\section{Physics Motivation}
\label{physics}
In this Section 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.
\subsection{Neutrino Oscillation Physics}
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. 
\subsubsection{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 $\Delta m^2_{\rm sol}$ of the order $10^{-5}$ eV$^2$ for
solutions involving the Mikheyev-Smirnov-Wolfenstein (MSW) resonant matter
oscillations~\cite{wolf}--\cite{ms} 
or of the order of $10^{-10}\,$eV$^2$ for vacuum
oscillations~\cite{just-so}.  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 $\Delta m^2_{\rm atm}\sim 3 \times 10^{-3}\rm\,eV^2$~\cite{sk} and maximal mixing $\sin^2 2 \theta_{\rm atm} = 1$.  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
$\Delta m^2_{\rm LSND} \sim 0.1\mbox{--}1\rm~eV^2$ 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 at least one further
 neutrino.
Since it is known 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.

\subsubsection{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}\\
-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}\\
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 $K^\prime =
{\rm diag}(1,e^{i\phi_1},e^{i\phi_2})$.  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, $\Delta m^2_{\rm atm}$, which is $\Delta m^2_{32} =
m(\nu_3)^2-m(\nu_2)^2$ in the favored fit, and $\Delta m^2_{\rm sol}$, which may be taken to be $\Delta m^2_{21}=m(\nu_2)^2- m(\nu_1)^2$.  Note that these
$\Delta m^2$ 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, which 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, $\Delta m^2_{\rm atm} \sim {\rm few} \times 10^{-3}\rm\,eV^2$, and one possible neutrino mass spectrum is the hierarchical one 
\begin{equation} 
\Delta m^2_{21}
= \Delta m^2_{\rm sol} \ll \Delta m^2_{31} \approx \Delta m^2_{32}=\Delta m^2_{\rm atm} \,.
\label{hierarchy}
\end{equation}
In this case, $CP$ $(T)$ violation effects may be 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_{\rm atm}L}{4E} \Bigr ) \nonumber\\
& = & \sin^2(2\theta_{23})\cos^4(\theta_{13})
\sin^2 \Bigl (\frac{\Delta m^2_{\rm 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_{\rm atm}L}{4E} \Bigr ) \nonumber\\
& = & \sin^2(2\theta_{13})\sin^2(\theta_{23})
\sin^2 \Bigl (\frac{\Delta m^2_{\rm 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_{\rm atm}L}{4E} \Bigr ) \nonumber\\
& = & \sin^2(2\theta_{13})\cos^2(\theta_{23})
\sin^2 \Bigl (\frac{\Delta m^2_{\rm 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
%
\begin{eqnarray}
P(\nu_e \to \nu_e) & = & 1 - \sum_x P(\nu_e \to \nu_x) \nonumber\\
                   & = & 1 - \sin^2(2\theta_{\rm reactor})
\sin^2 \Bigl (\frac{\Delta m^2_{\rm reactor}L}{4E} \Bigr ) \,,
\label{preactor}
\end{eqnarray}
%
where $\Delta m^2_{\rm reactor}$ is the squared mass difference relevant for
$\bar\nu_e \to \bar\nu_x$.  In particular, in the upper range of values of
$\Delta m^2_{\rm atm}$, 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_{\rm atm}L}{4E} \Bigr ) \,,
\label{preactoratm}
\end{equation}
%
i.e., $\theta_{\rm reactor}=\theta_{13}$, and, for the value $|\Delta m^2_{32}| = 3 \times 10^{-3}\rm\,eV^2$ 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 ``$\sin^2(2\theta_{\rm atm})$'' 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_{\rm 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 
$\nu_\mu \to \nu_\tau$ oscillations with maximal mixing, and hence
$\sin^2(2\theta_{23})=1$ and $|\theta_{13}| \ll 1$.  The various solutions of
the solar neutrino problem involve quite different values of $\Delta m^2_{21}$
and $\sin^2(2\theta_{12})$: (i)~large mixing angle solution, LMA: $\Delta
m^2_{21} \simeq {\rm few} \times 10^{-5}\rm\,eV^2$ and $\sin^2(2\theta_{12})
\simeq 0.8$; (ii)~small mixing angle solution, SMA: $\Delta m^2_{21} \sim
10^{-5}\rm\,eV^2$ and $\sin^2(2\theta_{12}) \sim 10^{-2}$, (iii)~LOW: $\Delta m^2_{21}
\sim 10^{-7}\rm\,eV^2$, $\sin^2(2\theta_{12}) \sim 1$, and (iv)~``just-so'': $\Delta
m^2_{21} \sim 10^{-10}\rm\,eV^2$, $\sin^2(2\theta_{12}) \sim 1$.  The SuperK experiment
favors the LMA solutions~\cite{sol}; for other global fits, see, e.g.,
Ref.~\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.
\subsubsection{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, Borexino, and MONOLITH~\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 2005 and, after some
years of running, to obtain higher statistics than the K2K experiment and to
achieve a sensitivity down to the level $|\Delta m^2_{32}| \sim
10^{-3}\rm\,eV^2$.  
The CERN--Gran Sasso program will also come on in
2005.  It will use  a higher-energy neutrino beam, $E_\nu\sim17$~GeV,
 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}.  For the joint capabilities of
MINOS, ICARUS and OPERA experiments see Ref.~\cite{minicop}.
Plans for the Japan Hadron Facility (JHF),
also called the High Intensity Proton Accelerator (HIPA), include the use of 
a 0.77~MW
proton driver to produce a high-intensity conventional neutrino beam with a
path length of 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 neutrino experiments relevant to the solar neutrino anomaly.  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. However, the uncertain normalization of the $^8$B flux makes it difficult to constrain a possible sterile neutrino component in the oscillations~\cite{unknowns}.
The KamLAND experiment~\cite{kamland} in Japan started taking data
in January 2002.  This is a reactor antineutrino experiment using
baselines of  100--250 km. It will search for $\bar\nu_e$ disappearance
and is sensitive to the solar neutrino oscillation scale.
KamLAND can provide precise measurements of the LMA solar parameters~\cite{bmw-kamland}.
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. 
\subsubsection{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, 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 ({\cal 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}{|llcc|}
\hline
& & Conventional & Neutrino \\[-2ex]
\raisebox{1ex}[0pt]{Measurement } & \raisebox{1ex}[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 the knowledge of $|\Delta m^2_{32}|$ and
$\sin^2(2\theta_{23})$ that will be available 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
path lengths 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 path lengths
(e.g.,~\cite{zp}). 
We anticipate 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 10\% level
(while recognizing that future projections such as this are obviously uncertain).
The Neutrino Factory will significantly improve precision in these parameters,
as can be seen from Fig.~\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.

