\chapter{Physics Motivation}
\label{physics}
In this chapter we cover the physics potential of the neutrino factory
accelerator complex, which includes superbeams of conventional
neutrinos that are possible using  the proton driver needed for the
factory,  and intense beams of cold
muons that become available once the muon cooling and collection
systems for the factory are in place. Once the cold muons are
accelerated and stored in the muon storage ring, we realize the full
potential of the factory in both neutrino oscillation and
non-oscillation physics.

 Cooling muons will be a learning experience. We hope that the
 knowledge gained in constructing a neutrino factory can be used to
 cool muons sufficiently to produce the first muon collider operating
 as a Higgs factory. We examine the physics capabilities of  such a
 collider, which if realized, will invariably lead to higher energy
 muon colliders with exciting physics opportunities.
%
\section{Neutrino Oscillation Physics}
%5 pages
Here we discuss~\cite{study2} the current evidence for neutrino 
oscillations, and hence
neutrino masses and lepton mixing, from solar and atmospheric data.  A review
is given of some theoretical background including models for neutrino masses
and relevant formulas for neutrino oscillation transitions.  We next mention
the near-term and mid-term experiments in this area and comment on what they
hope to measure.  We then discuss the physics potential of a muon storage ring
as a neutrino factory in the long term. 

\subsection{Evidence for Neutrino Oscillations} 

In a modern theoretical context, one generally expects nonzero neutrino masses
and associated lepton mixing.  Experimentally, there has been accumulating
evidence for such masses and mixing.  All solar neutrino experiments
(Homestake, Kamiokande, SuperKamiokande, SAGE, GALLEX and SNO) 
show a significant
deficit in the neutrino fluxes coming from the Sun~\cite{sol}. This deficit
can be explained by oscillations of the $\nu_e$'s into other weak
eigenstate(s), with $\Delta m^2_{sol}$ of the order $10^{-5}$ eV$^2$ for
solutions involving the Mikheev-Smirnov-Wolfenstein (MSW) resonant matter
oscillations~\cite{wolf}-\cite{ms} 
or of the order of $10^{-10}$ eV$^2$ for vacuum
oscillations.  Accounting for the data with vacuum oscillations (VO) requires
almost maximal mixing.  The MSW solutions include one for small mixing angle
(SMA) and one for large mixing angle (LMA).

Another piece of evidence for neutrino oscillations is the atmospheric neutrino
anomaly, observed by Kamiokande~\cite{kam}, IMB~\cite{imb}, SuperKamiokande
~\cite{sk} with the highest statistics, and by Soudan~\cite{soudan2} and MACRO
~\cite{macro}.  These data can be fit by the inference of $\nu_{\mu} \rightarrow
\nu_x$ oscillations with $\Delta m^2_{atm}\sim 3.5 \times 10^{-3}$ eV$ ^2$
~\cite{sk} and maximal mixing $\sin^2 2 \theta_{atm} = 1$.  The identification
$\nu_x = \nu_\tau$ is preferred over $\nu_x=\nu_{sterile}$, and the
identification $\nu_x=\nu_e$ is excluded by both the Superkamiokande data and
the Chooz experiment~\cite{chooz}.

In addition, the LSND experiment~\cite{lsnd} has reported 
$\bar\nu_\mu \to \bar \nu_e$ and $\nu_{\mu} \to \nu_e$ oscillations with
$\Delta m^2_{LSND} \sim 0.1 - 1$ eV$^2$ and a range of possible mixing angles.
This result is not confirmed, but also not completely ruled out, by a similar
experiment, KARMEN~\cite{karmen}.  The miniBOONE experiment at Fermilab is
designed to resolve this issue, as discussed below.

If one were to try to fit all of these experiments, then, since they involve
three quite different values of $\Delta m^2_{ij}=m(\nu_i)^2-m(\nu_j)^2$, which
could not satisfy the identity for three neutrino species, 
%
\begin{equation}
\Delta m^2_{32} + \Delta m^2_{21} + \Delta m^2_{13}=0
\label{mident}
\end{equation}
%
it would follow that one would have to introduce further neutrino(s).
Since one knows from the measurement of the $Z$ width that there are
only three leptonic weak doublets with associated light neutrinos, it
follows that such further neutrino weak eigenstate(s) would have to be
electroweak singlet(s) (``sterile'' neutrinos).  Because the LSND
experiment has not been confirmed by the KARMEN experiment, we choose
here to use only the (confirmed) solar and atmospheric neutrino data
in our analysis, and hence to work in the context of three active
neutrino weak eigenstates.


\subsection{Neutrino Oscillation Formalism} 

     In this theoretical context, consistent with solar and atmospheric data,
there are three electroweak-doublet neutrinos and the neutrino mixing 
matrix is described by,

%
\begin{equation}
U=\left(
\begin{array}{ccc}
c_{12} c_{13}&c_{13} s_{12}&s_{13} e^{-i\delta}\cr
-c_{23}s_{12}-s_{13}s_{23}c_{12}e^{i\delta}
&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\cr
s_{12}s_{23}-s_{13}c_{12}c_{23}e^{i\delta}
&-s_{23}c_{12}-s_{12}c_{23}s_{13}e^{i\delta}&c_{13}c_{23}
\end{array}
\right)K^\prime
\end{equation}
%
where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$, and $K^\prime =
diag(1,e^{i\phi_1},e^{i\phi_2})$.  The phases $\phi_1$ and $\phi_2$ do not
affect neutrino oscillations.  Thus, in this framework, the neutrino mixing
relevant for neutrino oscillations depends on the four angles $\theta_{12}$,
$\theta_{13}$, $\theta_{23}$, and $\delta$, and on two independent differences
of squared masses, $\Delta m^2_{atm.}$, which is $\Delta m^2_{32} =
m(\nu_3)^2-m(\nu_2)^2$ in the favored fit, and $\Delta m^2_{sol.}$, which may
be taken to be $\Delta m^2_{21}=m(\nu_2)^2- m(\nu_1)^2$.  Note that these
quantities involve both magnitude and sign; although in a two-species neutrino
oscillation in vacuum the sign does not enter, in the 
three-species-oscillation, that includes  both matter effects and CP violation,
the signs of the $\Delta m^2$ quantities enter and can, in principle, be
measured.

For our later discussion it will be useful to record the formulas for the
various neutrino-oscillation transitions.  In the absence of any matter effect, the probability that a (relativistic) weak neutrino eigenstate
$\nu_a$ becomes $\nu_b$ after propagating a distance $L$ is
%
\begin{eqnarray}
P(\nu_a \to \nu_b) &=& \delta_{ab} - 4 \sum_{i>j=1}^3
Re(K_{ab,ij}) \sin^2  \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr ) +
\nonumber\\&+& 4 \sum_{i>j=1}^3 Im(K_{ab,ij})
 \sin \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr )
\cos \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr )
\label{pab}
\end{eqnarray}
where
\begin{equation}
K_{ab,ij} = U_{ai}U^*_{bi}U^*_{aj} U_{bj}
\label{k}
\end{equation}
and
\begin{equation}
\Delta m_{ij}^2 = m(\nu_i)^2-m(\nu_j)^2
\label{delta}
\end{equation}
Recall that in vacuum, CPT invariance implies
$P(\bar\nu_b \to \bar\nu_a)=P(\nu_a \to \nu_b)$ and hence, for $b=a$,
$P(\bar\nu_a \to \bar\nu_a) = P(\nu_a \to \nu_a)$.  For the
CP-transformed reaction $\bar\nu_a \to \bar\nu_b$ and the T-reversed
reaction $\nu_b \to \nu_a$, the transition probabilities are given by the
right-hand side of (\ref{pab}) with the sign of the imaginary term reversed.
(Below we shall assume CPT invariance, so that CP violation is equivalent to T
violation.) 

In most cases there is only one mass scale
relevant for long-baseline neutrino oscillations, $\Delta m^2_{atm} \sim {\rm
few} \times 10^{-3}$ eV$^2$, and one possible neutrino mass spectrum is the
hierarchical one 
\begin{equation} 
\Delta m^2_{21}
= \Delta m^2_{sol} \ll \Delta m^2_{31} \approx \Delta m^2_{32}=\Delta m^2_{atm}
\label{hierarchy}
\end{equation}
In this case, CP (T) violation effects are negligibly small, so that in
vacuum
\begin{equation}
P(\bar\nu_a \to \bar\nu_b) = P(\nu_a \to \nu_b)
\label{pcp}
\end{equation}
and
\begin{equation}
P(\nu_b \to \nu_a) = P(\nu_a \to \nu_b)
\label{pt}
\end{equation}
In the absence of T violation, the second equality (\ref{pt}) would still hold
in uniform matter, but even in the absence of CP violation, the first equality
(\ref{pcp}) would not hold.  With the hierarchy (\ref{hierarchy}), the
expressions for the specific oscillation transitions are
\begin{eqnarray}
P(\nu_\mu \to \nu_\tau) & = & 4|U_{33}|^2|U_{23}|^2
\sin^2 \Bigl ( \frac{\Delta m^2_{atm}L}{4E} \Bigr ) \cr\cr
& = & \sin^2(2\theta_{23})\cos^4(\theta_{13})
\sin^2 \Bigl (\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{pnumunutau}
\end{eqnarray}
\begin{eqnarray}
P(\nu_e \to \nu_\mu) & = & 4|U_{13}|^2 |U_{23}|^2
\sin^2 \Bigl ( \frac{\Delta m^2_{atm}L}{4E} \Bigr ) \cr\cr
& = & \sin^2(2\theta_{13})\sin^2(\theta_{23})
\sin^2 \Bigl (\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{pnuenumu}
\end{eqnarray}

\begin{eqnarray}
P(\nu_e \to \nu_\tau) & = & 4|U_{33}|^2 |U_{13}|^2
\sin^2 \Bigl ( \frac{\Delta m^2_{atm}L}{4E} \Bigr ) \cr\cr
& = & \sin^2(2\theta_{13})\cos^2(\theta_{23})
\sin^2 \Bigl (\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{pnuenutau}
\end{eqnarray}

In neutrino oscillation searches using reactor antineutrinos,
i.e. tests of $\bar\nu_e \to \bar\nu_e$, the two-species mixing hypothesis used
to fit the data is
%
\beqs 
P(\nu_e \to \nu_e) & = & 1 - \sum_x P(\nu_e \to \nu_x) \cr\cr
                   & = & 1 - \sin^2(2\theta_{reactor})
\sin^2 \Bigl (\frac{\Delta m^2_{reactor}L}{4E} \Bigr )
\label{preactor}
\eeqs
%
where $\Delta m^2_{reactor}$ is the squared mass difference relevant for
$\bar\nu_e \to \bar\nu_x$.  In particular, in the upper range of values of
$\Delta m^2_{atm}$, since the transitions $\bar\nu_e \to \bar\nu_\mu$ and
$\bar\nu_e \to \bar\nu_\tau$ contribute to $\bar\nu_e$ disappearance, one has
\begin{equation}
P(\nu_e \to \nu_e) = 1 - \sin^2(2\theta_{13})\sin^2 \Bigl
(\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{preactoratm}
\end{equation}
%
i.e., $\theta_{reactor}=\theta_{13}$, and, for the value $|\Delta m^2_{32}| = 3
\times 10^{-3}$ eV$^2$ from SuperK, the CHOOZ experiment on $\bar\nu_e$ disappearance
yields the upper limit~\cite{chooz}
%
\begin{equation}
\sin^2(2\theta_{13}) < 0.1
\label{chooz}
\end{equation}
which is also consistent with conclusions from the SuperK data analysis
~\cite{sk}.

