%modified by R. Raja Referee's comments 2-mar-99
%modified by R. Raja and D.Finley for style 18-Dec-98
%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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\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}}}}
%\newcommand{\Afb}       {A_{\mathrm{FB}}}



\section{COLLIDER STORAGE RING}

\subsection{Introduction}

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.
%
Parameters for several possible collider storage rings are given 
in table~\ref{sum}. Collider ring lattices have been developed for two of
the collision energies in this table: 100~GeV and 3~TeV in the 
center of mass. 
%In addition, a preliminary interaction region for a 
%500~GeV \cite{ref33} lattice has been designed.

Three operational modes are proposed in the above table for the 100~GeV
collider, each requiring  different machine optics. The following sections
discuss a 100~GeV collider lattice for two of the modes, the broad momentum
spread case ($\Delta p/p$ of $0.12\%$, rms)  and the narrow momentum spread 
case ($\Delta p/p$ of $0.003\%$), as well as a 3~TeV collider lattice.

\subsection{Collider Lattices}

\subsubsection{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 consistent with acceptable dynamic aperture.
The required $\beta^*$ values for the 100~GeV collider are
4~cm for the broad momentum spread case and 14~cm for the narrow momentum spread case.
For the 3~TeV machine, $\beta^*$ is only 3~mm.  These $\beta^*$ values
were tailored to match the longitudinal bunch lengths in order 
to avoid luminosity dilution from the hour-glass effect.  
Achieving this requirement in the 3~TeV lattice is complicated by
the high peak beta function values in the final focus quadrupoles
requiring 8-10~cm radial apertures.
The correspondingly weakened gradients
combined with the ultra-high energy make for a long final focus structure.
(In contrast, the lower energy and larger $\beta^*$ values
in the 100~GeV collider lead to an efficient, compact final focus telescope.)
Compounding the problem, particularly for the 3~TeV design, is 
the need to protect the superconducting coils
from the decay products of the muons.
Placing a tungsten shield between the vacuum
chamber and the coils can increase the radial aperture in
the 3~TeV quadrupoles by as much as 6~cm,
lowering  available gradients still further.
Final focus designs must also include 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.
In the lattices presented here, control over the momentum compaction 
is achieved through
appropriate design of the arcs.

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. Achieving 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 pole tip fields have been assumed 
for all superconducting magnets, with
the exception of the 3~TeV final focus quadrupoles, whose pole tips  are assumed
to be as high as 12~T.  In addition, design studies for still higher field dipoles
are in progress.


\subsubsection{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 negligible 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 \cite{refbns,ref36} 
damping of the transverse head-tail instability.

\subsubsection{3~TeV CoM lattice}
The 3~TeV 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 (CCS) and a matching section. 
The chromatic 
correction section is  optimized to correct the ring's linear chromaticity,
which is mostly generated by the low beta quadrupoles 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,chromatica,chromaticb,chromaticc}, as was found to be the case here.
The 3~TeV IR and CCS are displayed in Fig.~\ref{3_tev_ir}.
The accompanying 3~TeV arc module in Fig~\ref{3_tev_arc}
is an example of a module which controls momentum compaction
(i.e. isochronicity) of the entire ring. 

\begin{figure*}[thb!]
\begin{center}
\includegraphics*[width=4.5in,height=5.0in,bb=88 230 450 625,angle=-90]{ca11.ps}
\end{center}
\caption{Example~(a):
3~TeV IR and chromatic correction section.}
\label{3_tev_ir}
\end{figure*}

\begin{figure*}[bth!]
\begin{center}
\includegraphics*[width=4.5in,height=5.0in,bb=88 130 450 625,angle=-90]{ca22.ps}
\end{center}
\caption{Example~(a): 3~TeV arc module.}
\label{3_tev_arc}
\end{figure*}

\subsubsection{100~GeV CoM lattices}
For the 100~GeV CoM collider \cite{ref33}, 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 ring geometry is highly compact and
more complicated than a racetrack, but the lattice
has regions with the same functions as those of the 3~TeV ring.
 
