%\documentstyle[12pt,epsf,epsfig,rotating]{article}
\documentstyle[12pt,epsf,epsfig]{article}
%\documentstyle[aps,epsf,epsfig,preprint,tighten,rotating]{revtex}
%\documentstyle[aps,preprint,epsf,tighten]{revtex}
\special{header=Draft.ps}
\parskip= 4pt plus 1pt
\textwidth=5.65in
\textheight=23.0cm
\oddsidemargin=0.4in
\evensidemargin=0.4in
\headsep=0.1mm
\topmargin=0.001in
\input hep_macro
%\input epsf
\newcommand{\centii}{\hbox{\rm cm$^2$}}
%%% Chris Quigg
%\usepackage{hyperref}
%\newcommand{\hepex}[1]{(hep-ex/#1)}
\newcommand{\hepex}[1]{\mbox{\href{http://xxx.lanl.gov/abs/hep-ex/#1}{(hep-ex/
#1)}}}
%\newcommand{\hepph}[1]{(hep-ph/#1)}
\newcommand{\hepph}[1]{\mbox{\href{http://xxx.lanl.gov/abs/hep-ph/#1}{(hep-ph/
#1)}}}
%\newcommand{\hepth}[1]{(hep-th/#1)}
\newcommand{\hepth}[1]{\mbox{\href{http://xxx.lanl.gov/abs/hep-th/#1}{(hep-th/
#1)}}}
%\newcommand{\heplat}[1]{(hep-lat/#1)}
\newcommand{\heplat}[1]{\mbox{\href{http://xxx.lanl.gov/abs/hep-lat/#1}{(hep-
lat/#1)}}}
%\newcommand{\astro}[1]{(astro-ph/#1)}
\newcommand{\astroph}[1]{\mbox{\href{http://xxx.lanl.gov/abs/astro-ph/
#1}{(astro-ph/#1)}}}
\def\url#1{\mbox{\href{#1}{\sf #1}}}
\def\urll#1#2{\mbox{\href{#1}{\sf #2}}}
\def\urlp#1#2{\mbox{\href{#1}{#2}}}
\newcommand{\CP}{\textsf{CP}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% \def\hepph#1{(hep-ph/#1)} %
% \def\hepex#1{(hep-ex/#1)} %
% \def\astroph#1{(astro-ph/#1)} %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\pr#1#2#3{\frenchspacing{\it Phys. Rev. D}{\bf #1} (19#3) #2}
\def\prstab#1#2#3{\frenchspacing{\it Phys. Rev. ST Accel. Beams\ }{\bf #1}
(19#3) #2}
\def\prl#1#2#3{\frenchspacing{\it Phys. Rev. Lett.\ }{\bf #1} (19#3) #2}
\def\pl#1#2#3{\frenchspacing{\it Phys. Lett.\ }{\bf #1} (19#3) #2}
\def\prc#1#2#3{\frenchspacing{\it Phys. Rev. C}{\bf #1} (19#3) #2}
\def\np#1#2#3{\frenchspacing{\it Nucl. Phys. }{\bf #1} (19#3) #2}
\def\ib#1#2#3{\frenchspacing{\it ibid. }{\bf #1} (19#3) #2}
\def\jhep#1#2#3{\frenchspacing{\it J. High Energy Phys.\ }{\bf #1} (19#3) #2}
\def\npbps#1#2#3{{\em Nucl. Phys. B (Proc. Supp.)\/} {\bf #1} (19#3) #2}
\def\rmp#1#2#3{\frenchspacing{\it Rev. Mod. Phys. }{\bf #1}, #2 (19#3)}
%%%%%%%%%%%%%%%%%%String definitions%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\xw}{\ensuremath{\sin^2\theta_W}}
\newcommand{\ev}{\ensuremath{\hbox{ eV}}}
\newcommand{\evcc}{\ensuremath{\hbox{ eV}\!/\!c^2}}
\newcommand{\mevcc}{\ensuremath{\hbox{ MeV}\!/\!c^2}}
\newcommand{\mev}{\ensuremath{\hbox{ MeV}}}
\newcommand{\gevcc}{\ensuremath{\hbox{ GeV}\!/\!c^2}}
\newcommand{\gev}{\ensuremath{\hbox{ GeV}}}
%\newcommand{\cm}{\ensuremath{\hbox{ cm}}}
%\newcommand{\etal}{et al.}
\def\ltap{\mathop{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}}}
\def\gtap{\mathop{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}}
\newcommand{\km}{\hbox{ km}}
\newcommand{\m}{\hbox{ m}}
%%% Steve Parke
\def\Bra#1{\left\langle #1\right|}
\def\Ket#1{\left| #1\right\rangle}
\def\bra#1{\langle #1|}
\def\ket#1{| #1\rangle}
\def\aprle{\buildrel < \over {_{\sim}}}
\def\aprge{\buildrel > \over {_{\sim}}}
\newcommand{\gtwid}{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex
\hbox{$\sim$}}}}
\newcommand{\ltwid}{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex
\hbox{$\sim$}}}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\begin{document}
\title{Physics at a Neutrino Factory}
\author{lots of names}
\date{Draft 0}
\maketitle
\newpage
\tableofcontents
\clearpage
\section{Introduction}
New accelerator technologies offer the possibility that it may be
possible, not too many years in the future, to accumulate
$10^{19\mathrm{ - }20}$ (or even $10^{21}$ or more) muons per year.
If the challenge of producing, capturing, storing, and replenishing
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 delicious possibilities for the study of neutrino interactions
and neutrino properties \cite{geer,abp,bgw,suite}. In a
\textit{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 metamorphose 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 may be a mixture of different masses. 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 metamorphose into a
$\nu_{e}$,
\begin{equation}
P_{\nu_{e}\leftarrow\nu_{\mu}} = \sin^{2}2\theta \sin^{2}\left(1.27
\frac{\delta m^{2}}{1\rm{eV}^{2}} \cdot \frac{L}{1\rm{km}} \cdot
\frac{1\rm{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}\leftarrow\nu_{\mu}}(L) =
1 - \sin^{2}2\theta \sin^{2}\left(1.27 \frac{\delta m^{2}}{1\rm{eV}^{2}}
\cdot \frac{L}{1\km} \cdot \frac{1\rm{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 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 $|\delta m^{2}|_{\mathrm{solar}} \approx
10^{-5}$eV$^{2}$ and $\sin^{2}2\theta_{\mathrm{solar}}\approx 1\hbox{
or a few}\times 10^{-3}$, or $|\delta m^{2}|_{\mathrm{solar}} \approx
10^{-10}$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 Super-Kamiokande using
water-Cerenkov techniques, and Soudan II 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 $|\delta
m^{2}|_{\mathrm{atm}} = 10^{-3}\hbox{ to }10^{-4}$eV$^{2}$.
\item The LSND experiment \cite{LSND} reports the observation of
$\bar{\nu}_{e}$-like events is 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
$(\sin^{2}2\theta_{\mathrm{LSND}} = 10^{-3},|\delta
m^{2}|_{\mathrm{LSND}} \approx 1$eV$^{2})$ to
$(\sin^{2}2\theta_{\mathrm{LSND}} = 1,|\delta
m^{2}|_{\mathrm{LSND}} \approx 7 \times 10^{-2}$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.
From 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 storied 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.
\textbf{Oscillation questions and payoff go here.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% \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.
\textbf{Non-oscillation experiments and payoff summarized here.}
Perhaps we want to conclude this introduction with a recommendation for action.
\section{Beam properties}
Consider an ensemble of polarized negatively-charged muons.
When the muons decay they produce muon neutrinos with a distribution
of energies and angles in the muon rest--frame described
by~\cite{gaisser}:
%
\begin{eqnarray}
\frac{d^2N_{\nu_\mu}}{dxd\Omega} &\propto& {2x^2\over4\pi}
\left[ (3-2x) + (1-2x) P_\mu \cos\theta \right] \, ,
\label{eq:n_numu}
\end{eqnarray}
where $x\equiv 2E_\nu/m_\mu$, $\theta$ is the angle between the neutrino
momentum vector and the muon spin direction, and $P_\mu$ is the average muon
polarization along the beam direction.
The electron antineutrino distribution is given by:
\begin{eqnarray}
\frac{d^2N_{\bar\nu_e}}{dxd\Omega} &\propto& {12x^2\over4\pi}
\left[ (1-x) + (1-x) P_\mu\cos\theta \right] \, ,
\label{eq:n_nue}
\end{eqnarray}
%
and the corresponding distributions for
$\bar\nu_\mu$ and $\nu_e$ from $\mu^+$ decay are obtained by
the replacement $P_{\mu} \to -P_{\mu}$.
Only neutrinos and antineutrinos emitted in the forward
direction ($\cos\theta\simeq1$) are relevant to the neutrino flux for
long-baseline experiments; in this limit
$E_\nu = x E_\mu$ and at high energies the maximum $E_\nu$ in the
laboratory frame is given by
$E_{max} = \gamma (1 + \beta \cos\theta_{cm})m_{\mu}/2 $.
The $\nu_\mu$ and $\overline{\nu}_{e}$ distributions are then given by:
\begin{eqnarray}
\frac{d^2N_{\nu_{\mu}}}{dxd\Omega_{lab}} &\propto&
{1\over \gamma^2 (1- \beta\cos\theta_{lab})^2}\frac{2x^2}{4\pi}
\left[ (3-2x) + (1-2x)P_{\mu}\cos\theta_{cm} \right] ,
\label{eq:numu}
\end{eqnarray}
and
\begin{eqnarray}
\frac{d^2N_{\overline{\nu}_{e}}}{dxd\Omega_{lab}} &\propto&
{1\over \gamma^2 (1- \beta\cos\theta_{lab})^2}\frac{12x^2}{4\pi}
\left[ (1-x) + (1-x)P_{\mu}\cos\theta_{cm} \right] \; .
\label{eq:nue}
\end{eqnarray}
Thus, for a high energy muon beam with no beam divergence,
the neutrino and antineutrino energy-- and angular--
distributions depend upon the parent muon
energy, the decay angle, and the direction of the
muon spin vector.
With the muon beam intensities that could be provided by a
muon--collider type muon source the resulting neutrino fluxes
at a distant site would be large. For example, Fig.~\ref{fluxes} shows
as a function of muon energy and polarization,
the computed fluxes per $2\times 10^{20}$ muon decays at a site on the
other side of the Earth ($L = 10000$~km).
Note that the $\nu_e$ and $\overline{\nu}_e$ fluxes are suppressed
when the muons have $P = +1$ (-1). This can be understood by examining
Eq.~\ref{eq:nue} and noting that for $P = -1$ the two terms cancel
in the forward direction for all $x$.
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{fluxes_fig.eps}}
%\centerline{\epsffile{cc_rates_fig.ps}}
%\renewcommand{\baselinestretch}{2}
\caption{Calculated $\nu$ and $\overline{\nu}$ CC rates at a far site located
10000 km from a neutrino factory in which
$2 \times 10^{20}$ muons have decayed in the beam--forming straight section.
The fluxes are shown as a function of the energy of the stored
muons for negative muons (top two plots)
and positive muons (bottom two plots), and for three muon polarizations
as indicated.}
\label{fluxes}
\end{figure}
\subsection{Interaction rates}
Neutrino CC scattering cross-sections are shown as a function of
energy in Fig.~\ref{tau_ratio}.
%
\begin{figure}
\epsfxsize3.in
\centerline{\epsffile{tau_ratio.eps}}
\caption{a) The total cross section for charged current neutrino scattering
by muon and tau neutrinos. b) the ratio of tau to muon neutrino cross sections
as a function of neutrino energy.}
\label{tau_fig}
\end{figure}
%
At low energies the neutrino scattering cross section is dominated by
quasi-elastic scattering and single pion production.
However, if $E_\nu$ is greater than $\sim10$~GeV,
the total cross section is dominated by deep inelastic scattering
and is approximately\cite{CCFRsigma}:
%
\begin{eqnarray}
\sigma(\nu +N \gt \muminus + X) &\approx& 0.67\times 10^{-38} \;
\centi^2\times (E_{\nu},
\GeV) \, , \\
%\end{eqnarray}
%\begin{eqnarray}
\sigma(\antinu +N \gt \muplus + X) &\approx& 0.34\times
10^{-38} \; \centi^2\times (E_{\antinu},
\GeV) \; .
\end{eqnarray}
%
The number of $\nu_e$ and $\nu_\mu$ CC events observed per incident neutrino
on an isoscalar target is given by:
\begin{eqnarray}
N(\nu +N \gt \muminus + X)
&=& 4.0 \times 10^{-15} (E_{\nu}, \GeV) \;
\hbox{events per gr/cm$^2$} \; , \\
N(\antinu +N \gt \muplus + X)
&=& 2.0 \times 10^{-15}( E_{\antinu}, \GeV) \;
\hbox{events per gr/cm$^2$} \; .
\end{eqnarray}
Using this simple form for the energy dependence of the cross section,
the predicted energy distributions for $\nu_e$ and $\nu_\mu$
interacting in a far detector ($\cos\theta = 1$) at a neutrino
factory are shown in Fig.~\ref{polarization}. The interacting $\nu_\mu$
energy distribution is compared in Fig~\ref{minos_wbb}
with the corresponding distribution arising from the high--energy
NUMI wide band beam. Note that neutrino beams from a
neutrino factory can be considered narrow band beams.
%
\begin{figure}
\epsfxsize3.in
\centerline{\epsffile{polarization.eps}}
%\renewcommand{\baselinestretch}{2}
\caption{Charged current event spectra at a far detector. The solid lines
indicate zero polarization, the dotted lines indicate polarization of $\pm 0.3$ and
the dashed lines indicate full polarization. The $P=1$ case for electron neutrinos
results in no events and is hidden by the x axis.}
\label{polarization}
\end{figure}
%%
\begin{figure}
\epsfxsize3.in
\centerline{\epsffile{minos_wbb.ps}}
\caption{Comparison of interacting $\nu_\mu$ energy distributions for
a 20~GeV neutrino factory beam and the NUMI high energy wide band beam.
}
\label{minos_wbb}
\end{figure}
%
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{elept.ps}}
\caption{Lepton energy spectra for CC $\overline{\nu}_\mu$ (top left),
$\nu_\mu$ (top right), $\nu_e$ (bottom left), and $\overline{\nu}_e$
(bottom right) interactions. Note that Z is the normalized energy $E_L/E_\mu$.
Calculation from Ref.\ref{BGW99}.}
\label{fig:elept}
\end{figure}
%
In practice, CC interactions can only be cleanly identified when the final
state lepton exceeds a threshold energy. The calculated final state lepton
distributions are shown in Fig.~\ref{fig:elept}.
Integrating over the energy distribution, the total $\nu$ and $\overline{\nu}$
interaction rates per incident particle are given by:
%
\begin{eqnarray}
N_\nu &=& 1.2 \times 10^{-14} \; \biggr[{(E_{\mu}, \GeV)^3\over (L, km)^2}\biggl]
\times C(\nu) \;\; \hbox {events per kt}
\end{eqnarray}
and
\begin{eqnarray}
N_{\overline{\nu}}&=&0.6\times10^{-14} \;
\biggr[{(E_{\mu}, \GeV)^3\over (L, km)^2}\biggl]
\times C(\nu) \;\; \hbox{events per kt} \, ,
\end{eqnarray}
%
where
%
\begin{eqnarray}
C(\nu_{\mu})&=& {7\over 10} + P_{\mu} {3\over 10}, \ \ \ \
C(\nu_{e}) ={6\over 10} - P_{\mu} {6\over 10}\\
\end{eqnarray}
The calculated $\nu_e$ and $\nu_\mu$ CC interaction rates resulting from
$10^{20}$ muon
decays in the beam--forming straight--section of a neutrino factory
are compared in Table~\ref{rates_tab} with
expectations for the corresponding rates at the next generation
of accelerator--based neutrino experiments, and the radial dependence
of the event rate is shown in Fig.~\ref{fig:radial} for a 20~GeV
neutrino factory and three baselines. Note that event rates
at a neutrino factory increase as $E_\mu^3$, and are significantly
larger than expected for the next generation of approved experiments
if $E_\mu > 20$~GeV.
\begin{table}
%\renewcommand{\baselinestretch}{2}
\caption{\label{compare_tab}
Muon neutrino and electron antineutrino CC interaction rates in the
absence of oscillations, calculated for baseline length $L = 732$~km
(FNAL $\rightarrow$ Soudan), for MINOS using the wide band beam and a
muon storage ring with $E_\mu=10, 20, 50$ and $250$~GeV at 3 baselines.
}
\begin{center}
%\vspace{0.6 cm}
\begin{tabular}{|ccc|cc|cc|cc|cc|cc}
%\hline \hline
\hline
& &Baseline & $\langle E_{\nu_\mu} \rangle$ & $\langle E_{\bar \nu_e} \rangle$
& N($\nu_\mu$ CC) & N($\bar\nu_e$ CC) \\
Experiment & &(km) & (GeV) & (GeV) & (per kt--yr) & (per kt--yr) \\
\hline
MINOS& Low energy &732& 3 & -- & 458 & 1.3 \\
& Medium energy &732& 6 & -- & 1439 & 0.9 \\
& High energy &732& 12 & -- & 3207 & 0.9 \\
\hline
Muon ring & $E_\mu$ (GeV) & & & & & \\
\hline
& 10 &732& 7 & 6 & 3,000 & 1,300 \\
& 20 &732& 14 & 12 & 24,000 & 11,000\\
& 50 &732& 35 & 30 & 3.8$\times$10$^5$ & 1.7$\times$10$^5$ \\
%& 250 &732& 175 & 150 & 4.7$\times$10$^7$ & 2.1$\times$10$^7$ \\
\hline
Muon ring& $E_\mu$ (GeV)& & & & & \\
\hline
& 10 &2900& 7 & 6 & 190 &84\\
& 20 &2900& 14 & 12 & 1,500 & 670\\
& 50 &2900& 35 & 30 & 24,000& 11,000 \\
%& 250 &2900& 175 & 150 & 3.0$\times$10$^6$ & 1.3$\times$10$^6$ \\
\hline
Muon ring& $E_\mu$ (GeV)& & & & & \\
\hline
& 10 &7300& 7 & 6 & 30 & 13 \\
& 20 &7300& 14 & 12 & 240 & 110 \\
& 50 &7300& 35 & 30 & 3,800 & 1,700 \\
%& 250 &7300& 175 & 150 & 4.8$\times$10$^5$ & 2.1$\times$10$^5$ \\
\hline
%\hline\hline
\end{tabular}
\end{center}
\end{table}
%
\begin{figure}
\epsfxsize 3.in
\centerline{\epsffile{20gev.eps}}
\caption{Events/kT of detector as a function of distance from the beam center
for a 20 GeV muon beam.}
\label{fig:radial}
\end{figure}
Finally, for an isoscalar target the neutral current (NC) cross sections
are approximately 1/3 of the CC cross sections, and are given by:
\begin{eqnarray}
\sigma(\nu +N \gt \nu + X) &\approx& 0.21\times 10^{-38} \;
\centi^2\times (E_{\nu},
\GeV) \, , \\
%\end{eqnarray}
%\begin{eqnarray}
\sigma(\antinu +N \gt \overline{\nu} + X) &\approx& 0.12\times
10^{-38} \; \centi^2\times (E_{\antinu},
\GeV) \; .
\end{eqnarray}
\subsection{Tau neutrino interactions}
%
Tau neutrino interaction rates are substantially less than the corresponding
$\nu_e$ and $\nu_\mu$ rates,
especially near the production threshold of $\sim 3.3$~GeV.
Above threshold the $\tau$--lepton mass terms in the leptonic
current cannot be ignored, allowing the axial vector structure
functions $W_4$ and $W_5$ to play a non-negligible role.
Figure~\ref{tau_fig} shows the calculated~\cite{goodman}
ratio of $\nu_\tau / \nu_\mu$ CC
interaction rates as a function of the neutrino energy.
Near threshold, contributions
from quasi--elastic and single--pion production dominate.
If the $\nu_\tau$ cross sections from
Ref.~\cite{casper} are used, the predicted event rates are 20--30\%
higher.
\begin{table}
\renewcommand{\baselinestretch}{1}
\begin{center}
\label{tab:com}
\vspace{0.6 cm}
\caption{Dependence of predicted charged current event rates on muon
beam properties at a neutrino factory. The last column
lists the required precisions with which each beam property must
be determined if the uncertainty on the neutrino flux at the
far site is to be less than $\sim1$\%. Here $\Delta$ denotes uncertainty
while $\sigma$ denotes the spread in a variable.}
\vspace{0.2cm}
\begin{tabular}{c|c|cc} \hline
Muon Beam & Beam & Rate & Target\\
property & Type & Dependence & Precision \\
\hline
Energy ($E_\mu$) & $\nu$ (no osc)
& $\Delta N / N = 3 \; \Delta E_\mu/E_\mu$
& $\Delta(E_\mu)/E_\mu < 0.003$ \\
& $\nu_{e} \rightarrow \nu_{\mu}$
&$\Delta N / N = 2 \; \Delta E_\mu/E_\mu$
& $\Delta(E_\mu)/E_\mu < 0.005$ \\
\hline
Direction ($\Delta\theta$) & $\nu$ (no osc)
& $\Delta N/N \leq 0.01$
& $\Delta\theta < 0.6 \; \sigma_\theta$ \\
& & (for $\Delta\theta < 0.6\; \sigma_\theta$) & \\
\hline
Divergence ($\sigma_\theta$)
& $\nu$ (no osc)
& $\Delta N / N \sim 0.03 \; \Delta\sigma_\theta / \sigma_\theta$
& $\Delta\sigma_\theta / \sigma_\theta < 0.2$ \\
& & (for $\sigma_\theta \sim 0.1/\gamma$)
& (for $\sigma_\theta \sim 0.1/\gamma$)\\
\hline
Momentum spread ($\sigma_p$)
& $\nu$ (no osc)
& $\Delta N / N \sim 0.06 \; \Delta\sigma_p / \sigma_p$
& $\Delta\sigma_p / \sigma_p < 0.17$ \\
\hline
Polarization ($P_\mu$)
& $\nu_e$ (no osc)
& $\Delta N_{\nu_e} / N_{\nu_e} = \Delta P_\mu$
& $\Delta P_\mu < 0.01$ \\
& $\nu_{\mu}$ (no osc)
& $\Delta N_{\nu_\mu} / N_{\nu_\mu} = 0.4 \; \Delta P_\mu$
& $\Delta P_\mu < 0.025$ \\
\hline
\end{tabular}
\label{tab:flux}
\end{center}
\end{table}
\subsection{Systematic uncertainties on the muon beam and neutrino flux}
In the neutrino beam--forming straight section the muon beam
is expected to have an average divergence given by
$\sigma_\theta =$~O($0.1/\gamma$). The neutrino beam divergence will
therefore be dominated by muon decay kinematics, and uncertainties
on the beam direction and divergence will yield only small
uncertainties in the neutrino flux at a far site. However, if
very precise knowledge of the flux is required, the uncertainties
on $\theta$ and $\sigma_\theta$ must be taken into account, along
with uncertainties on the flux arising from uncertainties on
the muon energy distribution and polarization.
The relationships between the uncertainties on the muon beam
properties and the resulting uncertainties on the neutrino flux
are summarized in Table~\ref{tab:flux}. If, for example, we wanted
to know the $\nu_e$ and $\nu_mu$ fluxes at a far site with a
precision of 1\%, we would need to know the beam divergence $\sigma_\theta$
to 20\% (Fig.~\ref{fig:flux_xy}), and ensure that the beam direction
was within $0.6\;\sigma_\theta$ of the nominal direction
(Fig.~\ref{fig:flux_d}).
%
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{flux_xy.ps}}
\vspace{-2.3cm}
\caption{
Dependence of CC interaction rates on the muon beam divergence
for a detector located at
$L = 2800$~km from a muon storage ring containing 30~GeV unpolarized muons.
Rates are shown
for $\nu_e$ (triangles) and $\nu_\mu$ (circles) beams
in the absence of oscillations,
and for $\nu_e \rightarrow \nu_\mu$ oscillations (boxes) with the
three--flavor oscillation parameters IA1.
The calculation is from Ref.~\ref{geer00}.
}
\label{fig:flux_xy}
\end{figure}
%
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{flux_d.ps}}
\vspace{-2.0cm}
\caption{
Dependence of CC interaction rates on the neutrino beam direction.
Relative rates are shown for
a detector at a far site located downstream of
a storage ring containing 30~GeV unpolarized muons, and
a muon beam divergence of 0.33~mr. Rates are shown
for $\nu_e$ (triangles) and $\nu_\mu$ (circles) beams
in the absence of oscillations,
and for $\nu_e \rightarrow \nu_\mu$ oscillations (boxes) with the
three--flavor oscillation parameters IA1.
The calculation is from Ref.~\ref{geer00}.
}
\label{fig:flux_d}
\end{figure}
%
\subsection{Event distributions at a near site}
The event distributions measured in a detector close to the neutrino
factory will be quite different from the corresponding distributions
at a far site. There are two main reasons for this difference.
First, the near detector accepts neutrinos over a large range of
muon decay angles $\theta$, not just those neutrinos traveling in the
extreme forward direction. This results in a broader neutrino energy
distribution that is sensitive to the radial size of the
detector (Fig.~\ref{nearspectra}).
Second, if the distance of the detector from the end of the beam forming
straight section is of the order of the straight section length,
then the $\theta$ acceptance of the detector varies with the position of the
muon decay along the straight section. This results in a more complicated
radial flux distribution than expected for a far detector
(Fig.~\ref{xplot}).
\begin{figure}
\epsfxsize 3.in
\centerline{\epsffile{nearspectra.ps}}
\caption{Events per gr/cm$^2$ per GeV for a detector 40~m from a
muon storage ring with a 600 m straight section. The 3 curves show
all events and those falling within 50 and 20~cm of the beam center. }
\label{nearspectra}
\end{figure}
%
\begin{figure}
\epsfxsize 3.in
\centerline{\epsffile{x.eps}}
\caption{Events per gr/cm$^2$ as a function of the
transverse coordinate x 50~m downstream of a 50~GeV neutrino
factory providing $10^{20}$ muon decays.
The central peak is mainly due to decays
in the last hundred meters of the decay pipe
while the large tails are due to upstream decays.}
\label{xplot}
\end{figure}
Note that,even in a limited angular range, the event rates in a
near detector are very high. Figure~\ref{eventrates} illustrates the
event rates per gram/cm$^2$ as a function of energy. Because
most of neutrinos produced forward in the center of mass are detected,
the factor of $\gamma^2$
present in the flux for $\theta \sim 0$ is lost and the event rate
increases linearly with $E_{\mu}$. For a 50~GeV
muon storage ring, the interaction rate per 10$^{20}$ muon decays is
7~million events/gram/cm$^2$. Rate calculations are discussed further
in the context of specific experiments in the section on
non--oscillation experiments.
Finally, in the absence of special magnetized shielding,
the high neutrino event rates in any material
upstream of the detector will cause substantial backgrounds.
The event rate in the last 3 interaction lengths (300~gr/cm$^2$)
of the shielding between
the detector and the storage ring would be 140 interactions per beam
spill at a 15 Hz
machine delivering $2\times 10^{20}$ muon decays per year.
These high background rates will require clever magnetized shielding
designs and fast detector
readout to avoid overly high accidental rates in low mass experiments.
\begin{figure}
\epsfxsize 3.in
\centerline{\epsffile{eventrates.eps}}
\caption{Events per gr/cm$^2$ at a near detector as a function of muon beam energy. The curves
indicate (solid) all events, the dashed and dotted curves show the effects of
radial position cuts. REPLACE BY TABLE?}
\label{eventrates}
\end{figure}
\clearpage
\section{Oscillation physics}
The recent impressive atmospheric neutrino results from the SuperK
experiment have gone a long way towards establishing the existence
of neutrino oscillations. Up to the present era, neutrino oscillation
experiments at accelerators were searches for a phenomenon that might
or might not be within experimental reach. The situation now is quite
different. The atmospheric neutrino deficit defines for us the
$\delta m^2$ and oscillation amplitude that future long baseline oscillation
experiments must be sensitive to. Experiments that achieve these
sensitivities are guaranteed an excellent physics program that addresses
fundamental physics questions. We can hope that future neutrino
oscillation experiments will provide the keys we need to understand
really fundamental questions, for example: the origin of the minute
neutrino masses and the reason why there are three lepton families ?
We cant guarantee that these insights will be forthcoming from
neutrino oscillation measurements, but they might be ! For this reason
it is important to take a hard look at how our community can get
detailed experimental information on the neutrino oscillation scheme,
the mass splittings between the neutrino mass eigenstates, and the
leptonic mixing matrix that controls the oscillation probabilities.
A neutrino factory would be a new tool, providing a beam of energetic
electron neutrinos. In this section we address how this new tool might
be exploited to go well beyond the capabilities of the next generation
of neutrino oscillation experiments.
\subsection{Theoretical framework}
There exists three known flavors of active neutrinos which
form left-handed doublets with their associated charged leptons.
The interaction of these active neutrinos with the
electroweak gauge bosons is described by the Standard Model (SM).
In principle there can be additional flavors of neutrino
which are singlets under the electroweak gauge group.
These electroweak singlet neutrinos do not have electroweak
couplings, and their interactions are not described by
the SM. Let us denote the flavor
vector of the SU(2) $\times$ U(1) active neutrinos as $\nu =
(\nu_e,\nu_\mu,\nu_\tau)$ and the vector of electroweak-singlet neutrinos
as $\chi = (\chi_1,..,\chi_{n_s})$. The Dirac
and Majorana neutrino mass terms can then be written compactly as
\beq
-{\cal L}_m =
{1 \over 2}(\bar\nu_L \ \overline{\chi^c}_L) \left( \begin{array}{cc}
M_L & M_D \\ (M_D)^T & M_R \end{array} \right )\left( \begin{array}{c}
\nu^{c}_R \\ \chi_R \end{array} \right ) + h.c.
\label{numass}
\eeq
where $M_L$ is the $3 \times 3$ left-handed Majorana mass matrix, $M_R$ is a
$n_s \times n_s$ right-handed Majorana mass matrix, and $M_D$ is the 3-row by
$n_s$-column Dirac mass matrix. In general, all of these are complex, and
$(M_L)^T = M_L \ , \quad (M_R)^T = M_R$. Without further theoretical input,
the number $n_s$ of ``sterile" electroweak-singlet neutrinos is
not determined.
For example, in SM, minimal supersymmetric standard model (MSSM),
or minimal SU(5) grand unified theory (GUT), $n_s=0$, while in the SO(10)
GUT, $n_s=3$. (This is true for both the original non-supersymmetric and the
current supersymmetric versions of these GUTs.)
%Within this theoretical context, s
Since the terms $\chi_{jR}^TC \chi_{k R}$ are electroweak singlets, the
%associated coefficients, which comprise the
elements of the matrix $M_R$, would not be expected to be related
to the electroweak symmetry breaking scale, but instead, would be expected to
be much larger, plausibly of the order of the GUT scale.
Mechanisms involving $M_L$ only for the generation of neutrino masses without
the presence of electroweak-singlet neutrinos exist. However, since Higgs
triplets are not observed, these mechanisms involve other extensions of the
SM, for example the addition of one or more Higgs singlets, non-renormalizable
terms involving a large mass scale such as the GUT scale, or R-parity-violating
terms in the context of supersymmetry.
The most natural explanation for the three known ultra-light neutrino
masses is generally regarded to be the seesaw mechanism~\cite{seesaw},
which involves $M_R$, and arises from Eq.~\ref{numass}
in the case of $n_s = 3$ electroweak singlet neutrinos. This leads to
neutrino masses generically of order
\beq
m_\nu \sim \frac{m_D^2}{M_R}
\label{seesaw}
\eeq
where $m_D$ and $M_R$ denote typical elements of the corresponding
matrices. With $m_D \sim 10$~GeV and $M_R \sim 10^{14}$~GeV, a scale of
$m_\nu \sim 10^{-3}$~eV is readily obtained.
%For the full $6 \times 6$
%dimensional mass matrix of interest in this case,
In this case
the three light neutrino
masses are obtained by diagonalization of the effective $3 \times 3$
light neutrino mass matrix
\beq
M_\nu = - M_D M_R^{-1} M_D^T
\label{meffective}
\eeq
while the super-heavy neutrinos are determined from the right-handed Majorana
matrix $M_R$.
% itself and are of no interest here.
Additional electroweak-singlet neutrinos may arise in string theory
with the existence of supersymmetric partners of moduli fields,
resulting in the appearance of $n_\ell$ light sterile neutrinos.
But the presence of these light sterile neutrinos may undermine the
seesaw mechanism and, for this reason, is not very appealing theoretically.
However, if one tries to fit all of the data from the oscillation experiments,
to obtain a reasonable $\chi^2$ it is necessary to include light sterile
neutrinos. We shall illustrate some of the effects of sterile neutrinos with a
toy model in which one studies the minimal number, $n_\ell=1$.
\subsubsection{Neutrino Oscillations in Vacuum}
The presence of non-zero masses for the light neutrinos introduces a leptonic
mixing matrix, $U$, which is the analogue of the CKM quark mixing matrix,
and which in general is not expected to be diagonal.
The matrix $U$ connects the flavor eigenstates
with the mass eigenstates:
\begin{equation}
|\nu_\alpha\rangle = \sum_i U_{\alpha i}|\nu_i\rangle,
\end{equation}
\noindent
where $\alpha$ denotes one of the active neutrino flavors, $e,\ \mu$ or $\tau$
or one of the $n_s$ sterile flavors, while $i$ runs over
the light mass eigenstate labels. The number of flavor states considered
here is equal to the number of light mass eigenstates, so $U$ is a square
unitary matrix.
The tiny neutrino mass differences and the mixing parameters can be probed by
studying oscillations between different flavors of neutrinos,
as a function of the neutrino energy $E$ and the distance
traversed $L$.
The oscillation probability $P(\nu_\alpha \rightarrow \nu_\beta)$
is given by the absolute square of the overlap of
the observed flavor state, $|\nu_\beta\rangle$, with the time-evolved
initially-produced flavor state, $|\nu_\alpha\rangle$. In vacuum, the
evolution operator involves just the Hamiltonian $H_0$ of a free particle,
yielding the well-know result:
%
\begin{equation}
\begin{array}{rl}
P(\nu_\alpha \rightarrow \nu_\beta) =&\left|\langle\nu_\beta |
e^{-iH_0L}|\nu_\alpha\rangle\right|^2
= \sum_{i,j} U_{\alpha i}U^*_{\beta i}U^*_{\alpha j}U_{\beta j}
e^{-i\delta m^2_{ij}L/2E}\\[0.1in]
=&P_{\rm CP-even}(\nu_\alpha \rightarrow \nu_\beta)
+ P_{\rm CP-odd}(\nu_\alpha \rightarrow \nu_\beta) \; . \\[0.1in]
\end{array}
\end{equation}
\noindent
The CP-even and CP-odd contributions are
\begin{equation}
\begin{array}{rl}
P_{\rm CP-even}(\nu_\alpha \rightarrow \nu_\beta) =&P_{\rm CP-even}(
\bar{\nu}_\alpha \rightarrow \bar{\nu}_\beta)\\[0.1in]
=&\delta_{\alpha\beta} -4\sum_{i>j}\ Re\ (U_{\alpha i}
U^*_{\beta i}U^*_{\alpha j}U_{\beta j})\sin^2
({{\delta m^2_{ij}L}\over{4E}})\\[0.1in]
P_{\rm CP-odd}(\nu_\alpha \rightarrow \nu_\beta) =&-P_{\rm CP-odd}(
\bar{\nu}_\alpha \rightarrow \bar{\nu}_\beta)\\[0.1in]
=&2\sum_{i>j}\ Im\ (U_{\alpha i}U^*_{\beta i}U^*_{\alpha j}
U_{\beta j})\sin ({{\delta m^2_{ij}L}\over{2E}})\\[0.1in]
\end{array}
\label{cprels}
\end{equation}
so that
\beq
P(\bar\nu_\alpha \to \bar\nu_\beta)= P(\nu_\beta \to \nu_\alpha) =
P_{\rm CP-even}(\nu_\alpha \rightarrow \nu_\beta) -
P_{\rm CP-odd}(\nu_\alpha \rightarrow \nu_\beta)
\label{cprels2}
\eeq
where, by CPT invariance, $P(\nu_\alpha \to \nu_\beta) =
P(\bar\nu_\beta \to \bar\nu_\alpha)$.
