\section{Muon Colliders} 
\label{higgsfact}

The primary advantage of using  muons in a lepton collider arises from the fact that the muon is
$\approx$~200 times heavier than the electron. It is thus 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  the problem of
cooling a muon beam to sufficiently small emittances 
is solved and the beams  can be accelerated, higher energies 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
the beam before a substantial number have decayed.  The number of
turns in a muon lifetime is independent of the muon momentum for a
given magnetic field, since both the circumference and the muon
lifetime in the laboratory frame scale with muon momentum.  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.



 Muon decay also gives rise to large numbers of electrons that can
affect the cryogenics of the magnets in the machine as well as 
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. Some of the ideas needed to obtain
longitudinal cooling necessary for the Muon Collider are discussed in
section~\ref{long-cool} and some of the parameters of the accelerator
system for higher energy colliders are discussed in section~\ref{high-acc}
below.

%
\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
$\approx$ 10$^6$ from production~\cite{INTRO:ref5} to the point of
collision for there to be significant luminosity for
experiments. Table~\ref{emitable} lists the transverse and
longitudinal emittances at the end of the decay channel,
Study-II~\cite{EPP:studyii} cooling channel and those needed for a
0.1~TeV Center of Mass Energy Muon Collider, also known as a Higgs Factory. 
It can be seen that one needs to cool by a factor of
$\approx$ 20 in the transverse dimension and $\approx$ 3 in the
longitudinal dimension from the Study-II emittances to achieve the
emittances necessary for a Higgs factory.
%
\begin{table*}[thb!]
\centering  
\caption[Emittances at the end of various machines. ]
{Transverse and longitudinal emittances at the end of the decay
channel, Study-II cooling channel and the Higgs factory cooling
channel.\label{emitable}}
\begin{tabular}{|c|c|c|}
\hline
Emittance at end of & Transverse emittance ($\pi$ mm) & Longitudinal
emittance ($\pi$ mm) \\
\hline
Decay Channel & 17 & 150 \\ Study-II cooler & 2.6 & 30 \\
Higgs Factory cooler & 0.14 & 9 \\
\hline
\end{tabular}
\end{table*}
%
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 energy spread of the beam increases, even if the average
longitudinal energy of the beam is kept constant. The longitudinal
emittance thus grows in a linear cooling channel.  In order to cool
longitudinally, one needs to create dispersion in the system and have
wedge absorbers at the point of maximum dispersion so that the faster
particles go through the thicker parts of the wedge. This results in a
reduction in longitudinal emittance accompanied by an increase in
transverse emittance and is thus called emittance exchange.

The status report~\cite{INTRO:ref5} outlines the details of the
acceleration and collider ring for the 0.1~TeV Higgs factory, shown 
schematically in Fig.~\ref{plan1}. 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, also known as a Higgs Factory.}
\label{plan1}
\end{figure*}
%

\subsection{Longitudinal Cooling}
\label{long-cool}
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 three types 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. Figure~\ref{dipole} shows a
configuration of such a dipole and the resulting magnetic field
components calculated using a 3D field calculation program.  
%
There are
4 long solenoids containing rf cavities and liquid hydrogen absorbers
for transverse cooling.  A magnetic field of 2.06~T at the end regions
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. This field flip causes the dispersion
in the long solenoids housing the rf cavities to be zero while
permitting a non-zero dispersion at the lithium hydride wedge
absorbers at the centers of the short solenoids which then produce
longitudinal cooling via emittance exchange.


  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 & \\  straight section & 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[htb!] 
\includegraphics[width=2.5in]{wedge_dipole.eps}
\includegraphics[width=2.5in]{dipole_fields.eps}
\caption{Figure shows a computer model of a 52cm radius dipole with 
index $n=-\frac{1}{2}$ and the calculated field components radially  10cm off axis.
\label{dipole}}
\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 possibility and is being
investigated.