\subsubsection{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 that 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 $E
\sim {\cal O}(10)$ 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 Barger, Pakvasa, Phillips and Whisnant~\cite{bppw-1980}. 
(See the papers~\cite{dgh}--\cite{cpv} for details  of the matter effects
and their relevance to neutrino factories.) In brief,
assuming a normal hierarchy, the transition
probabilities for propagation through
matter of constant density are~\cite{golden,formcon}
%
\begin{eqnarray}
P(\nu_e \to \nu_\mu) &=&
x^2 f^2 + 2 x y f g (\cos\delta\cos\Delta + \sin\delta\sin\Delta)
+ y^2 g^2\,,\\
P(\nu_e \to \nu_\tau) &=& {\rm cot}^2 \theta_{23} x^2 f^2 - 2 x y f g
(\cos\delta\cos\Delta + \sin\delta\sin\Delta)
+ {\rm tan}^2 \theta_{23} y^2 g^2\,,\\
P(\nu_\mu \to \nu_\tau) &=& \sin^2 2\theta_{23} \sin^2\Delta
 \\ & + &\alpha
 \sin 2\theta_{23} \sin 2\Delta \bigg({\hat A \over 1-\hat A}
\sin \theta_{13} \sin 2\theta_{12} \cos 2\theta_{23} \sin\Delta-\Delta
\cos^2 \theta_{12} \sin 2\theta_{23}\bigg)\,, \nonumber
\end{eqnarray}
where
%
\begin{eqnarray}
\Delta &\equiv& |\delta m_{31}^2| L/4E_\nu
= 1.27 |\delta m_{31}^2/{\rm eV^2}| (L/{\rm km})/ (E_\nu/{\rm GeV}) \,,
\label{eq:D}\\
\hat A &\equiv& |A/\delta m_{31}^2| \,,
\label{eq:Ahat}\\
\alpha &\equiv& |\delta m^2_{21}/\delta m^2_{31}|  \,,\\
x &\equiv& \sin\theta_{23} \sin 2\theta_{13} \,,
\label{eq:x}\\
y &\equiv& \alpha \cos\theta_{23} \sin 2\theta_{12} \,,
\label{eq:y}\\
f &\equiv& \sin((1\mp\hat A)\Delta)/(1\mp\hat A) \,,
\label{eq:f}\\
g &\equiv& \sin(\hat A\Delta)/\hat A \,.
\label{eq:alpha}
\end{eqnarray}
%
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 $Y_e
\simeq 0.5$.  
For neutrinos with $\delta
m^2_{31} > 0$ or anti-neutrinos with $\delta m^2_{31} < 0$, $\hat A =
1$ corresponds to a matter resonance.
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 $\sim 10^{-3}$.
\subsubsection{CP Violation}
$CP$ violation is measured by the (rephasing-invariant) product
%
\begin{eqnarray}
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  \,.
\end{eqnarray}
%
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: $J_{\rm CKM} \sim {\cal O}(10^{-5})$.  
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, 
%
\begin{eqnarray}
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
& = & -16J \sin \phi_{32} \sin \phi_{13} \sin \phi_{21} \,,
\label{pnuenumudif}
\end{eqnarray}
%
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 $\Delta
m^2_{\rm sol} = \Delta m^2_{21}$ not be too small. From atmospheric neutrino data,
we have $\theta_{23}\simeq \pi/4$ and $\theta_{13} \ll 1$.  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 $\gg J_{CKM}$.  Furthermore, for
 parts of  the LMA phase space where  
$\Delta m^2_{\rm sol} \sim 4 \times 10^{-5}$ 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~\cite{bmw-kamland} 
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 either $\mu^+$
or $\mu^-$ particles and measure the ratio of the number of events
$\bar\nu_e\rightarrow \bar\nu_\mu$/$\nu_e\rightarrow\nu_\mu$.
Figure~\ref{cpfig} shows this ratio for a Neutrino Factory with
10$^{21}$ decays and a 50~kiloton 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. The curves are based on calculations
presented in~\cite{barger-entry}. }
\end{figure}

