%\subsection[Neutrino Oscillation Experiments ]{Neutrino Oscillation Experiments with a Muon Storage Ring/Neutrino Factory} 
Here we discuss the current evidence for neutrino oscillations, and hence
neutrino masses and lepton mixing, from solar and atmospheric data.  A review
is given of some theoretical background including models for neutrino masses
and relevant formulas for neutrino oscillation transitions.  We next mention
the near-term and mid-term experiments in this area and comment on what they
hope to measure.  We then discuss the physics potential of a muon storage ring
as a Neutrino Factory in the long term. 

\subsection{Evidence for Neutrino Oscillations} 

In a modern theoretical context, one generally expects nonzero neutrino masses
and associated lepton mixing.  Experimentally, there has been accumulating
evidence for such masses and mixing.  All solar neutrino experiments
(Homestake, Kamiokande, SuperKamiokande (SuperK), SAGE, and GALLEX) show a significant
deficit in the neutrino fluxes coming from the Sun \cite{sol}. This deficit
can be explained by oscillations of the $\nu_e$'s into other weak
eigenstate(s), with $\Delta m^2_{sol}$ of the order $10^{-5}$ eV$^2$ for
solutions involving the Mikheev-Smirnov-Wolfenstein (MSW) resonant matter
oscillations \cite{wolf,ms} or of the order of $10^{-10}$ eV$^2$ for vacuum
oscillations.  Accounting for the data with vacuum oscillations (VO) requires
almost maximal mixing.  The MSW solutions include one for small mixing angle
(SMA) and one with large mixing angle (LMA).

Another piece of evidence for neutrino oscillations is the atmospheric neutrino
anomaly, observed by Kamiokande \cite{kam}, IMB \cite{imb}, SuperKamiokande
\cite{sk} with the highest statistics, and also by Soudan \cite{soudan2} and MACRO
\cite{macro}.  These data can be fit by the inference of $\nu_{\mu} \rightarrow
\nu_x$ oscillations with $\Delta m^2_{atm}\sim 3.5 \times 10^{-3}$ eV$ ^2$
\cite{sk} and maximal mixing, \textit{i.e.}, $\sin^2 2 \theta_{atm} = 1$.  The identification
$\nu_x = \nu_\tau$ is preferred over $\nu_x=\nu_{sterile}$, and the
identification $\nu_x=\nu_e$ is excluded by both the SuperKamiokande data and
the Chooz experiment~\cite{chooz}.

In addition to the above results, the LSND experiment \cite{lsnd} has reported observing
$\bar\nu_\mu \to \bar \nu_e$ and $\nu_{\mu} \to \nu_e$ oscillations with
$\Delta m^2_{LSND} \sim 0.1 - 1$ eV$^2$ and a range of possible mixing angles,
depending on $\Delta m^2_{LSND}$. This result is not confirmed, but also not
completely ruled out, by a similar experiment, KARMEN \cite{karmen}.  The
miniBOONE experiment at Fermilab is designed to resolve this issue, as
discussed below. 

With only three neutrino species, it is not possible to fit all of these experiments. They involve
three quite different values of $\Delta m^2_{ij}=m(\nu_i)^2-m(\nu_j)^2$ which
could not satisfy the identity for only three neutrino species that 
%
\beq
\Delta m^2_{32} + \Delta m^2_{21} + \Delta m^2_{13}=0.
\label{mident}
\eeq
It would follow then, that one would have to introduce further neutrino(s).  As we know that there are only three leptonic weak doublets, and associated light 
neutrinos, with weak isospin $T=1/2$ and $T_3=1/2$ from the measurement of the 
$Z$ width, it follows that additional neutrino weak eigenstates would have to
be electroweak singlets (that is, ``sterile'' neutrinos).  Because the LSND experiment
has not been confirmed  by the KARMEN experiment, we choose here to use only
the (confirmed) solar and atmospheric neutrino data in our analysis, and hence
to work in the context of three active neutrino weak eigenstates. 

