\section{Ionization Cooling}
\label{subsec-compcool}
The very intense and very diffused in phase space muon beam at the end of the decay channel will require the development of
a new method for beam cooling in order to be able to manipulate the beam and accelerate it to the final energy, let alone to achieve the desire beam brightness.
For a high luminosity collider, the phase-space volume of the muon beam must be reduced within a time of the order of the
$\mu$ 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. Ionization cooling\cite{ref24,ref24a,ref24b} of muons seems relatively straightforward and it is unique for muons.
Ionization cooling involves passing the beam through some
material in which the muons lose both transverse and longitudinal momentum
by ionization loss (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 also be reduced (longitudinal cooling)
by introducing a transverse variation in the absorber density or
thickness (e.g. a wedge)
at a location where there is dispersion (the position is energy dependent). The appropriate figure of merit for cooling is the final value of the 6-D relativistically invariant emittance $\epsilon_6$ -- the area in the 6-dimensional 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.
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 $10^5$ -- $10^6$.
Initial theoretical design studies have shown that a complete
cooling channel might
consist of 20 -- 30 cooling stages, each stage yielding about a factor of
two in phase-space reduction.
It is
recognized that the feasibility of constructing a muon
ionization cooling channel is on the critical path to understanding the
viability of the whole muon collider concept.
In the following parts of this section we will briefly describe the physics
underlying the process of ionization cooling.
\subsection{Ionization Cooling Theory}
In ionization cooling, the beam loses both transverse and longitudinal momentum
as it passes through a material medium. At the same time its emittance is increased due to stochastic multiple scattering and Landau straggling. Subsequently, the longitudinal
momentum can be restored by coherent reacceleration, leaving a net loss of
transverse momentum.
The approximate equation for transverse cooling (with energies in GeV) is
\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)^2}{2\beta^3 E_{\mu}m_{\mu}\ L_R}, \label{eq1}
\end{equation}
where $\epsilon_n$ is the normalized transverse emittance, $\beta_{\perp}$ is the betatron
function at the absorber, $dE_{\mu}/ds$ is the energy loss, and $L_R$ is the
radiation length of the material. The first term in this equation is the coherent
cooling term, and the second is the heating due to multiple scattering.
This heating term is minimized if $\beta_{\perp}$ is small (strong-focusing)
and $L_R$ is large (a low-Z absorber).
The equation for energy spread (longitudinal emittance) is:
\begin{equation}
{\frac{d(\Delta E)^2}{ds}}\ =
-2\ {\frac{d\left( {\frac{dE_\mu}{ds}} \right)} {dE_\mu}}
<(\Delta E_{\mu})^2 >\ +
{\frac{d(\Delta E_{\mu})^2_{{\rm straggling}}}{ds}}\label{eq2}
\end{equation}
where the first term is the cooling (or heating) due to energy loss,
and the second term is the heating due to straggling.
Energy spread can be reduced by artificially increasing
${d(dE_\mu/ds)\over dE_{\mu}}$ by placing a transverse variation in absorber
density or thickness at a location where position is energy dependent, i.e. where there is
dispersion. The use of such wedges can reduce energy spread, but it
simultaneously increases transverse emittance in the direction of the
dispersion. Six dimensional phase space is not reduced, we only exchange longitudinal phase space with transverse phase space.
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 have
been incorporated into a computer code. Allowance for the shifts in the
betatron phase advance due to space charge and aberrations were 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, rf, and focusing lattice at each stage\cite{coolbob}.
Finally, a number of tracking codes were either written or modified to study
the cooling process in detail. Three new codes (MUONMC\cite{mcbob}, SIMUCOOL\cite{van}, and ICOOL\cite{ref25})
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 to use Maxwellian models of the focusing fields. They do not
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 GEANT\cite{ref40,paul} has been changed to double precision. All of these codes are continually being
updated and optimized for studying the cooling problem.
\subsection{Cooling Components}
We require a reduction of the normalized transverse emittance by almost three
orders of magnitude (from $1\times 10^{-2}$ to $5\times 10^{-5}\,$m-rad for the high energy collider), and a
reduction of the longitudinal emittance by one order of magnitude.
This cooling is obtained in a series of cooling stages. In general, each stage
consists of a succession of two components:
\begin{enumerate}
\item material in a strong focusing (low$\beta_\perp$) environment alternated with linac accelerators. These components will cool the transverse phase space.
\item lattice that generates dispersion, with absorbing material wedges introduced to interchange
longitudinal and transverse emittance.
\end{enumerate}
\begin{figure}[tbh!]