In vacuum the CP-even and CP-odd contributions are even
and odd, respectively, under time reversal: $\alpha \leftrightarrow \beta$.
In Eq. (\ref{cprels}),
$\delta m^2_{ij} = m(\nu_i)^2 - m(\nu_j)^2$, and the combination
$\delta m^2_{ij}L/(4E)$ in $\hbar = c = 1$ units can be replaced
by $\sim1.27\delta m^2_{ij}L/E$ with $\delta m^2_{ij}$
in ${\rm eV^2}$ and $(L,\ E)$ in $({\rm km,\ GeV})$.
In disappearance experiments $\beta = \alpha$ and
no CP-violation can appear since the product of the mixing matrix
elements is inherently real. At distances $L$ large compared to the
oscillation length, $\lambda_{\rm osc} \sim E/\delta m^2_{ij}$, the sine
squared terms in $P_{\rm CP-even}$ average to 0.5 whereas the sine terms in
$P_{\rm CP-odd}$ average to zero.
% CHRIS HAS THIS STUFF IN INTRO
%Of special interest to date are the experimental searches for solar,
%atmospheric, reactor, and accelerator neutrino oscillations. Strong
%evidence for solar $\nu_e$ disappearance \cite{sol} and atmospheric
%$\nu_\mu$ disappearance \cite{kam} - \cite{macro} has been observed, while
%reactor $\bar{\nu}_e$ disappearance~\cite{chooz} has not been seen.
%In fact, the data for the atmospheric oscillations is best fit with an
%$\nu_\mu \rightarrow \nu_\tau$ appearance interpretation~\cite{learned}.
%Accelerator $\nu_\mu \rightarrow \nu_e$ and $\bar{\nu}_\mu \rightarrow
%\bar{\nu}_e$ appearance transitions have been observed by the LSND
%collaboration \cite{lsnd} but have not been confirmed or ruled out by the
%KARMEN group \cite{karmen}. Scenarios will be presented
%later which take into account some or all of the observed oscillations with
%three active neutrinos or one sterile and three active neutrinos.
\subsubsection{Three Active Neutrinos Only}
With three neutrinos, the mixing matrix $U$ is the $3\times3$
unitary Maki-Nagawa-Sakata (MNS) matrix\cite{mns}. We
parameterize $U$ by
%
\begin{equation}
U
= \left( \begin{array}{ccc}
c_{13} c_{12} & c_{13} s_{12} & s_{13} e^{-i\delta} \\
- c_{23} s_{12} - s_{13} s_{23} c_{12} e^{i\delta}
& c_{23} c_{12} - s_{13} s_{23} s_{12} e^{i\delta}
& c_{13} s_{23} \\
s_{23} s_{12} - s_{13} c_{23} c_{12} e^{i\delta}
& - s_{23} c_{12} - s_{13} c_{23} s_{12} e^{i\delta}
& c_{13} c_{23} \\
\end{array} \right) \,,
\end{equation}
%
where $c_{jk} \equiv \cos\theta_{jk}$ and $s_{jk} \equiv \sin\theta_{jk}$.
For Majorana neutrinos, $U$ contains two further multiplicative phase
factors, but these do not enter in oscillation phenomena.
With the plausible hierarchical neutrino mass spectrum
$m_1 < m_2 \ll m_3$, we can identify the largest $\delta m^2$ scale
with the atmospheric neutrino deficit:
$\delta M^2 = \delta m^2_{atm}
= \delta m^2_{32} \simeq \delta m^2_{31}$.
In the approximation that we neglect oscillations
driven by the small $\delta m^2$ scale, the probability for $\nu_e$
survival can be written
%
\begin{equation}
\begin{array}{rl}
P(\nu_e \rightarrow \nu_e) \simeq&1 - 4|U_{e1}|^2 |U_{e2}|^2
\sin^2 ({{\delta m^2_{21} L}\over{4E}})
-4|U_{e3}|^2 (1 - |U_{e3}|^2)\sin^2 ({{\delta M^2 L}
\over{4E}})\\[0.1in]
\end{array}
\end{equation}
\noindent
and
\beq
\begin{array}{rl}
P(\nu_e \to \nu_\mu) =& 4|U_{13}|^2 |U_{23}|^2
\sin^2 \Bigl ( \frac{\delta m^2_{atm}L}{4E} \Bigr ) \\[0.05in]
=& \sin^2(2\theta_{13})\sin^2(\theta_{23})
\sin^2 \Bigl (\frac{\delta m^2_{atm}L}{4E} \Bigr ) \; ,
\end{array}
\label{pnuenumu}
\eeq
\beq
\begin{array}{rl}
P(\nu_e \to \nu_\tau) =& 4|U_{33}|^2 |U_{13}|^2
\sin^2 \Bigl ( \frac{\delta m^2_{atm}L}{4E} \Bigr ) \\[0.05in]
=& \sin^2(2\theta_{13})\cos^2(\theta_{23})
\sin^2 \Bigl (\frac{\delta m^2_{atm}L}{4E} \Bigr ) \; .
\end{array}
\label{pnuenutau}
\eeq
\begin{equation}
P(\nu_\mu \rightarrow \nu_\tau) \simeq
4|U_{\mu 3}|^2 |U_{\tau 3}|^2 \sin^2 ({{\delta m^2_{atm} L}
\over{4E}}) \; .
\end{equation}
If the neutrinos propagate through matter, these expressions must
be modified.
The propagation of neutrinos through matter is described by the evolution
equation
%
\begin{equation}
i{d\nu_\alpha\over dt} = \sum_\beta \left[ \left( \sum_j U_{\alpha j} U_{\beta
j}^* {m_j^2\over 2E_\nu} \right) + {A\over 2E_\nu} \delta_{\alpha e}
\delta_{\beta e} \right] \nu_\beta \,, \label{eq:prop}
\end{equation}
%
where $A/(2E_\nu)$ is the amplitude for
coherent forward charged-current scattering of $\nu_e$ on electrons,
%
\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}) \,.
\end{equation}
%
Here $Y_e$ is the electron fraction and $\rho(x)$ is the matter density.
Density profiles through the
earth can be calculated using the preliminary Earth model, and are shown
in Fig.~\ref{profiles}. For
neutrino trajectories through the earth's crust, the density is typically of
order 3~gm/cm$^3$, and $Y_e \simeq 0.5$.
For very long baselines a constant density approximation is not good, and
oscillation calculations must explicitly take account of $\rho(x)$.
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{prof.ps}}
%\centerline{\epsffile{cc_rates_fig.ps}}
%\renewcommand{\baselinestretch}{2}
\caption{Density profiles for trajectories through the Earth.}
\label{profiles}
\end{figure}
The propagation Eq.~(\ref{eq:prop})
can be re-expressed in terms of mass-squared differences:
%
\begin{equation}
i{d\nu_\alpha\over dt} = \sum_\beta {1\over2E_\nu} \left[
\delta m_{31}^2 U_{\alpha 3} U_{\beta 3}^*
+ \delta m_{21}^2 U_{\alpha 2} U_{\beta 2}^*
+ A \delta_{\alpha e} \delta_{\beta e} \right]
\nu_\beta\,. \label{eq:prop2}
\end{equation}
%
This evolution equation can be solved numerically for given input values of the
$\delta m^2$ and mixing matrix elements.
In the approximation where we neglect oscillations
driven by the small $\delta m^2$ scale, the evolution equations are:
%
\begin{equation}
i {d\over dt}
\left( \begin{array}{c} \nu_e \\ \nu_\mu \\ \nu_\tau \end{array} \right)
= {\delta m^2\over 2E}
\left( \begin{array}{ccc}
{A\over \delta m^2} + |U_{e3}|^2 & U_{e3}U_{\mu3}^* & U_{e3}U_{\tau3}^* \\
U_{e3}^*U_{\mu3} & |U_{\mu3}|^2 & U_{\mu3}U_{\tau3}^* \\
U_{e3}^*U_{\tau3} & U_{\mu3}^*U_{\tau3} & |U_{\tau3}|^2
\end{array} \right)
\left( \begin{array}{c} \nu_e \\ \nu_\mu \\ \nu_\tau \end{array} \right)
\,.
\end{equation}
%
For propagation through matter of constant density, the flavor eigenstates are
related to the mass eigenstates $\nu_j^m$ by
%
\begin{equation}
\nu_\alpha = \sum U_{\alpha j}^m \nu_j^m \,,
\end{equation}
%
where
%
\begin{equation}
U^m = \left( \begin{array}{ccc}
0 & c_{13}^m & s_{13}^m \\
-c_{23} & -s_{13}^m s_{23} & c_{13}^m s_{23} \\
s_{23} & -s_{13}^m c_{23} & c_{13}^m c_{23}
\end{array} \right)
\end{equation}
%
and $\theta_{13}^m$ is related to $\theta_{13}$ by
%
\begin{equation}
\tan 2\theta_{13}^m = {\sin\theta_{13}\over \cos2\theta_{13}
- {A\over \delta m^2}} \,. \label{eq:tan}
\end{equation}
%
We note that $U^m$ has the form of the vacuum $U$ with the substitutions
%
\begin{equation}
\theta_{13}\to\theta_{13}^m\,, \quad \theta_{23} \to\theta_{23}^m\,, \quad
\theta_{12}\to\pi/2\,,\quad \delta = 0 \,.
\end{equation}
%
Equation~(\ref{eq:tan})
implies that
%
\begin{equation}
\sin^2 2\theta_{13}^m = {\sin^22\theta_{13}\over
\left({A\over\delta m^2} - \cos 2\theta_{13} \right)^2
+ \sin^2 2\theta_{13}} \,. \label{eq:sin}
\end{equation}
%
Thus there is a resonant enhancement for
%
\begin{equation}
A = \delta m^2 \cos2\theta_{13}
\end{equation}
%
or equivalently
%
\begin{equation}
E_\nu \approx 15{\rm\ GeV} \left(\delta m^2 \over 3.5\times
10^{-3}{\rm\,eV^2}\right) \left( 1.5{\rm\ g/cm^3}\over \rho Y_e \right)
\cos2\theta_{13} \,. \label{eq:Enu}
\end{equation}
The resonance occurs only for positive $\delta m^2$. For negative
$\delta m^2$ the oscillation amplitude in (\ref{eq:sin}) is smaller than the
vacuum oscillation amplitude. Thus the matter effects give us a way in
principle to determine the sign of $\delta m^2$.
It is instructive to look at the dependence of the oscillation probabilities
on the neutrino energy as a function of the oscillation parameters and
the baseline. Some examples from Ref.~\cite{shrock} are shown in
Fig.~\ref{fig:shrock} for $\nu_e\to\nu_\mu$ oscillations.
Note that for parameters corresponding to the
large mixing angle MSW solar solution, maximal CP violation results in
a small but visible effect.
Matter effects, which have been computed using the density profile
from the preliminary Earth model, can have substantial effects, and
are very sensitive to $\sin^22\theta_{13}$.
\begin{figure}
\begin{center}
\mbox{
\epsfxsize=6.5truecm
\epsfysize=5.6truecm
\epsffile{slacmue.eps}}
\mbox{
\epsfxsize=6.5truecm
\epsfysize=5.6truecm
\epsffile{cp.eps}}
\mbox{\epsfxsize=6.5truecm
\epsfysize=6truecm
\epsffile{nuvbarnu.eps}}
\mbox{\epsfxsize=6.5truecm
\epsfysize=6truecm
\epsffile{snuvbarnu.eps}}
\mbox{\epsfxsize=6.5truecm
\epsfysize=6truecm
\epsffile{fgsme1.eps}}
\mbox{\epsfxsize=6.5truecm
\epsfysize=6truecm
\epsffile{sme1.eps}}
\end{center}
\caption{Dependence of $\nu_e\to\nu_\mu$ oscillation probability on neutrino
energy for some representative oscillation parameters. Plots are from
Ref.~xxx and show the effects
of varying $\delta$ (top plots), matter effects (middle plots), and
$\sin^22\theta_{13}$ dependence (bottom plots).}
\label{fig:shrock}
\end{figure}
%
\subsubsection{Three Active Flavor Oscillation Scenarios}
We now define some representative three--flavor neutrino
oscillation parameter sets that can be used
to establish how well experiments at a neutrino factory
could determine the oscillation parameters. We begin be
considering constraints from existing experiments.
For the reactor antineutrino
experiments, the oscillation probability for $\bar{\nu}_e \rightarrow
\bar{\nu}_e$ is equal to that for $\nu_e \rightarrow \nu_e$.
The CHOOZ results~\cite{chooz} imply:
%
\begin{equation}
\sin^2 2\theta_{reac} \equiv 4|U_{e3}|^2 (1 - |U_{e3}|^2)
= \sin^2 2\theta_{13} \le 0.1
\end{equation}
\noindent
for the range $\delta M^2 \aprge 10^{-3}\ {\rm eV^2}$.
On the other hand, for the solar neutrino experiments, with $|U_{e3}|^2 \ll 1$,
one finds
\begin{equation}
\sin^2 2\theta_{solar} \equiv 4|U_{e1}|^2|U_{e2}|^2 =
\sin^2 2\theta_{12}\cos^4 \theta_{13} \sim \sin^2 2\theta_{12}
\end{equation}
\noindent
with $\sin^2 2\theta_{12} \sim 0.006$ in the case of the small angle MSW
solution with $\delta m^2_{21} \sim 6 \times 10^{-6}\ {\rm eV^2}$ or
$\sim 1.0$ in the case of the large angle MSW solution with $\delta m^2_{21}
\sim 5 \times 10^{-5}\ {\rm eV^2}$, the LOW solution with $\delta m^2_{21}
\sim 10^{-7}\ {\rm eV^2}$, or the vacuum solutions with $\delta m^2_{21}
\sim 4 \times 10^{-10}\ {\rm eV^2}$ or $\delta m^2_{21} \sim 8 \times
10^{-11}\ {\rm eV^2}$.
The atmospheric neutrino oscillation experiments favor $\nu_\mu \rightarrow
\nu_\tau$ \cite{learned}, and in the one-mass-scale-dominant
approximation the
best fit from the SuperK experiment \cite{sk} yields
%
\begin{equation}
\sin^2 2\theta_{atm} \equiv 4|U_{\mu 3}|^2 |U_{\tau 3}|^2
= \sin^2 2\theta_{23} \cos^4 \theta_{13} = 1.0
\end{equation}
\noindent
with $\delta m^2_{atm} = 3.5 \times 10^{-3}\ {\rm eV^2}$. Unpublished
analyses of a substantially enlarged data set by the SuperK experiment have yielded
the same central value for $\sin^2 2\theta_{atm}$ and essentially the same
value of $\delta m^2_{atm}$, $2.5 \times 10^{-3}$ \cite{sk}; we shall use
the published fits in the following.
Based on these considerations we define the representative three--flavor
parameter sets shown in Table~\ref{table:3flav}.
The first three scenarios do not attempt to fit the LSND anomaly.
These scenarios have
the Atmospheric anomaly explained by $\nu_{\mu} \rightarrow \nu_{\tau}$
oscillation with maximal mixing and the Solar Anomaly explained by
one of the MSW Solar solutions:
\begin{description}
\item{Scenario IA1} - Large Angle MSW
\item{Scenario IA2} - Small Angle MSW
\item{Scenario IA3} - LOW MSW.
\end{description}
Alternatively we can keep the LSND anomaly, and either drop the
solar neutrino deficit, or attempt to find a ``fit" (necessarily
with a poor $\chi^2$) that explains all three neutrino anomalies:
\begin{description}
\item{Scenario IB1} - Atmospheric and LSND
\item{Scenario IC1} - Atmospheric, Solar and LSND
\end{description}
For scenario IC1 the Atmospheric anomaly is a mixture of
$\nu_{\mu} \rightarrow \nu_{\tau}$ and $\nu_{\mu} \rightarrow \nu_{e}$
and the solar electron neutrino flux is reduced by a factor two
independent of energy. There are large contributions to the $\chi^2$ for
this scenario coming from the Atmospheric Neutrino Anomaly as well as the
Homestake (Chlorine) Solar neutrino experiment.
\begin{table}
\caption{Parameters for the three-flavor oscillation scenarios
defined for the study.}
\vspace{0.1cm}
\begin{tabular}{c|ccccc}
\hline
parameter & IA1 & IA2 & IA3 & 1B1 & 1C1 \\
\hline
$\delta m^2_{32}$ (eV$^2$)&$3.5\times10^{-3}$&$3.5\times10^{-3}$&
$3.5\times10^{-3}$&$3.5\times10^{-3}$&0.3 \\
$\delta m^2_{32}$ (eV$^2$)&$5\times10^{-5}$&$6\times10^{-6}$&
$1\times10^{-7}$&0.3&$7\times10^{-4}$\\
$\sin^22\theta_{23}$ &1.0&1.0&1.0&1.0&0.53 \\
$\sin^22\theta_{13}$ &0.04&0.04&0.04&0.015&0.036 \\
$\sin^22\theta_{12}$ &0.8&0.006&0.9&0.015&0.89 \\
$\delta$ &0,$\pm\pi/2$&0,$\pm\pi/2$&0,$\pm\pi/2$&
0,$\pm\pi/2$&0,$\pm\pi/2$ \\
\hline
$\sin^22\theta_{atm}$ &0.98&0.98&0.98&0.99& - \\
$\sin^22\theta_{reac}$ &0.04&0.04&0.04&0.03& - \\
$\sin^22\theta_{solar}$ &0.78&0.006&0.88& - & - \\
$\sin^22\theta_{LSND}$ & - & - & - &0.03&0.036\\
$J$ &0.02&0.002&0.02&0.002&0.015\\
\hline
\end{tabular}
\label{table:3flav}
\end{table}
Note that the Jarlskog J-factor~\cite{jarlskog}
$2J = 2c_{12}c^2_{13}c_{23}s_{12}s_{13}s_{23}(\sin \delta)$
is small for all scenarios.
It is clear
that CP violation will be very difficult to observe.
\subsubsection{Three Active and One Sterile Neutrinos}
In order to incorporate the observed
$\nu_\mu \rightarrow \nu_e$ and $\bar{\nu}_\mu \rightarrow\bar{\nu}_e$
LSND appearance results~\cite{lsnd}
and achieve an acceptable $\chi^2$ in the fit, it is necessary to
introduce at least one light sterile neutrino. As discussed earlier, the
theoretical case for sterile neutrinos is unclear, and various neutrino mass
schemes predict anything from $n_s = 0$ to many.
To admit just one must be
regarded as a rather unnatural choice.
We consider
this case because it allows us to explain the Atmospheric,
Solar and LSND anomalies with the fewest number of new parameters.
Scenarios with three nearly degenerate neutrinos (for example
$m_1 \leq m_2 \leq
m_3 \ll m_4$ or $m_1 \ll m_2 \leq m_3 \leq m_4$) are essentially ruled out
by a Schwarz inequality on the leptonic mixing elements \cite{bilenky}:
$|U_{\mu 4}U^*_{e4}|^2 \leq |U_{\mu 4}|^2|U_{e4}|^2 \leq 0.008$ which fails
to be satisfied in the allowed LSND region. Of the two scenarios with
$m_1 < m_2 \ll m_3 < m_4$, the one with $\delta m^2_{21} \sim \delta
m^2_{solar},\ \delta m^2_{43} \sim \delta m^2_{atm}$ is preferred over
the other arrangement which is on the verge of being ruled out by the
Heidelberg-Moscow $\beta\beta_{0\nu}$ decay experiment \cite{h-m} giving
$\langle m \rangle \leq 0.2$ eV.
With the three relevant mass scales given by\\
$$\delta m^2_{sol} = \delta m^2_{21} \ll \delta m^2_{atm}
= \delta m^2_{43} \ll \delta m^2_{LSND} = \delta m^2_{32}$$
and the flavors ordered according to $\{s,\ e,\ \mu,\ \tau\}$, the
$4 \times 4$ neutrino mixing matrix depends on six angles and three phases
and is conveniently chosen to be \cite{donini}
\begin{equation}
\begin{array}{rl}
U =& \left(\matrix{U_{s1} & U_{s2} & U_{s3}, & U_{s4}\cr
U_{e1} & U_{e2} & U_{e3}, & U_{e4}\cr
U_{\mu 1} & U_{\mu 2} & U_{\mu 3}, & U_{\mu 4}\cr
U_{\tau 1} & U_{\tau 2} & U_{\tau 3}, & U_{\tau 4}\cr}
\right)\\[0.4in]
=& R_{14}(\theta_{14})R_{13}(\theta_{13})R_{24}(\theta_{24})
R_{23}(\theta_{23},\delta_3)
R_{34}(\theta_{34},\delta_2)R_{12}(\theta_{12},\delta_1)\\
\end{array}
\end{equation}
\noindent
where, for example,
$$R_{13}(\theta_{13},\delta) = \left(\matrix{c_{13} & 0 &
s_{13}e^{-i\delta}\cr
0 & 1 & 0\cr -s_{13}e^{i\delta} & 0 & c_{13}\cr}
\right) \; .\\$$
\noindent
In the limit where the $m_1 - m_2$ and $m_3 - m_4$ pairs are considered
degenerate, $R_{12}(\theta_{12},\delta_1) = R_{34}(\theta_{34},\delta_{34})
= I$, and only four angles and one phase appear in the mixing matrix
\begin{equation}
U = \left(\matrix{c_{14}c_{13} & -c_{14}s_{13}s_{23}e^{i\delta_3}
-s_{14}s_{24}c_{23} & c_{14}s_{13}c_{23} -s_{14}s_{24}s_{23}
e^{-i\delta_3} & s_{14}c_{24}\cr
0 & c_{24}c_{23} & c_{24}s_{23}e^{-i\delta_3} & s_{24}\cr
-s_{13} & -c_{13}s_{23}e^{i\delta_3} & c_{23}c_{13} & 0\cr
-s_{14}c_{13} & s_{14}s_{13}s_{23}e^{i\delta_3} -c_{14}s_{24}c_{23}
& -s_{14}s_{13}c_{23} -c_{14}s_{24}s_{23}e^{-i\delta_3} & c_{14}c_{24}
\cr}\right)
\end{equation}
\noindent
with the same angle and phase rotation convention adopted as before.
In this one-mass-scale-dominant approximation with the large mass gap labeled
$\delta M^2 = \delta m^2_{LSND}$, the oscillations are again CP-conserving,
and a short baseline experiment is needed to determine the extra
relevant mixing angles and phase. The oscillation probabilities of interest
are:
\begin{equation}
\begin{array}{rl}
P(\nu_e \rightarrow \nu_e) =& 1 - 4c^2_{24}c^2_{23}(s^2_{24} +
s^2_{23}c^2_{24})\sin^2\left(1.27{{\delta M^2 L}\over{E}}\right),
\\[0.1in]
P(\nu_e \rightarrow \nu_\mu) =& P(\nu_\mu \rightarrow \nu_e) =
4c^2_{13}c^2_{24}c^2_{23}s^2_{23}
\sin^2\left(1.27{{\delta M^2 L}\over{E}}\right),\\[0.1in]
P(\nu_e \rightarrow \nu_\tau) =& 4c^2_{23}c^2_{24}
\left[(s^2_{13}s^2_{14}s^2_{23} + c^2_{14}c^2_{23}s^2_{24})\right.
\\
& \left. -2c_{14}s_{14}c_{23}s_{23}s_{13}s_{24}\cos \delta_3\right]
\sin^2\left(1.27{{\delta M^2 L}\over{E}}\right),\\[0.1in]
P(\nu_\mu \rightarrow \nu_\mu) =& 1 - 4c^2_{13}c^2_{23}(s^2_{23} +
s^2_{13}c^2_{23})\sin^2\left(1.27{{\delta M^2 L}\over{E}}\right),
\\[0.1in]
P(\nu_\mu \rightarrow \nu_\tau) =& 4c^2_{13}c^2_{23}
\left[(s^2_{13}s^2_{14}c^2_{23} + c^2_{14}s^2_{23}s^2_{24})\right.
\\
& \left. +2c_{14}s_{14}c_{23}s_{23}s_{13}s_{24}\cos \delta_3\right]
\sin^2\left(1.27{{\delta M^2 L}\over{E}}\right) \; . \\
\end{array}
\end{equation}
In order to search for CP violation, at least two mass scales must be
relevant.
For simplicity consider the pair of mass states relevant for solar
oscillations degenerate, and set
\begin{equation}
\begin{array}{rl}
\delta m^2_{21} =& 0, \qquad \delta m^2_{43} = \delta m^2,\\[0.1in]
\delta m^2_{32} =& \delta m^2_{31} = \delta M^2,\\[0.1in]
\delta m^2_{42} =& \delta m^2_{41} = \delta M^2 + \delta m^2\\
\end{array}
\end{equation}
\noindent
with five angles and two phases present, since $U_{12}(\theta_{12},\delta_1)
= I$. The CP-odd parts of the relevant probabilities are:
%
\begin{equation}
\begin{array}{rl}
P_{\rm CP-odd}(\nu_e \rightarrow \nu_\mu) =& 8c^2_{13}c^2_{23}c_{24}c_{34}
s_{24}s_{34} \sin (\delta_2 + \delta_3)\left({{\delta m^2 L}
\over{4E}}\right) \sin^2 \left({{\delta M^2 L}\over{4E}}\right)
\\[0.1in]
P_{\rm CP-odd}(\nu_e \rightarrow \nu_\tau) =& 4c_{23}c_{24}\left\{2c_{14}
s_{14}c_{23}s_{23}s_{13}s_{24}(s^2_{13}s^2_{14}-c^2_{14})
\sin(\delta_2 + \delta_3)\right.\\[0.05in]
& +c_{14}c_{34}s_{13}s_{14}s_{34}\left[(s^2_{23} - s^2_{24})
\sin \delta_2 + s^2_{23}s^2_{24}\sin (\delta_2 + 2\delta_3)\right]
\\[0.05in]
& + \left. c_{14}c_{24}s_{13}s_{14}s_{23}s_{24}(c^2_{34} - s^2_{34})
\sin \delta_3 \right\}\\[0.05in]
& \times \left({{\delta m^2 L}\over{4E}}\right)
\sin^2 \left({{\delta M^2 L}\over{4E}}\right)\\[0.1in]
P_{\rm CP-odd}(\nu_\mu \rightarrow \nu_\tau) =& 8c^2_{13}c^2_{23}c_{24}
c_{34}s_{34}\left[c_{14}c_{23}s_{13}s_{14} \sin \delta_2 +
c^2_{14}s_{23}s_{24} \sin (\delta_2 + \delta_3)\right]\\[0.05in]
& \times \left({{\delta m^2 L}\over{4E}}\right)
\sin^2 \left({{\delta M^2 L}\over{4E}}\right)\\
\end{array}
\end{equation}
\noindent
where only the leading order term in $\delta m^2$ has been kept. The
CP-even expressions also have such additional small corrections.
The present atmospheric neutrino data favors the $\nu_\mu \rightarrow
\nu_\tau$ oscillation over the $\nu_\mu \rightarrow \nu_{s}$ oscillation.
On the other hand, if a solar neutrino oscillates significantly into a
sterile neutrino, only the small angle MSW solution is viable since
the large angle solutions fail to provide enough $\nu + e^- \rightarrow
\nu + e^-$ elastic scattering to be consistent with SuperK measurements~\cite{sk}.
Hence
if it turns out that one of the large angle solar mixing solutions is correct,
it is unlikely that a light sterile neutrino will play any role in explaining
the solar, atmospheric and LSND results.
\subsubsection{Scenarios with Three Active plus One Sterile Neutrino}
We now consider some representative four--flavor neutrino oscillation
parameter sets that can be used to establish how well experiments
at a neutrino factory could determine the oscillation parameters.
As was noted earlier, the only viable solutions with one sterile
and three active neutrinos require that there be two sets of
almost degenerate neutrinos separated by the largest $\delta m^2$.
We begin by considering the constraints from CHOOZ and LSND.
Note that the effective two-component atmospheric
and solar mixing angles are:
%
\begin{equation}
\begin{array}{rl}
\sin^2 2\theta_{atm} =&4|U_{\mu 3}|^2 |U_{\mu 4}|^2
= c^4_{23}c^4_{13}\sin^2 2\theta_{34}\\[0.1in]
\sin^2 2\theta_{sol} =&4|U_{e1}|^2 |U_{e2}|^2
= c^4_{24}c^4_{23}\sin^2 2\theta_{12}\\
\end{array}
\end{equation}
%
The CHOOZ constraint~\cite{chooz} from $P(\bar{\nu}_e \rightarrow
\bar{\nu}_e)$ is:
\begin{equation}
c^2_{23}\sin^2 2\theta_{24} + c^4_{24}\sin^2 2\theta_{23} \le 0.2
\end{equation}
\noindent
while the LSND constraint \cite{lsnd} from $P(\nu_\mu \rightarrow \nu_e)$ is:
\begin{equation}
10^{-3} \leq c^2_{13}c^2_{24}\sin^2 2\theta_{23} \le 10^{-2} \; .
\end{equation}
With this in mind,
the parameter sets we have defined are summarized in Table~\ref{table:4flav}.
They are:
\begin{description}
\item{Scenario IIA1} - Low Mass LSND
\item{Scenario IIB1} - High Mass LSND
\end{description}
\begin{table}
\caption{Parameters for the four-flavor oscillation scenarios
defined for the study. Note that for these parameter sets
$\delta m^2_{41}\sim\delta m^2_{31}\sim\delta m^2_{42}
\sim\delta m^2_{32}\equiv\delta M^2$,
and
$\sin^22\theta_{14}=\sin^22\theta_{13}=\sin^22\theta_{24}=\sin^22\theta_{23}$}
\bigskip
\begin{center}
\begin{tabular}{c|cc}
\hline
parameter & IIA1 & IIB1 \\
\hline
$\delta m^2_{43}$ (eV$^2$)&$3.5\times10^{-3}$&$3.5\times10^{-3}$ \\
$\delta m^2_{21}$ (eV$^2$)&$6\times10^{-6}$&$6\times10^{-6}$ \\
$\delta M^2$ (eV$^2$) &0.3&1.0 \\
$\sin^22\theta_{34}$ &1.0&1.0\\
$\sin^22\theta_{12}$ &0.006&0.006 \\
$\sin^22\theta_{14}$ &0.03&0.003 \\
$\delta_1$ &0&0\\
$\delta_2$ &0,$\pm\pi/2$&0,$\pm\pi/2$ \\
$\delta_3$ &0&0\\
\hline
\end{tabular}
\end{center}
\label{table:4flav}
\end{table}
\subsection{Where will we be in 5-10 years ?}
In this section, we briefly discuss the prospects for currently
operating, planned
and discussed experiments exploring the neutrino oscillation phenomena.
The discussion will be broken down according to the various oscillation modes.
The current limits and the expected reach of some of the future
experiments are summarized in Fig.~\ref{fig:everything}, and
Tables~\ref{expt_table} and \ref{3nu_table}.
\begin{figure}
\epsfxsize=0.8\textwidth
\centerline{\epsfbox{everything.eps}}
\caption{The current and expected limits at some of the future
neutrino oscillation experiments. Note that different
oscillation modes are shown together.}
\label{fig:everything}
\end{figure}
\subsubsection{$\nu_\mu \rightarrow \nu_\tau$, $\nu_s$}
The evidence for $\nu_\mu$ disappearance in atmospheric neutrinos at
SuperK is convincing \cite{evidence}.
The preferred region of parameter space is
$\delta m^2 \sim 10^{-3}\mbox{--}10^{-2}~{\rm eV}^2$
at near maximal mixing, $\sin^2 2\theta \sim 1$ \cite{Kajita}.
The $\nu_\mu \leftrightarrow \nu_e$ possibility
is disfavored by the SuperK data and is ruled out by the CHOOZ~\cite{CHOOZ}
and PaloVerde \cite{PaloVerde} experiments.
The two central issues here are (1)
the precise determination of $\delta m^2$ and $\sin^2 2\theta$, and (2)
discrimination between $\nu_\tau$ and $\nu_s$. On the first issue, we
believe SuperK will probably not shrink the current preferred
region by very much.
The fully-contained events are most
sensitive to the preferred $\delta m^2$ close to the
horitonal direction, but such
studies are limited both by the angular resolution and statistics.
On the discrimination between $\nu_\tau$ and $\nu_s$, there will be
great progress in the next few years given the many handles
on that exist~\cite{Learned,Kajita}.
The $\pi^0/e$ ratio is currently limited by
systematic uncertainty in the NC single $\pi^0$
production cross section, which will be improved by the currently running
K2K near detector.
The up/down ratio of high-energy $\pi^0$ events are limited
solely by statistics.
The unsuccessful search for matter effects in
up-going muons, partially-contained events as well as
the NC-enriched multi-ring event sample already disfavor the
oscillation into $\nu_s$ at the 99\% confidence level. All of these will
likely be much stronger in the next few years.
Long-baseline experiments will cover this oscillation mode very well.
K2K \cite{K2K} will cover $\delta m^2 > 2 \times 10^{-3}~{\rm
eV}^2$ after 3--5 years of running, and MINOS \cite{MINOS} will
cover $\delta m^2 > 0.6 \times 10^{-3}~{\rm eV}^2$ (both at 90\%
CL and for maximal mixing). They are expected to confirm the
neutrino oscillation interpretation of the atmospheric neutrino data
with man-made
beams by about 2005. K2K can also look for a possible distortion in
the neutrino energy spectrum using quasi-elastic events, and MINOS can
study NC/CC and CC event energy distributions.
They can also measure the neutrino
parameters quite well, especially at MINOS. With CNGS beam (from
2005), OPERA \cite{OPERA} and ICANOE \cite{ICANOE} both aim primarily
at $\tau$-appearance and will cover $\delta m^2 > 2 \times
10^{-3}~{\rm eV}^2$ after about 5 years of running. Except for the
unfortunate case of $\delta m^2 < 2 \times 10^{-3}~{\rm eV}^2$
(allowed at 99\% CL at SuperK), we will basically have a
complete accelerator based experimental
confirmation of atmospheric neutrino oscillations by 2010.
In the unfortunate case, some of the
currently discussed experiments may be necessary, such as MONOLITH
(30~kt calorimeter)~\cite{MONOLITH} or AQUARICH (novel 1~Mt Water
Cherenkov)~\cite{AQUARICH}. They may study atmospheric neutrinos and
see dips in the zenith angle (or $L/E$) distribution.
