%Written by C. Johnstone
%Written by Bill Ng in collaboration with D. Trbojevic
%additional editing by B. Ng (9/98)
%editing by GRF
%additions from Ng, edited by JCG.
\section{COLLIDER STORAGE RING}
\subsection{Introduction}
After acceleration, both $\mu^+$
and $\mu^-$ bunches are injected into a separate storage ring. The highest
possible average bending field is desirable, to maximize the number of
revolutions before decay, and thus maximize the luminosity. Collisions would
occur in one, or perhaps two, low $\beta^*$ interaction areas. Parameters
of the rings are given in table~\ref{sum}.
\subsection{Lattice design}
In order to
maintain the required short bunches without excessive rf, the ring must be approximately, or fully,
isochronous.
The required $\beta$ functions at the interaction point are small for a 3~TeV collider, $\beta^*=3\,{\rm mm}$, and the quadrupoles needed to generate them are large (20-30~cm diameter). At 100~GeV CoM, the $\beta$s at the IP are more than an order of magnitude larger (see table~\ref{sum}), and the quadrupoles are more conventional. In designs of e$^+$e$^-$ colliders, it has been found that local chromatic correction is essential \cite{chromatic,chromatica,chromaticb,chromaticc} for final focus requirements similar to the cases we consider here.
The rings are
racetracks, with two circular arcs
separated by an experimental insertion on one side, and a utility insertion
for injection, extraction, and beam scraping on the other.
The experimental insertion includes the interaction region (IR)
followed by a local chromatic correction section (CCS) and a matching section. The chromatic correction section is optimized to correct the ring's linear chromaticity, which is almost completely generated by the IR.
The superconducting bending magnets must be shielded from the electrons emitted by decay of the muon beam. Fields of 8~T have been assumed in the 100~GeV lattices described below, but higher field dipoles would reduce the ring diameters and increase luminosity. Studies of higher field dipoles have been started.
The rf requirements depend on the lattice momentum compaction and bunch parameters. For the very low momentum spread Higgs collider parameters, synchrotron motion does not occur and rf is used solely to correct the impedance generated momentum spread in the bunch. For the higher momentum spread cases, there are two options. If the momentum compaction can be corrected to high order, then synchrotron motion can still be eliminated and the rf is again only used for energy spread correction. Alternatively, if some
momentum compaction is retained, then significant rf is needed to maintain the specified short bunches.
In either case rf quadrupoles will be required to generate BNS \cite{refbns,ref36} damping of the transverse head-tail instability.
\subsection{Lattices}
Preliminary lattices have been designed for both 4~TeV and 0.5~TeV machines \cite{ref33}, and several designs now exist for the 100~GeV case.
For
the 100~GeV CoM collider, two operating cases are being considered: a high-luminosity case with
broad momentum acceptance
to accommodate a beam with a $\delta p/p$ of $\pm 0.12\%$ (rms), and one with a much narrower momentum acceptance
and lower luminosity for beam with $\delta p/p$ of $\pm 0.003\%$ (rms).
For the broad momentum acceptance case, $\beta^*$ must be 4~cm
and for the narrow momentum acceptance case, 13~cm.
In either case, the bunch
length must be held comparable to the value of $\beta^{*}$.
It is assumed that a single lattice can be designed that can operate in either mode by adjusting settings.
We describe below two examples of 100 GeV designs. They were generated by two different semi-independent, but converging, studies. The primary difference between the two is in the flexible momentum compaction arc modules.
\subsubsection{100~GeV CoM first example}
The first lattice design, denoted as Example~(a), shown in Fig.~\ref{50_gev_hlfring}, has a total circumference of about $350\,{\rm m}$ with
arc modules accounting for only about a quarter of the ring circumference.
\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_hlfring.ps}}
\centerline{
\epsfig{figure=cfg1_new.ps,height=4.0in,width=4.5in,bbllx=30bp,bblly=116bp,bburx=545bp,bbury=675bp,clip=}
}
\caption{Example~(a):
Half of the IR, local chromatic correction, and one of three arc modules.}
\label{50_gev_hlfring}
\end{figure*}
The low beta function values at the IP
are mainly produced by three strong superconducting quadrupoles
in the Final Focus Telescope (FFT) with pole-tip fields
of 8~T. The full interaction region is symmetric with reflection about the interaction point (IP).
Because of significant, large-angle backgrounds from muon decay,
a background-sweep
dipole must be included in the final focus telescope
and placed near the IP to protect the detector and low $\beta$ quadrupoles \cite{carnik96}.
It was found
that 2.5~m of an 8~T field is required to attain sufficient background
suppression.
The first quadrupole is located 5~m away from the interaction point, and the value of the beta function reaches a maximum of $1.5\,{\rm km}$ in the final focus telescope
when the beta functions are equalized
in both planes. For this value of beta, quadrupole apertures must be at least 15~cm in
radius to accommodate $5\,{\rm \sigma}$ of a $90~\pi$~mm-mrad,
50-GeV muon beam (normalized rms emittance) plus a 2 to 3~cm thick
tungsten liner \cite{scraping}. The natural chromaticity
of this interaction region is about $-60.$
Local chromatic correction of all muon collider interaction regions is
required to achieve broad momentum acceptance.