\subsection{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,superbeam-peak}. 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{jhf}, (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,superbeam-peak}.

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,superbeam-peak}, 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 labeled). 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 more 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 ${\cal O}(0.0001)$. 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
0.001 to 0.1, 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} 
%% restart here 1.16.02, noon
\subsection{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}.  A Neutrino Factory
can measure individual parton distributions within the proton for all
light quarks and anti-quarks.  It could improve valence distributions
by an order of magnitude in the kinematical range $x\gsim 0.1$ in the
unpolarized case.  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
$0.1\lsim x \lsim 0.6$. A full exploitation of the Neutrino Factory
potential for polarized measurements of the shapes of individual
partonic densities requires an {\it a priori} knowledge of the
polarized gluon density.  The forthcoming set of polarized deep
inelastic scattering experiments at CERN, DESY and RHIC may provide
this information.  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 that 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.  A measurement of
$\as(M_Z)$ and $\sin^2\theta_W$ will involve different systematics
from current measurements and will therefore provide an important
consistency check of current data, although the accuracy of these
values is not expected to be improved.  The weak mixing angle can be
measured in both the hadronic and leptonic modes with a precision of
approximately $2\times 10^{-4}$, dominated by the statistics and the
luminosity measurement.  This determination would be sensitive to
different classes of new-physics contributions.  Neutrino interactions
are a very good source of clean, sign-tagged charm particles. A
Neutrino Factory can measure charm production with raw event rates up
to 100 million charm events per year with $\simeq$ 2 million
double-tagged events.  (Note that charm production becomes significant
for storage ring energies above 20~GeV).  Such large samples are
suitable for precise extractions of branching ratios and decay
constants, the study of spin-transfer phenomena, and the study of
nuclear effects in deep inelastic scattering.  The ability to run with
both hydrogen and heavier targets will 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 clarify the intriguing problem of spin transfer.

