\section{IONIZATION COOLING} \label{subsec-compcool} \subsection{Introduction} The design of an efficient and practical cooling system is one of the major challenges for the muon collider project. For a high luminosity collider, the 6-D phase space volume occupied by the muon beam must be reduced by a factor of $10^5 - 10^6.$ Furthermore, this phase space reduction must be done within a time that is not long compared to the muon lifetime ($\mu$ lifetime $\approx 2~\mu s$). Cooling by synchrotron radiation, conventional stochastic cooling and conventional electron cooling are all too slow. Optical stochastic cooling \cite{ref21}, electron cooling in a plasma discharge \cite{ref22}, and cooling in a crystal lattice \cite{ref23,ref23a} are being studied, but appear technologically difficult. The new method proposed for cooling muons is ionization cooling. This technique \cite{Budker78,ref1a,ref1,ref24} is uniquely applicable to muons because of their minimal interaction with matter. It is a method that seems relatively straightforward in principle, but has proven quite challenging to implement in practice Ionization cooling involves passing the beam through some material in which the muons lose both transverse and longitudinal momentum by ionization energy loss, commonly referred to as $dE/dx.$ The longitudinal muon momentum is then restored by reacceleration, leaving a net loss of transverse momentum (transverse cooling). The process is repeated many times to achieve a large cooling factor. The energy spread can be reduced by introducing a transverse variation in the absorber density or thickness (e.g.\ a wedge) at a location where there is dispersion (the transverse position is energy dependent). This method results in a corresponding increase of transverse phase space and is thus an exchange of longitudinal and transverse emittances. With transverse cooling, this allows cooling in all dimensions. We define the root mean square rms normalized emittance as \begin{equation} \epsilon_{i,N}=\sqrt{\langle\delta r_i^2\rangle \langle\delta p_i^2\rangle-\langle\delta r_i\delta p_i\rangle^2}/m_{\mu}c \end{equation} where $r_i$ and $p_i$ are the beam canonical conjugate variables with $i=1,2,3$ denoting the x, y and z directions, and $\langle...\rangle$ indicates statistical averaging over the particles. The operator $\delta$ denotes the deviation from the average, so that $\delta r_i=r_i-\langle r_i\rangle$ and likewise for $\delta p_i.$ The appropriate figure of merit for cooling is the final value of the 6-D relativistically invariant emittance $\epsilon_{6,N},$ which is proportional to the area in the 6-D phase space $(x,y,z,p_x,p_y,p_z)$ since, to a fairly good approximation, it is preserved during acceleration and storage in the collider ring. This quantity is the square root of the determinant of a general quadratic moment matrix containing all possible correlations. However, until the nature and practical implications of these correlations are understood, it is more conservative to ignore the correlations and use the following simplified expression for 6-D normalized emittance, \begin{equation} \epsilon_{6,N}\approx \epsilon_{x,N}\times \epsilon_{y,N}\times \epsilon_{z,N} %<<< %\left(\sigma_z^{rms}\sigma_E^{rms}\gamma \beta\right) >>> \end{equation} Theoretical studies have shown that, assuming realistic parameters for the cooling hardware, ionization cooling can be expected to reduce the phase space volume occupied by the initial muon beam by a factor of $10^5$ -- $10^6$. A complete cooling channel would consist of 20 -- 30 cooling stages, each stage yielding about a factor of two in 6-D phase space reduction. It is recognized that understanding the feasibility of constructing a muon ionization cooling channel is on the critical path to determining the viability of the whole muon collider concept. The muon cooling channel is the most novel part of a muon collider complex. Steady progress has been made both in improving the design of sections of the channel and in adding detail to the computer simulations. A vigorous experimental program is needed to verify and benchmark the computer simulations. The following parts of this section briefly describe the physics underling the process of ionization cooling. We will show results of simulations for some chosen examples, and outline a six year R\&D program to demonstrate the feasibility of using ionization cooling techniques. \subsection{Cooling theory} In ionization cooling, the beam loses both transverse and longitudinal momentum as it passes through a material. At the same time its emittance is increased due to stochastic multiple scattering and Landau straggling. The longitudinal momentum can be restored by reacceleration, leaving a net loss of transverse momentum. The approximate equation for transverse cooling in a step $ds$ along the particle's orbit is \cite{ref2b,ref1a,ref1,ref2a,ref24b,ref24c} \begin{equation} \frac{d\epsilon_N}{ds} = -{1\over \beta^2}\frac{dE_{\mu}}{ds}\ \frac{\epsilon_N}{E_{\mu}} + \frac{\beta_{\perp} (0.014 GeV)^2}{2\beta^3 E_{\mu}m_{\mu}\ L_R}, \label{eq1} \end{equation} where $\beta$ is the normalized velocity, $E_{\mu}$ is the total energy, $m_{\mu}$ is the muon mass, $\epsilon_N$ is the normalized transverse emittance, $\beta_{\perp}$ is the betatron function at the absorber, $dE_{\mu}/ds$ is the energy loss per unit length, and $L_R$ is the radiation length of the material. The betatron function is determined by the strengths of the elements in the focusing lattice \cite{pdgaccel}. Together with the beam emittance this function determines the local size and divergence of the beam. Note that the energy loss $dE_{\mu}/ds$ is defined here as a positive quantity, unlike the convention often used in particle physics. The first term in this equation is the cooling term, and the second describes the heating due to multiple scattering. The heating term is minimized if $\beta_{\perp}$ is small (strong-focusing) and $L_R$ is large (a low-Z absorber). The minimum, normalized transverse emittance that can be achieved for a given absorber in a given focusing field is reached when the cooling rate equals the heating rate in Eq.~\ref{eq1} \begin{equation}\epsilon_{N,min} = {\beta_{\perp} (14 MeV)^2 \over 2 \beta m_{\mu} {dE_{\mu} \over ds} L_R }\label{equi} \end{equation} For a relativistic muon in liquid hydrogen with a betatron focusing value of 8 cm, which corresponds roughly to confinement in a 15~T solenoidal field, the minimum achievable emittance is about 340~mm-mrad. The equation for energy spread is \cite{ref1a,ref2a,ref2c} \begin{equation} {\frac{d(\Delta E_\mu)^2}{ds}}\ = -2\ {\frac{d\left( {\frac{dE_\mu}{ds}} \right)} {dE_\mu}} \langle(\Delta E_{\mu})^2 \rangle\ + {\frac{d(\Delta E_{\mu})^2_{{\rm stragg.}}}{ds}}\label{eq2} \end{equation} where the first term describes the cooling (or heating) due to energy loss, and the second term describes the heating due to straggling. $\Delta E_{\mu}$ is the rms spread in the energy of the beam. Ionization cooling of muons seems relatively straightforward in theory, but will require extensive simulation studies and hardware development for its optimization. There are practical problems in designing lattices that can transport and focus the large emittance beam. There will also be effects from space charge and wake fields. We have developed a number of tools for studying the ionization cooling process. First, the basic theory was used to identify the most promising beam properties, material type and focusing arrangements for cooling. Given the practical limits on magnetic field strengths, this gives an estimate of the minimum achievable emittance for a given configuration. Next, the differential equations for cooling and heating described above were incorporated into a computer code. Allowance for the shifts in the betatron phase advance due to space charge and aberrations was included. This code was used to develop an overall cooling scenario, which broke the cooling system into a number of stages, and determined the properties of the beam, radio frequency (rf) cavities, and focusing lattice at each stage. Finally, several tracking codes were either written or modified to study the cooling process in detail. Two new codes (SIMUCOOL \cite{van}, and ICOOL \cite{ref25,ref25a}) use Monte Carlo techniques to track particles one at a time through the cooling system. All the codes attempt to include all relevant physical processes to some degree, (e.g. energy loss, straggling, multiple scattering) and use Maxwellian models of the focusing fields. They do not yet take into account any space charge or wake field effects. In addition, we have also used a modified version of PARMELA \cite{parmela} for tracking, which does include space charge effects, and a double precision version of GEANT \cite{ref40,paul}. We have recently developed \cite{envelope} a model of beam cooling based on a second order moment expansion. A computer code solving the equations for transverse cooling gives results that agree with tracking codes. The code is being extended to include energy spread and bends. It is very fast and is appropriate for preliminary design and optimization of the cooling channel. All of these codes are actively being updated and optimized for studying the cooling problem. \subsection{Cooling system} The cooling is obtained in a series of cooling stages. Each stage consists of a succession of the following components: \begin{itemize} \item Transverse cooling sections using materials in a strong focusing (low $\beta_\perp$) environment alternated with linear accelerators. \item Emittance exchange in lattices that generate dispersion, with absorbing wedges to reduce momentum spread. \item Matching sections to optimize the transmission and cooling parameters of the following section. \end{itemize} The question of the best energy to use for cooling has been discussed in detail in reference \cite{palmer_lectures}. For the set of parameters