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\title{50 on 50-GeV Muon Collider Storage Ring}

\author{C. Johnstone$^*$ and A. Garren$^{\dagger}$}
\address{$^*$FNAL, \thanks{Work supported by the U.S. Department of Energy 
 under contract No. DE-AC02-76HO3000.}\\
Batavia, IL 60555, USA\\
$^{\dagger}$UCLA, Los Angeles, CA 90024,USA}

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\begin{abstract}
Two modes are being considered for a 50 on 50-GeV muon collider: one being
a high-luminosity ring with broad momentum acceptance (dp/p of
$\sim 0.12\%$, rms) and the other lower luminosity with narrow momentum
acceptance (dp/p of $\sim 0.003\%$, rms), or Higgs Factory. To reach the design luminosities, the
value of beta at collision in the two rings must be 4~cm and 14~cm, respectively.
In addition, the bunch length must be held comparable to the value of the
collision beta to avoid luminosity dilution due to the hour-glass
effect.  To assist the rf system in preventing the bunch from
spreading in time, the constraint of isochronicity is also
imposed on the lattice.  Finally, the circumference must be kept as
small as possible to minimize luminosity degradation due to
muon decay.  Two lattice designs will be presented which meet
all of these conditions.  Furthermore, the high-luminosity and Higgs Factory
lattice designs have been
successfully merged into one physical ring with mutual components;
the only difference being a short chicane required to match dispersion
and floor coordinates from one lattice into the other.
\end{abstract}

\section*{Introduction}
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After one $\mu^+$ bunch and one $\mu^-$ bunch have been accelerated 
to collision energy, the two bunches
are injected into the collider ring, which is a fixed-field storage ring.
Two cases are being considered for
a 50 on 50-GeV collider:  a ring with
broad momentum acceptance ($dp/p_{rms}$ of $\pm .12\%$)
and high luminosity,
and one with a much narrower momentum acceptance
($dp/p_{rms}$ of $\pm .003\%$) and lower luminosity.
The narrow-band machine is
intended to resolve the width of the Higgs mass to high precision.

The two operational modes for the 100-GeV collider
require different machine optics.
The following sections discuss collider lattices for
both
the broad momentum  application
and the monochromatic mode.

\section*{Design Criteria}

Stringent criteria have been imposed on the collider lattice designs in order to attain
the specified luminosities.  The first and most difficult criterion to satisfy is
provision of an Interaction Region (IR) with extremely low
$\beta^*$ values at the collision point consistant with acceptable dynamic aperture.
The required $\beta^*$ values for the 100-GeV collider are
4~cm for the broad momentum-width case and 14~cm for the narrow-width case.
These $\beta^*$ values
were tailored to match the longitudinal bunch lengths in order 
to avoid luminosity dilution from the hour-glass effect.
Final-focus designs must also provide collimators and background
sweep dipoles, and other provisions for protecting the magnets
and detectors from muon-decay electrons.  Effective schemes have
been incorporated into the current lattices.

Another difficult constraint imposed on the lattice is that of isochronicity.
A high degree of isochronicity is required
in order to maintain the short bunch structure without
excessive rf voltage.
A final criterion especially important in
the lower-energy colliders, 
is that the ring circumference be
as small as feasible in order to minimize luminosity degradation through decay
of the muons. To achieve small circumference requires high fields in the bending 
magnets as well as a compact, high dipole packing-fraction design.
(To meet the small circumference demand, 8~T poletip fields have been assumed 
for all superconducting magnets.) 
  
Some of these criteria conflict with one another.
For example, the small value of $\beta^*$ leads to large peak beta values in the
final-focus quadrupoles and correspondingly large linear chromaticities in the
IR.  For the high-luminosity machine, local correction of the linear part of the IR
chromaticity is required to achieve adequate momentum acceptance.  Efficient
chromatic correction in turn requires large positive values for dispersion
in the correction sextupoles.  Because of the short circumference
condition, high dipole packing fractions must be maintained
not only in the arcs, but in the local Chromatic Correction
Section (CC) as well. One consequence of the high dipole concentration
in the CC is that a small  momentum compaction becomes difficult
to maintain because of the large number
of dipoles in regions of high positive dispersion, in conflict with
the need for isochronicity.  Control over the momentum compaction
is achieved through appropriate design of the arcs.  The
following sections discuss a base ring design which approaches
the limit of compactness for a 50-GeV collider lattice under
isochronous conditions and with strong local chromatic correction. 


