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\title[Muon Collider and Higgs Factory]{Muon Collider and Higgs Factory}

\author{Allen Caldwell \dag \footnote[3]{Email: caldwell@nevis.columbia.edu}}

\address{\dag\ Nevis Laboratories, Columbia University, New York, New
York, USA}

\begin{abstract}
Particle physics is at a stage where new, innovative, techniques are
required if the push to higher energies is to continue.  A Muon Collider
facility offers some tantalizing features, but the muon cooling issue
must first be solved.  An approach based on frictional cooling is
discussed here.
\end{abstract}

\section{Introduction}
The reasons for interest in a Muon Collider are well known: the large 
muon mass 
relative to the electron mass removes the concerns with energy loss due to
synchrotron radiation and beamstrahlung, 
and the point nature of the muon is highly favorable in comparison with a
complex proton.  We therefore dream of building a high 
energy Muon Collider.  
Parameter sets have been considered up to 100~TeV+100~TeV~\cite{ref:King}.
A lower energy Muon Collider would be interesting as a Higgs Factory,
since the Higgs production cross section would be 40000 times higher than
at an electron collider.  Also, very fine energy scans would be possible
since the beam energy is well-defined.  Other
physics research possibilities 
stem from the high energy neutrino beams resulting from the decay of
stored muons, and
low energy neutrinos from the target. There is also a wide range of 
applications
for low energy muons, which could be a by-product of the muon production
procedure~\cite{ref:Morenzoni}.  The development of a Muon Collider would
therefore open up a new realm for physics experimentation, providing
opportunities for a broad spectrum of physics topics.

\section{Parameters of a Muon Collider}
Figure~\ref{fig:layout} shows a conceptual layout for the Muon Collider 
facility.  Briefly, an intense, bunched, 
proton beam is directed on a target in a
region with strong solenoidal magnetic field.  The resulting pions are
collected and allowed to decay in a drift region, yielding muons in a 
large phase space.  The muons are squeezed in
longitudinal momentum with a phase rotation (time varying electric field).  
This is followed by a cooling
section, intended to reduce the muon `beam' phase space by $10^6$.  Once the
beams are cooled, they can then be accelerated to high energies.  A parameter
set for a 3 TeV CoM energy
Muon Collider is given in \cite{ref:Ankenbrandt}, and is
summarized in Table~\ref{tab:pars}.
The required normalized 6-D emittance for this  parameter set is 
$\epsilon_{6,N}=1.7\cdot10^{-10} ({\pi m})^3$.  
At the end of the drift
region the rms dimensions of the muon cloud  in
x,y,z is approximately 0.05,0.05,10~m, and in $P_x,P_y,P_z$ 
$50,50,100$~MeV/c, yielding $\epsilon_{6,D}=1.7\cdot 10^{-4} ({\pi m})^3$.
The required emittance reduction is therefore of order $10^6$, and this
must be done with a reasonable efficiency. 


\begin{figure}[hbpt]
\begin{center}
\epsfig{figure=scheme.eps,height=15.cm}
\end{center}
\caption[]{Possible layout for a Muon Collider facility~\cite{ref:Ankenbrandt}.}
\label{fig:layout}
\end{figure}

\begin{table}[hbpt]
\begin{center}
\begin{tabular}{lc}
\hline 
CoM energy (TeV) & 3 \\
proton energy (GeV) & 16 \\
protons/bunch & $2.5\cdot 10^{13}$ \\
bunches/fill & 4 \\
Rep. rate (Hz)  & 15 \\
$\mu$/bunch & $2 \cdot 10^{12}$ \\
collider circ. (m) & 6000 \\
$\epsilon_{6,N} (\pi m)^3$ & $ 1.7 \cdot 10^{-10} $ \\
Luminosity (cm$^{-2}$s$^{-1}$) & $7 \cdot 10^{34}$  \\
\hline
\end{tabular}
\caption{A set of parameters for a 3 TeV CoM energy Muon 
Collider~\cite{ref:Ankenbrandt}.}
\label{tab:pars}
\end{center}
\end{table}