Further, the quantity ``$\sin^2(2\theta_{atm})$'' often used to fit
the data on atmospheric neutrinos with a simplified two-species mixing
hypothesis, is, in the three-generation case,
%
\begin{equation}
\sin^2(2\theta_{atm}) \equiv \sin^2(2\theta_{23})\cos^4(\theta_{13})
\label{thetaatm}
\end{equation}
%
The SuperK experiment finds that the best fit to their data is to infer
$\nu_\mu \to \nu_\tau$ oscillations with maximal mixing, and hence
$\sin^2(2\theta_{23})=1$ and $|\theta_{13}| << 1$.  The various solutions of
the solar neutrino problem involve quite different values of $\Delta m^2_{21}$
and $\sin^2(2\theta_{21})$: (i) large mixing angle solution, LMA: $\Delta
m^2_{21} \simeq {\rm few} \times 10^{-5}$ eV$^2$ and $\sin^2(2\theta_{21})
\simeq 0.8$; (ii) small mixing angle solution, SMA: $\Delta m^2_{21} \sim
10^{-5}$ and $\sin^2(2\theta_{21}) \sim 10^{-2}$, (iii) LOW: $\Delta m^2_{21}
\sim 10^{-7}$, $\sin^2(2\theta_{21}) \sim 1$, and (iv) ``just-so'': $\Delta
m^2_{21} \sim 10^{-10}$, $\sin^2(2\theta_{21}) \sim 1$.  The SuperK experiment
favors the LMA solutions~\cite{sol}; for other global fits, see, e.g.,
Gonzalez-Garcia et al.~\cite{sol}. 

We have reviewed the three neutrino oscillation phenomenology that is
consistent with solar and atmospheric neutrino oscillations. In what
follows, we will examine the neutrino experiments planned for the
immediate future that will address some of the relevant physics. We
will then review the physics potential of the Neutrino Factory.

\subsection{Relevant Near- and Mid-Term Experiments} 

There are currently intense efforts to confirm and extend the evidence for
neutrino oscillations in all of the various sectors -- solar, atmospheric, and
accelerator.  Some of these experiments are running; in addition to
SuperKamiokande and Soudan-2, these include the Sudbury Neutrino Observatory,
SNO, and the K2K long baseline experiment between KEK and Kamioka.  Others are
in development and testing phases, such as 
miniBOONE, MINOS, the CERN - Gran Sasso
program, KamLAND, and Borexino~\cite{anl}.  Among the long baseline neutrino
oscillation experiments, the approximate distances are $L \simeq 250$ km for
K2K, 730 km for both MINOS (from Fermilab to Soudan) and the proposed CERN-Gran
Sasso experiments.  

K2K is a $\nu_\mu$ disappearence experiment with a
conventional neutrino beam having a mean energy of about 1.4 GeV, going from
KEK 250 km to the SuperK detector.  It has a near detector for beam
calibration.  It has obtained results consistent with the SuperK experiment,
and has reported that its data disagree by $2\sigma$ with the no-oscillation
hypothesis~\cite{k2k}.  

MINOS is another conventional neutrino beam experiment
that takes a beam from Fermilab 730 km to a detector in the Soudan mine in
Minnesota.  It again uses a near detector for beam flux measurements and has
opted for a low-energy configuration, with the flux peaking at about 3 GeV.
This experiment is scheduled to start taking data in early 2004 and, after some
years of running, to obtain higher statistics than the K2K experiment and to
achieve a sensitivity down to the level $|\Delta m^2_{32}| \sim
10^{-3} $eV$^2$.  

The CERN - Gran Sasso program will come on later, around
2005.  It will use  a higher-energy neutrino beam from CERN to the
Gran Sasso deep underground laboratory in Italy.  This program will emphasize
detection of the $\tau$'s produced by the $\nu_\tau$'s that result from the
inferred neutrino oscillation transition $\nu_\mu \to \nu_\tau$.  The OPERA
experiment will do this using emulsions~\cite{opera}, while the ICARUS proposal
uses a liquid argon chamber~\cite{icanoe}.  

Plans for the Japan Hadron Facility (JHF),
also called the High Intensity Proton Accelerator (HIPA), include the use of 
a 1 MW
proton driver to produce a high-intensity conventional neutrino beam with a
pathlength 300 km to the SuperK detector~\cite{jhf}.  Moreover, at Fermilab,
the miniBOONE experiment is scheduled 
to start data taking in the near future and to confirm or
refute the LSND claim after a few years of running.

There are several relevant solar neutrino experiments.  The SNO
experiment is currently running and has recently reported their first
results that confirm solar neutrino oscillations~\cite{snolatest}.
These involve measurement of the solar neutrino flux and energy
distribution using the charged current reaction on heavy water, $\nu_e
+ d \to e + p + p$.  They are expected to report on the neutral
current reaction $\nu_e + d \to \nu_e + n + p$ shortly. The neutral
current rate is unchanged in the presence of oscillations that involve
standard model neutrinos, since the neutral current channel is equally
sensitive to all the three neutrino species.  If however, sterile
neutrinos are involved, one expects to see a depletion in the neutral
current channel also.

The KamLAND experiment in Japan is scheduled to  begin
taking data in late 2001.  This is a reactor antineutrino experiment using
baselines of  100 - 250 km and will search for $\bar\nu_e$ disappearance
and is sensitive to the solar neutrino oscillation scale.
On a similar time scale, the Borexino experiment in Gran Sasso is scheduled 
to turn
on and  measure the $^7$Be neutrinos from the sun.  These experiments
should help us determine which of the various solutions to the solar neutrino
problem is preferred, and hence the corresponding values of $\Delta m^2_{21}$
and $\sin^2(2\theta_{12})$.

This, then, is the program of relevant experiments during the period
2000-2010.  By the end of this period, we may expect that much will be learned
about neutrino masses and mixing.  However, there will remain several
quantities that will not be well measured and which can be measured by a
neutrino factory. 

\subsection{Oscillation Experiments at a Neutrino Factory } 
\label{neuf}
Although a neutrino factory based on a muon storage ring will turn on
several years after this near-term period in which K2K, MINOS, and the
CERN-Gran Sasso experiments will run, we believe that it has a
valuable role to play, given the very high-intensity neutrino beams of
fixed flavor-pure content, including, uniquely, $\nu_e$ and
$\bar\nu_e$ beams in addition to $\nu_\mu$ and $\bar\nu_\mu$ beams. A
conventional positive charge selected neutrino beam  is primarily
$\nu_\mu$ with some admixture of $\nu_e$'s and other flavors from $K$
decays(O(1\%) of the total charged current rate) and  the fluxes of these
neutrinos can only be fully understood after measuring the charged
particle spectra from the target with high accuracy.  In contrast,
the potential of
the neutrino beams from a muon storage ring is that the neutrino beams
would be of extremely high purity: $\mu^-$ beams would yield 50 \%
$\nu_\mu$ and 50 \% $\bar\nu_e$, and $\mu^+$ beams, the charge
conjugate neutrino beams.  Furthermore, these could be produced with
high intensities and low divergence that make it possible to go 
to longer baselines.


In what follows, we shall take the design values from Study-II
of $10^{20}$ $\mu$ decays per ``Snowmass year'' ($10^7$ sec) as being typical.
The types of neutrino oscillations that can be searched for with the Neutrino
Factory based on the muon storage ring are listed in 
table~\ref{tab:nu-osc-ratings} for the 
case of $\mu^-$ which decays to $ \nu_\mu e^- \bar\nu_e$: 
\begin{table}
\caption[Neutrino Oscillation Modes]{Neutrino-oscillation modes that can be studied with conventional
neutrino beams or with beams from a Neutrino Factory, with ratings as to
degree of difficulty in each case; * = well or easily measured, $\surd$ =
measured poorly or with difficulty, --- = not
measured.\label{tab:nu-osc-ratings}}
\begin{center}
\begin{tabular}{|llll|}
\hline
& & Conventional & Neutrino \\
\raisebox{1.5ex}[0pt]{Measurement } & \raisebox{1.5ex}[0pt]{Type} & beam &
Factory \\
\hline
$\nu_\mu\to\nu_\mu,\,\nu_\mu\to\mu^-$ & survival & $\surd$ & *\\
$\nu_\mu\to\nu_e,\,\nu_e\to e^-$ & appearance & $\surd$ & $\surd$\\
$\nu_\mu\to\nu_\tau,\,\nu_\tau\to\tau^-,\,\tau^-\to(e^-,\mu^-)...$  &
appearance & $\surd$ & $\surd$ \\ \hline
$\bar \nu_e\to\bar \nu_e,\,\bar\nu_e\to e^+$ & survival  & --- & $*$\\
$\bar\nu_e\to\bar\nu_\mu,\,\bar\nu_\mu\to\mu^+$  & appearance & --- &  $*$
\\
$\bar\nu_e\to\bar\nu_\tau,\,\bar\nu_\tau\to\tau^+,\,\tau^+\to(e^+,\mu^+)...$
& appearance & ---& $\surd$ \\
\hline
\end{tabular}
\end{center}
\end{table}

It is clear from the 
processes listed that since the beam contains both neutrinos and
antineutrinos, the only way to determine  the flavor of the 
parent neutrino  is to
determine the identity of the final state charged lepton and measure its
charge.  

