\section{ACCELERATION}
Following cooling and initial bunch compression, the beams must be rapidly
accelerated. A sequence of linacs would work, but would
be expensive, so some form of circulating acceleration is preferred. At lower energies, the acceleration time is so short that any form of magnet ramping is probably impractical. The conservative option
is to use
a sequence of recirculating accelerators (similar to that used at TJNL), but fixed frequency alternating gradient acceleration (FFAG) is also being studied\cite{ref28}. At higher energies, it is probably
more economical to use fast rise time pulsed magnets in
more conventional synchrotrons\cite{ref29}.
The accelerator for the muons is physically the largest component of
a muon collider and is also expected to be the most expensive component
for higher energy colliders.
\subsection{Scenarios}
Tbs.~\ref{acceleration1} and \ref{acceleration2} give an example of possible sequences of
accelerators for a 100 GeV Higgs Factory and a 3 TeV collider. In both cases, following initial linacs, recirculating accelerators
are used. Designs\cite{ref30} have been made of multiple aperture superconducting magnets for use in recirculating acceleration. The use of such magnets was not assumed in the scenarios, but they would reduce the diameter of the recirculating accelerator ring and lower particle loss from decay.
In the high energy case, the final three stages use pulsed magnet synchrotrons.
If only pulsed magnets were used the power
consumed by a ring would be high and its circumference large, but a hybrid ring with alternating
pulsed warm magnets and fixed superconducting magnets appears practical. In the example, the last two such rings are located in the same tunnel with differing ratios of pulsed to fixed magnets. The fixed magnets are superconducting at 8 T; the pulsed magnets are warm with fields that swing from -~2~T to +~2~T.
In both cases, except for the earliest stages, superconducting rf is employed. The reason for this, in the earlier stage, is that the instantaneous acceleration power requirement is very high, and the use of superconducting cavities allows a longer rf fill time and a reduced rf power source requirement. For the higher energy accelerators the use of superconducting cavities is dictated by the need to achieve high wall to beam efficiency.
A study\cite{ref31} tracked particles through
a similar sequence of recirculating accelerators and found a dilution of
longitudinal phase space of the order of 10\% and negligible particle loss.
\begin{table}[bth!]
\caption{ Parameters Higgs Factory (100 GeV) Accelerators}
\label{acceleration1}
\begin{tabular}{llcccccc}
acc type & &linac &linac &recirc&recirc&recirc& sums\\
rf type & &sled Cu&sled Cu&sled Cu&sled Cu&SC Nb & \\
\hline % & & & & & & & \\
$E_{init}$ & GeV & 0.10& 0.20& 0.70& 2 & 7 & \\
\rr $E_{final}$ &\rr (GeV) &\rr 0.20&\rr 0.70&\rr 2 &\rr 7 &\rr 50 & \\
\hline % & & & & & & & \\
circ & km & 0.04& 0.07& 0.06& 0.18& 1.21& 1.57\\
turns & & 1 & 1 & 8 & 10 & 11 & \\
\rr decay loss &\rr \% &\rr 2.31&\rr 3.98&\rr 6.74&\rr 7.77&\rr 9.88&\rr 27.29\\
\rr decay heat & \rr W/m &\rr 0.85&\rr 1.88&\rr 10.50&\rr 12.39&\rr 12.14& \\
\hline % & & & & & & & \\
$B_{fixed}$ & T & & & 2 & 2 & 2 & \\
pipe width & cm & & & 30.66& 21.22& 10.44& \\
pipe ht & cm & & & 10 & 8 & 4.30& \\
\hline % & & & & & & & \\
rf freq & MHz & 90 & 90 & 120 & 170 & 400 & \\
acc/turn & GeV & 0.20& 0.40& 0.17& 0.50& 4 & \\
acc time & $\mu s$ & & & 1 & 6 & 43 & \\
acc Grad & MV/m & 8 & 8 & 8 & 10 & 15 & \\
%synch rot's & & 0.42& 0.42& 0.64& 0.92& 8.59& \\
%phase slip & deg & & & 17.43& 9.23& 11.85& \\
grad sag & \% & & & 13.08& 16.82& 27.15& \\
rf time & ms & 0.55& 0.56& 0.37& 0.24& 2.04& \\
peak rf /m & MW/m & 2.72& 2.56& 2.21& 4.40& 0.20& \\
ave rf power& MW & 0.61& 1.10& 0.28& 0.88& 1.99& 4.88\\
%rf wall & MW & \rr 4.71&\rr 8.50&\rr 1.67&\rr 5.20&\rr 5.87&\rr 25.94\\
\hline % & & & & & & & \\
\rr total wall p& \rr MW & \rr 4.71&\rr 8.50&\rr 1.67&\rr 5.20&\rr 5.87&\rr 25.94\\
beam power & MW & 0.00& 0.01& 0.03& 0.12& 0.92& 1.08\\
wall-beam eff& \% & 0.06& 0.15& 1.93& 2.22& 15.62& 4.16\\
\end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\squeezetable{
\begin{table}[tbh!]