There are severe difficulties in the way of a Muon Collider:
\begin{itemize}
\item Muons are not readily available, requiring a multi-megawatt proton 
source.  In addition to a large starting cost, there are crucial targetry
issues.
\item Muons decay in $2.2\cdot 10^{-6}$~s, so the time available for cooling,
bunching and accelerating the muons is very limited.  New techniques and
technologies are needed to address these problems.
\item There are large experimental backgrounds resulting from muon decays, so
detector design and operation will not be as straightforward as for an 
$e^+e^-$ collider.
\item High energy muon beams with large currents will face critical limitations
due to radiation resulting from the high neutrino flux.  The neutrinos will
be strongly beamed, producing regions with high radiation levels.
\end{itemize}

\section{Muon Cooling}

Muon cooling is the signature challenge for the Muon Collider.  A novel
cooling scheme is required to reach the factor $10^6$ reduction in emittance
required for the desired luminosity.  The short time available means no
existing scheme is applicable. An even larger emittance reduction is
desireable, as cooler beams would allow operation
with fewer muons for a given luminosity, thereby reducing the experimental
backgrounds and neutrino induced radiation, and would allow for smaller
accelerator element apertures, reducing the cost.  

The concept of ionization
cooling~\cite{ref:ioncooling} has been studied in some 
depth~\cite{ref:nufact_studies}.  In this scheme, the muons are maintained
at ca.~200~MeV while passed successively through an energy loss medium 
followed by an acceleration stage.  Cooling factors of order
20 appear feasible for the transverse emittance, 
but longitudinal cooling is difficult. In fact, the beam tends to increase
in longitudinal phase space during the process.  The possibility of exchanging
longitudinal and transverse degrees (emittance exchange) has long been
studied as a solution to this problem. Recent progress
with ring coolers has been reported at this meeting~\cite{ref:Palmer}, but
the goal of $10^6$ reduction in emittance still appears far away.  
Given the importance of cooling
for the Muon Collider, it is important that other schemes be consider.  One
such approach is `Frictional Cooling'~\cite{ref:Kottmann}.  The rest of this
talk is devoted to studies of this alternative.

\section{Frictional Cooling Studies}

The main idea of frictional cooling is to bring the muons to a velocity
regime where the energy loss while traversing a medium increases with velocity.
A constant accelerating force on the muons then yields an equilibrium energy,
as shown in Fig.~\ref{fig:dEdx}.  This occurs in the kinetic energy range 
$T< 10$~KeV.  

\begin{figure}[hbpt]
\begin{center}
\epsfig{figure=../study/cooling.eps,height=8.cm}
\end{center}
\caption[]{Stopping power for muons in Helium as a function of
kinetic energy.  Muons with energies below the peak of the $dE/dx$ curve
are brought to equilibrium with an external electric field.}
\label{fig:dEdx}
\end{figure}

At low kinetic energies, the stopping power
is extremely high, such that
low average densities are needed for realizeable electric fields.  Furthermore,
the electric field can only take effect once the muons are at low energies,
or the equilibrium regime will never be reached.  
Given the wide span of muon momenta, even after the phase rotation, the
low kinetic energies are reached at very different times and for
different ranges, such that a time dependent or local electric field
will not work.  We therefore
consider a crossed electric and magnetic field configuration.  The magnetic
field guides the muons while they are 
higher energies, and the electric field extracts
them from the slowing medium at small energies.  This allows 
the distance travelled by the muons while at low energies to be small, as 
discussed below.  Other difficuties arise from Muonium formation ($\mu^+$ captures an
electron) and muon capture ($\mu^-$ is capture in an atomic orbit).  Muonium
formation is limited by the use of Helium, due to its strong binding of 
orbital electrons.  Muon capture can only be suppressed by keeping the
kinetic energy much higher than typical binding energies.  Low Z materials are
adviseable.  We therefore consider using Helium gas as a stopping medium for
$\mu^+$ and Helium or Hydrogen gas for $\mu^-$.  A muon trajectory in a
Helium volume with density  $\rho=1.\cdot 10^{-4}$~gm/cm$^3$, with 
$B_z=5$~T and $E_x=5$~MV/m is shown in Fig.~\ref{fig:trajectory}. The main
features of the approach can be seen.  The muon follows the magnetic field
direction, with some drift, while at higher momentum.  Once it has reached a
low velocity, the magnetic force is considerably reduced and the electric
field pulls the muon out of the gas cell.  By keeping the transverse
dimensions of the cell small, the distance travelled at the small kinetic
energies can be kept within an allowable range.  


