\section[PION PRODUCTION AND CAPTURE]{PION PRODUCTION, CAPTURE AND PHASE ROTATION CHANNEL} 
\label{subsec-comppion}

This section first discusses the choice of target technology and optimization of the target geometry, and then describes design studies for the
pion capture and phase rotation channel.  Prospects for polarized muon beams
are discussed in detail.  The section concludes with an outline of
an R\&D program for target and phase rotation issues.

Figure~\ref{capture} gives an overview of the
configuration for production of pions by a 
proton beam impinging on a long, transversely thin target, followed by capture of
low-momentum, forward pions in a channel of solenoid magnets with rf
cavities to compress the bunch energy while letting the bunch length grow. This arrangement performs the desired rotation of the beam. 

\begin{figure*}[thb!]
\begin{center}
\includegraphics[width=5.5in]{capture.ps} 
\end{center}
\caption[Schematic view of pion production, capture and initial phase
rotation  ]{Schematic view of pion production, capture and initial phase
rotation. A pulse of 16-30~GeV
protons is incident on a skewed target inside a high-field solenoid magnet
followed by a decay and phase rotation channel. }
 \label{capture}
 \end{figure*}
 
\subsection{Pion production}

To achieve the luminosities for muon colliders presented in Table~\ref{sum},
$2\times 10^{12}$ (or $4\times 10^{12}$ in the 100~GeV CoM case) muons of each 
sign must be delivered to the collider ring in  
each pulse.  We estimate that a muon has a probability of only 1/4 of 
surviving the processes of cooling and acceleration, due to losses in
beam apertures or by decay.  
Thus, $0.8\times 10^{13}$ muons (1.6 $\times 10^{13}$ at 100~GeV) must exit the
 phase rotation channel each pulse.  For
pulses of $2.5\times 10^{13}$ protons ($5\times 10^{13}$ for 100~GeV), this 
requires 0.3~muons per initial proton.  Since
the efficiency of the phase rotation channel is about 1/2, this is equivalent 
to a capture of about 0.6~pions per proton: a very high efficiency. 

The pions are produced in the interaction of the
proton beam with the primary target.  
Extensive simulations have been performed for pion production from 
8-30~GeV proton beams on different target materials in a high-field 
solenoid \cite{ref6a,tar-snake,ref15,Ehst97,Takahashi97}.  %tarPAC97,ref15}. %,tarmok98}. 
Three different Monte Carlo codes \cite{arc,mars,ref16,dpmjet} predict 
similar
pion yields despite significant differences in their physics models.
Some members of the Collaboration are involved in an AGS 
experiment BNL E-910 \cite{exp910} to measure 
the yield of very low momentum pions, which will validate the 
codes in the critical kinematic region. This experiment ran for 14 weeks during the Spring of 1996 and has collected over 20 million events, of which about a quarter are minimum bias triggers for inclusive cross section measurements. The targets were varied in material (Be, Cu, Au, U) and thickness (2--100\% interaction length ($\lambda_{I}$)) and three different beam momenta were used (6, 12.5, 18~GeV/$c).$ Presently, the E910 collaboration is doing a careful analysis of the large data sample obtained. Figure~\ref{figdedx_exp} shows the dE/dx energy \vs~momentum for reconstructed tracks in the TPC; there is clear particle species separation \cite{refdedx_exp}. 
\begin{figure*}[thb!]
\begin{center}
\includegraphics[width=5in,clip=]{dedx_exp_new.eps} 
\end{center}
\caption[Ionization energy loss \vs~momentum. $dE/dx$ curve  ]{$dE/dx$ curve in arbitrary units for low momentum tracks; the ionization energy loss is for tracks with 30 or more hits in the TPC. The incident beam momentum is 18~GeV/$c.$ From left to right the bands correspond to muons, pions, kaons, protons and deuterium, respectively. Note the overlap of the (nearly horizontal) electron band with other species.}
 \label{figdedx_exp}
 \end{figure*}

