\subsection{Overview of a Muon Collider} The high-energy frontier of elementary-particle physics presently is pursued with two types of colliding beam facilities: \begin{itemize} \item {\bf Proton-proton machines}, in which storage rings contain the colliding beams, but for which the constituent (primarily gluon-gluon) center-of-mass (CoM) energy is poorly defined and effectively about 1/10 of the $p$-$p$ CoM energy. \item {\bf Electron-positron linear colliders}, in which all of the CoM energy is available for fundamental processes, but for which the effects of quantum beamstrahlung limit the precision of the initial state at high energies and luminosities. An $e^+e^-$ storage ring of CoM energy higher than LEP is not foreseen, due to the high power lost to synchrotron radiation. \end{itemize} Future examples of such machines will be rather costly. In this context a new type of high-energy particle accelerator, \begin{itemize} \item A {\bf muon collider} \cite{status,Snowmass}, \end{itemize} has many attractive features. Muons are leptons, and so, like electrons, couple directly to fundamental processes. But muons are 200 times heavier than electrons, so initial-state radiation (beamstrahlung and synchrotron radiation) are negligible for CoM energies up to ${\cal O}$(100~TeV). Hence, recirculating accelerators and storage rings \cite{Oneill} can be used at a muon collider \cite{ref01,ref01a,Skrinsky71}, offering cost advantages over both electron-positron and proton-proton colliders. \begin{figure}[bth!] \centerline{\epsfig{file=machine_comparison_new.ps,width=6in}} \caption{Comparison of footprints of possible future colliders, including muon colliders at CoM energies 0.1, 0.4 and 3 TeV. The energy shown in parentheses for lepton machines is the CoM energy, while for hadron machines it is the effective energy for parton-parton collisions. } \label{plan2} \end{figure} A physics advantage of muons over electrons and protons (\ie, gluons) is that muons have the largest $s$-channel coupling (which is proportional to $m^2$) to Higgs-type bosons, permitting precision studies of such particles \cite{ref2b,mup}. A first muon collider might focus on this unique opportunity. A 3-4-TeV muon collider would fit on the existing sites of BNL or FNAL, as shown in Fig.~\ref{plan2}. The front end (Fig.~\ref{plan1}) of such a machine would be suitable for studies of $s$-channel production of a light Higgs boson. \begin{figure}[tbh!] \centerline{\epsfig{file=fnalfg2.ps,width=5in}} \caption{Plan of a 100-GeV CoM Muon Collider, as an example of a first muon collider, to study $s$-channel light-Higgs production.} \label{plan1} \end{figure} Muons are unstable, with a lifetime of 2.2 $\mu$s in the muon's rest frame. Therefore, copious quantities of muons are required for high luminosity. For this, the muons will be collected over a relatively large volume of phase space, which must quickly be reduced by a cooling process. As muons have little interaction with matter, the cooling can be accomplished by a technique not suitable for other particles, namely ionization cooling \cite{Oneill,mura126,Kolomensky,Ado}. In this process, muons lose their transverse and longitudinal momentum during passage through bulk matter, after which only longitudinal momentum is restored via acceleration in rf cavities, as sketched in Fig.~\ref{transcool}. \begin{figure}[bth!] \centerline{\epsfig{file=transcool.eps,width=3.5in}} \caption{The concept of ionization cooling.} \label{transcool} \end{figure} The muon-decay products are electrons and neutrinos, so a muon collider would be the premier source of future neutrino beams \cite{mup,cline80}. Indeed, some care as to personnel safety from neutrino-induced radiation will be required in multi-TeV muon colliders \cite{status}. % \subsection{Activities of the Muon Collider Collaboration} A collaboration \cite{Muoncollab} of over 100 members\footnote{Spokesperson: R.B. Palmer.} has been active for several years in exploring the concept of a muon collider via analytic studies and computer simulation. This effort is summarized in refs.~\cite{status,Snowmass} and references therein. A key conclusion is that the concepts of a muon collider should be validated by a program of research and development in two areas: \begin{itemize} \item Ionization cooling. \item Targetry. \end{itemize} A proposal for an R\&D program on ionization cooling \cite{coolrnd} has been submitted to Fermilab recently. The present document proposes a companion program for targetry issues: the production of pions by protons hitting a target, and the subsequent capture and decay of the pions to produce the initial muon beam. \subsection{A Targetry Scenario for a Muon Collider} \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!] \centerline{\epsfig{file=capture.eps,width=6.5in}} \caption{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} \subsubsection{Pion production} To achieve the present design luminosity of $7 \times 10^{34}$ cm$^{-2}$s$^{-1}$ for a 3-TeV CoM muon collider (or $10^{31}$ cm$^{-2}$s$^{-1}$ at 100-GeV CoM), $2\times 10^{12}$ (or $4\times 10^{12}$ at 100-GeV CoM) muons of each sign must be delivered to the collider ring in each pulse at 15~Hz. 