\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~\cite{superk}.
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, namely $\delta m^2 =
$~\cal{O}($10^{-3}$)~eV$^2$
and $\sin^2 2\theta =$~\cal{O}(1).
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 cannot guarantee that these insights will be forthcoming from
neutrino oscillation measurements, but they might be. For this reason
it is important to understand 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 the following we address how this new tool might
be exploited to go well beyond the capabilities of the next generation
of neutrino oscillation experiments.
%In the following sub--sections we begin by introducing the theoretical
%concepts and framework used to describe neutrino oscillations, and
%then describe what is likely to learnt from the next generation
%of neutrino oscillation experiments that will be completed before
%a neutrino factory can be built. We then consider the characteristics and
%capabilities of detectors at a neutrino factory, and finally describe
%the physics program that could be pursued with these detectors
%as a function of the muon beam intensity and energy,
%and the baseline.
In this section we begin by describing the theoretical basis for
neutrino oscillations, and then define a selection of oscillation
parameter sets that can be used in assessing the physics program
at a neutrino factory. This is followed by a summary of the current
experimental status and how
it can be expected to change in the next few years.
We then discuss the parameters and the performance
of candidate detectors at a neutrino factory.
The section is completed
with a survey of the physics measurements that can be
performed at a neutrino factory as a function of beam energy,
intensity, and baseline, and finally, a summary of our conclusions.
\subsection{Theoretical framework}
\label{theory}
There exist 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 the 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.
The simplest scenarios, in which one or more Higgs triplets are introduced to
couple to a pair of left-handed neutrinos,
are excluded by measurements of the $\rho$ parameter.
Therefore, other extensions of the SM must be considered,
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 $m_D \sim m_t$ and $m_R \sim 10^{16}$~GeV,
as suggested in a (supersymmetric) SO(10) grand unified theory framework,
a scale of
$m_\nu \sim 10^{-3}$~eV is readily obtained.
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.
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 $n_\ell$ light 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 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-known 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 $1.2669 \cdots \delta m^2_{ij}L/E$ 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 all the
individual oscillation lengths,
$\lambda_{ij}^{\rm osc} \sim E/\delta m^2_{ij}$, 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.
Therefore CP violating effects are largest and hence easiest to observe
at distances between the smallest and largest oscillation lengths.
% 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
$m_1 < m_2 \ll m_3$
and the assumption that the LSND effect is not a neutrino oscillation
phenomena,
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_{e3}|^2 (1 - |U_{e3}|^2)
\sin^2 \Bigl (\frac{\delta m^2_{atm}L}{4E} \Bigr ) \\[0.05in]
= & 1 - \sin^2(2\theta_{13})
\sin^2 \Bigl (\frac{\delta m^2_{atm}L}{4E} \Bigr ) \; ,
\end{array}
\end{equation}
\beq
\begin{array}{rl}
P(\nu_e \to \nu_\mu) \simeq& 4|U_{e3}|^2 |U_{\mu 3}|^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) \simeq& 4|U_{\tau 3}|^2 |U_{e3}|^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
\noindent
and
\beq
\begin{array}{rl}
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}}) \\[0.05in]
=& \sin^2(2\theta_{23})\cos^4(\theta_{13})
\sin^2 \Bigl (\frac{\delta m^2_{atm}L}{4E} \Bigr ) \; .
\end{array}
\eeq
The CP-odd contribution to the atmospheric neutrino oscillation probability
vanishes in the one-mass-scale-dominant approximation.
However if we include the effects of the small mass scale, $\delta m^2_{21}$,
then
\begin{equation}
\begin{array}{rl}
P_{\rm CP-odd}(\nu_\mu \rightarrow & \nu_\tau)
= -4c_{12}c^2_{13}c_{23}s_{12}s_{13}s_{23}(\sin \delta)
\\[0.05in]
& \left [ \sin(\frac{\delta m^2_{21}L}{2E})
\sin^2(\frac{\delta m^2_{atm}L}{4E})
+ \sin(\frac{\delta m^2_{atm}L}{2E})
\sin^2(\frac{\delta m^2_{21}L}{4E}) \right ] .
\end{array}
\end{equation}
%
At distances significantly larger than the atmospheric neutrino oscillation
length, $ E/\delta m^2_{atm}$, the second term in brackets
averages to zero whereas the $\sin$ squared part of the first term
averages to one half, leaving
\begin{equation}
\begin{array}{rl}
P_{\rm CP-odd}(\nu_\mu \rightarrow & \nu_\tau)
\simeq 2c_{12}c^2_{13}c_{23}s_{12}s_{13}s_{23}(\sin \delta)
\sin(\frac{\delta m^2_{21}L}{2E}).