\caption{ Parameters 3 TeV Collider Accelerators}
\label{acceleration2}
\begin{tabular}{llcccccccc}
acc type & &linac &recirc&recirc&recirc&pulsed&pulsed &pulsed & sums\\
magnet type & & &warm &warm &warm &warm &hybrid&hybrid& \\
rf type & &sledCu&sledCu&sledCu&SC Nb &SC Nb &SC Nb &SC Nb & \\
\hline % & & & & & & & & & \\
$E_{init}$ & GeV & 0.10& 0.70& 2 & 7 & 50 & 200 & 1000 & \\
\rr $E_{final}$ &\rr GeV &\rr 0.70&\rr 2 &\rr 7 &\rr 50 &\rr 200 &\rr 1000 &\rr 1500 & \\
\hline % & & & & & & & & & \\
circ & km & 0.07& 0.12& 0.25& 1.16& 4.65& 11.30& 11.36& 28.93\\
turns & & 2 & 8 & 10 & 11 & 15 & 27 & 17 & \\
\rr decay loss &\rr \% &\rr 6.11&\rr 12.11&\rr 10.38&\rr 9.53&\rr 10.68&\rr \rr 10.07&\rr 2.65&\rr 47.68\\
decay heat & W/m & 3.46& 14.20& 16.03& 15.49& 19.44& 30.97& 18.09& \\
\hline % & & & & & & & & & \\
pulsed $B_{max}$ & T & & & & & 2 & 2 & 2 & \\
$B_{fixed}$ & T & & 0.70& 1.20& 2 & 2 & 8 & 8 & \\
ramp freq & kHz & 900 & 109 & 40.02& 7.99& 1.43& 0.33& 0.53& \\
\hline % & & & & & & & & & \\
sig beam & cm & 0.59& 0.51& 0.39& 0.25& 0.11& 0.08& 0.06& \\
sig width & cm & & 3.03& 3.65& 3.85& 1.36& 0.59& 0.18& \\
mom compactn& \% & & -1 & -2 & -2 & -1 & -1 & -1 & \\
pipe width & cm & & 30.31& 36.49& 38.53& 13.63& 5.86& 3 & \\
pipe ht & cm & & 10 & 8 & 4.30& 3 & 3 & 3 & \\
\hline % & & & & & & & & & \\
rf freq & MHz & 90 & 50 & 90 & 200 & 800 & 1300 & 1300 & \\
acc/turn & GeV & 0.40& 0.17& 0.50& 4 & 10 & 30 & 30 & \\
acc time & $\mu s$ & & 3 & 8 & 41 & 232 & 1004 & 631 & \\
eta & \% & 0.73& 0.22& 0.33& 0.44& 10.15& 14.37& 12.92& \\
acc Grad & MV/m & 8 & 8 & 10 & 15 & 15 & 25 & 25 & \\
synch rot's & & 0.54& 0.82& 1.91& 9.16& 27.07& 76.78& 31.30& \\
phase slip & deg & & 6.90& 4.62& 5.35& 1.64& & & \\
cavity rad & cm & 122 & 220 & 134 & 76.52& 19.13& 11.77& 11.77& \\
loading & \% & 4.23& 6.22& 11.98& 16.54& 210 & 527 & 296 & \\
grad sag & \% & & 3.16& 6.18& 8.65& & & & \\
rf time & ms & 0.56& 1.35& 0.59& 2.04& 0.40& 1.25& 0.96& \\
peak rf /m & MW/m & 2.56& 3.43& 6.05& 0.81& 0.91& 0.56& 0.50& \\
ave rf power& MW & 1.11& 1.54& 2.84& 7.20& 6.32& 21.91& 15.07& 55.99\\
\rr rf wall &\rr MW &\rr 8.50&\rr 9.05&\rr 16.69&\rr 21.18&\rr 18.59&\rr 44.72&\rr 30.76&\rr 149 \\
\hline % & & & & & & & & & \\
magnet ps & MJ & & & & & & 34.31& 13.19& 47.51\\
\rr magnet wall &\rr MW & & & & & &\rr 3.7&\rr 1.4& \rr 5.1\\
\hline % & & & & & & & & & \\
\rr total wall &\rr MW &\rr 8.50&\rr 9.05&\rr 16.69&\rr 21.18&\rr 18.59&\rr 48.4& \rr 32.2&\rr 155 \\
beam power & MW & 0.02& 0.04& 0.15& 1.17& 3.68& 17.54& 9.86& 32.47\\
wall-beam eff& \% & 0.26& 0.49& 0.91& 5.51& 19.81& 39.23& 32.06& 21.72\\
% & & & & & & & & & \\
\end{tabular}
\end{table}
}
\section{COLLIDER STORAGE RING}
After acceleration, both $\mu^+$
and $\mu^-$ bunches are injected into a separate storage ring. The highest
possible average bending field is desirable, to maximize the number of
revolutions before decay, and thus maximize the luminosity. Collisions would
occur in one, or perhaps two, low-$\beta^*$ interaction areas. Parameters
of the rings were given earlier in Tb.~\ref{sum}.