\section*{Overview}
For the 100~GeV CoM collider, two operating modes are contemplated:
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 a 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, 14~cm.
In either case, the bunch
length must be held comparable to the value of $\beta^{*}$.

The 100-GeV CoM ring has a roughly racetrack design 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 and a matching section. 
The chromatic 
correction section is  optimized to correct the ring's linear chromaticity,
which is almost completely generated by the low beta quadurupoles in the IR. 
In designs of e$^+$e$^-$ colliders, it has been found that local chromatic correction
of the final focus is essential
\cite{chromatic}, as was found to be the case here.

Two 100 GeV lattice designs have been made; these are described below.
The desig has two optics modes:
one mode has a $\beta^*$ value of 4~cm with small transverse and 
large momentum acceptance;
a second mode has a $\beta^*$ value of 14~cm with large transverse 
and small, approximately monochromatic, momentum acceptance.
Both lattices were merged into one physical, highly compact ring design with a total circumference
of only about 345 m.
The arc modules account for only about a quarter of the ring circumference.

\subsection*{The Interaction Region}

Because of the dynamics of the cooling process, $\mu^+$ and $\mu^-$
emerge from the cooling stage with roughly equal emittances.  Initially
unequal $\beta^*$s, or elliptical beams, were explored at the collision
point.  From an optics standpoint, elliptical beams are more manageable and
less nonlinear than round beams in the design of Interaction Regions.
Using a $\beta^*$ ratio of 1:4 for the horizontal to vertical (factor of 2 in the relative
beam sizes), however, causes a decrease in the luminosity of a factor of 2
and this was felt to be unacceptable.  Therefore, the condition of round beams at the
Interaction Point (IP) has been imposed in all current collider designs.

The need for different collision modes in the 100-GeV machine
led to an Interaction Region design with two optics modes: 
one with broad
momentum acceptance (dp/p of $0.12\%$, rms) and a collision $\beta$ of 4~cm
(Fig.~\ref{50_gev_4cm}),
and the other basically monochromatic (dp/p of $0.003\%$, rms) and a
larger collision $\beta$ of 14~cm (Fig.~\ref{50_gev_14cm}).
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. 
Because of significant, large-angle backgrounds from muon decay,
a background-sweep dipole is included in the final-focus telescope
and placed near the IP to protect the detector and the low-$\beta$ 
quadrupoles \cite{carnik96}.
It was found that this sweep dipole, 2.5~m long with an 8~T field,
provides sufficient background suppression.
The first quadrupole is located 5~m away from the interaction point, 
and the beta functions reach a maximum value of $1.5\,{\rm km}$ 
in the final focus telescope, when the maxima of the beta functions 
in both planes are equalized. For this maximum beta value, 
the quadrupole apertures must be at least 11~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 $-60.$

The proximity of the final-focus quadrupoles to the IP determines the
maximum beta and this value combined with the quadrupole strengths
and lengths
determine the natural chromaticity and, ultimately, the nonlinear
behavior of the lattice.
With poletip fields reaching 8T,
the final-focus triplet in the 100-GeV collider remains short:
quadrupole lengths range from .6 to 1.5 m.
With such short quadrupoles, the peak beam size in the 100-GeV
machine and, therefore, the natural chromaticity of its
interaction region is almost completely a property of the IP to
quadrupole spacing.

\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_4cm.ps}}
\centerline{
\epsfig{figure=hf_ring50_gev_4cm.ps,height=4.0in,width=4.5in,bbllx=60bp,
bblly=110bp,bburx=545bp,bbury=685bp,clip=}
}
\caption{4~cm $\beta^*$ Mode showing
half of the IR, local, chromatic correction, and one of three arc modules.}
\label{50_gev_4cm}
\end{figure*}