\subsubsection{$\nu_\mu \leftrightarrow \nu_e$}
For large $\delta m^2 \sim 1 ~{\rm eV}^2$ suggested by the
LSND experiment \cite{LSND}, Mini-BooNE
\cite{Mini-BooNE} is expected to cover the entire preferred region of
LSND with a wide safety margin. In the event of a positive signal,
they plan to build another detector (BooNE) and will be able to
measure the parameters with high precision.
For $\delta m^2 \sim 10^{-3}\mbox{--}10^{-2}~{\rm eV}^2$, we expect
some $\nu_\mu \leftrightarrow \nu_e$ if the heavier of two mass eigenstates
involved in the atmospheric neutrino oscillation contains any
admixture of $\nu_e$ ({\it i.e.}\/, if $U_{e3} \neq 0$). Current
limits from CHOOZ~\cite{CHOOZ} and Palo Verde \cite{PaloVerde} require
$|U_{e3}| < 0.1$. SuperK by itself is unlikely to
improve this limit. K2K can, however, look for $\nu_e$ appearance and
improve the limit in some $\delta m^2$ range. MINOS and ICANOE claim to
further improve the limit on $\sin^2 2\theta$ to ${\cal{O}}(10^{-2})$ in
$\nu_e$ appearance with the $\delta m^2$ region of interest.
At this time it is not clear what is the interesting range
for $\sin^2 2\theta$.
If this mixing angle is not too small then K2K/MINOS/ICANOE
can make a measurement.
For very small mixing angles, comparable
with the Small Mixing Angle MSW solution for the solar anamoly (see
\cite{Valle}),
something beyond these experiments is probably required.
For MINOS and ICANOE the sensitivity to the matter effect in the
Earth is expect to be minimal.
\subsubsection{$\nu_e \rightarrow \nu_\mu$, $\nu_\tau$, $\nu_s$}
Most reactor and solar neutrino experiments effectively look for this
oscillation in the disappearance mode.
The SNO \cite{SNO} detector
should discrimminate between $\nu_e \rightarrow \nu_{\mu,\tau}$
and $\nu_e \rightarrow \nu_s$ solutions to the solar neutrino
anamoly by studying the distortion in the electron neutrino energy spectrum
of the $^8$B flux and measuring the NC/CC ratio.
The spectral distortion should occur for the SMA solution
and in some regions of the VAC solution.
Borexino \cite{Borexino}
(or possibly KamLAND) will study $^7$Be solar neutrinos, and
should see day/night effect for the LOW scenario and seasonal effects for the VAC
solution. The absence of $^7$Be electron neutrino
flux would strongly suggest the SMA
solution. There are more experiments discussed to study lower energy
neutrinos (esp. $pp$): HELLAZ, HERON, LENS, etc (see \cite{Lanou} for
a recent overview). We will certainly
know whether or not any of the current neutrino oscillation
solutions to the solar neutrino problem are correct in the next 5--10 years.
None of the solar neutrino experiments, however, discriminate $\nu_e
\rightarrow \nu_\mu$ and $\nu_\tau$. In principle, MINOS, OPERA,
ICANOE can look for $\tau$ appearance from $\nu_e$, but will suffer
from the ``background'' of the $\nu_\mu \rightarrow \nu_\tau$ signal.
KamLAND \cite{KamLAND} will look for
$\bar{\nu}_e$ disappearance in the reactor signal down to $\delta m^2
> 10^{-5}~{\rm eV}^2$ for large mixing angles.
EDITORS NOTE: IN THE NEXT FEW DAYS WE NEED TO COMPLETE THE TABLES,
ADD A FEW LINES SUMMARIZING THE PICTURE IN 5-10 YEARS, AND THEN
MASSAGE THE NEXT SUB-SECTION TO INTERFACE WITH THIS.
\begin{table}
\caption{Experimental neutrino oscillation observations
expected in the next 5--10~years at accelerator based experiments.}
\bigskip
\begin{center}
\begin{tabular}{cc|ccc|ccc}
\hline
% & \multicolumn{6}{c}{Channel} \\
Scenario&Experiment&$\nu_\mu$ Disap. & $\nu_\mu\to\nu_e$&$\nu_\mu\to\nu_\tau$&
$\nu_e$ Disap. & $\nu_e\to\nu_\mu$&$\nu_e\to\nu_\tau$ \\
\hline
IA1 & K2K&yes& no& no& no& no& no\\
& MINOS&yes& no&yes& no& no& no\\
&ICANOE&yes&yes$^\star$&yes& no& no& no\\
& OPERA& no& no&yes& no& no& no\\
& BooNE& no& no& no& no& no& no\\
\hline
IA2 & K2K &yes& & & & & \\
& MINOS&yes& & & & & \\
&ICANOE&yes& & & & & \\
& OPERA& no& no&yes& no& no& no\\
& BooNE& no& no& no& no& no& no\\
\hline
IA3 & K2K &yes& & & & & \\
& MINOS&yes& & & & & \\
&ICANOE&yes& & & & & \\
& OPERA& no& no&yes& no& no& no\\
& BooNE& no& no& no& no& no& no\\
\hline
IB1 & K2K& & & & & & \\
& MINOS& & & & & & \\
&ICANOE& & & & & & \\
& OPERA& no& no&yes& no& no& no\\
& BooNE& no&yes& no& no& no& no\\
\hline
IC1 & K2K& & & & & & \\
& MINOS& & & & & & \\
&ICANOE& & & & & & \\
& OPERA& no& no&yes& no& no& no\\
& BooNE&yes&yes& no& no& no& no\\
\hline
IIA1 & K2K& & & & & & \\
& MINOS& & & & & & \\
&ICANOE& & & & & & \\
& OPERA& no&yes&yes& no& no& no\\
& BooNE& no&yes& no& no& no& no\\
\hline
IIB1 & K2K& & & & & & \\
& MINOS& & & & & & \\
&ICANOE& & & & & & \\
& OPERA& no& no&yes& no& no& no\\
& BooNE& no&yes& no& no& no& no\\
\hline
\end{tabular}
\end{center}
$\star \sin^2 2\theta_{13} > 0.01$
\label{expt_table}
\end{table}
\begin{table}
\caption{Neutrino oscillation measurement limits and uncertainties
expected in the next 5--10~years at accelerator based experiments.
}
\bigskip
\begin{center}
\begin{tabular}{cc|ccccc}
\hline
& & \multicolumn{5}{c}{Parameter} \\
Scenario&Experiment&$\sin^2 2\theta_{12}$&$\sin^2 2\theta_{23}$&
$\sin^2 2\theta_{13}$& $\delta$ & $\delta m^2$ (eV$^2$)\\
\hline
IA1 & K2K& & & & & \\
& MINOS& & & & & \\
&ICANOE& &13\%&60\%& &11\%\\
& OPERA& &20\%& & &14\%\\
& BooNE& & & & & \\
\hline
IA2 & K2K& & & & & \\
& MINOS& & & & &\\
&ICANOE& &13\%&60\%& &11\%\\
& OPERA& &20\%& & &14\%\\
& BooNE& & & & &\\
\hline
IA3 & K2K& & & & &\\
& MINOS& & & & &\\
&ICANOE& &13\%&60\%& &11\%\\
& OPERA& &20\%& & &14\%\\
& BooNE& & & & &\\
\hline
IB1 & K2K& & & & &\\
& MINOS& & & & &\\
&ICANOE& & & & &\\
& OPERA& &20\%& & &14\%\\
& BooNE&10\%& & & &10\%\\
\hline
IC1 & K2K& & & & &\\
& MINOS& & & & &\\
&ICANOE& & & & &\\
& OPERA& & 5\%& & & 7\%\\
& BooNE& &10\%&15\%& &10\%\\
\hline
IIA1& K2K& & & & &\\
& MINOS& & & & &\\
&ICANOE& & & & &\\
& OPERA& &xx & & &xx\\
& BooNE& &10\%& & &10\%\\
\hline
IIB1& K2K& & & & &\\
& MINOS& & & & &\\
&ICANOE& & & & &\\
& OPERA& &xx & & &xx\\
& BooNE& &10\%& & &10\%\\
\hline
\end{tabular}
\end{center}
\label{3nu_table}
\end{table}
\clearpage
\subsection{The neutrino factory oscillation physics program}
We now consider the program of neutrino oscillation measurements
at a neutrino factory in the era beyond the next generation of
long baseline experiments. The main goals in this era are likely
to be to precisely establish the oscillation framework
(for example, three--flavor mixing ?),
determine the pattern of neutrino masses,
measure matter effects to confirm the MSW phenomenon,
make precise measurements or place stringent limits
on all of the mixing--matrix elements (and hence mixing--angles),
and observe or place stringent limits on CP violation
in the lepton sector.
A neutrino factory can address each of these goals:
\begin{description}
\item{(i)} Establishing the oscillation framework.
This requires measuring as a function of $L/E$,
or putting stringent limits on,
all of the oscillation probabilities $P(\nu_e\to\nu_x)$
and $P(\nu_\mu\to\nu_x)$. The oscillation framework
can be established by summing the probabilities
(a) $P(\nu_e\to\nu_e) + P(\nu_e\to\nu_\mu) + P(\nu_e\to\nu_\tau)$, and
(b) $P(\nu_\mu\to\nu_e) + P(\nu_\mu\to\nu_\mu) + P(\nu_\mu\to\nu_\tau)$.
In a three--flavor mixing framework, both sums should be unity for all
$L/E$. If there are sterile neutrinos participating in the oscillations
one or both of the sums will be less than unity. Part (b) of the test
will almost certainly be made with conventional neutrino beams,
although with a precision that will be limited by the
$\nu_\mu\to\nu_\tau$ statistics and
by the uncertainty
on the $P(\nu_\mu\to\nu_e)$ measurement arising from the O(1\%) $\nu_e$
contamination in the beam. Part (a) of the test, which includes
the first observation of (or stringent limits on) $\nu_e\to\nu_\tau$
oscillations, can only be made
with a $\nu_e$ (or $\overline{\nu}_e$) beam, and will therefore
be a unique part of the neutrino factory physics program.
\item{(ii)} Determining the pattern of neutrino masses.
New experimental spectra have often led to significant theoretical insights.
The present experimental data suggests that, within a three--flavor
mixing framework, there are two neutrino mass eigenstates separated
by a small mass difference, and a third state separated from the pair
by a ``large" mass difference $\delta M^2$. What is unknown is whether there
is one low state plus two high states, or two low states
plus one high state. This can be determined by measuring the sign of
$\delta M^2$. The only way we know of making this
measurement is to exploit matter effects which, in a very long baseline
experiment, alter the probabilities for oscillations
that involve electron neutrinos; the modification being dependent
on the sign of $\delta M^2$. In principle the measurement could
be made using a conventional neutrino beam and measuring
$\nu_\mu\to\nu_e$ and $\overline{\nu}_\mu\to\overline{\nu}_e$
transitions over a baseline of several thousand km. However,
the O(1\%) $\nu_e$ ($\overline{\nu}_e$)
contamination in the beam will introduce an
irreducible background that is comparable to, or larger than,
the $\nu_e$ signal.
In contrast, at a neutrino factory it appears that
the measurement can be done with great precision.
Hence, determining the
sign of $\delta M^2$ and the pattern of neutrino masses
would be a key measurement at a neutrino factory.
\item{(iii)} Measuring matter effects to confirm the MSW phenomenon.
The same technique used to determine the sign of $\delta m^2_{32}$
can, with sufficient statistics, provide a precise quantitative
confirmation of the MSW effect for neutrinos passing through the
Earth. The modification to $P(\nu_e\to\nu_\mu)$, for example,
depends upon the density $\rho$ of the matter traversed. Global fits to
appearance and disappearance spectra that are used to
determine the oscillation parameters can include $\rho$ as a
free parameter. The quantitative MSW test would be to recover the
known $\rho$. This measurement exploits the clean $\nu_e\to\nu_\mu$
signal at a neutrino factory, and would be a unique part of the
neutrino factory physics program.
\item{(iv)} Making precise measurements or placing stringent limits
on all of the mixing--matrix elements.
In practice the measured oscillation probability amplitudes
are used to determine the mixing angles. If any of the angles
are unmeasured or poorly constrained the relevant entries in
the mixing matrix will also be poorly determined. At present
there is only an upper limit on $\theta_{13}$, the angle that
essentially determines the $\nu_e\to\nu_\mu$ oscillation amplitude.
A neutrino factory would provide a precise measurement of, or
stringent limit on, this difficult angle. In fact, because
all of the $\nu_\mu\to\nu_x$ and $\nu_e\to\nu_x$ oscillation
amplitudes can be measured at a neutrino factory, global fits
can be made to the measured spectra to provide a very precise
determination of the mixing angles. This exploits
the $\nu_e$ component in the beam. Finally, it should be
noted that it is important to test the overall consistency
of the oscillation framework by determining the mixing
angles in more than one way,
i.e. by using more than one independent set of measurements.
Clearly the $\nu_e$ beam is an asset for this check.
\item{(v)} Observing or placing stringent limits on CP violation
in the lepton sector.
Most of the oscillation scenarios defined for the study
predict very small CP violating amplitudes. An important test of
these scenarios would be to place stringent experimental limits
on CP violation in the lepton sector. The LAM scenario IA1
might result in sufficiently large CP violating effects to
be observable at a neutrino factory. The CP test involves
comparing $\nu_e\to\nu_\mu$ with $\overline{\nu}_e\to\overline{\nu}_\mu$
oscillation rates, possible at a neutrino factory because backgrounds
are very small. A search for CP violation in the lepton sector with
the required precision cannot be done with a conventional neutrino
beam, and is therefore a unique part of the neutrino factory
physics program.
\end{description}
Note that it is the $\nu_e$ ($\overline{\nu}_e$)
component in the neutrino factory beam
that drives the oscillation physics program.
A $\nu_e$ beam would
(a) enable a basic test of the oscillation
framework that cannot be made with a $\nu_\mu$ beam,
(b) enable the first observation of (or stringent limits on)
$\nu_e\to\nu_\tau$ oscillations,
(c) make a convincing determination of the pattern of neutrino
masses that would be difficult or impossible with a conventional
neutrino beam,
(d) make a quantitative check of the MSW effect
only possible with a neutrino factory beam,
(e) enable measurements or stringent limits on all of the
(three--flavor)
mixing angles with a precision that requires both $\nu_e$
and $\nu_\mu$ beams, and
(f) measure or put meaningful limits on CP violation
in the lepton sector, which requires a signal purity only available
at a neutrino factory.
A neutrino factory operating in the
next decade, after the next generation of long baseline experiments,
would appear to be the right tool at the right time.
However, before we can quantitatively assess how well a
neutrino factory might realize the physics program we
have listed, we must first understand the capabilities
of neutrino detectors in the neutrino factory era.
%\clearpage
\subsection{Detector considerations}
We would like to measure the oscillation probabilities
$P(\nu_\alpha \rightarrow \nu_\beta)$ as a function of the
baseline $L$ and neutrino energy $E$ (and hence $L/E$)
for all possible initial and final flavors $\alpha$ and $\beta$.
This requires a beam with a well known initial flavor content,
and a detector that can identify the flavor of the interacting
neutrino. The neutrinos interact in the detector via charged
current (CC) and neutral current (NC) interactions to produce
a lepton accompanied by a hadronic shower arising
from the remnants of the struck nucleon.
In CC interactions the final state lepton
tags the flavor ($\beta$) of the interacting neutrino.
At a neutrino factory in which, for example, positive
muons are stored, the initial beam consists of 50\% $\nu_e$ and
50\% $\overline{\nu}_\mu$.
In the absence of oscillations, the $\nu_e$ CC interactions
produce electrons and the $\overline{\nu}_\mu$ CC interactions
produce positive muons.
Note that the charge of the final state lepton tags the flavor
($\alpha$) of the initial neutrino or antineutrino.
In the presence of
$\nu_e \rightarrow \nu_\mu$ oscillations the $\nu_\mu$ CC interactions
produce negative muons (i.e. wrong--sign muons). Similarly,
$\overline{\nu}_\mu \rightarrow \overline{\nu}_e$ oscillations
produce wrong--sign electrons,
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ oscillations
produce events tagged by a $\tau^+$ and
$\nu_e \rightarrow \nu_\tau$ oscillations
produce events tagged by a $\tau^-$.
Hence, there is a variety of information that can be used
to measure or constrain neutrino oscillations at a neutrino factory,
namely the rates and energy distributions of events tagged by
(a) right--sign muons, (b) wrong--sign muons, (c) electrons
or positrons (their charge is difficult to determine in a massive
detector),
(d) positive $\tau$--leptons, (e) negative $\tau$--leptons,
and (f) no charged lepton. If these
measurements are made when there are alternately positive and negative
muons decaying in the storage ring, there are a total of 12 spectra
that can be used to extract information about the oscillations.
Some examples of the predicted measured spectra are shown as a function of the
oscillation parameters in Figs.~\ref{fig:m1} and
\ref{fig:m2} for a 10~kt detector sited 7400~km
downstream of a 30~GeV neutrino factory.
Clearly, the high intensity $\nu_e$, $\overline{\nu}_e$, $\nu_\mu$, and
$\overline{\nu}_\mu$ beams at a neutrino factory would provide a wealth of
precision oscillation data.
\begin{figure}
\epsfxsize3.4in
\centerline{\epsffile{mum.eps}}
%\vspace{-1.0cm}
\caption{Visible energy spectra for four event classes when
$10^{21} \mu^-$
decay in a 30~GeV neutrino factory at $L = 7400$~km.
Three cases are considered. Black histogram: no oscillations.
Blue dotted histogram: $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$,
$\sin^2\theta_{23}=1$.
Red dashed histogram: $\delta m^2_{32}=7\times 10^{-3}$~eV$^2$/c$^4$,
$\sin^2\theta_{23}=1$.
The distributions are for an ICANOE-type detector, and are
from Ref.~\ref{camp00}.}
\label{fig:m1}
\end{figure}
%
\begin{figure}
\epsfxsize3.4in
\centerline{\epsffile{mup.eps}}
%\vspace{-1.0cm}
\caption{Same as previous figure, but with positive muons circulating in the
storage ring. The difference between the two figures is due to the different
cross section for neutrinos and antineutrinos, and to matter effects.}
\label{fig:m2}
\end{figure}
The detectors required at a neutrino factory will have many similarities
to the detectors that have been designed
for the next generation of experiments at conventional neutrino beams.
However, there are some important differences.
First, we can anticipate more massive detectors. The sensitivity of
a neutrino factory oscillation experiment is proportional to the
product of the detector mass and beam intensity. It is likely that
the cost of increasing the MINOS detector fiducial
mass (for example) by a factor of a few
is smaller than the cost of increasing the neutrino factory beam intensity
by a factor of a few. Therefore, we believe that it is reasonable to assume
that detectors at a neutrino factory would be a factor of a few to
a factor of 10 more massive than the generation of neutrino
detectors presently under construction.
Second, the presence of both neutrinos and antineutrinos in the same
beam at a neutrino factory places a premium on measuring the sign
of the charge of the lepton produced in CC interactions. Charge--sign
determination may
not be practical for electrons, but is mandatory for muons and
highly desirable for $\tau$--leptons.
Finally, a relatively low energy threshold for the detection and
measurement of wrong--sign muons is very desirable. This is because
high muon detection thresholds require high energy interacting
neutrinos, and hence a high energy neutrino factory. Since the muon
acceleration system at a neutrino factory is likely to be expensive,
low energies are preferable.
In the following sections
we begin by considering general detector issues for the
measurement of final state muons and $\tau$--leptons, and then consider
some specific candidate detectors for a neutrino factory.
Some of these detector types are quite new and
are just beginning to be studied; for the more mature detectors
the ``neutrino'' energy resolution,
the signal efficiency, background rejection,
and fiducial mass are discussed.
%\clearpage
\subsubsection{Muon identification and measurement}
\label{bkgds}
The detection and measurement of muons (especially those of
opposite sign to the muons in the storage ring) is
crucial for many of the key oscillation physics measurements
at a neutrino factory. Before considering some specific
neutrino factory detectors it is useful to consider more generally
muon backgrounds and related issues.
Background muons can be produced in NC and CC interactions by:
\begin{description}
\item{(i)} Pions or kaons from the hadronic shower that decay to produce a
muon.
\item{(ii)} Non-interacting pions which fake a muon signature (punch-through).
\item{(iii)} Charm meson production and muonic decay.
\end{description}
A background muon event can be produced when
a background ``muon" of the appropriate sign is recorded in (a)
a NC event or (b) a CC event in which the primary lepton has been lost.
If the background muon has the same charge sign as that in the storage ring
the resulting event will be a background
for disappearance measurements, but more importantly, if it has
the opposite sign then the event will be a background for
wrong--sign muon appearance measurements.
The integrated wrong-sign
background fraction from the hadronic shower
is shown in Fig.~\ref{punch-charm}
as a function of the minimum muon energy accepted
for Steel/Scintillator and water detectors downstream of 20~GeV
and 50~GeV neutrino factories. The charm background comes from
$\nu_\mu$ CC events where the primary muon was less than XX GeV.
The peak at low muon energies is from
the hadron shower itself and from punch through, while the long tail is from
shower particles decaying to muons.
\begin{figure}[h]
\begin{center}
\epsfxsize=0.75\textwidth
\epsfbox[0 0 570 280]{punch-charm-20.ps}
\epsfxsize=0.75\textwidth
\epsfbox[0 0 570 280]{punch-charm-50.ps}
%\epsfig{file=punch-charm-20.ps,width=7.5cm}
%\epsfig{file=punch-charm-50.ps,width=7.5cm}
\end{center}
\caption{Background levels from punch through, pion/kaon decay, and
charm backgrounds for 20~GeV (top) and 50~GeV (bottom) neutrino
factories. The fraction of neutrino interactions that produce a
wrong--sign muon background event is shown as a function of
the minimum muon energy accepted.}
\label{punch-charm}
\end{figure}
In general there are two different standards for background levels which
are relevant: that of a disappearance experiment and that of an
appearance experiment.
Background estimates are not trivial, but if the backgrounds for a
disappearance measurement
are at the one per cent level, then the uncertainties on those
backgrounds can be expected to be small compared to the flux uncertainty.
On the other hand, wrong-sign muon appearance measurement uncertainties
are expected to be dominated by the statistics. An extremely aggressive
background
level requirement would be to have less than of the order of one
background event.
If there are several thousand CC events expected, then this
would require a minimum background rejection factor of $10^{4}$.
Backgrounds can be suppressed by imposing a minimum energy requirement
on the measured muon.
Figure ~\ref{enucut} shows the effect of several different minimum muon
energy cuts on a simulated oscillation signal observed in
a steel-scintillator type detector at a 20~GeV muon storage
ring, at a baseline length of 2800km \cite{bgrw00}.
A muon threshold energy of 4~GeV for example depletes the
low energy part of observed measured ``neutrino energy" distribution,
degrading but not completely removing the information about the
neutrino oscillation parameters that is encoded in the shape of
the distribution. A 4~GeV threshold at a 20~GeV neutrino factory
is probably tolerable. If higher thresholds are needed to reject
backgrounds, then a higher energy neutrino factory is desirable.
If a lower energy neutrino factory is to be viable, then lower
muon thresholds are desirable.
\begin{figure}
%\epsfig{file=steveplot.ps,width=8.cm}
\epsfxsize3.0in
\centerline{\epsffile{steveplot.ps}}
\bigskip
\caption{Reconstructed neutrino energy distribution for
several different minimum muon energy cuts for a 20GeV ring.
Result is from Ref.~\ref{bgrw00}.
}
\label{enucut}
\end{figure}
As is shown in figure \ref{punch-charm}, to get to a background level
of $10^{-4}$ one would need a 5 (6.5) GeV muon momentum cut in
Steel/scintillator (Water) for a 20 GeV muon storage ring, and a 10 (12) GeV
muon momentum cut in Steel/Scintillator (Water) for a 50 GeV muon storage
ring. Clearly more background rejection is desirable.
Fortunately muons from hadron decay in the hadronic shower
are likely to be more aligned with the shower direction than
muons from the leptonic vertex of the CC interaction.
This provides another handle on the background.
A useful variable to cut on is the momentum of the muon in the direction
transverse to the hadronic shower ($p_t$). Figure \ref{fig:pt2gen}
shows the generated $p_t^2$ distribution for background and signal
events, with no cut on the final state muon momentum. Note that
requiring $p_t^2 > 1$ the background is extremely low, while the signal
efficiency is high. The resolution with which $p_t^2$ is determined
is extremely detector dependent, and for detectors with reasonable
transverse and
longitudinal segmentation is dominated by the hadronic energy resolution.
\begin{figure}[tb]
\centering
\epsfig{file=pt2_gen.eps,width=\textwidth}
\caption{Transverse momentum distributions for $\nu_\mu$ charged
current events compared to background muons for a 20
and 50 GeV muon storage ring. }
\label{fig:pt2gen}
\end{figure}
%\clearpage
\subsubsection{$\tau$--lepton identification and measurement}
The detection and measurement of $\tau$--leptons is crucial
for $\nu_\mu \rightarrow \nu_\tau$ and $\nu_e \rightarrow \nu_\tau$
measurements at a neutrino factory. Note that $\nu_e \rightarrow \nu_\tau$
oscillations will be of special interest since they will not have been
previously
observed. The $\nu_e \rightarrow \nu_\tau$ signal can be separated from
$\nu_\mu \rightarrow \nu_\tau$ ``background" if
the sign of the $\tau$--lepton charge is measured.
The majority of $\tau$--lepton decays produce either one charged
track (electron, muon, of hadron) or three charged tracks (hadrons).
There are two general techniques that can be used to identify
$\tau$--leptons. The first technique exploits the one-prong and
three-prong topologies, and uses kinematic cuts to
suppress backgrounds. The second technique uses a detector with a high
spatial resolution to look for the displaced vertex or kink resulting
from $\tau$--lepton decay.
The advantage of the displaced vertex or kink detection $\tau$--lepton
technique is that the detailed $\tau$--lepton decay is measured and
background suppression is therefore large. The disadvantage is that
detectors that have sufficient spatial resolution are necessarily
less massive than coarse--grained detectors.
The advantage of the kinematic technique is that a very massive
detector can be used. If the $\tau$--leptons decay muonically
(BR = 17\%) a measurement of the muon charge--sign determines
the sign of the $\tau$ charge.
However, there are substantial backgrounds
that must be reduced. In the case of muonic $\tau$ decays,
the backgrounds are from
(a) $\nu_\mu$ (or $\bar\nu_\mu$) CC interactions
which typically produce muons at high
momentum and high $p_t^2$, and
(b) meson decays (discussed earlier)
which are at low momentum and low $p_t^2$.
For $\tau \to e$ decays, the main background comes from
$\nu_e$ and $\bar\nu_e$ CC interactions.
Fortunately the undetected neutrinos from
$\tau$ decays result in a larger missing transverse momentum
than expected for background events.
Exploiting these kinematic characteristics the backgrounds
can be reduced by a large factor.
For example, for an ICANOE--type detector a background rejection
factor of 200 has been estimated, with a corresponding signal efficiency
of 30\%. In the electron channel background can also come from
NC interactions which produce photon conversions
or Dalitz $\pi^0$ decays. These backgrounds can be suppressed
in detectors with good pattern recognition allowing conversions,
for example, to be identified and rejected.
The analysis of hadronic $\tau$ decays requires the identification of the
$\tau$ decay product inside a jet.
This can only be done with a detector having good pattern recognition.
It has been demonstrated that with an ICANOE--type detector a
background rejection factor of 200 can be expected for
$\tau\to$ 1 prong, $\tau\to\rho$, and $\tau\to 3\pi$ decays, with
a signal efficiency of 8\%.
%\begin{figure}[tb]
%\centering
%%\epsfig{file=CIN_HB_pr1.eps,width=5.5cm,angle=-90}
%\epsfig{file=Rubbia-Plenary.2.1-fig1.eps,width=7.5cm,angle=-90}
%\caption{Perspective view of the Liquid Argon TPC detector with
%4 supermodules. HOW ABOUT REPLACING THIS WITH AN ELECTRON EVENT PIC}
%\label{icanoe_pr}
%%\epsfig{file=overfig/NICm_HB.eps,width=12cm,angle=90}
%%\caption{Perspective view of the baseline detector with
%%8 supermodules. \ednote{SIZE NEED TO BE FIXED!}}
%\end{figure}
\begin{figure}[tb]
\centering
%\epsfig{file=icanoe_event.ps,width=7.5cm,angle=-90}
%\rotate[r]{
\epsfxsize=\textwidth
\epsfbox[60 150 800 490]{icanoe_event.ps}
%}
\label{fig:icanoe_elev}
\caption{Example of a $\nu_e$ Charged current event from the
full simulation of the ICANOE detector.}
\end{figure}
%\clearpage
\subsubsection{A Liquid Argon neutrino detector}
We have studied the performance of a large Liquid Argon
neutrino detector at a neutrino factory
using the ICANOE monte carlo program.
One ICANOE detector unit
consists of a liquid argon TPC followed by a magnetic spectrometer.
%The layout is shown in Fig.~\ref{icanoe_pr}.
The Liquid Argon TPC has extremely fine granularity, producing
bubble chamber like event images. Figure \ref{fig:icanoe_elev}
shows an example of an electron neutrino charged current event--note
the separation between the electromagnetic shower and the
hadronic shower of the nucleon remnant.
The TPC is instrumented with 3~mm pitch wires which allow tracking,
$dE/dx$ measurements, electromagnetic, and hadronic calorimetry.
Electrons and photons can be identified event by event, and their
energies are measured with a resolution given by
$\sigma_E/E = 0.03/\sqrt{E} \oplus 0.01$.
The hadron energy resolution is given by
$\sigma_E/E = 0.2/\sqrt{E} \oplus 0.05$.
The magnetic spectrometer is primarily needed to measure muon energy
and charge, but it is assumed that it will
also be instrumented as a calorimeter
to allow the hadron energy of showers
which leak into the spectrometer to be correctly measured
(albeit with worse resolution).
The muon momentum resolution is expected to be $20\%$.
In the design we have simulated, the
liquid Argon module is 18~m deep with a cross section of
$11.3m\times11.3m$. The active (total) mass of one Liquid Argon module
is 1.4~kt (1.9~kt). The magnetized calorimeter module is 2.6~m deep
with a cross section of $9m\times 9m$, and has a mass of 0.8~kt.
It consists of 2~m of steel, corresponding
to $7.4\ \lambda_{int}$ and $59\ X_0$, interleaved with
tracking chambers. Four Super-Modules are assumed, yielding
a total detector length of $82.5$~m and a total active mass
of 9.3~kt that is fully instrumented.
ICANOE can fully reconstruct
neutrino (and antineutrino) events of all active flavors, and with an
energy ranging from tens of MeV to tens of GeV, for the relevant
physics analyses. The unique imaging capabilities of the liquid
argon TPC allow one to cleanly determine whether a given
event is a $\nu_\mu$ CC event, a $\nu_e$ CC event, or a NC
event.
For our studies
the ICANOE fast simulation was used. Neutrino interaction
events are generated, with a proper treatment of quasi-elastic interactions,
resonance and deep-inelastic processes. The 4-vectors for all the
particles generated are smeared, according to the resolutions derived from the
full simulation. Muonic decays of pions and kaons are also considered, for a
proper wrong- and right-sign muon background treatment. Once a
2-GeV cut is placed on the outgoing muon momentum, the background levels
tend to be about $10^{-5}$ times the actual charged current event rate,
and are dominated by meson decay in the hadronic shower.
Examples of simulated oscillation signals in an ICANOE--type detector
at a neutrino factory are shown in Figs.~\ref{fig:m1} and \ref{fig:m2}.
More detailed results from a study of the sensitivity that might be
achieved using an ICANOE--type detector are discussed in the
oscillation measurements section of this report.
%\clearpage
\subsubsection{A magnetized Steel/Scintillator neutrino detector}
Steel/Scintillator calorimeters have been used extensively in past neutrino
experiments. Their performance is well understood
and well simulated. Typically a magnetized Steel/Scintillator (MINOS--like)
neutrino detector consists of iron plates
interspersed with scintillator planes. To obtain transverse position
information the scintillator can be
segmented transversely, or a separate detector system (e.g.
drift chambers) used.
Penetrating charged particles (muon candidates) can then be
reconstructed. With a reasonable transverse segmentation,
the transverse position resolution is dominated by multiple
coulomb scattering.
The detector performance depends primarily on its
longitudinal segmentation. The
segmentation needs to be fine enough to determine whether a
charged track has penetrated beyond
the region of the accompanying hadronic shower. If it has,
then the penetrating track is a muon candidate.
The muon momentum resolution is
determined by the magnetic field and the thickness
of the steel plates.
Neutrino CC and NC interactions have well defined signatures.
In a MINOS--like detector NC interactions produce a hadronic shower
reconstructed as a large energy deposition in a small
number of scintillator units. A
$\nu_\mu$ or $\bar{\nu_\mu}$ CC interaction will produce
a muon in the final state, characterized by a long
track in addition to the hadronic shower. These events can be
identified provided the muon penetrates well beyond the hadronic
shower. This imposes a minimum track-length, and hence minimum
energy, requirement on muons that can be identified. If the muon
is not identified the CC interaction will look like a NC event.
A $\nu_e$ or $\bar{\nu_e}$ CC interaction, will produce an
electron in the final state which cannot be resolved, so these events look
similar to NC interactions.
A $\nu_\tau$ or $\bar{\nu_\tau}$ CC interaction will also look like a NC
interaction unless the $\tau$--lepton decays muonically.
%The second consideration comes from the
%possibility of reconstructing as primary muons non-prompt muons or
%punchthrough pions in NC-like events ($\nu_mu$ NC, all $\nu_e$,
%and most $\nu_\tau$). The sources of these muons were listed in
%section \ref{bkgds}.
%
%The experimental technique proposed for this study is the observation of
%``wrong-sign'' muons. Given the above discussion, the issues which we need
%to address in order to understand the sensitivity of the magentized Fe/Sci
%detector are: the efficiency to identify and measure the charge of the primary
%muon (from the leptonic vertex) and the efficiency to remove secondary
%muons (from the hadronic vertex) without eliminating the signal.
To study the performance of a magnetized Steel/Scintillator detector at a
neutrino factory we have considered a detector geometry similar to the
CCFR/NuTeV calorimeter~\cite{nutevdet}, but with the addition of a toroidal
magnetic field of 1T.
The detector is constructed from $3\times 3 \times 0.3$~m$^3$
modules (see Fig.~\ref{fig:ccfr}). The 0.7~kt CCFR detector consists
of 42 modules. A neutrino factory detector with a mass of 50~kt
(10 $\times$ the MINOS detector) would require 3000 of these
modules.
The ultimate transverse size (and hence module mass) that is
practical is probably determined by the largest size over which
a large magnetic field can be generated.
\begin{figure} [h]
\begin{center}
% \epsfxsize=0.6\textwidth
% \epsfysize=0.3\textheight
% \epsfbox{unit-target-metric.eps}
\epsfxsize3.0in
\centerline{\epsffile{unit-target-metric.eps}}
\caption{Schematic of a CCFR/NuTeV calorimeter module.}
\end{center}
\label{fig:ccfr}
\end{figure}
In the following we consider how well a magnetized Steel/Scintillator detector
can identify and measure wrong--sign muon events at a neutrino factory.
For our simulations, we used the parameterized Monte Carlo developed by the
NuTeV
collaboration~\cite{nutevpub}, modified to include particle tracing in the
magnetic field.