The basic approach developed by Brown \cite{chromatica} and others \cite{donald},
is used. Two pairs
of sextupoles, one pair in each
plane, are separated by a phase advance of $\pi$, all in high dispersion, and each being
$(2n+1){\pi\over 2}$ from the IP. The geometric aberrations of each sextupole are canceled by its companion while the chromaticity corrections add.
In our case, this chromatic correction section design (Fig.~\ref{50_gev_ccs}) was optimized to be as short as possible.
The $\beta_{\textrm{max}}$
is only $100~\textrm{m}$ and the $\beta_{\textrm{min}}=0.7~\textrm{m}$, giving
a $\beta_{\textrm{ratio}}$ between planes of
about 150 without compromising aperture
by a large amplitude dependent tuneshift.
The sextupoles are centered
about the opposite plane minimum ($\beta_{min}<1$),
which
provides a chromatic correction
with minimal cross correlation between planes.
A further advantage to locating the opposite plane
minimum at the center of the sextupole is that this point is
${\pi\over 2}$ away from, or ``out of phase" with the chromatic effects
from the final focus quadrupoles; \ie the plane not being chromatically corrected is
treated like the IP in terms of phase
to eliminate a second order chromatic aberration arising
from the sextupole.
The large beta ratio combined with the opposite plane phasing
allows sextupoles
for opposite planes to be interleaved
without significantly impacting lattice nonlinearity.
In point of fact, interleaving improved lattice performance
over a non-interleaved correction scheme, primarily due to a shortening
of the chromatic correction section, which is accompanied by a lowering of its contribution to the overall chromaticity \cite{wan1}.
Shallower minima and less variation in beta
were also applied to soften chromatic aberrations, but at
the expense of an exact $\pi$ phase advance separation between sextupole partners.
The retention of an exact $\pi$ phase advance difference between sextupoles
was found
to be less important to the dynamic aperture than elimination of
minima with $\beta_{\textrm{min}}<0.5~\textrm{m}$.
\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_ccs.ps}}
\centerline{
\epsfig{figure=cfg2_new.ps,height=4.0in,width=4.5in,bbllx=60bp,bblly=110bp,bburx=545bp,bbury=685bp,clip=}
}
\caption{Example~(a): The Chromatic Correction Module.}
\label{50_gev_ccs}
\end{figure*}
The momentum compaction of the IR, chromatic correction section, and matching sections
is about $0.04$. Since the length of this part is $173\,{\rm m}$,
the 93~m arc length must have a negative momentum compaction of
about $-0.09$ in order to offset the positive portion of the ring.
The arc module is shown in Fig.~\ref{50_gev_arc}. It has the small beta functions characteristic
of FODO cells, yet a large, almost separate, variability
in the momentum compaction of the module which is a characteristic
associated with the flexible momentum compaction module \cite{ref32}.
The small beta functions are achieved through the
use of a doublet focusing structure which produces
a low beta simultaneously in both planes.
At the dual minima, a strong focusing quadrupole
is placed to control the derivative of dispersion with
little impact on the beta functions.
%(The center
%defocusing quadrupole is used only to clip the
%point of highest dispersion.)
%Ultimately a dispersion derivative can be generated which is
%negative enough to drive the dispersion negative
%through the doublet and the intervening waist.
Negative values of momentum compaction as low as
$\alpha=-0.13$ have been achieved, and $\gamma_t=2~i$, has been achieved
with modest values of the beta function.
This arc module was able to generate the needed negative
momentum compaction with beta functions of $40\,{\rm m}$ or less.
The peak dispersion is somewhat high, $4.6\,{\rm m},$ and needs to be reduced.
\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_arc.ps}}
\centerline{
\epsfig{figure=cfg3_new.ps,height=4.0in,width=4.5in,bbllx=48bp,bblly=115bp,bburx=545bp,bbury=695bp,clip=}
}
\caption{Example~(a): A Flexible Momentum Compaction Arc Module.}
\label{50_gev_arc}
\end{figure*}
%
\begin{figure*}[tbh!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_ap.ps}}
\centerline{
\epsfig{figure=carolfg4_april9.ps,height=3.0in,width=3.5in,bbllx=50bp,bblly=347bp,bburx=579bp,bbury=732bp,clip=}
}
\caption{Example~(a): A preliminary dynamic aperture.}
\label{50_gev_ap}
\end{figure*}
A very preliminary calculation of the dynamic aperture \cite{wan} without optimization
of the lattice nor inclusion of errors and end effects
is given in Fig.~\ref{50_gev_ap}. For the very low momentum spread operation, a larger acceptance is required. This should be possible by turning the sextupoles off. The result of doing this is also shown
in the figure (dashed line). The momentum acceptance is lost, as expected, but the gain in transverse acceptance, though real, is disappointing. More work is needed.
\subsubsection{100~GeV CoM second example}
%\mbox{~~}\\[-0.95in]
The second lattice design, Example~(b),
is shown in Fig.~\ref{f1} starting from the IP.
\begin{figure*}[bht!]
