\chapter{Muon Colliders} 
\label{higgsfact}

The lure of muon colliders arises from the
fact that the muon is $\approx$ 200 times heavier than the electron
and this makes it possible to accelerate the muon using circular
accelerators that are compact and fit on existing accelerator
sites. See Figure~\ref{compare} for a comparison of relative sizes of
muon colliders ranging from 500 GeV to 3 TeV center of mass energies
with respect to the LHC, SSC, and  NLC. Once we have solved the
problem of cooling a muon beam so that it can be accelerated, higher
enegies are much more easily obtained in a muon collider than in the
linear collider.
%
Because the muon is unstable, it becomes necessary to cool and
accelerate it before a substantial number have decayed. With typical
bending magnetic fields($\approx$ 5~Tesla) 
available with today's technology,  
the muons last $\approx$ 1000 turns before half
of them have decayed in the collider ring. This is a statement that is
independent of the energy of the collider to first order due to
relativistic time dilatation.

The muon decay also gives rise to large numbers of electrons that can
pose serious background problems for detectors in the collision
region. The 1999 Status Report~\cite{INTRO:ref5} contains an excellent
summary of the problems and possible solutions one faces on the way to
a muon collider.


%
\begin{figure}[bth!]
\includegraphics[width=6in,height=4.75in]{machine_comparison_new.ps}
%\centerline{\epsfig{file=machine_comparison_new.ps,height=4.75in,width=6.in}}
\vspace{0.5cm}
\caption[Sizes of various proposed high energy colliders]
{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}
%
Figure~\ref{schematic} shows  a schematic of such a muon
collider, along with a depiction of the possible physics that 
can be addressed with each stage of the facility.

%
\begin{figure}[bth!]
\centerline{\includegraphics[width=0.6\linewidth]{higgs_schematic.eps}}
\vspace{0.5cm}
\caption[Schematic of a  muon collider]{Schematic of a  muon collider.}
\label{schematic}
\end{figure}
%
\section{Higgs Factory Requirements}
The emittance of the muon beam needs to be
reduced by a factor 10$^6$ from production~\cite{INTRO:ref5} 
to the point of collision
for there to be significant luminosity for experiments. This can be
achieved by ionization cooling similar to the scheme described in
chapter~\ref{neufact}. The transverse emittance is reduced during
ionization cooling, since only the longitudinal energy loss is replaced
by $rf$ acceleration. However, due to straggling, the longitudinal
emittance grows. In order to cool longitudinally, one exchanges
longitudinal and transverse emittances and proceeds to cool the
transverse emittance.

The Status report~\cite{INTRO:ref5} outlines the details of the
acceleration and collider ring for the Higgs factory. Table \ref{sum}
gives a summary of the parameters of various muon colliders including
three different modes of running the Higgs Collider that have varying
beam momentum spreads. Additional information  about the 
Muon Collider can be found at~\cite{gail,higgsreport}.


\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}{|l|c|c|c|c|c|}
\hline
\rr CoM energy~ TeV   &\rr 3 &\rr 0.4 &
\multicolumn{3}{c}{\rr 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  & $2\times 10^{12}$ & $2\times10^{12}$ &\multicolumn{3}{c}{$4\times 10^{12}$ }  \\
\rr $\mu$ power~MW     & \rr 28 &\rr 4 & \multicolumn{3}{c}{\rr 1 }  \\
\rr Wall power~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 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~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!]
\includegraphics[height=2.9in,width=5.7in]{fnalfg2.ps}
\caption[Plan of a 0.1-TeV-CoM muon collider]
{Plan of a 0.1-TeV-CoM muon collider.}
\label{plan1}
\end{figure*}
%

\section{Longitudinal Cooling}

Currently there is no satisfactory solution for
emittance exchange and this remains a major stumbling block towards
realizing a muon collider. Figure~\ref{cell_exch} shows one of
the schemes that are under consideration to solve the emittance
exchange problem.
%
\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.in,height=4in]{double_bent_sol.ps}}
\caption[Bent Solenoid emittance exchange example]
{Representation of a bent solenoid longitudinal emittance exchange section.}
\label{cell_exch}
\end{figure}
%
 Ring Coolers have been introduced by Balbekov~\cite{BalbekovRing}.
 Figure~\ref{ring-cooler} shows a ring design currently under study.  The
 advantage of ring coolers is that one can circulate the muons in a
 cooling ring of circumference $\approx$ 30 m and circulate the muons
 many turns, thereby reusing the cooling channel elements.  It can be
 shown that such devices can cool in 6-D space. Figure~\ref{6dcooling}
 shows a simulation of the ring cooler that demonstrates cooling in 6
 dimensions. It should be emphasized that this simulation uses
 idealized magnetic fields. A study is currently under way to see if
 these results hold up with realistic magnetic fields.