A capability unique to the Neutrino Factory will be the 
measurement of the oscillation $\bar\nu_e \to \bar\nu_\mu$,
giving a wrong-sign $\mu^+$.  Of greater difficulty would be the measurement of
the transition $\bar\nu_e \to \bar\nu_\tau$, giving a $\tau^+$ which will decay
part of the time to $\mu^+$.  These physics goals mean that a detector must
have excellent capability to identify muons and measure their charges.
Especially in a steel-scintillator detector, the oscillation $\nu_\mu \to
\nu_e$ would be difficult to observe, since it would be 
difficult to distinguish
an electron shower from a hadron shower.  From the above formulas for
oscillations, one can see that, given a knowledge of $|\Delta m^2_{32}|$ and
$\sin^2(2\theta_{23})$ that one will have by the time a neutrino factory is
built, the measurement of the $\bar\nu_e \to \bar\nu_\mu$ transition yields the
value of $\theta_{13}$.

To get a rough idea of how the sensitivity of 
an oscillation experiment would scale with energy and baseline length, 
recall that the event rate in the absence of oscillations is 
simply the neutrino flux times the cross section.  
First of all, neutrino cross sections in the region above
about 10 GeV (and slightly higher, for $\tau$ production) grow linearly with
the neutrino energy.  Secondly, the beam divergence is
a function of the initial muon storage ring energy; 
this divergence yields a flux, as a
function of $\theta_d$, the angle of deviation from the forward direction, that
goes like $1/\theta_d^2 \sim E^2$.  Combining this with the linear $E$
dependence of the neutrino cross section 
and the overall $1/L^2$ dependence of the flux far from the
production region, one finds that the event rate goes like 
\begin{equation} 
\frac{dN}{dt} \sim \frac{E^3}{L^2}
\label{eventrate}
\end{equation}
We base our discussion on the event rates given in the 
Fermilab Neutrino Factory study~\cite{INTRO:ref9}. For
a stored muon energy of 20 GeV,  and a distance of 
$L=2900$ to the WIPP Carlsbad site in New Mexico, these event rates amount to 
several thousand events per kton of detector per year, i.e. they are
satisfactory for the physics program.   This is also true for the other
pathlengths under consideration, namely $L=2500$ km from BNL to Homestake and
$L=1700$ km to Soudan.  A usual racetrack design would only allow a single
pathlength $L$, but a bowtie design could allow two different pathlengths
(e.g.,~\cite{zp}). 

One could estimate that at a time when the neutrino factory turns on, $|\Delta
m^2_{32}|$ and $\sin^2(2\theta_{23})$ would be known at perhaps the 20 \% level
(we emphasize that future projections such as this are obviously uncertain).
The neutrino factory will significantly improve precision in these parameters,
as can be seen from figure~\ref{fig:30gev_disap_fit} which shows the error ellipses possible for a 30~GeV muon storage ring.
\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.0in]{30gev_disap_fit.eps}}
\bigskip
\caption[Error ellipses in  $\delta m^2$ sin$^2 2\theta$ space for a neutrino factory]
{ \label{fig:30gev_disap_fit}
Fit to muon neutrino survival distribution for $E_\mu=30$ GeV and $L=2800$~km for 10
pairs of sin$^2 2\theta$, $\delta m^2$ values. For each fit, the
1$\sigma$,\ 2$\sigma$
and 3$\sigma$ contours are shown. The generated points are indicated by the
dark
rectangles and the fitted values by stars. The SuperK 68\%, 90\%, and 99\% 
confidence
levels are superimposed. Each point is labelled by the predicted number of 
signal events for that point.}
\end{figure}
In addition the neutrino factory can contribute to the measurement of: (i) 
$\theta_{13}$, as discussed above; (ii) measurement of the sign of $\Delta
m^2_{32}$ using matter effects; and (iii) possibly a measurement of CP
violation in the leptonic sector, if $\sin^2(2\theta_{13})$,
$\sin^2(2\theta_{21})$, and $\Delta m^2_{21}$ are sufficiently large.  To
measure the sign of $\Delta m^2_{32}$, one uses the fact that matter effects
reverse sign when one switches from neutrinos to antineutrinos, and carries out
this switch in the charges of the stored $\mu^\pm$.  We elaborate on this next.

\subsection{Matter Effects} 

With the advent of the muon storage ring, the distances at which one 
can place detectors
are large enough so that for the first time matter effects can be exploited in
accelerator-based oscillation experiments.  Simply put, matter effects are the
matter-induced oscillations which neutrinos undergo along their flight path
through the Earth from the source to the detector.  Given the typical density
of the earth, matter effects are important for the neutrino energy range $E
\sim O(10)$ GeV and $\Delta m^2_{32} \sim 10^{-3}$ eV$^2$ values relevant for
the long baseline experiments. Matter effects in neutrino propagation were 
first pointed out by Wolfenstein~\cite{wolf} and Mikheyev and Smirnov~\cite{ms}. 
(See the papers~\cite{dgh}--\cite{cpv} for details  of the MSW effect
and its relevance to neutrino factories.) In brief,
the transition
probabilities
in the leading oscillation approximation for propagation through
matter of constant density are
%
\begin{eqnarray}
P(\nu_e\to \nu_\mu) &=& s_{23}^2 \sin^2 2\theta_{13}^m
\sin^2\Delta_{32}^m \,,
\nonumber\\
P(\nu_e\to \nu_\tau) &=& c_{23}^2 \sin^2 2\theta_{13}^m
\sin^2\Delta_{32}^m \,,
\label{eq:probs}\\
P(\nu_\mu\to \nu_\tau) &=& \sin^2 2\theta_{23} \left[
(\sin\theta_{13}^m)^2 \sin^2\Delta_{21}^m +(\cos\theta_{13}^m)^2
\sin^2\Delta_{31}^m -(\sin\theta_{13}^m\cos\theta_{13}^m)^2
\sin^2\Delta_{32}^m \right]
\,. \nonumber
\end{eqnarray}
%
 The oscillation arguments are given by
%
\begin{equation}
\Delta_{32}^m = \Delta_0 S \,,\qquad
\Delta_{31}^m = \Delta_0 {1\over2} \left[ 1+{A\over\delta m^2_{32}}+S \right]
\,, \qquad
\Delta_{21}^m = \Delta_0 {1\over2} \left[ 1+{A\over\delta m^2_{32}}-S \right]
\,, \label{eq:arg}
\end{equation}
%
where $S$ is given by
%
\begin{equation}
S \equiv \sqrt{ \left( {A\over\delta m^2_{32}}-\cos2\theta_{13}
\right)^2 + \sin^2 2\theta_{13}} \,,
\label{eq:S}
\end{equation}
%
and
%
\begin{equation}
\Delta_0 = {\delta m^2_{32} L\over 4E} = 1.267 {\delta m^2_{32}
{\rm\,(eV^2)} \; L {\rm\ (km)} \over E_\nu {\rm\ (GeV)}}
\,,
\label{eq:arg0}
\end{equation}
%
\begin{equation}
\sin^2 2\theta_{13}^m = {\sin^2 2\theta_{13}\over
\left({A\over\delta m^2_{32}} - \cos 2\theta_{13} \right)^2
+ \sin^2 2\theta_{13}} \,. \label{eq:sin}
\end{equation}
%
The amplitude $A$ for $\nu_e e$ forward scattering in matter is given
by
%
\begin{equation}
A = 2\sqrt2 G_F N_e E_\nu = 1.52 \times 10^{-4}{\rm\,eV^2} Y_e
\rho({\rm\,g/cm^3}) E({\rm\,GeV}) \,.
\label{eq:A}
\end{equation}
%
Here $Y_e$ is the electron fraction and $\rho(x)$ is the matter
density. For neutrino trajectories that pass through the earth's
crust, the average density is typically of order 3~gm/cm$^3$ and $Y_e
\simeq 0.5$.  The oscillation probability $P(\nu_e\to \nu_\mu)$ is
directly proportional to $\sin^2 2\theta_{13}^m$, which is
approximately proportional to $\sin^2 2\theta_{13}$. There is a
resonant enhancement for
%
\begin{equation}
  \cos2\theta_{13} = \frac{A}{\delta m^2_{32}} \,.
\end{equation}
%
For electron neutrinos, $A$ is positive and the resonance enhancement
occurs for positive values of $\delta m^2_{32}$ for $\cos2\theta_{13}>0$.
 The reverse is true
for electron anti-neutrinos and the enhancement occurs for negative
values of $\delta m^2_{32}$.  Thus for a neutrino factory operating
with positive stored muons (producing a $\nu_e$ beam) one expects an
enhanced production of opposite sign ($\mu^-$) charged-current events
as a result of the oscillation $\nu_e\to \nu_\mu$ if $\delta m^2_{32}$
is positive and vice versa for stored negative
beams.



Figure~\ref{fig:hists}~\cite{barger-raja} shows the wrong-sign 
muon appearance spectra 
 as function of $\delta m^2_{32}$ for both $\mu^+$ and
$\mu^-$ beams for both signs of $\delta m^2_{32}$ at a baseline of
2800~km. The resonance enhancement in wrong sign muon production is
clearly seen in Fig.~\ref{fig:hists} (b) and (c).

