,ref01a}, and developed further by A.~N.~Skrinsky \etal \cite{Skrinsky71,Budker78,Skrinsky80,ref1a,Skrinsky82,ref1,Skrinsky96a,%
Skrinsky96b} and D.~Neuffer \cite{ref2b,ref2,ref2a,ref2c}. 
They pointed out the significant challenges in
designing an accelerator complex that can  make, accelerate, and collide
$\mu^+$ and $\mu^-$ bunches all within the muon lifetime of $2.2\,\mu$s 
($c\tau=659$~m). A concerted study of a muon collider design has been underway in the U.S. since 1992 \cite{PortJeff,Napa,ref3a,Fontana,ref3b,Tamura94,ref3c,Snowmass96,SantaB96,LBL97,Vancouver97,BNLlattice97,FNALcool97,mup,sanfrisco,Alabama98,BNLcool98}.
By the Sausalito
workshop \cite{ref3b} in 1995 it was  realized that with new ideas and modern
technology, it may be feasible to make muon bunches containing a few  times
$10^{12}$ muons, compress their phase space and  accelerate them 
%the resulting still intense bunches 
up to the multi-TeV energy scale before more than about 3/4 of them have
decayed. With careful design of the collider ring and  shielding it  appears possible to
reduce to acceptable levels the backgrounds  within the detector that arise
from the very large flux of electrons produced  in muon decays. These
realizations led to  an intense activity,  which resulted in the muon-collider 
feasibility study report \cite{status96,ref6a} prepared for the 
1996 DPF/DPB Summer Study on High-Energy Physics (the Snowmass'96  
workshop).   Since then, the physics 
prospects at a muon collider have been studied extensively \cite{Barger,gunion,ref100}, and the potential physics program at a muon 
collider facility has been explored in workshops \cite{mup} and 
conferences \cite{sanfrisco}.
%Studies \cite{ref4a,ref4,ref4b,ref5,ref6,ref0,ref0a,ref0b}  have been made for 
%muon colliders at 3-4~TeV, 0.4-0.5~TeV and $\approx 100$~GeV.

 Encouraged by further progress in developing the 
muon-collider concept, together with the growing interest and involvement of 
the high-energy-physics community, the {\it Muon Collider Collaboration} 
became a formal
entity in May of 1997. The collaboration is led
by an executive board with members from Brookhaven National Laboratory (BNL), 
Fermi National Accelerator Laboratory (FNAL), 
Lawrence Berkeley National Laboratory (LBNL), 
Budker Institute for Nuclear Physics (BINP), 
University of California at Los Angeles (UCLA),
University of Mississippi and Princeton University.
The goal of the collaboration is to complete  within a
few years the R\&D needed to determine whether a Muon Collider is technically
feasible, and if it is, to design the First Muon Collider.