\begin{figure}[hbpt]
\begin{center}
\epsfig{figure=../study/along.eps,height=6.cm}
\epsfig{figure=../study/transverse.eps,height=6.cm}
\end{center}
\caption[]{Trajectory for a muon with $P_z=10$~MeV and $P_T=10$~MeV in
Helium using a continuous energy loss approximation.}
\label{fig:trajectory}
\end{figure}

We have performed a detailed simulation of some of the elements needed in
a frictional cooling scheme\footnote{This work was performed in collaboration
with H.~Abramowicz, Tel Aviv University, R.~Galea, Columbia University,
and S.~Schlenstedt, DESY/Zeuthen.}.  A brief review of some of the results
from the simulation (see also the presentation
by Rapheal Galea~\cite{ref:Raphe}) is given in the following sections.

\subsection{Target System}
The proton beam energy as well as the target geometry, orientation, material
composition, location in magnetic field, etc. have been studied in detail using
a combination of the MARS code~\cite{ref:MARS} and GEANT~\cite{ref:GEANT}.
We have concluded that a low proton beam energy 
(ca. 2~GeV) is optimal for our needs.
The target is oriented perpendicularly to the magnetic field to allow 
simultaneous capture of low energy $\pi^+$ and $\pi^-$ in two different
channels.  This is possible since low energy
pions are produced almost isotropically from the target.

%  The geometry
%is depicted in Fig.~\ref{fig:target}.

%\begin{figure}[hbpt]
%\begin{center}
%\epsfig{figure=geant1.eps,height=10.cm,angle=90.}
%\end{center}
%\caption[]{Schematic of the target region along with some sample
%pion and muon trajectories.}
%\label{fig:target}
%\end{figure}

\subsection{Drift, Phase Rotation, and Cooling Cell}

The drift length was optimized to maximize the yield of muons, resulting
in a drift region 28~m long.  
The muons then entered an 11~m long Helium cell (Hydrogen for $\mu^-$) region,
where gaps were included to allow flipping of the electric field
direction. The
muons not stopped by the first pass through the cell were reflected with a
time varying electric field, which provided some bunching of muons towards the
lower momenta. 

A longitudinal and transverse view of the 
cooling cell is seen in Fig.~\ref{fig:cell}.  The cell radius is $20$~cm,
and the cell resides in a solenoid of radius 1~m.  A uniform $E_x=5$~MV/m
field is assumed in the region of the cell, while an additional 
$E_z=100$~KV/m field is present outside the cell to provide a first
reacceleration of the muons.

\begin{figure}[hbpt]
\begin{center}
\epsfig{figure=simplercool.eps,height=6.cm}
\end{center}
\caption[]{Longitudinal and transverse views of the cooling cell and
phase rotation sections.}
\label{fig:cell}
\end{figure}

The energy loss and scattering simulation in the gas cell included the
following effects:
\begin{itemize}
\item
Electronic energy loss, treated as a continuous process.
\item
Nuclear scatters, treated as individual events for kinetic energies below
10~KeV as these are critical for reproducing the expected fluctuations in
trajectory and energy.  A semiclassical approach was used based on
algorithms described in ~\cite{ref:ICRU}.
\item
The Barkas effect~\cite{ref:Barkas} 
(reduced energy loss for $\mu^-$ relative to $\mu^+$) 
was parametrized and included.
\item
The $\mu^-$ capture process was implemented using a parametrization of
the calculations by Cohen~\cite{ref:Cohen}.
\item
The muon lifetime was implemented.
\end{itemize}
The simulation was cross checked with the NIST tables~\cite{ref:ICRU} 
and found to give excellent agreement.


