\section{CONCLUSIONS}

Unlike protons, muons are point-like but, unlike electrons, they emit
relatively little synchrotron radiation and therefore can be accelerated and collided in rings. 

Another advantage resulting from the low synchrotron radiation is
the lack of beamstrahlung and the possibility of very small collision energy
spreads. A beam energy spread of $\Delta$E/E of  0.003\%  
 is considered feasible for a 100~GeV machine. It has been shown that by observing spin precession, the absolute energy could
be determined to a small fraction of this width. These features become 
important in conjunction  with the large s-channel Higgs production ($\mu^+\mu^-\rightarrow h$, 
43000~times larger than for $e^+e^-\rightarrow h$), allowing precision
measurements of the Higgs mass, width and branching ratios. A higher energy muon collider can also distinguish the nearly degenerate heavy
Higgs bosons $H^0$ and $A^0$ of the minimal supersymmetric extension of the
standard model, since these states can also be produced in the $s$ channel.
We have also examined the ability of the muon collider to study
techni-resonances, do a high luminosity study of $Z$ boson physics, scan the
$W$ and $t\bar{t}$ thresholds to make precision mass measurements as well as
SUSY and strongly interacting W boson physics. The high luminosity proton driver
and  the cold low energy muons permit the study of rare kaon and muon decays.
Muon storage rings will permit low-systematics  studies of neutrino oscillations
for a wide range of mixing angle and $\delta m^2$ phase space with
hitherto unattainable sensitivity.

Such machines are clearly desirable. The issues are:
\begin{itemize} 
\item { whether they can be built and physics done with them}
\item { what they will cost}.
\end{itemize}
 Much progress has
been made in addressing the first question and the answer, so far, appears to
be yes. It is too early to address the second. 

We have studied
machines with CoM energies of 0.1, 0.4 and 3~TeV, defined
parameters and simulated many of their components. Most recent work has been done on
the 0.1~TeV \textit{First Muon Collider}, the energy taken to be representative
of the actual mass of a Higgs particle. A summary of progress and challenges follows:
\paragraph{Proton  driver}
The specification of the proton driver for the three machines is assumed the
same: $10^{14}$  protons/pulse at an energy above 16~GeV and 1-2~ns \textit{rms} bunch
lengths. There have been three studies of how to achieve these parameters. The most
conservative, at 30~GeV, is a generic design. Upgrades of the FNAL (at 16~GeV)
and BNL (at 24~GeV) accelerators have also been studied. Despite the very short bunch
requirement, each study has concluded that the specification is attainable.
Experiments are planned to confirm some aspects of these designs.

\paragraph{Pion production and capture}
Pion production has been taken from the best models available, but an
experiment (BNL-E910) that has taken data, and is being analyzed, will refine these
models. The assumed 20~T capture solenoid will require state-of-the-art technology. Capture, decay and phase rotation have been simulated, and
have achieved the specified production of 0.3~muons per initial proton. The
most serious remaining issues for this part of the machine are:
\begin{itemize}
\item { The nature and material of the target:}
The baseline assumption is that a liquid metal jet will be used, but the
effects of shock heating by the beam, and of the eddy currents induced in the
liquid as it enters the solenoid, are not yet fully understood.  
\item { The
maximum rf field in the phase rotation:} For the short pulses used, the current
assumptions would be reasonably conservative under normal operating conditions,
but the effects of the massive radiation from the nearby target are not known.
\end{itemize}
Both these questions can be answered in a target experiment planned to start within the next two years at the BNL AGS. 

Polarization of the muon beams represents a significant physics
advantage and is an important feature of a muon collider. Polarized muon beams are possible. Muons are produced with $100\%$ polarization
in the rest frame of the pion, but they travel in all directions. By accepting
the forward going muons, it is easy to obtain $25\%$ polarization in
either beam easily. The amount of polarization can be increased with an
accompanying price in luminosity.
\paragraph{Cooling}
The required ionization cooling is the most difficult and least understood
element in any of the muon colliders studied.
 Ionization cooling is a phenomenon that occurs whenever there is energy
loss in a strong focusing environment. 

 But achieving the
nearly $10^6$ reduction required is a challenge. Cooling over a wide range has
been simulated using lithium lenses and ideal (linear matrix) matching and
acceleration.  Examples of limited sections of solenoid lattices with
realistic accelerating fields have now been simulated, but the specification
and simulation of a complete system has not yet been done. Much theoretical
work remains: space  charge and wakefields must be included; lattices at the
start and end of the cooling sequences must be designed; lattices including
liquid lithium lenses must be studied, and the sections must be
matched together and simulated as a full sequence. The tools for this work
are nearly ready, and this project should be completed within two years.