\begin{table*}[thb!]
\centering  
\caption[Baseline parameters for high- and low-energy muon colliders. ]
{Baseline parameters for high- and low-energy muon colliders.
Higgs/year assumes a cross section $\sigma=5\times 10^4$~fb; a Higgs width 
$\Gamma=2.7$~MeV; 1~year = $10^7$~s.}
\label{sum}
\begin{tabular}{llccccc}
%\hline
\rr CoM energy         &\rr TeV   &\rr 3 &\rr 0.4 &
\multicolumn{3}{c}{0.1 }  \\
% & & & & & & \\
$p$ energy       & GeV        &  16  & 16 & \multicolumn{3}{c}{16}\\
$p$'s/bunch      &    &  $2.5\times 10^{13}$  & $2.5\times 10^{13}$  &
\multicolumn{3}{c}{$5\times 10^{13}$  }  \\  
Bunches/fill   &           & 4 & 4 & \multicolumn{3}{c}{2 }  \\
Rep.~rate  & Hz     &  15 & 15 & \multicolumn{3}{c}{15 }  \\
$p$ power        & MW         &  4   & 4 & \multicolumn{3}{c}{4}  \\ 
$\mu$/bunch  &    & $2\times 10^{12}$ & $2\times 10^{12}$ &
\multicolumn{3}{c}{$4\times 10^{12}$ }  \\
\rr $\mu$ power  &\rr MW     & \rr 28 &\rr 4 & \multicolumn{3}{c}{\rr 1 }  \\
\rr Wall power    &\rr MW    &  \rr  204 &\rr 120  & \multicolumn{3}{c}{\rr
81 }  \\
Collider circum.   & m          &  6000 & 1000 & \multicolumn{3}{c}{350 }  \\
Ave bending field & T       & 5.2 & 4.7 &\multicolumn{3}{c}{3 }  \\
%Depth   & m          &  500 & 100 & \multicolumn{3}{c}{10 }  \\
%\hline
\rr Rms ${\Delta p/p}$       &\rr  \%          &\rr 0.16 &\rr 0.14 &\rr
0.12 &\rr 0.01&\rr
0.003 \\
%\hline
6-D $\epsilon_{6,N}$    &  $(\pi \textrm{m})^3$&$1.7\times 10^{-10}$&$1.7\times
10^{-10}$&$1.7\times 10^{-10}$&$1.7\times 10^{-10}$&$1.7\times 10^{-10}$\\
Rms $\epsilon_n$     &$\pi$ mm-mrad     &  50 & 50 & 85 & 195 & 290\\
$\beta^*$         & cm          & 0.3 & 2.6 & 4.1 &  9.4 & 14.1\\
$\sigma_z$         & cm          & 0.3 & 2.6 & 4.1 &  9.4 & 14.1 \\
$\sigma_r$ spot    &$\mu$m     & 3.2 & 26 & 86 & 196 & 294\\
$\sigma_{\theta}$ IP    &mrad     & 1.1 & 1.0 & 2.1 & 2.1 & 2.1\\
Tune shift     &             &0.044 &0.044 & 0.051 &0.022 & 0.015\\
$n_{\rm turns}$ (effective) &     &  785 & 700 & 450 & 450 & 450 \\
%\hline
\rr Luminosity     &\rr cm$^{-2}$s$^{-1}$&\rr $7\times 10^{34}$ & $10^{33}$ &\rr
$1.2\times 10^{32}$ &\rr $2.2\times 10^{31}$&\rr $10^{31}$ \\
 & & & & & & \\
Higgs/year    &  &  & & $1.9\times 10^3$ & $4\times 10^3$ & $3.9\times 10^3$ \\
%\hline
\end{tabular}
\end{table*}

\begin{figure*}[tbh!]
%\centerline{\epsfig{file=fnalfg2.ps,height=2.9in,width=5.7in}}
\centerline{\epsfig{file=march8.ps,height=2.9in,width=5.7in}}
\caption{Plan of a 0.1-TeV-CoM muon collider.}
\label{plan1}
\end{figure*}
 
\begin{figure*}[bth!]
\centerline{\epsfig{file=fnalfg1.ps,height=5.0in,width=5.45in}}
\caption{Plan of a 3-TeV-CoM muon collider shown on the Fermi National
Laboratory site as an example.}
\label{plan2}
\end{figure*}

Table~\ref{sum} gives the parameters of the muon colliders under study \cite{ref4a,ref4,ref4b,ref5,ref6,ref0,ref0a,ref0b}, which have CoM energies of
0.1~TeV, 0.4~TeV and 3~TeV and
Figs.~\ref{plan1} and \ref{plan2} show possible outlines of the  0.1~TeV and 
3~TeV  machines. In the former case, parameters are given in the table  for
operation with three  different beam-energy spreads: ${\Delta p/p} = 0.12$,
0.01, and 0.003\%.  In all cases, proton bunches containing 2.5-$5\times
10^{13}$  particles are accelerated to energies of 16~GeV. The protons interact
in a target to  produce  ${\cal O}(10^{13})$ charged pions of each sign.   A
large fraction of these pions can be captured in a high-field solenoid.  Muons
are produced by allowing the pions to decay into a lower-field solenoidal
channel. To  collect as many particles as possible within a useful energy
interval, rf  cavities are used to accelerate the lower-energy particles and
decelerate the higher-energy particles (so-called phase rotation).  With two
proton bunches every  accelerator cycle, the first used to make and collect
positive muons and the  second to make and collect negative muons, there are
about $10^{13}$ muons of  each charge available at the end of the decay channel
per accelerator cycle.  If the proton accelerator is cycling at 15~Hz, then in
an operational year  ($10^7$~s), about $10^{21}$ positive and negative muons
would be produced and collected.

