\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. 

\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 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 the 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 ({ 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}.