Although the neutrino beam energies are well below any reasonable
threshold for new physics, the large statistics makes it possible to
search for physics beyond the Standard Model. The high intensity
neutrino beam allows a search for the production and decay of neutral
heavy leptons with mixing angle sensitivity two orders of magnitude
better than present limits in the 30--80 MeV range.  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.  A novel
neutrino magnetic moment search technique that uses oscillating
magnetic fields at the neutrino beam source could discover large
neutrino magnetic moments predicted by some theories.  Rare
lepton-flavor-violating decays of muons in the ring could be tagged in
the deep inelastic scattering final states through the detection of
wrong-sign electrons and muons, or of prompt taus.
%
% below modified K.J. 28.jul.2002
%
\subsection{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~\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 a rich variety of precision
measurements. In the area of low energy muon physics a majority of
experiments with a high physics potential is limited at present by
statistics.  The list of conceivable projects includes 
(see Table~\ref{tab:LEexpts}):
\begin{itemize}
\item 
precise determinations of the properties characterizing the muon,
which are the mass $m_{\mu}$, magnetic moment $\mu_{\mu}$, magnetic
anomaly $a_{\mu}$, charge $q_{\mu}$ and lifetime $\tau_{\mu}$,
\item 
measurements the muon decay parameters (Michel parameters),
\item
CPT tests from a comparison of $\mu^-$ and $\mu^+$ properties,
\item 
measurements of fundamental constants of general importance (e.g. the
electromagnetic fine structure constant $\alpha$ or the weak
interaction Fermi constant $G_F$)
\item sensitive searches for physics beyond the Standard Model either
through measuring differences of muon parameters from Standard Model
predictions or in dedicated searches for rare and forbidden processes,
such as $\mu \rightarrow e \gamma$, $\mu \rightarrow eee$, $\mu^-N
\rightarrow e^-N$ conversion and muonium-antimuonium (${\rm
M}-\overline{\rm M}$) conversion or searches for a permanent electric
dipole moment $d_{\mu}$ of the particle,
\item searches for $P$ and $T$ violation in muonic atoms,
\item precise determinations of nuclear properties in muonic
(radioactive) atoms,
\item  applications in condensed matter, thin films and at surfaces,
\item  applications in life sciences, and
\item  muon catalyzed fusion($\mu$CF).
\end{itemize}

A detailed evaluation of the possibilities has recently been made by a
CERN study group, which assumed a facility  with a 4 MW proton 
driver\cite{Aysto_01}.

In the search for ``forbidden'' decays,
Marciano~\cite{Marciano97} has suggested that muon Lepton Flavor Violation
(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. The MECO experiment \cite{MECO}  proposed at BNL
offers, through a novel detector concept, very high sensitivity and some
four orders of magnitude improvement over the current limits from
PSI \cite{SINDRUM}.  At a future high muon flux facility, such as the Neutrino Factory, this could be  improved further by 1-2 orders of magnitude.

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 not only been discussed~\cite{Cooper97}
but also have lead to an active proposal at PSI \cite{Mori_99}. The
experiment aims for an improvement of three orders of magnitude over MEGA
which had systematics limitations.  The $\mu$-to-$e$-conversion approach
has the additional virtue of sensitivity to new physics that
does not couple to the photon.

An observation of a non-zero Electric Dipole Moment (EDM) of the muon, 
$d_{\mu}$, could prove equally exciting; This has generated a Letter of Intent~\cite{EDMLOI} to observe $d_\mu$, which  proposes to use the
the large electric fields associated with 
relativistic particles in a magnetic storage ring.  
As CP violation enters in the quark sector
starting with the second generation, the muon is a particularly
valuable probe in this regard, 
despite the already low limits for electrons. Moreover,
there exist some models in which the electric dipole moment 
scales stronger than linearly\cite{Ellis_01}.

It is worth noting that for  searches of rare muon decays 
and for $d_{\mu}$ that
the standard model predictions are orders of magnitude below
the presently established limits. Any observation which can be shown
to be not an artefact of the experimental method or due to background
would therefore be a direct sign of new physics.

There are  three experiments going on currently to improve the
muon lifetime $\tau_\mu$ \cite{tau_mu}. Note that the 
Fermi coupling constant $G_F$ is derived from a measurement of $\tau_\mu$. The
efforts are therefore worthwhile whenever experimental conditions allow
substantial improvement. One should however be aware that a comparison
with theory in this channel is presently dominated by 
theoretical uncertainties.