Two independent 100~GeV lattice designs have evolved; these are described below
in separate sections and denoted Example (a) and Example (b), respectively.
The first design described is a lattice which has two optics modes.
In the high luminosity mode, the $\beta^*$ value is 4~cm with a
transverse and
momentum aperture sufficient to accept a normalized beam emittance
of $90\pi$ (rms) and a $\Delta p/p$ of $\pm 0.12\%$ (rms).
The second, lower luminosity mode has a $\beta^*$ value of 14~cm 
with a very large transverse acceptance,
but small, approximately monochromatic, momentum acceptance.

The second 100~GeV lattice described 
is another collider design 
with a 4~cm $\beta^*$ optics mode.
Although the number of magnets differ between the two lattices,
 the most important optics difference between the two 
is in the modules used in the arcs.  

\subsubsection{100~GeV CoM. Example~(a)}

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 ($\Delta p/p$ of $0.12\%$, rms) and a collision $\beta^*$ of 4~cm,
and the other basically monochromatic ($\Delta p/p$ of $0.003\%$, rms) and a
larger collision $\beta^*$ of 14~cm.  
The first lattice design, denoted Example~(a), shown in 
Fig.~\ref{50_gev_4cm} and Fig.~\ref{50_gev_14cm}, 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!]
\begin{center}
\includegraphics*[width=4.5in,height=5.0in,bb= 17 30 350 537,angle=-90]{ca33.ps}
\end{center}
\caption[Example~(a):  4~cm $\beta^*$ ]{Example~(a):  4~cm $\beta^*$ Mode showing
half of the IR, local chromatic correction section and one of three arc modules.}
\label{50_gev_4cm}
\end{figure*}

\begin{figure*}[bth!]
\begin{center}
\includegraphics*[width=4.5in,height=5.0in,bb= 17 30 350 537,angle=-90]{ca44.ps}
\end{center}
\caption[Example~(a): 14~cm $\beta^*$  ]{Example~(a): 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 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 under 
reflection about the interaction point (IP).
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$~\sigma$ 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 the muon collider interaction region is
required to achieve broad momentum acceptance.
The basic approach developed by Brown \cite{chromatica} and others \cite{donald},
is implemented in the Chromatic Correction Region (CC). 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 very low beta waist in the other plane. Moreover, the two sextupoles of each pair
are separated by a betatron phase advance of near $\pi$, 
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.

The sextupoles of each pair are centered 
about a minimum in the opposite plane ($\beta_{min}<1m$), 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.


In this lattice example, the CC 
(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, 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 \cite{wan1}.
The use of somewhat shallower beta minima with less variation in beta through the
sextupoles was made 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 $\beta_{\textrm{min}}<0.5~\textrm{m}$.
%%% here I delete this figure
\begin{figure*}[thb!]
%\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_ccs.ps}}
%\centerline{
%\epsfig{figure=cfg2_new.eps,height=4.0in,width=4.5in,bbllx=0bp,
%bblly=110bp,bburx=600bp,bbury=685bp,clip=}
%}
\begin{center}
\includegraphics*[width=4.5in,height=5.0in,bb=17 30 350 537,angle=-90]{ca55.ps}
\end{center}
\caption{Example~(a): The chromatic correction module.}
\label{50_gev_ccs}
\end{figure*}

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~m.
From these numbers, it follows that this arc must have a negative momentum
compaction of about $-0.09$ in order to offset the positive contributions
from the rest of the ring.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 12/13 15:40 PST

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,ref32a}.
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.  
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.