\subsection{Neutrino Oscillation Formalism} 

     In this simplest theoretical context, there are three electroweak-doublet neutrinos.  Although electroweak-singlet neutrinos may be present in the theory, one expects that, since their
bare mass terms are electroweak-singlet operators, the associated masses should
not have any close relation with the electroweak symmetry breaking scale. Indeed, from a top-down point of view, such as a grand unified theory, the masses should be much
larger than this scale.  If this is the case, then the neutrino mixing can be 
described by the matrix 
%
\beq
U=\left(
\begin{array}{ccc}
c_{12} c_{13}&c_{13} s_{12}&s_{13} e^{-i\delta}\cr
-c_{23}s_{12}-s_{13}s_{23}c_{12}e^{i\delta}
&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\cr
s_{12}s_{23}-s_{13}c_{12}c_{23}e^{i\delta}
&-s_{23}c_{12}-s_{12}c_{23}s_{13}e^{i\delta}&c_{13}c_{23}
\end{array}
\right)K'
\end{equation}
%
where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$, $K'$ is a diagonal matrix with elements $diag(1,e^{i\phi_1},e^{i\phi_2}).$ The phases $\phi_1$ and $\phi_2$ do not affect neutrino oscillation. 
Thus, in this framework, the neutrino mixing depends on the four angles
$\theta_{12}$, $\theta_{13}$, $\theta_{23}$, and $\delta$, 
and on two independent
differences of squared masses, $\Delta m^2_{atm}$, which is 
$\Delta m^2_{32} = m(\nu_3)^2-m(\nu_2)^2$ in the favored fit, and 
$\Delta m^2_{sol}$, which may be taken to be $\Delta m^2_{21}=m(\nu_2)^2-
m(\nu_1)^2$.  Note that these quantities involve both magnitude and sign;
although in a two-species neutrino oscillation in vacuum the sign does not
enter, in the three species oscillations relevant here, and including both
matter effects and CP violation, the signs of the $\Delta m^2$ quantities do
enter and can, in principle, be measured.  

For our later discussion it will be useful to record the formulas for the
various relevant neutrino oscillation transitions.  In the absence of any
matter effect, the probability that a (relativistic) weak neutrino eigenstate
$\nu_a$ becomes $\nu_b$ after propagating a distance $L$ is
\begin{eqnarray}
P(\nu_a \to \nu_b) &=& \delta_{ab} - 4 \sum_{i>j=1}^3
Re(K_{ab,ij}) \sin^2  \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr ) \nonumber\\
&+& 4 \sum_{i>j=1}^3 Im(K_{ab,ij})
 \sin \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr )
\cos \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr )
\label{pab}
\end{eqnarray}
where
\begin{equation}
K_{ab,ij} = U_{ai}U^*_{bi}U^*_{aj} U_{bj}
\label{k}
\end{equation}
Note that, in vacuum, CPT invariance implies
$P(\bar\nu_b \to \bar\nu_a)=P(\nu_a \to \nu_b)$ and hence, for $b=a$,
$P(\bar\nu_a \to \bar\nu_a) = P(\nu_a \to \nu_a)$.  For the
CP-transformed reaction $\bar\nu_a \to \bar\nu_b$ and the T-reversed
reaction $\nu_b \to \nu_a$, the transition probabilities are given by the
right-hand side of (\ref{pab}) with the sign of the imaginary term reversed.
(Below, we shall assume CPT invariance, so that CP violation is equivalent to T
violation.) 

In most cases there is only one mass scale
relevant for long baseline neutrino oscillations, $\Delta m^2_{atm} \sim {\rm
few} \times 10^{-3}$ eV$^2$ and one possible neutrino mass spectrum is the
hierarchical one 
\begin{equation} 
\Delta m^2_{21}
= \Delta m^2_{sol} \ll \Delta m^2_{31} \approx \Delta m^2_{32}=\Delta m^2_{atm}
\label{hierarchy}
\end{equation}
In this case, CP (T) violation effects are negligibly small, so that in
vacuum
\begin{equation}
P(\bar\nu_a \to \bar\nu_b) = P(\nu_a \to \nu_b)
\label{pcp}
\end{equation}
\begin{equation}
P(\nu_b \to \nu_a) = P(\nu_a \to \nu_b)
\label{pt}
\end{equation}
In the absence of T violation, the second equality Eq.~(\ref{pt}) would still hold
in matter, but even in the absence of CP violation, the first equality
Eq.~(\ref{pcp}) would not hold.  With the hierarchy (\ref{hierarchy}), the
expressions for the specific oscillation transitions are
\begin{eqnarray}
P(\nu_\mu \to \nu_\tau) & = & 4|U_{33}|^2|U_{23}|^2
\sin^2 \Bigl ( \frac{\Delta m^2_{atm}L}{4E} \Bigr ) \cr\cr
& = & \sin^2(2\theta_{23})\cos^4(\theta_{13})
\sin^2 \Bigl (\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{pnumunutau}
\end{eqnarray}
\begin{eqnarray}
P(\nu_e \to \nu_\mu) & = & 4|U_{13}|^2 |U_{23}|^2
\sin^2 \Bigl ( \frac{\Delta m^2_{atm}L}{4E} \Bigr ) \cr\cr
& = & \sin^2(2\theta_{13})\sin^2(\theta_{23})
\sin^2 \Bigl (\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{pnuenumu}
\end{eqnarray}