\subsubsection{RFOFO ring coolers}

\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]{ringfig.eps} 
\caption{Layout of an RFOFO cooling ring. \label{rforing}}
\end{figure}

The basic 33 m circumference 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, in plan (a) and side (b) views.                    

\begin{figure}[tbh] 
%\includegraphics {rfofo_h2reg1.ps}                                      
\begin{center} 
a)
\includegraphics[width=3.5in] {scott32a.ps}                              

b)
%\includegraphics {rfofo_h2reg1.ps} 
\includegraphics[width=3.5in] {scott32b.ps}                              
\end{center}
\caption{Three cells of the RFOFO lattice; a) plan, b) side. \label{cells}} 
\end{figure}

The longitudinal field on-axis has an approximately sinusoidal
dependence on position. The beam axis is displaced laterally 
with respect to the coil centers (as shown in figure \ref{cells}a) to 
minimize horizontal fields that cause vertical beam deviations.
The lattice transmits
particles in the momentum band from 150 to 250 MeV/c. 
The minimum value of the beta
function at the central momentum is 40 cm.


Bending round the ring, and the required dispersion, are provided by applying an approximately 0.125 T vertical bending field generated by alternately tilting
the solenoids (as shown in figure \ref{cells}b).  There is no attempt to set the
field index $n$ (where $B\propto r^n$) to the 0.5 value;  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 is
approximately 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. 

The liquid-hydrogen wedge has a central thickness of 28.6 cm and a
total wedge angle of 100 degrees and is rotated 30 degrees from the
vertical.  The rf
cavities are at a frequency of 201.25 MHz and have an accelerating 
gradient of 12 MV/m. 

The ICOOL~\cite{icool} simulation (with results shown in shown Figure~\ref{all})
used fields calculated from the specified coils, and thus 
neglects no end field effects. But in this simulation,
no absorber, or rf, windows, are included; nor did it include
the injection/extraction insertion.
The rf was represented by the fields in perfect pillbox
cavities. The input tracks were taken from a Study-II~\cite{EPP:studyii} 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 this beam to the ring
dispersion or slight differences in the transverse beta functions.

Figure~\ref{all} shows the simulated transmission, transverse emittance,
 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}
\includegraphics[height=3.6in]{scott33.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 $y$, 
because it is the $y$ emittance that is exchanged with the
longitudinal. 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 238, with a transmission of 48 \%
(66\% without decay). The merit factor is 136. The same factor for the
Study-II cooling lattice, also without windows, is 13. Studies with realistic
windows and the injection/extraction insertion added, show lower merit 
factors, but always far better than the Study-II example.

The design of the injection/ejection channels and kickers will be challenging, 
and this ring could not be used, as is, to replace the Study-II cooling
channel because the bunch train in that case is too long to fit in
the ring. Both problems would be removed in a helical cooling channel. 
The merit factor 
for such a channel could be even better than that of the ring
because it would be possible to ``taper'' the optics, as a function 
of distance down the channel, and thus
lower the final equilibrium emittance. 


\subsubsection{Quadrupole Ring Coolers}

Alternative ring designs have been explored which are based on storage
rings which consist of conventional quadrupole and dipole magnetic elements
instead of solenoids~\cite{ucla}.  The strategy has been to utilize the SYNCH
storage ring design code~\cite{synch} to develop linear lattice solutions
and then transfer the lattice parameters into the ray-tracing tracking 
code ICOOL~\cite{icool} in which absorbers and energy recovery with rf cavities
can be included for full simulation.  An example of such a ring is shown
in Figure~\ref{fig:half} where the elements of a half cell for a 22.5$^\circ$
bending cell are depicted schematically.  The correspondence between the beam
envelope beta and dispersion functions resulting from a simulation with the 
ICOOL tracking code and the SYNCH calculated values are shown in 
Figure~\ref{fig:cell}.   The full sixteen cell ring is shown in 
Figure~\ref{fig:ringucla}.