\end{array}
\end{equation}
The Jarlskog factor~\cite{jarlskog}, J, is given by
$J=c_{12}c^2_{13}c_{23}s_{12}s_{13}s_{23}(\sin \delta)$
and is a convenient measure of the size of the CP violation.
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}) \,
\label{eq:defnA}
\end{equation}
(for $\bar{\nu_e}$ A is replaced with -A).
%
Here $Y_e$ is the electron fraction and $\rho(t)$ is the matter density.
Density profiles through the
earth can be calculated using the Earth Model~\cite{prem},
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 sufficient
and oscillation calculations must explicitly take account of $\rho(t)$.
However the constant density approximation is very useful to understand
the physics of neutrinos propagating through the earth since the variation
of the earth's density is not large.
\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.
Calculation from Ref.~\ref{bgw99}.
}
\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 \rangle \,,
\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)
\label{eq:matter}
\end{equation}
%
and $\theta_{13}^m$ is related to $\theta_{13}$ by
%
\begin{equation}
\tan 2\theta_{13}^m = {\sin2\theta_{13}} / \left( {\cos2\theta_{13}
- {A\over \delta m^2}}\right) \,. \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}\,, \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} /
\left({ \left( {A\over\delta m^2} - \cos 2\theta_{13} \right)^2
+ \sin^2 2\theta_{13}} \right) \,. \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 neutrinos
and only for negative $\delta m^2$ for
anti-neutrinos.\footnote{If the LSND effect is due to neutrino oscillations
then $\delta m^2 >>$~O($10^{-3}$)~eV$^2$ and
the resonance occurs at energies much higher than those of interest
at a neutrino factory.}
For negative
$\delta m^2$ the oscillation amplitude in Eq.~(\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{lb} 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 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.~\ref{lb} 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 by
considering constraints from existing experiments.
If we assume CPT invariance then
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
$\sin^2 2\theta_{12} \sim 1.0$ 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 \\
$\delta m^2_{21}$ (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{s1LSND}
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,
$$R_{23}(\theta_{23},\delta_3) = \left(\matrix{ 1 & 0 & 0 & 0 \cr
0 & c_{23} & s_{23}e^{-i\delta_3} & 0 \cr
0 & -s_{23}e^{i\delta_3} & c_{23} & 0 \cr
0 & 0 & 0 & 1 \cr}\right).$$
%$$R_{13}(\theta_{13},\delta) = \left(\matrix{c_{13} & 0 &
% s_{13}e^{-i\delta} & 0 \cr
% 0 & 1 & 0 & 0 \cr
% -s_{13}e^{i\delta} & 0 & c_{13} & 0 \cr
% 0 & 0 & 0 & 1 }
% \right) \; .\\$$
\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({{\delta M^2 L}\over{4E}}\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({{\delta M^2 L}\over{4E}}\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({{\delta M^2 L}\over{4E}}\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({{\delta M^2 L}\over{4E}}\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({{\delta M^2 L}\over{4E}}\right) \; . \\
\end{array}
\end{equation}
If the neutrinos propagate through matter, these expressions
must be modified.
Matter effects for the three active and one sterile neutrino scenario
are similar in nature to those for the three active neutrino case,
Eq.~(\ref{eq:prop}).
However in Eq.~(\ref{eq:prop}) a flavor diagonal term
that only contributes to an overall phase has been discarded.
This term comes from the coherent forward scattering amplitude for the
active flavors scattering from the electrons, protons and neutrons
in matter via the exchange of a virtual Z-boson.
Since the sterile neutrino does not interact with the Z-boson this
term must be added to the diagonal terms for the active
neutrinos (or equivalently
subtracted from the diagonal part for the sterile neutrino).
That is in Eq.~(\ref{eq:prop})
%
\begin{equation}
{A \over 2E_\nu} \delta_{\alpha e} \delta_{\beta e}
\rightarrow
{A \over 2E_\nu} \delta_{\alpha e} \delta_{\beta e}
-{A^\prime \over 2E_\nu} \delta_{\alpha s} \delta_{\beta s}
\end{equation}
\noindent
where $A^\prime$ is given by Eq.~(\ref{eq:defnA}) with
$Y_e$ replaced by $-\frac{1}{2}(1-Y_e)$
for electrically neutral matter.
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 to be 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 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{s1LSND} 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}