The hadron energy resolution of this detector is described in detail in
~\cite{nutevdet}, and is approximately given by
$\sigma_E / E = 0.85 / \sqrt{E}$.
The muon momentum resolution
depends on the track length in the steel, and whether the muon is contained
within the detector. For muons which range out in the detector the
effective
momentum resolution is $\sigma_P/P = 0.05$, while for tracks which leave the
fiducial volume of the detector the resolution is described by
$\sigma_P/P \sim \theta_{MCS}\theta_{BdL}$, where the angles
$\theta_{MCS}$ and $\theta_{BdL}$ describe respectively
the change in direction
due to multiple scattering and curvature in the magnetic field.
The simulation includes a detailed parameterization of the hadron-shower
development, with the inclusion of charm production and $\pi$, $K$
decays (the data set on which the decay probability parameterization
was tuned contained muons with momentum higher than 4~GeV/c).
Note that $\pi$ punchthrough was not included in the parameterization,
but is expected to make only a small contribution to background
muons above 4~GeV.
\begin{figure} [h]
\epsfig{file=pttalk20.eps,width=7.cm}
\epsfig{file=pttalk50.eps,width=7.cm}
\caption{Reconstructed $\mu^-$ $P_t^2$ with respect to the shower direction
for
20 GeV and 50 GeV $\mu^+$ decaying in a neutrino factory.}
\label{pt}
\end{figure}
To be conservative, and reduce the dependence of our study on low energy
processes that may not be adequately described by the Monte Carlo program,
in our analysis all muons with generated energy below
4~GeV are considered lost. Muons with
track length in steel less than 50~cm past the hadronic shower
are considered also considered lost. All other muons are assumed
to be identified
with 100\% efficiency, and measured sufficiently well to determine
their charge sign.
For the background events we considered (i) all the $\pi$, $K$ decay events
producing ``wrong--sign'' muons in NC interactions,
and (ii) all the charm production
and $\pi$, $K$ decay events producing ``wrong--sign" muons
in CC events where the primary muon was considered lost.
To reduce the backgrounds, we cut on $P_t^2$.
The reconstructed $P_t^2$ distribution is shown in figure~\ref{pt}
for signal and background muons in a 10~kt detector
2800~km downstream of 20~GeV and 50~GeV neutrino factories which
provide $10^{20}$ $\mu^+$ decays. The oscillation parameters
corresponding to the LAM scenario IA1.
As expected, background wrong--sign muons,
tend to have smaller $P_t^2$ than genuine wrong--sign muons from
the leptonic vertex.
The reconstructed wrong-sign muon spectrum is shown in Fig.~\ref{data}
for a 20~GeV storage ring before (top plot) and after (bottom plot)
muon energy, track length and $P_t^2 > 2$~GeV$^2$ cut were applied.
Signal and background rates are summarized in Table~\ref{thetable}.
After the cuts the signal/background ratio is above 10 to 1 in
scenario IA1 for a detector 2800~km away, while $40-50\%$ of
the $\nu_e \rightarrow \nu_\mu$ signal events are retained.
\begin{figure} [h]
\begin{center}
\epsfxsize=0.75\textwidth
\epsfbox{{signalwriteplus20.eps}}
\caption{Reconstructed wrong-sign muons as a function of the muon energy for
a $\mu+$ 20 GeV ring. Top plot accepted events for the signal
($\nu_e \rightarrow \nu_\mu$--stars-- and
$\nu_e \rightarrow \nu_\tau \rightarrow \mu + X$--crosses) and the potential
backgrounds (x). The bottom plot shows the signal and the
background after cuts.}
\label{data}
\end{center}
\end{figure}
\begin{table}
\caption{Wrong-sign muon rates after all cuts for a 10~kt
steel-scintillator detector downstream of a neutrino factory providing
$10^{20}$ muon decays. The oscillation parameters correspond to
scenario IA1. The loss of signal acceptance is due solely to the
kinematic and reconstruction cuts, as is the background rejection.}
\bigskip
\begin{tabular}{cc|ccccc}
\hline
\multicolumn{2}{c}{$\mu$ Ring} & &$\nu_e \rightarrow \nu_\tau$ & & & \\
Energy & Charge & $\nu_e \rightarrow \nu_\mu$ &$\rightarrow \mu + X$ &
background & signal & background \\
GeV & & events & events & events & acceptance & rejection \\
\hline
50 & + & 268.5 & 15.4 & 21.6 & 0.50$\pm$0.02 & 0.013$\pm$0.006 \\
50 & $-$ & 55.2 & 4.7 & 3.5 &0.48$\pm$0.02& 0.006$\pm$0.008 \\
20 & + & 85.7 & 3.5 & 0.7 & 0.41$\pm$0.02 & 0.021$\pm$0.008 \\
\hline
\end{tabular}
\label{thetable}
\end{table}
%\clearpage
\subsubsection{A Water Cerenkov detector}
Preliminary studies have explored the possibility of using a large water
Cerenkov detector as a distant target for a neutrino factory beam.
Traditionally this type of detector has
been used for measuring much lower energy neutrinos than expected
at a muon storage ring, but to date water Cerenkov
neutrino detectors are the only existing neutrino detectors with
masses already in the 50~kt range. Water is of course the
lightest target material under consideration
in this report, but this type of detector has several advantages
when extrapolating to large masses, namely
(i) low cost target material, (ii) only the surface of a very large volume
needs to be instrumented, and (iii) good calorimetry.
A large volume guarantees containment of hadronic and
electromagnetic showers (as well as muons up to a certain energy).
The low density of the target and good
angular resolution from the Cerenkov cone might yield an
overall hadron angle resolution that is as good as or better than
the corresponding resolution obtained with
steel-scintillator calorimeters.
Water Cerenkov devices as large as 50~kt (SuperK) are already in
operation and may continue data-taking for ten years or more. Therefore, the
response of the existing SuperK detector at a baseline
distance of 9100~km has been
studied as a test case. Next generation detectors, up to 1~Mton in mass, are
technically feasible and are currently under consideration for proton decay
and neutrino measurements, sited perhaps at the Kamioka mine or elsewhere.
For this initial study, the primary question is the suitability of a water
Cerenkov detector for the higher energy neutrino beam produced by a 10-50
GeV muon storage ring. At these energies, the multiplicity of hadrons is
greater than for typical atmospheric neutrino interactions, and event
topologies are correspondingly more complex. Figure \ref{fig:f2kevt}
shows the Cerenkov light produced in a
typical neutrino event from a 50 GeV muon storage ring at the SuperK
detector: the circles in the display are estimates of the outgoing
angles of different charged particles produced in the hadronic
shower. Some particle identification is possible from the pulse-height
information. Reconstruction software from the
SuperK experiment must be further optimized to study the detector response to
neutrinos from 10~GeV and 50~GeV muon storage rings. It is worth noting
that neutrinos produced by a 50~GeV muon beam induce a large number of events
in the material (rock) surrounding any detector (producing an entering muon),
and for a SuperK sized device these events outnumber the
those produced in the detector's water volume. Both contained and
entering events have therefore been studied.
\begin{figure} [h]
\begin{center}
\epsfxsize=0.75\textwidth
\epsfbox[0 310 550 530]{f2kevt.ps}
\caption{Simulated neutrino event from a 50~GeV muon storage ring
in the SuperKamiokande detector. The rings indicate where the
reconstruction software found charged particles in the hadronic shower,
as well as the exiting muon.}
\label{fig:f2kevt}
\end{center}
\end{figure}
The response of a detector the size of SuperK changes drastically as a
function of beam energy. At a 10~GeV neutrino factory, 57\% of the muon CC
events are fully contained in the inner water volume, whereas only
11\% are fully contained at a 50~GeV neutrino factory. This large difference only
exists for events containing penetrating muons; at 10~GeV (50~GeV) both
${\bar \nu_e}$ and NC events are contained greater than 98\%
(90\%) of the time. The existing $\mu$-like particle identification
algorithm works to produce a reasonably pure (89\%) $\nu_\mu$-CC sample for
fully contained events in the 10~GeV beam, but $e$-like events are a mixture
of of ${\bar \nu_e}$-CC, NC and $\nu_\mu$-CC contamination. Exiting and
entering events are pure samples of $\nu_\mu$-CC simply because of
their penetrating
nature. The muon angular resolution (3${}^{\circ}$) is much less than the
muon-neutrino angular correlation (15${}^{\circ}$). With $2 \times 10^{20}$
decays at a 50~GeV neutrino factory and a baseline of 9100~km,
approximately $200,000$
$\nu_\mu$ CC events would be observed entering or exiting the
current SuperK detector. Combined with muon charge identification
this sample should be able to provide good oscillation measurements.
Implementing charge identification in a water Cerenkov
detector is not trivial, however. Two possibilities
have been proposed: (i) several large water targets,
each one followed by a thin external muon spectrometer,
and (ii) a magnetic field
introduced into the water volume itself. Although the first
design would have lower geometrical acceptance and a higher muon
energy threshold, it would pose much less of a problem for the
phototubes since the magnetic field would presumably be well-contained
in the spectrometer. The second proposal could in principle have
good low energy muon momentum acceptance, but the resolution on the
muon and hadron shower angles might be compromised.
For a magnetic field internal to the target,
0.5-1~kG is sufficient to visually determine
the charge of a several meter-long ($>1$~GeV)
muon, but no automated algorithms have
yet been developed. A number of conceptual magnet designs have been
studied: solenoidal, toroidal, and concentric current loops in the center or
at the ends of the detector.
A detailed study of one particular design has shown that one can
immerse the central volume of a SuperK sized
detector in a 0.5~kG magnetic field while leaving only a 0.5~G fringe field
(which may be acceptable with shielding and/or local compensation) in the
region of the PMTs. Many of the difficulties inherent in placing a field
inside a water detector would be avoided if an alternative light collector
(insensitive to the field) were used. Work on magnet design and
alternative light collection is ongoing, but the internal magnetic field
option must be considered speculative at this point.
The results we
will describe in the remainder of this section
are for a water Cerenkov detector with an external
magnetic field, because
neutrino event reconstruction is more straightforward to simulate and the
spectrometer technology is well-understood.
Although the studies of this detector are very preliminary,
they look promising and warrant further investigation.
We have used a
LUND/GEANT Monte Carlo program which uses as its geometry a
$40\times40\times100$~m$^3$
box of water, followed by a 1~m long muon spectrometer. This
simulation can be used to study
acceptance issues and background contamination for a range of
geometries and storage ring energies.
%The background levels for this
%type of detector compared to that of steel-scintillator detectors
%are discussed above.
Figure~\ref{fig:wateracc} shows the geometrical acceptance for the box-like
water Cerenkov detector as a function of distance of the neutrino
interaction vertex from the spectrometer, for CC
$\nu_\mu$ events from 20 and 50~GeV storage rings. The
loss in acceptance close to the spectrometer is due to rejection of
events where there is more than one muon which traverses the
spectrometer (where the extra muon comes from background processes).
It is clear that for a 20~GeV muon storage ring one would want
a muon spectrometer much more frequently than once every 100~m. Of
course, noting that steel has a density of 8 times that of water,
the smaller the ratio of water thickness to steel thickness
the more it approximates a magnetized steel/scintillator target
interspersed with water volumes with fine granularity.
\begin{figure} [h]
\begin{center}
\epsfxsize=0.75\textwidth
\epsfbox[0 0 520 530]{muon-acc.ps}
\caption{Acceptance in a water target for
charged current $\nu_\mu$ (solid) and $\overline\nu_\mu$ (dashed)
events in a 20 and 50 GeV storage ring, as a function of distance of
the neutrino interaction vertex from the muon spectrometer.}
\end{center}
\label{fig:wateracc}
\end{figure}
Clearly more work is needed to optimize the design for this
kind of detector, but
it might be an inexpensive
compromise between a coarse-grained sampling calorimeter
and a very fine-grained liquid argon TPC.
%\clearpage
\subsubsection{Specialized $\tau$--lepton detectors}
The measurement of $\tau$--lepton appearance in large mass
neutrino detectors is challenging. There are several ideas
that might lead to viable new $\tau$--appearance detectors
within the next 5--10~years, and that might be suitable for use at a
neutrino factory. We briefly describe three examples in the
following: (i) a perfluorohexane Cerenkov detector,
(ii) a hybrid emulsion detector, and (iii) a very fine--grained
micro--strip gas chamber target.
Consider first a Cerenkov detector filled with perfluorohexane ($C_6F_{14}$),
which has a density 1.7 times that of water.
This has been proposed~\cite{forty} for use
in the CERN to Gran Sasso beamline.
The detector geometry consists of several target volumes followed by short
muon spectrometer modules.
A 1~Ton perfluorohexane detector
(with a very different geometry) exists at DELPHI.
The $\tau$--lepton signature in this type of detector consists of
a sparsely populated Cerenkov ring from the $\tau$ before it decays,
together with a more densely populated ring from the daughter muon.
The two rings would have offset centers. Figure \ref{fig:forty} shows a
simulated quasi-elastic $\nu_\tau$ event (no hadron energy)
from this kind of detector.
This technique would probably
not work for events with high energy hadron showers
because of the large
number of charged particles that would result in overlapping rings near
the initial $\tau$--lepton ring.
\begin{figure}[h]
\epsfxsize=\textwidth
\epsfbox[50 400 580 690]{forty_fig3.ps}
\caption{Quasi-elastic $\nu_\tau$ event in a perfluoroHexane Cerenkov
detector: the ring described by about eight hits
on the left is from the tau before it decays.}
\label{fig:forty}
\end{figure}
Next consider a hybrid emulsion detector consisting of, for example,
thin ($\sim 100\mu$m) sheets of emulsion
combined with low-density ($\sim 300\mu$m) spacers.
The signature for a $\tau$--lepton 1--prong decay would be
a change in direction of the track measured before and after the
spacer~\cite{strolin}.
For charge identification the
detector could be within a large magnetic field volume.
With an emulsion track angular resolution of 2~mrad, a
5$\sigma$ charge--sign determination of a 10~GeV/c
charged particle could be achieved with a
2~T field and a 1.2~mm thick spacer \cite{para}.
An $\sim 20$~kt hybrid emulsion detector of this type might
consist of 20~kt of steel segmented into 1~mm thick sheets,
and an equal volume of thin emulsion layers plus low density spacers.
The resulting detector would fit into the ATLAS barrel toroid
magnet, which has a magnetic field ranging from 2 to 5 Tesla~\cite{para}.
A hybrid emulsion detector with an external downstream muon spectrometer
will be used by
the OPERA experiment, which is to be put in the Cern to Gran Sasso beam.
The muon spectrometer will determine the charge sign for
$\tau\to\mu$ decays provided the muon
reaches the spectrometer.
According to the OPERA studies~\cite{opera},
with an average neutrino energy of 20~GeV
the total efficiency for
seeing the $\tau$ decays is 29\% (including the branching
ratios).
%for $\delta m_{23}^2 = 2.5x10^{-3}$ in the CERN to
The efficiency is largely geometric
and should not be compromised by the addition of a magnetic field,
provided the bend in the spacer due to the magnetic field is much less
than the "apparent bend" due to the $\tau$--lepton decay.
%However, it should be noted that for an average neutrino energy of
%20GeV the muon storage ring would have to be roughly 30GeV, and
%the efficiency of $\tau$ detection and charge measurement
%will deteriorate severely as the energy of the taus decreases.
Finally, consider a target consisting of a tracking
chamber constructed from micro--strip gas chambers (MSGCs) and a low $Z$
material (for example, nylon) in a large magnetic field volume.
This would be a NOMAD--like detector with a much larger
O(1~kt) fiducial mass and an improved spatial resolution. Because of the
low $Z$ of the material electrons can travel a long distance in the
detector before showering, and with a high enough field their charge
can therefore be measured. Although again a kink is not seen, the
tau decay could be distinguished kinematically.
For example, nylon has a radiation length of 37 cm. With a $B$ field of
1~Tesla and
MSGC's every 10~cm one would have a XX sigma measurement of a 50~GeV
electron's charge. This idea is worthy of further consideration,
particularly if the LSND signal is confirmed and lower-mass tau
detectors are warranted.
%\clearpage
\subsubsection{Detector summary}
In our initial studies we have simulated the performance of
steel/scintillator and liquid Argon detectors at
a neutrino factory. Results are encouraging. These technologies could
provide detectors with masses of order 10~kt (liquid Argon) to
a few $\times 10$~kt (steel/scintillator) that
yield good wrong--sign muon identification and adequate
background rejection. Our simulations of the capabilities
of water Cerenkov detectors at a neutrino factory are less
advanced, but initial results are encouraging, and this
detector technology might permit very large detector masses
to be realized. Some relevant characteristics of
steel/scintillator, liquid Argon, and water Cerenkov detectors
are listed in Table~\ref{dettab}. It is premature to choose
between detector types at this early stage. However, some general
points are worth noting:
\begin{description}
\item{(i)} We believe that a cost optimization of detector mass (cost)
versus neutrino factory beam intensity (cost) will probably favor
detectors that are at least a factory of a few to a factor of 10
more massive than, for example, ICANOE or MINOS. A detector mass
in the range 10~kt to 50~kt does not seem unreasonable.
\item{(ii)} The minimum energy a muon must have for good identification
and measurement may well determine the minimum viable
muon storage ring energy. This threshold is a few GeV, and is
detector technology dependent. With a steel/scintillator detector
and a threshold of 4~GeV, for example, the minimum acceptable neutrino
factory energy appears to be in the neighborhood of 20~GeV.
\end{description}
In this initial study we have not comprehensively considered
to what extent massive detectors at a neutrino factory need to
be deep underground. It seems very likely that detectors with
low detection thresholds (water Cerenkov and liquid argon)
will need to be well protected from cosmic ray backgrounds,
regardless of the neutrino factory energy. For the
steel/scintillator detector, the cosmic ray backgrounds for
charged current events with muons in them are likely to be small
for a detector at the surface of the earth, but
there will be substantial background to neutral current
or $\nu_e$ charged current interactions.
Finally, we note that the development of a new generation of
very massive detectors
capable of identifying and measuring the charge--sign of muons and
$\tau$--leptons, would be of great benefit to a neutrino factory.
There is a possible area of mutual interest with the nucleon decay
community in developing the technology for a really massive
1~Mton scale water Cerenkov detector. This possibility deserves
further investigation.
\begin{table}[h]
\caption{Comparison of detector parameters for candidate
detectors at a neutrino factory.}
\bigskip
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Characteristic & \multicolumn{3}{|c|}{Detector Technology } \\
& Steel/Scint & Liquid Argon TPC & Water Cerenkov \\\hline
Resolutions of: & & & \\
Electron Energy& $50\%/\sqrt{E}$ & $3\%/\sqrt{E}\oplus 1\%$ & $0.6\oplus 2.6\%/\sqrt{E}$ \\
Hadron Energy & $85\%/\sqrt{E}$ & $20\%/\sqrt{E}\oplus 5\%$ & 20-30\% \\
Muon Energy &$5\%$ & $20\%$ & $20\%^\dagger$ \\
Hadron Shower Angle & &$.02 \oplus .21/\sqrt{p} rad $& \\
& & (each hadron)& \\
Muon Angle & 5\% for & & $3^\circ$ \\
& 50cm track & $.02\oplus .21/\sqrt{p}$
& \\
Maximum mass &50 kton & 30 kton & 1Mton?\\
What limits size? & & safety, tunnel & tunnel\\
Required Overburden$^{**}$ & 0 m &50 m & 50-100m\\
Analysis Cuts & $P_{\mu}>4 GeV$ &$P_{\mu}>2 GeV$& \\
& $P_t^2 > 4GeV^2$ & & \\
%Signal acceptance from these cuts & 50\% &$>99\%$ & 50\% \\
Background level & $10^{-4}$ &$2\times 10^{-5}$ & \\
%Maximum B field& 1T & 2 T & 1T \\
%Transverse segmentation&& 3 mm wire pitch & xxxpmts/4$\pi$\\
\hline\hline
\end{tabular}
\end{center}
$^{**}$ The overburden required for all technologies
depends on the neutrino factory duty factor.
The overburden required for a steel-scintillator
calorimeter also depends on the energy of the muon storage ring; but
in the past this type of detector has been used
at ground level with minimal
contamination in the $\nu_\mu$ charged current sample
above a neutrino energy of 5~GeV.$^{\dagger}$ The
muon momentum resolution would be comparable to that of an
ICANOE detector if the muon spectrometer were separated from the
water tank volume.
\label{dettab}
\end{table}
\clearpage
\subsection{Oscillation measurements}
Using the oscillation scenarios described in section A
as examples, we can now assess how well the neutrino oscillation
physics program outlined in section C can be pursued
at a neutrino factory with the detectors described in section D.
In the following sub-sections the oscillation measurements
that can be made at a neutrino factory are discussed as a function of
baseline,
muon beam energy, and muon beam intensity.
In particular we consider
the first observation of $\nu_e \rightarrow \nu_\mu$
oscillations, the measurement of the sign of $\delta m^2$ and hence the
pattern of neutrino masses,
the first observation of $\nu_e \rightarrow \nu_\tau$ oscillations,
the measurement of $\nu_\mu \rightarrow \nu_\tau$ oscillations,
precisions measurements of the oscillation parameters,
and the search for CP violation in the lepton sector.
These results are based on the calculations described in more detail in
Refs.~\cite{cerv00, bgrw00, bern00, camp00}.
%\clearpage
\subsubsection{Observation of $\nu_e \rightarrow \nu_\mu$ oscillations and
the pattern of neutrino masses}
At a neutrino factory $\nu_e \rightarrow \nu_\mu$ oscillations would be
signaled by the appearance of CC interactions tagged by a
wrong--sign muon~\cite{geer98}.
Within the framework of three--flavor
oscillations the $\nu_e \rightarrow \nu_\mu$ oscillation amplitude is
approximately proportional to $\sin^2 2\theta_{13}$.
At the present time only an upper limit exists on $\sin^2 2\theta_{13}$.
The next generation long-baseline oscillation experiments are expected
to be able to improve the sensitivity to
$\sin^2\theta_{13}\approx 10^{-2}$, i.e. about one
order of magnitude below the present bound. If $\sin^2 2\theta_{13}$
is in this range we would expect to observe
$\nu_e \rightarrow \nu_\mu$ oscillations at a relatively low intensity
neutrino factory, measure matter effects,
and determine the pattern of neutrino masses~\cite{bgrw99}.
This is discussed further in the remainder of this sub--section.
%
\begin{figure}[h]
\epsfxsize3.0in
\centerline{\epsffile{fig_v1.ps}}
\vspace{0.3cm}
\caption{Reach in $\sin^22\theta_{13}$ for the observation of
10 $\mu^-$ events from $\nu_e \rightarrow \nu_\mu$ oscillations,
shown versus baseline for three $\delta m^2_{32}$
spanning the favored SuperK range. The other oscillation parameters
correspond to the LAM scenario IA1.
The curves correspond to
$10^{19} \mu^+$ decays in a 20~GeV neutrino factory with
a 50~kt detector, and a minimum muon detection threshold of 4~GeV.
Result are from Ref.~\ref{bgrw00}.}
\label{fig:v1}
\end{figure}
It is useful to define~\cite{bgrw00} the $\sin^22\theta_{13}$
``reach" for an
experiment as the value of $\sin^22\theta_{13}$ for which a given physics
goal would be met. We take as our initial goal the observation of 10
$\nu_e \rightarrow \nu_\mu$ events tagged by a wrong--sign muon.
Consider first the
$\sin^22\theta_{13}$ reach for a 50~kt detector sited a distance $L$
from a 20~GeV neutrino factory in which there are $10^{19} \mu^+$ decays
in the beam--forming straight section. The baseline--dependent
$\sin^22\theta_{13}$ reach is shown in Fig.~\ref{fig:v1} for a three-flavor
oscillation scenario in which $\delta m^2_{21}, \sin^22\theta_{12}$,
and $\sin^22\theta_{23}$ correspond to the LAM scenario IA1,
and the value of $\delta m^2_{32}$ is varied over the favored SuperK range.
If $\delta m^2_{32}$ is in the center of the SuperK range,
the $\sin^22\theta_{13}$ reach is about an order of magnitude below
the currently excluded region, improving slowly with decreasing $L$.
The reach improves (degrades) by a about a factor of 2 (3) if
$\delta m^2_{32}$ is at the upper (lower) end of the current SuperK
range. If the oscillation parameters correspond to the LAM scenario IA1
($\sin^22\theta_{13} = 0.04$),
then only $2 \times 10^{18}$ muon decays are required at a 20~GeV
neutrino factory to observe 10~signal events in a 50~kt detector at
$L = 2800$~km. The calculation~\cite{bgrw00}
assumes that CC events producing
muons with energy less than (greater than) 4~GeV are detected with
an efficiency of 0 (1).
The number of muon decays needed to observe 10 $\nu_e \rightarrow \nu_\mu$
events
is shown in Fig.~\ref{fig:v2} as a function of $E_\mu$ for the LAM
scenario IA1, the SAM scenario IA2, and the LOW scenario IA3.
The required muon beam intensities decrease with increasing $E_\mu$,
and are approximately proportional to $E_\mu^{-1.6}$. Compared with
the SAM and LOW scenarios, slightly less
intensity is needed for the LAM scenario, showing the small but
finite contribution to the signal rate from the sub--leading $\delta m^2$
scale. In all three scenarios (LAM, SAM, LOW) a 20~GeV neutrino factory
providing $10^{19}$ decays in the beam--forming straight section
would enable the first observation of $\nu_e \rightarrow \nu_\mu$
oscillations in a 50~kt detector provided $\sin^22\theta_{13} > 0.01$.
%
\begin{figure}
\epsfxsize5.0in
\centerline{\epsffile{fig_v2.ps}}
\vspace{0.3cm}
\caption{The required number of muon decays needed in the beam--forming
straight section of a neutrino factory to achieve the physics goals described
in the text, shown as a function of storage ring energy for the
LAM scenario IA1, SAM scenario IA2, LOW scenario IA3, and a bimaximal
mixing scenario BIMAX.
The baseline is taken to be 2800~km, and
the detector is assumed to be a 50~kt wrong--sign muon
appearance device with a muon detection threshold of 4~GeV or, for
$\nu_e \rightarrow \nu_\tau$ appearance, a 5~kt detector.
Result are from Ref.~\ref{bgrw00}.}
\label{fig:v2}
\end{figure}
%
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{fig_v3.ps}}
\caption{Predicted measured energy distributions for CC events tagged by a
wrong-sign (negative) muon from $\nu_e \rightarrow\nu_\mu$ oscillations, shown
for various $\delta m^2_{32}$, as labeled. The predictions correspond to $2
\times 10^{20}$ decays, $E_\mu = 30$~GeV, $L = 2800$~km, with the values for
$\delta m^2_{12}$, $\sin^22\theta_{13}$, $\sin^22\theta_{23}$,
$\sin^22\theta_{12}$, and $\delta$ corresponding to
the LAM scenario IA1. Result are from Ref.~\ref{bgrw00}.}
\label{fig:v3}
\end{figure}
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{fig_v4.ps}}
\caption{Same as previous figure, for CC events
tagged by a wrong-sign (positive) muon from $\bar{\nu}_e \rightarrow
\bar{\nu}_\mu$ oscillations.
}
\label{fig:v4}
\end{figure}
Having established $\nu_e \rightarrow \nu_\mu$ oscillations,
further data taking would facilitate the
measurement of matter effects and the determination of the sign of
$\delta m^2$, and hence the pattern of neutrino masses.
To illustrate the effect of matter on the
$\nu_e \rightarrow \nu_\mu$ oscillation probability,
the predicted measured energy distributions 2800~km downstream of
a 30~GeV neutrino
factory are shown in Figs.~\ref{fig:v3} and \ref{fig:v4} for respectively
$\nu_e \rightarrow \nu_\mu$ and
$\overline{\nu}_e \rightarrow \overline{\nu}_\mu$ wrong--sign muon
events.
The distributions are shown for
a range of positive and negative values of $\delta m^2_{32}$.
Note that for a given $|\delta m^2_{32}|$, if $\delta m^2_{32} < 0$
we would expect to observe a lower wrong--sign muon event rate and
a harder associated spectrum when positive muons are stored in
the neutrino factory than when negative muons are stored.
On the other hand, if $\delta m^2_{32} > 0$
we would expect to observe a higher wrong--sign muon event rate and
a softer associated spectrum when positive muons are stored in
the neutrino factory than when negative muons are stored. Hence,
measuring the differential spectra when positive and negative muons are
alternately stored in the neutrino factory can enable the sign of
$\delta m^2_{32}$ to be unambiguously determined~\cite{bgrw99}.
\begin{table}[t]
\caption{Wrong-sign muon rates for a 50~kt detector
(with a muon threshold of 4~GeV) a distance $L$ downstream
of a neutrino factory (energy $E_\mu$) providing $10^{19}$ muon
decays. Rates are shown for LAM scenario IA1 with both signs of
$\delta m^2_{32}$ considered separately. The background rates
listed correspond to an assumed background level of
$10^{-4}$ times the total CC rates. Results are from Ref.~\ref{bgrw00}
}
\bigskip
\begin{tabular}{cc|ccc|ccc}
\hline
$E_\mu$&$L$&\multicolumn{3}{c}{$\mu^+$ stored}&\multicolumn{3}{c}{$\mu^-$
stored}\\
GeV & km & $\delta m^2_{32} > 0$ & $\delta m^2_{32} < 0$ & Backg &
$\delta m^2_{32} > 0$ &$\delta m^2_{32} < 0$ & Backg \\
\hline
20 & 732 & 52. & 36. & 7.3 & 32. & 26. & 6.5 \\
&2800 & 46. & 9.2 &0.43 & 7.1 & 26. & 0.36\\
&7332 & 33. & 0.97&0.063&0.55 & 19. & 0.05\\
\hline
30 & 732 &100. & 72. & 25. & 58. & 45. & 24.\\
&2800 & 90. & 26. & 1.6 & 19. & 43. & 1.4\\
&7332 & 43. & 3.3 & 0.19& 2.1 & 33. & 0.17\\
\hline
40 & 732 &150. &110. & 60. & 83. & 65. & 58.\\
&2800 &140. & 48. & 4.0 & 36. & 64. & 3.8\\
&7332 & 54. & 5.6 & 0.49& 3.1 & 28. & 0.43\\
\hline
50 & 732 &200. & 140.& 120.& 110.& 84. & 120.\\
&2800 &180. & 71. & 7.9& 53.& 82. & 7.7\\
&7332 & 56. & 8.0 & 1.1& 5.0 & 34. & 1.0\\
\hline
\end{tabular}
\label{dm2table}
\end{table}
The expected number of wrong--sign muon events are listed in
Table~\ref{dm2table} for the LAM scenario IA1, and a 50~kt detector
downstream of a neutrino factory providing $10^{19} \mu^+$ decays
and the same number of $\mu^-$ decays. The event rates are shown
for both signs of $\delta m^2_{32}$, and for various storage ring
energies and baselines. Even at a 20~GeV neutrino factory the
signal rates at $L = 7332$ and 2800~km are large enough to permit the
sign of $\delta m^2_{32}$
to be determined with a few years of data taking.
We conclude that for the LAM, SAM, and LOW
three--flavor mixing scenarios we have considered,
a 20~GeV neutrino factory providing $10^{19}$ decays in the beam--forming
straight section would be a viable entry--level facility. In particular,
with a 50~kt detector and a few years of data taking either
$\nu_e \rightarrow \nu_\mu$ oscillations would be observed and the sign of
$\delta m^2_{32}$ determined or a very stringent upper limit on
$\sin^2 2\theta_{13}$ will have been obtained (discussed later).
Long baselines ($>2000$~km) are preferred. The longest baseline we
have considered (7332~km) has the advantage of lower total event rates
and hence lower background rates.
%\clearpage
\subsubsection{Observation of $\nu_e \rightarrow \nu_\tau$ oscillations}
We begin by considering the LAM scenario IA1, and ask: What
beam intensity is needed to make the first
observation of $\nu_e \rightarrow \nu_\tau$ oscillations in a detector
that is 2800~km downstream of a 20~GeV neutrino factory ?
The $\nu_e \rightarrow \nu_\tau$
and the accompanying $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$
event
rates are shown in Fig.~\ref{fig:t1} as a function of the
oscillation parameters
$\sin^2 2\theta_{13}$ and $\delta m^2_{32}$. The $\nu_e \rightarrow \nu_\tau$
signal rate is sensitive to both of these parameters, and hence provides an
important consistency check for three-flavor mixing: the observation or
non--observation of a $\nu_e \rightarrow \nu_\tau$ signal must be consistent
with the oscillation parameters measured from, for example,
$\nu_e \rightarrow \nu_\mu$, $\nu_\mu \rightarrow \nu_\tau$,
and $\nu_\mu$ disappearance measurements.
For the LAM scenario IA1 the observation of
10 signal events in a 5~kt detector (with 100\% $\tau$--lepton efficiency)
would require $7 \times 10^{19} \mu^+$ decays in the beam forming straight
section. Very similar beam intensities are required for the SAM and LOW
scenarios (IA2 and IA3).
Note that, over the $\sin^22\theta_{13}$ range shown in Fig.~\ref{fig:t1},
the $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rates are one to two
orders of magnitude higher than the $\nu_e \rightarrow \nu_\tau$ rates.
Hence, we will need a detector that can determine the sign of
the tau--lepton charge at the $2\sigma-3\sigma$ level, or better.
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{fig_t1.ps}}
\caption{$\nu_\tau$ CC appearance rates in a 5~kt detector 2800~km
downstream of a 20~GeV neutrino factory in which there are
$10^{20} \mu^+$ decays in the beam--forming straight section.
The rates are shown as a function of
$\sin^22\theta_{13}$ and $\delta m_{32}^2$ with the other oscillation
parameters corresponding to the LAM scenario IA1.
The top 3 curves are the predictions for
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$
events and the lower curves are for $\nu_e \rightarrow \nu_\tau$ events.
Result are from Ref.~\ref{bgrw00}.
}
\label{fig:t1}
\end{figure}
Let us define the $\sin^22\theta_{13}$ ``reach" for an experiment as
the value of
$\sin^22\theta_{13}$ for which we would observe 10
$\nu_e \rightarrow \nu_\tau$ events when there are $10^{20}$ muon decays
in the beam--forming straight section of a neutrino factory.
The $\sin^22\theta_{13}$ reach is shown as a function of the baseline and
storage ring energy in Fig.~\ref{fig:t2} for a 5~kt detector and an
oscillation scenario in which all of the
parameters except $\sin^22\theta_{13}$ correspond to scenario IA1.
The reach improves with energy (approximately $\sim E^{-1.6}$~\cite{bgrw00})
and is almost independent of baseline
except for the highest energies and baselines considered.
We conclude that within the LAM, SAM, and LOW scenarios,
a 20~GeV storage ring in which there are O($10^{20}$) muon decays per year
would begin to permit an observation of, or meaningful limits on,
$\nu_e \rightarrow \nu_\tau$ oscillations provided a multi-kt detector
with good tau--lepton identification and charge discrimination is
practical.
\begin{figure}
\epsfxsize3.3in
\centerline{\epsffile{fig_t2.ps}}
\caption{Reach in $\sin^22\theta_{13}$ for the observation of
10 $\nu_e \rightarrow \nu_\tau$ oscillation events,
shown as a function of baseline for four storage ring energies.
The oscillation parameters correspond to the LAM scenario IA1.