\centerline{\epsfig{file=fnalfg6.ps,height=3.0in,width=4.0in}}
\caption{The cross section of one cell of an alternating solenoid cooling system. }
\label{altsol}
\end{figure}
Simulations have been performed on examples of each component using the program ICOOL\cite{ref25} which includes Vavilov distributions (with Landau and Gaussian limits) for dE/dx, and Moliere scattering distributions (with Rutherford limit). The only mayor effects which are not yet included are space-charge and wake-field effects.
\subsubsection{Transverse Cooling}
The baseline solution for the first component involves the use of liquid hydrogen absorbers in strong solenoid focusing fields, interleaved with short linac sections. The solenoidal fields in successive absorbers must be reversed to avoid build up of the canonical angular momentum. Fig.~\ref{altsol} shows the cross section of one cell of such a system. The top plot in Fig.~\ref{coolingeg} shows the reduction of transverse emittance in 10 such cells (20 m); the middle one shows the increase in longitudinal emittance induced by straggling and the adverse dependence of dE/dx with energy; while the bottom one shows the overall reduction in 6-dimensional emittance. This simulation has been confirmed, with minor differences by the codes double precision GEANT\cite{paul} and PARMELA\cite{parmela}.
\begin{figure}[tbh!]
\centerline{\epsfig{file=fnalfg7.ps,height=5.0in,width=5.0in}}
\caption{Emittance vs. length in 10 alternating solenoid cells; top: transverse emittance; middle: longitudinal emittance; and bottom: 6-dimensional emittance.}
\label{coolingeg}
\end{figure}
Using 30 T solenoids at the end of a cooling sequence can attain a transverse emittance of 190 mm mrad and a six dimensional emittance of $30\times 10^{-12}$ m$^3$ (cf. 280 mm mrad and $170\times 10^{-12}$ m$^3$ respectively, required for a Higgs factory).
Other solutions, e.g. rapidly alternating solenoids and LiH absorbers\cite{fofo} and current carrying Li rods have been and will continue to be studied, but do not appear to be required to meet the baseline parameters (see below).
\subsubsection{Linac}
The linacs used in the above simulations have a frequency of $800\,$MHz and required an accelerating gradient (peak phase) of $29\,$MV/m. The current designs use cavities separated by thin Be foils, ${2\pi \over 3}$ or ${2\pi \over 4}$ phase advanced per cavity, and powered in 3 or 4 separate interleaved side-coupled standing-wave systems\cite{zhao,moretti}. In order to reduce power source requirements the cavities may be operated at liquid nitrogen temperatures.
\subsubsection{Longitudinal-Transverse Exchange}
The exchange of longitudinal and transverse emittance requires dispersion in a large acceptance channel. One way of achieving this is in a bent solenoid. Fig.~\ref{exch} shows transverse positions vs. their momenta: a) before the bend, b) after the bend, and c) after hydrogen wedges. The \textit{rms} momentum spread in this example is reduced from $8\,$MeV/c to $4.6\,$MeV/c with an accompanying approximately equivalent increase in the emittances $\epsilon_x$ and $\epsilon_y.$
Emittance exchange in solid wedges in the presence of ideal dispersion has also been simulated using SIMUCOOL\cite{dave}. Dispersion generation by weak focusing spectrometers\cite{balbekov} and dipoles with solenoids\cite{wan} have also been studied.
\begin{figure}[tbh!]
\centerline{\epsfig{file=fnalfg8.ps,height=3.5in,width=5.5in}}
\vspace{0.5cm}
\caption{Transverse trajectory positions vs.their momenta: a) before the bend, b) after the bend, and c) after hydrogen wedges.}
\label{exch}
\end{figure}
\subsection{Cooling System}
The required total 6 dimensional cooling is about 10$^6$. Since a single stage, as illustrated above, gives a factor of 2 reduction, about 20 such stages are required.
The total length of the system would be of the order of 500 m, and the total acceleration would be of the order of
6 GeV. The fraction of muons remaining at the end of the cooling system
is estimated to be $\approx 60 \%.$
In a few of the later stages, current carrying lithium rods might replace the solenoid and material in the cell. In this case the rod serves simultaneously to maintain the low
$\beta_{\perp}$, and attenuate the beam momenta. Similar lithium rods, with
surface fields of $10\,$T, were developed at Novosibirsk (BINP) and have been used as focusing elements at FNAL and CERN\cite{ref26,ref26a,ref26b,ref26c,ref27}. Cooling in beam recirculators could lead to reduction of costs of the cooling section\cite{balbekov}.
\subsection{Outlook}
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 the channel
and in adding detail to the computer simulations. We are now very close
to the goal of having a detailed and complete simulation of an entire
cooling channel. A vigorous experimental program is needed to verify and benchmark the computer simulations.