%
\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.0in]{plt_paper_503.eps}}
\caption[Wrong sign muon appearance rates and sign of  $\delta m^2_{32}$]
{The wrong sign muon appearance rates for a 20 GeV muon storage ring at
a baseline of 2800~km with 10$^{20}$ decays and a 50 kiloton detector
for (a)~$\mu^+$ stored and negative $\delta m^2_{32}$\,, (b)~$\mu^-$ stored
and negative $\delta m^2_{32}$\,, (c)~$\mu^+$ stored and positive $\delta
m^2_{32}$\,,
(d)~$\mu^-$ stored and positive $\delta m^2_{32}$. The values of $|\delta
m^2_{32}|$ range from 0.0005 to 0.0050 eV$^2$ in steps of 0.0005~eV$^2$.  
Matter enhancements are evident in (b) and (c).
\label{fig:hists}}
\end{figure}
%
By comparing these (using first a stored $\mu^+$ beam and then a stored $\mu^-$
beam) one can thus determine the sign of $\Delta m^2_{32}$ as well as the value
of $\sin^2(2\theta_{13})$.  
Figure~\ref{fig:sigmas}~\cite{barger-raja} shows the difference in negative
log-likelihood between a
correct and wrong-sign mass hypothesis expressed as a number of
equivalent Gaussian standard deviations versus baseline length for
muon storage ring energies of 20, 30, 40 and 50~GeV. The values of the
oscillation parameters are for the LMA scenario with 
$\sin^22\theta_{13}=0.04$. 
Figure~\ref{fig:sigmas}(a) is for 10$^{20}$ decays
for each sign of stored energy and a 50 kiloton detector and positive
$\delta m^2_{32}$ , (b) is for negative $\delta m^2_{32}$ for various
values of stored muon energy. Figures~\ref{fig:sigmas} ;(c) and (d)
show the corresponding curves for 10$^{19}$ decays and a 50 kiloton
detector. An entry-level machine would permit one to perform a
5$\sigma$ differentiation of the sign of $\delta m^2_{32}$ at a
baseline length of $\sim$2800~km.
%3
\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.0in]{letter_plots.eps}}
\caption[$\delta m_{32}^2$ sign determination at a Neutrino Factory]
{The statistical significance (number of standard deviations) 
with which the
sign of $\delta m_{32}^2$ can be determined versus baseline length for
various muon storage ring energies. The results are shown for 
a 50~kiloton detector, and (a)~10$^{20}$
$\mu^+$ and $\mu^-$ decays and positive values of $\delta m_{32}^2$;
(b)~10$^{20}$ $\mu^+$ and $\mu^-$ decays and
negative values of $\delta m_{32}^2$; (c)~10$^{19}$ $\mu^+$ and 
$\mu^-$ decays and positive values of $\delta
m_{32}^2$; (d)~10$^{19}$ $\mu^+$ and $\mu^-$
decays and negative values of $\delta m_{32}^2$.
\label{fig:sigmas}}
\end{figure}

For the Study II design, in accordance with the previous
Fermilab study~\cite{INTRO:ref9}, 
one estimates  that it is possible to determine the sign of $\delta m^2_{32}$
even if 
$\sin^2(2\theta_{13}) is as small as \sim 10^{-3}$.

\subsection{CP Violation}

CP violation is measured by the (rephasing-invariant) Jarlskog product
%
\beqs
J & =& Im(U_{ai}U_{bi}^* U_{aj}^* U_{bj}) \cr\cr
& & = \frac{1}{8}
\sin(2\theta_{12})\sin(2\theta_{13})\cos(\theta_{13})\sin(2\theta_{23})\sin
\delta 
\eeqs
%
Leptonic CP violation also requires that each of the leptons in each charge
sector be nondegenerate with any other leptons in this sector; this is,
course, true of the charged lepton sector and, for the neutrinos, this requires
$\Delta m^2_{ij} \ne 0$ for each such pair $ij$.  In the quark sector, $J$ is 
known to be small: $J_{CKM} \sim O(10^{-5})$.  
A promising asymmetry to measure is $P(\nu_e \to \nu_\mu)-P(\bar\nu_e - 
\bar\nu_\mu)$.  As an illustration, in the absence of matter effects, 
%
\beqs
P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu) & = & -4J(\sin 2\phi_{32}+
\sin 2\phi_{21} + \sin 2\phi_{13}) \cr\cr
& = & -16J \sin \phi_{32} \sin \phi_{13} \sin \phi_{21} 
\label{pnuenumudif}
\eeqs
%
where
%
\begin{equation}
\phi_{ij} = \frac{\Delta m^2_{ij}L}{4E}
\label{phiijdef}
\end{equation}
%
In order for the CP violation in eq. (\ref{pnuenumudif}) to be large enough to
measure, it is necessary that $\theta_{12}$, $\theta_{13}$, and $\Delta
m^2_{sol} = \Delta m^2_{21}$ not be too small. From atmospheric neutrino data,
we have $\theta_{23}\simeq \pi/4$ and $\theta_{13} << 1$.  If LMA describes
solar neutrino data, then $\sin^2(2\theta_{12}) \simeq 0.8$, so $J \simeq
0.1\sin(2\theta_{13})\sin \delta$.  For example, if
$\sin^2(2\theta_{13})=0.04$, then $J$ could be $>> J_{CKM}$.  Furthermore, for
 parts of  the LMA phase space where  
$\Delta m^2_{sol} \sim 4 \times 10^{-5}$ eV$^2$ 
the CP violating effects might be observable. In the absence of matter, one
would measure the asymmetry
%
\begin{equation}
\frac{P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu)}{
P(\nu_e \to \nu_\mu) + P(\bar\nu_e \to \bar\nu_\mu)} =
-\frac{\sin(2\theta_{12})\cot(\theta_{23})\sin\delta \sin \phi_{21}}{
\sin \theta_{13}}
\end{equation}
%
However, in order to optimize this ratio, because of the smallness of $\Delta
m^2_{21}$ even for the LMA, one must go to large pathlengths $L$, and here
matter effects are important.  These make leptonic CP violation challenging to
measure, because, even in the absence of any intrinsic CP violation, these
matter effects render the rates for $\nu_e \to \nu_\mu$ and $\bar\nu_e \to
\bar\nu_\mu$ unequal since the matter interaction is opposite in sign for $\nu$
and $\bar\nu$.  One must therefore subtract out the matter effects in order to
try to isolate the intrinsic CP violation.  Alternatively, one might think of
comparing $\nu_e \to \nu_\mu$ with the time-reversed reaction $\nu_\mu \to
\nu_e$.  Although this would be equivalent if CPT is valid, as we assume, and
although uniform matter effects are the same here, the detector response is
quite different and, in particular, it is quite difficult to identify $e^\pm$.
Results from SNO and KamLAND testing the LMA will help further planning.

The Neutrino Factory provides an ideal set of controls to measure CP
violation effects since we can fill the storage ring with both $\mu^+$
and $\mu^-$ particles and measure the ratio of the number of events
$\anti\nu_e\rightarrow \anti\nu_\mu$/$\nu_e\rightarrow\nu_\mu$.
Figure~\ref{cpfig} shows this ratio for a Neutrino Factory with
10$^{21}$ decays and a 50~kilo-ton detector as a function of the
baseline length. The ratio depends on the sign of $\delta
m^2_{32}$. The shaded band around either curve shows the variation of
this ratio as a function of the CP violating phase $\delta$. The
number of decays needed to produce the error bars shown is directly
proportional to $sin^2\theta_{13}$, which for the present example is
set to 0.004. Depending on the magnitude of $J$, one may be driven to
build a Neutrino Factory just to understand CP violation in the lepton
sector, which could have a significant role in explaining the baryon
asymmetry of the Universe~\cite{yanag}.

\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.0in]{cp_fig.eps}}
\bigskip
\caption[CP violation effects in a neutrino factory]
{ \label{cpfig}
Predicted ratios of wrong-sign muon event rates when positive and
negative muons are stored in a 20~GeV neutrino factory, shown as a
function of baseline.  A muon measurement threshold of 4~GeV is
assumed. The lower and upper bands correspond respectively to negatve
and positive $\delta m^2_{32}$. The widths of the bands show how the
predictions vary as the CP violating phase $\delta$ is varied from
$-\pi$/2 to $\pi$/2, with the thick lines showing the predictions for
$\delta$=0. The statistical error bars correspond to a
high-performance neutrino factory yielding a data sample of 10$^{21}$
decays with a 50~kiloton detector. Figure is based on calculations
presented in~\cite{barger-entry} }

\end{figure}



\section{Physics Potential of Superbeams}

It is possible to extend the reach of the current conventional
neutrino experiments by enhancing the capabilities of the proton
sources that drive them. These enhanced neutrino beams have been
termed ``superbeams'' and form an intermediate step on the way to a
neutrino factory. Their capabilities have been explored in  recent
papers~\cite{superbeams,bargersuperbeam}. These articles consider
the capabilities of enhanced proton drivers at (i) the proposed
0.77~MW 50~GeV proton synchrotron at the Japan Hadron Facility
(JHF)~\cite{jhfloi}, (ii) a 4~MW upgraded version of the JHF, (iii) a
new $\sim 1$~MW 16~GeV proton driver~\cite{brighter} that would replace
the existing 8~GeV Booster at Fermilab, or (iv) a fourfold intensity
upgrade of the 120~GeV Fermilab Main Injector (MI) beam (to 1.6~MW)
that would become possible once the upgraded (16~GeV) Booster was
operational.  Note that the 4~MW 50~GeV JHF and the 16~GeV upgraded
Fermilab Booster are both suitable proton drivers for a neutrino
factory. The conclusions of both reports are that superbeams will
extend the reaches in the oscillation parameters of the current
neutrino experiments but ``the sensitivity at a neutrino factory to
$CP$ violation and the neutrino mass hierarchy extends to values of
the amplitude parameter $\sin^2 2\theta_{13}$ that are one to two
orders of magnitude lower than at a superbeam''~\cite{bargersuperbeam}.



