\section{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  muons 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
electron-positron collider.
%
Because the muon is unstable, it is 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.

 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}
%
\subsection{Higgs Factory Requirements}
The emittance of the muon beam needs to be reduced by a factor of 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
the section~\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 in~\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*}
%

\subsection{Longitudinal Cooling}

At the time of writing of the status report~\cite{INTRO:ref5} 
there was no satisfactory solution for the  emittance exchange problem and
this remained a major stumbling block towards realizing a muon
collider. However, ring coolers have been found to hold significant
promise in cooling in 6-dimensional phase space. Another advantage of
ring coolers is that one can circulate the muons many turns, thereby
reusing the cooling channel elements.  
Several meetings on emittance exchange were held~\cite{eemeets} and a 
successful workshop~\cite{eework} was held in 2001, where we
explored in some depth several kinds of ring coolers.
These options differ primarily in the type of focusing used to 
contain the beam. We describe the current status of our
understanding of ring coolers here.

%
\subsubsection{Solenoidal Ring Coolers}



The basic design of the solenoidal ring cooler~\cite{balb1} is
presented in Figure~\ref{ring}.  Eight focusing dipole magnets with an
index $n=-\frac{1}{2}$ are used for bending and focusing of the beam. Each of
these dipoles bends the beam through 45 degrees with a central orbit
bending radius of 52 cm. We have done calculations to show that such
dipoles are buildable. There are 4 long solenoids containing RF
cavities and liquid hydrogen absorbers for transverse cooling.  A
magnetic field of 2.06~T on the edges of the solenoids provides the same
transverse focusing as the bending magnets.  The magnetic field
adiabatically increases to 5.15 T towards the center of the solenoid
in order 
to produce a small $\beta$ function (25-30 cm) at the absorbers.  The
short solenoids are designed to create an appropriate dispersion
function that is zero at the long solenoids, which house the 200 MHz
rf cavities.  Their field is $\pm 2.06~T$ at the edges
and $\pm 2.75$~T centrally.  A symmetric field flip is required in the 
short solenoids
to prevent the build up of canonical angular momentum. Lithium hydride wedge
absorbers covering  half of the vertical aperture at the centers of the
short solenoids are used for emittance exchange to produce 
longitudinal cooling.

  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbh!]
\vspace{0mm}
\begin{minipage}[tbh!]{0.47\linewidth}
\includegraphics[width=\linewidth]{02_ringA.eps}
%\centerline{\epsfig{file=02_ringA.eps,width=\linewidth}}
\end{minipage}
\begin{minipage}[tbh!]{0.47\linewidth}
\begin{tabular}{ll}
Circumference & 36.963 m \\ Nominal energy & \\ at short SS & 250 MeV
\\ Bending field & 1.453 T \\ Norm. field gradient & 0.5 \\
Max. solenoid field & 5.155 T \\ RF frequency & 205.69 MHz \\
Accelerating gradient & 15 MeV/m \\ Main absorber length & 128 cm \\
LiH wedge absorber & 14 cm \\ Grad. of energy loss & 0.75 MeV/cm \\
\end{tabular}
\end{minipage}
\caption{Layout and parameters of the solenoid based ring cooler
\label{ring}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbh!]
\begin{minipage}[t!]{0.47\linewidth}
\includegraphics[width=1.15\linewidth]{10_evol.eps}
%\centerline{\epsfig{file=10_evol.eps,width=1.15\linewidth}}
\end{minipage}
\begin{minipage}[t!]{0.47\linewidth}
\vspace{-5mm}
\begin{tabular}{|l|c|c|c|}
\hline
Number of turns & 0 & 10 & 15 \\
\hline
$x$ emittance (cm) & 1.2 & 0.24 & 0.21 \\ $y$ emittance (cm) & 1.2 & 0.24
& 0.21 \\ $z$ emittance (cm) & 1.5 & 0.79 & 0.63 \\ 6-D emitttance (cm$^3$) &
2.2 & 0.045 & 0.028 \\ 6-D cooling factor & 1 & 49 & 79 \\ Trans. w/o
decay & 1 & 0.72 & 0.71 \\ Trans. with decay & 1 & 0.56 & 0.48 \\
Merit factor & 1 & 27 & 38 \\
\hline
\end{tabular}
\end{minipage}
\vspace{-3mm}
\caption{Evolution of the beam emittance/transmission at the ring cooler.
\label{evol}}
\vspace{-3mm}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Evolution of the beam emittance and transmission is
shown in Figure~\ref{evol} as a function of the number of turns in the
ring.  In 15 turns, the transverse emittance decreases from 1.2
cm to 0.21 cm yielding a cooling factor of 5.7, the longitudinal
emittance decreases from 1.5 cm to 0.63 cm (cooling factor 2.4), and the 6-D
emittance decreases from 2.2 cm$^3$~to 0.028 cm$^3$, with an overall cooling
factor $\approx$~79.  The transmission is 0.71 without decay and 0.48
with decay. We define a merit factor for cooling that is 
the total transmission including decay times the 6-D cooling factor.
The  merit factor for this ring is then 38.  This implies that transverse
emittance at the ring cooler is about the same as at a linear SFOFO
cooling channel employed in Study-II~\cite{EPP:studyii}, whereas the
longitudinal emittance is noticeably less.