of interest, a kinetic energy close to 100~MeV appears optimal. At higher energies, weaker focusing raises the heating term from Coulomb scattering, and more acceleration is required for a given amount of cooling. At lower energies, the beam divergence become large, and the rise of ${dE\over dx}$ with falling energy causes a greater increase in energy spread. There can be an advantage, initially, of using a somewhat higher energy to reduce the beam dimensions and bucket length; at the end, the energy can be dropped to attain the lowest transverse emittances at the expense of longitudinal emittance. \begin{figure} \begin{center} \includegraphics[width=3in]{sf_fg5.ps} \end{center} \caption{Transverse ($\epsilon_{x,y}$), longitudinal emittance ($\epsilon_L$) and beam energy {\it vs.} stage number in the cooling sequence. } \label{bnl_62740} \end{figure} In the examples that follow it is seen that the simulated transverse cooling stages lower the 6-D emittance by a factor of about 2. Since the required total 6-D cooling is $O(10^6)$, about 20 such stages are required. We have performed calculations of complete cooling systems for the Higgs factory and for a high energy collider. These calculations are based on theoretical models of the expected cooling performance. They give an indication of the system dimensions, magnet strengths, rf frequencies and gradients, and beam parameters that will be required in a cooling system. The calculations suggest that the required cooling for a Higgs factory could be achieved in 25~stages, while the high energy collider would require 28~stages. Emittances and energies as a function of stage are shown in Fig.~\ref{bnl_62740}. The sequence can be considered to consist of 3~parts: \begin{enumerate} \item For the first 12~stages the primary effort is to cool in the longitudinal direction in order to reduce the bunch lengths and allow higher frequency rf to be employed. Some transverse cooling is needed in order to reduce the transverse dimensions of the beam and allow it to fit through the smaller irises at the higher frequencies. In this example, for the first stage, an energy of 300~MeV was used to calculate the reduction on the momentum spread and transverse beam dimensions. It is recognized that designing an emittance exchange at the beginning of the system that can simultaneously accept the large initial phase space of the beam and reduce the longitudinal emittance will be a major challenge. In later stages the energy is closer to 100~MeV. Solenoid focusing was assumed in all stages, with an initial field of the order of 1~T rising to about 3~T at the end. \item In the second part ( in this example, stages 13 - 25) the 6-D emittance is reduced as far as possible. For the case of a low momentum spread Higgs collider, the required beam parameters are now achieved and the third part is not required. In this example, 80~MeV energy was used for all stages. Solenoid focusing was used in all but the last two stages, where lithium lenses were assumed. \item For the higher luminosity and higher energy colliders, further reduction in transverse emittance is required, but this can be obtained without reduction of the 6-D phase space, by allowing the longitudinal phase space to grow. This exchange of emittances is, in this example, achieved by reducing the energy to near 10~MeV in two long lithium lens cooling stages. The same effect could probably be achieved at similar energy, by using a hydrogen absorber with solenoid focusing. It might also be possible by using wedges. \end{enumerate} The total length of the system would be of the order of 600~m, and the total acceleration required would be approximately 6 GeV. The fraction of muons remaining at the end of the cooling system is estimated to be $\approx 60\%$. It must be emphasized that this sequence was derived without simulation of the individual stages. It serves however to guide the choice of stages to study in detail. Three transverse cooling examples have been designed and simulated. The first uses 1.25~T solenoids to cool the very large emittance beam coming from the phase rotation channel. The muon beam at the end of the decay channel is very intense, with approximately $7.5\times 10^{12}$ muons/bunch, but with a large normalized transverse emittance $(\epsilon_{x,N}(\textrm{rms})\approx 15\times 10^3~\pi \textrm{mm-mrad})$ and a large normalized longitudinal emittance $(\epsilon_{z,N}(\textrm{rms})\approx 612~\pi \textrm{mm}).