To illustrate these points, we choose one of the most favorable superbeam 
scenarios 
studied: a 1.6~MW NuMI--like high energy beam with $L = 2900$~km, detector 
parameters corresponding to the liquid argon scenario in 
~\cite{bargersuperbeam}, and oscillation parameters 
$|\delta m^2_{32}| = 3.5 \times 10^{-3}$~eV$^2$ and 
$\delta m^2_{21} = 1 \times 10^{-4}$~eV$^2$. 
The calculated three--sigma error ellipses in the 
$\left(N(e^+), N(e^-)\right)$--plane are shown in Fig.~\ref{fig:signdm2}
for both signs of $\delta m^2_{32}$, with the curves corresponding to 
various CP--phases $\delta$ (as labelled). The magnitude of 
the $\nu_\mu \to \nu_e$ oscillation amplitude parameter 
$\sin^2 2\theta_{13}$ varies along each curve, as indicated. The 
two groups of curves, which correspond to the two signs of $\delta m^2_{32}$, 
are separated by more than $3\sigma$ provided 
$\sin^2 2\theta_{13} \gsim 0.01$. Hence the mass heirarchy can be determined 
provided the $\nu_\mu \to \nu_e$ oscillation amplitude is not less than an 
order of magnitude below the currently excluded region. Unfortunately, within 
each group of curves, the CP--conserving predictions are separated from the 
maximal CP--violating predictions by at most $3\sigma$. Hence, it will 
be difficult to conclusively establish CP violation in this scenario.

Note for comparison that a very long baseline experiment at a neutrino 
factory would be able to observe $\nu_e \to \nu_\mu$ oscillations and 
determine the sign of $\delta m^2_{32}$ for values of $\sin^2 2\theta_{13}$ 
as small as O(0.0001)~! This is illustrated in Fig.~\ref{fig:nufact}.
A Neutrino Factory, thus outperforms a conventional superbeam in its ability to
determine the sign of  $\delta m^2_{32}$.
Comparing Fig.~\ref{fig:signdm2} and Fig.~\ref{fig:nufact} one sees that 
the value of   $\sin^2 2\theta_{13}$, which has yet to be measured, 
will determine the parameters of the first neutrino factory.
%
\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.0in]{fig15_superbeams.ps}}
\caption[Error ellipses for superbeams for electron appearance]
{Three--sigma error ellipses in the 
$\left(N(e^+), N(e^-)\right)$--plane, shown for 
$\nu_\mu \to \nu_e$ and $\bar\nu_\mu \to \bar\nu_e$ oscillations 
in a NuMI--like 
high energy neutrino beam driven by a 1.6~MW proton driver. 
The calculation assumes a liquid argon detector with the parameters 
listed in \cite{superbeams}, a baseline of 2900~km, 
and 3~years of running with neutrinos, 6~years running 
with antineutrinos. 
Curves are shown for different CP--phases $\delta$ (as labelled), and 
for both signs of $\delta m^2_{32}$ with 
$|\delta m^2_{32}| = 0.0035$~eV$^2$, and 
the sub--leading scale $\delta m^2_{21} = 10^{-4}$~eV$^2$. 
Note that $\sin^22\theta_{13}$ varies along the curves from
0.0001 to 0.01, as indicated~\cite{bargersuperbeam}.
}
\label{fig:signdm2}
\end{figure}
%
\begin{figure}[tbh!]
%\epsfxsize=3.1in\epsffile{bgrwfig12.eps}
\centerline{\includegraphics[width=4.0in]{fig16_superbeams.ps}}
\caption[Error ellipses for neutrino factory for muon appearance]
{Three--sigma error ellipses in the 
$\left(N(\mu+), N(\mu-)\right)$--plane, shown for a 20~GeV neutrino 
factory delivering $3.6\times10^{21}$ useful muon decays and
$1.8\times10^{21}$ antimuon decays, with a 50~kt
detector at $L = 7300$~km, $\delta m^2_{21} = 10^{-4}$~eV$^2$, 
and $\delta = 0$. Curves are shown for both signs of
$\delta m^2_{32}$; $\sin^22\theta_{13}$ varies along the curves from
0.0001 to 0.01, as indicated~\cite{bargersuperbeam}.
}
\label{fig:nufact}
\end{figure}


Finally, we compare the superbeam $\nu_\mu \to \nu_e$ reach with the 
corresponding neutrino factory $\nu_e \to \nu_\mu$ reach in 
Fig.~\ref{fig:reach}, which shows the $3\sigma$ sensitivity contours in 
the $(\delta m^2_{21}, \sin^2 2\theta_{13})$--plane. The superbeam 
$\sin^2 2\theta_{13}$ reach of a few $\times 10^{-3}$ is almost independent 
of the sub--leading scale $\delta m^2_{21}$. However, since the neutrino 
factory probes oscillation amplitudes $O(10^{-4})$ the sub--leading effects 
cannot be ignored, and $\nu_e \to \nu_\mu$ events 
would be observed at a neutrino factory 
over a significant range 
of $\delta m^2_{21}$ even if $\sin^2 2\theta_{13} = 0$.
%
\begin{figure}[tbh!]
\centerline{\includegraphics[width=4.0in]{fig20_superbeams.ps}}
\caption[Comparison of superbeams and neutrino factories]
{Summary of the $3\sigma$ level sensitivities for the 
observation of $\nu_\mu \to \nu_e$ at various MW--scale superbeams 
(as indicated) with liquid argon ``A'' and water cerenkov ``W'' detector 
parameters, and the observation of $\nu_e \to \nu_\mu$ in a 50~kt detector 
at 20, 30, 40, and 50~GeV neutrino factories delivering $2 \times 10^{20}$ 
muon decays in 
the beam forming straight section. The limiting $3\sigma$ contours are 
shown in the $\delta m^2_{21}, \sin^2 2\theta_{13}$--plane. All curves 
correspond to 3~years of running. The grey shaded 
area is already excluded by current experiments.
}
\label{fig:reach} 
\end{figure} 


\section{Non-oscillation physics at a Neutrino Factory}
The study of the utility of intense neutrino beams from a muon storage ring in
determining the parameters governing non-oscillation physics was begun
in 1997~\cite{rajageer}. More complete studies can be found in
~\cite{INTRO:ref9} and recently a European group has brought out an
extensive study on this topic~\cite{cern-nonosc}. We quote their
conclusions here verbatim.  

``In the case of determinations of the
partonic densities of the nucleon, we proved that the \nufact\ could
significantly improve the already good knowledge we have today. In the
unpolarized case, the knowledge of the valence distributions would
improve by more than one order of magnitude, in the kinematical region
$x\gsim 0.1$, which is best accessible with 50~GeV muon beams. The
individual components of the sea ($\bar{u}$, $\bar{d}$, ${s}$ and
$\bar{s}$), as well as the gluon, would be measured with relative
accuracies in the range of 1--10\%, for $0.1\lsim x \lsim 0.6$. The
high statistics available over a large range of $Q^2$ would
furthermore allow the accurate determination of higher-twist
corrections, strongly reducing the theoretical systematics that affect
the extraction of $\as$ from sum rules and global fits.


``In the case of polarized densities, we stressed the uniqueness of the
\nufact\ as a means of disentangling quark and antiquark distributions,
and their first moments in
particular. These can be determined at the level of few per cent for
up and down, and 10\% for the strange, sufficient to distinguish
between theoretical scenarios, and thus allowing a full understanding
of the proton spin structure. A potential ability
to pin down the shapes of individual flavour components with accuracies at the
level of few per cent is  limited by the mixing with the 
polarized gluon. To identify this possible
weakness of the \nufact\ polarized-target programme, it was crucial to
perform our analysis at the NLO; we showed in fact that any study
based on the LO formalism would have resulted in far too optimistic
conclusions. This holds true both in the
case of determinations based on global fits and on direct
extractions using flavour tagging in the final state. 
Our conclusion here is that a full exploitation of the \nufact\
potential for polarized measurements of the shapes of
individual partonic densities requires an a-priori knowledge of
the polarized gluon density. It is hoped that the new information
expected to arise from the forthcoming set of polarized DIS
experiments at CERN, DESY and RHIC will suffice.

``The situation is also very bright for measurements of C--even
distributions. Here, the first moments of singlet, triplet and octet
axial charges can be measured with
accuracies which are up to one order of magnitude better than the
current uncertainties. In particular, the improvement in the
determination of the singlet axial charge would allow a definitive
confirmation or refutation of the anomaly scenario compared to the
`instanton' or `skyrmion' scenarios, at least if the theoretical
uncertainty originating from the small--$x$ extrapolation can be kept under
control. The measurement of the octet axial charge with a few percent
uncertainty will allow a determination of the strange contribution to
the proton spin better than 10\%, and allow stringent tests of models
of $SU(3)$ violation when compared to the direct determination from
hyperon decays.

``The measurement of two fundamental constants of nature, $\as(M_Z)$ and
$\sin^2\theta_W$, will be possible using a variety of techniques. At
best the
accuracy of these measurements will match or slightly improve
the accuracy available
today, although the
measurements at the \nufact\ are subject to different systematics and
therefore provide an important consistency check of current data.
In the case of $\as(M_Z)$, the dependence of the results on the
modeling of higher-twist corrections both in the structure function
fits and in the GLS sum rule is significantly reduced relative to
current measurements, as mentioned above. 
In the case of $\sin^2\theta_W$, its determination via $\nu e$
scattering at the \nufact\ has an uncertainty of approximately
$2\times 10^{-4}$, dominated by the statistics and the luminosity
measurement. This error is comparable to what already known
today from EW measurements in $Z^0$ decays. Compared to these,
however, this determination would
improve current low-energy extractions, and be 
subject to totally different systematic uncertainties. It would also be
sensitive to different classes of new-physics contributions.
The extrapolation
to $Q=M_Z$ is affected, at the same level of uncertainty, by the
theoretical assumptions used in the evaluation of the hadronic-loop
corrections to $\gamma$-$Z$ mixing. The determination via DIS, on the
other hand, is limited by the uncertainties on the heavy-flavour
parton densities. As shown earlier, these should be significantly
reduced using the \nufact\ data themselves.