In the case of precision measurements ($\tau_\mu$, $a_\mu$, etc.),
new-physics effects appear  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 would constitute evidence for new physics. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand\mtcol[2]{\parbox[c]{#1}{\centering #2}}
\newcommand\mtcola[1]{\mtcol{0.73in}{#1}}
\newcommand\mtcolb[1]{\mtcol{1.54in}{#1}}
\begin{table*}[htbp]
  \centering
  \makeatletter
  \preprintsty@sw{\scriptsize}{\squeezetable}\makeatother
  \setlength\slideback{2pt}
  \addtolength\slideback{-\baselineskip}
% Delete next line
  \resizebox{\textwidth}{!}{
  \begin{tabular}{|c|c||c|}
    \hline
    \mtcola{Type of Experiment}&
    \mtcolb{Physics Issues}&
    \begin{subtable}
      \stcola{Possible experiments}&
      \stcolb{Previously established accuracy}&
      \stcolc{Present activities\\(proposed accuracy)}&
      \stcold{Projected for SMS @ CERN}
    \end{subtable}\\
    \hline\hline
    \mtcola{``Classical'' rare \& forbidden decays}&
    \mtcolb{Lepton number violation; searches for new physics: SUSY, L-R
      Symmetry, R-parity violation,\ldots}&
    \begin{subtable}
      \stcola{$\mu^-N \rightarrow e^-N$}&
      \stcolb{$6.1\cdot 10^{-13}$}&
      \stcolc{PSI, proposed BNL ($5 \cdot 10^{-17}$)}&
      \stcold{$<10^{-18}$}\\
      \stcola{$\mu\rightarrow e\gamma$}&
      \stcolb{$1.2\cdot10^{-11}$}&
      \stcolc{Proposed PSI ($1 \cdot 10^{-14}$)}&
      \stcold{$ < 10^{-15}$}\\
      \stcola{$\mu \rightarrow eee$}&
      \stcolb{$1.0 \cdot 10^{-12}$}&
      \stcolc{completed 1985 PSI}&
      \stcold{$ < 10^{-16}$}\\
      \stcola{$\mu^+e^- \rightarrow \mu^-e^+$}&
      \stcolb{$8.1 \cdot 10^{-11}$}&
      \stcolc{completed 1999 PSI}&
      \stcold{$ < 10^{-13}$}
    \end{subtable}\\
    \hline
    \mtcola{Muon Decays}&
    \mtcolb{$G_F$; searches for new physics; Michel parameters}&
    \begin{subtable}
      \stcola{$\tau_{\mu}$}&
      \stcolb{$18\cdot 10^{-6}$}&
      \stcolc{PSI (2x), RAL         ($1 \cdot 10^{-6}$)}&
      \stcold{$ < 10^{-7}$}\\
      \stcola{non $(V-A)$}&
      \stcolb{typ. few $10^{-3}$}&
      PSI, TRIUMF           ($1 \cdot 10^{-3}$)&
      $ < 10^{-4}$
    \end{subtable}\\
    \hline
    \mtcola{Muon Moments}&
    \mtcolb{Standard model tests; new physics; CPT tests
      T- resp. CP-violation in 2nd lepton generation}&
    \begin{subtable}
      &&&\\
      \stcola{$g_{\mu}-2$}&
      \stcolb{$1.3 \cdot 10^{-6} $}&
      \stcolc{BNL ($3.5\cdot10^{-7}$)}&
      \stcold{$ < 10^{-7}$}\\
      \stcola{$edm_{\mu}$}&
      \stcolb{$3.4 \cdot 10^{-19} e$ cm}&
      \stcolc{proposed BNL           ($10^{-24} e\,cm$)}&
      \stcold{$ < 5 \cdot 10^{-26} e$ cm}\\
      &&&
    \end{subtable}\\
    \hline
    \mtcola{Muonium Spectroscopy}&
    \mtcolb{Fundamental constants, $\mu_{\mu}$, $m_{\mu}$, $\alpha$;
      weak interactions; muon charge}&
    \begin{subtable}
      &&&\\[-0.5\baselineskip]
      \stcola{$M_{HFS}$}&
      \stcolb{$12 \cdot 10^{-9}$}&
      \stcolc{completed 1999 LAMPF}&
      \stcold{$ 5 \cdot 10^{-9}$}\\
      \stcola{$M_{1s2s}$}&
      \stcolb{$1 \cdot 10^{-9}$}&
      \stcolc{completed 2000 RAL}&
      \stcold{$ < 10^{-11}$}\\[-0.5\baselineskip]
      &&&
    \end{subtable}\\
    \hline
    \mtcola{Muonic Atoms}&
    \mtcolb{Nuclear charge radii;\\weak interactions}&
    \begin{subtable}
      \stcola{$\mu^-$ atoms}&
      \stcolb{depends}&
      \stcolc{PSI, possible CERN\\($<r_p>$ to $10^{-3}$)}&
      \stcold{new nuclear structure}
    \end{subtable}\\
    \hline
    \mtcola{Condensed Matter}&
    \mtcolb{surfaces, catalysis, bio sciences\ldots}&
    \begin{subtable}
      &&&\\[-0.5\baselineskip]
      \stcola{surface $\mu$SR}&
      \stcolb{n/a}&
      \stcolc{PSI, RAL (n/a)}&
      \stcold{high rate}\\[-0.5\baselineskip]
      &&&
    \end{subtable}\\
    \hline
  \end{tabular}
% Delete next line
  }
  
