\section{Muon Colliders} \label{higgsfact} The lure of muon colliders arises from the fact that the muon is $\approx$~200 times heavier than the electron and this makes it possible to accelerate muons using circular accelerators that are compact and fit on existing accelerator sites. See Figure~\ref{compare} for a comparison of relative sizes of muon colliders ranging from 500 GeV to 3 TeV center of mass energies with respect to the LHC, SSC, and NLC. Once we have solved the problem of cooling a muon beam so that it can be accelerated, higher enegies are much more easily obtained in a muon collider than in the linear electron-positron collider. % Because the muon is unstable, it is necessary to cool and accelerate it before a substantial number have decayed. With typical bending magnetic fields($\approx$~5~Tesla) available with today's technology, the muons last $\approx$~1000 turns before half of them have decayed in the collider ring. This is a statement that is independent of the energy of the collider to first order due to relativistic time dilatation. Muon decay also gives rise to large numbers of electrons that can pose serious background problems for detectors in the collision region. The 1999 Status Report~\cite{INTRO:ref5} contains an excellent summary of the problems and possible solutions one faces on the way to a muon collider. % \begin{figure}[bth!] \includegraphics[width=6in,height=4.75in]{machine_comparison_new.ps} %\centerline{\epsfig{file=machine_comparison_new.ps,height=4.75in,width=6.in}} \vspace{0.5cm} \caption[Sizes of various proposed high energy colliders] {Comparative sizes of various proposed high energy colliders compared with the FNAL and BNL sites. The energies in parentheses give for lepton colliders their CoM energies and for hadron colliders the approximate range of CoM energies attainable for hard parton-parton collisions.} \label{compare} \end{figure} % Figure~\ref{schematic} shows a schematic of such a muon collider, along with a depiction of the possible physics that can be addressed with each stage of the facility. % \begin{figure}[bth!] \centerline{\includegraphics[width=0.6\linewidth]{higgs_schematic.eps}} \vspace{0.5cm} \caption[Schematic of a muon collider]{Schematic of a muon collider.} \label{schematic} \end{figure} % \subsection{Higgs Factory Requirements} The emittance of the muon beam needs to be reduced by a factor of 10$^6$ from production~\cite{INTRO:ref5} to the point of collision for there to be significant luminosity for experiments. This can be achieved by ionization cooling similar to the scheme described in the section~\ref{neufact}. The transverse emittance is reduced during ionization cooling, since only the longitudinal energy loss is replaced by rf acceleration. However, due to straggling, the longitudinal emittance grows. In order to cool longitudinally, one exchanges longitudinal and transverse emittances and proceeds to cool the transverse emittance. The status report~\cite{INTRO:ref5} outlines the details of the acceleration and collider ring for the Higgs factory. Table \ref{sum} gives a summary of the parameters of various muon colliders including three different modes of running the Higgs Collider that have varying beam momentum spreads. Additional information about the Muon Collider can be found in~\cite{gail,higgsreport}. \begin{table*}[thb!] \centering \caption[Baseline parameters for high- and low-energy muon colliders. ] {Baseline parameters for high- and low-energy muon colliders. Higgs/year assumes a cross section $\sigma=5\times 10^4$~fb; a Higgs width $\Gamma=2.7$~MeV; 1~year = $10^7$~s.} \label{sum} \begin{tabular}{|l|c|c|c|c|c|} \hline \rr CoM energy~ (TeV) &\rr 3 &\rr 0.4 & \multicolumn{3}{c|}{\rr 0.1 } \\ % & & & & & & \\ $p$ energy~(GeV) & 16 & 16 & \multicolumn{3}{c|}{16}\\ $p$'s/bunch & $2.5\times 10^{13}$ & $2.5\times 10^{13}$ & \multicolumn{3}{c|}{$5\times 10^{13}$ } \\ Bunches/fill & 4 & 4 & \multicolumn{3}{c|}{2 } \\ Rep.