\begin{figure*}[thb!]
%\epsfxsize4.0in
%%\centerline{\epsffile{50_gev_arc.ps}}
%\centerline{
%\epsfig{figure=cfg3_new.eps,height=4.0in,width=4.5in,bbllx=0bp,
%bblly=115bp,bburx=600bp,bbury=695bp,clip=}
%}
\begin{center}
\includegraphics*[width=4.5in,height=5.0in,bb=17 30 350 537,angle=-90]{ca66.ps}
\end{center}
\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}}
%\includegraphics[width=3.0in,height=3.5in,angle=-90]{50_gev_ap.ps}
%\centerline{
%\epsfig{figure=50_gev_ap.eps,height=3.0in,width=3.5in,angle=-90}
%}
\begin{center}
\includegraphics*[width=4.5in,height=4.5in,angle=-90]{new_50_gev_ap.ps}
\end{center}
\caption[Example~(a): A preliminary dynamic aperture ]{Example~(a): A preliminary dynamic aperture for the 4~cm
$\beta^*$ mode where $\sigma_\textrm{rms} = 82~\mu m$ (solid) and the
14~cm $\beta^*$ mode where $\sigma_\textrm{rms} = 281~\mu m$ (dashed).}
\label{50_gev_ap}
\end{figure*}

A very preliminary calculation of the dynamic aperture \cite{wan1} without optimization 
of the lattice or inclusion of errors and end effects
is given in Fig.~\ref{50_gev_ap}.  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 was found to be strongly
nonlinear with limited on-momentum dynamic acceptance.

A normal form analysis
using COSY INFINITY \cite{cosy} showed that the variation of 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 confirmed  similar dynamic aperture and
variation of tune shift with amplitude.
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. These findings were also verified \cite{ohnuma} using 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  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 
despite various versions and 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 variation of tune shift with amplitude was less by an order of magnitude;
hence the large transverse acceptance shown in Fig.~\ref{50_gev_ap} (dashed).

\subsubsection{100~GeV CoM. Example~(b)}
%\mbox{~~}\\[-0.95in]
The second lattice design, Example~(b), 
is shown in Fig.~\ref{f1} starting from the IP.
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.  Because of the small low betatron
functions in both transverse planes, the betatron functions at the 
final focusing triplets increase to $\sim 1550$~m.  The natural chromaticities,
of order $\sim -40$, are high, requiring local correction.
Due to the size limitation of the collider ring, it appears that we have room
for only two pairs of interleaved sextupoles on each side of the IP, each
pair correcting chromaticity in one transverse plane.
The correction section on each side of the IP spans
a distance of roughly 61.3~m.
\begin{figure*}[bht!]
\centering{\epsfig{figure=fig47.ps,width=6.0in
%,bbllx=73bp,bblly=342bp,bburx=530bp,bbury=658bp
,clip=}}
\caption[Example (b): Lattice structure of the IR including 
local chromaticity corrections]{Example (b): Lattice structure of the IR including local chromaticity corrections. $\beta_x$ (solid); $\beta_y$ (dashed); dispersion (dotted). 
The maximum and minimum $\beta_x$ are 1571.74 and 0.040~m,
the maximum and minimum $\beta_y$ are 1550.94 and 0.040~m, while
the maximum and minimum dispersions are  4.31 and $-3.50$~m.
The natural horizontal and vertical chromaticities are $-41.46$ and $-39.90$,
giving a transition gamma of $\gamma_t=5.52$.  
The total module length is 85.32~m
with a total bend angle of 1.307~rad.}
\label{f1}
\end{figure*}

The SX1's are the two horizontal
correction sextupoles.  They should be placed at positions with the
same betatron functions and dispersion function, and
separated horizontally and vertically by phase advances $\Delta\psi_x$
and $\Delta\psi_y=\pi$ 
so that their nonlinear effect
will be confined in the region between the two sextupoles.  Their horizontal
phase advances should also be  
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 which should be 
placed similarly at designated
locations.
In general, it will be difficult to satisfy all the requirements mentioned;
especially in this situation, luminosity arguments limit the lattice size.
For this lattice, 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.
An attempt was made to satisfy all the requirements at the expense of
having $\Delta\psi_y/(2\pi)=0.60$ instead of 0.50 for the SX1's.
This trade-off is explained below.