The curves correspond to
$10^{20}$ $\mu^+$ decays in a 20~GeV neutrino factory with
a 5~kt detector.
Result are from Ref.~\ref{bgrw00}.
}
\label{fig:t2}
\end{figure}
Next, consider the oscillation scenarios IB1 (atmospheric + LSND scales) and
IC1 (three--flavor with atmospheric, solar, and LSND data stretched).
In these cases the leading $\delta m^2$ is large (0.3~eV$^2$/c$^4$)
and medium baseline experiments ($L =$ 10-100~km) become interesting.
As an example, consider a medium baseline experiment a few $\times 10$~km
downstream of a 20~GeV neutrino factory in which there are $10^{20} \mu^+$
decays. The $\nu_e \rightarrow \nu_\tau$
and accompanying $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ event
rates are shown in Fig.~\ref{fig:t3} as a function of the
baseline and the phase $\delta$ with the other oscillation parameters
corresponding to scenario IB1. In contrast to the
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rates,
the $\nu_e \rightarrow \nu_\tau$ rates are very sensitive to $\delta$,
and for $|\delta| > 20^\circ$ can be very large, yielding thousands of
events per year in a 1~kt detector at $L = 60$~km, for example.
Note that the corresponding
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rate is
of order 100~events. For small $|\delta|$ the
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rate
will dominate the $\tau$ appearance event sample. For
larger $|\delta|$ the $\nu_e \rightarrow \nu_\tau$ rate dominates.
Good $\tau$ charge determination will therefore be important
to measure both $\nu_e \rightarrow \nu_\tau$ and
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ oscillations.
\begin{figure}
\epsfxsize3.3in
\centerline{\epsffile{fig_t3.ps}}
\caption{$\nu_\tau$ CC appearance rates in a 1~kt detector
downstream of a 20~GeV neutrino factory in which there are
$10^{20} \mu^+$ decays.
Rates are shown as a function of the baseline $L$ and
phase $\delta$, with the other oscillation
parameters corresponding to the LSND + Atmospheric scenario IB1.
Predictions for $\nu_e \rightarrow \nu_\tau$ and
$\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$
are shown separately, as labeled.
Result are from Ref.~\ref{bgrw_prep}.
}
\label{fig:t3}
\end{figure}
Now consider the $\tau$ appearance rates in scenario IC1.
In this case the rates are not sensitive to $\delta$ and,
for a 1~kt detector at $L = 60$~km, there are about 8000
$\nu_e \rightarrow \nu_\tau$ events and
93000 $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ events.
A detector with $3\sigma$ (or better) $\tau$-lepton charge discrimination
would enable these two rates to be separately measured.
We conclude that measurements of the $\nu_e \rightarrow \nu_\tau$ oscillation
rate at a neutrino factory would provide an important test of the
oscillation scenario. In LAM, SAM, and LOW three-flavor oscillation
scenarios, a 20~GeV neutrino factory providing O($10^{20}$) muon decays
could permit an observation of, or meaningful limits on,
$\nu_e \rightarrow \nu_\tau$ oscillations. In LSND-type scenarios where the
leading $\delta m^2$ scale is large, a 20~GeV neutrino factory providing
O($10^{19}$) muon decays might already permit hundreds of
$\nu_e \rightarrow \nu_\tau$ events to be measured.
It should be noted that the feasibility of a multi-kt detector with
good $\tau$ identification and charge sign determination has not been
explored in detail at this stage, and further work is required to
identify the best detector technology for this, and determine the
expected resolutions and efficiencies.
%\clearpage
\subsubsection{Measurement of $\nu_\mu \rightarrow \nu_\tau$ oscillations}
The present SuperK data suggests that the atmospheric neutrino deficit
is due to $\nu_\mu \rightarrow \nu_\tau$ oscillations. If this is
correct the next generation of accelerator based long baseline experiments
are expected to measure these oscillations. Hence
$\nu_\mu \rightarrow \nu_\tau$ appearance measurements at a neutrino
factory are likely to be less interesting than $\nu_e \rightarrow \nu_\mu$
or $\nu_e \rightarrow \nu_\tau$ appearance measurements. Nevertheless,
for a fixed neutrino factory energy and baseline, it is of interest
to measure or put stringent constraints on all of the appearance channels
so that the sum of the appearance modes can be compared with
the disappearance measurements. Hence, we
briefly consider $\nu_\mu \rightarrow \nu_\tau$ rates
at a neutrino factory. Note that at a 20~GeV neutrino factory the
average interacting neutrino energy is of order 15~GeV, and for
$\delta m^2$ within the favored SuperK range, the first oscillation
maximum occurs at baselines of $7000 \pm 3000$~km. At shorter baselines
the oscillation probabilities are lower and hence the signal/background
ratio is lower, although the signal rate can be higher.
Consider first a 5~kt detector 2800~km downstream of a 20~GeV
neutrino factory in which there are $10^{20}$ muon decays.
The $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ event
rates are shown in Fig.~\ref{fig:t1} as a function of the
$\sin^2 2\theta_{13}$ and $\delta m^2_{32}$, with the other oscillation
parameters corresponding to the LAM scenario IA1. If negative muons
are stored in the neutrino factory, the resulting
$\nu_\mu \rightarrow \nu_\tau$ event rates would be about a factor of two
higher than the $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rates
shown in the figure. A neutrino factory providing
O($10^{20}$) muon decays would enable $\nu_\mu \rightarrow \nu_\tau$
appearance
data samples of a few hundred to a few thousand events to be obtained.
Similar rates
are expected in SAM and LOW three-flavor mixing scenarios.
Next consider a longer baseline example in which a 10~kt ICANOE--type
detector is 7400~km downstream of a 30~GeV neutrino factory which
provides $10^{20}$ muon decays in the beam forming straight
section~\cite{camp00}. The main advantage of
a longer baseline is that the total interaction rate, and hence the
$\tau$--lepton background, is reduced.
The energy distribution for
events in which there is no charged lepton can directly reflect
the presence of a $\nu_\tau$ signal (see Fig.~\ref{fig:m1}).
The non--$\tau$ events in this event sample can be suppressed
using topology--dependent kinematic cuts. It is desirable that
the $\tau$ charge--sign also be determined
which, with an external muon spectrometer, will be
possible for the $\tau \rightarrow \mu$ subsample.
We conclude that the measurement of
$\nu_\mu \rightarrow \nu_\tau$ oscillations with high statistical
precision will be possible at a neutrino factory in long and
very long baseline experiments. A more complete study is warranted.
%\clearpage
\subsubsection{Determination of $\sin^2 2\theta_{13}$, $\sin^2 2\theta_{23}$,
and $\delta m^2_{32}$}
Consider first the determination of $\sin^2 2\theta_{13}$.
The most sensitive way to measure $\sin^2 2\theta_{13}$ at a
neutrino factory is to measure the
$\nu_e \rightarrow \nu_\mu$ oscillation amplitude, which is
approximately proportional to $\sin^2 2\theta_{13}$.
More explicitly, the value of $\sin^2 2\theta_{13}$ is extracted
from a fit to the spectrum of CC interactions tagged by a wrong--sign muon.
Background contributions from, for example, muonic decays of charged mesons
must be kept small, which favors small total event samples and hence
long baselines.
\begin{figure}[th]
\epsfxsize3.3in
\centerline{\epsffile{s2_contrib_to_ws_muons.eps}}
\caption{Visible energy spectrum for events tagged by wrong-sign muons
in an ICANOE--type detector (full histogram). The oscillation parameters
are $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$,
$\sin^2\theta_{23}=1$, and
$\sin^2 2\theta_{13}=0.05$. Also shown are the contributions from
$\nu_e\to\nu_\mu$ oscillations (black dashed curve),
$\nu_e\to\nu_\tau$, with a subsequent muonic decay of the $\tau$ lepton
(red curve), and background from
muonic decays of pions or kaons in neutral current or charged current
events (blue dot-dashed curve). Result are from Ref.~\ref{camp00}.
}
\label{fig:m3}
\end{figure}
%
As an example, consider a 10~kt ICANOE--like detector that is 7400~km
downstream of a 30~GeV neutrino factory~\cite{camp00}.
The simulated energy spectrum of wrong-sign muon events
is shown in Fig.~\ref{fig:m3} for three--flavor oscillations with
the parameters
$\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$,
$\sin^2 2\theta_{23}=1$, and
$\sin^2 2\theta_{13}=0.05$.
Note that the backgrounds predominantly contribute to the low energy
part of the spectrum.
To fit the observed spectrum and extract $\sin^2 2\theta_{13}$
matter effects must be taken into account. The modification of the
oscillation probability due to matter effects is a function of the
profile of the matter density $\rho$ between the neutrino source and
the detector. The density profile is known from geophysical measurements,
and this knowledge can either be used in the fit, or alternatively $\rho$
can be left as a free parameter. It has been shown that both methods
give consistent results~\cite{camp00}, and that the uncertainties
on the fitted values of $\rho$ and $\sin^2 2\theta_{13}$ are not
strongly correlated. However, the fitted value for
$\sin^2 2\theta_{13}$ does depend on the assumed values for
$\sin^2 \theta_{23}$ and $\delta m^2_{32}$.
The measured right--sign muon ($\nu_\mu$ disappearance) distribution,
together with the distributions of events tagged by electrons,
$\tau$--leptons, or the absence of a lepton,
can be used to constrain these additional oscillation parameters.
Hence, the best way to extract $\sin^2 2\theta_{13}$ is from a global
fit to all of the observed event distributions, with the
oscillation parameters (and optionally $\rho$) left as free parameters.
If the density profile is left as a free
parameter, the fit determines its value with an uncertainty of about
10\%~\cite{camp00,cerv00}. This provides a quantitative test of the
MSW effect ! Examples of fit results in the
($\sin^2 2\theta_{13}$,~$\sin^2 \theta_{23}$)--plane are shown in
Fig.~\ref{fig:m7} for $10^{19}$, $10^{20}$, and $10^{21}$ muon decays
in the neutrino factory. As the beam intensity increases the measurements
become more precise. With $10^{19} \mu^+$ and $\mu^-$ decays
$\sin^2 2\theta_{13}$ and $\sin^2 2\theta_{23}$ are determined with
precisions of 40\% and 20\% respectively. With $10^{21}$ decays
these precisions have improved to $\sim5$\%.
If the baseline is decreased from 7400~km to 2900~km the oscillation
parameters are determined with comparable (although slightly worse)
precisions (Fig.~\ref{fig:m7a}).
We conclude that within the framework of three--flavor mixing,
provided $\sin^2 2\theta_{13}$ is not too small,
a global fit to the observed oscillation distributions would enable
$\sin^2 2\theta_{13}$, $\sin^2 \theta_{23}$, and $\delta m^2_{32}$
to be simultaneously determined, and the MSW effect to be measured.
\begin{figure}[t]
\epsfxsize3.3in
\centerline{\epsffile{contgeers.eps}}
\caption{Results from a global fit to the visible energy distributions for
various event classes recorded in a 10~kt ICANOE--type detector
7400~km downstream of a 30~GeV neutrino factory.
The 68\% CL contours correspond to
experiments in which there are $10^{19}$, $10^{20}$, and
$10^{21} \mu^+$ decays in the neutrino factory (as labeled)
followed by the same number of $\mu^-$ decays.
Upper plot: density fixed to its true value. Lower plot: density is a free
parameter of the fit. Result are from Ref.~\ref{camp00}.
}
\label{fig:m7}
\end{figure}
\begin{figure}[t]
\epsfxsize3.3in
\centerline{\epsffile{contgeers2.eps}}
\caption{Results from a global fit to the visible energy distributions for
various event classes recorded in a 10~kt ICANOE--type detector
downstream of a 30~GeV neutrino factory in which there are
$10^{20} \mu^+$ decays in the neutrino factory
followed by the same number of $\mu^-$ decays.
The 68\% CL contours correspond to baselines of 7400~km and 2900~km,
as labeled. Result are from Ref.~\ref{camp00}.
}
\label{fig:m7a}
\end{figure}
\begin{figure}
\epsfxsize3.3in
\centerline{\epsffile{s15.ps}}
\caption{Fit results in the (density parameter, $\theta_{13}$)--plane
for a simulated experiment in which a 40~kt
Fe-scintillator detector is a distance $L$~km downstream
of a 50~GeV neutrino factory in which there are
$10^{21} \mu$ decays. The curves are 68.5, 90, and 99\% CL contours.
Result are from Ref.~\ref{cerv00}.}
\label{fig:s15}
\end{figure}
\begin{figure}
\epsfxsize3.3in
\centerline{\epsffile{s2_l7400_exclu_t13.eps}}
\caption{Allowed regions in oscillation parameter space
calculated for a simulated experiment in which $10^{20} \mu^+$ followed
by $10^{20} \mu^-$ decay in a 30~GeV neutrino factory that is 7400~km
from a 10~kt ICANOE--type detector.
This result is two orders of
magnitude better than the analogous for ICANOE at the CNGS.
Result are from Ref.~\ref{camp00}.}
\label{fig:m4}
\end{figure}
\begin{figure}
\epsfxsize3.3in
\centerline{\epsffile{s13.ps}}
\caption{Allowed regions in oscillation parameter space
calculated for a simulated experiment in which a 40~kt
Fe-scintillator detector is a distance $L$~km downstream
of a 50~GeV neutrino factory in which there are
$10^{21} \mu$ decays. The curves are 90\% CL contours for
$L = 732$~km (dashed), 3500~km (solid), and 7332~km (dotted).
Result are from Ref.~\ref{cerv00}.}
\label{fig:s13}
\end{figure}
\begin{figure}
\epsfxsize3.0in
\centerline{\epsffile{mu3-30gev_2800km_1234.eps}}
\caption{Visible energy distributions for events tagged by a right--sign muon
in a MINOS--type detector 2800~km downstream of a 20~GeV neutrino factory
in which there are $2 \times 10^{20} \mu^-$ decays. Predicted distributions
are shown for four values of $\delta m^2_{32}$, with the other parameters
corresponding to the LAM scenario IA1. For each panel, the points with
statistical error bars show an example of a simulated experiment. The
light shaded histograms show the predicted distributions in the
absence of oscillations. Results are from Ref.~\ref{bgrw00}.
}
\label{fig:v5}
\end{figure}
\begin{figure}
\epsfxsize2.8in
\centerline{\epsffile{30gev_disap_fit.eps}}
\caption{Fit results for simulated $\nu_\mu$ disappearance
measurements with a 10~kt MINOS-type
detector 2800~km downstream of a 30~GeV neutrino factory in which
there are $2 \times 10^{20} \mu^-$ decays.
For each trial point the $1\sigma$, $2\sigma$,
and $3\sigma$ contours are shown for a perfect detector
(no backgrounds) and no systematic uncertainty on the beam flux.
The 68\%, 90\% and 95\% SuperK regions are indicated.
Result are from Ref.~\ref{bgrw00}.
}
\label{fig:v6}
\end{figure}
\begin{figure}
\epsfxsize2.8in
\centerline{\epsffile{newbob.eps}}
\caption{Fit results (1~$\sigma$ contours)
for (a) simulated $\nu_\mu$ disappearance
measurements with a 10~kt MINOS-type
detector 2800~km downstream of a 30~GeV neutrino factory in which
there are $2 \times 10^{20} \mu^-$ decays, and
(b) wrong--sign muon appearance measurements.
A 2\% systematic uncertainty on the
flux are included.
The acceptance for a muon is
zero for $p_{\mu} < 4$ GeV and unity for $p_{\mu} \geq 4$ GeV. Backgrounds
are included but no $p_{\perp}^2$ cut has been used.
Result are from Ref.~\ref{bern00}.
}
\label{fig:b1}
\end{figure}
To illustrate the ultimate sensitivity to the oscillation parameters
that might be achievable at a high intensity neutrino factory,
consider next a 40~kt Fe-scintillator
detector downstream of a 50~GeV
neutrino factory in which there are $10^{21} \mu^+$ decays followed
by $10^{21} \mu^-$ decays~\cite{cerv00}. Fit results in the
(matter density, $\sin^2 2\theta_{13}$)--plane
are shown in Fig.~\ref{fig:s15} for three baselines.
The precision on the $\sin^2 2\theta_{13}$ determination is a few percent.
At the shortest baselines (732~km) matter effects are too
small to obtain a good determination of the matter density parameter.
Consider next the precision with which the oscillation parameters
can be determined if $\sin^2 2\theta_{13}$ is very small, and hence
no $\nu_e \rightarrow \nu_\mu$ oscillation signal is observed.
The resulting limits on $\sin^2 2\theta_{13}$ are shown as a
function of $\delta m^2_{32}$ in Fig.~\ref{fig:m4}
for a 10~kt ICANOE type detector 7400~km downstream of a 30~GeV
neutrino factory in which there are $10^{20} \mu^+$ decays followed
by $10^{20} \mu^-$ decays~\cite{camp00}.
The resulting upper limit on $\sin^2 2\theta_{13}$
would be O($10^{-3}-10^{-4}$),
about three orders
of magnitude below the present experimental bound, and
2 orders of magnitude below the bound that would be expected at
the next generation of long--baseline experiments.
The limit would become even more stringent at a higher intensity
neutrino factory. As an example of the ultimate sensitivity that
might be achievable, in Fig.~\ref{fig:s13}
the limits on $\sin^2 2\theta_{13}$ are shown as a
function of $\delta m^2_{32}$ and baseline for a 40~kt Fe-scintillator
detector downstream of a 50~GeV
neutrino factory in which there are $10^{21} \mu$ decays~\cite{cerv00}.
The non--observation of $\nu_e \rightarrow \nu_\mu$ oscillations could
result in an upper limit on $\sin^2 2\theta_{13}$ below $10^{-5}$ !
With this level of sensitivity $\nu_e \rightarrow \nu_\mu$ oscillations
driven by the sub--leading $\delta m^2$ scale might be observed~\cite{bgrw00}.
The number of muon decays required to produce 10
$\nu_e \rightarrow \nu_\mu$ events in a 50~kt detector 2800~km
downstream of a neutrino factory is shown for a bimaximal mixing
scenario ($\sin^22\theta_{13} = 0$)
in Fig.~\ref{fig:v2} as a function of the stored muon energy.
Approaching $10^{21}$ muon decays might be sufficient to observe
oscillations driven by the sub--leading scale.
With a vanishing or very small $\sin^2 2\theta_{13}$
only the $\nu_\mu\to\nu_\tau$
oscillations will have a significant rate, and the oscillation
parameters $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$ can be
determined by fitting the right--sign muon ($\nu_\mu$ disappearance)
spectrum. Good sensitivity can be obtained provided the baseline
is chosen such that the first oscillation maximum occurs in the
middle of the visible energy spectrum.
As a first example,
spectra of events tagged by right--sign muons are shown in
Fig.~\ref{fig:v5} as a function of $\delta m^2_{32}$ for a 10~kt MINOS--type
detector 2800~km downstream of a 30~GeV neutrino factory in which
there are $2 \times 10^{20} \mu^-$ decays in the beam--forming straight
section~\cite{bgrw00}.
The position of the oscillation maximum (resulting in a dip in
the observed distributions) is clearly sensitive to $\delta m^2_{32}$.
The depth of the observed dip is sensitive to the oscillation amplitude,
and hence to $\sin^2 2\theta_{23}$.
The visible energy spectrum of right--sign muon events can be fit
to obtain $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$.
We begin by considering the statistical precision that could be
obtained with a perfect detector having MINOS--type resolution functions,
no backgrounds, and no systematic uncertainty on the neutrino flux.
Fit results are shown in Fig.~\ref{fig:v6}.
For $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$
the fit yields a statistical
precisions of a few percent on the the values of the oscillation parameters.
If $L$ is increased to 7332~km, the statistical precision improves to
about 1\%.
With this level of statistical precision it is likely that
systematic uncertainties will be significant~\cite{bern00}.
To illustrate this in Fig.~\ref{fig:b1} the $1\sigma$ contours are shown
in the ($\delta m^2_{32}$, $\sin^2 2\theta_{23}$) from fits which
include backgrounds together with 1\% and 2\% systematic uncertainties
on the beam flux. With a 1\% flux uncertainty the precision on
$\delta m^2_{32}$ and $\sin^2 2\theta_{23}$ are respectively
x\% and y\%.
\begin{figure}[t]
\epsfxsize3.5in
\centerline{\epsffile{s1_syst_err_732.eps}}
\caption{Fit results for simulated $\nu_\mu$ disappearance
measurements with a 10~kt ICANOE type
detector 732~km downstream of a 30~GeV neutrino factory in which
there are $10^{20} \mu$ decays. The effect
of a systematic uncertainty on the neutrino flux is shown.
Result are from Ref.~\ref{camp00}.
}
\label{fig:m8}
\end{figure}
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{s1_comp_fit_mix23_diff_munorm.eps}}
\caption{Fit results for simulated $\nu_\mu$ disappearance
measurements with a 10~kt ICANOE type
detector 2900~km (top plot) and 7400~km (bottom plot)
downstream of a 30~GeV neutrino factory in which
there are (a) $10^{20} \mu$ decays and (b) $10^{21} \mu$ decays.
Result are from Ref.~\ref{camp00}.
}
\label{fig:m9}
\end{figure}
\begin{figure}[t]
\epsfxsize3.5in
\centerline{\epsffile{s1_diff_dm_3500.eps}}
\caption{Fit results for simulated $\nu_\mu$ disappearance
measurements with a 10~kt ICANOE type
detector 2900~km
downstream of a 30~GeV neutrino factory in which
there are $10^{20} \mu^+$ decays followed by $10^{20} \mu^+$ decays.
Result are shown for 3 values of $\delta m^2_{32}$, and are from
Ref.~\ref{camp00}.
}
\label{fig:m10}
\end{figure}
\begin{figure}
\epsfxsize3.5in
\centerline{\epsffile{s1_diff_dm_7344.eps}}
\caption{Fit results for simulated $\nu_\mu$ disappearance
measurements with a 10~kt ICANOE type
detector 7400~km
downstream of a 30~GeV neutrino factory in which
there are $10^{20} \mu^+$ decays followed by $10^{20} \mu^+$ decays.
Result are shown for 3 values of $\delta m^2_{32}$, and are from
Ref.~\ref{camp00}.
}
\label{fig:m11}
\end{figure}
As a second example, consider a 10~kt ICANOE--type detector
that is downstream of a 30~GeV neutrino factory in which there are
$10^{20} \mu^+$ decays in the beam--forming straight section
followed by $10^{20} \mu^-$ decays~\cite{camp00}. The sensitivity to the
oscillation parameters has been studied by fitting simulated
visible energy distributions for events tagged by a right--sign muon.
The analysis includes a 2\% bin-to-bin uncorrelated systematic error on
the number of neutrino interactions which takes into account
the uncertainties on neutrino flux, the cross section,
and the selection efficiency. To reduce background from charged meson decays,
the events entering the fit are those with muons having momenta $>2$~GeV.
Figures~\ref{fig:m8}-\ref{fig:m11} show fit results in the
($\sin^2 2\theta_{23}$,~$\delta m^2_{32}$)--plane as a function
of the oscillation parameters and baseline.
Note that for the ``short'' baseline ($L =732$~km)
the first oscillation maximum for the reference value of
$\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$
occurs at a neutrino energy of about 2~GeV.
This is too low to produce a clear dip in the visible energy spectrum,
and as a result $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$ can only
be determined with relatively low precision (Fig.~\ref{fig:m8}), and
the fit results are sensitive to systematic uncertainties on the
neutrino flux.
At the longer baselines ($L = 2900$~km and 7400~km)
the oscillation dip is visible, and the oscillation
parameters can be measured with a precision that is mostly determined
by the statistical uncertainty (Fig.~\ref{fig:m9}).
For a 30~GeV neutrino factory and
$\delta m^2_{32} = 3.5 \times 10^{-3}$~eV$^2$/c$^4$
the longer baseline (7400~km) yields the most precise result.
Specifically, for $10^{20} \mu$ decays
the statistical precisions on $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$
are respectively about 10\% and 1\%.
With $10^{21} \mu$ decays the $\sin^2 2\theta_{23}$
precision improves by about a factor of 2.
It should be noted that the best
baseline choice depends on $\delta m^2_{32}$
(Figs.~\ref{fig:m10}-\ref{fig:m11}),
or more specifically $\delta m^2_{32}/E$.
We conclude that, within the framework of three--flavor mixing,
the oscillation parameters $\sin^2 2\theta_{13}$, $\sin^2 2\theta_{23}$,
and $\delta m^2_{32}$ can be determined at a neutrino factory
by fitting the observed visible energy distributions for various
event types. A comprehensive study of the expected precisions of
the measurements as a function of the oscillation parameters, baseline,
and neutrino factory parameters has not yet been undertaken.
However, detailed studies have been made for some examples in
which there are $10^{20} \mu^+$ decays
followed by $10^{20} \mu^-$ decays in a 30~GeV neutrino factory.
For these examples
we find that (i) if $\sin^2 2\theta_{13} >$~O($10^{-2}$) global fits
can be used to determine its value, (ii) if $\sin^2 2\theta_{13}$
is too small to observe $\nu_e \rightarrow \nu_\mu$ oscillations
then we would expect to place the very stringent upper limit on its
value of $10^{-3}$ or better, and (iii) the values of
$\sin^2 2\theta_{23}$, and $\delta m^2_{32}$ could be determined
with precisions of respectively better than or of order 10\% and
of order 1\%, provided the baseline is chosen so that the dip corresponding
to the first oscillation maximum is in the middle of
the visible energy distribution.
At a high--intensity neutrino factory (for example with $10^{21}$ decays
of 50~GeV muons) the mixing angles could be measured with a precision
of a few percent, and
if $\sin^2 2\theta_{13}$ is vanishingly small, the
resulting upper limit could be at the O($10^{-5}$)--level.
%\clearpage
\subsubsection{Search for CP violation}
\begin{figure}
\epsfxsize2.8in
\centerline{\epsffile{kerry.eps}}
\caption{The ratio $R$ of
$\bar\nu_e \to \bar\nu_\mu$ to $\nu_e \to \nu_\mu$ event
rates at a 20~GeV neutrino factory
for $\delta = 0$ and $\pm\pi/2$. The upper group of curves
is for $\delta m^2_{32} < 0$, the lower group is for
$\delta m^2_{32} > 0$, and the statistical errors correspond to
$10^{21}$ muon decays of each sign and a 50~kt detector.
The oscillation parameters correspond to the LAM scenario IA1.
Result are from Ref.~\ref{bgrw00}.
}
\label{fig:cp1}
\end{figure}
\begin{figure}
\epsfxsize3.2in
\centerline{\epsffile{cp_sensitivity_6c.ps}}
\caption{Reach in $\sin^22\theta_{13}$ that yields a $3\sigma$
discrimination between
(a) $\delta = 0$ and $\pi/2$ with $\delta m^2_{32} > 0$,
(b) $\delta = 0$ and $\pi/2$ with $\delta m^2_{32} < 0$,
(c) $\delta = 0$ and $-\pi/2$ with $\delta m^2_{32} > 0$, and
(d) $\delta = 0$ and $-\pi/2$ with $\delta m^2_{32} < 0$.
The discrimination is based on a comparison of wrong--sign
muon CC event rates in a 50~kt detector when $10^{21}$ positive and
negative muons alternately decay in the neutrino factory.
The reach is shown versus baseline for four storage ring energies.
The oscillation parameters correspond to the LAM scenario IA1.
Result are from Ref.~\ref{bgrw00}.
}
\label{fig:cp2}
\end{figure}
\begin{figure}
\epsfxsize3.2in
\centerline{\epsffile{s18.ps}}
\caption{Fit results in the CP phase $\delta$ versus $\theta_{13}$ plane
for a LMA scenario with $\delta m^2_{21} = 1 \times 10^{-4}$~eV$^2$/c$^4$.
The 68.5, 90, and 99\% CL contours are shown for a 40~kt detector a distance
$L$~km
downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu^+$
and $10^{21} \mu^-$ decays.
Result are from Ref.~\ref{cerv00}.
}
\label{fig:cp3}
\end{figure}
\begin{figure}
\epsfxsize3.2in
\centerline{\epsffile{s22.ps}}
\caption{The lowest value of $\delta m^2_{21}$, shown as a function of
$\theta_{13}$, for
which the maximal CP phase $\delta = \pi/2$ can be distinguished
from a vanishing phase in a LMA oscillation scenario.
The curve corresponds to a 40~kt detector 3500~km
downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu^+$
and $10^{21} \mu^-$ decays.
Result are from Ref.~\ref{cerv00}.
}
\label{fig:cp4}
\end{figure}
In the majority of the three--flavor oscillation scenarios described in
section A the CP violating amplitude is expected to be too small to
produce an observable effect. Nevertheless, in these cases stringent limits
on CP violation would provide an important check of the overall interpretation
of the oscillation data. If however the LAM scenario provides the
correct description of neutrino oscillations, CP violating effects
might be sufficiently large to be observable at a high--intensity
neutrino factory~\cite{cerv00,bgrw00}.
This is illustrated in Fig.~\ref{fig:cp1} which shows,
as a function of baseline at a 20~GeV neutrino factory, the ratio $R$
for $\delta = 0$ and $\pm\pi/2$, where $R$ is defined as the
$\overline{\nu}_e \rightarrow \overline{\nu}_\mu$ event rate divided by
the $\nu_e \rightarrow \nu_\mu$ event rate.
The upper group of curves
is for $\delta m^2_{32} < 0$, the lower group is for
$\delta m^2_{32} > 0$, and the statistical errors correspond to
$10^{21}$ muon decays of each sign with a 50~kt detector.
If $L$ is a few thousand~km a non--zero $\delta$ can produce a
modification to $R$ that is sufficiently large to
be measured !
Since the $\nu_e \rightarrow \nu_\mu$ oscillation rates
are to a good approximation proportional to $\sin^22\theta_{13}$, it
is useful to define the $\sin^22\theta_{13}$ reach as that value of
$\sin^22\theta_{13}$ that will produce a 3$\sigma$
change in the predicted ratio $R$ when
$\delta$ is changed from $0$ to $\pm\pi/2$.
The $\sin^22\theta_{13}$ reach is shown as a function of baseline and
stored muon energy in Fig.~\ref{fig:cp2} for a 50~kt detector
at a neutrino factory in which there are $10^{21} \mu^+$ decays followed by
$10^{21} \mu^-$ decays. With an optimum baseline of about
3000~km (for $\delta m^2_{32} = 3.5 \times 10^{-3}$~eV$^2$/c$^4$)
the $\sin^22\theta_{13}$ reach is approximately $10^{-2}$, an order
of magnitude below the current experimental bound.
Thus, in a LAM scenario, CP violation in the lepton sector might
be measurable at a neutrino factory providing O($10^{21}$) muon decays.
As an example, consider a 40~kt Fe-scintillator detector downstream
of a 50~GeV neutrino factory providing $10^{21} \mu^+$ decays followed
by $10^{21} \mu^-$ decays in the beam--forming straight section~\cite{cerv00}.
The results of fits to the simulated wrong--sign muon event distributions,
with $\delta$ and $\sin^22\theta_{13}$ left as free parameters, are
shown in Fig.~\ref{fig:cp3} for various baselines, with the
sub--leading scale $\delta m^2_{21} = 1 \times 10^{-4}$~eV$^2$/c$^4$.
The analysis includes
the detector resolutions, reasonable event selection criteria, and
backgrounds. As might be expected from Fig.~\ref{fig:cp1} at $L = 7332$~km
there is little sensitivity to $\delta$, and at the ``short" baseline
$L = 732$~km the fit has difficulty untangling $\delta$ from
$\sin^22\theta_{13}$. However, at a baseline of $L = 3500$~km for the
example shown $\delta$ and $\sin^22\theta_{13}$ can be determined with
precisions of respectively about $15^\circ$ and a few percent. Note
that a combination of baselines can yield a modest improvement in
the precision of the measurement.
The sensitivity to CP violation decreases with decreasing
$\delta m^2_{21}$. Figure~\ref{fig:cp4} shows as a function of
$\sin^22\theta_{13}$ the lowest value of $\delta m^2_{21}$ for
which the maximal CP phase $\delta = \pi/2$ can be distinguished
from a vanishing phase at $L = 3500$~km. This limiting $\delta m^2_{21}$
is below the current central value for the LAM parameter space suggested
by solar neutrino deficit, and is about $2 \times 10^{-5}$~eV$^2$/c$^4$,
independent of $\sin^22\theta_{13}$.
%\clearpage
\subsection{Summary}
The oscillation physics that could be pursued at a neutrino factory
appears to be compelling. In particular, experiments at a neutrino factory
would be able to simultaneously measure, or put stringent limits on,
all of the appearance modes $\nu_e \rightarrow \nu_\tau$,
$\nu_e \rightarrow \nu_\mu$, and $\nu_\mu \rightarrow \nu_\tau$.
Comparing the sum of the appearance modes with the disappearance
measurements would provide a unique basic check of the candidate
oscillation scenario that cannot be made with a conventional neutrino
beam.
In addition, for all of the specific oscillation
scenarios we have studied, the
$\nu_e$ component in the beam can be exploited to enable
crucial physics questions to be addressed. These include
(i) the pattern of neutrino masses (sign of $\delta m^2$) and
a quantitative test of the MSW effect,
(ii) the precise determination of (or stringent limits on) all of the
leading oscillation parameters, which in a three--flavor mixing
scenario would be $\sin^22\theta_{13}$, $\sin^22\theta_{23}$,
and $\delta m^2_{32}$, and
(iii) the observation of, or stringent limits on, CP violation in
the lepton sector, and a corresponding measurement of the phase $\delta$.
To be more quantitative in assessing the beam energy, intensity, and
baseline required to accomplish a given set of physics goals it is
necessary to consider two very different experimental possibilities:
(a) the LSND oscillation results are confirmed by the MiniBooNE
experiment, or (b) they are not confirmed.
\begin{description}
\item{(a) LSND not confirmed.}
Fairly extensive neutrino factory studies have been made within
the framework of three--flavor oscillation scenarios in which there is
one ``large"
$\delta m^2$ scale identified with the atmospheric neutrino deficit
results, and one small $\delta m^2$ identified with the solar neutrino
deficit results. A summary of the energy dependent beam intensities
required to cross a variety of ``thresholds of interest" is provided by
Fig.~\ref{fig:v2}. A 20~GeV neutrino factory providing $10^{19}$ muon
decays per year is a good candidate ``entry--level" facility which would
enable either (i) the first observation of $\nu_e \rightarrow \nu_\mu$
oscillations, the first direct measurement of matter effects,
and a determination of the sign of $\delta m^2_{32}$ and
hence the pattern of neutrino masses, or (ii) a very stringent limit
on $\sin^22\theta_{13}$ and a first comparison of the sum of all
appearance modes with the disappearance measurements.
The optimum baselines for this entry--level
physics program appears to be of the order of 3000~km or greater,
for which matter effects are substantial.
A 20~GeV neutrino factory providing $10^{20}$ muon
decays per year is a good candidate upgraded neutrino factory (or
alternatively a higher energy facility providing a few $\times 10^{19}$ decays
per year). This would enable the first observation of, or meaningful
limits on, $\nu_e \rightarrow \nu_\tau$ oscillations, and precision
measurements of the leading oscillation parameters. In the more distant
future, a candidate for a second (third ?) generation neutrino factory might
be a facility that provides O($10^{21}$) decays per year and enables the
measurement of, or stringent limits on, CP violation in the lepton sector.