$ The second example would lie toward the end of a full cooling sequence and uses 15~T solenoids. The third example, using 31~T solenoids, meets the requirements for the Higgs factory and could be the final cooling stage for this machine. The baseline solution for emittance exchange involves the use of bent solenoids to generate dispersion and wedges of hydrogen or LiH to reduce the energy spread. A simulated example is given for exchange that would be needed after the 15~T transverse cooling case. A lithium lens solution may prove more economical for the final stages, and might allow even lower emittances to be obtained. In this case, the lithium lens serves simultaneously to maintain the low $\beta_{\perp}$, and provide $dE/dx$ for cooling. Similar lenses, with surface fields of $10\,$T, were developed at Novosibirsk (BINP) and have been used, at low repetition rates, as focusing elements at FNAL and CERN \cite{reviewtev,ref26,ref26a,ref26b,ref27}. Lenses for the cooling application, which would operate at 15~Hz, would need to employ flowing liquid lithium to provide adequate thermal cooling. Higher surface fields would also be desirable. Studies have simulated cooling in multiple lithium lenses, and have shown cooling through several orders of magnitude \cite{Balbekov96}. But these studies have, so far, used ideal matching and acceleration. Cooling is also being studied in beam recirculators, which could lead to reduction of costs of the cooling section \cite{balbekov,balbekov1}, but full simulations with all higher order effects have not yet been successfully demonstrated. \subsection{15~T solenoid transverse cooling example} The lattice consists of 11 identical 2~m long \textit{cells}. In each cell there is a liquid hydrogen absorber (64~cm long, 10~cm diameter) in the 15~T solenoid focusing magnet (64~cm long, 12~cm diameter). The direction of the fields in the magnets alternates from one cell to the next. Between the 15~T solenoids there are magnetic matching sections (1.3~m long, 32~cm inside diameter) where the field is lowered and then reversed. Inside the matching sections are short, 805~MHz, high gradient (36~MeV/m) linacs. \begin{figure*}[thb!] \centerline{\epsfig{file=radii_beta_axial.eps,height=5.0in,width=5.0in}} \caption[Cross section of one half period of an alternating solenoid cooling lattice ]{(a) Cross section of one half period of an alternating solenoid cooling lattice; (b) axial magnetic field \vs~z; (c) $\beta_{\perp} $~function \vs~z. } \label{altsol} \end{figure*} Figure~\ref{altsol} shows the cross section of one cell of such a system, together with the betatron function, and the magnetic field along the axis. For convenience in modeling, the section shown in Fig.~\ref{altsol}(a) starts and ends symmetrically in the middle of hydrogen absorber regions at the location of the peaks in the axial magnetic field. In practice each cell would start at the beginning of the hydrogen region and extend to the end of the rf module. A GEANT simulation of muons traversing a section of the cooling channel is shown in Fig.~\ref{dpgeant}. Additional simulations were performed \cite{angmom,aac_paper} using the program ICOOL. The only likely significant effects which are not yet included are space charge and wakefields. Analytic calculations for particle bunches in free space indicate that these effects should, for the later stages, be significant but not overwhelming. \begin{figure*}[thb!] \centerline{\epsfig{file=dpgeant_new.eps,height=3.0in,width=6.0in,clip=}} \caption[GEANT simulation of muons traversing a section of cooling channel]{GEANT simulation of muons traversing a section of the alternating solenoid cooling channel. The variation of the magnetic field $B_z$ is shown for $1{1\over 2}$ cells of the figure.} \label{dpgeant} \end{figure*} A full simulation must be done before we are assured that no problems exist. Particles are introduced with transverse and longitudinal emittance (186~MeV/$c,$ 1400~$\pi$~mm-mrad transverse, and 1.1~$\pi$~mm~longitudinal), together with a number of naturally occurring correlations. Firstly, the particles are given the angular momentum appropriate for the starting axial magnetic field. Secondly, particles with large initial radius $r_o$ and/or divergence $\theta_o$ have longer pathlengths in a solenoidal field and tend to spread out with time. This can be parameterized by defining an initial transverse amplitude \begin{equation} A^2={r_o^2\over \beta_{\perp}^2 }+\theta_o^2. \end{equation} The temporal spreading can be minimized by introducing an initial correlation between $p_z$ and $A^2$ that equalizes the forward velocity of the initial particles. This correlation causes the average momentum of the beam to grow from the reference value of 186~MeV/$c$ to $\approx 195$~MeV/$c.$ Lastly, a distortion of the longitudinal bunch distribution can be introduced to reflect the asymmetric nature of the ``alpha"-shaped rf bucket. Figure~\ref{simu8}(a) shows the average momentum of the beam as a function of distance along the channel. The momentum drops as the beam crosses the liquid hydrogen absorbers. The gradient and phase of the rf cavities have been adjusted so that the reacceleration given to the reference particle equals the mean energy loss. This causes the average momentum of the beam to remain in a narrow band around 195~MeV/$c.