The pion yield is greater for relatively high $Z$ materials, and for these,
the pion yield is maximal for
longitudinal momenta of the same order as the average transverse momentum
($\approx 200$ MeV/$c$).  
Targets of varying composition ($6<Z<82$), radii (0.2-3\,cm) and thicknesses 
(0.5-3~nuclear interaction lengths) have been explored using a Monte Carlo simulation \cite{ref15}.
For a fixed number of interaction lengths, the pion yield per proton rises 
almost linearly with proton energy, and hence almost proportional 
to the energy deposited in the target.
The yield is higher for medium- and high-$Z$ target materials,
with a noticeable gain at $Z>26$ for 30~GeV proton beams, but
with only a minor effect for $E \leq 16$~GeV. This is  shown in
Fig.~\ref{yield-E-A} where results of detailed MARS13(98) \cite{mars} 
simulations are presented.
The curves show the meson yield ($\pi + K$) from the targets in the
momentum interval $0.05 \leq P \leq 0.8$~GeV/$c$  (labeled Y) and 
the number of mesons that are both
captured in the high field solenoid and transported into the decay channel 
(labeled YC). The typical statistical error is a few percent.

\begin{figure*}[thb!]
%\noindent
\begin{minipage}{.50\linewidth} % fig 4a
\centering\epsfig{figure=mok4a.eps,width=\linewidth}
\end{minipage}\hfill
\begin{minipage}{.50\linewidth} % fig 4b
\centering\epsfig{figure=mok4b.eps,width=\linewidth}
\end{minipage}
\begin{minipage}{.50\linewidth} % fig 4c
\centering\epsfig{figure=mok4c.eps,width=\linewidth}
\end{minipage}\hfill
\begin{minipage}{.50\linewidth} % fig 4d
\centering\epsfig{figure=mok4d.eps,width=\linewidth}
\end{minipage}
%\vspace{10pt}
\caption[Meson yield ($\pi + K$) from different targets tilted by angle 
$\alpha$.]
{Meson yield ($\pi + K$) from different targets tilted by angle 
$\alpha$ in a solenoidal field $B$ of aperture $R_a$ as calculated with the 
MARS13(98) code. The target is aligned along the beam.
The curves labeled YC show mesons that are transported into the decay channel.
(a) Yield from a $1.5~\lambda_{I}$, 1 cm radius target irradiated
with 8, 16 and 30~GeV proton beams ($\sigma_x = \sigma_y =4$~mm) 
as a function of target atomic mass ($B =20$~T, $R_a = 7.5$~cm, $\alpha=0$);
(b) Yield from a $3\,\lambda_{I}$, 1 cm radius gallium target 
tilted at $\alpha=150$~mrad in a 16~GeV proton beam 
($\sigma_x = \sigma_y = 4$~mm)
{\it vs.}\ solenoid field for a fixed adiabatic invariant $B R_a^2$;
(c) Yield as a function of radius of  a $3~\lambda_{I}$ gallium target in 
a 16~GeV proton beam ($\sigma_x =  \sigma_y = 4$~mm, $B = 20$~T, 
$R_a = 7.5$~cm, $\alpha = 100$~mrad);
(d) Yield from a $3~\lambda_{I}$, 1 cm radius gallium target {\it vs.}\
tilt angle 
between the axis of the capture solenoid and the proton beam for a 16~GeV 
proton beam ($\sigma_x = \sigma_y = 4$~mm, $B = 20$~T, $R_a = 7.5$~cm).}
\label{yield-E-A}
\end{figure*}


\subsection{Target}

The target should be 2-3 interaction lengths long to maximize
pion production.  A high-density material is favored to minimize the
size and cost of the capture solenoid magnet.  Target radii larger than about 
1~cm lead to
lower pion rates due to reabsorption, while smaller diameter targets have less production from secondary interactions.  Tilting the target by 
 100-150~mrad minimizes loss of pions by absorption in the target after one 
 turn on their helical trajectory \cite{ref4b,mcprod}. 
Another advantage of the tilted target 
 geometry is that the high energy and neutral components of the shower can be 
 absorbed in a water-cooled beam dump to the side of the focused beam.