%\vspace{-0.15in}
\centering{\psfig{figure=figjuan1.ps,height=3.5in,bbllx=72bp,bblly=342bp,bburx=530bp,bbury=658bp,clip=}}
\scriptsize
\parbox{3.5in}{\vspace{-0.05in}
Dispersion max/min= $4.31345/-3.49914$~m,\hspace{0.1in}
$\gamma_t= ( 5.52, 0.00)$\\
$\beta_x$ max/min= 1571.74/0.0400~m, $\nu_x= 1.55305$, $\xi_x= -41.46$,
Module length= 85.3238~m\\
$\beta_y$ max/min= 1550.94/0.0400~m, $\nu_y=2.12758$, $\xi_y=-39.90$,
Total bend angle= 1.30703508~rad
\vspace{0.05in}}
\small
\caption{Example~(b): The lattice structure of the IR including local chromaticity corrections.}
\label{f1}
%\vspace{-.15in}
\end{figure*}
The 1.5~m background clearing dipole is 2.5~m away from the IP and is
followed by the triplet
quadrupoles with the focusing quadrupole in the center.
The interaction region (IR) stops at
about 24~m from the IP and the local correction
section on each side of the IP spans
a distance of roughly 61.3~m. The Twiss properties at the centers of
the four correction sextupoles are
listed in table~\ref{t1}, where all the figures given by the lattice code
are displayed.
\begin{table*}[bht!]
%\vskip -0.05in
\caption{Twiss properties of the IR correction sextupoles.}
\label{t1}
\begin{tabular}{ccccddc}
%\vspace{-0.25in}
%\mbox{~~~~}&&&&&&\\
&Distance & \multicolumn{2}{c}{~~~Phase Advances} &
\multicolumn{2}{c}{~~~~~Betatron Functions (m)} & Dispersion \\
&(m) &$\nu_x$ &$\nu_y$ &~~~$\beta_x$ &~~~$\beta_y$ &(m)\rule[-0.08in]{0in}{-0.08in} \\
\tableline
SX2 & 33.5061 & 0.48826 & 0.74953 & 1.00000 & 100.00012 & 2.37647\\
SX2 & 62.3942 & 0.98707 & 1.24953 & 1.00000 & 100.00009 & 2.37651\\
SX1 & 49.3327 & 0.74892 & 0.87703 & 100.00023 & 1.00000 & 2.66039\\
SX1 & 74.6074 & 1.24892 & 1.47987 & 99.99967 & 0.99992 & 2.65817\\
\end{tabular}
%\vskip -.22in
\end{table*}
The SX1's are the two horizontal
correction sextupoles. They are placed at positions with the
same betatron functions and dispersion function, and are
separated horizontally and vertically by phase advances of $\pi$
so that their nonlinear effect
will be confined in the region between the two sextupoles. Their horizontal
phase advances are also
integral numbers of $\pi$ from the triplet focusing F-quadrupole
so that the chromaticity compensation for that quadrupole will be most
efficient \cite{donald}.
The SX2's are the
two vertical correction sextupoles and similarly they should also be placed at designated locations. In general, it will be difficult to satisfy all these requirements; in addition, luminosity arguments limits the lattice size. As a consequence, the phase advance difference in the vertical plane of the SX1's is 0.6 instead of 0.5.
Flexible momentum compaction (FMC) modules \cite{ref32a}
are used in the arc. The momentum compaction of the arc has to be
made negative in order
to cancel the positive momentum compaction of the IR, so that the whole ring
becomes quasi-isochronous.
This is accomplished in three ways:
by removing the central dipole of the usual FMC
module; by increasing the length of the first and last dipoles; and by
increasing the negative dispersion at the entrance.
Two such modules will be
required for half of the
collider ring, one of which is shown in Fig.~\ref{f2}.
To close the ring
geometrically, there will be a $\sim72.0$~m straight section between the
two sets of FMC modules. The total length of this compact collider ring is $C=354.3$~m.
%This is a nice feature, since a small ring allows larger number of
%collisions before the muons decay appreciably.
Note that the IR and local
correction sections take up 48.2\% of the whole ring.
The momentum compaction factor
of this ring is now $\alpha_0=-2.77\times10^{-4}$.
The rf voltage required to maintain a bunch with rms length
$\sigma_{\rule[0.05in]{0in}{0.05in}\ell}$
and rms momentum spread $\sigma_{\rule[0.05in]{0in}{0.05in}\delta}$ is
%$V_{\rm rf}=(C\sigma_\delta/\sigma_\ell)^2[|\eta|E/(2\pi h)]$,
$V_{\rm rf}=|\eta|EC^2\sigma_\delta^2/(2\pi h\sigma_\ell^2)$,
where $\eta$ is the slippage factor and $E$ the muon energy.
On the other hand,
if the bucket height
is taken as $k$ times the rms momentum spread of the bunch,
the rf harmonic is given by
$h=C/(k\pi\sigma_{\rule[0.05in]{0in}{0.05in}\ell})$. Thus, for
$\sigma_{\rule[0.05in]{0in}{0.05in}\ell}=4$~cm and
$\sigma_{\rule[0.05in]{0in}{0.05in}\delta}=0.0012$ (see table~\ref{sum}), this lattice requires
an rf voltage
of $V_{\rm rf}\approx 88$~kV. Since $\alpha_0$ is
negative already, its absolute value can be further lowered easily
if needed. However,
we must make sure that the contributions from the higher order momentum
compaction are small also.
\begin{figure*}[hbt!]
