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\begin{document}

\title{Development of Muon Ring Coolers, Neutrino
Factories and Supersymmetric Higgs Factory}

\author{David B. Cline}

\address{Center for Advanced Accelerators
Department of Physics and Astronomy University of California, Los
Angeles Los Angeles, California 90095-1547}

\begin{abstract}
Over the past few years or so a key new development is the
invention of Ring Coolers for muon cooling. In particular these
rings demonstrate robust cooling of the longitudinal phase space.
We discuss the Quadrupole or UCLA Ring Cooler and the prospects to
make this a final cooler to reduce the tranceiver emittance to the
value required for a $\mu ^{+} \mu ^{-}$ collider. This will lead
to a Higgs Factory for the $A_{0}$ /$H_{0}$ in supersymmetry
models.
\end{abstract}

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\section{Introduction}
Ever since the recent start of the $\mu ^{+} \mu ^{-}$ collider in
December 1992 at a meeting in Napa, the issue of longitudinal
phase space cooling has been an issue [1]. In Table 1 we give the
formulas for transverse and longitudinal cooling. It now appears
that the emphasis on transverse cooling for all of these years may
have missed the key point: longitudinal cooling may be easier to
achieve as we now see in the Ring Cooler studies.

On the other hand a $\mu ^{+} \mu ^{-}$ collider requires a strong
reduction in transverse emittance to the level of 0.3p mm rad. We
show here how this may be achieved in a Ring Cooler.

The transverse cooling formula is given in (1).

\begin{equation}
\frac{d \varepsilon_{\perp}}{ds} = \frac{1}{\beta
^{2}}\frac{dE_{\mu}}{ds}\frac{\varepsilon _{\perp}}{E _{\mu}} +
\frac{\beta ^{*}_{\perp}}{2 \beta^{2}} \frac{(0.014)^2}{E_{\mu}
M_{\mu}L_{R}}
\end{equation}
\begin{equation}
\frac{d(\Delta E_{\mu})^2}{ds} = {}^{-}2d
[dE_{\mu}/ds]/dE_{\mu}\langle(\Delta E_{\mu})^{2}\rangle
\end{equation}

We note the need for a very small $\beta ^{*}_{\perp }$  and large
LR, radiation length, to reduce the heating term. The study of the
ring cooler now shows that longitudinal or energy cooling may be
easier than transverse cooling due to the heating term (equation
2). We believe a new scheme for muon colliders and neutrino
factory using ring coolers (as shown in Figure 1) could be
constructed in the future [2][3][4].

\section{Supersymmetric Higgs Factory}

In the model of supersymmetry there will likely be one low-mass
Higgs ( $h^{0}$ ) and two high mass (or supersymmetric) Higgs A
and H. For the parameter tan $\beta $, larger values lead to a
near mass degenerate system of H and A states, most likely in the
300 - 500 GeV mass range. Current evidence on SUSY suggests a
large value of tan $\beta $. In this case the coupling of H and A
to $t \overline{t} $ and gauge bosons is sharply reduced, making
them difficult to produce and study at the LHC or NLC [5].

In some ways a Muon Collider is the ideal method to produce (in
the S channel) and study these states. Since one will be a scalar
and the other pseudoscalar, the use of partially polarized $
\mu^{+}$ beams will be of great usefulness. The mass splitting of
the A and H is of order GeV and much easier to observe than the
tiny width of the $h^{0}$.

The physics goals of a supersymmetric Higgs Factory are shown in
Figure 2.

\begin{itemize}
    \item The detector of these states of $e^{+} e^{-}$ or pp
colliders depends on the coupling to $t \overline{t} $ states.
This coupling falls rapidly with tan $\beta $ ; current estimates
suggest large values of tan $\beta $ and weak production by these
machines. The $\mu ^{+} \mu ^{-}$ coupling will remain large.
    \item The states could be nearly mass degenerative as shown
in Figure 2 for large tan $\beta $.  Again this makes the study of
the states by associated production at $e^{+} e^{-} $ or $pp $
machines very difficult.
\end{itemize}

The study of the A and H is much less demanding on a  collider
than for the lower mass Higgs. Table 1 gives the machine
parameters. The use of ring coolers could be crucial to the
development of  such a collider. This will be studied by this
group in the next three years.