\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_14cm.ps}}
\centerline{
\epsfig{figure=hf_ring50_gev_14cm.ps,height=4.0in,width=4.5in,bbllx=60bp,
bblly=110bp,bburx=545bp,bbury=685bp,clip=}
}
\caption{14~cm $\beta^*$ Mode showing
half of the IR, local, chromatic correction, and one of three arc modules.}
\label{50_gev_14cm}
\end{figure*}



The optimum design of a very low-beta IR is to make the imaging
as point to parallel as is practical to soften chromatic aberrations.  The less
the applied chromatic correction, the larger, in general, is the dynamic
aperture.  In the 100-GeV machine, circumference constraints require
the IP to be imaged in a short distance; implying stronger than optimal
focussing from the high-beta triplet.  The IP image distance can be reduced by
as much as 35 meters on either side of the IP; or about a 30\% decrease in the
ring circumference.  The stronger quadrupole strengths do increase the linear chromaticity
of the IR from about 60 to 85 in the vertical with
little effect on the horizontal (assuming the triplet powering is FDF).
In practive, the demagnification is about halfway between
a compact and an optimal, or soft-focussing IR.  Some deterioration
in dynamic aperture is evident with stronger focussing, although
studies of high-order and phase dependencies are underway and careful
tuning appears to meliorate these effects.

Initially, the powering of the triplet was chosen such that the vertical
apertures in the near dipoles were minimized.  This requires placing
the vertical high-beta peak at the center of the triplet, so that the
triplet sequence is FDF.  This has the disadvantage in that the local
chromatic correction is not as efficient (the higher the dispersion, the more
efficient the correction).  Higher values of dispersion are usually obtained
at peaks in the horizontal beta function than in vertical beta peaks.  The plane
corrected first should be the one with the highest chromaticity; in this case
the vertical.  If the dispersion is lower, then the chromatic correction, even
with $\pi$ pairs of sextupoles, is not as efficient and generates stronger
nonlinearities.  These nonlinearities propagate and appear to be enhanced
by the sextupoles of the opposite plane and can be correlated to an
observed decrease in dynamic aperture in this plane.

In a test lattice, the triplet was powered in a DFD configuration out
of concern for the dynamic aperture.  The plane with the highest chromaticity
and the highest achievable dispersion at the sextupoles was corrected closest
to the source, effecting a more efficient chromatic correction.  Nonlinear terms were
amplified less by sextupoles in the opposite plane as was evidenced by a slight
improvement in dynamic aperture.  A questionable consequence of installing the
horizontal high-dispersion peak nearest the IP was the unavoidable
application of reverse bends to create a dispersion plateau (D'=0)
after a defocussing quadrupole.  (These reverse bends are not needed
if vertical chromaticity correction is performed first because a dispersion plateau
can follow a focussing quadrupole.)  The net increase in circumference due
to reverse bends and less efficient dipole packing in general brought
the circumference up by at least 50 m; making the circumference more than
400 m when injection and scraping are included.  The loss in muon lifetime
was felt to outweight the small advantage to the optics of the ring.  The final
triplet powering remains as FDF with the vertical chromaticity beign corrected
closest to the IP.  

\subsection*{Chromatic Correction}

Local chromatic correction of the muon collider interaction region is
required to achieve broad momentum acceptance.
With such a large aperture in the final-focus quadrupoles,
adding dispersion to the final focus is not reasonable
and therefore chromatic correction must take place in a
specialized section.
The basic approach developed by Brown \cite{chromatic} and others
is implemented in the Chromatic Correction Region (CC) used here. The CC contains
two pairs of sextupoles, one pair for each transverse
plane, all located at locations with high dispersion. 
The sextupoles of each pair are located at positions of equal, high beta value
in the plane (horizontal or vertical) whose chromaticity is to be corrected,
and low beta value in the other plane. Moreover, the two sextupoles of each pair
are separated by a betatron phase advance of $\pi$, 
and each sextupole has a phase separation of $(2n+1){\pi\over 2}$ from the IP,
where $n$ is an integer.
The result of this arrangement is that
the geometric aberrations of each sextupole is canceled by its 
companion while the chromaticity corrections add.