About 30~kJ of energy is deposited in the target by each proton pulse (10\% of
the beam energy). Hence, the target absorbs 400~kW of power at the 15-Hz pulse 
rate.  Cooling of the target via 
contact with a thermal bath would lead to unacceptable
absorption of pions, and radiative cooling is inadequate for such high
power in a compact target.  Therefore,
the target must move so as to carry the energy
deposited by the proton beam to a heat exchanger 
outside the solenoid channel. 
 
Both moving solid metal and flowing liquid targets have been considered, with
the latter as the currently preferred solution.  A liquid is relatively
easy to move, easy to cool, can be readily removed and replaced, and is the
preferred target material for most spallation neutron sources under study. A
liquid flowing in a pipe was considered, but experience at ISOLDE
with short proton pulses \cite{Lettry} as well as simulations \cite{ref11,Bauer} 
suggest serious problems in shock damage to the pipe.
An open liquid jet is thus proposed.

A jet of liquid mercury has been demonstrated \cite{ref11} 
but not exposed to a beam. For our application, safety and other
considerations favor the use of a low melting point lead alloy rather than 
mercury. Gallium alloys, though with lower density, are also being considered.
Experimental and theoretical studies are underway to determine the 
consequences of beam shock heating of the liquid. 
%, and an experimental program is being planned (see section ***).
It is expected that the jet will disperse after being
exposed to the beam.  The target station must survive
damage resulting from the violence in this dispersion. This consideration will
determine the minimum beam, and thus jet, radius.
% (1 cm radius appears conservative, but 5~mm would be preferred - see below).

For a conducting liquid jet in a 
strong magnetic field, as proposed, strong eddy currents will be
induced in the jet, causing reaction forces that may disrupt its flow \cite{ref12,ref13}. The forces induced
are proportional to the square of the jet radius, and set
a maximum for this radius of order 5-10~mm.
If this maximum
is smaller than the minimum radius set by shock considerations, then multiple
smaller beams and jets could be used; {\it e.g.}, four jets of 5~mm radius with 
four beams with $2.5 \times 10^{13}$ protons per bunch.
Other alternatives include targets made from insulating materials such as 
liquid PtO$_2$ or Re$_2$O$_3$, slurries 
({\it e.g.},  Pt in water), or powders \cite{mumu98-10}.

A moving solid metal target is not the current baseline solution, but is a 
serious possibility. In this case, the target could consist of a long flat band
or hoop of copper-nickel that moves along its length (as in a
band saw) \cite{cunitarget}. The band  would be many meters in length, would be
cooled by gas jets away from the target area, and would be supported and
moved by rollers, as shown in Fig.~\ref{bandsaw}.  

\begin{figure*}[thb!]
%\centering
\begin{center}
\includegraphics[width=5in,clip=]{test_chainsaw.eps} 
\end{center}
\caption{Alternative concept of a solid metal target in the form of a
rotating Cu-Ni band.}
 \label{bandsaw}
 \end{figure*}

The choice and parameters of the target are critical issues that need resolution. These can be resolved by experiments with a strong magnetic field and a beam, as discussed in  section~IV.H.

\subsection{Capture}

To capture all pions with transverse momenta $p_T$ less than their
typical value of 200~MeV/$c$, the product of the capture solenoid field $B$ 
and its radius $R_a$
must be greater than 1.33~T~m. The use of a high field and small radius
is preferred to minimize the corresponding transverse
emittance, which is proportional to $BR^2$: for a fixed transverse momentum 
capture, this emittance is thus proportional to $R$. A field of 20~T
and 7.5~cm radius was chosen on the basis of simulations described below. 
This gives $BR$ = 1.5~T~m, $BR^2 = 0.1125$~T~m$^2$ and 
a maximum transverse momentum capture of $p_T = 225$ MeV/$c$.

A preliminary design \cite{ref18} of the capture solenoid 
has an inner 6~T, 4~MW, water cooled, hollow conductor magnet with an inside 
diameter of 24~cm and an outside diameter of 60~cm.   
There is space for a 4~cm thick, water cooled, heavy metal shield 
inside the coil.  The outer superconducting magnet has
three coils, with inside diameters of 60 to 80~cm. It generates
an additional 14~T of field  at the target and provides the required tapered
field to match into the decay channel. Such a hybrid
solenoid has parameters compatible with those of existing 
magnets \cite{ref17}.       