\begin{table}
\caption{} \lineup
\begin{tabular}{@{}ll}
\br
CoM energy (TeV)&{\rm 0.4}\\
\mr
P energy (GeV)&16\\
P/bunch&$2.5 x 10^{13} $\\
Bunches/fill&4\\
Rep. rate (Hz)&15\\
1/$\tau _{\mu} $&240\\
$P$ power (MW)&4\\
$\mu $/bunch&$2 x 10^{12} $\\
$\mu $ power (MW)&4\\
Wall power (MW)&120\\
Collider cirumference (m)&1000\\
$<B> (T)$ &4.7\\
$\delta p$ (\%) &0.14\\
6-D $\eta _{6,N} (\pi m ^3 )$ &$1.7 x 10^{-10} $\\
Rms $\eta _{n} $ ($\pi $ mm-mrad)&50\\
$\beta $ * (cm)&2.6\\
$\sigma _{z}$ (cm)&26\\
$\sigma _{r}$ spot ($\mu m $)&1.0\\
$\sigma _{\theta} $ IP (mrad)&0.044\\
Tune shift&\\
$n^{effective}_{turns} $&700\\
\mr
Luminosity $(cm^{-2}s^{-1}) $&$10^{33} $\\
Higgs/year&\\
\br
\end{tabular}
\end{table}

\section{THE STUDY OF THE UCLA QUADRUPOLES RING COOLER}
In Figure 3 we show the UCLA Ring Cooler worked out with Al
Garren, P. He, Y. Fukui (UCLA), H. Kirk (BNL) and the author [2].
Clear evidence for GD cooling has been obtained in our simulators
[3][4].

In Figure 4 we show the early evidence for GD cooling. By now the
figure of merit is nearly 30 [6] (30 times more GD cooling
compared to losses).

\section{REQUIREMENT FOR THE SUPERSYMMETRIC HIGGS FACTORY}

In order to construct a $\mu ^+ \mu ^- $  collider to study the
supersymmetric Higgs particles $H^0 $ and $A ^0 $ the 6D cooling
must be extremely good. Figure 5 shows the requirement on
longitudinal and transverse emittance (from Rich Fernow and D.
Neuffer). The UCLA Ring Cooler has reached nearly the required
longitudinal emittance reduction but is far from the transverse
requirement.

\section{A LI LENS INSERT INTO THE RING COOLER}

Going back to formular (2) we see that a very low $\beta ^* $  is
needed to reduce the heating term. Such a low $\beta ^* $ is
possible in a Li Lens (pulsed current of 500 K amp). $ \beta ^* $
could reach less than 1 cm in this case. We propose to put such a
lens as an insert into the UCLA Ring Cooler. Thus the muon beam
would make many passes through the lens and the result could be
strong transverse cooling in the ring. The insert beam envelope
could look like that in Figure 6. Figure 7 shows an example of a
red Li Lens at FNAL now that we are using for the study.

\smallskip

\section{References}
\begin{thebibliography}{5}

\item Ankenbrandt, C M et al, 1999, ``Status of Muon Collider
Research and Development and Future Plans'', NuMu Collaboration
{\it Phys.Rev. ST  Accel.Beams} {\bf 2} 081001. \item Kirk, H et
al, M1 Working Group Paper, Snowmass 2001 Conference, Snowmass,
Colorado, 2001, BNL-68735. \item Fukui, Y et al, Workshop book,
UCLA Ring Cooler Workshop, Tuscon, Arizona, December 2001. \item
Cline, D B et al, 2002, Mini Workshop at UCLA: The Use of a Ring
Cooler for a Neutrino Factory/Muon Collider, UCLA, Los Angeles,
California, March 2002. \item Fukui, Y  et al, Contribution Paper
to the Overarching Document of the Snowmass 2001 Conference,
Snowmass, Colorado, 2001.

\end{thebibliography}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig1.eps}
\end{center}
\caption{\label{1.}Schematic for a new design of the Muon Collider
and Neutrino Factory using rings compared to the earlier design.}
\end{figure}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig2.eps}
\end{center}
\caption{\label{2.}Hard A particles with small mass difference.}
\end{figure}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig3.eps}
\end{center}
\caption{\label{3.}Schematics of the UCLA Ring Cooler.}
\end{figure}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig4.eps}
\end{center}
\caption{\label{4.}The Evolution of x, y, z normalized emittances
in 30 full turns.}
\end{figure}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig5.eps}
\end{center}
\caption{\label{5.}The transverse  and longitudinal emittance
reduction needed for a supersymmetric Higgs Factory.}
\end{figure}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig6.eps}
\end{center}
\caption{\label{6.}Beam envelope for a Li Lens insert in the UCLA
Ring Cooler.}
\end{figure}

\input{epsf}
\begin{figure}
\begin{center}
\epsffile{devfig7.eps}
\end{center}
\caption{\label{7.}Li Lens now at FNAL.}
\end{figure}

\end{document}