\begin{eqnarray}
P(\nu_e \to \nu_\tau) & = & 4|U_{33}|^2 |U_{13}|^2
\sin^2 \Bigl ( \frac{\Delta m^2_{atm}L}{4E} \Bigr ) \cr\cr
& = & \sin^2(2\theta_{13})\cos^2(\theta_{23})
\sin^2 \Bigl (\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{pnuenutau}
\end{eqnarray}

In neutrino oscillation searches using reactor antineutrinos,
\textit{i.e.}, tests of $\bar\nu_e \to \bar\nu_e$, the two-species mixing hypothesis used
to fit the data is
%
\beqs 
P(\nu_e \to \nu_e) & = & 1 - \sum_x P(\nu_e \to \nu_x) \cr\cr
                   & = & 1 - \sin^2(2\theta_{reactor})
\sin^2 \Bigl (\frac{\Delta m^2_{reactor}L}{4E} \Bigr )
\label{preactor}
\eeqs
%
where $\Delta m^2_{reactor}$ is the squared mass difference relevant for
$\bar\nu_e \to \bar\nu_x$.  In particular, in the upper range of values of
$\Delta m^2_{atm}$, since the transitions $\bar\nu_e \to \bar\nu_\mu$ and
$\bar\nu_e \to \bar\nu_\tau$ contribute to $\bar\nu_e$ disappearance, one has
\begin{equation}
P(\nu_e \to \nu_e) = 1 - \sin^2(2\theta_{13})\sin^2 \Bigl
(\frac{\Delta m^2_{atm}L}{4E} \Bigr )
\label{preactoratm}
\end{equation}
\textit{i.e.}, $\theta_{reactor}=\theta_{13}$, and for the value $|\Delta m^2_{atm}|=3\times 10^{-3}\textrm{eV}^2$ from SuperK, the Chooz reactor experiment yields
the bound \cite{chooz}
\begin{equation}
\sin^2(2\theta_{13}) < 0.1
\label{chooz}
\end{equation}
which is also consistent with conclusions from the SuperK data analysis
\cite{sk}.

Further, in the three-generation case, the quantity ``$\sin^2(2\theta_{atm})$'' often used to fit
the data on atmospheric neutrinos with a simplified two-species mixing
hypothesis, is, 
%
\beq
\sin^2(2\theta_{atm}) \equiv \sin^2(2\theta_{23})\cos^4(\theta_{13})
\label{thetaatm}
\eeq
%
The SuperK experiment finds that the best fit to their data is to infer
$\nu_\mu \to \nu_\tau$ oscillations with maximal mixing, and hence
$\sin^2(2\theta_{23})=1$ and $|\theta_{13}| << 1$. The various solutions of 
the solar neutrino
problem involve quite different values of $\Delta m^2_{21}$ and
$\sin^2(2\theta_{21})$: (i) large mixing angle solution, LMA: $\Delta m^2_{21}
\simeq {\rm few} \times 10^{-5}$ eV$^2$ and $\sin^2(2\theta_{21}) \simeq 0.8$;
(ii) small mixing angle solution, SMA: $\Delta m^2_{21} \sim 10^{-5}$ and 
$\sin^2(2\theta_{21}) \sim 10^{-2}$, (iii) LOW: $\Delta m^2_{21} \sim 10^{-7}$,
$\sin^2(2\theta_{21}) \sim 1$, and (iv) ``just-so'': $\Delta m^2_{21} \sim
10^{-10}$, $\sin^2(2\theta_{21}) \sim 1$.  The SuperK experiment favors the LMA solution~\cite{sol}; for other global fits, see, \textit{e.g.}, Gonzalez-Garcia \textit{et al.} in ~\cite{sol}. 