``In several other areas, the data from the \nufact\ will allow 
quantitative studies to be made of phenomena that, so far have only
been explored at a mostly qualitative level. This is the case of the
exclusive production of charmed mesons and baryons (leading to very
large samples, suitable for precise extractions of branching ratios
and decay constants), of the study of spin-transfer
phenomena, and of the study of nuclear effects in DIS. While nuclear
effects could be bypassed at the \nufact\ by using  hydrogen
targets directly, the flavour separation of partonic densities will require
using also targets containing neutrons. This calls for an accurate
understanding of nuclear effects. The ability to run with both $H$ and
heavier targets will in turn provide rich data sets useful for
quantitative studies of nuclear models.
The study of $\Lambda$ polarization both in the target and in the
fragmentation regions, will help clarifying the intriguing problem of
spin transfer. We reviewed several of the existing models, and
indicated how semi-inclusive neutrino DIS will allow the
identification of the
right ones, as well as providing input for the measurement of
polarized fragmentation functions.

``Finally, we presented some cases of exploration for physics beyond the
SM using the \nufact\ data. Although the neutrino beam energies
considered in our work are well below any reasonable threshold for new
physics, the large statistics makes it possible to search for
manifestations of virtual effects. The exchange of new gauge bosons
decoupled from the first generation of quarks and leptons can be seen
via enhancements of the inclusive charm production rate, with a
sensitivity well beyond the present limits. Rare
lepton-flavour-violating decays of muons in the ring could be tagged
in the DIS final states
through the detection of wrong-sign electrons and muons, or of prompt
taus. Once again, the sensitivity at the \nufact\ goes well beyond
existing limits...''


\section{Physics that can be done with Intense Cold Muon Beams}
Experimental studies of muons at low and medium energies have had a
long and distinguished history, starting with the first search for
muon decay to electron plus gamma-ray~\cite{Hincks-Pontecorvo}, and
including along the way the 1957 discovery of the nonconservation of
parity, in which the $g$ value and magnetic moment of the muon were
first measured~\cite{Garwinetal}.  The years since then have brought
great progress: limits on the standard-model-forbidden decay $\mu\to
e\gamma$ have dropped by nine orders of magnitude, and the muon
anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ has yielded one of the
more precise tests ($\approx1$ ppm) of physical theory, as well as a
possible hint of physics beyond the standard model~\cite{BNLg-2}.

The front end of a neutrino factory has the potential to provide
$\sim10^{21}$ muons per year, five orders of magnitude beyond the most
intense beam currently available.\footnote{The $\pi$E5 beam at PSI,
Villigen, providing a maximum rate of $10^9$ muons/s~\cite{Edgecock}.}
Such a facility could enable precision measurements of the muon
lifetime $\tau_\mu$ and Michel decay parameters as well as sensitive
searches for lepton-flavor nonconservation (LFV), a possible ($P$- and
$T$-violating) muon electric dipole moment (EDM)
$d_\mu$~\cite{HIMUS99}, and $P$ and $T$ violation in muonic atoms. It
could also lead to an improved direct limit on the mass of the muon
neutrino~\cite{numass}. Of these possibilities,
Marciano~\cite{Marciano97} has suggested that muon LFV (especially
coherent muon-to-electron conversion in the field of a nucleus) is the
``best bet" for discovering signatures of new physics using low-energy
muons; measurement of $d_\mu$ could prove equally exciting but is not
yet as well developed, being only at the Letter of Intent stage at
present~\cite{EDMLOI}.\footnote{Experimentalists might argue that
extending such measurements as $\tau_\mu$ and the Michel parameters is
worthwhile whenever the state of the art allows substantial
improvement. However, their comparison with theory is dominated by
theoretical uncertainties. Thus, compared to Marciano's ``best bets,"
they represent weaker arguments for building a new facility.}

The search for $\mu\to e \gamma$ is also of great interest. The MEGA
experiment recently set an upper limit $B(\mu^+\to
e^+\gamma)<1.2\times10^{-11}$~\cite{MEGA}. Ways to extend sensitivity
to the $10^{-14}$ level have been
discussed~\cite{Cooper97}. Sensitivity greater than this may be
possible but will be difficult since at high muon rate there will be
background due to accidental coincidences; a possible way around this
relies on the correlation between the electron direction and the
polarization direction using a polarized muon beam. The
$\mu$-to-$e$-conversion approach does not suffer from this drawback
and has the additional virtue of sensitivity to possible new physics
that does not couple to the photon.

In the case of precision measurements ($\tau_\mu$, $a_\mu$, etc.),
new-physics effects can appear only as small corrections arising from
the virtual exchange of new massive particles in loop diagrams. In
contrast, LFV and EDMs are forbidden in the standard model, thus their
observation at any level constitutes evidence for new physics. The
current status and prospects for advances in these areas are
summarized in Table~\ref{tab:LEmuons}. It is worth recalling that LFV
as a manifestation of neutrino mixing is suppressed as $(\delta
m^2)^2/m_W^4$ and is thus entirely negligible. However, a variety of
new-physics scenarios predict observable effects.
Table~\ref{tab:newmuphys} lists some examples of limits on new physics
that would be implied by nonobservation of $\mu$-to-$e$ conversion
($\mu^-N\to e^-N$) at the $10^{-16}$ level~\cite{Marciano97}.

\begin{table}
\caption[Current and future tests in low energy muons] {Some current
and future tests for new physics with low-energy muons
(from~\protect\cite{Marciano97}, \protect\cite{PDG}, and
\protect\cite{Aoki01}). Note that the ``Current prospects" column
refers to anticipated sensitivity of experiments currently approved or
proposed; ``Future" gives estimated sensitivity with Neutrino Factory
front end. (The $d_\mu$ measurement is still at the Letter of Intent
stage and the reach of experiments is not yet entirely
clear.)\label{tab:LEmuons}}
\begin{center}
\begin{tabular}{|lccc|}
\hline
Test & Current bound & Current prospects & Future \\
\hline
$B(\mu^+\to e^+\gamma)$ & $<1.2\times10^{-11}$ &
$\approx5\times10^{-12}$ & $\sim10^{-14}$\\ $B(\mu^-{\rm Ti}\to
e^-{\rm Ti})$ & $<4.3\times10^{-12}$ & $\approx2\times10^{-14}$ &
$<10^{-16}$\\ $B(\mu^-{\rm Pb}\to e^-{\rm Pb})$ & $<4.6\times10^{-11}$
& & \\ $B(\mu^-{\rm Ti}\to e^+{\rm Ca})$ & $<1.7\times10^{-12}$ & & \\
$B(\mu^+\to e^+e^-e^+)$ & $<1\times10^{-12}$ & & \\ $d_\mu$ &
$(3.7\pm3.4)\times10^{-19}\,e\cdot$cm & $10^{-24}\,e\cdot$cm? & ? \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption[New physics probed by $\mu\rightarrow e$ experiments]
{Some examples of new physics probed by the nonobservation of 
$\mu\rightarrow e$ conversion at the $10^{-16}$ level 
(from~\protect\cite{Marciano97}).\label{tab:newmuphys}}
\begin{center}
\begin{tabular}{|lc|}
\hline
New Physics & Limit \\
\hline
Heavy neutrino mixing & $|V_{\mu N}^*V_{e N}|^2<10^{-12}$\\ Induced
$Z\mu e$ coupling & $g_{Z_{\mu e}}<10^{-8}$\\ Induced $H\mu e$
coupling & $g_{H_{\mu e}}<4\times10^{-8}$\\ Compositeness &
$\Lambda_c>3,000\,$TeV\\
\hline
\end{tabular}
\end{center}
\end{table}

Precision studies of atomic electrons have provided notable tests of
QED ({\it e.g.}\ the Lamb shift in hydrogen) and could in principle be
used to search for new physics were it not for nuclear corrections.
Studies of muonium ($\mu^+e^-$) are free of such corrections since it
is a purely leptonic system. Muonic atoms also can yield new
information complementary to that obtained from electronic atoms. A
number of possibilities have been enumerated by Kawall {\it et
al.}~\cite{Kawall97} and Molzon~\cite{Molzon97}. As an example we
consider the hyperfine splitting of the muonium ground state, which
has been measured to 36 ppb~\cite{Mariam} and currently furnishes the
most sensitive test of the relativistic two-body bound state in
QED~\cite{Kawall97}. The precision could be further improved with
increased statistics.  The theoretical error is 0.3 ppm but could be
improved by higher-precision measurements in muonium and muon spin
resonance, also areas in which the Neutrino Factory front end could
contribute. Another interesting test is the search for
muonium-antimuonium conversion, possible in new-physics models that
allow violation of lepton family number by two units. The current
limit is $R_g \equiv G_C / G_F< 0.0030$~\cite{PDG}, where $G_C$ is the
new-physics coupling constant and $G_F$ is the Fermi coupling
constant. This sets a lower limit of $\approx 1 \,$TeV$/c^2$ on the
mass of a grand-unified dileptonic gauge boson and also constrains
models with heavy leptons~\cite{Abela}.





\section[Physics Potential of a Higgs Factory Muon Collider]
{Physics potential of a Low energy Muon Collider 
operating as a Higgs Factory}

Muon colliders~\cite{bargersnow,clinehanson} have a number of unique
features that make them attractive candidates for future
accelerators~\cite{INTRO:ref5}.  The most important and fundamental of
these derive from the large mass of the muon in comparison to that of
the electron.  This leads to: a) the possibility of extremely narrow
beam energy spreads, especially at beam energies below $100\gev$; b)
the possibility of accelerators with very high energy; c) the
possiblity of employing storage rings at high energy; d) the
possibility of using decays of accelerated muons to provide a high
luminosity source of neutrinos as discussed in section~\ref{neuf}; e)
increased potential for probing physics in which couplings increase
with mass (as does the SM $\hsm f\anti f$ coupling).


The relatively large mass of the muon compared to the mass of the electron
means that the coupling of Higgs bosons to $\mu^+\mu^-$ is very 
much larger than to $e^+e^-$, implying much larger $s$-channel Higgs
production rates at a muon collider as compared to an electron collider.
For Higgs bosons with a very small (MeV-scale) width, 
such as a light SM Higgs boson,
production rates in the $s$-channel 
are further enhanced by the 
muon collider's ability to achieve beam energy spreads
comparable to the tiny Higgs width. 
In addition, there is little bremsstrahlung, 
and the beam energy can be tuned to one part
in a million through continuous spin-rotation measurements~\cite{Raja:1998ip}.
Due to these important qualitative difference
between the two types of machines, only muon colliders can be 
advocated as potential $s$-channel
Higgs factories capable of determining the mass and decay width
of a Higgs boson to very high precision~\cite{Barger:1997jm,Barger:1995hr}.
High rates of Higgs production at $\epem$ colliders rely on
substantial $VV$Higgs coupling for the
$Z+$Higgs (Higgstrahlung) or $WW\to$Higgs ($WW$ fusion) reactions.
In contrast, a $\mupmum$ collider can provide a factory for producing
a Higgs boson with little or no $VV$ coupling so long as it
has SM-like (or enhanced) $\mupmum$ couplings.