  \caption{
    Experiments which could beneficially take advantage of the intense
    future stopped muon source. The numbers were worked out for
    scenarios at a future Stopped Muon Source (SMS) of a neutrino
    factory at CERN \cite{Aysto_01}. They are based on a muon flux of
    $10^{21}$ particles per annum in which beam will be available for
    $10^7$ s. Typical beam requirements are given in
    Table~\ref{tab:LE_beams}.
  }
  \label{tab:LEexpts}
\end{table*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{table}[tbh]
\caption[]{
Beam requirements for new muon experiments.  We show  the needed
muonic charge $q_{\mu}$ and the minimum of the total muon number
$\int I_{\mu}dt$ above which significant progress can be expected in
the physical interpretation of the experiments. Measurements which
require pulsed beams are sensitive to the muon suppression $I_0/I_{m}$
between pulses of length $\delta T$ and separation $\Delta T$.  Most
experiments require energies up to 4 MeV corresponding to 29 MeV/c
momentum. Thin targets, respectively storage ring acceptances, demand
rather small momentum spreads $\Delta p_{\mu}/p_{\mu}$
\cite{Aysto_01}.   
\label{tab:LE_beams} 
} 
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
%
&&&&&&&\\
%
Experiment & $q_{\mu}$ &$\int I_{\mu}dt$&$I_0/I_{\mu}$&$\delta
T$&$\Delta T$&$E_{\mu}$&$\Delta p_{\mu}/p_{\mu}$\\ & & & & [ns] & [ns]
& [MeV] & [\%] \\
%
\hline
%
$\mu^-N \rightarrow e^-N$ &-- &$10^{19}$&$<10^{-9}$&$\leq 100$&$\geq
1000$ &$<20$ &1...5 \\ $\mu \rightarrow e \gamma$ &+ &$10^{16}$& n/a
&continuous &continuous &1...4 &1...5 \\ $\mu \rightarrow eee$ &+
&$10^{15}$& n/a &continuous &continuous &1...4 &1...5 \\ $\mu^+e^-
\rightarrow \mu^-e^+$&+ &$10^{16}$&$<10^{-4}$&$<1000$s &$\geq 20000$
&1...4 &1...2 \\
%
\hline
%
$\tau_{\mu}$ &+ &$10^{13}$&$<10^{-4}$&$<100 $ &$\geq 20000$ &4 &1...10
\\ $non (V-A)$ &$\pm$&$10^{13}$&$ n/a $ &continuous &continuous &4
&1...5 \\
%
\hline
%
$g_{\mu}-2$ &$\pm$&$10^{15}$&$<10^{-7}$&$\leq 50 $ &$\geq 10^6$ &3100
&$10^{-4}$ \\ $d_{\mu}$ &$\pm$&$10^{16}$&$<10^{-6}$&$\leq 50 $
&$\geq 10^6 $ &$\leq$1000&$\leq 10^{-5}$\\
%
\hline
%
$M_{HFS}$ &+ &$10^{15}$&$<10^{-4}$&$\leq 1000$ &$\geq 20000$ &4 &1...3
\\ $M_{1s2s}$ &+ &$10^{14}$&$<10^{-3}$&$\leq 500 $ &$\geq 10^6$ &1...4
&1...2 \\
%
\hline
%
$\mu^- atoms$ &-- &$10^{14}$&$<10^{-3}$&$\leq 500 $&$\geq 20000$
&1...4 &1...5 \\
%
\hline
%
$condensed$ $matter$ &$\pm$&$10^{14}$&$<10^{-3}$&$< 50 $ &$\geq 20000$
&1...4 &1...5 \\ $(incl.$$bio$ $ sciences)$ &&&&&&&\\
%
\hline
\end{tabular}
\end{center}
\end{table}
%
The current status and prospects for advances in these areas are
shown in Table~\ref{tab:LEexpts}, which lists present efforts in the
field and possible improvements at a Neutrino Factory or Muon
Collider facility. The beam parameters necessary for the expected
improvements are listed in Table~\ref{tab:LE_beams}.