Some results of the simulation for a single particle are shown
in Fig.~\ref{fig:track} for illustration.  As can be seen in this example,
the energy fluctuations resulting from nuclear scatters is substantial.

\begin{figure}[hbpt]
\begin{center}
\epsfig{figure=ani820.eps,height=12.cm,angle=0.}
\end{center}
\caption[]{The simulation results for the last stages of one track.  The
upper left plot shows the radius as a function of time, the upper right the
y coordinate as a function of time, the lower left the z coordinate as a
function of time, and the lower right the kinetic energy (in KeV) as a
function of time.  The effect of the individual nuclear scatters are
visible, and produce significant fluctuations in the kinetic energy.}
\label{fig:track}
\end{figure}


\subsection{Yield and Cooling}
A full simulation was performed, starting with 2~GeV protons on target,
and simulating all decays, aperture limitations, magnetic field nonuniformities
in the target region, etc.  All pions were tracked and decayed, and
the individual muons were tracked through the gas cell
using the full simulation described above.
The yield and emittance of the muons coming out after the
11~m cooling cell region was calculated.  Note that an initial
reacceleration has already taken place in the 11~m region.  The yield was
0.002 $\mu^{\pm}$ per 2~GeV proton, or about a factor four smaller than 
necessary to match the parameters given earlier.  The rms of the beam
in x,y,z (or rather $\beta c t$) was 0.015, 0.036 and 30~m, respectively.
In momentum space, the rms was 0.18, 0.18, 4.0~MeV/c.  These values 
result in $\epsilon_{6,N}=5.7\cdot 10^{-11}$~$(\pi m)^3$, which is somewhat
better than the desired value.  The results of the simulation are thus far
quite encouraging.  However, there are many important issues remaining, such
as the understanding of the effect of windows, a realistic design of
electric and magnetic field configurations in the gas cell region, 
a concept for the reacceleration
and bunching of the muons, and experimental data on muon capture and Muonium
formation cross sections.  An experimental program is underway to address the
latter issues.

\section{Conclusions}
It is clear that a Muon Collider facility would be a boon
for many areas of physics.  To build such a complex, including a high
energy frontier collider, requires to solve the muon cooling problem.  This
is the key issue faced by those interested in the Muon Collider, and the
importance of the problem dictates that different schemes be investigated.
One alternative, frictional cooling, has been discussed in this presentation.

\section*{References}
\begin{thebibliography}{99}

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B.~J.~King,
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\bibitem{ref:Morenzoni}
E.~Morenzoni,
``Physics and applications of low energy muons,''
PSI-PR-98-23
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Scotland, 17-28 Aug 1998}.

\bibitem{ref:Ankenbrandt}
C.~M.~Ankenbrandt {\it et al.},
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\bibitem{ref:nufact_studies}
M.~M.~Alsharoa {\it et al.}  [Muon Collider/Neutrino Factory Collaboration],
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\bibitem{ref:Palmer}
See talk by R. Palmer, these Proceedings

\bibitem{ref:Kottmann}
M. Muhlbauer, et al., Hyperfine Interact. {\bf 119}, 305 (1999) .

\bibitem{ref:Raphe}
See talk by R. Galea, these Proceedings

\bibitem{ref:MARS}
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\bibitem{ref:GEANT}
R.~Brun, R.~Hagelberg, M.~Hansroul and J.~C.~Lassalle,
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\bibitem{ref:Cohen}  J. Cohen, Phys. Rev. A Vol. 62 022512-1.
\end{thebibliography}    

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