%\vspace{-0.15in}
\centering{\psfig{figure=figjuan2.ps,height=3.2in,bbllx=72bp,bblly=342bp,bburx=530bp,bbury=658bp,clip=}}
\scriptsize
\parbox{3.5in}{\vspace{-0.05in}
Dispersion max/min= 1.35084/$-3.50000$~m,\hspace{0.1in}
$\gamma_t= ( 0.00, 4.43)$\\
$\beta_x$ max/min= 19.57/0.28915~m, $\nu_x= 0.75506$, $\xi_x= -1.77$,
Module length= 27.9062~m\\
$\beta_y$ max/min= 23.63/7.80157~m, $\nu_y= 0.36290$, $\xi_y=-0.92$,
Total bend angle= 0.9170180~rad
\vspace{0.05in}}
\caption{Example~(b): Lattice structure of the flexible momentum-compaction module.}
\label{f2}
\end{figure*}
The dynamical aperture of the lattice is computed by tracking particles with
the code COSY \cite{cosy}.
Initially 16 particles with the same momentum offset and
having vanishing $x'$ and $y'$ are placed uniformly on a circle in
the $x$-$y$ plane.
The largest radius that provides survival of the 16 particles
in 1000 turns is defined here as the dynamical aperture at this momentum
offset and is plotted in solid in Fig.~\ref{f3} (left plot) in units of the rms radius of the beam.
(At the 4~cm low beta IP,
the beam has an rms radius of $82~\mu$m.) As a reference,
the 7-sigma aperture
spanning $\pm6$~sigmas of momentum offset is also displayed as a
semi-ellipse in \textcolor{red}{dashdot}.
To maximize the aperture, the tunes must be chosen to avoid
parametric resonances. The on-momentum
amplitude dependent horizontal and vertical tunes
are
\begin{eqnarray}
\nu_x\!&=\! 8.126337\! -\!100~\epsilon_x \! -\! 4140~\epsilon_y,\\
\nu_y\!&=\!6.239988\!-\! 4140~\epsilon_x\! -\! 54.6~\epsilon_y,
\end{eqnarray}
where $\epsilon_x$ and $\epsilon_y$ are the horizontal and vertical
unnormalized emittance in $\pi$~m. With the designed rms
$\epsilon_x= \epsilon_y=0.169\times10^{-6}~\pi$~m,
the on-momentum tune variations are at most 0.0007. Next, the chromaticity variations with momentum must be as small as
possible. This is shown in Fig.~\ref{f3} (right plot). Note that there are no families
of sextupoles to correct for the higher order chromaticities in this
small ring with only four FMC modules.
For momentum spread varying from $-1$ to 0.9\%,
$\nu_x$ varies from 8.16698 to 8.07459, and $\nu_y$ from 6.28305 to 6.22369
for the center of the beam.
During aperture tracking, we notice that
particle loss occurs mostly in the horizontal direction.
We are convinced that the small momentum aperture is a result of the
large dispersion swing in the lattice
from $+4.5$ to $-3.5$~m.
For example, $4.5$~m dispersion and 0.6\%
momentum offset translates into a 2.7~cm off-axis motion.
The nonlinearity of the lattice will therefore diminish the dynamical
aperture. A resonant strength study using, for example, swamp plots and
normalized resonance basis coefficient analysis \cite{yan} actually reveals that
this lattice and some of its variations are unusually nonlinear.
Recently, we made modifications to the FMC arc modules resulting in an smaller
dispersion swing from $-2.6$ to $+2.0$~m. The IR
was not changed except for matching to the arc modules.
The aperture has been tracked with TEAPOT \cite{teapot}
in the same way as COSY
and is plotted as \textcolor{magenta}{dashes} in Fig.~\ref{f3} (left plot). We see that the momentum
aperture has widened considerably. The near on momentum dynamical aperture,
however, is one sigma less than the lattice presented here.
Nevertheless, it is not clear that this decrease
is significant because all
trackings have been performed in steps of one sigma only.
However this type of aperture is still far from satisfactory, because
so far we have been studying
a bare lattice. The aperture will be reduced
when fringe fields, field errors, and misalignment
errors are included.
It has been argued that the small momentum spread aperture is limited by
the dramatic changes in betatron functions near the IP \cite{ohnuma}.
These changes
are so large that Hill's equation is no longer adequate and the
exact equation for beam transport must be used. This equation brings in
nonlinearity and limits the aperture, which can easily be demonstrated
by turning off all the sextupoles.
In other words, although the momentum
aperture can be widened by suitable deployment of sextupoles, the
on-momentum dynamical aperture is determined by the triplet quadrupoles and
cannot be increased significantly by the sextupoles.
Some drastic changes in the low beta design may be necessary.
%\begin{figure*}[hbt!]