 Ring coolers hold the promise to solve the emittance exchange
 problem. However, it has yet to be demonstrated that it is possible
 to inject into and extract from them high emittance beams. If these
 problems can be solved, it may be possible to cascade a number of
 ring coolers each providing cooling by a factor $\approx$ 30 to
 achieve the needed factor of 10$^6$.
%
Another ring cooling design by Al Garren~\cite{hansoncline} operates 
at higher energies and also holds the promise of 6-D cooling. 
%
\begin{figure}[bth!]
\centerline{\includegraphics[width=0.5\linewidth]{ringcooler.eps}}
\caption[Example of a ring cooler]{Example of a Ring Cooler.}
\label{ring-cooler}
\end{figure}
%
%
\begin{figure}[bth!]
\centerline{\includegraphics[width=0.5\linewidth]{6dcooling.eps}}
\caption[Cooling in a ring cooler]
{First results from a ring cooler showing 6 dimensional cooling.
 Both the transverse and longitudinal emittance plots show cooling. 
The transmission with and without muon decay is shown as a function of turn.
The horizontal axis shows period number, 4 periods making one turn.}
\label{6dcooling}
\end{figure}
If one can solve the longitudinal cooling problem,
 both neutrino factories and muon colliders will benefit. 
% 
\section{Higher Energy Muon colliders}
Once the cooling problems have been solved to realize the first muon
collider, acceleration to higher energies becomes possible.  Colliders
with 4 TeV center of mass energy have been studied~\cite{INTRO:ref5}
and Table \ref{dntable} lists the parameters for such a collider. The
radiation from the neutrinos from the muon decay begins to become a
problem at CoM energies of 3 TeV. One may attempt to solve this by a number of
means, including optical stochastic cooling of muons in the collider, whereby
one can get the same luminosity with less intensity.

\begin{table*}[tbh!]
 \caption[Parameters of Acceleration for 4~TeV Collider]
{Parameters of Acceleration for 4~TeV Collider.}
\label{dntable}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
                       & Linac & RLA1 & RLA2 & RCS1 & RCS2 \\
\hline
E (GeV) & 0.1$\rightarrow$ 1.5 & 1.5 $\rightarrow$ 10 & 10
$\rightarrow$ 70 & 70 $\rightarrow$ 250 & 250 $\rightarrow$ 2000 \\
f$_{rf}$ (MHz) & 30 $\rightarrow$ 100 & 200 & 400 & 800 & 1300 \\
N$_{turns}$ & 1 & 9 & 11 & 33 & 45 \\ V$_{rf}$(GV/turn) & 1.5 & 1.0 &
6 & 6.5 & 42 \\ C$_{turn}$(km) & 0.3 & 0.16 & 1.1 & 2.0 & 11.5 \\ Beam
time (ms) & 0.0013 & 0.005 & 0.04 & 0.22 & 1.73 \\
$\sigma_{z,beam}$(cm) & 50 $\rightarrow$ 8 & 4 $\rightarrow$ 1.7 & 1.7
$\rightarrow$ 0.5 & 0.5 $\rightarrow$ 0.25 & 0.25 $\rightarrow$ 0.12
\\ $\sigma_{E,beam}$(GeV) & 0.005 $\rightarrow$ 0.033 & 0.067
$\rightarrow$ 0.16 & 0.16 $\rightarrow$ 0.58 & 0.58 $\rightarrow$ 1.14
& 1.14 $\rightarrow$ 2.3 \\ Loss (\%) & 5 & 7 & 6 & 7 & 10 \\
\hline
\end{tabular}
\end{table*}
%
\section{Muon Collider Detectors} 
%
Figure~\ref{geant} shows a strawman
muon collider detector for a Higgs factory simulated in Geant. The
background from muon decay sources has been extensively 
studied~\cite{INTRO:ref5}. At the Higgs factory, 
the main sources of background are from photons generated by the
showering of muon decay electrons. At the higher energy colliders,
Bethe-Heitler muons produced in electron showers become a problem.
Work was done to optimize the shielding by using specially shaped
tungsten cones~\cite{INTRO:ref5}. 
The background rates obtained were shown to be
similar to those predicted for the LHC experiments. It still needs to
be established whether pattern recognition is possible in the presence
of these backgrounds.
%
\begin{figure}[bth!]
\centerline{\includegraphics[width=0.5\linewidth]{mu_geant_cut.eps}}
\caption[Strawman Geant detector for a muon collider]
{Cut view of a strawman detector in GEANT for the Higgs factory with
a Higgs$\rightarrow b\bar b$ event superimposed. No backgrounds
shown. The tungsten cones on either side of the interaction region
mask out a 20$\deg$ area.}
\label{geant}
\end{figure}
%