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. 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 by 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{Snowmass,tar-snake,Ehst97,Takahashi97,Mokhov98-41}. Three different Monte Carlo codes \cite{arc,mars,Mokhov98-53,dpmjet} predict similar pion yields despite significant differences in their physics models. The Collaboration is involved in an AGS experiment (see Appendix A and \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}a. \begin{figure}[thb!] %\noindent \begin{minipage}{.50\linewidth} % fig 4b \centering\epsfig{figure=mok4a.eps,width=\linewidth} \end{minipage}\hfill \begin{minipage}{.50\linewidth} % fig 5a \centering\epsfig{figure=mok4b.eps,width=\linewidth} \end{minipage} \begin{minipage}{.50\linewidth} % fig 5b \centering\epsfig{figure=mok4c.eps,width=\linewidth} \end{minipage}\hfill \begin{minipage}{.50\linewidth} % fig 5yvsang \centering\epsfig{figure=mok4d.eps,width=\linewidth} \end{minipage} %\vspace{10pt} \caption{a) Meson yield ($\pi + K$) from a 1.5-$\lambda_{I}$, 1-cm-radius target irradiated with 8, 16 and 30-GeV proton beams as a function of target atomic mass; b) Meson yield from a 3-$\lambda_{I}$, 1-cm-radius gallium target tilted at angle 150~mrad in a 16-GeV proton beam {\it vs.} solenoid field for a fixed adiabatic invariant $B R_a^2$; c) Meson yield as a function of target radius; d) Meson yield {\it vs.} tilt angle between the axis of the capture solenoid and the proton beam. The target is aligned along the beam. The curves labeled YC show mesons that are transported into the decay channel.} \label{yield-E-A} \end{figure} \subsubsection{Target} The target should be 2-3 interaction lengths long to maximize pion production. A high-density material is favored to reduce the target length, thereby minimizing 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 the loss of pions by absorption in the target after one or more turns 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 experience at CERN \cite{Lettry} and Novosibirsk \cite{Silvestrov} indicates that shock damage to the pipe is a serious problem. Therefore, an open liquid jet is proposed. A jet of liquid mercury has been demonstrated \cite{Johnson} (see Fig.~\ref{hgjetphoto}) 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, despite their lower density, are also being considered. \begin{figure}[thb!] \begin{center} \includegraphics[width=5.25in,clip=]{hg-jet4.eps} \end{center} \caption{Photographs of a 3-mm-diameter mercury jet.} \label{hgjetphoto} \end{figure} %Experimental and theoretical studies are underway to determine the %consequences of beam shock heating of the liquid. It is expected that the jet will disperse after being struck by 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{mumu97-3,Weggela}. 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 consideration. 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!] \begin{center} \includegraphics[width=5in,clip=]{bandsaw.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 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. \subsubsection{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{Weggelb} 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 will call for state-of-the-art magnet technology \cite{Miller}. 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. Results of a MARS simulation of pion production and capture \cite{Mokhov98-41} are shown in Fig.~\ref{yield-E-A}. The curves labeled YC in Fig.~\ref{yield-E-A}b show, as a function of field in the capture solenoid, the yield of pions that are both captured in the high-field solenoid and transported into the decay channel. The radius of the capture solenoid maintains the same $BR_a^2$ as in the decay channel. The optimum field is 20~T in the capture solenoid. %In this simulation, the %beam and target were at angle~150 mrad to the axis of the solenoid. Figure~\ref{yield-E-A}c shows that pion production {\it vs.}\ target radius has a maximum near $r = 0.85$~cm. This is the result of the competition between secondary production and absorption. %, both of which increase with target radius. 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 at an angle of 100-150~mrad with respect to the field axis, % \cite{Mokhov98-41}, %\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 0.45 pions per proton will enter the pion decay channel \cite{Mokhov98-41}. %tarmok98}. \subsubsection{Phase-Rotation Linac} The pions, and the muons into which they decay, have an energy spread with an rms value of approximately 100\% and a peak value near 200~MeV/$c$. It would be difficult to handle such a wide spread in any subsequent system. Therefore, a linac is 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, 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 among them. Independent of the above choices is the question of the location of the focusing solenoid coils and rf cavity design. The closer the first rf cavity is to the target, the less the muon bunch lengthens due to the variation in velocity of the pions. We discuss here only the case where muons are captured at a mean kinetic energy of 130~MeV. See ref.