\item{(b) LSND confirmed.}
Less extensive studies have been made for the class of scenarios that
become of interest if the LSND oscillation results are confirmed.
However, in the scenarios we have looked at (IB1 and IC1) we find that
the $\nu_e \rightarrow \nu_\tau$ rate is sensitive to the
oscillation parameters
and can be substantial. With a large leading $\delta m^2$ scale medium
baselines (for example a few $\times 10$~km) are of interest, and
the neutrino factory intensity required to effectively exploit the
$\nu_e$ beam component might be quite modest ($< 10^{19}$ decays per
year).
\end{description}
The neutrino factory oscillation physics study we have pursued
goes beyond previous studies. In particular we have explored the
physics capabilities as a function of the muon beam energy and
intensity, and the baseline. Based on the representative
oscillation scenarios and parameter sets defined for the study,
it would appear that a 20~GeV neutrino factory providing
O($10^{19}$) decays per year would be a viable entry--level
facility for experiments at baselines of $\sim3000$~km or greater.
There are still some basic open questions that deserve further
study:
(1) We have sampled, but not fully explored, the beam
energy and intensity required to explore the scenarios that become
relevant if the LSND oscillation results are confirmed.,
(2) Possible routes to a very massive neutrino factory detector
have been considered, but these considerations deserve to be
pursued further. The chosen detector technology will determine
the need to go deep underground.
(3) We have developed tools that can explore the utility of having
polarized muon beams. The physics payoff with polarization is a
detailed issue. It deserves to be studied in the coming months.
Based on our study, we believe that a neutrino factory in 5--10~years
from now would be the right tool for oscillation physics
at the right time.
\newcommand{\ignore}[1]{}
\section*{Introduction}
{\it version from Eric Hawker Mar 14, 2000}\\
{\it small mods by Mayda Velasco Mar 15, 2000}\\
{\it rewrite of DIS sections, Heidi Schellman, Mar 16, 2000}\\
{\it rewrite the introduction, structure fuctions, ew and charm, Mar 21, 2000\\}
Due to the theoretically clean nature of weak interactions, conventional
neutrino scattering experiments have always provided very precise
measurements of fundamental parameters. These include
measurements of the Weinberg angle and the strong coupling constant
$\alpha_s$ which are competitive with any other methods. Perhaps because
of this success, we forget how crude existing neutrino experiments are.
The high statistics experiments such as CDHS and CCFR/NuTeV, in order
to obtain statistical samples of more that 10$^5$ events, rely on
coarsely segmented massive iron/scintillator calorimeters weighing
close to 1000 tons. Measurements on proton targets and detailed studies
of the final state have been confined to very low statistics bubble chamber
and electronic detectors. As a result we have virtually no precise measurements
of neutrino-proton scattering and no measurements on polarized targets
which offer totally new insights into the spin structure of the nucleon.
The advent of a high flux neutrino factory, with neutrino fluxes of
10$^{20}$/year instead of the 10$^{15-16}$ at existing facilities would
open a new era in conventional neutrino physics. We would be able
to use low mass targets and high resolution detection technologies and
still achieve better statistical power than present-day experiments.
For example a 50 GeV muon storage ring would produce around 400 M events
neutrino charged current interactions per year in a cubic meter of
water,
5-10 times the statistics of the CCFR/NuTeV experiments with 600 ton detectors
Better understanding of neutrino fluxes from the decay of monochromatic
muons will also reduce many of the dominant systematic errors.
In this study we have concentrated on new measurements only possible
with higher fluxes rather than repeating older measurements with 1000
times the statistics. As a result, the statistical errors shown are
often not negligible, but without the high flux at a neutrino factor the
measurements themselves would
be impossible.
\subsection*{Outline}
Due to the breadth of the field we were unable to give a complete
survey and have instead highlighted a few
of the areas where the high flux beam at a neutrino factory allows
new measurements.
\begin{itemize}
\item A description of a low mass target/detector and typical rates in such a detector.
\item Conventional deep inelastic scattering measurements and a proposed
detector design.
\item Measurements of neutrino cross sections on nuclear targets, a topic
of great interest to the nuclear physics community and also needed to
understand normalization at a far neutrino oscillation detector.
\item Spin structure functions, which have never been measured in neutrino
beams.
\item The potential of the neutrino factory as a clean source
of single tagged charm mesons and baryons.
\item Electroweak measurements in both the hadronic and purely leptonic sectors.
\item Use of the very clean initial state to search for exotic interactions
\item Searches for anomalous neutrino interactions with electromagnetic fields.
\end{itemize}
%\subsection*{Structure Function Measurements}
\subsection*{Possible detector configurations and statistics}
For studies of charged current deep-inelastic scattering on proton
targets, the optimal detector system is probably a target followed by
precision magnetic tracking sytems, an electromagnetic calorimeter
and a muon detection system. Such detectors have been used in
muon scattering experiments at CERN and FNAL and in the new generation
neutrino scattering experiments CHORUS and NOMAD. A low mass target
followed by tracking and electromagnetic calorimetry makes the
electron anti-neutrinos in the beam a source of additional statistics
rather than a backround, except in the case of neutral current studies.
The numerical estimates in this study use, unless otherwise noted,
$10^20$ 50 GeV muon decays in a 600 m straight section.
The target itself should be thin enough that particles produced within
it have a small probability of interacting before they reach the tracking
systems.
In this study we considered liquid hydrogen and deuterium targets -
both polarized and unpolarized - and heavier solid nuclear targets.
The hydrogen and deuterium targets are 1m long while the polarized target
is 50 cm long. All targets
are 20 cm in radius, to fit the central beam spot at 50 GeV.
Nuclear targets are scaled so that the interaction length
in the material is constant at 14\%. The charge current muon neutrino
interaction rates are summarized in table \ref{rates}.
\begin{table}[b]
\caption{\label{rates} Charged current muon-neutrino scattering
rates in a small target located near a muon storage ring. Rates
are per $10^{20}$ muon decays. The detector is located ($E_{\mu}$, GeV)
meters away from the ring to assure that primary muons have ranged out
before the detector.}\begin{center}
\begin{tabular}{|c|c|r|r|}
\hline
Machine& Target & Thickness,cm & Events \\
\hline
50 GeV neutrino factory &Liquid H$_2$& 100 & 12.1M\\
&Liquid D$_2$& 100 & 29.0M\\
&solid HD & 50 &9.3M\\
&C&5.3&20.7M\\
&Si&6.3&25.4M\\
&Fe&2.3&31.6M\\
&Sn&3.1&39.1M\\
&W&1.3&44.3M\\
&Pb&2.4&46.5M\\
\hline
CCFR/NuTeV&Fe& 600& $\sim$ 2M\\
\end{tabular}\end{center}
\end{table}
%
%
\begin{figure}
%\epsfysize=2.5in
\epsfxsize=3.0in
\centerline{
\epsffile{diskin.eps}}
\caption{Kinematics of neutrino scattering in the parton model.
The energy-momentum tranfer from the leptons to the proton
is $\fv{q}$ and the fraction of the proton momentum carried
by the struck quark is approximately $x$.
\label{fig:comparejlab}}
\end{figure}
These are the total event rates for muon-neutrino scattering. The anti-neutrino
rates are half as large. Kinematic cuts reduce the statistics
by less than a factor of two.
We have only considered muon-neutrino charge current scattering for
structure function measurements, although for such thin targets, electron
neutrino scatters should also be reconstructable with high precision.
\subsection*{Neutrino Scattering Kinematics}
\newcommand{\Elep}[0]{E_{\lepton}}
The kinematic variables for neutrino deep inelastic scattering are
illustrated in figure \ref{diskin}:
\newcommand{\Enu}[0]{E_{\nu}}
\newcommand{\mlep}[0]{m_{\lepton}}
\begin{eqnarray}
\fourv{q} &=&{\fvk{\nu} - \fvk{\lepton}}, \hskip .4 in Q^2 = -\fourv{q}^2 = 2
\Elep\Enu -\mlep^2 - 2 \Enu p_{\lepton} \cos\theta_{lab},\\
\nu &=&(\fv{\quark}\fourv{q})/M \Elep - \Elep^{\prime},\\
x &=&Q^2/2 \mtarget \nu,\\
y &=& \mtarget \nu/(\fvk{\nu} \fv{\quark}) = (1 + cos\theta_{CM})/2 \approx
\nu/\Elep,\\
W^2 &=& 2 \mtarget \nu + \mtarget^2 -Q^2,
\end{eqnarray}
where the $\fvk{}$ are the neutrino and final state four vectors,
$\fv{p}$ is the proton four-vector, $M$ is the target nucleon mass, $\Enu$ is the incoming neutrino
energy $\Elep, p_{\lepton}$ are the outgoing lepton energy and momentum
$\theta_{lab}$ is
the lepton angle with respect to the incoming beam. $\fourv{q}$ is the
four-momentum transfer to the target, $\nu$
is the energy transfer, $x$ is the Bjorken $x$ variable, $y$ is the scaled
energy transfer and $W^2$ is the invariant mass of the final state hadronic
system
squared.
\begin{figure}
%\epsfysize=2.5in
\epsfxsize=5.0in
\centerline{
\epsffile{10_comparejlab.eps}}
\caption{Comparison of kinematic ranges for present DIS experiments
with a 50 GeV Neutrino factory. }
\label{fig:comparejlab}
\end{figure}
Fig. \ref{fig:comparejlab} show the kinematic region for a neutrino
factory as compared to other deep-inelastic scattering experiments.
%\end{document}
%\subsubsection{Unpolarized Neutrino Scattering}\label{DIScross}
% The kinematic variables are defined in section \ref{DIS}.
For $Q << E$ and $s << M_W$ the the unpolarized neutrino(anti-neutrino) scattering cross section
is:
\begin{eqnarray}
{d\sigma^{\nu(\antinu)}\over dx dy} =
{ G_F^2 M E_{\nu} \over 2\pi }\biggr[ [F_2(x,Q^2) \pm xF_3] + [F_2(x,Q^2)\mp xF_3(x,Q^2)] (1-y)^2 \\\nonumber
- 2 y^2 F_L(x,Q^2).
\end{eqnarray}
where the $F_i$ are
Structure Functions. $F_L = F_2 - 2xF_1 $ is a purely longitudinal
structure function.
$F_3$ changes sign for anti-neutrino scattering.
There are additional structure functions $F_4$ and $F_5$ which are
suppressed by factors of the lepton mass squared. For $\nutau$ and $\numu$
scattering at very low energies, these terms can become
quite important.
%\subsection{Polarized Scattering}
If the target is longitudinally polarized with respect to the
neutrino polarization, then the cross section difference:
\begin{eqnarray}
{d^2(\sigma_{\Rightarrow}^{\leftarrow}
-\sigma_{\Leftarrow}^{\leftarrow})^{\nu(\bar\nu) } \over dxdy} =
{G_F^2 M E_{\nu} \over \pi }
\bigl\{\pm
y(1-{y \over 2}-{xyM \over 2E})xg_1
\mp
{x^2yM \over E}g_2
+y^2x(1+{xM \over E})
g_3\\\nonumber
+(1-y-{xyM \over 2E})
[(1+{xM \over E})g_4+g_5]
\bigr\},
\label{pol_lon}
\end{eqnarray}
\noindent
is described by two parity conserving Polarized Structure
Functions $g_1$ and $g_2$, and by three parity
violating Polarized Structure Functions $g_3, g_4$ and $g_5$.
However, if the nucleon is transversely polarized:
\begin{eqnarray}
{d^2(\sigma_{\Uparrow}^{\leftarrow}
-\sigma_{\Downarrow}^{\leftarrow})^{\nu(\bar\nu) } \over dxdy} =
\frac{ G_F^2 M}{ 16 \pi^2 }
\sqrt{xyM \left[ 2(1-y)E-xyM \right] }
\bigr\{\mp2xy({y\over 2}g_1+g_2)\\\nonumber
+xy^2g_3
+ (1-y-{xyM \over 2E})g_4
-{y\over 2}g_5\bigr\} \/.
\label{pol_tra}
\end{eqnarray}
%{\it is there a factor of 2 missing for the subtraction here}
\noindent The transverse cross section is suppressed by
${M/Q}$ with respect to the longitudinal cross section.
\subsection{Structure function measurements at a neutrino factory}
In principle, the structure functions can be extracted by fits to the $y$ dependence of the
cross section. To date this
has proven very difficult as the data must be binned in $x$, $y$ and $Q^2$ and
no experiment has had sufficient statistics to perform such an analysis with high accuracy\cite{VallageThesis}.
Instead, high statistics experiments such as CHARM, CCFR, CDHSW have relied on massive targets (Iron, Calcium) which are
approximately iso-scalar and have combined neutrino and anti-neutrino information in
order to extract average structure functions.
The structure functions averages have leading-order parton model interpretations:
%
\begin{eqnarray}
\overline{F}_2^N(x,Q^2) &\simeq &\sum (x\quark(x,Q^2) + x\antiquark(x,Q^2)),\\
\overline{F}_3^N(x,Q^2) &\simeq &\sum(x\quark(x,Q^2) - x\antiquark(x,Q^2)),\\\nonumber
\end{eqnarray}
%
where $\overline{F}_2(x,Q^2)$ and $\overline{F}_3(x,Q^2) $ are the average of neutrino
and antineutrino structure functions measured on a target which is an average of neutron and proton and $\quark(x,Q^2)$ and $\antiquark(x,Q^2)$ represent the
parton distribution functions or total probability of finding a quark
or antiquark in the proton:
\begin{eqnarray}
\quark(x,Q^2) &=& \uquark(x,Q^2) + \dquark(x,Q^2)+\squark(x,Q^2)+\cquark(x,Q^2)...\\
\antiquark(x,Q^2) &=& \antiuquark(x,Q^2) + \antidquark(x,Q^2)+\antisquark(x,Q^2)+\anticquark(x,Q^2)...\\\nonumber
\end{eqnarray}
%\subsubsection{Measurements with high statistics}
Given the expectation of 12 M events/year in a 1 m hydrogen or deuterium target at a 50 GeV
muon collider we can do a complete analysis of each channel $\nu p, \nu n, \antinu p,
\antinu n$ without averaging. Such an analysis allows a unique extraction of individual quark flavor parton
distribution functions.
Consider $\nu$p scattering, a $W^{+}$ is exchanged and the reaction can only be
sensitive to negatively charged quarks. Due to the helicity dependence of the interaction
only left-handed $\dquark$ type and right handed $\antiuquark$ quarks will be involved.
The leading order parton model cross section is simply
\begin{eqnarray}
{d\sigma^{\nu p}\over dx dy} &\simeq& {4 G_F^2 M E_{\nu}\over \pi }x [(\dquark_L (x,Q^2) + \squark_L(x,Q^2)) +(\antiuquark_R(x,Q^2) +\\
&&\hskip 2 in \anticquark_R(x,Q^2))(1-y)^2],\\\nonumber
\label{partonformula}\end{eqnarray}
%
and the different contributions can be extracted from the $y$ dependence of this
cross section and the corresponding anti-neutrino cross section. The relative
$\squark$ and $\dquark$ contributions can be measured in charm production.
For an unpolarized target
$\quark_L(x) = \quark_R(x) = \onehalf \quark(x)$. For
a polarized quark $\quark_L(x) = \onehalf (\quark(x) + \delta\quark(x))$
and $\quark_R(x) = \onehalf(\quark(x) - \delta\quark(x))$ where
$\delta\quark(x)$ is the degree to which the quark spin
is aligned with the proton spin\footnote{ The traditional $\Delta q$ spin distributions from
electron and muon scattering measure the sum $\Delta \quark = \delta \quark + \delta \antiquark$ as
photon probes cannot tell quarks and anti-quarks apart.}.
Thus a $\sigma_{\nu p}$ measurement on an unpolarized target can determine $\dquark+\squark$ and $\antiuquark +
\anticquark$ by averaging over the proton spin, while by measuring the polarization
asymmetry one can measure $\delta \dquark+\delta\squark$ and $\delta\antiuquark +
\delta\anticquark$.
Scattering on neutrons can be related to scattering on protons by an isospin
transformation which exchanges $\uquark$ and $\dquark$ quarks and anti-quarks.
Differences of neutron and proton cross sections can then be used to cancel the
$\uquark$ and $\dquark$ components leaving observables
sensitive only to $\squark$ and $\cquark$
distributions.
\subsection{Perturbative QCD}
Neutrinos do not couple directly to gluons. As a result, QCD effects
appear in neutrino scattering as well-understood higher order corrections to
the leading order parton model. Measurements
of the $\qsq$ dependence of neutrino cross sections are one of the most
sensitive measurements of the strong coupling constant $\alpha_s$ and
some information on the gluon distribution can be obtained from its coupling
to the structure functions via the DGLAP evolution equations.
The neutrino structure functions can be divided into two types, singlet
and non-singlet, depending on their sensitivity to gluon effects in
their evolution.
The structure functions $2x F_1$, $F_2$ and $g_1$ are singlet functions
and are directly coupled to the gluon distribution via the evolution equations.
The structure functions $x F_3 + x F_3$, $2x g_3,g_4$ and $g_5$
averaged over neutrino and anti-neutrino are non-singlet
functions and their evolution is independent of the gluon distribution.
The combinations $F_2^p - F_2^n$ and $g_1^p - g_1^n$ also cancel the
gluon contributions and are thus non-singlet in nature.
To date, extractions of $\alpha_s$ from non-singlet distributions have
been statistics limited and strongly affected by flux uncertainties.
The additional factor of 10-100 in statistics
and improved flux understanding
at a neutrino factory should allow vastly improved measurements of strong
interaction parameters in this very clean channel.
Once the quark distributions and strong interaction effects have been thoroughly studied in the non-singlet structure function, that knowledge can be used for
improved constraints on the gluon distributions via the evolution of the
singlet structure functions.
%Jorge's section
\subsection{Nuclear Effects}
Experiments at a neutrino factory of nuclear effects in the distribution of partons within nuclei relative
to protons and deuterons are
of interest to both the nuclear and high energy communities.
These nuclear effects have been studied extensively using muon
and electron beams but have only been glanced at in low-statistics bubble
chamber experiments using neutrinos. If we consider the behavior of the
structure functions $F_{2}(x,Q^{2})$ measured on a nucleus (A) to
$F_{2}(x,Q^{2})$ measured on a nucleon as a function of $x$ we pass
through four distinct regions in going from $x$ = 0 to $x$ =
1.0:
\subsubsection*{Shadowing Region $x < 0.1$}
In the shadowing region ($x < 0.1$) there are several
effects that should yield a different ratio $R_{A}\equiv F_{2(A)}/F_{2(N)}$
when using neutrinos as
the probe. In the limit $Q^{2} \gt 0$, the vector current is conserved and
goes to 0, however the axial-vector part of the weak current is only
partially conserved (PCAC) and $F_{2}(x,Q^{2}) \rightarrow$ a non-zero constant as
$Q^{2} \gt 0$. According to the Adler theorem \cite{Adler} the cross
section of $\nu_{\mu}$ - N can be related to the cross section for $\pi$ - N
at $Q^{2}$ = 0. What effect does a nuclear environment have on the Adler
theorem?
As we increase $Q^{2}$ from 0 but keep it under 10 GeV$^{2}$ in the
shadowing region we enter the region of vector meson dominance (VMD) in
$\mu$/e - A scattering. The physics concept of VMD is the dissociation of
the virtual boson into a quark/antiquark pair, one of which interacts
strongly with the "surface" nucleons of the target nucleus (thus the
"surface" nucleons "shadow" interior nucleons). In $\nu$ - A scattering
there is an additional contribution from axial-vector mesons that is not
present in $\mu$/e - A scattering. Boros et al \cite{Boros} predict
that the resulting shadowing effects in $\nu$ - A scattering will be
roughly 1/2 that measured in $\mu$/e - A scattering.
In a more quantitative analysis, Kulagin \cite{Kulagin} used a
non-perturbative parton model to predict shadowing effects in $\nu$ - A
scattering. At 5 GeV$^{2}$ he predicts equal or slightly more shadowing in
$\nu$ - A scattering than in $\mu$/e - A scattering. He also attempts to
determine quark flavor dependence of shadowing effects by separately
predicting the shadowing observed in $F_{2}(x,Q^{2})$ (sum of all quarks)
and $xF_{3}(x,Q^{2})$ (valance quarks only). Fig.\ \ref{fig:shadow} shows
the results of a 2 year run using predictions of Kulagin's model for
$F_{2}$ and $xF_{3}$. As can be seen,
the predicted difference between the shadowing on sea and valence quarks is
clearly visible down to $x$ = 0.02 - 0.03.
\begin{figure}
\epsfysize=2.5in
\epsfxsize=5.0in
\centerline{
\epsffile{Kulagin.eps}}
\caption{$R_{Ca:D_{2}}$ for both $F_{2}$ and $xF_{3}$ as measured with
14 M events on each target. }
\label{fig:shadow}
\end{figure}
\subsection*{Anti-shadowing Region (0.1 < x < 0.2) }
Drell-Yan experiments have also measured nuclear effects. Their results
are quite similar to DIS experiments in the shadowing region. However, in
the anti-shadowing region where $R_{A}$ makes a brief but statistically
significant excursion above 1.0 in DIS, Drell-Yan experiments see no
effect. This could be an indication of difference in nuclear effects
between valence and sea quarks. Eskola et al \cite{Eskola} quantified this
difference by using a leading order/leading twist DGLAP model
.% which used
%initial nuclear parton distributions from CTEQ4L and GRV-LO and assumed
%scale evolution of nuclear parton distribution is perturbative. The model
%predicts that the difference between nuclear effects in $xF_{3}(x,Q^{2})$
%and $F_{2}(x,Q^{2})$ persist through the anti-shadowing region as well.
Taking the work of Kulagin and Eskola together implies that nuclear effects
in $xF_{3}(x,Q^{2})$ should be quite dramatic with more shadowing than
$F_{2}(x,Q^{2})$ at lower $x$ and then $R_{A}$ rising fairly rapidly to yield
significant antishadowing around $x = 0.1$. With 14 M events on each target
we should be able to measure antishadowing effects and the
difference between shadowing effects in $F_{2}(x,Q^{2})$ and
$xF_{3}(x,Q^{2})$ to the 6 $\sigma$ statistical level.
\subsubsection*{EMC-effect Region($ 0.2 < x < 0.7$)}
To determine individual quark contributions to the EMC-effect will be
challenging since the participation of sea quarks, and thus the difference
between $F_{2}(x,Q^{2})$ and $xF_{3}(x,Q^{2})$, shrinks rapidly with
increasing $x$. However, Eskola's predictions for this region indicate
that the contribution of $\overline{u}$ and $\overline{d}$ to $R^{(2)}_{A}$ in the
$Q^{2}$ range of this experiment remains well below 1 so that the quantity
$R^{(2)}_{A}$ - $R^{(3)}_{A}$ should remain negative well into the EMC-effect
region.
\subsection*{Behavior of $F_{2}(x,Q^{2})$ as $x \rightarrow 1$ in a Nuclear
Environment}
When working in the "fermi-motion" region it has been shown that we need to
add more than the Fermi gas model to a simple nucleon to reproduce the
behavior of $F_{2}(x,Q^{2})$ at high $x$. Few-nucleon-correlation
models and multi-quark cluster models allow quarks to have a higher
momentum which translates into a high-$x$ tail. In this region
$F_{2}(x,Q^{2})$ should behave as $e^{-ax}$. There have been analysis
of this behavior in similar kinematic domains using $\mu$ + C and $\nu$ +
Fe interactions. The muon experiment finds a = 16.5~$\pm$~0.5 while the neutrino
experiment finds a = 8.3~$\pm$~0.7~$\pm$~0.7 (systematic). Is the value of a
dependent on the nucleus? One would expect any few nucleon correlation or
multi-quark effects to have already saturated by Carbon. Is a dependent
on the probe?
\subsubsection*{Summary}
There is a very rich program of studying nuclear effects with a neutrino
probe in a high statistics neutrino factory experiment. The effects could
be measured to statistically significant accuracy in a 2 year exposure to
the beam in the near-detector experiment described above. The data
gathered would allow separate measurements of the effects on valence quarks
and sea quarks across much of the $x$ range.
The nuclear community should be informed of what a valuable tool for
nuclear research awaits them in the neutrino factory facility.
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bmat}{\left(\ba}
\newcommand{\emat}{\ea\right)}
\def\3{\ss}
\def\p{p\llap{/}}
\def\d{\delta}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\s{s\llap{/}}
\def\k{k\llap{/}}
\def\g5{\gamma_5}
\def\mn{\mu\nu}
\def\rs{\rho\sigma}
\def\b{\beta}
\def\a{\alpha}
\def\ve{\varepsilon}
\def\r{\rho}
\def\si{\sigma}
\def\as2{\alpha^2_s}
\def\ha{{1\over 2}}
\def\pa{\partial}
\def\du{\delta u}
\def\GeV{{\rm GeV}}
\def\Pon{P^{(0)n}}
\def\hPon{\hat P^{(0)n}}
\def\Q2{(Q^2_0)}
\def\zweib{\frac{2}{\beta_0}}
\def\vph{\varphi}
\def\nspm{NS\pm}
\def\gen{\gamma^{(1)n}}
\def\aspi{\frac{\a_s}{2\pi}}
\def\Pen{P^{(1)n}}
\def\hPen{\hat P^{(1)n}}
\def\tolimit_#1{\mathrel{\mathop{\longrightarrow}\limits_{#1}}}
\def\tosim_#1{\mathrel{\mathop{\thicksim}\limits_{#1}}}
\subsection{Spin Structure}%Mayda's section
An intense neutrino beam at a neutrino factory would create significant
event rates in very compact detectors. This opens the possibility of
using a polarized target, and hence a completely new class of neutrino
measurements becomes possible. At present we know
very little about the spin structure functions $g_1^\nu - g_5^\nu$
introduced in Eq.s \ref{pol_lon} and \ref{pol_tran}. In particular,
the parity violating functions have only been explored via weak-interference
measurements of proton form factors by the SAMPLE collaboration
\cite{SAMPLE} with very low statistics. A neutrino factory
would allow direct high-statistics measurements of all of these
structure functions and should be able to answer many
unresolved questions about the spin structure of the nucleon.
\ignore{
In the naive parton model,
\begin{eqnarray}
g_1^{\nu p}(x,Q^2)&=&\delta \dquark(x,Q^2) +\delta \squark(x,Q^2) + \delta
\antiuquark(x,Q^2) + \delta \anticquark(x,Q^2),\\
g_1^{\bar\nu p}(x,Q^2)&=&\delta \uquark(x,Q^2) +\delta \cquark(x,Q^2) + \delta
\antidquark(x,Q^2) + \delta \antisquark(x,Q^2).
\label{pol_g1}
\end{eqnarray}
\nonindent Note that $g_2$ has a a twist--2
($g_2^{WW}$) and a twist--3 ($\bar g_2$) contribution and has no
simple parton model interpretation,
\begin{eqnarray}
g_2&=&g_2^{WW}+\bar g_2\\
g_2^{WW}(x,Q^2)&=&-g_1(x,Q^2)+ \int_x^1 {dy
\over y}g_1(y,Q^2).
\label{pol_g2}
\end{eqnarray}
For $g_3$ and $g_4+g_5$ the parton model predictions are:
}
The nucleon spin ($\frac{1}{2}$)
can decomposed in terms of quark and gluon contributions:
%
\begin{equation}
\frac{1}{2}= \frac{1}{2}\Delta\Sigma + \Delta g + L_q + L_g,
\end{equation}
%
where $\Delta\Sigma \equiv \Delta u+ \Delta d+ \Delta s+\Delta c $
is the net quark helicity and
$\Delta g$ is the net gluon helicity along the nucleon
spin direction, while $L_i$ are their relative orbital angular
momentum.( We use $\Delta \quark$ as a shorthand for the integral $\int \Delta \quark(x) dx$.)
To date, the only experiments which have studied the
spin structure of the nucleon are low energy charged
lepton polarized deep-inelastic scattering experiments (PDIS) where only
the parity conserving polarized structure functions $g_1^l$ and $g_2^l$
can be measured.
$g_1^\lepton$ can be written in the leading order parton model as a sum of a nonsinglet and singlet part:
\begin{eqnarray}
g_1^\lepton(x,Q^2) &=& g_{1,NS}^\lepton(x,\qsq) + g_{1,S}^\lepton(x,\qsq) \\
&=&\onehalf \sum (e_i^2 - )\Delta \quark_i(x,Q^2) +
\onehalf \sum \Delta \quark_i(x,Q^2)\\
\end{eqnarray}
The first non-singlet term evolves independently of the gluonic spin contribution while the second is coupled to, and thus depends on the gluon spin
contribution $\Delta g$.
The integral structure functions have the following relation
to the parton spin contributions:
\begin{eqnarray}
\Gamma_1^{\lepton}(Q^2)& =& \int dx g_1^{l p/n} (x,Q^2)\\
&=& \Gamma^{\lepton }_{1,NS}(\qsq) + \Gamma^{\lepton }_{1,S}(Q^2)\\
\Gamma_1^{\lepton p}(Q^2)&=& \frac{C_1^{NS}(\qsq)}{6}\biggr [\onehalf a_3 + \frac{1}{6} a_8\biggl ] +
\frac{C_1^S}{9} a_0\\
\Gamma_1^{\lepton n}(Q^2)&=& \frac{C_1^{NS}(\qsq)}{6}\biggr [-\onehalf a_3 + \frac{1}{6} a_8\biggl ] +
\frac{C_1^S}{9} a_0\\\end{eqnarray}
Where the $C_1$ are coefficient functions and the axial charge matrix elements
\begin{eqnarray}
a_3 &\equiv& F+D \simeq \Delta \uquark - \Delta \dquark \\
a_8 &\equiv& 3F-D \simeq \Delta \uquark + \Delta \dquark - 2 \Delta \squark \\
a_0 &\equiv& \Delta \uquark + \Delta \dquark + \Delta \squark = \Delta \Sigma \\
\end{eqnarray}
can be expressed in terms of coupling constants $F$ and $D$ obtained from
neutron and hyperon beta decays \cite{betadecays}.
Because the interaction between $\Delta g $ and $\Delta \Sigma $
in the evolution of the singlet ($a_0$) component, interpretation of
$\Gamma_1^\lepton$ in terms of the quark spin is problematic.
Fig. \label{gluon} shows NLO QCD predictions for $\Delta \Sigma$ as a function of $\Delta g$.
\begin{figure}
\centerline{\epsffile{delta.eps}}
\caption{
Model dependent decomposition of singlet term into quarks and gluon based
on PDIS data,
$a_0 \rightarrow \Delta q - 3{\alpha_s \over 2 \pi} \Delta g$.
The QPM expectation and the results from a NLO fit of the
$Q^2$ evolution of most of the available data on $g_1^l$
are also shown. From the fit it was found that under this singlet
%decomposition, $\Delta g = 1.6 \pm 0.3\pm 1.0$, where
the error is dominated by theoretical
uncertainties.
\label{gluon}}
\end{figure}
Neutrino beams introduce both additional parity violating spin
structure functions $g_3, g_4 $ and $g_5$ and new combinations based
on sums and differences of neutrino and anti-neutrino scattering.
For example the sum
$$\Gamma_1^{\nu } + \Gamma_1^{\nu } = \int dx (g_1^{\nu } + g_1^{\nubar }) $$
for both proton and neutron targets is only sensitive to the singlet $a_0$ term and no input from
beta decay is necessary.
The parton model interpretation of these new structure functions is:
\begin{eqnarray}
g_{4+5}^{\nu p}(x,Q^2)&=&2xg_3^{\nu p}(x,Q^2)\\\nonumber&=&-x[\delta d(x,Q^2)+\delta
s(x,Q^2)-\delta\antiuquark(x,Q^2)-\delta \anticquark(x,Q^2)],\\
g_{4+5}^{\bar\nu p}(x,Q^2)&=&2xg_3^{\bar\nu p}(x,Q^2)\\\nonumber&=&-x[\delta
u(x,Q^2)+\delta c(x,Q^2)-\delta \antidquark(x,Q^2)-\delta \antisquark(x,Q^2)].
\label{pol_g3}
\end{eqnarray}
On a deuterium target, the $\uquark$ and $\dquark$ contributions
to $g_3$ can be cancelled leading to a direct measurement of
the strange sea contribution to the nucleon spin
$$g_3^{\nu (np)} - g_3^{\nubar (np)} = -2 (\delta \squark + \delta \antisquark)+2 (\delta \cquark + \delta \anticquark) $$.
which can also be studied via polarization asymmetries in charm production
from strange quarks.
The structure functions $g_3 -- g_5$, like $F_3$ are non-singlet functions
in which contribution from gluons cancel. Comparison of the non-singlet
functions with the single functions $g_1$ and $F_2$ is an indirect way
of measuring the contribution of gluons $\Delta g$.
%
\subsection{Experimental Setup at the Neutrino Factory}
A very promising target technology is the 'ICE' target \cite{ICE},
a solid HD compound in which the protons or the
deuterons can be polarized independently.
The expected polarization and dilution are
$P_H$=80\%, and $f_H=1/3$ for H, and $P_D$=50\% and $f_D=2/3$ for deuteron.
A 7 kg ($\rho_t$=1.1gr/cm$^2$) polarized target with the qualities mentioned
above would be 20~cm in radius and 50~cm thick, similar to
the other light targets proposed for structure
function studies. Raw event rates in
such a detector would be around 20M per 10$^{20}$ muon decays.
If such a data sample is analyzed in 10 in $x$ bins,
the error
in each $x$ bin would be:
$\delta {g_1}\simeq (fP_T\sqrt{N})^{-1} \sim 1\%$.
%measurement of the strange polarization from final charm state
%(di-lepton events), instead of the evaluation of Eq.~\ref{g3}.
%The total di-lepton cross-section is approximately $2\%$.
If the neutrino beam intensities and polarized target described above are
feasible, the physics motivations would be very strong. We
will be able to do high precision
measurements where we can cleanly separate singlet
(gluon-dependent) from non-singlet (gluon-free) terms. Furthermore,
due to the nature of the neutrino charged interactions it will be
possibility to perform a measurement of the polarization of the
proton's quarks by flavor, with sea and valence contributions separated.
