\section{PION PRODUCTION, CAPTURE AND PHASE-ROTATION CHANNEL} \label{subsec-comppion} This subsection first discusses the choice of target technology, optimization of the target geometry, and then describes design studies for the pion capture and phase-rotation channel. Figure~\ref{capture} gives an overview of the configuration for production of pions by a proton beam impinging on a long, 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: thus rotating the bunch in phase space. \begin{figure}[thb!] %\centering \begin{center} %\includegraphics[width=5.5in]{fnalfg3_new_magic.ps} \includegraphics[width=5.5in]{capture.eps} \end{center} %%\centerline{\epsfig{file=fnalfg3.ps,width=5.5}} %\vspace{0.5cm} \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 Tb.~\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 ${1\over 4}$ 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. And 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}. %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~\cite{exp910} to measure the yield of very low momentum pions, which will validate the codes in the critical kinematic region. 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 ($626$ for 30-GeV proton beams, but with only a minor effect for $E \leq 16$~GeV, as shown in Fig.~\ref{yield-E-A} where results of detailed MARS13(98)~\cite{mars} simulations are presented. The curves show both the meson yield ($\pi + K$) from the targets in the 0.05$\leq$p$\leq$0.8~GeV/c momentum interval (labeled $Y$) and the number of mesons that are both captured in the high-field solenoid and transported into the decay channel (labeled $YC$). 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) 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) 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 a 3-$\lambda_{I}$ gallium target radius in a 16-GeV proton beam ($\sigma_x$=$\sigma_y$=4~mm, $B$=20~T, $R_a$=7.5~cm, $\alpha$=100); (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 reduce the added 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. 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 (see Fig.~\ref{capture} ). 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 calculations done at CERN\cite{ref11} 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 10$^{13}$ protons per bunch. 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\cite{cunitarget} the target could consist of a long flat band or hoop of copper-nickel that moves along its length (as in a band saw). 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=5.5in,clip=]{test_chainsaw.eps} \end{center} %%\centerline{\epsfig{file=chainsaw.e.ps,width=5.5in}} %\vspace{0.5cm} \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 a critical question that needs resolution. Only an experiment in a magnetic field and in a beam will settle it, and this is being planned. \subsection{Capture} In order to capture all pions with transverse momenta $P_\perp$ less than their typical values 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 $P_\perp = 225$ 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 an 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 $BR^2 \propto RP_\perp$. Thus, for a 1.25-T decay channel, $B$ drops by a factor of 1/16 between the target and decay channel, $R$ and $P_\perp$ 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 increased by 60 \%, as shown in Fig.~\ref{yield-E-A}(d). In sum, 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 spread with an rms value of approximately 100~\% and a peak value 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 deaccelerate 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. These will be discussed below in the subsection on Phase Rotation RF. \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. It is seen that the frequencies vary between 60 and 30~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. \begin{table}[bth!] %\centering \caption{Parameters of Phase Rotation Linacs} \label{rot} %\centering \protect \begin{tabular}{cccc} %\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 \end{tabular} \end{table} Figure~\ref{Evsctpol2} shows the energy {\it vs.}\ $ct$ at the end of the decay and phase rotation channel. The abcissa is a measure of bunch length at the end of the channel: total transit time of each $\pi/\mu$ is multiplied by the velocity of light with total length of the channel ($D$) subtracted. Thus a photon at the center of the proton bunch at the target arrives at 0~cm. 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 0.385~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 0.305~muons per incident proton. \begin{figure}[hbt!] \begin{center} \includegraphics[width=4.5in]{new_fg4_may13.ps} \end{center} %\centering %\centerline{\epsfig{file=fnalfg4.ps,height=4.0in,width=4.0in}} \caption[Energy {\it vs.}\ $ct$ of $\mu$'s at end of the Option-1 phase rotation channel ]{Energy {\it vs.