~rate~(Hz) & 15 & 15 & \multicolumn{3}{c|}{15 } \\ $p$ power~(MW) & 4 & 4 &\multicolumn{3}{c|}{4} \\ $\mu$/bunch & $2\times 10^{12}$ & $2\times10^{12}$ &\multicolumn{3}{c|}{$4\times 10^{12}$ } \\ \rr $\mu$ power~(MW) & \rr 28 &\rr 4 & \multicolumn{3}{c|}{\rr 1 } \\ \rr Wall power~(MW) & \rr 204 &\rr 120 & \multicolumn{3}{c|}{\rr 81 } \\ Collider circum.~(m) & 6000 & 1000 & \multicolumn{3}{c|}{350 }\\ Ave bending field~(T) & 5.2 & 4.7 &\multicolumn{3}{c|}{3 } \\ %Depth~ m & 500 & 100 & \multicolumn{3}{c}{10 } \\ \hline \rr Rms ${\Delta p/p}$~\% &\rr 0.16 &\rr 0.14 &\rr 0.12 &\rr 0.01&\rr 0.003 \\ \hline 6-D $\epsilon_{6,N}$~$(\pi \textrm{m})^3$&$1.7\times 10^{-10}$&$1.7\times 10^{-10}$&$1.7\times 10^{-10}$&$1.7\times 10^{-10}$&$1.7\times 10^{-10}$\\ Rms $\epsilon_n$~($\pi$ mm-mrad) & 50 & 50 & 85 & 195 & 290\\ $\beta^*$~(cm) & 0.3 & 2.6 & 4.1 & 9.4 & 14.1\\ $\sigma_z$~(cm) & 0.3 & 2.6 & 4.1 & 9.4 & 14.1 \\ $\sigma_r$spot~$(\mu$m) & 3.2 & 26 & 86 & 196 & 294\\ $\sigma_{\theta}$ IP~(mrad) & 1.1 & 1.0 & 2.1 & 2.1 & 2.1\\ Tune shift &0.044 &0.044 & 0.051 &0.022 & 0.015\\ $n_{\rm turns}$ (effective) & 785 & 700 & 450 & 450 & 450 \\ \hline \rr Luminosity~cm$^{-2}$s$^{-1}$&\rr $7\times 10^{34}$ & $10^{33}$ &\rr $1.2\times 10^{32}$ &\rr $2.2\times 10^{31}$&\rr $10^{31}$ \\ & & & & & \\ Higgs/year & & & $1.9\times 10^3$ & $4\times 10^3$ & $3.9\times 10^3$ \\ \hline \end{tabular} \end{table*} % \begin{figure*}[tbh!] \includegraphics[height=2.9in,width=5.7in]{fnalfg2.ps} \caption[Plan of a 0.1-TeV-CoM muon collider] {Plan of a 0.1-TeV-CoM muon collider.} \label{plan1} \end{figure*} % \subsection{Longitudinal Cooling} At the time of writing of the status report~\cite{INTRO:ref5} there was no satisfactory solution for the emittance exchange problem and this remained a major stumbling block towards realizing a muon collider. However, ring coolers have been found to hold significant promise in cooling in 6-dimensional phase space. Another advantage of ring coolers is that one can circulate the muons many turns, thereby reusing the cooling channel elements. Several meetings on emittance exchange were held~\cite{eemeets} and a successful workshop~\cite{eework} was held in 2001, where we explored in some depth several kinds of ring coolers. These options differ primarily in the type of focusing used to contain the beam. We describe the current status of our understanding of ring coolers here. % \subsubsection{Solenoidal Ring Coolers} The basic design of the solenoidal ring cooler~\cite{balb1} is presented in Figure~\ref{ring}. Eight focusing dipole magnets with an index $n=-\frac{1}{2}$ are used for bending and focusing of the beam. Each of these dipoles bends the beam through 45 degrees with a central orbit bending radius of 52 cm. We have done calculations to show that such dipoles are buildable. There are 4 long solenoids containing RF cavities and liquid hydrogen absorbers for transverse cooling. A magnetic field of 2.06~T on the edges of the solenoids provides the same transverse focusing as the bending magnets. The magnetic field adiabatically increases to 5.15 T towards the center of the solenoid in order to produce a small $\beta$ function (25-30 cm) at the absorbers. The short solenoids are designed to create an appropriate dispersion function that is zero at the long solenoids, which house the 200 MHz rf cavities. Their field is $\pm 2.06~T$ at the edges and $\pm 2.75$~T centrally. A symmetric field flip is required in the short solenoids to prevent the build up of canonical angular momentum. Lithium hydride wedge absorbers covering half of the vertical aperture at the centers of the short solenoids are used for emittance exchange to produce longitudinal cooling. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[tbh!] \vspace{0mm} \begin{minipage}[tbh!]{0.47\linewidth} \includegraphics[width=\linewidth]{02_ringA.eps} %\centerline{\epsfig{file=02_ringA.eps,width=\linewidth}} \end{minipage} \begin{minipage}[tbh!]