As stated before, the muons exiting the decay channel populate a very diffuse phase space. The
next step in the muon-collider complex is to \textit{cool} the muon bunch, \ie,
to turn the diffuse muon cloud into a very \textit{bright} bunch with small  
longitudinal
and transverse dimensions, suitable for accelerating and  injecting into a
collider. The cooling must be done within a time that is short compared to
the muon lifetime. Conventional cooling techniques  (stochastic cooling \cite{Ruggiero92} and electron cooling \cite{Budker78}) take too long. 
The technique proposed for cooling muons is called ionization cooling \cite{Kolomensky,Ado,Balbekov96}, and 
will be discussed in detail in sec.~V.  Briefly, the muons traverse 
some material in which they  lose
both longitudinal and transverse momentum by ionization losses ($dE/dx$).  The
longitudinal momentum is then replaced using an rf accelerating cavity,  and
the process is repeated many times until there is a large reduction in  the
transverse phase space occupied by the muons. The energy spread within  the
muon beam can also be reduced by using a wedge-shaped absorber in a region of 
dispersion (where the transverse position is momentum dependent). 
The wedge is arranged so that the higher-energy particles
pass through more material than lower-energy particles.  Initial calculations
suggest that the 6-D phase space occupied by the initial muon
bunches can be reduced by a factor of $10^5$-$10^6$ before multiple Coulomb
scattering and energy straggling limit further reduction. We reiterate that ionization
cooling is uniquely suited to muons because of the absence of strong nuclear
interactions and electromagnetic shower production for these particles at energies around 200~MeV/$c.$

Rapid acceleration to the collider beam energy is needed to avoid % further
excessive particle loss from decay. It can be achieved, initially in a linear
accelerator, and later in  recirculating linear accelerators, rapid-cycling
synchrotron, or fixed-field-alternating-gradient (FFAG) accelerators. Positive 
and negative muon bunches are then injected in opposite directions into a 
collider storage ring and brought into collision at the interaction point.  The
bunches circulate and collide for many revolutions before decay has  depleted
the beam intensities to an uninteresting level. Useful luminosity can be delivered for about 800 revolutions for the high-energy collider and 450 revolutions for the low-energy one.

There are many interesting and challenging problems that need to be  resolved
before the feasibility of building a muon collider can be  demonstrated. For
example, (i) heating from the very intense proton bunches may  require the
use of  of a liquid-jet target,  and (ii) attaining the desired cooling factor
in the ionization-cooling channel may require the development of
rf cavities with thin beryllium windows operating at liquid-nitrogen
temperatures in high solenoidal fields. In addition, the development of long
liquid-lithium lenses  may be desirable to provide  stronger radial focusing 
for the final cooling stages. 

This article describes  the status of our muon-collider feasibility
studies, and is organized as follows. Section~\textbf{II} gives a brief summary of the
physics potential of muon colliders, including physics at the accelerator
complex required for a muon collider. Section~\textbf{III} describes the proton-driver
specifications for a muon collider, and two site-dependent
examples that have been studied in some detail. 
Section~\textbf{IV} presents pion production,
capture, and the pion-decay channel, and section~\textbf{V} discusses the design of the
ionization-cooling channel needed to produce an intense muon beam suitable
for acceleration and injection into the final collider. Sections~\textbf{VI} and \textbf{VII} 
describe the acceleration scenario and collider ring, respectively. 
Section~\textbf{VIII}
discusses backgrounds at the collider interaction point and section~\textbf{IX} deals with possible detector scenarios.  A summary of the conclusions is given in section~\textbf{X}.