An innovative module was developed specifically for chromatic correction
(Fig.~\ref{50_gev_ccs})
and implemented first in the 4-TeV muon collider~\cite{IR_snow}.
Its characteristics include
a high-dispersion and high-beta plateau in one plane
coincident with a deep minimum in beta in the opposite plane.
The high-beta plateaus alternate between planes, with
the single intervening deep minimum establishing a $\pi$
phase advance between plateaus in the same plane.
The sextupoles of each pair are centered 
about a minimum in the opposite plane ($\beta_{min}<1$), which
provides chromatic correction
with minimal cross correlation between the planes.
A further advantage to locating the opposite plane's
minimum at the center of the sextupole, is that this point is
${\pi\over 2}$ away from, or ``out of phase" with, the source of chromatic effects
in the final focus quadrupoles; i.e. the plane not being chromatically corrected is
treated like the IP  in terms of phase to eliminate a second order 
chromatic aberration generated by an ``opposite-plane''
sextupole.

\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_ccs.ps}}
\centerline{
\epsfig{figure=hf_ring50_gev_ccs.ps,height=4.0in,width=4.5in,bbllx=60bp,
bblly=110bp,bburx=545bp,bbury=685bp,clip=}
}
\caption{The Chromatic Correction Module.}
\label{50_gev_ccs}
\end{figure*}

In this lattice example, the CC was optimized to be as short as possible
with a smooth transition designed to place the first chromatic correction
sextupole at the same phase as the high-beta point~\cite{IR_snow}.
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, so the dynamic aperture is not compromised
by a large amplitude-dependent tuneshift.

This large beta ratio, combined with the opposite-plane phasing,
allows the sextupoles for the opposite planes to be interleaved,
without significantly increasing the nonlinearity of the lattice.
In fact, interleaving improved lattice performance compared to that of
a non-interleaved correction scheme, due to a shortening
of the chromatic correction section, which 
lowers its chromaticity contribution.
The use of somewhat shallower beta-minima with less variation in beta through the
sextupoles were also applied to soften the chromatic aberrations, although this caused a 
slight violation of the 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 $\beta_{\textrm{min}}<0.5~\textrm{m}$.


This module, specifically optimized to perform chromatic correction, is
particularly powerful in that it can accomodate long sextupoles
without beta and phase changes taking place in the plane being
corrected.  However, because of finite element lengths and changes in the
phase advance between sextuple as a function of energy, a tuneshift
with amplitude is unavoidable, and depends most sensitively on the
beta amplitude in the sextupole, but also on the length of the sextupole
and the tune of the ring.  
Ultimately, a tuneshift with amplitude constricts the dynamic aperture
and a tradeoff exists between momentum acceptance and transverse
dynamic aperture.  Lattice parameters, especially the beta values
at the sextupoles and the phase advance around the ring, must be
carefully tuned to optimize both acceptances simultaneously.

For the narrow band acceptance, local chromatic correction; i.e. the
sextupoles are turned off.  The momentum acceptance narrows to about
a $\delta p/p$ of about $\pm0.2\%$, while the transverse dynamic
aperture increases rapidly to over 10$\sigma$ at the central momentum.

\subsection*{The Arc}

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{FMC}.
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.


\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_arc.ps}}
\centerline{
\epsfig{figure=hf_ring50_gev_arc.ps,height=4.0in,width=4.5in,bbllx=60bp,
bblly=110bp,bburx=545bp,bbury=685bp,clip=}
}
\caption{The 100-GeV CoM collider arc: a new flexible momentum compaction module.}
\label{50_gev_arc}
\end{figure*}



The entire ring was designed to control momentum compaction, even in the match
section with connects the CC to the arc.  This careful attention
to momentum compaction for the isochronicity condition resulted in
a circumference which is just under 350 m, as opposed to rings which
were greater than the 400 m characteristic of earlier designs.
The total momentum compaction contributions of the IR, CC, and matching sections
is about $0.04$.  The total length of these parts is $173\,{\rm m}$, while that
of the the momentum-compaction-correcting arc is $93\,{\rm m}.$
From these numbers, it follows that this arc must and does have a negative momentum
compaction of about $-0.09$ in order to offset the positive contributions
from the rest of the ring.