{0.47\linewidth} \begin{tabular}{ll} Circumference & 36.963 m \\ Nominal energy & \\ at short SS & 250 MeV \\ Bending field & 1.453 T \\ Norm. field gradient & 0.5 \\ Max. solenoid field & 5.155 T \\ RF frequency & 205.69 MHz \\ Accelerating gradient & 15 MeV/m \\ Main absorber length & 128 cm \\ LiH wedge absorber & 14 cm \\ Grad. of energy loss & 0.75 MeV/cm \\ \end{tabular} \end{minipage} \caption{Layout and parameters of the solenoid based ring cooler \label{ring}} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[tbh!] \begin{minipage}[t!]{0.47\linewidth} \includegraphics[width=1.15\linewidth]{10_evol.eps} %\centerline{\epsfig{file=10_evol.eps,width=1.15\linewidth}} \end{minipage} \begin{minipage}[t!]{0.47\linewidth} \vspace{-5mm} \begin{tabular}{|l|c|c|c|} \hline Number of turns & 0 & 10 & 15 \\ \hline $x$ emittance (cm) & 1.2 & 0.24 & 0.21 \\ $y$ emittance (cm) & 1.2 & 0.24 & 0.21 \\ $z$ emittance (cm) & 1.5 & 0.79 & 0.63 \\ 6-D emitttance (cm$^3$) & 2.2 & 0.045 & 0.028 \\ 6-D cooling factor & 1 & 49 & 79 \\ Trans. w/o decay & 1 & 0.72 & 0.71 \\ Trans. with decay & 1 & 0.56 & 0.48 \\ Merit factor & 1 & 27 & 38 \\ \hline \end{tabular} \end{minipage} \vspace{-3mm} \caption{Evolution of the beam emittance/transmission at the ring cooler. \label{evol}} \vspace{-3mm} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Evolution of the beam emittance and transmission is shown in Figure~\ref{evol} as a function of the number of turns in the ring. In 15 turns, the transverse emittance decreases from 1.2 cm to 0.21 cm yielding a cooling factor of 5.7, the longitudinal emittance decreases from 1.5 cm to 0.63 cm (cooling factor 2.4), and the 6-D emittance decreases from 2.2 cm$^3$~to 0.028 cm$^3$, with an overall cooling factor $\approx$~79. The transmission is 0.71 without decay and 0.48 with decay. We define a merit factor for cooling that is the total transmission including decay times the 6-D cooling factor. The merit factor for this ring is then 38. This implies that transverse emittance at the ring cooler is about the same as at a linear SFOFO cooling channel employed in Study-II~\cite{EPP:studyii}, whereas the longitudinal emittance is noticeably less. This cooler provides mainly transverse cooling and can be used as a part of Neutrino Factory or a muon collider. A cooler specially designed for strong longitudinal cooling (``bunch compressor'') can also be created using a similar scheme. Such a compressor would be a part of a muon collider to shorten muon bunches from 6-8 m (minimal length after $\pi - \mu$~decay and phase rotation, see Ref.~\cite{INTRO:ref5}) to 0.6-0.8 m acceptable for further cooling by a 200 MHz channel. Two options for the bunch compressor are considered in Ref.~\cite{balb4}. The first one is a two-step cooler where each step is very similar to the ring cooler shown in Figure~\ref{ring}. The main difference is that the primary goal in the first cooler is the longitudinal bunching of the beam. This leads to a uniform magnetic field in the long solenoids and lower frequency/voltage of the accelerating rf system (15.6 MHz/4 MeV/m at the first stage vs. 62.5 MHz/8 MeV/m at the second one). Another option is a 15 MHz octagonal cooler composed of the same cells as in Figure~\ref{ring}, but with half the bending magnet angle. Decrease of longitudinal emittance from 43 cm to 2.5-3 cm, as required for muon collider, is obtained in both cases. We are proceeding with a realistic simulation of this system using Geant and ICOOL that employs realistic magnetic fields~\cite{kahn} produced by field calculation programs. After the two stage cooler, we still need a factor of $\approx$~30 in transverse cooling, but we are within a factor of 4 in longitudinal cooling relative to the Higgs factory goals. Lithium lens cooling, which with its strong focusing will cool transversely further while degrading longitudinally due to straggling, is a posibility and is being investigated. \subsubsection{RFOFO ring coolers} The