\newcommand{\stw}{\mbox{$\sin^2\theta_W$}}
\newcommand{\nub}{\overline{\nu}}
\newcommand{\qbar}{\overline{q}}
%\newcommand{\nue}{\nu_{e}}
%\newcommand{\numu}{\nu_{\mu}}
\newcommand{\nubmu}{\overline{\nu_{\mu}}}
\newcommand{\nube}{\overline{\nu_{e}}}
\newcommand{\muebar}{\numu\nube}
\newcommand{\mubare}{\nubmu\nue}
\newcommand{\ubar}{\antiuquark}
\newcommand{\dbar}{\antidquark}
\newcommand{\alps}{\mbox{$\alpha_s$}}
\newcommand{\asop}{\mbox{$\frac{\alpha_s}{\pi}$}}
%\newcommand{\qsq}{\mbox{$Q^2$}}
\newcommand{\qnsq}{\mbox{$Q_0^2$}}
\newcommand{\mztwo}{\mbox{$M_Z^2$}}
\newcommand{\mz}{\mbox{$M_Z$}}
\newcommand{\mw}{\mbox{$M_W$}}
\newcommand{\mtop}{\mbox{$M_{\rms top}$}}
\newcommand{\mhiggs}{\mbox{$M_{\rms Higgs}$}}
\newcommand{\lmsb}{\mbox{$\Lambda_{\overline{MS}}$}}
\newcommand{\avgth}{\left< \theta_\nu\right> }
\subsection{Charm Production and $\dzero- \dzerobar$ Mixing}
\begin{figure}[tpb]
\begin{center}
\epsfxsize=5 in
\epsfbox{heavy-flavor-grv.eps}
%\epsfxsize=0.8\textwidth\epsfbox{heavy-flavor-grv.ps}
\end{center}
\caption{Charm and bottom quark production as a fraction of the total
cross-section as a function of $E_\nu$. }
\label{fig:charmrate}
\end{figure}
\begin{figure}[tpb]
\begin{center}
\epsfxsize=5 in
\epsfbox{charm-spect-grv.eps}
\end{center}
\caption{Charmed hadron spectra from neutrino interactions in a near detector
from a $50$~GeV muon storage ring.}
\label{fig:charmspect}
\end{figure}
Neutrino interactions are a very good source of clean, sign-tagged charm
particles. Single charm quarks are produced
via the processes
\begin{eqnarray}
\nu \squark &\gt& \lminus \cquark \hbox{\ \ \ Cabbibo favored} \\
\nu \dquark &\gt& \lminus \cquark \hbox{\ \ \ Cabbibo suppressed} \\
\nu \antisquark &\gt& \lplus \anticquark \hbox{\ \ \ Cabbibo favored} \\
\nu \antidquark &\gt& \lplus \anticquark \hbox{\ \ \ Cabbibo suppressed} \\
\end{eqnarray}
The fraction of heavy flavor produced as
a function of $E_\nu$ is shown in Fig.~\ref{fig:charmrate}.
An experiment
at a 50 GeV muon storage ring with 10$^{20}$ muon decays and a
a one ton (fiducial) target made up of silicon strip detectors
interleaved with heavier material would observe $\approx 3\times10^{9}$ muon-neutrino charged-current
interactions and around $1.2\times 10^8$ charm hadrons with energies
above 4 GeV/year.
All of these charmed
hadrons are flavor tagged at the point of production by the charge of the
outgoing primary lepton ($c$ production with $\ell^-$ and $\overline{c}$
production with $\ell^+$).
There are several interesting physics motivations for charm studies at muon
storage rings, including measurements of the strange contribution
to proton structure and spin; however, the primary motivation for producing
charm by this method is the cleanliness of the final state relative to
hadroproduction and the flavor tagging in production. This experimental fact
compliments the theoretically ``clean laboratory'' of charm in searches for
FCNC, CP asymmetries and ${\rm D^0 \rightarrow \overline{D^0}}$ oscillations,
all of which are very small in the standard model because of the lack of
coupling of charm to the heavy top quark.
As an example of the physics reach of a neutrino charm factory, consider the
example of $D^0-\overline{D^0}$ mixing measurements. The most sensitive
current searches for time-integrated mixing place limits on the process of
$\sim 5\times 10^{-3}$ \cite{E791,CLEO}. BaBar expects to have sensitivity to
mixing at the $\sim 5\times10^{-4}$ level after several years at design
luminosity \cite{BaBar}. These measurements are ultimately limited by
tagging mistakes and backgrounds to final state $D^0$ or $\overline{D^0}$
identification from doubly-Cabibbo suppressed decays, such as
$D^0\to K^+\pi^-$ which occur at the few part per thousand level.
At a $50$~GeV muon storage ring, with a high mass detector,
one could probe $D^0-\overline{D^0}$ mixing
{\em via}
\begin{eqnarray*}
\nu N \to & c \ell^- X\hspace*{8ex} \\
& \hookrightarrow \ell^+ X\hspace*{5ex} \\
& \hookrightarrow \overline{c} \to \ell^- X,
\end{eqnarray*}
and its charge conjugates. The appearance of like-signed leptons would
indicate mixing, where opposite-signed leptons are expected. Assuming $50\%$ of the charm produces hadronizes as a $D^0$ or
$\overline{D^0}$, this would result in the observation of $2\times 10^6$
tagged neutral charm meson semi-leptonic decays in either the muon or electron
channel.% These should allow limits on D meson mixing at the $
\approx\sqrt{\frac{\pi^2}{16\gamma_\mu^2}+\frac{m_eE_\nu}{2}}.
\end{equation}
For a $50$~GeV storage ring, this factor is dominated by the fundamental
$p_t$ of the interaction and is typically $\sim90$~MeV. For a lower
energy storage ring of about $15$~GeV, these factors become equal.
\begin{figure}
\begin{center}
\mbox{\epsfxsize=5.0in\epsffile{nue-sn.eps}}
\caption{Signal to noise in the low $p_t$ region
($p_t^2<\frac{\pi^2}{16\gamma_\mu^2}+\frac{m_eE_\nu}{2}$) as
a function of $E_\nu$.}\label{fig:nue-sn}
\end{center}
\end{figure}
The primary background
to this measurement is from quasi-elastic $\nue-N$ or
$\bar{\nu}_e-N$ scattering events which occur at $p_t$ up to $\sqrt{m_N E_\nu}$.
Fig.~\ref{fig:nue-sn} shows the estimated signal to background
ratios expected in the low $p_t$ region.
Because of the exceptionally low cross section, the target must be
very massive.
The detector must therefore be capable of resolving the
$p_t$ with much better resolution than the background spread.
This favors the use of a fully active, high resolution tracking detector with
sub-radiation length sampling in order to resolve the $p_t$ of the single
electron before it is significantly broadened by shower development. A
liquid Argon TPC, such as the one proposed for the ICANOE
experiment\cite{ICANOE} might be ideal for such a measurement. Another
possibility would be a scintillating fiber/tungsten calorimeter.
% Bruce had L of 10^46 and E_\nu=100 GeV which is out of the energy
% range! So I decreased the beam energy by three and upped the
% luminosity by 10 to keep the same sensitivity.
\begin{figure}
\begin{center}
\mbox{\epsfxsize=5.0in\epsffile{nue-beam-sens.eps}}
\caption{Statistical uncertainty in $\siniiW$ for a luminosity of
$10^{46}$~cm$^{-2}$ as a function of $y_{\rm cut}$ for a $30$~GeV
neutrino beam. Note that the $\mu^-$ produced beam is much less sensitive
to $\siniiW$ due to nearly exactly cancellation in the $\siniiW$ dependence of
the two neutrino species in the beam.}\label{fig:nue-sens}
\end{center}
\end{figure}
The largest experimental challenge
for measurement is likely to be the normalization of the absolute
neutrino flux. Despite the precise knowledge of muon decays, it
would be extremely difficult to predict the precise neutrino flux
at the $10^{-4}$ level merely from monitoring the parent muon beam.
Instead, the signal processes will probably have to be normalized
to the theoretically predictable processes of inverse muon decay,
$\numu e^-\to\nue \mu^-$, and muon production through annihilation,
$\antinu_e e^-\to \antinu_{\mu}\mu^-$, both of which occur only in the
$\muebar$ beam. Normalization of the $\nue$ beam may be possible
through comparison neutrino-nucleon
scattering, $\nu N \to \l^\pm N'$, in the $\mubare$ and $\muebar$
beam.
For a 20 ton detector at a 50 GeV muon storage ring,
with $2\times 10^{20}$ $\muminus$ decays/year there will be approximately
$1.5\times 10^{10}$ DIS charged current events and 8.5M $\numu/antinue$-electron
scatters per year. These lead to an estimated sensitivity of $\delta\siniiW^{\rm (stat)}\sim0.0002$
% Jae Yu's section *******************************************************
\subsection*{Electro-weak measurements in neutrino-nucleon scattering}
There were two dominant systematic uncertainties in present-day meaurements
of the weak mixing angle in neutrino nucleon scatering,
$\nu_{e}$ contamination in the $\nu_{\mu} $ beam and the kinematic
suppression of scattering from strange quarks in the charged current channel.
For an isoscalar target, the neutral current rates can be related to the charged current rates via \cite{LL-SMITH}:
\begin{eqnarray}\label{NCCC}
R_{\nu} - \Delta R_s &=& (\onehalf - \siniiW +{5\over 9} \sin^4\theta_W)[ 1 - \Delta R_c] + \\
&&\hskip 2 in{5\over 9} \sin^4\theta_W [r - r\Delta \overline{R}_c]\\
%
R_{\nubar} - \Delta \overline{R}_s&=& (\onehalf - \siniiW +{5\over 9} \sin^4\theta_W)[ 1 - \Delta\overline{R}_c]+\\
& & \hskip 2 in{5\over 9}\sin^4\theta_W[ {1\over r}- {1\over r} \Delta R_c]
%\nonumber &\simeq& 0.12\times 10^{-38} (E_{\nu}, \GeV), \centii\\
\end{eqnarray}
where $R_{\nu/\nubar}$ is the ratio of neutral to charged current
cross sections, $r \sim 0.5$ is the ratio of charged current anti-neutrino to neutrino cross sections and $\Delta R_s$ and $\Delta R_c$ are small corrections for
the kinematic suppression of $\squark \gt \cquark$ in charged current scattering where the neutral current process $\squark\gt \squark$ has no suppression.
The charm corrections can be eliminated by a judicious subtraction
of neutrino and anti-neutrino rates \cite{ref:paschos} but with a consequent
reduction in statistical power.
Experiments CITE(CDHSW, CCFR, NUTEV) to date have had integrated fluxes of 10$^{15}$-10$^{16}$ neutrinos
and have relied on dense iron targets. In such targets neutral current events are distinguished from charged current events by
the presence or absence of a muon in the final state. But in a
calorimeter, electron neutrino
charged current
induced events look very similar to neutral current events and form a significant background for precision measurements with conventional beams
produced by pion and kaon decay.
The most precise measurement to date is from the NuTeV collaboration \cite{NuTeV:prelim} of
\begin{equation}\label{eq:nutev-stw}
\siniiW=0.2253\pm0.0019{ (stat)}\pm0.0010{ (syst)}.
\end{equation}
At a neutrino factory, the neutrino flux will be several orders of
magnitude higher but the beam will consist of approximately equal numbers
of $\numu$ and $\antinue$. This makes a detector capable of
distinguishing electron charged current events from neutral current
events desirable and implies a low density detector such as those
considered for the deep-inelastic scattering studies.
We have considered several possible observables for a neutrino factory
measurement and propose:
\begin{eqnarray}\label{eq:Rmuebar}
R_e^{\muminus}=\frac{\sigma(\nu_{\mu},NC)+\sigma(\nubar_{e},NC)}
{\sigma(\nu_{\mu},CC)-\sigma(\nubar_{e},CC)}
&=&{R^{\nu}+grR^{\nubar}\over {1-gr}}
\end{eqnarray}
or
\begin{eqnarray}\label{eq:Rmuebarhat}
\hat{R}^{\muminus}=\frac{\sigma(\nu_{\mu},NC)+\sigma(\nubar_{e},NC)+\sigma(\nubar_{e},CC)}
{\sigma(\nu_{\mu},CC)}
&=&{R^{\nu}+grR^{\nubar} + gr}
\end{eqnarray}
for the $\muebar$ beam, and
\begin{eqnarray}\label{eq:Rmubare}
R_e^{\muplus}=\frac{\sigma(\nubar_{\mu},NC)
+\sigma(\nu_{e},NC)}
{\sigma(\nu_{e},CC)-\sigma(\nubar_{\mu},CC)}
={R^{\nu}+g^{-1}rR^{\nubar}\over{1-g^{-1}r}}
\end{eqnarray}
or
\begin{eqnarray}\label{eq:Rmubarehat}
\hat{R}^{\muplus}&=&{\sigma(\numubar,NC)
+\sigma(\nue,NC)+\sigma(\nue),CC)
\over\sigma(\numubar,CC)}\\
&=&{{g\over r}R^{\nu}+R^{\nubar} + {g\over r}}
\end{eqnarray}
for the $\mubare$ beam, where
$R_{\nu/\nubar}$ is the ratio of neutral to charged current cross sections
from Eq. \ref{NCCC}.
The variable $g$ is the energy-weighted flux ratio between $\nu_{\mu}$
and $\overline{\nu}_{e}$ or, equivalently, between $\overline{\nu}_{\mu}$
and $\nu_{e}$:
The observable $R_e^{\mu}$ requires electron identification while
$\hat{R}^{\mu}$ requires only muon identification.
The flux ratio for neutrinos and anti-neutrinos $g$ is:
\begin{eqnarray}\label{eq:little-g}
g\equiv\frac{\int \Phi(E_{\overline{\nu}_{e}})E_{\overline{\nu}_{e}}dE_{\overline{\nu}_{e}}}
{\int\Phi(E_{{\nu}_{\mu}})E_{{\nu}_{\mu}}dE_{{\nu}_{\mu}}}
&=&\frac{\int\Phi(E_{{\nu}_{e}})E_{{\nu}_{e}}dE_{{\nu}_{e}}}
{\int \Phi(E_{\overline{\nu}_{\mu}})E_{\overline{\nu}_{\mu}}
dE_{\overline{\nu}_{\mu}}} \simeq {6\over7 }.
\end{eqnarray}
Due to the nature of muon decay kinematics, the flux ratio should
be quite well known. However, the relative detection efficiencies for muons
and electrons must be known at the $2\times 10^{-4}$ level in order to
determine $\siniiW$ to 10$^{-3}$ by the first method.
In addition, the charm contributions are not cancelled in this observable
and must be measured directly in the same experiment.
%\end{itemize}
For the $R_e$ measurement, which requires electron identification,
an active target of 20~cm radius, 10 gr/cm$^2$ thick consisting of either CCD's or silicon strip detectors ($\sim$ 140 300-$\mu$m detectors)
spaced over a meter and
followed by the
tracking, electromagnetic and hadron calorimetry and muon identification proposed above
for structure function measurements
would yield 15M muon and 8M electron charged current
deep-inelastic scattering events/10$^{20}$ $\muminus$ decays and would yield a statistical precision
of 0.0004 in $\siniiW$. The charm corrections partially cancel in this
observable and would also be measured directly
via the 2M charm events/year produced in such a detector.
The $\hat{R}$ measurement, which relies only on muon identification
can be done with a much denser target, perhaps an iron/silicon sandwich
calorimeter. Such a calorimeter 200 gr/cm$^2$ thick would have a
statistical sensitivity of $\Delta \siniiW ~ 0.0001$ per year at a 50 GeV machine. This method is quite similar to the method used in the NuTeV \ref{nutev-stw}
measurement and would be dominated by systematic errors.
%Joe and Eric's section ***************************************************
\def\pl#1#2#3 {{ Phys. Lett.} {\bf#1}, #2 (#3). }
\def\prev#1#2#3 {{ Phys. Rev. } {\bf#1}, #2 (#3). }
\section*{Heavy Lepton Mixing}
A muon storage ring offers ample opportunities to search for new
phenomena in yet unexplored physical regions. One such opportunity
is the ability to search for the
existence of neutral heavy leptons. Several models describe heavy isospin
singlets
that interact and decay by
mixing with their lighter neutrino counterparts \cite{GLR,Shrock}. The
high intensity neutrino beam created by the muon storage ring provides an
ideal setting to search for neutral heavy leptons with a mass below
the muon mass, 105.6 MeV$/c^2$.
It is postulated that neutral heavy leptons ($L_0$) could be produced
from muon decay when one of the neutrinos mixes with its heavy,
isospin singlet partner. Neutral heavy leptons can be produced via one of
two channels:
\begin{equation}
\mu^- \rightarrow L_0 + \overline{\nu}_{e} + e^-
\end{equation}
\begin{equation}
\mu^- \rightarrow \nu_{\mu} + L_0 + e^-
\end{equation}
The branching ratio for each of these reactions is given by:
\begin{equation}
BR(\mu\rightarrow L_0 \mu e) = |U_i|^2 (1 - 8x_m^2 + 8x_m^6 - x_m^8 +
12x_m^4\ln{x_m^2})
\end{equation}
\noindent Here $x_m \equiv m_{L_0}/m_{\mu}$ and $|U_i|^2$ is
the mixing constant between the specific type of neutrino
and the neutral heavy lepton: $U_i \equiv \langle L_0 | \nu_i \rangle$.
Note that $|U_{\mu}|^2$ and $|U_{e}|^2$ need not be identical.
Once produced, a neutral heavy lepton of such low mass will either
decay via $L_0 \rightarrow \nu \nu \nu$, $L_0 \rightarrow \nu e e$, or
$L_0 \rightarrow \gamma \nu$. The most viable mode for detection is
the two-electron channel. For this particular decay mode, the
$L_0$ can decay either via charged current or charged and neutral
current interactions. The branching ratio for this decay process has
been previously calculated \cite{Bolton}. Since the decay
width is proportional to $U_j^2$, the number of $L_0$'s detectable is
proportional to $U_i^2\cdot U_j^2$ in the limit where the
distance from the source to the detector is short compared to the
lifetime of the $L_0$.
Using the above model, one can estimate the number of neutral heavy
leptons produced at the muon storage ring which later decay within a
given detector:
\begin{equation}
N_{L_0} = N_{\nu}*
BR(\mu\rightarrow L_0 \nu e)*\epsilon
*e^{-L / \gamma c \tau}*
BR(L_0 \rightarrow detectable)*
(1 - e^{- \delta l / \gamma c \tau})
\end{equation}
\noindent Here $N_{\nu}$ is the number of neutrinos produced from muon
decay, $BR(\mu\rightarrow L_0 \nu e)$ is
the branching ratio of muons decaying into neutral heavy leptons
versus ordinary muon decay, $L$ is the distance from the beamline to
the detector, $\delta l$ is the length of the detector, $\epsilon$ is
the combined detector and geometric efficiency, $\tau$ is the $L_0$
lifetime, and $BR(L_0 \rightarrow \hbox{detectable})$
is the branching ratio for the neutral heavy lepton decaying via a
detectable channel (presumably $L_0 \rightarrow \nu e e$).
In estimating the sensitivity to $L_0$ production at the muon storage
ring, we make a few underlying assumptions. We assume that the
storage ring utilizes a pure, unpolarized muon beam with straight
sections such that 25 percent of the muons will decay to neutrinos
pointing towards the detector. We assume that the fiducial volume is 3 meters
in diameter and 30 meters in length, and that the detector has sufficient
tracking resolution to detect
the $e^+e^-$ vertex from the $L_0$ decay. We assume for now that the
background is negligible. These parameters correspond to the
fiducial volume of the decay channel used for the $L_0$ search at
E815 (NuTeV) \cite{NuTeVNHL,NuTeVQ0}.
The sensitivity of the detector has been calculated for a number of
different muon energies and beam intensities. Fig. \ref{nhl_mustore} shows
limits on the $L_0$-$\nu_\mu$ mixing as a function of $L_0$ mass. One
achieves the best limits from using relatively low energy/high
intensity muon beams. This is a major improvement over previous
neutral heavy lepton searches, where limits do not reach below
$6.0\times 10^{-6}$ in the low mass region \cite{PDB,Shrock}.
The single event sensitivity quoted here depends on having
minimal background levels in the signal region. Part of this can be
achieved by kinematic cuts which discriminate against neutrino interactions
in the detector material. However, it will probably be necessary to
reduce the amount of material in the fiducial region compared
to NuTeV. We estimate that even if the
decay region is composed only of helium gas, the number of
neutrino interactions will approach a few thousand. The ideal
detector, therefore, would consist of a long vacuum or
quasi-vacuum pipe with appropriate segmentation for tracking. The
decay pipe could be used in conjunction with larger neutrino detectors
adapted for the muon storage ring.
The muon storage ring may prove to be an ideal location to continue
the search for neutral heavy leptons. The high intensity neutrino
beam allows for a neutral heavy lepton search to be sensitive to
the 10 -- 100~MeV/$c^2$ mass range. In addition, such a neutral
heavy lepton program could easily interface with a
neutrino detector which uses the same neutrino beam. It is also clear,
however, that a neutral heavy lepton search would receive the most
benefit at lower muon energies, and thus would yield best results at
the earlier stages of the muon storage ring program.
\begin{figure}
\begin{center}
\mbox{\epsfxsize=6.0in\epsffile{mulimits.eps}}
\caption{Limits on $|U_{\mu L}|^2$ as a function of $L_0$ mass for one
year of running. The curves show sensitivities for 20 GeV and 50 GeV
muon energies. Sensitivities assume no background events in signal
region.}
\label{nhl_mustore}
\end{center}
\end{figure}
%%% John Krane, March 2000
\subsection{Neutrino Magnetic Moments}
Although neutrino oscillation searches focus on the mass differences between
neutrino eigenstates, neutrinos can possess other observable properties
such as a magnetic moment. A measurement of the neutrino magnetic
moment (NMM) would not only have great impact in the field of cosmology,
particularly the development of stellar models, but would also help
constrain several Standard Model extensions. An important experimental
advantage is that a NMM search can run parasitically as the front--end to a
typical long baseline detector.
Despite their lack of charge, neutrinos can possess a non-zero magnetic
moment that can arise through loop diagrams. In the Standard Model, extended
to include a right--handed neutrino or with left--handed neutrinos which have mass, the expected magnitude of the \cite{shrock} neutrino magnetic moment
is given by
\begin{equation}
\label{krane}
\mu _{\nu }\simeq 3\times 10^{-19}\;\mu _{B}\cdot ( \frac{m_{\nu }}{%
1\;\hbox{eV}}) ,
\end{equation}
where $\mu _{B}$ is the Bohr magneton. Although quite minuscule, several
extensions to the Standard Model could dramatically increase $\mu_\nu$ : supersymmeteric
models can produce $10^{-14}\mu_{B}$ to $10^{-12}\mu _{B}$
\cite{frank} and calculations that invoke large extra dimensions easily
yield $10^{-11}\mu _{B}$ or larger \cite{ng}.
Relative to the Standard Model expectation, the excluded values of NMM\ are
not at all stringent, being seven to nine orders of magnitude larger. The
current limits on neutrino magnetic moment from laboratory experiments are
$\mu _{\nu }\leq 1.5$ to $1.8\times 10^{-10}\mu _{B}$ for electron
neutrinos \cite{beacom}\cite{mu_e_limit} and $\mu _{\nu }\leq 7.4\times
10^{-10}\mu _{B}$ for muon neutrinos \cite{mu_mu_limit}. Astrophysical
limits are stronger:\ the slow rate of plasmon decay in horizontal branch
stars \cite {star1} implies $\mu _{\nu }\le 10^{-11}\mu _{B}$,
while the neutrino energy loss rate from supernova 1987a \cite{star2}
yields $\mu _{\nu }\le $ $10^{-12}\mu _{B}$. Note, however, that
several assumptions are implicit to the astrophysics limits, including the
core temperature of the stars; if stellar models omit important processes,
these limits might be overestimates. Also, the supernova limit applies only
to Dirac neutrinos and not to the Majorana case.
Existing search schemes possess a weakness that sharply limits their
ultimate sensitivity: the formulae for the hypothesized effect are quadratic
in $\mu _{\nu }$ but linear in terms of the experimenter--controlled
parameters. In contrast, the following scheme is quadratic in terms of the
product of the NMM$\;$and a magnetic field strength, $\mu _{\nu }\cdot $B;
hence a carefully designed and executed experiment could improve the limits
from current experiments and possibly the limits from astrophysics
calculations, or actually detect a NMM.
The energy $E$ of a neutrino with a magnetic moment in a magnetic field B
gains a new term $\mu _{\nu }\cdot $B. Consider a B field along the
$x$-axis, and a neutrino with momentum and helicity along the $z$-axis at
$t=0$. The eigenstates of the neutrino are projections along the $x$-axis,
and the state of the neutrino is expressed as:
\begin{equation}
\left| \uparrow \right\rangle =\frac{e^{-i\left( E+\mu _{\nu }B\right) t}}{%
\sqrt{2}}\left| \leftarrow \right\rangle +\frac{e^{-i\left( E-\mu _{\nu
}B\right) t}}{\sqrt{2}}\left| \rightarrow \right\rangle . \label{eq_split}
\end{equation}
As the neutrino propagates, the relative phase of the two components
changes, corresponding to a rotation to a sterile state in the case of a
Dirac neutrino or to an antineutrino in the Majorana case. At a far
detector, the signal would be a deficit in the number of neutrinos detected
or increase in the number of antineutrinos detected with the B field in
place compared to the sample detected with no B field turned on.
In this phase rotation scheme, the energy splitting occurs as the neutrino
passes through a field gradient and experiences a force $F=\nabla (\mu _{\nu
}\cdot $B$)$. To preserve this energy difference, which drives the phase
difference in the absence of the B-field, the field must be turned off
instead of allowing the neutrino to experience the reverse gradient as it
exits the field region.
There are thus two basic requirements for the magnetic field:
\begin{itemize}
\item[1) ] The magnetic field must oscillate such that the neutrino
experiences only one sign of the gradient. This study assumes that the
neutrino bunch length is small compared to the oscillation length. If this
assumption is not true, the effects discussed here will be diluted but the
basic conclusions will still apply.
{\it Editor's note - is this possible or does the neutrino see
an effective spatial gradient in all case}
\item[2) ] The magnetic field must be as strong as possible.
\end{itemize}
We have explored the possibility \cite{norbert} of using two existing
technologies for the B field: resonant cavities and kicker magnets. In both
cases the maximum magnetic field is too small to yield improved magnetic
moment limits given realistic equipment. We are exploring configurations
involving current sheets.
The formula for the number of events lost to sterile states may be expressed
very simply as:
\begin{equation}
N_{lost}=N*\sin ^{2}\left( \mu _{\nu }\hbox{B}t\right) \label{eq_nlost}
\end{equation}
\begin{figure}
\begin{center}
\mbox{\epsfxsize=4.5in\epsfbox{mag_mom_signif.eps}}
\caption{Significance for several scenarios.}
\label{mag_mom_signif}
\end{center}
\end{figure}
\noindent
where $t$ is the neutrino flight time from entering the magnetic field to
detection. We note that, in contrast to an oscillation disappearence
signal, this effect is explicitly independant of the neutrino energy. Fig.
\ref{mag_mom_signif} compares the number of events that vanish
because of phase rotations to the expected statistical fluctuations in the
number of CC events ($ N$) observed in a 50 kton \cite{geer_2day} far
detector. We see that for a 10 T field gradient and $10^{19}$ muon decays
we expect a $> 10 \sigma$ significance for a NMM of
$10^{-11}\mu _{B}$. With a 3T gradient, the limit drops below
two sigma. The sensitivity can be increased very strongly by increasing the
field strength and more weakly by
increasing the number of events in the far detector. Because the detector
distance deterimines both $t$ and $N$ in Eqn. \ref {eq_nlost}, the
``significance'' in the figure is linearly dependent on distance.
To conclude, we have discussed a novel neutrino magnetic moment search
technique that uses oscillating magnetic fields at the source of a long
baseline detector's neutrino beam. This is the only technique we know of
that is quadratic in both $\mu _{\nu }$ and a controllable parameter, and
thus has the potential for improved sensitivity as we improve our ability
to create oscillating magnetic field gradients.
%Fritz's section *********************************************************
\section*{Anomolous Lepton Production}
The neutrino beam from a muon storage ring would consist of a virtually pure
combination of $\bar{\nu}_e$ and $\nu_\mu$ (or charge-conjugate). Also, at the
source of the neutrino beam, the $\bar{\nu}_e$ and $\nu_\mu$ will not have
had time to oscillate into other flavors: For a
20 GeV muon storage ring with a
700 m straight section, and neutrino oscillations with
$\Delta m^2 \ = \ 3.5 \times 10^{-3} \ \rm eV^2$, the oscillation probability
is $\le 5 \times 10^{-9}$. Furthermore, the neutrino flux is highest
at the source. Thus, an experiment at the neutrino source could be highly
sensitive to exotic processes resulting in production of $e^-$, $\mu^+$,
or $\tau$ of either charge.
While such a search is interesting in its own right, it is also useful
to rule out exotic contributions to long-baseline neutrino oscillation
signals.
These exotic processes would probably have a flat or rising dependence on
the neutrino energy $E_\nu$. In contrast, neutrino oscillations
have a $1/E_\nu^2$ dependence. Also, if the distance $L$ of the experiment
changes, the rate of exotic events would decrease with the flux as $1/L^2$.
In contrast, the neutrino oscillation probability would increase as $L^2$
(for $L$ small compared to the oscillation period),
and so the rate of oscillated events would be independent of $L$. Thus,
one could distinguish between exotic processes and a
neutrino oscillation.
Anomalous lepton production could occur if muons decay to neutrino flavors
other than those in the usual decay $\mu \to e \bar{\nu}_e \nu_\mu$, and the
anomalous neutrinos then interact in the target. Alternatively, they could
be produced if a $\bar{\nu}_e$ or $\nu_\mu$ interacts with the target
via an exotic process.
The only direct experimental limit on exotic
$\mu \to e \bar{\nu}_x \nu_y$ decays is
$BR(\mu \to e \bar{\nu}_\mu \nu_e) < 1.3\%$\cite{PDG}. Indirect limits are
also very weak. The contribution of non- $V-A$ interactions to the muon
decay rate has been limited to 8\%\cite{PDG}.
Also, the total muon decay rate is one of the main measurements used to
constrain electroweak parameters\cite{PDG}. To first order,
\begin{equation}
\frac{1}{\tau_\mu} = \frac{G_F m_\mu^5}{192\pi^3} .
\end{equation}
Assuming the standard model, $G_F$ is determined to 1 part in $10^5$
from muon lifetime measurements. If there are exotic contributions to
the muon lifetime, the measured value of $G_F$ would be shifted from
the true value.
Since
\begin{equation}
m_W \propto G_F^{-1/2} ,
\end{equation}
the 0.1\%
uncertainty on $m_W$ corresponds to a 0.4\%
shift in the muon lifetime.
Finally, the CKM matrix element $V_{ud}$ is determined from the rate of
nuclear $\beta$-decays relative to the muon lifetime. The assumption
of unitarity on the CKM matrix gives us the following constraint on
the first row:
\begin{equation}
|V_{ud}|^2 + |V_{us}|^2 + |V_{ud}|^2 = 1 .
\end{equation}
The experimental determination is\cite{PDG}:
\begin{equation}
|V_{ud}|^2 + |V_{us}|^2 + |V_{ud}|^2 = 0.991 \pm 0.005 .
\end{equation}
The uncertainty on this constraint corresponds to a 0.5\%
shift in the muon lifetime. Additional contributions to the muon
decay rate would lead to a downward shift in
the determined value of $|V_{ud}|^2$ from the true
value. We conclude that exotic decay modes of the muon with branching
ratios totaling 0.5\%
are possible without contradicting current measurements or tests of the
standard model.
As a concrete example of such an exotic process we consider
R-parity-violating supersymmetric models. These models lead to
lepton-number-violating vertices with couplings $\lambda$, and muon
decay processes such as $\mu \to e \bar{\nu_\tau} \nu_\tau$ as shown
in Fig.~1. The matrix element for these decays turns out to have
the same form as for the standard W-exchange. The current constraints
on the couplings $\lambda$ are reviewed in Ref.~\cite{dreiner}.
These constraints allow a branching ratio of 0.4\%
for the process in Fig.~\ref{exotics:decay}.
Similar processes are allowed for anomalous
lepton production as shown for example in
Fig.~\ref{exotics:decay}.
Estimates for allowed rates are in progress~\cite{quigg}.
These diagrams
involve the $\lambda '$ couplings. Currently, the best limit on one of
these couplings,
$\lambda'_{231}$, is from $\nu_\mu$ deep-inelastic scattering, so existing
neutrino data is already providing constraints!
As a start on estimating the capabilities of an experiment at the
neutrino source, we consider the detector concept illustrated in
Fig.~\ref{exotics:detector}.
This concept consists of a repeating sequence of 1.5 mm-thick Tungsten
sheets with Silicon tracking, separated by
4 mm. Tungsten, being dense, provides a high
target mass while being thin enough for a
produced $\tau$ to have a high probability
of hitting the Silicon. The impact parameter of the $\tau$ decay products
is typically 90 microns with a broad distribution, so we would like a
hit resolution of 5 microns or better. Although there is a lot of
multiple scattering in the tungsten, the short extrapolation distance
provides for a good impact parameter resolution on the $\tau$ decay
products. This configuration has been optimized for a 50 GeV muon beam.
For lower energy beams, the planes should be spaced more closely, and the
Tungsten thickness perhaps reduced.
Studies of detectors with passive target mass and tracking with
emulsion sheets~\cite{emulsion} suggest that we can expect $\tau$
reconstruction efficiencies as high as 30\%.
We would propose placing such a detector in a magnetic field, and
measuring the momentum of muons and hadrons should be straightforward.
However, each Tungsten sheet is 0.4 radiations lengths thick, and while
we should obtain good energy resolution for electromagnetic showers,
determining the electron charge will be difficult. We estimate it would
take roughly a 4 Tesla field transverse to the beam direction to have
a good chance of measuring the charge of a 50 GeV electron. Even then,
detailed studies are needed to determine if this can be done reliably.
A total mass of 6 tons of Tungsten corresponds to 200 $\rm m^2$ of Silicon
tracking. For $5\times 10^{20}$ muon decays at 50 GeV, we expect a total
of 35 billion neutrino interactions. Obviously there is much potential for
sensitivity to very rare exotic processes, but detailed simulations and
studies of possible Silicon tracking technologies are needed to quantify
this.
\begin{figure}[h]
\begin{center}
\mbox{\epsfxsize=2.8in\epsffile{exotic_decay.eps}}
\mbox{\epsfxsize=2.8in\epsffile{exotic_interaction.eps}}
\end{center}
\caption{Example of exotic muon decay in R-parity-violating SUSY (Left),
and an example of an exotic neutrino interaction in R-parity-violating
SUSY (Right).}
\label{exotics:decay}
\end{figure}
%\begin{figure}
%\epsfysize=1.5in
%\centerline{
%\epsffile{exotic_decay.eps}}
%\caption{Example of exotic muon decay in R-parity-violating SUSY.}
%\label{exotics:decay}
%\end{figure}
%\begin{figure}
%\epsfysize=1.5in
%\centerline{
%\epsffile{exotic_interaction.eps}}
%\caption{Example of an exotic neutrino interaction in R-parity-violating SUSY.}
%\label{exotics:interaction}
%\end{figure}
\begin{figure}
\epsfysize=1.5in
\centerline{
\epsffile{exotic_detector.eps}}
\caption{One plane of a detector for $\tau$ production.}
\label{exotics:detector}
\end{figure}
\section{Summary}
We have investigated possible conventional neutrino physics studies done
at a detector located near a muon storage ring. We emphasized novel methods
rather than extensions of existing experiments with additional statistics.
For a reference machine with 50 GeV stored muons and 10$^{20}$ muon decays
per year we find that it is possible to:
\begin{itemize}
\item Measure individual parton distributions within the proton for all
light quarks and anti-quarks.
\item Determine the effects of a nuclear environment on individual quark species.
\item Measure the spin contributions of individual quark species, including
strange quarks and do precision studies of the QCD evolution of spin
effects without a need for data from beta decay measurements.
\item Measure charm production with raw event rates of up to 100 million
charm events/year with $\simeq$ 2M double tagged events.
\item Measure the Weinberg angle in both hadronic and purely leptonic
modes with a precision of 0.0001 to 0.0002.
\item Search for the production and decay of neutral heavy leptons
with mixing angle sensitivity 2 orders of magnitude better than
present limits in the 30-80 MeV region.