Of course, there is a tradeoff between small beam energy spread,
$\delta E/E=R$, and luminosity. Current estimates for yearly
integrated luminosities (using
$\call=1\times 10^{32}$cm$^{-2}$s$^{-1}$ as implying $ L=1\fbi/{\rm yr}$) are:
$\lyear\gsim 0.1,0.22,1 \fbi$ at $\rts\sim 100\gev$
for beam energy resolutions of $R=0.003\%,0.01\%,0.1\%$, respectively;
$\lyear\sim 2,6,10 \fbi$ at $\rts\sim 200,350,400\gev$, respectively, for 
$R\sim 0.1\%$.
Despite this, studies show that for small Higgs width the $s$-channel
production rate (and statistical significance over background) is maximized
by choosing $R$ to be such that $\srts\lsim \gamhtot$. In particular,
in the SM context for $\mhsm\sim 110\gev$ this corresponds to $R\sim 0.003\%$.
 

If the $\mh\sim 115\gev$ LEP signal is real or if the 
interpretation of the precision
electroweak data as an indication of a light Higgs boson (with
substantial $VV$ coupling) is valid,
then both $\epem$ and $\mupmum$ colliders will be valuable.
In this scenario the Higgs boson would have been discovered at a previous 
higher energy collider (possibly a muon collider
running at high energy), and then the Higgs factory
would be built with a center-of-mass energy 
precisely tuned to the Higgs boson mass.
The most likely scenario is that the Higgs boson 
is discovered at the LHC via gluon fusion
($gg\to H$) or perhaps 
earlier at the Tevatron via associated production 
($q\bar{q}\to WH, t\overline{t}H$), and its mass is determined to an 
accuracy of about 100~MeV. If a linear collider has also observed the Higgs
via the Higgs-strahlung process ($e^+e^-\to ZH$), one might know the Higgs 
boson mass to better than 50~MeV with an integrated luminosity of 
$500$~fb$^{-1}$.
The muon collider would be optimized to run at $\sqrt{s}\approx m_H$, and this
center-of-mass energy would be varied over a narrow range
so as to scan over the Higgs resonance (see Fig.~\ref{mhsmscan} below). 

\subsection{Higgs Production}

The production of a Higgs boson (generically denoted $\h$)
in the $s$-channel with interesting rates is  
a unique feature of a muon collider~\cite{Barger:1997jm,Barger:1995hr}. 
The resonance cross section is
%
\begin{equation}
\sigma_h(\sqrt s) = {4\pi \Gamma(h\to\mu\bar\mu) \, \Gamma(h\to X)\over
\left( s - m_h^2\right)^2 + m_h^2 \left(\Gamma_{\rm tot}^h \right)^2}\,.
\label{rawsigform}
\end{equation}
In practice, however, there is a Gaussian spread ($\srts$) to
the center-of-mass energy and one must compute the
effective $s$-channel Higgs cross section after convolution 
assuming some given central value of $\rts$:
%
\begin{eqnarray}
\anti\sigma_h(\sqrt s) & =& {1\over \sqrt{2\pi}\,\srts} \; \int
\sigma_h  
(\sqrt{\what s}) \; \exp\left[ -\left( \sqrt{\what s} - \sqrt s\right)^2
\over  
2\sigma_{\sqrt s}^2 \right] d \sqrt{\what s}\\
&&\stackrel{\rts=\mh}{\simeq} {4\pi\over m_h^2} \; {\br(h\to\mu\bar\mu)
\,
\br(h\to X) \over \left[ 1 + {8\over\pi} \left(\srts\over\gamhtot 
\right)^2 \right]^{1/2}} \,.
%
%\anti\sigma_h(\sqrt s) & =& {1\over \sqrt{2\pi}\,\srts} \; \int \sigma_h  
%(\sqrt{\what s}) \; \exp\left[ -\left( \sqrt{\what s} - \sqrt s\right)^2 \over  
%2\sigma_{\sqrt s}^2 \right] d \sqrt{\what s}\\
%\stackrel{\rts=\mh}{\simeq} {4\pi\over m_h^2} \; {\br(h\to\mu\bar\mu) \,
%\br(h\to X) \over \left[ 1 + {8\over\pi} \left(\srts\over\gamhtot 
%\right)^2 \right]^{1/2}} \,.
\label{sigform}
\end{eqnarray}
%
\begin{figure}[tbh!]
\centering\leavevmode
%\epsfxsize=3.5in\epsffile{singlemhscan_mh110_r003_brem.ps}
\centerline{\includegraphics[width=4.0in]{singlemhscan_mh110_r003_brem.ps}}
\caption[Scan of the Higgs resonance using a muon collider]{
Number of events and statistical errors in the $b\overline{b}$
final state as a function
of $\protect\rts$ in the vicinity of $\mhsm=110\gev$,
assuming $R=0.003\%$,
and $\epsilon L=0.00125$~fb$^{-1}$ at each data point.
%The precise theoretical prediction is given by the solid line.
%The dotted (dashed) curve is the theoretical prediction
%if $\Gamma _{tot}$ is decreased (increased) by 10\%, {\it keeping
%the $\Gamma(h\to\mu^+\mu^-)$ and $\Gamma(h\to b\overline{b})$
%partial widths fixed at the predicted SM value.}
\label{mhsmscan}}
\end{figure}
%

It is convenient to express $\srts$ in 
terms of the root-mean-square (rms) Gaussian spread
of the energy of an individual beam, $R$: 
%
\begin{equation}
\srts = (2{\rm~MeV}) \left( R\over 0.003\%\right) \left(\sqrt s\over  
100\rm~GeV\right) \,.
\end{equation}
%
From Eq.~(\ref{rawsigform}), it is apparent that a
resolution $\srts \lsim \gamhtot$ is needed to be
sensitive to the Higgs width. Further, Eq.~(\ref{sigform}) implies that
$\anti\sigma_h\propto 1/\srts$ for $\srts>\gamhtot$ {\it and}
that large event rates are only possible if $\gamhtot$ is not so large
that $\br(\h\to \mu\anti\mu)$ is extremely suppressed.
The width of a light SM-like Higgs is very small (e.g. a few MeV
for $\mhsm\sim 110\gev$), implying the need for $R$
values as small as $\sim 0.003\%$ for studying a light SM-like $\h$.
Fig.~\ref{mhsmscan} illustrates the result for the SM Higgs boson 
of an initial centering scan over $\rts$ values
in the vicinity of $\mhsm=110\gev$.
This figure dramatizes: a)  that the beam energy spread must be very small
because of the very small $\gamhsmtot$ (when $\mhsm$ is small
enough that the $WW^\star$ decay
mode is highly suppressed); b) that we require
the very accurate {\it in situ} determination 
of the beam energy to one part in a million through the spin 
precession of the muon noted earlier in order to perform the scan
and then center on $\rts=\mhsm$ with a high degree of stability.

If the $\h$ has SM-like couplings to $WW$, its width will
grow rapidly for $\mh>2m_W$ and its $s$-channel production cross
section will be severely suppressed by the resulting 
decrease of $\br(\h\to\mu\mu)$. 
More generally, any $\h$ with SM-like or larger $\h\mu\mu$ coupling
will retain a large $s$-channel production rate when 
$\mh>2m_W$ only if the $\h WW$ coupling becomes 
strongly suppressed relative to the $\hsm WW$ coupling.


The general theoretical prediction within supersymmetric models is that the 
lightest supersymmetric Higgs boson $\hl$ will
be very similar to the $\hsm$ when the other Higgs bosons are
heavy.  This `decoupling limit' is very likely to arise if the
masses of the supersymmetric particles are large (since the Higgs
masses and the superparticle masses are typically similar in
size for most boundary condition choices).
Thus, $\hl$ rates will be very similar to $\hsm$ rates.
In contrast, the heavier Higgs bosons in a typical supersymmetric model
decouple from $VV$ at large mass  and remain reasonably
narrow. As a result, their $s$-channel production rates remain large.

%% fig 3 used to be here

For a SM-like $\h$, at $\sqrt s = \mh \approx 115$~GeV
and $R=0.003\%$, the $b\bar b$ rates are
\vspace{-.05in}
\begin{eqnarray}
\rm signal &\approx& 10^4\rm\ events\times L(fb^{-1})\,,\\
\rm background &\approx& 10^4\rm\ events\times L(fb^{-1})\,.
\end{eqnarray}


\subsection{What the Muon Collider Adds to LHC and LC Data}
An assessment of the need for a Higgs factory requires that one detail the 
unique capabilities of a muon collider versus the other possible future 
accelerators as well as comparing the abilities of all the machines to 
measure the same Higgs properties. 
Muon colliders and a Higgs factory in particular
would only become operational after the LHC physics program is well-developed 
and quite possibly after a linear collider program is mature as well. So one
important question is the following: if
a SM-like Higgs boson and, possibly, important
physics beyond the Standard Model have been discovered at the LHC and perhaps 
studied at a linear collider, what new information could a Higgs factory 
provide?

The $s$-channel production process allows one to determine the mass, 
total width, and the cross sections
$\overline \sig_h(\mupmum\to\h\to X)$ 
for several final states $X$ 
to very high precision. The Higgs mass, total width and the cross sections 
can be used to constrain the parameters of the Higgs sector. 
For example, in the MSSM their precise values will
constrain the Higgs sector parameters
$\mha$ and $\tanb$ (where $\tanb$ is 
the ratio of the two vacuum expectation values (vevs) of the 
two Higgs doublets of the MSSM). The main question is whether these
constraints will be a valuable addition to LHC and LC constraints.