\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) &$\psi_x/(2\pi)$ &$\psi_y/(2\pi)$ &~~~$\beta_x$ &~~~$\beta_y$ &(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}
\end{table*}

The second order effects of the sextupoles
contribute to the amplitude dependent tune spreads, which, if too large,
can encompass
resonances leading to dynamical aperture limitation.  For example, in
this lattice,
\begin{eqnarray}
\nu_x\!&=8.126337 - ~100~\epsilon_x - \,4140~\epsilon_y,
\nonumber \\
\nu_y\!&=6.239988 - 4140~\epsilon_x - ~54.6~\epsilon_y,
\label{amp-tune}
\end{eqnarray}
where $\epsilon_x$ and $\epsilon_y$ are the horizontal and vertical
unnormalized emittances in $\pi$m.
In order to
eliminate these tune spreads due to the sextupole nonlinearity, 
the sufficient conditions are \cite{nonlinear}:
\begin{displaymath}
\sum_k\frac{S_ke^{i\psi_{xk}}}{\sin\pi\nu_x}=0\,,~~~~
\sum_k\frac{S_ke^{i3\psi_{xk}}}{\sin3\pi\nu_x}=0\,,~~~~
\end{displaymath}
\begin{equation}
\sum_k\frac{\bar S_ke^{i\psi_{xk}}}{\sin\pi\nu_x}=0\,,~~~~
\sum_k\frac{\bar S_ke^{i\psi_{\pm k}}}{\sin\pi\nu_\pm}=0\,,
\label{restriction}
\end{equation}
where for the $k$th thin normal sextupole with strength 
$S_{Nk}=
~\raisebox{-1.3ex}{$\stackrel
{\textstyle\lim}{\scriptstyle\ell\rightarrow0}$}~
%\stackk{\lim}{\scriptstyle\ell\rightarrow0}
[B''\ell/(B\rho)]_k$,
\begin{equation}
S_k=\left[S_{_N}\beta_x^{3/2}\right]_k\,,
\quad
\bar S_k=\left[S_{_N}\beta_x^{1/2}\beta_y\right]_k\,,
\end{equation}
$\psi_{\pm k}=(2\psi_y\pm\psi_x)_k$, 
and $\nu_{\pm k}=(2\nu_y\pm\nu_x)_k$.
The 5 requirements come about because there are 5 first order resonances
driven by the sextupoles when the residual tunes of the ring satisfy
$[3\nu_x]=0$, $[\nu_\pm]=0$ and two $[\nu_x]=0$.
The nominal tunes shown in Eq.~(\ref{amp-tune}) are far from these
resonances.  Therefore, the sines in the 
denominators of Eq.~(\ref{restriction})
can be omitted in this discussion.  
Since the strengths of SX1 and SX2 are similar, we have
$S_{\rm SX2}\ll\bar S_{\rm SX1}\ll\bar S_{\rm SX2}\ll S_{\rm SX1}$.
In fact, they are roughly in the ratios of
$1:(\beta_{\rm max}/\beta_{\rm min})^{1/2}
:\beta_{\rm max}/\beta_{\rm min}:(\beta_{\rm max}/\beta_{\rm min})^{3/2}$,
which amount roughly to 1:10:100:1000 in this lattice.
In above, $\beta_{\rm max}$ represents either $\beta_x$ at the SX1's or
$\beta_y$ at the SX2's, and 
$\beta_{\rm min}$ represents either $\beta_y$ at the SX1's or
$\beta_x$ at the SX2's.
Thus, the first two restrictions in Eq.~(\ref{restriction}) are the most
important, implying that all  $\beta_{\rm max}$ and $\beta_{\rm min}$
for each pair of SX1's must be made equal  and 
$\Delta\psi_x=\pi$ between them must be strictly obeyed.
The third restriction is the next important one, for which $\bar S_{\rm SX2}$
must be made equal for each pair of SX2's and their horizontal phase
difference must equal $\pi$.
The only two parameters left are
$\Delta\psi_y$ 
between a pair of SX1's and  $\Delta\psi_y$ between a pair of 
SX2's.  They affect
the restrictions for the
$\nu_\pm$ resonances only,
where the effective sextupole strengths $\bar S_{\rm SX1}$
and $\bar S_{\rm SX2}$ are involved.  Thus if we allow one restriction
to be relaxed,
the relaxation of $\Delta\psi_y=\pi$ for the SX1's 
will be least harmful.