The 20~T capture solenoid is matched via a transfer solenoid \cite{tar-snake} into a decay channel consisting 
of a system of superconducting solenoids 
with the same adiabatic invariant $BR^2 \propto R p_T$.
Thus, for a 1.25~T decay channel, $B$ drops by a factor of 16
between the target and decay channel, 
$R$ and $p_T$ change by factors of
4 and 1/4, respectively.  This permits improved acceptance
of transverse
momentum within the decay channel, at the cost of an increased spread
in longitudinal momentum.
Figure~\ref{yield-E-A}(b) shows the meson yield as a
function of field in the capture solenoid, with the radius of the capture
solenoid adjusted to maintain the same $BR_a^2$ as in the decay channel.
The optimum field is 20~T in the capture solenoid.  

If the axis of the target is coincident with that of the solenoid field, then
there is a relatively high probability that pions re-enter the target after
one cycle on their helical trajectory and are lost due to nuclear interactions.
When the target and proton beam 
are set at an angle of 100-150~mrad with respect to the field 
axis \cite{ref15}, %\cite{ref14,ref15}, 
the probability for such
pion interactions at the target is reduced, and the overall production rate 
is increased by 60\%, as shown in Fig.~\ref{yield-E-A}(d). 

In summary, the simulations indicate that a 20~T solenoid of 16~cm inside diameter 
surrounding a tilted target will capture about
half of all produced pions.  With target efficiency included, about 0.6
pions per proton will enter the pion decay channel \cite{ref15}. %tarmok98}. 


\subsection{Phase rotation linac}

The pions, and the muons into which they decay, have a momentum distribution with an rms spread of approximately 100\% and a peak at about 200~MeV/$c$. 
It would be difficult to handle such a wide
spread in any subsequent system. A linac is thus introduced along the decay
channel, with frequencies and phases chosen to decelerate the fast particles
and accelerate the slow ones; {\it i.e.}, to phase rotate the muon bunch. 
Several studies have been made of the design of this system, using differing 
ranges of rf frequency, delivering different final muon momenta, and differing 
final bunch lengths. In all cases, muon capture efficiencies of close to 
0.3 muons per proton are obtained. Until the early stages of the ionization 
cooling have been designed, it is not yet possible to choose between them.
Independent of the above choices is a question of the location of the focusing 
solenoid coils and rf cavity design, as discussed below in the 
section~IV.F.
\begin{table*}[htb!]
\begin{center}
{\caption{Parameters of the Lower-Energy Phase Rotation Linacs}
\label{rot}}
\begin{tabular}{cccc}
%\hline\hline
Linac     & Length & Frequency & Gradient  \\
          &  (m)     &   (MHz)     &  (MeV/m)    \\
\hline
1         &  3    &   60     &   5      \\
2         &  29   &   30     &   4       \\
3         &  5    &   60     &   4      \\
4         &  5    &   37     &   4       \\
%\hline\hline
\end{tabular}
\end{center}
\end{table*}
\subsubsection{Lower energy, longer bunch example}

This example captures muons at a mean kinetic energy of 130~MeV.
Table~\ref{rot} gives parameters of the linacs used. The gradients listed are relatively high for continuous low frequency systems, but far below the surface fields achieved in short pulses ($\approx 75$ MV/m at 202~MHz for $1\,ms$ and an effective acceleration gradient of 10.7~MV/m on tank2 of the CERN Lead injector \cite{gerhard}. GSI is also testing  36~MHz linac cavities for its injector and the expected peak gradient is $\approx 15$~MV/m  \cite{gsi}. We expect that the greatest problem will be the development of sufficiently high power low frequency rf sources. 
%It is seen that the 
%frequencies vary between 30 and 60~MHz, and that the overall length is 42~m.
Monte Carlo simulations \cite{ref6}, with the program MUONMC \cite{mcbob},  
were done using pion production calculated by ARC \cite{arc} for a copper target
of 1-cm radius at an angle of 150~mrad. 
A uniform solenoidal field was assumed in
  the phase rotation, and the rf was approximated by a series of kicks. 