\subsection{rf system}

The rf requirements depend on the momentum compaction of the lattice
and on the parameters of the muon bunch. 
For the case of very low momentum spread,
synchrotron motion is negligibe and the rf system is used solely to correct 
an energy spread generated through the
impedance of the machine. For the cases of higher momentum spreads,
there are two approaches. One is to make the momentum compaction zero
to high order through lattice design.
Then the synchrotron motion can be eliminated,
and the rf is again only needed to compensate the induced 
energy spread correction. Alternatively, if some
momentum compaction is retained, then a more powerful rf system is needed to maintain 
the specified short bunches.
In either case, rf quadrupoles will be required to generate BNS 
damping of the transverse head-tail instability.

\section*{Performance}


A very preliminary calculation of the dynamic aperture without optimization 
of the lattice nor inclusion of errors and end effects
is given in Fig.~4.  One would expect that simply turning
off the chromatic correction sextupoles in the 4~cm $\beta^*$ mode would  
result in a linear lattice with a large transverse aperture.
With only linear elements, the 4~cm $\beta^*$ optics showed to be strongly
nonlinear with limited on-momentum dynamic acceptance.

A normal form analysis
using COSY INFINITY showed that the tune-shift-with-amplitude was large, which was the source
of the strong nonlinearity in the seemingly linear lattice.
To locate the source of this nonlinearity, a lattice 
consisting of the original IR
and arcs only (no CC), was studied. Numerical
studies showed that a similar dynamic aperture and
tune-shift-with-amplitude terms.
This ruled out the possibility that the dynamic aperture was limited by
the low beta points in the local chromatic correction section and points 
to the IR as the source of the
nonlinearity. (The findings were verified by S. Ohnuma who used
a Runge-Kutta integrator to track through the IR and a linear matrix for the rest of the
lattice.) Further analytical study using perturbation theory showed that the first-order
contribution to the tune shift with amplitude is proportional to $\gamma^2_{x,y}$ and
$\gamma_x \gamma_y$, which are large in this IR. These terms come from the nonlinear
terms of $p_x/p_0$ and $p_y/p_0$, which, to the first order, equal the angular
divergence of a
particle. As a demonstration, a comparison to the LHC low-beta IR was done. Taking into
account only the drift from the IP to the first quadrupole, the horizontal detuning
at 10$\sigma$ of the present IR ($\beta^{*}$ $=$ 4 cm) is 0.01, whereas the detuning of
the entire LHC lattice is below 1e-4. This also explains the fact that the on-momentum
aperture of the wide momentum spread mode remains roughly constant among different
versions despite various correction attempts.                                              

It was therefore concluded and later shown that the dynamic aperture of the more
relaxed $\beta^*$ of 14~cm would not have the same strong nonlinearities
due to the reduced angular terms.
In fact, the tune shift with amplitude was less by an order of magnitude;
hence the large transverse acceptance shown in Fig.~\ref{50_gev_ap} (dashed line).

\begin{figure*}[thb!]
\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_ap.ps}}
\centerline{
\epsfig{figure=hf_ring50_gev_ap.ps,height=3.5in,width=4.0in,bbllx=60bp,
bblly=10bp,bburx=545bp,bbury=885bp,angle=-90,clip=}
}
\caption{A preliminary dynamic aperture for the 4~cm
$\beta^*$ mode where $\sigma$ (rms) = 82$\mu m$ (solid line) and the
14~cm $\beta^*$ mode where $\sigma$ (rms) = 281$\mu m$ (dashed line).}
\label{50_gev_ap}
\end{figure*}




\begin{references}
\bibitem{chromatic}K. Brown, ``A Conceptual Design of Final Focus Systems
for Linear Colliders,''PUB-4159, (1987).
\bibitem{carnik96}C. Johnstone and N. Mokhov, FERMILAB-Conf-96-366(1996).
\bibitem{scraping}A. Drozhdin, et al, AIP Conf. Proc. {\bf 441}, p. 242 (1998).
\bibitem{IR_snow}C. Johnstone and A. Garren, ``An IR and Chromatic Correction Design
for a 2-TeV Muon Collider,''{\it Proceedings of the Workshop on New Directions
for High-Energy Physics, Snowmass 96, Jun.-Jul., 1996}, pp. 222.
\bibitem{FMC}K. Ng, S. Lee, and D. Trbojevic, FNAL-FN595 (Fermilab, 1992).
\end{references}
 
\end{document}