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:
1) removing the central dipole of the usual FMC
module; 2) increasing the length of the first and last dipoles, and 3) 
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 the collider ring is now only
$C=354.3$~m.  
This is a nice feature, since a small ring allows a 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 
$\sigma_{\rule[0.05in]{0in}{0.05in}\delta}=0.0012$, this lattice requires 
an rf voltage
of $V_{\textrm {rf}}\approx 88k$~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 in addition.
\begin{figure*}[bht!]
\centering{\psfig{figure=fig48.ps,width=6.0in,
%bbllx=73bp,bblly=342bp,bburx=530bp,bbury=658bp,
clip=}}
\caption[Lattice structure of the flexible momentum compaction module]{Example (b):
Lattice structure of the flexible momentum compaction module. $\beta_x$ (solid); $\beta_y$ (dashed); dispersion (dotted).
The maximum and minimum $\beta_x$ are 19.57 and 0.29~m, 
the maximum and minimum $\beta_y$ are 23.63 and 7.80~m,
the maximum and minimum dispersions are  1.35 and $-3.50$~m.
The natural horizontal and vertical chromaticities are $-1.77$ and $-0.92$,
giving a transition gamma of $\gamma_t=i4.43$.  
The total module length is 27.91~m
with a total bend angle of 0.917~rad.}
\label{f2}
\end{figure*}


The dynamical aperture of the lattice is computed by tracking particles with 
the code COSY INFINITY \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} 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}{dot-dashed}.
To maximize the aperture, first, the tunes must be chosen to avoid
parametric resonances.  The on-momentum 
amplitude dependent horizontal and vertical tunes
are given in Eq.~(\ref{amp-tune}).
With the designed rms
$\epsilon_x= \epsilon_y=0.169\times10^{-6}~\pi$m, 
the on-momentum tune variations are at most 0.0007. 
Second, the chromaticity variations with momentum must be as small as 
possible.  This is shown in Fig.~\ref{f3} (right hand side 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.
As the  momentum spread varies 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 make a modification of the FMC arc modules which have
a smaller dispersion swing from $-2.6$ to $+2.0$~m only.  The IR  
has not been changed except for the 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}{dashed} 
in Fig.~\ref{f3} (left hand side plot).  We see that the momentum
aperture has widened appreciably.  The dynamical aperture near on-momentum, however, is one sigma less than the lattice presented here.  
Nevertheless, it is not clear that this decrease
is significant because all
tracking has 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.

We suspect that the aperture for small momentum spread is limited by
the dramatic changes in betatron functions near the IP \cite{ohnuma}.  
These changes
are so large that Hill's equation would no longer be 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.
%\psfrag{Aperture in Sigmas}{Aperture in No. $\sigma_{\textrm{rms}}$}
%\psfrag{Momentum Spread in \%}{dp/p (\%)}
\begin{figure*}[hbt!]
\dofigs{3.25in}{new_fig49left.ps}{3.25in}{fig49rigth.ps}
%\dofigs{3.25in}{fig49left.ps}{3.25in}{fig49rigth.ps}
\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}, $7~\sigma_{\textrm{rms}}$ in \textcolor{red}{dot-dashed}, and 
TEAPOT calculation with modified FMC modules in \textcolor{magenta}{dashed}. 
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 the  scraping power of a factor of 1000; that is, for every 1000 halo muons, one remains.

\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 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*}[bth!]
\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, 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  
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) \textit{cos 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}
\epsfig{figure=costheta.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 midplane; 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=gupta22.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=0]{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 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.

\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}(a) 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. Figure~\ref{efig4}(b) shows the fractional error in
$\delta\gamma/\gamma$ obtained by fitting the rate of decay of the muons in the
collider. The accuracy using this method is much worse than that from fitting 
the polarization oscillation and cannot be used for precision measurement of
the energy. Figure~\ref{efig4}(c) shows the $\chi^2$ of the fits. 
No significant fitting biases are evident.

% 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.