cooling lattice for the Neutrino Factory (see section~\ref{neufact}) employs a configuration of fields known as an SFOFO lattice (super-FOFO) where the axial magnetic field profile changes polarity in alternate cells of the lattice. For the ring cooler design under consideration here, we employ an RFOFO lattice (regular-FOFO) where the axial field profile changes polarity in the middle of a cell. As a result all cells in an RFOFO lattice are identical. The ring cooler design employs a single cell for both transverse cooling and emittance exchange. It uses solenoids for focusing, giving large angular and momentum acceptances. The cell includes dispersion, acceleration, and energy loss in a single thick hydrogen wedge. Figure \ref{rforing} shows the layout of the cooling ring drawn to scale. The RFOFO lattice was chosen because, unlike in the SFOFO case used in Study-II, all cells are strictly identical, and the presence of an integer betatron resonance within the momentum acceptance is eliminated. \begin{figure}[htb!] \includegraphics[width=\linewidth]{rfofo_ring3.ps} \caption{Layout of an RFOFO cooling ring. \label{rforing}} \end{figure} The basic ring is made up of 12 identical 2.75-m long cells. In the figure, this symmetry is broken for injection and extraction, but the magnetic fields in this insertion are nearly identical to those in the rest of the ring. Figure \ref{cells} shows a detailed view of three cells of the lattice. \begin{figure}[tbh] \includegraphics {rfofo_h2reg1.ps} \caption{Three cells of the RFOFO lattice. \label{cells}} \end{figure} The longitudinal field on-axis has an approximately sinusoidal dependence on position. The actual coils to generate the axial fields, in the presence of the bending fields, would have to be slightly different from those used in the simulation, but since the 3D fields used are consistent with Maxwell's equations, there is no question that suitable coil positions can be found. The lattice transmits particles in the momentum band from 150 to 250 MeV/c. The average momentum for a small emittance beam varied from 191 to 201 MeV/c across each cell of the lattice. The minimum value of the beta function at the central momentum is 40 cm. Dispersion is provided by applying an approximately 0.125 T transverse bending field generated by alternately tilting the vertical plane of the solenoids by 1.5 degrees. There is no attempt to control the field index $n$ (where $B\propto r^n$). So the focusing in x and y are not identical. It is found that the acceptance is reduced as the bending field is increased. We thus use a wedge with maximum possible angle (giving zero thickness on one side), and the lowest bending field consistent with adequate emittance exchange. The dispersion at the absorber of -8 cm in a direction 30 degrees from the $y$ axis, The dispersion at the center of the rf is of the opposite sign, and also mostly in the $y$ direction. Its direction is Larmor rotated by the axial fields. The liquid-hydrogen wedge has a central thickness of 28.6 cm and a total wedge angle of 76.93 degrees and is rotated 30 degrees from the vertical. No absorber windows are included in this simulation. The RF cavities had a frequency of 201.25 MHz and a gradient of 16 MV/m. No RF windows were included. The ICOOL simulations shown do not include the injection/extraction insertion, and use axial and transverse magnetic fields generated by a truncated Fourier decomposition of the fields from a straight solenoid lattice. The RF is represented as fields in perfect pillbox cavities. The input tracks are taken from a Study-II simulation, using distributions from just upstream of the transverse cooling system. The use of Study-II simulated distributions is intended to allow a more realistic estimate of the ring's performance. No attempt was made to match the ring dispersion or slight differences in the transverse beta functions. Figure~\ref{all} shows the transmission, transverse emittance (in $x$, $y$), longitudinal emittance, 6-dimensional emittance, and a merit factor $M$ vs. length in the ring. $M$ is given by: $$M~=~{\epsilon_6(initial) \over \epsilon_6(final)}~\times~{\rm Transmission}$$ \begin{figure}[tbh] \includegraphics{rfofo_all.ps} %\vskip.5in \caption{Transmission, normalized transverse emittance, normalized longitudinal emittance, normalized 6-dimensional emittance, and the merit factor, as a function of distance. \label{all}} \end{figure} Initially, the $x$ emittance falls more rapidly than the $y$. This is expected because it is the $y$ emittance that is exchanged with the longitudinal emittance, but the Larmor rotations soon mix the $x$ and $y$ emittances bringing them to a common value. After a distance of 400 m ($\approx$~12 turns), the 6-dimensional emittance has fallen by a factor of 290, with a transmission of 44 \% (61\% without decay). The merit factor is 130. The same factor for the Study-II cooling lattice, with no windows, is 13. With realistic windows and the injection/extraction insertion added, the merit factor will be much less than 130, but is likely to remain far better than the Study-II example. This ring could not be used, as is, to replace the Study-II cooling channel because the bunch train in this case is far too long to fit in the ring. But a spiral 3D cooling channel could be used and an even greater performance gain could be expected if the spiral were also tapered. This approach seems very attractive, but it is still far from fully realistic, and much work needs to be done. \subsubsection{Quadrupole Ring Coolers} Another type of ring cooler has been studied that uses quadrupole focusing~\cite{ucla}. The SYNCH storage ring design program code~\cite{synch} was used to design this ring, which uses conventional magnet elements. Figure~\ref{fig:ringucla} shows such a ring that has 16 cells. Elements in a half lattice cell are shown schematically in Figure~\ref{fig:half}. \begin{figure}[tbh!] %\includegraphics[width=8cm]{fig1} \includegraphics[width=8cm]{ucla_ring} \caption{\label{fig:ringucla} Top view of the 16 cell muon cooling ring.} \end{figure} % \begin{figure}[tbh!] %includegraphics[height=6cm,width=8cm]{fig2} \includegraphics[height=6cm,width=8cm]{ucla_1125deg} \caption{\label{fig:half} Schematic diagram of half of the 22.5 degree bending cell. A wedge absorber is located in the middle of the cell.} \end{figure} % The SYNCH design parameters were then transferred to the ICOOL~\cite{icool} ray tracing code that simulates ionization cooling. Figure~\ref{fig:cell} shows the $\beta_{x}$, $\beta_{y}$, and $D$(dispersion) from SYNCH as a function of arc length in a lattice cell. Superimposed are the same quantites derived from beam behavior in the ICOOL simulation, showing consistency between the two programs. \begin{figure}[tbh!] %\includegraphics[height=5cm,width=8cm]{fig3} \includegraphics[height=5cm,width=8cm]{ucla_16cell} \caption{\label{fig:cell}The $\beta_{x}$, $\beta_{y}$, and $D$(dispersion) in a 22.5 degree bending cell. SYNCH input(solid curves) and ICOOL simulation(marked points) are compared.} \end{figure} Figure~\ref{fig:figaa} shows the transverse and longitudinal normalized emittances as a function of the number of turns. The average muon beam momentum is 500 MeV/c, and liquid H$_2$ absorbers with wedge opening angles of 40 degrees are used. The path length of the central trajectory through the liquid H$_2$ wedge absorbers is 25 cm per 22.5 degree bending cell. Average energy loss in the wedge absorbers is compensated in the 201 MHz rf cavities. The equilibrium normalized emittances are about 1 mm$\cdot$rad in $x$ and $y$, and around 10 mm in $z$. \begin{figure}[tbh!] \includegraphics[height=5cm,width=8cm]{ucla_turns} %\includegraphics[height=5cm,width=8cm]{figa} \caption{\label{fig:figaa} The evolution of x, y, z normalized emittances in 30 full turns.