 Figure~\ref{Evsctpol2} shows the energy {\it vs.}\ $ct$ at the end 
of the decay and phase rotation channel. 
The abscissa $ct$ is a measure of bunch length at the end of the
channel: the total transit time of each $\pi/\mu$ is multiplied by the velocity
of light and the total length of the channel is subtracted. Thus a ficticious reference particle at 
the center of the incident bunch at the target arrives at $ct= 0$~m.
A loose final bunch selection was defined with an energy $130 \pm 70$~MeV and 
bunch $ct$ from 3 to 
11~m. With this selection, the rms energy spread is 16.5\%, the rms $ct$
 is 1.7~m, and there are 0.39~muons per incident proton.
A tighter selection with an energy $130 \pm 35$~MeV and bunch $ct$ from 4 to 
10~m gave an rms energy spread of 11.7\%, rms $ct$ of 1.3~m, and contained 
0.31~muons per incident proton.

\begin{figure*}[hbt!]
\begin{center}
\includegraphics[width=5in]{new_fg4_may13_new.ps} 
\end{center}
\caption[Energy {\it vs.}\ $ct$ of $\mu$'s at end of the lower-energy 
phase rotation channel.  ]
{Energy {\it vs.}\ $ct$ of $\mu$'s at end of the lower-energy
phase rotation channel.  The symbols +, o and $-$ denote muons with 
polarization $P>{1\over 3},\ -{1\over 3} < P < {1\over 3}$ and 
$P <-{1\over 3}$, respectively.}
 \label{Evsctpol2}
 \end{figure*}

\subsubsection{Higher energy, shorter bunch example}

  In this example the captured muons have a mean kinetic energy close to
  320~MeV.  It is based on a Monte Carlo study which uses 
the updated
  MARS pion production model \cite{ref16} to generate pions created by 16~GeV 
protons
  on a 36~cm long, 1~cm radius coaxial gallium target. Figure~\ref{neu_van} 
shows the longitudinal phase space of the muons at the end of an 80~m long, 5~T solenoidal decay channel with
  cavities of frequency in the 30-90~MHz range and acceleration gradients of
  4-18~MeV/m.  A total of 0.33~muons per proton fall within the indicated
  cut (6~m$ \times $300~MeV).  The rms bunch length inside the cut is 148~cm
  and rms energy spread is 62~MeV.  The normalized six dimensional (6-D) emittance is 217~cm$^3$
  and the transverse part is 1.86~cm (the normalized 6-D emittance $\epsilon_{6,N}$ is defined in section \textbf{V}).

 A sample simulation with lithium hydride absorbers regularly spaced in the 
  last 60~m of a 120~m decay channel and with compensating acceleration 
  captures 0.3~muons with mean kinetic energy of about 380~MeV in a
  (6~m$\times $300~MeV) window. 
The longitudinal phase space is about the same as in the
  previous example but the transverse part shrinks to 0.95~cm due to ionization
  cooling which reduces the 6-D phase space to 73.5~cm$^3.$

\begin{figure*}[thb!]
\centerline{\epsfig{file=andy.eps,height=5in,width=5in}}
\vspace{0.5cm}
\caption[Longitudinal phase space at the end of decay channel with projections 
onto time and energy axes ]
{Longitudinal phase space at the end of decay channel with projections 
onto time and energy axes per incident proton.  The four dashed lines delineate the
region deemed acceptable for the cooling channel.  }
 \label{neu_van}
 \end{figure*}

\subsection{Use of both signs}

Protons on the target produce pions of both signs, and a solenoid will
capture both, but the subsequent rf systems will have opposite effects
on each sign. The proposed baseline approach uses two separate proton
bunches to create separate positive and negative pion bunches and accepts
the loss of half the pions/muons during phase rotation. 