%
\begin{figure}[tbh!]
\includegraphics[height=6cm,width=8cm]{ucla_1125deg}
\caption{\label{fig:half} Schematic diagram of half of a 22.5$^\circ$
bending cell. A wedge absorber is located in the middle of the cell.}
\end{figure}
%
\begin{figure}[tbh!]
\includegraphics[height=5cm,width=8cm]{ucla_16cell}
\caption{\label{fig:cell}The $\beta_{x}$, $\beta_{y}$, and $D$(dispersion)
in a 22.5$^\circ$ bending cell. SYNCH calculations(solid curves) and
beam parameters from an ICOOL simulation(marked points) are compared.}
\end{figure}
%
\begin{figure}[tbh!]
\includegraphics[width=8cm]{ucla_ring}
\caption{\label{fig:ringucla} Top view of a sixteen cell muon cooling ring.}
\end{figure}
%

In general, we find that the performance of the rings, as measured in emittance
reduction along with particle transmission and decay losses, improves when
more compact lattice designs are considered.  In Figure~\ref{fig:figaa}, the
variation of the normalized emittances as a function of ring turns
is shown for an eight cell ring.  A reduction of normalized emittance
is observed for all three dimensions.  This particular
ring has a total circumference of 30.9~m.  Each half cell contains
one 22.5$^\circ$ combined function dipole proceeded and followed by
a single horizontally focusing quadrupole.
The average muon beam momentum is 250~MeV/c and liquid hydrogen
absorbers with wedge opening angles of 40$^\circ$ are used. For each cell, the
central beam orbit traverses 24 cm of absorber.  The energy loss in the wedge
absorbers is compensated with 201~MHz rf cavities with peak on-axis
gradients of 16~MV/m.

\begin{figure}[tbh!]
\includegraphics[height=5cm,width=8cm]{chgr_norm.ps}
\caption{\label{fig:figaa} The evolution of x, y, z normalized emittances
in 32 full turns.}
\vskip0.5cm
\includegraphics[height=5cm,width=8cm]{chgr_merit.ps}
\caption{\label{fig:figbb}The transmission and the figure of merit factor
as a function of the arc length in 32 full turns.}
 
\end{figure}

The muon transmission efficiency and total merit factor (muon survival
rate times the ratio of initial to final 6-dimensional emittance) is shown
in Figure~\ref{fig:figbb} as a function of ring turns.
The merit factor reaches 18, while the muon transmission efficiency
including decay losses after eighteen full turns is 40\%.


Rings in which the focusing function is handled exclusively by the
dipole elements have also been explored.  In this case, the natural
focusing power of the dipole is utilized for horizontal focusing while
the entrance and exit dipole edge angles are adjusted to provide the
required vertical focusing.  Several examples of such lattices have
been examined.  Because these lattices are more compact than lattices
which also include quadrupoles, the performance of these rings tend to
be better.  As an example, performances with merit factors on the
order of 100 have been observed with rings based on four cell lattice
with eight 45$^\circ$ dipoles.  For one case, the entrance and exit
angles dipole faces are 7.4$^\circ$ and 21.8$^\circ$ relative to the
normal of the beam trajectory.  The ring for this example has a
circumference of only 9.8 m and the design of injection/ejection cells
may prove to be very challenging.

%
\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}
\label{high-acc}
Once the required cooling  has been achieved to make the first
muon collider feasible, 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 muons are accelerated initially by a linear
accelerator followed by a series of recirculating linear accelerators
(RLA's) followed by rapid cycling synchrotrons (RCS's).  The radiation
from the neutrinos from the muon decay begins to become a problem at
CoM energies of 3 TeV~\cite{kingnu}. 

 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} that reduce the backgrounds resulting from electomagnetic 
showers from entering the detector.  The occupancy levels due to background photons and neutrons in detectors
 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, especially the Bethe Heitler muons, which are a unique source of 
background to muon collider detectors..
%
\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}
%