%\vspace{-0.02in}
%\raggedright{\psfig{figure=figjuan3.ps,height=2.5in,width=2.85in,bbllx=80bp,bblly=275bp,bburx=515bp,bbury=625bp,clip=}}
%\vskip -2.5in
%\raggedleft{\epsfig{figure=figjuan4.ps,height=2.5in,width=3.20in,bbllx=32bp,bblly=275bp,bburx=505bp,bbury=625bp,clip=}}
%\vskip -.05in
%\parbox{5.5in}
%{\vspace{-0.6in}\hspace{1.1in}(a)\hspace{2.8in}(b)\vspace{-0.45in}}
%\small
%\caption{Example~(b):(a) Dynamical aperture of the lattice versus momentum offset.
%(b) Chromaticities versus momentum offset.}
%\label{f3}
%\end{figure*}
\begin{figure*}[hbt!]
\dofigs{3.25in}{figjuan3.ps}{3.25in}{figjuan4.ps}
%\small
\caption[Example~(b): Dynamical aperture and chromaticities {\it vs.} momentum offset]{Example~(b): Left-hand-side plot is dynamical aperture of the lattice {\it vs.} momentum offset. COSY calculation in \textcolor{blue}{solid}, $7~\sigma$ in \textcolor{red}{dot-dashes}, and TEAPOT calculation with modified FMC modules in \textcolor{magenta}{dashes}. Right-hand-side plot is chromaticities {\it vs.} momentum offset.}
\label{f3}
%\vspace{-.5in}
\end{figure*}
\subsection{Scraping}
It has been shown \cite{ref42} that detector backgrounds
originating from beam halo
can exceed those from decays in the vicinity
of the interaction point (IP). Only with a dedicated beam cleaning system
far enough from the IP can one mitigate this problem \cite{scraping}.
Muons injected with large momentum errors or betatron oscillations will be lost
within the first few turns. After that, with active scraping,
the beam halo generated through
beam-gas scattering, resonances and beam-beam
interactions at the IP
reaches equilibrium and beam losses remain constant throughout the
rest of the cycle.
Two beam cleaning schemes have been designed \cite{scraping}, one for muon colliders at
high energies, and one for those at low energies.
The studies \cite{scraping} showed that no absorber, ordinary or
magnetized, will suffice for beam cleaning at 2~TeV;
in fact the disturbed
muons are often lost in the IR, but a simple metal collimator was found to be satisfactory at 100 GeV.
\subsubsection{Scraping for high energy collider}
At high energies, a 3-m long electrostatic deflector (Fig.~\ref{scrap1}) separates
muons with amplitudes larger than 3$\sigma$ and deflects them into
a 3-m long Lambertson magnet, which extracts these downwards through a deflection
of 17~mrad. A vertical septum magnet is used in the vertical scraping section
instead of the Lambertson to keep the direction of extracted beam down.
The shaving process lasts for the first few turns.
To achieve practical distances and design apertures for the separator/Lambertson
combinations,
$\beta$-functions must reach a kilometer in the 2-TeV case, but only 100~m at 50 GeV.
The complete system consists of a vertical scraping
section and two horizontal ones for positive and negative momentum
scraping (the design is symmetric about the center, so
scraping is identical for both $\mu^+$ and $\mu^-$). The system provides
a scraping power of a factor of 1000; that is, for every 1000 halo muons, we retain 1.
\subsubsection{Scraping for low energy collider}
At 50~GeV,
collimating muon halos with a 5-m long
steel absorber (Fig.~\ref{scrap2}) in a simple compact utility section
does an excellent job. Muons lose a significant fraction of their energy in
such an absorber (8\% on average) and have broad angular and spatial distributions.
Almost all of these muons are then lost in the first 50-100~m downstream
of the absorber with only 0.07\% of the scraped muons reaching
the low $\beta$ quadrupoles in the IR, \ie a scraping power is 1500 in
this case, which is significantly better than with in an earlier septum scraping system design \cite{scraping} similar to that developed for the high energy collider.
\begin{figure*}[thb!]
\centering{{\epsfig{figure=scrap1.eps,width=1.6in,angle=270}}}
\vspace{10pt}
\caption{Schematic view of a \mumu collider beam halo extraction.}
\label{scrap1}
\end{figure*}
\begin{figure*}[thb!]
\centering{{\epsfig{figure=scrap2.eps,width=1.7in,angle=270}}}
\vspace{10pt}
\caption{Scraping muon beam halo with a 5-m steel absorber.}
\label{scrap2}
\end{figure*}
\subsection{Beam-beam tune shift}
Several studies have considered beam emittance growth due to the beam-beam tune shift and none have observed significant luminosity loss. For instance, a study \cite{furman}, using the high energy collider parameters (see table~\ref{sum}), in which particles were tracked
assuming Gaussian beam field distributions, and no muon decay,
%(see figure\ref{snowmass8.17})
showed a luminosity loss of only 4\%. With muon decay included, the loss contribution from beam-beam effects is even less. Another study \cite{chen} using a particle in cell approach with no assumptions about field symmetry obtained a similar result. Collisions between beams displaced by 10\% of their radius also gave little loss. But all these studies assumed an ideal lattice, and none considered whether small losses due to nonlinearities give rise to an
%a very small fraction of the beam may be lost, giving probably
unacceptable background.
%\begin{figure*}[thb!]