If the pions can be
charge separated with limited loss before the phase rotation cavities
are reached, then higher luminosity may be obtained. 
The separation of charged pions in a curved solenoid decay line was
studied in \cite{tar-snake}. Because
of the resulting dispersion in a bent solenoid, an initial beam
of radius $R$  with maximum-to-minimum momentum ratio $F$ will
require a large beampipe of radius $(1+F)R$ downstream to accommodate the
separated beams. A septum can then be used to capture the two beams into
separate channels. Typically the reduction in yield for a curved solenoid
compared  to a straight solenoid is about 25\% (due to the loss of
very low and very high momentum pions to the walls or septum), but this
must be weighed against the fact that both charge signs are captured for
an overall net gain. A disadvantage is that this charge separation takes
place over several meters of length during which time the beam spreads
longitudinally. This makes capture in an rf phase rotation system
difficult, although a large aperture cavity system could be incorporated
in the bent solenoid region to  alleviate this. The technique deserves
further study and may be useful to consider as an intensity upgrade to
a muon collection system.
  
\subsection{Solenoids and rf}

As noted above, capture using higher frequencies appears to be less efficient, 
and most studies now use frequencies down to 30~MHz. Such cavities, when 
conventionally designed, are very large (about 6.6~m diameter). 
In the Snowmass study \cite{snowp220} a reentrant design reduced this diameter 
to 2.52~m, but this is still large, and it was first assumed that the 
5~T focusing solenoids would, for economic reasons, be placed within the 
irises of the cavities (see Fig.~\ref{reentrant}).

\begin{figure*}[hbt!]
\begin{center}
\includegraphics[height=5in,angle=-90]{412.eps}
\end{center}
\caption[Schematic of capture and phase rotation using rf cavities ]
{Schematic of capture and phase rotation using rf cavities with 
superconducting solenoids (hatched) inside the irises. Three groups of three cavities operating at 90, 50, and 30~MHz are shown from left to right, respectively.}
 \label{reentrant}
 \end{figure*}

A study of transmission down a realistic system of iris located 
coils revealed betatron resonant excitation from the magnetic field 
periodicities, leading to significant particle loss. This was reduced by the 
use of more complicated coil shapes \cite{snowp220}, smaller gaps, and 
shorter cavities, but remained a problem. 

An alternative is to place continuous focusing coils outside the cavities 
 as shown in Fig.~\ref{capture}. In this case, cost will be minimized with 
lower magnetic fields (1.25-2.5~T) and correspondingly larger decay channel 
radii (21-30~cm). Studies are underway to determine the optimal solution.

\subsection{Polarization}

Polarization of the muon beams presents a significant physics
advantage over the 
unpolarized case, since signal and background of electroweak processes usually 
come predominantly from different polarization states. 

\subsubsection{Polarized muon production}

 In the center of mass of a decaying pion, the outgoing muon is fully 
polarized ($P = -1$ for $\mu^+$ and +1 for $\mu^-$). In the lab system the 
polarization depends on the decay angle $\theta_d$ and initial pion energy \cite{ref19,ref19a,ref19b}.  
For pion kinetic energy larger than the pion mass, the 
average polarization is about 20\%, and if nothing else is done, the 
polarization of the captured muons after the phase rotation system is 
approximately this value.

If higher polarization is required, some selection of muons from forward pion
decays  $(\cos{\theta_d} \rightarrow 1)$ is required. Figure~\ref{Evsctpol2}, 
above, showed the polarization of the phase rotated muons. The polarization 
\{P$>{1\over 3}$,
$-{1\over 3}< P<{1\over 3}$, and P$<-{1\over 3}$\} is marked by the symbols
$\mathbf{+,\,o\,}$ and $\mathbf{-}$ respectively. 
If a selection is made on the minimum energy of the muons, then greater 
polarization is obtained. The tighter the cut, the higher the 
polarization, but the less the fraction $F_{\rm surv}$  of muons that survive. 
 Figure~\ref{polvscutnew} gives the results of a Monte Carlo study.

\begin{figure}[bht!] 
\centerline{
\epsfig{file=pol_surv.ps,height=4.in,width=4.in}
}
\caption{Polarization {\it vs.}\ fraction $F_{\rm surv}$ of $\mu$'s accepted. 
\label{polvscutnew}}
\end{figure}

If this selection is made on both beams, and if the proton bunch intensity is 
maintained, then each muon bunch is reduced by the factor $F_{\rm surv}$ 
and the 
luminosity would fall by $F_{\rm surv}^2$. But if, instead, proton bunches are 
merged so as to obtain half as many bunches with twice the intensity, then the 
muon bunch intensity is maintained and the luminosity (and repetition rate)  
falls only as $F_{\rm surv}$.