This cooler provides mainly transverse cooling and can be used as a
part of Neutrino Factory or a muon collider.  A cooler specially designed for
strong longitudinal cooling (``bunch compressor'') can also be created using
a similar scheme.  Such a compressor would be a part of a muon collider
to shorten muon bunches from 6-8 m (minimal length after $\pi -
\mu$~decay and phase rotation, see Ref.~\cite{INTRO:ref5}) to 0.6-0.8
m acceptable for further cooling by a 200 MHz channel.

Two options for the bunch compressor are considered in
Ref.~\cite{balb4}.  The first one is a two-step cooler where each step
is very similar to the ring cooler shown in Figure~\ref{ring}.  The main
difference is that  the primary goal in the first cooler is the
longitudinal bunching of the beam. This leads to a uniform magnetic
field in the long solenoids and lower frequency/voltage of the
accelerating rf system (15.6 MHz/4 MeV/m at the first stage vs.  62.5
MHz/8 MeV/m at the second one).  Another option is a 15 MHz octagonal
cooler composed of the same cells as in Figure~\ref{ring}, but with half the
bending magnet angle.  Decrease of longitudinal emittance
from 43 cm to 2.5-3 cm, as required for muon collider, is obtained in
both cases.

We are proceeding with a realistic simulation of this system using
Geant and ICOOL that employs realistic magnetic fields~\cite{kahn} produced by
field calculation programs.

After the two stage cooler, we still need a factor of $\approx$~30 in
transverse cooling, but we are within a factor of 4 in longitudinal
cooling relative to the Higgs factory goals. Lithium lens cooling, which with
its strong focusing will cool transversely further while degrading
longitudinally due to straggling, is a posibility and is being
investigated.

\subsubsection{RFOFO ring coolers}
The cooling lattice for the Neutrino Factory (see section~\ref{neufact}) 
employs a
configuration of fields known as an SFOFO lattice (super-FOFO) where
the axial magnetic field profile changes polarity in alternate cells
of the lattice. For the ring cooler design under consideration here,
we employ an RFOFO lattice (regular-FOFO) where the axial field
profile changes polarity in the middle of a cell. As a result all
cells in an RFOFO lattice are identical.

The ring cooler design employs a single cell for both transverse
cooling and emittance exchange. It uses solenoids for focusing, giving
large angular and momentum acceptances. The cell includes dispersion,
acceleration, and energy loss in a single thick hydrogen wedge. Figure
\ref{rforing} shows the layout of the cooling ring drawn to scale.  The
RFOFO lattice was chosen because, unlike in the SFOFO case used in
Study-II, all cells are strictly identical, and the presence of an
integer betatron resonance within the momentum acceptance is
eliminated.


\begin{figure}[htb!] 
\includegraphics[width=\linewidth]{rfofo_ring3.ps} 
\caption{Layout of an RFOFO cooling ring. \label{rforing}}
\end{figure}

The basic ring is made up of 12 identical 2.75-m long cells. In the
figure, this symmetry is broken for injection and extraction, but the
magnetic fields in this insertion are nearly identical to those in the
rest of the ring.  Figure \ref{cells} shows a detailed view of three
cells of the lattice.