%\vspace{10pt}
%\centerline{\epsfig{file={miguel_fig1.ps},height=4in,width=4in}}
%\centering{{\epsfig{figure=snowmass817}}}
%\caption{Luminosity as a function of turn number assuming stable muons}
%\label{snowmass8.17}
%\end{figure*}
\subsection{Impedance/wakefield considerations}
A study \cite{ref35} has examined the resistive wall impedance longitudinal
instabilities in rings at several energies. At the higher energies and larger
momentum spreads, solutions were found with small but finite momentum
compaction and moderate rf voltages. For
the special case of the Higgs Factory, with its very low momentum spread, a
solution was found with no synchrotron motion, but rf was provided to correct
the first order impedance generated momentum spread. The remaining
off-momentum tails which might generate background could be removed by a
higher harmonic rf correction without affecting luminosity.
Solutions to the higher energy and larger momentum spread cases without synchrotron motion are also being considered.
Given the very slow, or nonexistent synchrotron oscillations, the transverse beam breakup instability is significant. This instability can
be stabilized using rf quadrupole \cite{ref36} induced BNS damping.
For instance, The required tune shift with position in the bunch, calculated using the two particle model approximation \cite{ref38}, is only $1.58\times10^{-4}$ for the 3~TeV case using a 1~cm radius aluminum pipe. This stabilizes the resistive wall instability. However, this application of BNS damping to a quasi-isochronous ring, and other head-tail instabilities due to the chromaticities $\xi$ and $\eta_1$, needs more study.
\subsection{Bending magnet design}
The dipole field assumed in the 100~GeV collider lattices described above was 8~T. This field can be obtained using $1.8^o$ niobium titanium (NbTi) cosine theta superconducting magnets similar to those developed for the LHC. The only complication is the need for a tungsten shield between the beam and coils to shield the latter from beam decay heating.
The $\mu$'s decay within
the rings ($\mu^- \rightarrow\ e^-\overline{\nu_e}\nu_{\mu}$), producing
electrons whose mean energy is approximately $0.35$ that of the muons. With no shielding, the average power deposited per unit length would be about 2 kW/m
in the 4 TeV machine, and 300 W/m in the 100 GeV Higgs factory.
Figure~\ref{shieldingnew} shows the power penetrating tungsten shields of different
thickness \cite{ref6a,carnik96,scraping,shield96}. One sees that 3 cm in the low energy case,
or 6~cm at high energy would reduce the power to below 10 W/m, which can reasonably
be taken by superconducting magnets.
\begin{figure*}[bht!]
\centerline{
\epsfig{file=fnalfg10.ps,height=3.95in,width=3.45in}
}
\caption{Power penetrating tungsten shields vs.\ their thickness for a) 4~TeV, and b) 0.1~TeV, colliders. \label{shieldingnew}}
\end{figure*}
Figure~\ref{costheta} shows the cross section of a baseline magnet suitable for the 100~GeV collider.
\begin{figure*}[bht!]
\centering{
\epsfig{figure=cos_theta1.ps, width=4in}
}
\caption{Cross Section of a Baseline Dipole Magnet Suitable for the 100 GeV Collider.
\label{costheta}}
\end{figure*}
The quadrupoles could use warm iron
poles placed as close to the beam as practical. The coils could then be either
superconducting or warm, placed at a greater distance from the beam and shielded from it by the poles.
The collider ring could be made smaller, and the luminosity increased, if higher field dipoles were used. In the low energy case, the gain would not be great since less than half the circumference is devoted to the arcs. For this reason, and to avoid yet another technical challenge, higher field magnets are not part of the baseline design of a 100 GeV collider. But they would give a significant luminosity improvement for the higher energy colliders, and would be desirable there. There have been several studies
of possible designs, three of which (two that are promising and one that appears not to work) are included below.
\subsubsection{Alternative racetrack Nb$_3$Sn dipole}
A higher field magnet based on Nb$_3$Sn conductor and racetrack coils is presently being designed. The Nb$_3$Sn conductor allows higher fields and provides a large temperature margin
over the operating temperature, but being brittle and sensitive to bending or other stress, presents a number of engineering challenges.
In this design, the stress levels in the conductor are reduced by the use of a rectangular coil block geometry and end support problems are reduced by keeping the coils flat.
In the more conventional \textit{cos theta} designs, the conductor is distributed around a cylinder and the forces add up towards the mid-plane; in addition, the ends, as they arc over the cylinder, are relatively hard to support.
The geometry of the cross section is shown in Fig.~\ref{g_mag}. It uses all 2-D flat racetrack
coils. Each quadrant of the magnet aperture has two blocks of
conductors. The block at
the pole in the first quadrant has a return block in the second
quadrant, similar to that in
a conventional design. The height of this block is such that it
completely clears the
bore. In a conventional design, the second block, the midplane block,
would also have a
return block in the second quadrant. That would, however, require the
conductor block to
be lifted up in the ends to clear the bore and thus would lose the
simple 2-D geometry. In
the proposed design, the return block retains the 2-D coil geometry,
as it is returned on the
same side (see Fig.~\ref{g_mag}) and naturally clears the bore. Since the return block does not
contribute to the field, this design uses 50\% more conductor. This,
however, is a small
penalty to pay for a few magnets where the performance and not the
cost is a major issue.
The field lines are also shown in Fig.~\ref{g_mag}.