$ Figure~\ref{simu8}(b) shows the mechanical and canonical angular momenta as a function of distance along the channel. The mechanical angular momentum shows the rotational motion of the beam around the axial solenoidal field. It periodically reverses sign when the solenoids alternate direction. The canonical angular momentum is defined such that it removes the axial field dependence \cite{aac_paper}. Without the absorbers, the beam would have a constant (0) value for the canonical angular momentum. However, the presence of absorbers causes the canonical angular momentum to grow and would lead to severe emittance growth by the end of a long channel. This growth is stopped by alternating the direction of the solenoid field, as shown in Fig.~\ref{simu8}(b) Simulations have shown that 2~m is a reasonable (half) period for the field, since the net growth in canonical angular momentum is small. In addition synchrobetatron resonances are avoided since the periodicity of the field forces the average betatron wavelength to be 2~m, whereas the synchrotron oscillation wavelength seen in the simulations for this arrangement is $\approx 14$~m. \begin{figure*}[htb!] \centerline{\epsfig{file=simu8.ps,height=4.0in,width=6.0in}} \caption[Average momentum \textit{vs.} z ]{a) Average momentum \textit{vs.} z; b) Average angular momentum: mechanical (solid curve) and canonical (dashed curve), \textit{vs.} z.} \label{simu8} \end{figure*} Figure~\ref{simu6}(a) shows the rms and maximum radius of any particle in the beam distribution as a function of distance along the channel. The rms radius shows that most of the beam is confined to within 2~cm of the axis. The peak rms radius decreases towards the end of the channel as a result of the cooling. The maximum particle radius is about 8~cm, which determines the radius of the windows required in the rf cavities. Figure~\ref{simu6}(b) shows the rms momentum spread corrected for the correlation between $p_z$ and transverse amplitude imposed on the initial particle distribution. The momentum spread grows as a function of distance since the alternating solenoid system only cools the transverse emittance. Figure~\ref{simu6}(c) shows the rms bunch length s a function of distance along the channel. Again this grows with distance since this channel does not cool longitudinally. \begin{figure*}[bht!] \centerline{\epsfig{file=simu6.ps,height=4.0in,width=6.0in}} \caption[rms and maximum beam radii \vs~z ]{a) rms and maximum beam radii, b) rms corrected momentum, c) rms bunch length; all \vs~z.} \label{simu6} \end{figure*} Figure~\ref{coolingeg}(a) shows the decrease in transverse normalized emittance as a function of distance along the channel. The system provides cooling by a factor of $\approx $~2 in both the $x$ and $y$ transverse phase spaces. From the changing slope of the curve we note that the rate of cooling is dropping. This sets $\approx$~22~m as the maximum useful length for this type of system. It must be followed by a longitudinal emittance exchange region to reduce the momentum spread and bunch length approximately back to their starting values. Figure~\ref{coolingeg}(b) shows the increase in longitudinal normalized emittance in the channel due to the increase in momentum spread and bunch length. Finally, Fig.~\ref{coolingeg}(c) shows the decrease in the 6-D normalized emittance as a function of distance along the channel. There is a net decrease in 6-D emittance by a factor of $\approx$~2 in the channel. Table~\ref{cooltab} gives the initial and final beam parameters. \begin{figure*}[bht!] \centerline{\epsfig{file=simu7.ps,height=4.0in,width=6.0in}} \caption[Emittance \vs~z ]{Emittance \vs~z: a) transverse emittance $\epsilon_{T,N} (\pi \textrm{mm-mrad})$; b) longitudinal emittance $\epsilon_L (10^{-3}\pi \textrm{mm})$; and c) 6-D emittance $\epsilon_6 (10^{-12} \pi \textrm{m}^3)$.} \label{coolingeg} \end{figure*} \begin{table*}[thb] \caption{Initial and final beam parameters in a 15 T transverse cooling section.} \label{cooltab} \begin{tabular}{llccc} & & initial & final & final/initial\\ \hline Particles tracked & & 1000 & 980 & 0.98 \\ Reference momentum & MeV/$c$ &186 & 186 & 1.0\\ Transverse Emittance & $\pi$ mm-mrad & 1400 & 600 & 0.43 \\ Longitudinal Emittance & $\pi$ mm & 1.1 & 2.3 & 2.09 \\ 6-D emittance~$\times 10^{-12}$& ($\pi$~m-rad)$^3$ & 2000 & 800 & 0.40 \\ rms beam radius in hydrogen & cm & 0.8 & 0.55 & 0.69 \\ rms beam radius in linac & cm & 2.0 & 1.4 & 0.70 \\ max beam radius in linac & cm & 7.0 & 7.0 & 1.0 \\ rms bunch length & cm & 1.5 & 2.2 & 1.5 \\ max bunch full width & cm & 13 & 19 & 1.5 \\ rms $\Delta p/p$ & \% & 3.8 & 5.6 & 1.5 \\ \end{tabular} \end{table*} This simulation has been confirmed, with minor differences, by double precision GEANT \cite{paul} and PARMELA \cite{parmela} codes. \subsection{31~T solenoid transverse cooling example} As in the preceding example, the lattice consists of 11 identical 2~m long cells with the direction of the fields in the solenoids alternating from one cell to the next. The maximum solenoidal field is higher (31~T) than in the previous example, but the bore can be smaller ($6