\begin{figure}[tbh] 
\includegraphics {rfofo_h2reg1.ps} 
\caption{Three cells of the RFOFO lattice. \label{cells}}
\end{figure}

The longitudinal field on-axis has an approximately sinusoidal
dependence on position. The actual coils to generate the axial fields,
in the presence of the bending fields, would have to be slightly
different from those used in the simulation, but since the 3D fields
used are consistent with Maxwell's equations, there is no question
that suitable coil positions can be found.  The lattice transmits
particles in the momentum band from 150 to 250 MeV/c. The average
momentum for a small emittance beam varied from 191 to 201 MeV/c
across each cell of the lattice.  The minimum value of the beta
function at the central momentum is 40 cm.


Dispersion is provided by applying an approximately 0.125 T transverse
bending field generated by alternately tilting the vertical plane of
the solenoids by 1.5 degrees.  There is no attempt to control the
field index $n$ (where $B\propto r^n$).  So the focusing in x and y
are not identical.

It is found that the acceptance is reduced as the bending field is
increased. We thus use a wedge with maximum possible angle (giving
zero thickness on one side), and the lowest bending field consistent
with adequate emittance exchange.  The dispersion at the absorber of
-8 cm in a direction 30 degrees from the $y$ axis, The dispersion at the
center of the rf is of the opposite sign, and also mostly in the $y$
direction. Its direction is Larmor rotated by the axial fields.

The liquid-hydrogen wedge has a central thickness of 28.6 cm and a
total wedge angle of 76.93 degrees and is rotated 30 degrees from the
vertical. No absorber windows are included in this simulation.  The RF
cavities had a frequency of 201.25 MHz and a gradient of 16 MV/m. No
RF windows were included.

The ICOOL simulations shown do not include the injection/extraction
insertion, and use axial and transverse magnetic fields generated by a
truncated Fourier decomposition of the fields from a straight solenoid
lattice. The RF is represented as fields in perfect pillbox
cavities. The input tracks are taken from a Study-II simulation, using
distributions from just upstream of the  transverse cooling system. 
The use of Study-II
simulated distributions is intended to allow a more realistic estimate
of the ring's performance.  No attempt was made to match the ring
dispersion or slight differences in the transverse beta functions.

Figure~\ref{all} shows the transmission, transverse emittance (in $x$,
$y$), longitudinal emittance, 6-dimensional emittance, and a merit
factor $M$ vs. length in the ring. $M$ is given by:

$$M~=~{\epsilon_6(initial) \over \epsilon_6(final)}~\times~{\rm
Transmission}$$

\begin{figure}[tbh] 
\includegraphics{rfofo_all.ps}
%\vskip.5in 
\caption{Transmission,
normalized transverse emittance, normalized longitudinal emittance,
normalized 6-dimensional emittance, and the merit factor, as a
function of distance.  \label{all}}
\end{figure}

Initially, the $x$ emittance falls more rapidly than the $y$. This is
expected because it is the $y$ emittance that is exchanged with the
longitudinal emittance, but the Larmor rotations soon mix the $x$ and $y$
emittances bringing them to a common value.

 After a distance of 400 m ($\approx$~12 turns), the 6-dimensional
emittance has fallen by a factor of 290, with a transmission of 44 \%
(61\% without decay). The merit factor is 130. The same factor for the
Study-II cooling lattice, with no windows, is 13. With realistic
windows and the injection/extraction insertion added, the merit factor
will be much less than 130, but is likely to remain far better than
the Study-II example.

This ring could not be used, as is, to replace the Study-II cooling
channel because the bunch train in this case is far too long to fit in
the ring. But a spiral 3D cooling channel could be used and an even
greater performance gain could be expected if the spiral were also
tapered.

This approach seems very attractive, but it is still far from fully
realistic, and much work needs to be done.


\subsubsection{Quadrupole Ring Coolers}

Another type of ring cooler has been studied that uses quadrupole 
focusing~\cite{ucla}.