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[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}

The muon magnetic anomaly (muon g-2 value \cite{Farley_90}) has been
measured recently at the Brookhaven National Laboratory (BNL) with 0.7
ppm accuracy \cite{BNLg-2}.  At present, no definite statement can
be made whether this result agrees or disagrees with standard theory, which is sensitive to electroweak corrections.
The theory has recently come under severe scrutiny and 
in particlar an error was found in the calculation of 
hadronic light by light scattering
\cite{Knecht_02}. The theoretical calculations are being improved upon, 
and with more data, there is a good chance that this might eventually lead to
evidence for beyond the standard model effects\cite{Czarnecki_01}. 
The final goal of the experimental precision is
0.35 ppm for the current set of experiments.  
This value could be improved  by an order of
magnitude at a Neutrino Factory, 
provided cold muons of energy 3.1 GeV  are  made available.  
This could  further spur more accurate theoretical calculations 
that improve upon contributions from 
hadronic vacuum polarization and hadronic light
by light scattering~\cite{Marciano_2001}.  
In addition, the  muon g-2 experiments at CERN have provided the best
test of CPT invariance at a level of $2\cdot10^{-22}$ which is  more
than three orders of magnitude better than the mostly quoted result ${\rm
K}^0-\overline{{\rm K}^0}$ mass difference
\cite{Kostelecki_00}. 
A $g-2$ measurement at the Neutrino Factory front end that uses muons 
of both charges would lead to  further improvement in these CPT limits.

Precision studies of atomic electrons have provided notable tests of
QED ({ 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  can also yield new
information complementary to that obtained from electronic atoms. A
number of possibilities have been enumerated by Kawall {\it et
al.}~\cite{Kawall97}, Jungmann \cite{Jungmann_01} and
Molzon~\cite{Molzon97}.

By making measurements on the muonium system, for instance, one can
produce precise measurements of the fundamental constants and also do
sensitive searches for new physics.
The muonium ground state hyperfine structure has been measured to 12
ppb~\cite{Liu_99} and currently furnishes the most sensitive test of
the relativistic two-body bound state in QED~\cite{Jungmann_01}. The
precision could be further improved significantly with increased
statistics.  The theoretical error is 120~ppb. The uncertainty arising
from the muon mass is five times larger than that from calculations.
If one assumes the theory to be correct, the muon-electron mass ratio
can be extracted to 27~ppb. A precise value for the electromagnetic
fine structure constant $\alpha$ can be extracted.  Its good agreement
with the number extracted from the electron magnetic anomaly must be
viewed as the best test of the internal consistency of QED, as one case
involves bound state QED and the other that of free particles. The
Zeeman effect of the muonium hyperfine structure allows the best
direct measurement of the muon magnetic moment, respectively its mass,
to 120~ppb, improved by higher-precision measurements in muonium and
muon spin resonance. These are  also areas in which the Neutrino Factory front
end could contribute.  Laser spectroscopy of the muonium 1s-2s
transition
\cite{Meyer_00} has resulted in a
precise value of the muon mass as well as  the testing of the muon-electron 
charge
ratio to about $2\cdot 10^{-9}$. This is by far the best
test of charge equality in the first two generations.

The search for muonium-antimuonium conversion had been proposed by
Pontecorvo three years before the systemwas first produced by
Hughes {\it et al.}~\cite{Hughes_60}. Several 
new-physics models allow violation of lepton family number by two
units. The current limit is $R_g \equiv G_C / G_F<
0.0030$~\cite{Willmann_99}, where $G_C$ is the new-physics coupling
constant,
and $G_F$ the Fermi coupling constant. 
This sets a lower limit of $2.6 \,$TeV$/c^2$ (90\% C.L.) on the mass
of a grand-unified dileptonic gauge boson and also strongly disfavours
models with heavy lepton seeded radiative mass
generation~\cite{Willmann_99}. The search for muonium-antimuonium
conversion has by far the strongest gain in sensitivity of all rare
muon decay experiments \cite{Jungmann_01}.

