\section{Introduction}
New accelerator technologies offer the possibility of building, not
too many years in the future, an accelerator complex to accumulate
more than $10^{19}$, and perhaps more than $10^{20}$,
muons per year~\cite{status_report}.
It has been proposed~\cite{geer98} to build a \textit{Neutrino Factory} by
accelerating the muons from this intense source to
energies of several GeV or more, injecting the muons into a storage ring
having long straight sections, and exploiting the
intense neutrino beams that are produced by muons decaying
in the straight sections.
If the challenge of producing,
capturing, accelerating, and storing a millimole of unstable muons can
be met, the decays
\begin{equation}
\mu^{-} \rightarrow e^{-}\nu_{\mu}\bar{\nu}_{e}\; , \qquad
\mu^{+} \rightarrow e^{+}\bar{\nu}_{\mu}\nu_{e}
\label{mumpdk}
\end{equation}
offer exciting possibilities for the study of neutrino interactions
and neutrino properties~\cite{geer98,abp,bgw,suite}. In a
Neutrino Factory the composition and spectra of intense
neutrino beams will be determined by the charge, momentum, and
polarization of the stored muons. The prospect of intense,
controlled, high-energy beams of electron neutrinos and
antineutrinos---for which we have no other plausible source---is very
intriguing.
Neutrinos---weakly interacting, nearly massless elementary
fermions---have long been objects of fascination, as well as reliable
probes. One of the most dramatic recent developments in particle
physics is the growing evidence that neutrinos may oscillate from
one species to another during propagation, which implies that
neutrinos have mass.
If neutrinos $\nu_{1},
\nu_{2}, \ldots$ have different masses $m_{1}, m_{2}, \ldots$ ,
each neutrino flavor state may be a mixture of different mass states. Let us
consider two species for simplicity, and take
\begin{equation}
\left(
\begin{array}{c}
\nu_{e} \\
\nu_{\mu}
\end{array}
\right) = \left(
\begin{array}{cc}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{array}
\right) \left(
\begin{array}{c}
\nu_{1}\\
\nu_{2}
\end{array}
\right)\; .
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%
%
% If neutrinos are emitted with a definite momentum $p$, the wave
%
% functions corresponding to the two mass eigenstates evolve with
%
% different frequencies. As a consequence, a beam born as pure
%
% $\nu_{\mu}$ may evolve a $\nu_{e}$ component with time. If the
%
% neutrino momentum is large compared with the neutrino masses, $p \gg
%
% m_{i}$, then the probability for a $\nu_{e}$ component to develop in a
%
% $\nu_{\mu}$ beam after a time $t$ is
%
% \begin{equation}
%
% P_{\nu_{e}\leftarrow\nu_{\mu}}(t) = \sin^{2}2\theta \sin^{2}\left(
%
% \frac{\Delta m^{2}\,t}{4p}\right)\; .
%
% \end{equation}
%
% Measuring the propagation distance $L = ct$, approximating the
%
% neutrino energy as $E \approx pc$, and using the conversion factor
%
% $\hbar c \approx 1.97 \times 10^{-13}\mev\m$, we can re-express
%
% \begin{equation}
%
% \sin^{2}\left(\frac{\Delta m^{2}t}{4p}\right) \approx
%
% \sin^{2}\left(1.27 \frac{\Delta m^{2}}{1\ev^{2}} \cdot \frac{L}{1\km}
%
% \cdot \frac{1\gev}{E}\right)\; .
%
% \label{eq:metamorph}
%
% \end{equation}
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
The probability for a neutrino born as $\nu_{\mu}$ to oscillate into a
$\nu_{e}$,
\begin{equation}
P(\nu_{\mu}\rightarrow\nu_{e}) = \sin^{2}2\theta \sin^{2}\left(1.27 \:
\frac{\delta m^{2}}{1\hbox{ eV}^{2}} \cdot \frac{L}{1\hbox{ km}} \cdot
\frac{1\hbox{ GeV}}{E}\right)\; ,
\end{equation}
depends on two parameters related to experimental conditions: $L$, the
distance from the neutrino source to the detector, and $E$, the
neutrino energy. It also depends on two fundamental neutrino
parameters: the difference of masses squared, $\delta m^{2} =
m_{1}^{2} - m_{2}^{2}$, and the neutrino mixing parameter,
$\sin^{2}2\theta$.
The probability that a neutrino born as $\nu_{\mu}$ remain a
$\nu_{\mu}$ at distance $L$ is
\begin{equation}
P(\nu_{\mu}\rightarrow\nu_{\mu}) =
1 - \sin^{2}2\theta \sin^{2}\left(1.27\: \frac{\delta m^{2}}{1\hbox{ eV}^{2}}
\cdot \frac{L}{1\hbox{ km}} \cdot \frac{1\hbox{ GeV}}{E}\right)\; .
\end{equation}
Many experiments have now used natural sources of neutrinos, neutrino
radiation from fission reactors, and neutrino beams generated in
particle accelerators to look for evidence of neutrino oscillation.
The positive indications for neutrino oscillations fall into three
classes:\cite{janetc}
\begin{enumerate}
\item Five solar-neutrino experiments report deficits with respect
to the predictions of the standard solar model: Kamiokande and
Super-Kamiokande (SuperK) using water-Cerenkov techniques, SAGE and GALLEX
using chemical recovery of germanium produced in neutrino
interactions with gallium, and Homestake using radiochemical
separation of argon produced in neutrino interactions with
chlorine. These results suggest the oscillation $\nu_{e}
\rightarrow \nu_{x}$, with $|\delta m^{2}|_{\mathrm{solar}} \approx
10^{-5}\hbox{ eV}^{2}$ and $\sin^{2}2\theta_{\mathrm{solar}}\approx 1\hbox{
or a few}\times 10^{-3}$, or $|\delta m^{2}|_{\mathrm{solar}} \approx
10^{-10}\hbox{ eV}^{2}$ and $\sin^{2}2\theta_{\mathrm{solar}}\approx 1$.