}\ $ct$ of $\mu$'s at end of the Option-1 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 taken from a Monte Carlo study\cite{neuffer_van} 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 co-axial gallium target. Figure~\ref{neu_van} shows the longitudinal phase space of the muons at the end of an 80~m long 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 (6 m$ \times $300~MeV) cut. The rms bunch length inside the cut is 148~cm and rms energy spread is 62~MeV. Normalized 6-D emittance is 217~cm$^3$ and the transverse part is 1.86~cm. A sample simulation with lithium hydride absorbers regularly spaced in the last 60~m of a 120~m decay channel and with compensating acceleration provided captures 0.297~muons with mean kinetic energy of about 380~MeV in a (6~m$\times $300~MeV) cut. Longitudinal phase space is about the same as in the previous example but the transverse part shrinks to 0.95~cm due to ionization cooling which reduces 6-D phase space to 73.5~cm$^3.$ \begin{figure}[hbt!] \centerline{\epsfig{file=andy.ps,height=6.in,width=4.5in, angle=90}} \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. The four dashed lines delineate 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 required subsequent phase rotation rf systems will have opposite %effects on each. The baseline approach is to use two proton bunches, aim them %at the same target one after the other, and adjust the rf phases such as to %act correctly on one sign of the first bunch and on the other sign of the %second. %A second possibility would be to separate the charges into two channels, and %phase rotate them separately. However, the separation, probably using a bent %solenoid, is not simple and would not be fully efficient. Whether a gain in %overall efficiency could be achieved is not yet known. 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 solenoid causes all charged particles to execute Larmor gyrations as they travel down the decay line. A gradient in the magnetic field or a curvature in the field produces drifts of the particle guide centers, and this can be used as the basis of a charge separation system. Drift directions are opposite for oppositely charged particles and perpendicular to the curvature or gradient. Drift velocities depend quadratically on particle velocity components. The separation of charged pions in a curved solenoid decay line was studied in \cite{tar-snake}. Because of the beam's 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 accomodate 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}). % But it has a problem that %is not included in the above simulations. \begin{figure}[hbt!] \begin{center} \includegraphics[height=5in,angle=-90]{412.eps} \end{center} %\centering %\centerline{\epsfig{file=fnalfg4.ps,height=4.0in,width=4.0in}} \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. Only three sections each are shown for cavities operating at 90, 50, and 30~MHz} \label{reentrant} \end{figure} A study\cite{harold} 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 advantage over the unpolarized case, since signal and background 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\cite{ref19,ref19a,ref19b} on the decay angle $\theta_d$ and initial pion energy. 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 $F_{loss}$ of muons that is selected. Figure~\ref{polvscutnew} gives the results of a Monte Carlo study. \begin{figure}[bht!] %\centering %\includegraphics[height=4.0in,width=3.5in]{fnalfg5.ps} \centerline{\epsfig{file=fnalfg5.ps,height=4.0in,width=4.0in}} \caption{Polarization {\it vs.}\ fraction $F_{loss}$ 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 $F_{loss}$ and the luminosity would fall by $F_{loss}^2$. 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 $F_{loss}$. One also notes that the luminosity could be maintained at the full %unpolarized value if the proton source intensity could be increased. Such an polarized value if the proton source intensity could be increased. Such an increase in proton source intensity in the unpolarized case would 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. \subsubsection{Polarization Preservation} A paper\cite{ref20} has discussed the preservation of muon polarization in some detail. 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 {dE \over E}\label{pol}, \end{equation} where $\beta_v$ is the muon velocity divided by $c$, and ${\Delta E\over E}$ 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 ${\Delta \mathcal{P}\over \mathcal{P}}$ 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 ${\Delta \gamma \over \gamma}$ 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 long. This might be a preferred solution at high energies but is not practical in the 100-GeV machine. 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. \subsection{R\&D Program} An R\&D program is underway to continue theoretical studies (optimization of pion production and capturing up to the cooling channel) and to clarify several critical issues related to targetry and phase rotation. 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 F.E.B. U-line of 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 first two studies should be accomplished during 1999, and the third and fourth in the years 2000/01.