Technically, one of the most challenging aspects of the cooling system appears
to be: 
\begin{itemize}
\item { High gradient rf} (e.g. 36~MV/m at 805~MHz) operating in strong
(5-10~T) magnetic field, with beryllium foils between the cavities. 
\end{itemize}
An experiment is planned that will test such a cavity, in the required fields,
in about two years time. On an approximately six year time scale, a \textit{Cooling Test Facility} is being proposed that could test ten meter lengths of different
cooling systems. If they are required, then an urgent need is to develop:
\begin{itemize}
\item { Lithium Lenses:} (e.g.\ 2~cm diameter, 70~cm long, liquid lithium
lenses with 10~T surface fields and a repetition rate of 15~Hz).
\end{itemize}
The use of 31~T solenoids could avoid their need, at least in the low energy \textit{First Muon Collider}, which would ease the urgency of this rather long term R\&D, but both options would require long-term R\&D. Meanwhile a short lithium lens
is under construction at BINP (Novosibirsk, Russia).

\paragraph{Acceleration}

The acceleration system is probably the least controversial, although
possibly the most expensive, part of a muon collider. Preliminary parameters
have been specified for acceleration sequences for a 100~GeV and a 3~TeV
machine, but they need refinement. In the low energy case, a linac is followed
by three recirculating or FFAG accelerators. In the high energy accelerator, the
recirculating or FFAG accelerators are followed by three fast ramping synchrotrons
employing alternating pulsed and superconducting magnets. The parameters do
not appear to be extreme, and it does not appear as if serious problems are
likely. 

\paragraph{Collider}

The collider lattices are challenging because of the requirement of  very low
beta functions at the interaction point, high single bunch intensities, and short bunch lengths.
 However, the fact that all muons will decay after about 800~turns means that
slowly developing instabilities  are not a problem. Feasibility lattices have
been generated for a 4~TeV case, and more detailed designs for 100~GeV machines are been studied. In the latter case, but still without errors, $5\sigma$~acceptances in
both transverse and longitudinal phase space have been achieved in tracking
studies. Beam scraping schemes have been designed for both the low energy
(collimators) and high energy (septum extractors) cases.

The short bunch length and longitudinal stability problems are avoided if the rings, as
specified, are sufficiently isochronous, but some rf is needed to remove the
impedance generated momentum spread. Transverse instabilities (beam breakup)
should be controlled by rf BNS damping.

The heating of collider ring superconducting magnets by electrons from muon
decay can be controlled by thick tungsten shields, and this technique also
shields the space surrounding the magnets from the induced radioactivity on the
inside of the shield wall. A conceptual design of magnets for the low energy
machine has been defined.

Although much work is yet to be done (inclusion of errors, higher order
correction, magnet design, rf design, etc), the collider ring does not appear
likely to present a serious problem.

\paragraph{Neutrino radiation and detector background}

Neutrino radiation, which rises as the cube of the energy, is not 
serious for machines with center of mass energies below about 1.5~TeV. It is
thus not significant for the First Muon Collider; but above  2~TeV, it
sets a constraint on the muon current and makes it harder to achieve desired
luminosities. However, advances in cooling and correction of tune shifts may
still allow a machine at 10~TeV with substantial luminosity ($>\ 10^{35}\
\textrm{cm}^{-2}\textrm{s}^{-1}$).

Background in the detector was at first expected to be a very serious
problem, but after much work, shielding systems have evolved that limit most
charged hadron, electron, gamma and neutron backgrounds to levels that are
acceptable. Muon background, in the higher energy machines, is a
special problem that can cause serious fluctuations in calorimeter
measurements. It has been shown that fast timing and segmentation can help
suppress this background, and preliminary studies of its effects on a physics
experiment are encouraging. The studies are ongoing.

\paragraph{Detector scenarios}
We have considered several options for the experimental detector components for
 various CoM energy colliders. Much work needs to be done to optimize the physics reach at each energy by feeding back the results of detailed simulations of backgrounds and signal to the detector design. Only then will the feasibility of doing physics with a muon collider be fully explored. 

\section{ACKNOWLEDGMENTS}
This research was supported by the U.S. Department of Energy under Contracts No.
DE-ACO2-98CH10886, DE-AC02-76CH03000 and DE-AC03-76SF00098.