~\cite{status} for an example of capture at a higher energy. Table~\ref{rot} gives parameters of the linacs used. The frequencies vary between 30 and 60~MHz, and the overall length is 42~m. Monte Carlo simulations \cite{Snowmass}, 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 channel, and the rf was approximated by a series of kicks. \begin{table}[bth] \begin{center} \caption{Parameters of Phase-Rotation Linacs} \label{rot} \vskip6pt \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} %\vspace{-0.4in} \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 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. Without energy selection, the average polarization of the muons is 20\%.} \label{Evsctpol2} \end{figure} Fig.~\ref{Evsctpol2} shows the energy {\it vs.}\ $ct$ at the end of the decay and phase-rotation channel. A loose final-bunch selection was defined with an energy 130 $\pm$ 70~MeV and bunch $ct$ between 3 and 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. \subsubsection{Use of Both Signs} Protons on the target produce pions of both signs, and a solenoid will capture both, but the subsequent phase-rotation rf systems will have opposite effects on opposite charges within the same bunch. The baseline solution is to use two proton bunches, separated in time by an odd number of rf half cycles. 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. \subsubsection{Solenoids and RF} Pion 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 2.5~m, but this is still large, and it was first assumed that 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[width=2.5in, 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 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} However, 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). Simulations are underway to determine the optimal solution. \subsubsection{Muon Polarization} Polarization of the muon beams presents a significant physics advantage over the unpolarized case, since signals and backgrounds of electroweak processes usually come predominantly from different polarization states. In the center-of-mass frame 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,ref20}. 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. \begin{figure}[bht!] \begin{center} \includegraphics[width=4.5in]{pol_loss.eps} \end{center} \caption{Polarization {\it vs.}\ fraction $F_{\rm loss}$ of $\mu$'s accepted. \label{polvscutnew}} \end{figure} If higher polarization is required, some selection of muons from forward pion decays $(\cos{\theta_d} \rightarrow 1)$ is required. Figure~\ref{Evsctpol2} shows 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 smaller the fraction $F_{\rm loss}$ of muons that remain. Figure~\ref{polvscutnew} gives the results of a Monte Carlo study. 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 $F_{\rm loss}$, and the luminosity would fall by $F_{\rm loss}^2$. 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 $F_{\rm loss}$. 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 power in the high-energy-muon beam, 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 at the pion-production target. \subsection{Summary of Critical Targetry Issues} There are several key issues which require laboratory studies before the targetry scenario can be placed on a firm basis: \begin{enumerate} \item What is the effect of the pressure wave induced in the target by the proton pulse? If the liquid target is dispersed by the beam, do the droplets damage the containment vessel? \item What is the effect of the magnetic field of the capture solenoid on the motion of the liquid-jet target? Is the jet badly distorted by Lorentz forces on the eddy currents induced as the jet enters the field? Does the magnetic field damp the effects of the beam-induced pressure wave? \item Can the first rf cavity of the phase-rotation channel operate viably in close proximity to the target? \item What is the yield of low-energy pions from 16-24-GeV protons incident on the target of the muon-collider source? \item Can numerical simulations of target behavior be developed that permit reliable extrapolation of the data we obtain? \end{enumerate} The present understanding of questions 1-3 is discussed in sec.~2. Data relevant to question 4 has been collected by BNL experiment E-910 \cite{exp910}, the status of which is summarized in Appendix A. The proposed eight-part R\&D program to address all of these questions is presented in sec.~3.