\item Search for a neutrino magnetic moment which may be much large
than the Standard Model prediction in
some supersymmetric theories.
\item Search for anomolous tau production predicted
by some R-parity violating supersymmetric models.
\end{itemize}
We note that the event rates at a near detector increase
linearly with neutrino energy. In addition, the acceptance
of small detectors is better for the narrower beam produced
by higher energy machines. Almost
all of the above measurements, with the exception of the neutral heavy
lepton search, lose sensitivity if the beam energy is less than 50 GeV
and gain if it is greater.
If the storage ring beam energy is lowered to 20 GeV, the statistical power of almost all of the
measurements considered here would drop a factor of 2.5 or more. The number
of deep-inelastic scattering events with $\qsq$ high enough for perturbative
QCD to be meaningful drops even further and the minimum $x$ rises to 0.05.
Measurements involving charm or tau production in the final
state would be severely limited, as would the inverse muon
decay normalization for $\nu-e$ scattering, which has a threshold of $\sim$ 11 GeV.
\clearpage
%\begin{references}
\begin{thebibliography}{99}
%%%%%%%% introduction
\bibitem{geer} S. Geer, \pr{57}{6989}{98}, \ib{59}{039903E}{99}.
\bibitem{abp} A. De R\'{u}jula, M. B. Gavela, and P. Hern\'{a}ndez,
%%%\np{B547}{21}{99} \hepph{9811390}.
\bibitem{bgw} V. Barger, S. Geer, and K. Whisnant, ``Long Baseline
Neutrino Physics with a Muon Storage Ring Neutrino Source''
%%%\hepph{9906487}.
\bibitem{suite} Within a growing literature, see, for example, A.
Bueno, M. Campanelli, and A. Rubbia, ``Long baseline neutrino
oscillation disappearance search using a $\nu$ beam from muon decays''
%%%\hepph{9808485};
``A medium baseline search for $\nu_\mu\to\nu_e$
oscillations at a $\nu$ beam from muon decays''
%%%\hepph{9809252}; \\
S. Dutta, R. Gandhi, and B. Mukhopadhyaya, ``Tau-neutrino
Appearance Searches using Neutrino Beams from Muon Storage Rings''
%%%\hepph{9905475}.
\bibitem{janetc} For summaries of the current evidence about neutrino
oscillations, see J. M. Conrad, in \textit{Proceedings of the 29th
International Conference on High Energy Physics,} Vancouver, edited
by A. Astbury, D. Axen, and J. Robinson (World Scientific, Singapore,
1999), p.~25
%%%(\hepex{9811009}),
and ``Where in the World Is the
Oscillating Neutrino?'' talk at Inner Space / Outer Space 1999 and
PANIC99, available at
%%%\textsf{http://portia.fnal.gov/\~{}jconrad/isos.html}; \\
P. Fisher, B. Kayser, and K. S. McFarland, ``Neutrino Mass and
Oscillation,'' \textit{Annu. Rev. Nucl. Part. Sci.} \textbf{49,}
481 (1999),
%%%\hepph{9906244};\\
L. DiLella, ``Accelerator and Reactor Neutrino Experiments,''
%%%\hepex{9912010}.
\bibitem{SKatm} Y. Fukuda, \etal\ (Super-Kamiokande Collaboration),
\prl{81}{1562}{98}.
\bibitem{SKLyon} W. A. Mann, ``Atmospheric Neutrinos and the
Oscillations Bonanza,'' Plenary talk at the XIX Int. Symposium on
Lepton and Photon Interactions at High Energies, Stanford, Aug. 1999,
%%%\hepex{9912007}.
\bibitem{LSND} C. Athanassopoulos, \etal\ (LSND Collaboration),
\prl{77}{3082}{96}; \ib{81}{1774}{98}; \prc{58}{2489}{98}.
\bibitem{sgeer} V. Barger, S. Geer, and K. Whisnant, ``Long Baseline Neutrino
Physics with a Muon Storage Ring Neutrino Source,''
%%%\hepph{9906487};
V. Barger, S. Geer, R. Raja, and K. Whisnant, ``Long Baseline Study of
the Leading Neutrino Oscillation at a Neutrino Factory,''
%%%\hepph{9911524}.
\bibitem{nucp} F. Vissani, ``Aspects of \CP\ violation in the leptonic
sector,'' \textit{These Proceedings.} See also A. Donini, M.B. Gavela,
P. Hernandez, and S. Rigolin, ``Neutrino mixing and \CP-violation,
%%%\hepph{9909254}.
\bibitem{twonu} Two-neutrino experiment \ldots
\bibitem{weaknc} Discovery of neutral currents \ldots
\bibitem{rmpnurev} J. M. Conrad, M. H. Shaevitz, and T. Bolton,
``Precision Measurements with High Energy Neutrino Beams,''
\rmp{70}{1341}{98},
%%%\hepex{9707015}.
%%%%%%%% theory section
\bibitem{seesaw} M. Gell-Mann, P. Ramond, R. Slansky, in {\it Supergravity},
edited by P. van Nieuwenhuizen and D. Freedman (North Holland, Amsterdam,
1979), p. 315; T. Yanagida in proceedings of {\it Workshop on Unified Theory
and Baryon Number in the Universe}, KEK, 1979.
\bibitem{sol} Fits and references to the Homestake, Kamiokande, GALLEX,
SAGE, and Super Kamiokande data include N. Hata and P. Langacker,
Phys. Rev. {\bf D56} 6107 (1997); J. Bahcall, P. Krastev, A. Smirnov,
Phys. Rev. {\bf D58}, 096016 (1998); J. Bahcall and P. Krastev,
Phys. Lett. {\bf B436}, 243 (1998) and
%%%http://www.sns.ias.edu/~jnb/.
Recent
Super Kamiokande data is reported in Super Kamiokande Collab., Y.Fukuda et al.,
Phys. Rev. Lett. {\bf 82}, 1810, 243 (1999).
\bibitem{kam}
Kamiokande Collab., K. S. Hirata, Phys. Lett. {\bf B205}, 416; {\it ibid.}
{\bf 280}, 146 (1992); Y.Fukuda et al., Phys. Lett. {\bf B335}, 237 (1994);
S. Hatakeyama et al. Phys. Rev. Lett. {\bf 81}, 2016 (1998).
\bibitem{imb}
IMB Collab., D. Casper et al., Phys. Rev. Lett. {\bf 66}, 2561 (1991);
R.Becker-Szendy et al., Phys. Rev. {\bf D46}, 3720 (1992); Phys. Rev. Lett.
{\bf 69}, 1010 (1992).
\bibitem{sk}
Super-Kamiokande Collab., Y. Fukuda et al., Phys. Lett. {\bf B433}, 9 (1998);
Phys. Rev. Lett. {\bf 81},1562 (1998); {\it ibid.}, {\bf 82}, 2644 (1999);
Phys. Lett. {\bf B467}, 185 (1999). The fit from the analysis of the current
61 kton-yrs of atmospheric data was reported by K. Scholberg (Stony Brook
seminar, Feb. 7, 2000).
\bibitem{soudan}
Soudan Collab., W. Allison et al, Phys. Lett. {\bf B391}, 491 (1997);
Soudan-2 Collab., Phys. Lett. {\bf B449}, 137 (1999);
A. Mann, in Proceedings of the 1999 Photon-Lepton Symposium,
hep-ex/9912007.
\bibitem{macro}
MACRO Collab., M. Ambrosio et al., Phys. Lett. {\bf B434}, 451 (1998);
hep-ex/0001044.
\bibitem{chooz} M. Apollonio et al., Phys. Lett. {\bf B420}, 397 (1998);
Phys. Lett. {\bf B466}, 415 (1999).
\bibitem{learned} J. Learned, in the Proceedings of the Workshop on
the Next Generation Nucleon Decay and Neutrino Detector NNN99, Stony Brook
(Sept. 1999).
\bibitem{lsnd} LSND Collab., C. Athanassopoulous et al., Phys. Rev. Lett.
{\bf 77}, 3082 (1996), LSND Collab., C. Athanassopoulous et al., Phys.
Rev. Lett. {\bf 81}, 1774 (1998).
\bibitem{karmen}KARMEN Collab., K. Eitel, B. Zeitnitz, in Proceedings of
Neutrino-98, Nucl. Phys. (Proc. Suppl.) {\bf 77}, 212 (1999).
\bibitem{jarlskog} C. Jarlskog, Phys. Rev. Lett. {\bf 55}, 1039 (1985).
\bibitem{bilenky} S. M. Bilenky, C. Giunti, and W. Grimus, hep-ph/9812360.
\bibitem{h-m} L. Baudis et al., Phys. Lett. B {\bf 407}, 219 (1997);
M. Gunther et al., Phys. Rev. D {\bf 55}, 54 (1997).
\bibitem{donini} A. Donini, M.B. Gavela, P. Hern\'{a}ndez, and S. Rigolin,
hep-ph/9910516.
\bibitem{wolf} L. Wolfenstein, Phys. Rev. {\bf D17}, 2369 (1978).
%\bibitem{ms}
S. P. Mikheyev and A. Smirnov, Yad. Fiz. {\bf 42}, 1441 (1985)
[Sov.J. Nucl. Phys. {\bf 42}, 913 (1986)], Nuovo Cim., {\bf C9}, 17 (1986).
%\bibitem{barger80}
V. Barger, K. Whisnant, S. Pakvasa, and R. J. N. Phillips, Phys. Rev.
{\bf D22}, 2718 (1980). See also V. Barger, K. Whisnant, and
R. J. N. Phillips, Rev. Rev. Lett. {\bf 45}, 2084 (1980).
%\bibitem{snowmass}
D. Ayres, T. Gaisser, A. K. Mann, and R. Shrock, in {\it Proceedings of the
1982 DPF Summer Study on Elementary Particles and Future Facilities}, Snowmass,
p. 590; D. Ayres, B. Cortez, T. Gaisser, A. K. Mann, R. Shrock, and L. Sulak,
Phys. Rev. {\bf D29}, 902 (1984).
%\bibitem{kp88}
P. Krastev, S. Petcov, Phys. Lett. {\bf B205}, 8 (1988).
%\bibitem{baltz}
A. J. Baltz, J. Weneser, Phys. Rev. {\bf D37}, 3364 (1988).
%\bibitem{petcov}
S. Petcov, Phys. Lett. {\bf B434}, 321 (1998).
M. Chizhov, M. Maris, S. Petcov, hep-ph/9810501;
M. Chizhov, S. Petcov, hep-ph/9903424;
M.Chizhov, S.Petcov, Phys. Rev. Lett. {\bf 83}, 1096 (1999).
%\bibitem{akh}
E. Akhmedov, A. Dighe, P. Lipari, A. Smirnov,
Nucl. Phys. {\bf B542}, 3 (1999);
E. Akhmedov, Nucl.Phys. {\bf B538}, 25 (1999); hep-ph/0001264.
%\bibitem{bernpark}
R. H. Bernstein and S. J. Parke, Phys. Rev. {\bf D44}, 2069 (1991).
%\bibitem{arubbia}
M. Campanelli, A. Bueno, A. Rubbia, hep-ph/9905240.
%\bibitem{gandhi}
S. Dutta, R. Gandhi, and B. Mukhopadhyaya, hep-ph/9905475.
%\bibitem{bargergeer}
V. Barger, S. Geer, K. Whisnant, Phys.Rev. {\bf D61}, 053004 (2000).
%\bibitem{kim}
D. Dooling, C. Giunti, K. Kang, C. W. Kim, hep-ph/9908513.
%\bibitem{bargergeer2}
V. Barger, S. Geer, R. Raja, K. Whisnant, hep-ph/9911524.
%\bibitem{fos}
M. Freund, T. Ohlsson, hep-ph/9909501;
T. Ohlsson, H. Snellman, hep-ph/9910546, hep-ph/9912295.
%\bibitem{lindneretal}
M. Freund, M. Lindner, S.T. Petcov, A. Romanino, hep-ph/9912457.
%\bibitem{hern} A. Cervera, A. Donini, M. B. Gavela, J. J. Gomez C\'adenas,
P. Hern\'andez, O. Mena, S. Rigolin, hep-ph/0002108
\bibitem{lipari} P. Lipari, hep-ph/9903481.
\bibitem{nnn99} I. Mocioiu, R. Shrock, in the Proceedings of the Workshop on
the Next Generation Nucleon Decay and Neutrino Detector NNN99, Stony Brook
(Sept. 1999), hep-ph/9910554.
\bibitem{cpv}
S.M. Bilenky, C. Giunti, W.Grimus, Phys.Rev.{\bf D58}, 033001
(1998); K. Dick, M. Freund, M. Lindner, A. Romanino,
Nucl. Phys. {\bf B562}, 29 (1999); M. Tanimoto,
Phys. Lett. {\bf B462}, 115 (1999);
A. Donini, M.B. Gavela, P. Hernandez, S. Rigolin, hep-ph/9909254;
M. Koike, J. Sato, hep-ph/9909469;
P.F. Harrison, W.G. Scott, hep-ph/9912435.
\bibitem{lb} I. Mocioiu, R. Shrock, hep-ph/0002149.
\bibitem{anl}
%References and websites for these experiments and future
%projects can be found, e.g., at\\
%%%http://www.hep.anl.gov/ndk/hypertext/nu\_industry.html.
%\bibitem{geer}
S. Geer, Phys.Rev. {\bf D57},\\ 6989 (1998).
%\bibitem{dgh}
De Rujula, M. B. Gavela, and P. Hernandez, Nucl. Phys.
{\bf B547}, 21 (1999).
%\bibitem{web}
%Some relevant websites at BNL, FNAL, and CERN containing
%further information are
%%%http://www.cap.bnl.gov/mumu.
%%%http://www.fnal.gov/projects/muon\_collider/physics.\\
%%%http://www.cern.ch/~autin/nufact99/whitepap.ps.\\
%\bibitem{bnl98}
Workshop on the Potential for Neutrino Physics at Future Muon
Colliders, BNL (Aug. 1998),
%%%http://pubweb.bnl.gov/people/bking/nushop/workshop.html.\\
%\bibitem{kirk}
Neutrino Factory and Muon Collider Collaboration, Expression of
Interest for R+D towards a Neutrino Factory Based on a Storage Ring and Muon
Collider, physics/9911009,
%%%http://puhep1.princeton.edu/mumu/NSFLetter/nsfmain.ps.
%\bibitem{bnlbook}
I. Bigi et al., The Potential for Neutrino Physics at Muon
Colliders and Other High-Current Muon Storage Rings, to appear in Phys. Rept.;
%see also http://pubweb.bnl.gov/people/bking/nushop.
\bibitem{prem} A.Dziewonski, Earth Structure, in: ''The Encyclopedia of Solid
Earth Geophysics'', D.E.James (Ed.), (Van Nostrand Reinhold, New York, 1989),
p.331.
%%%%%%%% where will we be in 5-10 years
\bibitem{evidence} Y.~Fukuda {\it et al.} [Super-Kamiokande
Collaboration],
%``Evidence for oscillation of atmospheric neutrinos,''
Phys.\ Rev.\ Lett.\ {\bf 81}, 1562 (1998) [hep-ex/9807003].
%%CITATION = HEP-EX 9807003;%%
%\href{\wwwspires?eprint=HEP-EX/9807003}{SPIRES}
\bibitem{Kajita} T. Kajita, talk at 7th International Symposium On
Particles, Strings And Cosmology (PASCOS 99) 10-16 Dec 1999,
Granlibakken, Tahoe City, California, http://pc90.ucdavis.edu/schedule.html
\bibitem{CHOOZ} M.~Apollonio {\it et al.},
%``Limits on neutrino oscillations from the CHOOZ experiment,''
Phys.\ Lett.\ {\bf B466}, 415 (1999)
[hep-ex/9907037].
%%CITATION = HEP-EX 9907037;%%
%\href{\wwwspires?eprint=HEP-EX/9907037}{SPIRES}
\bibitem{PaloVerde} F.~Boehm {\it et al.},
%``Results from the Palo Verde neutrino oscillation experiment,''
hep-ex/0003022.
%%CITATION = HEP-EX 0003022;%%
%\href{\wwwspires?eprint=HEP-EX/0003022}{SPIRES}
\bibitem{Learned} J. Learned, talk at International Workshop On Next
Generation Nucleon Decay And Neutrino Detector (NNN 99), 23-25 Sep
1999, Stony Brook, New York.
\bibitem{K2K} Nishikawa, talk at NNN 99.
\bibitem{MINOS} MINOS Technical Design Report NuMI-L-337 TDR,
{\tt http://www.hep.anl.gov/ndk/hypertext/minos\_tdr.html}
\bibitem{OPERA} OPERA Progress Report, LNGS-LOI 19/99,
{\tt http://www.cern.ch/opera/documents.html}
\bibitem{ICANOE} ICANOE proposal, LNGS-P21/99,
{\tt http://pcnometh4.cern.ch/}.
\bibitem{MONOLITH} A.Curioni, talk at NNN 99.
\bibitem{AQUARICH} T. Ypsilantis, talk at NNN 99.
\bibitem{LSND} C.~Athanassopoulos {\it et al.} [LSND Collaboration],
%``Evidence for anti-nu/mu --> anti-nu/e oscillation from the LSND experiment at the Los Alamos Meson Physics Facility,''
Phys.\ Rev.\ Lett.\ {\bf 77}, 3082 (1996)
[nucl-ex/9605003].
%%CITATION = NUCL-EX 9605003;%%
%\href{\wwwspires?eprint=NUCL-EX/9605003}{SPIRES};
C.~Athanassopoulos {\it et al.} [LSND Collaboration],
%``Evidence for nu/mu --> nu/e neutrino oscillations from LSND,''
Phys.\ Rev.\ Lett.\ {\bf 81}, 1774 (1998)
[nucl-ex/9709006].
%%CITATION = NUCL-EX 9709006;%%
%\href{\wwwspires?eprint=NUCL-EX/9709006}{SPIRES}
\bibitem{Mini-BooNE} {\tt http://www-boone.fnal.gov/}
\bibitem{Valle} M.~C.~Gonzalez-Garcia, P.~C.~de Holanda, C.~Pena-Garay
and J.~W.~Valle,
%``Status of the MSW solutions of the solar neutrino problem,''
hep-ph/9906469.
%%CITATION = HEP-PH 9906469;%%
%\href{\wwwspires?eprint=HEP-PH/9906469}{SPIRES}
\bibitem{SNO} Sudbury Neutrino Observatory, http://www.sno.phy.queensu.ca/
\bibitem{Borexino} Borexino experiment, http://almime.mi.infn.it/
\bibitem{Lanou} R.E.~Lanou, Jr., Invited talk at 18th International
Conference on Neutrino Physics and Astrophysics (NEUTRINO 98),
Takayama, Japan, 4-9 Jun 1998, {\sl Nucl. Phys. Proc. Suppl.}\/ {\bf
77}, 55 (1999), hep-ex/9808033.
\bibitem{KamLAND} J.~Busenitz {\it et al}\/., ``Proposal for US
Participation in KamLAND,'' March 1999, http://bkf0.lbl.gov/kamland/
%%%%%%%%% detector sub-section
\bibitem{forty}R. Forty, JHEP 9912:002,1999 \verb+hep-ex/9910061+
%\bibitem{geer}need other geer reference with muon energy cut...plus
% V. Barger, S. Geer, R. Raja, K. Whisnant, \verb+hep-ph/9911524+
\bibitem{strolin} P.~Strolin, Nufact'99 Workshop,July 5-9th, Lyon
\bibitem{para} D.A.~Harris, A. Para, \verb+hep-ex/0001035+
\bibitem{atlas} W.W. Armstrong {\em et al}, Atlas Technical Proposal.
\bibitem{nutevdet} D.A. Harris, J.Yu {\em et al}.,
NuTeV collaboration, hep-ex/9908056.
{\it To appear in Nucl.Instrum.Meth.A.}
\bibitem{nutevpub} Numonte Reference...TBA
\bibitem{LArdedx} Doke, {\em et al.}, Nucl. Instrum. Meth {\bf A237} 475 (1985)
\bibitem{opera} K.Kodama et al, CERN-SPSC-98-25, Oct 1998.
%%%%% measurements sub-section
\bibitem{cerv00}
A. Cervera et al., ``Golden measurements at a neutrino factory",
hep-ph/0002108.
\label{cerv00}
\bibitem{bgrw00}
V. Barger, S. Geer, R. Raja, K. Whisnant, ``Neutrino oscillations at an
entry--level neutrino factory and beyond", Fermilab-PUB 00/049-T.
\label{bgrw00}
\bibitem{camp00}
M. Campanelli et al; in preparation.
\label{camp00}
\bibitem{bern00}
R. Bernstein, in preparation.
\label{bern00}
\bibitem{bgrw_prep}
V. Barger, S. Geer, R. Raja, K. Whisnant; in preparation.
\label{bgrw_prep}
\bibitem{geer98}
S. Geer, Phys. Rev D57, 6989 (1998).
\bibitem{bgrw99}
V. Barger, S. Geer, R. Raja, K. Whisnant;
hep-ph/9911524, submitted to PRD (in press).
\input refs5
%Jorge's ref
% Jae Yu's ref
\bibitem{ccfr:bjking}
C.Arroyo, B.J.King {\it et. al.}, Phys. Rev. Lett. {\bf 72}, 3452 (1994).
\bibitem{ccfr:mcfarland}
K.S.McFarland {\it et al.}, CCFR, Eur. Phys. Jour. {\bf C1}, 509 (1998)
\bibitem{th:llsmith}
C.H.Llewellyn Smith, Nucl. Phys. {\bf B228}, 205 (1983)
\bibitem{ex:ssqt}
R.Bernstein {\it et. al.}, NuTeV Collaboration, Fermilab-TM-1088 (1994).
\bibitem{th:paschos}
E.A.Paschos and L. Wolfenstein, Phys. Rev. {\bf D7}, 91 (1973)
\bibitem{NuTeV:prelim}
K.S.McFarland, NuTeV collaboration, to be published in the proceedings of the XXXIIIrd Rencontres de Moriond (1998);
J. Yu, NuTeV collaboration, to be published in the proceedings of the DIS98, Brussel, Belgium, Fermilab-Conf-98/200-E (1998)
\bibitem{ex:heidi}
H. Schellman, to be published in the
proceedings of the Workshop on Physics at the First Muon Collider,
Fermilab, Batavia, IL, USA, Nov. 1997.
\bibitem{ex:dah_ksm}
D.A.Harris \& K.S.McFarland, MIT-LNS-98-276, to be published in the
proceedings of the Workshop on Physics at the First Muon Collider,
Fermilab, Batavia, IL, USA, Nov. 1997
\bibitem{ex:bjking_det}
B.J.King, to be published in the
proceedings of the Workshop on Physics at the First Muon Collider,
Fermilab, Batavia, IL, USA, Nov. 1997.
\bibitem{ph:arie-unki-ht}
U.K Yang and A. Bodek, {\it UR-1543}, Submitted to Phys. Rev. Lett,
hep-ex/9809480 (1998)
\bibitem{ex:tev33_mw}
U. Baur and M. Demarteau, ``Precision Electroweak Physics at Future Collider
Experiments,'' Fermilab-Conf-96/423 (1996).
\bibitem{CHARM2} P.~Villain {\em et al.\,}, Phys.\ Lett.\ {\bf B335}:246 (1994).
See also Phys.\ Lett.\ {\bf B302}:351 (1993) and
Phys.\ Lett.\ {\bf B281}:159 (1992).
%....additional from ew chapter (Bolton)...
\bibitem{bardin}D. Yu. Bardin and V. A. Dokuchaeva, JINR E2-86-260 (1986).
\bibitem{baur}U. Baur, ``Electroweak Radiative Corrections to $W$ Boson
Production at the Tevatron'', SUNY-Buffalo preprint UB-HET-98-02, Sep 1998
\ (e-Print Archive: hep-ph/9809327).
\bibitem{sather}E. Sather, Phys. Lett. \textbf{B274} (1992) 433.
%NHL ref
\bibitem{GLR}
M.Gronau, C.N. Leung, and J.L. Rosner, \prev{D29}{2539}{1984}
%
\bibitem{Shrock}
R. E. Shrock, \prev{D24}{1232}{1981}
%
\bibitem{Bolton}
T. Bolton, L. Johnson, and D. McKay, Phys. Rev. D {\bf 56} (1997) 2970.
%
\bibitem{NuTeVNHL}
A. Vaitaitis {\em et al.}, Phys. Rev. Lett. {\bf 83} (1999) 4943.
%
\bibitem{NuTeVQ0}
J. Formaggio {\em et al.}, Phys. Rev. Lett. {\bf XXX} (2000) XXX.
%
\bibitem{PDB}
{\it Review of Particle Physics} Eur. Phys. J. C. 3,320 (1998).
%
\bibitem{Shrock}
R. Shrock, Phys. Rev. D {\bf 24} (1981) 1232.
%John Krane's ref
\bibitem{shrock} B.W.Lee and R.E.Shrock, Phys. Rev. \textbf{D16} 1444
(1977).
\bibitem{frank} M.Frank, Phys. Rev. D60:093005 (1999).
\bibitem{ng} G.C.McLaughlin and J.N.Ng, hep-ph/9909558.
\bibitem{beacom} J.F.Beacom and P.Vogel, Phys. Rev. Lett. \textbf{83}, 5222
(1999).
\bibitem{mu_e_limit} A.V.Derbin, Phys. of Atomic Nuclei, \textbf{57}, 222
(1994) [Yad. Fiz. \textbf{57}, 236 (1994)], and A.I.Derbin \textit{et al., }%
JETP Lett. \textbf{57}, 768 (1993).
\bibitem{mu_mu_limit} D.A.Krakauer \textit{et al.}, Phys. Lett. \textbf{B252%
}, 177 (1990); R.C.Allen \textit{et al.}, Phys. Rev. \textbf{D47}, 11 (1993).
\bibitem{star1} MFukugita and S.Yazaki, Phys. Rev. \textbf{D36} (1987) 3817.
\bibitem{star2} R.Barbieri and R.N.Mohapatra, Phys. Rev. Lett. \textbf{61}
(1988) 27, and J.M. Lattimer and J.Cooperstein, Phys. Rev. Lett. \textbf{61}
(1988) 23.
\bibitem{norbert} Private discussions with Norbert Holtkamp, Fermilab.
\bibitem{geer_2day} See Steve Geer's talk at the 2-day meeting. Number of
events (5000) was quoted for a 50 kTon detector at L=2800 km and a 20 GeV
muon storage ring.
% Fritz's ref
\bibitem{PDG} Review of Particle Properties, C.~Caso {\it et. al.},
Euro. Phys. J. {\bf C}3, 1 (1998).
\bibitem{dreiner} H. Dreiner, hep-ph/9707435
\bibitem{quigg} Chris Quigg, private communication
\bibitem{emulsion}
K. Kodama {\it et. al.} (OPERA Collaboration), CERN/SPSC 98-25. \\
A. E. Asratyan {\it et. al.}, hep-ex/0002019
\bibitem{Adler} S. Adler, Phys.Rev.143, (1966), 1144-1155.
\bibitem{Boros} C. Boros et al., hep-ph/9810220, Oct. 1998.
\bibitem{Kulagin} S.A. Kulagin, hep-ph/9812532, Aug. 1998.
\bibitem{Eskola} K.J. Eskola et al. hep-ph/9906484, May, 1999.
\bibitem{SAMPLE} D.T. Spayde et al. nucl-ex/9909010, Phys.Rev.Lett.84:1106-1109,2000
\bibitem{nutev-wma} K.~McFarland {\em et al.\,}, Eur. Phys. J.
{\bf C1} (1998) 509.
\bibitem{CHARMII} P.~Vilain {\em et al\,}, Phys.\ Lett.\ {\bf B335} (1994) 248.
\bibitem{numcbook} "The Potential
for Neutrino Physics at Muon Colliders and Other Muon Storage
Rings", B.~King {\em et al}, to appear in {\em Physics Reports}.
\bibitem{ICANOE} ICANOE Proposal, CERN/SPSC 99-25.
\bibitem{bruce} Bruce King, ``High Rate Physics at Neutrino Factories'',
Proc. 23rd Johns Hopkins Workshop on Current Problems in Particle Theory,
\verb+hep-ex/9911008+.
\bibitem{E791} E.~Aitala {\em et al\,}, Phys.\ Rev.\ {\bf D57} (1998) 13.
\bibitem{CLEO} M.~Artuso {\em et al\,}, ``Search for $D^0-\overline{D^0}$
Mixing'', \verb+hep-ex/9908040+.
\bibitem{BaBar} P.~F.~Harrison and H.~R.~Quinn [BABAR Collaboration],
``The BaBar physics book: Physics at an asymmetric B factory,''
SLAC-R-0504.
\bibitem{brucehere} Bruce King, ``High Rate Neutrino Detectors for Neutrino
Factories'', Proc. ICFA/ECFA Workshop
"Neutrino Factories based on Muon Storage Rings"
(nuFACT'99), \verb+hep-ex/0001043+.
% Kevin's ref--- fist section
\bibitem{nutev-wma} K.~McFarland {\em et al.\,}, Eur. Phys. J.
{\bf C1} (1998) 509.
\bibitem{numcbook} "The Potential
for Neutrino Physics at Muon Colliders and Other Muon Storage
Rings", B.~King {\em et al}, to appear in {\em Physics Reports}.
\bibitem{CHARMII} P.~Vilain {\em et al\,}, Phys.\ Lett.\ {\bf B335} (1994) 248.
\bibitem{ICANOE} ICANOE Proposal, CERN/SPSC 99-25.
% Jae Yu's ref
\bibitem{ccfr:bjking}
C.Arroyo, B.J.King {\it et. al.}, Phys. Rev. Lett. {\bf 72}, 3452 (1994).
\bibitem{ccfr:mcfarland}
K.S.McFarland {\it et al.}, CCFR, Eur. Phys. Jour. {\bf C1}, 509 (1998)
\bibitem{th:llsmith}
C.H.Llewellyn Smith, Nucl. Phys. {\bf B228}, 205 (1983)
\bibitem{ex:ssqt}
R.Bernstein {\it et. al.}, NuTeV Collaboration, Fermilab-TM-1088 (1994).
\bibitem{th:paschos}
E.A.Paschos and L. Wolfenstein, Phys. Rev. {\bf D7}, 91 (1973)
\bibitem{NuTeV:prelim}
K.S.McFarland, NuTeV collaboration, to be published in the proceedings of the XXXIIIrd Rencontres de
Moriond (1998);
J. Yu, NuTeV collaboration, to be published in the proceedings of the DIS98, Brussel, Belgium,
Fermilab-Conf-98/200-E (1998)
\bibitem{ex:heidi}
H. Schellman, to be published in the
proceedings of the Workshop on Physics at the First Muon Collider,
Fermilab, Batavia, IL, USA, Nov. 1997.
\bibitem{ex:dah_ksm}
D.A.Harris \& K.S.McFarland, MIT-LNS-98-276, to be published in the
proceedings of the Workshop on Physics at the First Muon Collider,
Fermilab, Batavia, IL, USA, Nov. 1997
\bibitem{ex:bjking_det}
B.J.King, to be published in the
proceedings of the Workshop on Physics at the First Muon Collider,
Fermilab, Batavia, IL, USA, Nov. 1997.
\bibitem{ph:arie-unki-ht}
U.K Yang and A. Bodek, {\it UR-1543}, Submitted to Phys. Rev. Lett,
hep-ex/9809480 (1998)
\bibitem{ex:tev33_mw}
U. Baur and M. Demarteau, ``Precision Electroweak Physics at Future Collider
Experiments,'' Fermilab-Conf-96/423 (1996).
\bibitem{CHARM2} P.~Villain {\em et al.\,}, Phys.\ Lett.\ {\bf B335}:246 (1994).
See also Phys.\ Lett.\ {\bf B302}:351 (1993) and
Phys.\ Lett.\ {\bf B281}:159 (1992).
%....additional from ew chapter (Bolton)...
\bibitem{bardin}D. Yu. Bardin and V. A. Dokuchaeva, JINR E2-86-260 (1986).
\bibitem{baur}U. Baur, ``Electroweak Radiative Corrections to $W$ Boson
Production at the Tevatron'', SUNY-Buffalo preprint UB-HET-98-02, Sep 1998
\ (e-Print Archive: hep-ph/9809327).
\bibitem{sather}E. Sather, Phys. Lett. \textbf{B274} (1992) 433.
\bibitem{GLR}
M.Gronau, C.N. Leung, and J.L. Rosner, \prev{D29}{2539}{1984}
%
\bibitem{Shrock}
R. E. Shrock, \prev{D24}{1232}{1981}
%
\bibitem{Bolton}
T. Bolton, L. Johnson, and D. McKay, Phys. Rev. D {\bf 56} (1997) 2970.
%
\bibitem{NuTeVNHL}
A. Vaitaitis {\em et al.}, Phys. Rev. Lett. {\bf 83} (1999) 4943.
%
\bibitem{NuTeVQ0}
J. Formaggio {\em et al.}, Phys. Rev. Lett. {\bf XXX} (2000) XXX.
%
\bibitem{PDB}
{\it Review of Particle Physics} Eur. Phys. J. C. 3,320 (1998).
%
\bibitem{Shrock}
R. Shrock, Phys. Rev. D {\bf 24} (1981) 1232.
%John Krane's ref
\bibitem{shrock} B.W.Lee and R.E.Shrock, Phys. Rev. {\bf D16} 1444 (1977).
\bibitem{frank} M.Frank, Phys. Rev. D60:093005 (1999).
\bibitem{ng} G.C.McLaughlin and J.N.Ng, hep-ph/9909558.
\bibitem{beacom} J.F.Beacom and P.Vogel, Phys. Rev. Lett. {\bf 83}, 5222
(1999).
\bibitem{mu_e_limit} A.V.Derbin, Phys. of Atomic Nuclei, {\bf 57}, 222
(1994) [Yad. Fiz. {\bf 57}, 236 (1994)], and A.I.Derbin {\it et al., }JETP
Lett. {\bf 57}, 768 (1993).
\bibitem{mu_mu_limit} D.A.Krakauer {\it et al.}, Phys. Lett. {\bf B252},
177 (1990); R.C.Allen {\it et al.}, Phys. Rev. {\bf D47}, 11 (1993).
\bibitem{star1} MFukugita and S.Yazaki, Phys. Rev. {\bf D36} (1987) 3817.
\bibitem{star2} R.Barbieri and R.N.Mohapatra, Phys. Rev. Lett. {\bf 61}
(1988) 27, and J.M. Lattimer and J.Cooperstein, Phys. Rev. Lett. {\bf 61}
(1988) 23.
\bibitem{norbert} Private discussions with Norbert Holtkamp, Fermilab.
\bibitem{geer_2day} See Steve Geer's talk at the 2-day meeting. Number of
events (5000) was quoted for a 50 kTon detector at L=2800 km and a 20 GeV
muon storage ring.
% Fritz's ref
\bibitem{PDG} Review of Particle Properties, C.~Caso {\it et. al.},
Euro. Phys. J. {\bf C}3, 1 (1998).
\bibitem{dreiner} H. Dreiner, hep-ph/9707435
\bibitem{quigg} Chris Quigg, private communication
\bibitem{emulsion}
K. Kodama {\it et. al.} (OPERA Collaboration), CERN/SPSC 98-25. \\
A. E. Asratyan {\it et. al.}, hep-ex/0002019
\end{thebibliography}
%\end{references}
\end{document}