Preliminary design parameters for two cases are given in table~\ref{magnetdesign}. The first
case is one
where the performance of the cable used is the same that
is in the LBL D20
magnet, which created a central field of 13.5~T. The second case,
is the one where the
cable is graded and two types of cable are used, and
it is
assumed that a reported
improvement in cable performance is realized.
It is expected to produce a
central field of 14.7~T when operated at $4.2~{}^{o}$K.
%\begin{center}
\begin{table*}[tbh!]
\caption[Preliminary design parameters for a racetrack Nb$_3$Sn dipole]{Preliminary design parameters for a racetrack Nb$_3$Sn dipole with two different types of cable.}
\label{magnetdesign}
\begin{tabular}{ll}
\multicolumn {2}{l}{\emph{Case 1}:
Same conductor as in LBL 13.5~T D20 magnet without grading} \\ \hline
Central field at quench & 13~T at $4.2~^{o}\!K$\\
Coil dimensions & $25~\textrm{mm} \times 70$~mm\\
Total number of racetrack coils in whole magnet & 6\\
Total number of blocks per quadrant in aperture & 2 (+1 outside the aperture)\\
Yoke outer radius &500~mm (same as in D20)\\
Field harmonics & a few parts in $10^{-5}$ at 10~mm\\
Midplane gap (midplane to coil) & 5~mm (coil to coil 10~mm)\\
Minimum coil height in the end & 45~mm (Note: coils are not lifted up.)\\
% & \\
%\vspace{.1in}
\hline
\multicolumn {2}{l}{\emph{Case 2}: Newer conductor and graded}\\
\hline
Central field at quench & 14.7~T at $4.2~^{o}\!K$\\
Grading & 70~mm divided in two 35~mm layers \\
Overall current densities & 370~A/mm$^2$ and 600~A/mm$^2$\\
Peak fields & 16~T and 12.5~T\\
Copper current density & 1500~A/mm$^2$\\
Other features are the same as in \emph{Case 1} & \\
\end{tabular}
\end{table*}
%\end{center}
\begin{figure*}[hbt!]
%\vspace{-0.15in}
%\begin{center}
%\includegraphics[width=3.0in,height=4.0in,angle=-90]{dip-test.ps}
\centering{\psfig{figure=gupta1.eps,height=3.0in,clip=}}
\caption[Cross section of Alternative High Field Race Track Coil Dipole Magnet ]{Cross section of Alternative High Field ($\approx 15$~T)
Race Track Coil Dipole Magnet with Nb$_3$Sn conductor.}
\label{g_mag}
\end{figure*}
\subsubsection{Alternative \textit{Cos Theta} Nb$_3$Sn dipole}
\begin{figure*}[hbt!]
\begin{center}
\includegraphics[width=4.5in,angle=90]{slotmag.eps}
%\includegraphics[width=4.5in]{slotmagnew.eps}
\end{center}
%\centering{{\epsfig{figure=slotmag.eps},width=4in, height=4in,angle=90,clip=}}
\caption{Cross section of Alternative High Field Slot Dipole made with
Nb$_3$Sn conductor.}
\label{slotmag}
\end{figure*}
In this case the problem with the brittle and sensitive conductor is solved by winding the coil inside many separate slots cut in metal support cylinders. There is no build up of forces on the coil at the mid-plane. The slots continue around the ends, and
thus solve the support problem there too.
Figure~\ref{slotmag} shows this alternative Nb$_3$Sn dipole \textit{cos theta} design. It is an extension of the concept used to build helical magnets \cite{willenhelical} for the polarized proton program at RHIC \cite{rhic}.
The magnet is wound with pre-reacted, kapton-insulated, B-stage impregnated, low current cable. The build up of forces is controlled by laying the cables in machined slots in a metal support cylinder. After winding, the openings of the slots are bridged by metal spacers and the coils pre compressed inward by winding B-stage impregnated high tensile thread around the spacers. After curing, the outside of each coil assembly is machined prior to its insertion into an outer coil, or into the yoke. There are 3 layers. The inner bore is 55~mm radius, the outer coil radius approximately 118~mm, and the yoke inside radius is 127~mm. The maximum copper current density is 1300~A/mm$^2$.
Using the same material specifications as used in the above high field option, a central short sample field of 13.2 T was calculated. This is somewhat less than the block design discussed above, but could be improved by increasing the cable diameters to improve the currently rather poor (64\%) cable to cable-plus-insulator ratio.
\subsubsection{Study of C-magnet dipole}
\begin{figure*}[hbt!]
\centering{
\epsfig{figure=cmag.ps,width=3.5in}
}
\caption{Cross section of an unsuccessful alternative high field C magnet with
open mid-plane.}
\label{cmag}
%\end{center}
\end{figure*}
Figure~\ref{cmag} shows the cross section of a high field dipole magnet in which it was hoped to bring the coils in closer to the beam pipe without suffering excessive heating from beam decay. The coil design \cite{willencmag} appeared reasonable, but the required avoidance of coil heating was not achieved.
Decay electrons are generated at very small angles ($\approx \ 1/\gamma$) to the beam, and with an average energy about 1/3 of the beam.