The high intensity proton machine needed for the Neutrino Factory
can also find use as a new generation isotope facility which would
have much higher rates compared to the present ISOLDE
facility at CERN. Nucleids yet not studied  could be produced at quantities
which allow precision investigations of their properties
\cite{Aysto_01}.  The  measurements of muonic spectra can yield
most precise values for the charge radii of nuclei as well as other
ground state properties such as moments and even B(E2) transition
strengths for even-even nuclei. An improved understanding of nuclear
structure can be expected which may be of significance for
interpreting various neutrino experiments, rare decays involving
nuclei, and nuclear capture. An urgent need exists for accurate
charge and neutron radii of Francium and Radium isotopes which are of
interest for atomic parity violation research and $EDM$ searches in
atoms and nuclei.

Muonic x-ray experiments generally promise higher accuracy for most of
these quantities compared to electron scattering, particularly because
the precision of electron scattering data depends on the location of
the minimum of the cross section where rates are naturally low.  In
principle, for chains of isotopes charge radii can be inferred from
isotope shift measurements with laser spectroscopy. However, this
gives only relative information. For absolute values, calibration is
necessary and has been obtained in the past for stable nuclei from
muonic spectra.  In general, two not too distant nuclei are needed
for a good calibration.

The envisaged experimental approaches include i) the
technique pioneered by Nagamine and Strasser \cite{strasser_02}, which
involves cold films for keeping radioactive atoms and as a host
material in which muon transfer takes place; ii) merging beams if
radioactive ions and of muons; and iii) trapping of exotic isotopes in
a Penning trap which is combined with a cyclotron trap.  Large
formation rates can be expected from a setup containing a Penning
trap
\cite{Penning_trap},
the magnetic field of which serves also as a cyclotron muon trap
\cite{Simons}. 
For muon energies in the range of electron binding energies the muon
capture cross sections grow to atomic values, efficient atom
production results at the rate of approximately 50~Hz. 
It should be noted that antiprotonic atoms could be produced similarly
~\cite{Hayano_2001} 
and promise measurements of neutron distributions in nuclei.
%%
%%
\subsection[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.The synchrotron radiation loss in a circular accelerator
goes as the inverse fourth power of the mass and is two billion times
less for a muon than for an electron. Direct $s$ channel coupling to the
higgs boson goes as the mass squared and is 40,000 greater for the
muon than for 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 $\hsm f\bar f$
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 beamstrahlung, 
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 differences
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
$\call=1\times 10^{32}\rm\,cm^{-2}\ s^{-1}$ 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 (even 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). 
\subsubsection{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}
\bar\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}} \ .
%
%\bar\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
$\bar\sigma_h\propto 1/\srts$ for $\srts>\gamhtot$ {\it and}
that large event rates are only possible if $\gamhtot$ is not so large
that $\br(\h\to \mu\bar\mu)$ 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 $\h$.
Figure~\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.

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}

\subsubsection{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 and  couplings to heavy quarks
~\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 that 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 & $\mu^+\mu^-$\\
\hline
$\mh$ & $9\times 10^{-4}$ & $3\times 10^{-4}$ & $1-3\times 10^{-6}$ \\
$\gamhtot$ & $>0.3$ & 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,  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 $\mupmum\to\hl\to
b\bar b$ (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.

\subsection{Physics Potential of a High Energy Muon Collider}
Once one learns to cool muons, it becomes 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{bbgh-wtt,bergerw} and the top quark mass~\cite{bbgh-wtt,bergertop}
with high
accuracy, by scanning the thresholds of these particles and making use
of the excellent energy resolution of the beams. We consider 
further here 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.

\subsubsection{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 inferred.  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  $b\bar b \hh,b\bar b\ha$ production
for a wedge of moderate $\tanb$ 
values. (For large enough 
values of $\tanb$ the heavy Higgs bosons are expected to be observable
in $b\bar b \ha,b\bar b \hh$ 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 $b\bar b$ 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~\cite{kingnu}. 
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.