} \vskip0.5cm \includegraphics[height=5cm,width=8cm]{ucla_trans} %\includegraphics[height=5cm,width=8cm]{figb} \caption{\label{fig:figbb}The transmission and the figure of merit factor as a function of s ~in 16 full turns.} \end{figure} Figure~\ref{fig:figbb} shows the muon transmission efficiency and the merit factor, as a function of the number of turns. The merit factor reaches $\approx$~5 after 16 turns, where the transmission efficiency is $\approx$~0.55. At 500 MeV/c, the fraction of muons lost due to muon decay in 16 full turns is 0.57, yielding an overall transmission factor of 0.31. %\begin{figure}[tbh!] %\includegraphics[height=5cm,width=8cm]{figb} %\caption{\label{fig:figbb}The transmission and the figure of merit factor %as a function of s ~in 16 full turns.} %\end{figure} % \subsubsection{Injection into Ring Coolers} % The most serious technical problem facing the ring cooler approach is the injection system which may require a very powerful kicker magnet~\cite{balb5}. The energy stored in the injection kicker goes as the square of the emittance of the beam and inversely as the circumference of the ring. A promising injection scheme that does not use kicker magnets, but instead uses absorbers to degrade the beam energy and RF phase manipulations has been proposed~\cite{balb6} and is being studied. % \subsection{Higher Energy Muon colliders} Once the cooling problems have been solved for the design of the first muon collider, acceleration to higher energies becomes possible. Colliders with 4 TeV center of mass energy have been studied~\cite{INTRO:ref5} and Table \ref{dntable} lists the parameters for such a collider. The radiation from the neutrinos from the muon decay begins to become a problem at CoM energies of 3 TeV. One may attempt to solve this by a number of means, including optical stochastic cooling of muons in the collider, thus permitting the same luminosity with less intensity. \begin{table*}[tbh!] \caption[Parameters of Acceleration for a 4~TeV Muon Collider] {Parameters of Acceleration for a 4~TeV Muon Collider.} \label{dntable} \begin{tabular}{|l|c|c|c|c|c|} \hline & Linac & RLA1 & RLA2 & RCS1 & RCS2 \\ \hline E (GeV) & 0.1$\rightarrow$ 1.5 & 1.5 $\rightarrow$ 10 & 10 $\rightarrow$ 70 & 70 $\rightarrow$ 250 & 250 $\rightarrow$ 2000 \\ f$_{rf}$ (MHz) & 30 $\rightarrow$ 100 & 200 & 400 & 800 & 1300 \\ N$_{turns}$ & 1 & 9 & 11 & 33 & 45 \\ V$_{rf}$(GV/turn) & 1.5 & 1.0 & 6 & 6.5 & 42 \\ C$_{turn}$(km) & 0.3 & 0.16 & 1.1 & 2.0 & 11.5 \\ Beam time (ms) & 0.0013 & 0.005 & 0.04 & 0.22 & 1.73 \\ $\sigma_{z,beam}$(cm) & 50 $\rightarrow$ 8 & 4 $\rightarrow$ 1.7 & 1.7 $\rightarrow$ 0.5 & 0.5 $\rightarrow$ 0.25 & 0.25 $\rightarrow$ 0.12 \\ $\sigma_{E,beam}$(GeV) & 0.005 $\rightarrow$ 0.033 & 0.067 $\rightarrow$ 0.16 & 0.16 $\rightarrow$ 0.58 & 0.58 $\rightarrow$ 1.14 & 1.14 $\rightarrow$ 2.3 \\ Loss (\%) & 5 & 7 & 6 & 7 & 10 \\ \hline \end{tabular} \end{table*} \subsection{Muon Collider Detectors} % Figure~\ref{geant} shows a strawman muon collider detector for a Higgs factory simulated in Geant. The background from muon decay sources has been extensively studied~\cite{INTRO:ref5}. At the Higgs factory, the main sources of background are from photons generated by the showering of muon decay electrons. At the higher energy colliders, Bethe-Heitler muons produced in electron showers become a problem. Work was done to optimize the shielding by using specially shaped tungsten cones~\cite{INTRO:ref5}. The background rates obtained were shown to be similar to those predicted for the LHC experiments. It still needs to be established whether pattern recognition is possible in the presence of these backgrounds. % \begin{figure}[bth!] \centerline{\includegraphics[width=0.5\linewidth]{mu_geant_cut.eps}} \caption[Strawman Geant detector for a muon collider] {Cut view of a strawman detector in Geant for the Higgs factory with a Higgs$\rightarrow b\bar b$ event superimposed. No backgrounds shown. The tungsten cones on either side of the interaction region mask out a 20~$\deg$ area.} \label{geant} \end{figure} %