Such electrons initially spiral inward (to the right in Fig.~\ref{cmag}) bent by the high dipole field. In the high energy case, these electrons also radiate a significant fraction of their energy as ($\approx$ 1 GeV) synchrotron gamma rays, some of which end up on the outside (to the left in Fig.~\ref{cmag}).
The concept was to use a very wide beam pipe, allow the electrons to exit between the coils, and be absorbed in an external cooled dump. Unfortunately a preliminary study found that a substantial fraction of the electrons did not reach the dump. They were
bent back outward before reaching it by the return field of the magnet coils and the nature of the curved ring geometry. Such electrons were then trapped about the null in the vertical field and eventually hit the upper or lower face of the unshielded
vacuum pipe. They showered, and deposited unacceptable levels of heat in the coils.
Another idea called for collimators between each bending magnet that would catch such trapped electrons. This option has not been studied in detail, but the impedance consequences of such periodic collimators are expected to be unacceptable.
Further study of such options might find a solution, but the use of a thick cylindrical heavy metal shield appears practical, adequate, and is thus the current baseline choice.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\Afb} {A_{\mathrm{FB}}}
\newcommand{\AFB} {A_{\mathrm{FB}}}
\newcommand{\qqbar} {\ifmath{\mathrm{q\bar{q}}}}
\newcommand{\ttbar} {\ifmath{\mathrm{t\bar{t}}}}
\newcommand{\uubar} {\ifmath{\mathrm{u\bar{u}}}}
\newcommand{\ddbar} {\ifmath{\mathrm{d\bar{d}}}}
\newcommand{\ppbar} {\ifmath{\mathrm{p\bar{p}}}}
\subsection{Energy scale calibration}
In order to scan the width of a Higgs boson of mass around 100~GeV, one needs
to measure the energy of the individual muon stores to an accuracy of a few
parts per million, since the width of a Higgs boson of that mass is expected to
be a few MeV. Assuming that muon bunches can be produced with modest
polarizations of $\approx 0.25,$ and that the polarization can be maintained
from turn to turn in the collider, it is possible to use the precession of the
polarization in the ring to measure accurately the average energy of the muons
\cite{ref7}. The total energy of electrons produced by muon decay observed in
the calorimeter placed in the ring varies from turn to turn due to the $g-2$
precession of the muon spin, which is proportional to the Lorentz factor $\gamma$
of the muon beam. Figure~\ref{efig3} shows the result of a fit of the total
electron energy observed in a calorimeter to a functional form that includes
muon decay and spin precession. Figure~\ref{efig4} shows the
fractional error $\delta\gamma/\gamma$ obtained from a series of such fits
plotted against the fractional error of measurement in the total electron
energy that depends on the electron statistics. It has been shown that precisions of a few parts per million in $\gamma$ are
possible with modest electron statistics of $\approx 100,000$ detected.
It should be noted that there are $3.2\times 10^6$ decays per meter for a muon intensity of
$10^{12}$ muons.
% In order to maintain the precession of the polarization in a horizontal plane,
% it is necessary to compensate for the rotation in polarization
%introduced by the detector solenoid with opposing solenoids placed on either
%side of the interaction region.
\begin{figure*}[thb!]
\centering
\epsfxsize = 5in
\epsffile{rajafig3.eps}
\caption[ Energy detected in the calorimeter during the first 50 turns in a
50 GeV muon storage ring ]{a) Energy detected in the calorimeter during the first 50 turns in a
50 GeV muon storage ring (points). An average polarization value of ${\hat P}=-0.26$ is
assumed and a fractional fluctuation of $5\times 10^{-3}$ per point. The curve is the
result of a MINUIT fit to the expected functional form. b) The
same fit, with the function being plotted only at integer turn
values. A beat is evident. c) Pulls as a function of turn number. d) Histogram
of pulls. A pull is defined by (measured value-fitted value)/(error in
measured-fitted). }
\label{efig3}
\end{figure*}
%
\begin{figure*}[hbt!]
\centering
\epsfxsize = 5in
\epsffile{rajafig4.eps}
\caption[Fractional error in $\delta\gamma/\gamma$ ]{a) Fractional error in $\delta\gamma/\gamma$ obtained from the
oscillations as a function of polarization $\hat P$ and the fractional error in
the measurements PERR. b) Fractional error in $\delta\gamma/\gamma$ obtained
from the decay term as a function of polarization $\hat P$ and the fractional
error in the measurements PERR. c) The total $\chi^2$ of the fits for 1000
degrees of freedom. PERR is the percentage measurement error on the total electron energy in the
calorimeter measuring the decay electrons. }
\label{efig4}
\end{figure*}
%
%\subsubsection{Possible implementation strategy}
Our current plans to measure the energy due to decay electrons entail an electromagnetic calorimeter that is
segmented both longitudinally and transversely and placed inside an enlarged beam
pipe in one of the straight sections in the collider ring. The length of the
straight section upstream of the calorimeter can be chosen to control the total number of decays and hence the rate of energy deposition. The sensitive material can be
gaseous, since the energy resolution is controlled by decay fluctuations
rather than sampling error. In order to measure the total number of electrons
entering the calorimeter, we plan to include a calorimeter layer with little
absorber upstream of it as the first layer.
This scheme will enable us to calibrate and correct the energy of individual
bunches of muons and permit us to measure the width of a low mass Higgs boson.