The luminosity could be maintained at the full 
unpolarized value if the proton source intensity could be increased. Such an 
increase in proton source intensity in the unpolarized case might be 
impractical because of the resultant excessive high energy muon beam power, 
but this restriction does not apply if the increase is used to offset losses 
in generating polarization. 

Thus, the goal of high muon beam polarization may shift the parameters of
the muon collider towards lower repetition rate and higher peak currents
in the proton driver.

\subsubsection{Polarization preservation}

The preservation of muon polarization has been discussed in some detail in \cite{ref20}.
During the ionization cooling process the muons lose energy in material and 
have a spin-flip probability $\mathcal{P},$
\begin{equation}
{\mathcal {P} }\approx \int {m_e\over m_{\mu}}\beta_v^2 {\delta E \over E}\label{pol}, 
\end{equation} 
where $\beta_v$ is the normalized muon velocity and $\delta E/E$ is the fractional 
loss of energy due to ionization. In our case, the integrated energy loss 
is approximately 3~GeV and the typical energy is 150~MeV, so the integrated 
spin-flip probability is close to 10\%. The change in polarization 
$\delta \mathcal {P}/\mathcal {P}$ is 
twice the spin-flip probability, so the reduction in polarization is 
approximately 20\%.  This loss is included in Fig.~\ref{polvscutnew}.

   During circulation in any ring, the muon spin, if initially longitudinal, 
will precess by $\gamma(g-2)/2$ turns per revolution, where
$(g-2)/2$ is $1.166\times 10^{-3}$. A given energy spread 
${\Delta \gamma/\gamma}$ will 
introduce variations in these precessions and cause dilution of the 
polarization. 
But if the particles remain in the ring for an exact integer number of 
synchrotron 
oscillations, then their individual average $\gamma$'s will be the same and no 
dilution will occur. 

   In the collider,
 bending can be performed with the spin orientation in the vertical 
direction, and the spin rotated into 
the longitudinal direction only for the interaction region. The design of such 
spin rotators appears relatively straightforward, but they are too long. This might be a 
preferred solution at high energies but is not practical in the 100~GeV 
machine because of the constraint on the circumference of the ring imposed by the muon lifetime. An alternative is to use such a small energy spread, as in the 
Higgs factory, that although the polarization vector precesses, the beam 
polarization  does not become significantly diluted. In addition, calibration of the Higgs factory collider energy to 1 part in a million \cite{ref7} requires the spins to precess continuously from turn to turn.

\subsection{R\&D program}

An R\&D program is underway to continue theoretical studies (optimization
of pion production and capture) and to clarify 
several critical issues related to targetry and phase rotation
\cite{targetprop}.  A jet of the room temperature 
eutectic liquid alloy of Ga-Sn will be exposed to nanosecond pulses of 
$1.5 \times 10^{13}$ 24~GeV protons at the Brookhaven AGS 
to study the
effect of the resulting pressure wave on the liquid.  The same jet will
also be used in conjunction with a 20~T, 20~cm bore resistive magnet at the
National High Magnetic Field Laboratory (Tallahassee, FL) to study the
effect of eddy currents on jet propagation.  Then,
a pulsed, 20~T magnet will be added to the
BNL test station to explore the full configuration of jet, magnet and
pulsed proton beam.  Also, a 70~MHz rf cavity will be exposed to the
intense flux of secondary particles downstream of the target and 20~T magnet
to determine viable operating parameters for the first phase rotation cavity.
The complete configuration of the targetry experiment is sketched in
Fig.~\ref{targetexpt}.

\begin{figure*}[tbh!]
\begin{center}
\includegraphics[ width=5in]
      {tgtexpt5.eps}
\caption{Plan view of the full configuration of the targetry experiment.
\label{targetexpt}}
\end{center}
\end{figure*}

The first two studies should be accomplished during 1999, and the third and
fourth in the years 2000/01.