\item Five atmospheric-neutrino experiments report anomalies in the
arrival of muon neutrinos: Kamiokande, IMB, and SuperK using
water-Cerenkov techniques, and Soudan~2 and MACRO using sampling
calorimetry. The most striking result is the zenith-angle dependence
of the $\nu_{\mu}$ rate reported last year by SuperK
\cite{SKatm,SKLyon}. These results suggest the oscillation $\nu_{\mu}
\rightarrow \nu_{\tau}\hbox{ or }\nu_{s}$, with
$\sin^{2}2\theta_{\mathrm{atm}} \approx 1$ and $|\delta
m^{2}|_{\mathrm{atm}} = 10^{-3}\hbox{ to }10^{-2}\hbox{ eV}^{2}$. The
oscillation $\nu_{\mu} \rightarrow \nu_{\tau}$ is increasingly
the favored interpretation.
\item The LSND experiment~\cite{s1LSND} reports the observation of
$\bar{\nu}_{e}$-like events in what should be an essentially pure
$\bar{\nu}_{\mu}$ beam produced at the Los Alamos Meson Physics
Facility, suggesting the oscillation $\bar{\nu}_{\mu} \rightarrow
\bar{\nu}_{e}$. This result has not yet been reproduced by any other
experiment. The favored region lies along a band from
$(\sin^{2}2\theta_{\mathrm{LSND}} = 10^{-3},|\delta
m^{2}|_{\mathrm{LSND}} \approx 1\hbox{ eV}^{2})$ to
$(\sin^{2}2\theta_{\mathrm{LSND}} = 1,|\delta
m^{2}|_{\mathrm{LSND}} \approx 7 \times 10^{-2}\hbox{ eV}^{2})$.
\end{enumerate}
A host of other experiments have failed to turn up evidence for neutrino
oscillations in the regimes of their sensitivity. These results limit
neutrino mass-squared differences and mixing angles. In more than a
few cases, positive and negative claims are in conflict, or at least
face off against each other. Over the next five years, many
experiments will seek to verify, further quantify, and extend these
claims. If all of the current experimental indications of neutrino
oscillation survive, there are apparently three different
mass-squared-difference scales, which cannot be accommodated with
only three neutrino types. New \textit{sterile} neutrinos may be
required. This would be a profound discovery.
From the era of the celebrated two-neutrino experiment \cite{twonu} to modern
times, high-energy neutrino beams have played a decisive role in the
development of our understanding of the constituents of matter and the
fundamental interactions among them. Major landmarks include the
discovery of weak neutral-current interactions \cite{weaknc}, and
incisive studies of the structure of the proton and the quantitative
verification of perturbative quantum chromodynamics as the theory of
the strong interactions \cite{rmpnurev}. The determinations of the
weak mixing parameter $\sin^{2}\theta_{W}$ and the strong coupling
constant $\alpha_{s}$ in deeply inelastic neutrino interactions are
comparable in precision to the best current measurements. Though
experiments with neutrino beams have a long history, beams of
greatly enhanced intensity would bring opportunities for dramatic
improvements. Because weak-interaction cross sections are small,
high-statistics studies have required massive targets and
coarse-grained detectors. Until now, it has been impractical to
consider precision neutrino experiments using short liquid hydrogen
targets, or polarized targets, or active semiconductor
target-detectors. All of these options are opened by a muon storage
ring, which would produce neutrinos at approximately $10^{4}$ times
the flux of existing neutrino beams.
At the energies best suited for the study of neutrino
oscillations---tens of GeV, by our current estimates---the muon
storage ring is compact. We could build it at one laboratory, pitched
at a deep angle, to illuminate a laboratory on the other side of the
globe with a neutrino beam whose properties we can control with great
precision. By choosing the right combination of energy and
destination, we can tune future neutrino-oscillation experiments to
the physics questions we will need to answer, by specifying the ratio
of path length to neutrino energy and determining the amount of matter
the neutrinos traverse. Although we can use each muon decay only
once, and we will not be able to select many destinations, we may be
able to illuminate two or three well-chosen sites from a
muon-storage-ring neutrino source. That possibility---added to the
ability to vary the muon charge, polarization, and energy---may give
us just the degree of experimental control it will take to resolve the
outstanding questions about neutrino oscillations.
Experiments at a Neutrino Factory would seek to verify the number
of neutrino types participating in the oscillations, precisely
determine the mixing parameters that relate the flavor states
to the mass states, determine the pattern of neutrino masses,
and look for CP violation in the lepton sector.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% \cite{sgeer} %
% Under rather special circumstances, %
% it may be possible to observe \CP\ violation in neutrino %
% oscillations.\cite{nucp} %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The prodigious flux of neutrinos close to the muon storage ring raises
the prospect of neutrino-scattering experiments of unprecedented
sensitivity and delicacy.
Experiments that might be pursued at a Neutrino Factory include
precise measurements of the nucleon structure (including
changes that occur in a nuclear environment),
measurements of the spin structure of the nucleon using a
new and powerful technique,
charm measurements with several million tagged particles,
precise measurements of Standard Model parameters,
and searches for exotic phenomena.
We believe that the physics program at a Neutrino Factory is
compelling and encourage support for a vigorous R\&D program
to make neutrino factories a real option for the future.