\subsection{Lattice Design}
In order to
maintain the required short bunches, without excessive rf, approximately
isochronous Flexible Momentum Compaction lattices\cite{ref32,ref32a,ref32b} would be used. In the high energy cases, the required betas at the intersection point are very small (e.g. $\beta^*=3\,{\rm mm}$ for 4 TeV), and the quadrupoles needed to generate them are large (20-30 cm diameter). At 100 GeV, the betas at the IP are not that small and the quadrupoles are more conventional, but in both cases it has been found that local chromatic correction is essential\cite{chromatic,chromatica,chromaticb,chromaticc}.
\begin{figure}[bht!]
\centerline{\epsfig{file=fnalfg9.ps,height=4.0in,width=3.5in, angle=-90}}
\caption{Dynamical aperture after 1000 turns.}
\label{dynaper}
\end{figure}
Preliminary lattices have been designed for both 4 TeV and 0.5 TeV machines\cite{ref33}, and several designs now exist for the 100 GeV case. Fig.~\ref{dynaper} gives the dynamic aperture of one such lattice\cite{ref33a} for the required 1000 turns.
\subsection{Scraping}
Collimation schemes have been designed\cite{ref34} for colliders at both high and low energies. At low energies, as in the Higgs Factory, tungsten collimators have been shown to be effective. At higher energies, the muons are scattered, but not stopped,
by such collimators. For this case it has been shown that electrostatic septa followed by sweeping magnets could effectively extract the tail muons. Lattices\cite{ref33} have been designed incorporating these systems.
\subsection{Instabilities}
Studies\cite{furman,chen} have considered beam emittance growth due to beam-beam tune shift, and both, although some assumptions were made, predict negligible effects in 1000 tunes at the values shown in Tb.~\ref{sum}.
A study\cite{ref35} has examined the resistive wall impedance longitudinal instabilities in rings at several energies. At the higher energies and larger momentum spreads, solutions were found with small but finite momentum compaction, and moderate rf. For the special case of the Higgs Factory, with its very low momentum spread, a solution was found with no synchrotron motion, but rf provided to correct the first order impedance generated momentum spread. The remaining off-momentum tails, that would not affect the luminosity, but which might generate background, could be removed by a higher harmonic rf correction.
Given the very slow, or nonexistent synchrotron oscillations, the transverse beam breakup instability is significant. But this instability can
be stabilized using rf quadrupole\cite{ref36} induced BNS damping. For instance, in the 3 TeV case, to stabilize the resistive wall instability, the required tune
spread, calculated\cite{ref38} using the two particle
model approximation, for a 1 cm radius aluminum pipe, is only
$1.58\times10^{-4} $.
However, this application of the BNS damping to a quasi-isochronous ring, and other head-tail instabilities due to the chromaticities $\xi$ and $
\eta_1$, needs more careful study.
\subsection{Bending Magnet Design}
\begin{figure}[bht!]
\centerline{\epsfig{file=fnalfg10.ps,height=4.0in,width=3.5in}}
\caption{Power penetrating tungsten shields vs. their thickness for a) 4 TeV, and b) 100 GeV, collider. \label{shieldingnew}}
\end{figure}
The magnet design is complicated by the fact that the $\mu$'s decay within
the rings ($\mu^- \rightarrow\ e^-\overline{\nu_e}\nu_{\mu}$), producing
electrons whose mean energy is approximately $0.35$ that of the muons. With no shielding, the average power deposited per unit length would be about 2 kW/m
in the 4 TeV machine, and 300 W/m in the 100 GeV Higgs factory.
Fig.~\ref{shieldingnew} shows the power penetrating tungsten shields of different thickness. One sees that 3 cm in the low energy case, or 6 cm at high energy would reduce the power to below 10 W/m, which can reasonably be taken by superconducting magnets.
The quadrupoles could use warm iron
poles placed as close to the beam as practical. The coils could then be either
superconducting or warm, placed at a greater distance from the beam and shielded from it by the poles.