The SYNCH storage ring design program code~\cite{synch} was used to 
design this ring, which uses conventional magnet elements.
Figure~\ref{fig:ringucla} shows  such a ring that has 16 cells.
Elements in a half lattice cell are shown  schematically 
in Figure~\ref{fig:half}.

\begin{figure}[tbh!]
%\includegraphics[width=8cm]{fig1}
\includegraphics[width=8cm]{ucla_ring}
\caption{\label{fig:ringucla} Top view of the 16 cell muon cooling ring.}
\end{figure}
%
\begin{figure}[tbh!]
%includegraphics[height=6cm,width=8cm]{fig2}
\includegraphics[height=6cm,width=8cm]{ucla_1125deg}
\caption{\label{fig:half} Schematic diagram of  half of the 22.5
degree bending cell. A wedge absorber is located in the middle of the cell.}
\end{figure}
%
The SYNCH design parameters were then transferred to the
ICOOL~\cite{icool} ray tracing code that simulates ionization cooling.
Figure~\ref{fig:cell} shows the $\beta_{x}$, $\beta_{y}$, and
$D$(dispersion) from SYNCH as a function of arc length in a lattice
cell. Superimposed are the same quantites derived from beam behavior
in the ICOOL simulation, showing consistency between the two programs.

\begin{figure}[tbh!]
%\includegraphics[height=5cm,width=8cm]{fig3}
\includegraphics[height=5cm,width=8cm]{ucla_16cell}
\caption{\label{fig:cell}The $\beta_{x}$, $\beta_{y}$, and $D$(dispersion)
in a 22.5 degree bending cell. SYNCH input(solid curves) and ICOOL
simulation(marked points) are compared.}
\end{figure}


Figure~\ref{fig:figaa} shows the transverse and longitudinal normalized
emittances as a function of the number of turns. The average muon beam
momentum is 500 MeV/c, and liquid H$_2$ absorbers with wedge opening
angles of 40 degrees are used. The path length of the central trajectory
through
the liquid H$_2$
wedge absorbers is 25 cm per 22.5 degree bending cell.
Average energy loss in the wedge absorbers is compensated in the 201 MHz
rf cavities. The equilibrium normalized emittances are about 1 mm$\cdot$rad 
in $x$ and $y$, and around 10 mm in $z$.


\begin{figure}[tbh!]
\includegraphics[height=5cm,width=8cm]{ucla_turns}
%\includegraphics[height=5cm,width=8cm]{figa}
\caption{\label{fig:figaa} The evolution of x, y, z normalized emittances
in 30 full turns.}
\vskip0.5cm
\includegraphics[height=5cm,width=8cm]{ucla_trans}
%\includegraphics[height=5cm,width=8cm]{figb}
\caption{\label{fig:figbb}The transmission and the figure of merit factor
as a function of s ~in 16 full turns.}

\end{figure}


Figure~\ref{fig:figbb} shows the muon transmission efficiency 
and the  merit factor,   
as a function of the number of turns. The merit factor 
reaches $\approx$~5 after 16 turns, 
where the transmission efficiency is  $\approx$~0.55.
At 500 MeV/c, 
the fraction of muons lost due to muon decay in  16 full turns is
0.57, yielding an overall transmission factor of 0.31.

%\begin{figure}[tbh!]
%\includegraphics[height=5cm,width=8cm]{figb}
%\caption{\label{fig:figbb}The transmission and the figure of merit factor
%as a function of s ~in 16 full turns.}
%\end{figure}



%
\subsubsection{Injection into Ring Coolers}
%
The most serious technical problem facing the ring cooler approach is
the injection system which may require a very powerful kicker
magnet~\cite{balb5}. The energy stored in the injection kicker goes as
the square of the emittance of the beam and inversely as the
circumference of the ring. A promising injection scheme that does not
use kicker magnets, but instead uses absorbers to degrade the beam
energy and RF phase manipulations has been proposed~\cite{balb6} and
is being studied.
% 
\subsection{Higher Energy Muon colliders}
Once the cooling problems have been solved for the design of 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~\cite{kingnu}. 
One may attempt to solve this by a
number of means, including optical stochastic cooling of muons in the
collider, thus permitting the same luminosity with less intensity.

 There have been preliminary attempts to study colliders of
even higher energy, starting at 10~TeV all the way up to 
100~TeV in the center of mass and
we include the references to these studies~\cite{kinghi} for the sake 
of completeness.

\begin{table*}[tbh!]
 \caption[Parameters of Acceleration for a 4~TeV Muon Collider] {Parameters
 of Acceleration for a 4~TeV Muon 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*}

\subsection{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}
%