The expectations for the luminosity available at linear colliders has risen 
steadily. The most recent studies assume an integrated luminosity of some
$500$~fb$^{-1}$ corresponding to 1-2 years of running at a 
few$\times100$~fb$^{-1}$ 
per year. This luminosity results in the production of greater than $10^4$
Higgs bosons per year through the Bjorken Higgs-strahlung process, 
$e^+e^-\to Z\h$, provided the Higgs boson is kinematically accessible. This is 
comparable or even better than can be achieved with the current machine
parameters for a muon collider operating at the Higgs resonance; in fact, 
recent studies have described high-luminosity linear colliders as ``Higgs
factories,'' though for the purposes of this report, we will reserve this term
for muon colliders operating at the $s$-channel Higgs resonance. 

A linear collider with such high luminosity can certainly perform quite 
accurate measurements of certain Higgs parameters such as the Higgs mass, 
couplings to gauge bosons, couplings to heavy quarks, 
etc.~\cite{Battaglia:2000jb}.
Precise measurements of the couplings of the Higgs boson to the Standard 
Model particles is an important test of the mass generation mechanism.
In the Standard Model with one Higgs doublet, this coupling is proportional 
to the particle mass. In the more general case there can be mixing angles
present in the couplings. Precision measurements of the couplings can 
distinguish the Standard Model Higgs boson from one from a more general model
and can constrain the parameters of a more general Higgs sector.


\begin{table*}[h!]
\begin{center}
\caption[Comparison of a Higgs factory muon collider with LHC and LC]
{Achievable relative
uncertainties for a SM-like $\mh=110$~GeV for measuring the
Higgs boson mass and total width
for the LHC, LC (500~fb$^{-1}$), and the muon collider (0.2~fb$^{-1}$). 
}\label{unc-table}
\protect\protect
\begin{tabular}{cccc}
\hline
\ & LHC & LC & $\mu^+\mu^-$\\
$\mh$ & $9\times 10^{-4}$ & $3\times 10^{-4}$ & $1-3\times 10^{-6}$ \\
$\gamhtot$ & $>0.3$ & 0.17 & 0.2 \\
\hline
\end{tabular}
\end{center}
\end{table*}

The accuracies possible at different colliders
for measuring $\mh$ and $\gamhtot$ of
a SM-like $\h$ with $\mh\sim 110\gev$ are given in Table~\ref{unc-table}.
Once the mass is determined to about 1~MeV at the LHC and/or LC, 
the muon collider would employ a
three-point fine scan~\cite{Barger:1997jm} near the resonance peak.
Since all the couplings of the Standard Model are known, $\gamhsmtot$
is known. Therefore a precise determination of 
$\gamhtot$ is an important test of the Standard Model, and any deviation
would be evidence for a nonstandard Higgs sector. 
For a SM Higgs boson with a mass sufficiently below the $WW^\star$ 
threshold, the Higgs total width is very small (of order several MeV), and the 
only process where it can be measured {\it directly} is in the $s$-channel
at a muon collider. Indirect determinations at the LC can have
higher accuracy once $\mh$ is large enough that the $WW^\star$ mode
rates can be accurately measured, requiring $\mh>120\gev$.
This is because at the LC the total width must be determined 
indirectly by measuring a partial width and a branching fraction, and then 
computing the total width,
\begin{eqnarray}
&&\Gamma _{tot}={{\Gamma(h\to X)}\over {BR(h\to X)}}\;,
\end{eqnarray} 
for some final state $X$. For a Higgs boson so light that the $WW^\star$ decay
mode is not useful, then the total width measurement would probably require
use of the $\gamma \gamma $ decays~\cite{Gunion:1996cn}. This would require
information from a photon collider as well as the LC
and a small error is not possible.
The muon collider can measure the total width of the Higgs boson directly,
a very valuable input for precision tests of the Higgs sector. 


 
To summarize,
if a Higgs is discovered at the LHC or possibly earlier at the Fermilab 
Tevatron, attention will turn to determining  whether this Higgs has the 
properties expected of the Standard Model Higgs. If the Higgs is discovered
at the LHC, it is quite possible that supersymmetric states will be 
discovered concurrently. The next goal for a linear collider or a muon collider
will be to better measure the Higgs boson properties to determine if 
everything is consistent within a supersymmetric framework or consistent
with the Standard Model.
A Higgs factory of even modest luminosity can provide uniquely
powerful constraints on the parameter space of the supersymmetric
model via its very precise measurement of the light Higgs mass, the
highly accurate determination of the total rate for $\mupmum\to\hl\to
b\anti b$ (which has almost zero theoretical systematic uncertainty
due to its insensitivity to the unknown $m_b$ value) and the
moderately accurate determination of the $\hl$'s total width.  In
addition, by combining muon collider data with LC data, a completely
model-independent and very precise determination of the
$h^0\mu^+\mu^-$ coupling is possible. This will provide another strong
discriminator between the SM and the MSSM.  Further, the
$h^0\mu^+\mu^-$ coupling can be compared to the muon collider and LC
determinations of the $h^0\tau^+\tau^-$ coupling for a precision test
of the expected universality of the fermion mass generation mechanism.

\section{Physics Potential of a High Energy Muon Collider} Once one
learns to cool muons, it become possible to build muon colliders with
energies of $\approx$ 3 TeV in the center of mass that fit on an
existing laboratory site~\cite{INTRO:ref5,rajawitherell}. At
intermediate energies, it becomes possible to measure the W mass
\cite{bergerw} and the top quark mass~\cite{bergertop} with high
accuracy, by scanning the thresholds of these particles and making use
of the excellent energy resolution of the beams. We consider here
further the ability of a higher energy muon collider to scan higher
lying Higgs like objects such as the H$^0$ and the A$^0$ in the MSSM
that may be degenerate with each other.

\subsection{Heavy Higgs Bosons}
As discussed in the previous section, precision measurements of the
light Higgs boson properties might make it possible to not only
distinguish a supersymmetric boson from a Standard Model one, but also
pinpoint a range of allowed masses for the heavier Higgs bosons.  This
becomes more difficult in the decoupling limit where the differences
between a supersymmetric and Standard Model Higgs are
smaller. Nevertheless with sufficiently precise measurements of the
Higgs branching fractions, it is possible that the heavy Higgs boson
masses can be infered.  A muon collider light-Higgs factory might be
essential in this process.

In the context of the MSSM, $\mha$ can probably be restricted to
within $50\gev$ or better if $\mha<500\gev$.
This includes the $250-500\gev$
range of heavy Higgs boson masses for which discovery is not possible 
via $\hh\ha$ pair production 
at a $\rts=500\gev$ LC. Further, the $\ha$ and $\hh$
cannot be detected in this mass range at either the LHC or LC 
in  $b\anti b \hh,b\anti b\ha$ production
for a wedge of moderate $\tanb$ 
values. (For large enough 
values of $\tanb$ the heavy Higgs bosons are expected to be observable
in $b\anti b \ha,b\anti b \hh$ production
at the LHC via their $\tau ^+\tau ^-$ decays and also at the LC.)

A muon collider can fill some, perhaps all of this moderate $\tanb$ wedge.
If $\tanb$ is large the $\mupmum \hh$ and $\mupmum\ha$ couplings (proportional
to $\tanb$ times a SM-like value) are enhanced
thereby leading to enhanced production rates in $\mupmum$ collisions.
The most efficient procedure is to operate the muon collider
at maximum energy and produce the $\hh$ and $\ha$ (often as overlapping
resonances) 
via the radiative return mechanism. By looking for a peak
in the $b\anti b$ final state, the $\hh$ and $\ha$ 
can be discovered and, once discovered, the machine $\rts$
can be set to $\mha$ or $\mhh$ and factory-like precision studies pursued.
Note that the $\ha$ and $\hh$ are typically broad enough that $R=0.1\%$
would be adequate to maximize their $s$-channel production rates.
In particular, $\Gamma\sim 30$~MeV
if the $t\overline{t}$ decay channel is not open, and $\Gamma\sim 3$~GeV if it
is. Since $R=0.1\%$ is sufficient, much higher luminosity
($L\sim 2-10~{\rm fb}^{-1}
/{\rm yr}$) would be possible as compared to that 
for $R=0.01\%-0.003\%$ required for studying the $\hl$.
 
In short, for these moderate $\tanb$--$\mha\gsim 250\gev$
scenarios that are particularly difficult for both
the LHC and the LC, the muon collider would be the only 
place that these extra Higgs bosons can be discovered and their properties 
measured very precisely.  


In the MSSM, the heavy Higgs bosons are largely degenerate, especially in the 
decoupling limit where they are heavy. Large values of $\tan \beta$ heighten
this degeneracy.
A muon collider with sufficient energy resolution might be
the only possible means for separating out these states.
Examples showing the $H$ and $A$ resonances for $\tan \beta =5$ and $10$
are shown in Fig.~\ref{H0-A0-sep}. For the larger value of 
$\tan \beta$ the resonances are clearly overlapping. For the better energy 
resolution of $R=0.01\%$, the two distinct resonance peaks are still 
visible, but become smeared out for $R=0.06\%$. 


\begin{figure}[tbh!]
\centering\leavevmode
\centerline{\includegraphics[width=4.0in]{hh_ha_susy_rtsscan.ps}}

\caption[Separation of $A$ and $H$ signals for $\tan\beta=5$ and $10$]
{Separation of $A$ and $H$ signals for $\tan\beta=5$ and $10$. From  
Ref.~\cite{Barger:1997jm}. \label{H0-A0-sep}}
\end{figure}

Once muon colliders of these intermediate energies can be built,
higher energies such as 3-4~TeV in the center of mass become feasible.
Muon colliders with these energies will be complementary to hadron
colliders of the SSC class and above. The background radiation from
neutrinos from the muon decay becomes a problem at $\approx$~3~TeV in
the CoM. Ideas for ameliorating this problem have been discussed and include
optical stochastic cooling to reduce the number of muons needed for a
given luminosity, elimination of straight sections via wigglers or
undulators, or special sites for the collider such that the neutrinos
break ground in uninhabited areas.