\subsection{Types of Neutrino Masses, Seesaw Mechanism} 

We review here the theoretical background concerning neutrino masses and
mixing.  In the standard SU(3) $\times$ SU(2)$_L \times$ U(1)$_Y$ model (SM),  
neutrinos occur in SU(2)$_L$ doublets with $Y=-1$:
\begin{equation} 
{\cal L}_{L \ell} = \left ( \begin{array}{c}
                     \nu_\ell \\
                     \ell \end{array} \right )
 \ , \quad  \ell=e, \ \mu, \ \tau 
\end{equation}
There are no electroweak-singlet neutrinos (often called right-handed
neutrinos)  $\chi_{R,j}$, $j=1,...,n_s$.  Equivalently, these could be 
written as $\overline{\chi^c}_{L,j}$.  There are three types of 
possible Lorentz-invariant bilinear operator products that can be formed from 
 two Weyl fermions $\psi_L$ and $\chi_R$: 

\begin{itemize} 

\item 

Dirac: \ $m_D \bar \psi_L \chi_R + h.c.$ \ This connects opposite-chirality
fields and conserves fermion number. 

\item 

Left-handed Majorana: \ $m_L \psi_L^T C \psi_L + h.c.$ where
$C=i\gamma_2\gamma_0$ is the charge conjugation matrix.

\item 

Right-handed Majorana: \ $m_R  \chi_R^T C \chi_R + h.c.$

\end{itemize}
The Majorana mass terms connect fermion fields of the same chirality and
violate fermion number (by two units).  
Using the anticommutativity of fermion fields and the property $C^T = -C$, it
follows that a Majorana mass matrix appearing as 
\begin{equation}
\psi_i^T C (M_{maj})_{ij} \psi_j
\end{equation}
is symmetric in flavor indices: 
\begin{equation}
M_{maj}^T = M_{maj} 
\end{equation}
Thus, in the Standard Model (SM), there is 
no Dirac neutrino mass term because: i) it is forbidden as a bare
mass term by the gauge invariance; ii) it cannot occur, as do the quark and
charged-lepton mass terms, via spontaneous symmetry breaking (SSB) of the
electroweak (EW) symmetry starting from a Yukawa term, as there are no
EW-singlet neutrinos $\chi_{R,j}$.  There is also 
no left-handed Majorana mass term because: i) it is forbidden as a bare mass 
term and ii) it would
require a Higgs field with $T=1$, $Y=2$, but the SM has no such Higgs field.
Finally, there is 
no right-handed Majorana mass term because there is no
$\chi_{R,j}$.  The same holds for the minimal supersymmetric standard model 
(MSSM) and the minimal SU(5) grand unified theory (GUT), both for the original
and supersymmetric versions. 

However, it is easy to add EW-singlet neutrinos $\chi_R$ to the SM,
MSSM, or SU(5) GUT; these are gauge-singlets under the SM gauge group and
SU(5), respectively.  Denote these theories as the extended SM, etc. The extended theories give
rise to both Dirac and Majorana mass terms, the former via Yukawa terms and the
latter as bare mass terms.  In the extended SM:
\begin{equation}
-{\cal L}_{Yuk} = \sum_{i=1}^3 \sum_{j=1}^{n_s} h_{ij}^{(D)} \bar 
{\cal L}_{L,i} \chi_{R,j} \phi + h.c. 
\end{equation}
The electroweak symmetry breaking (EWSB), with 
\begin{equation}
\langle \phi \rangle_0 = \left( \begin{array}{c}
                                 0 \\
                                v/\sqrt{2} \end{array} \right )
\end{equation}
where $v = 2^{-1/4}G_F^{-1/2} \simeq 250$ GeV, yields the Dirac mass term 
\begin{equation}
\sum_{i=1}^3 \sum_{j=1}^{n_s} \bar\nu_{L,i} (M_D)_{ij} \chi_{Rj} + h.c.
\end{equation}
with
\begin{equation}
(M_D)_{ij} = h_{ij}^{(D)}\frac{v}{\sqrt{2}}
\end{equation}
The Majorana bare mass terms are
\begin{equation}
\sum_{i,j=1}^{n_s} \chi_{Ri}^T C (M_R)_{ij} \chi_{Rj} + h.c. 
\end{equation}
For compact notation, define the flavor vectors 
$\nu = (\nu_e,\nu_\mu,\nu_\tau)$ and $\chi = (\chi_1,..,\chi_{n_s})$ and
observe that one can equivalently write $\nu_L$ or $\nu^c_R$ and 
$\chi_R$ or $\chi^c_L$, where $\psi^c = C \overline{\psi}^T$, $\overline \psi
= \psi^\dagger \gamma^0$.  
The full set of Dirac and Majorana mass terms can then be written in the
compact matrix form 
\begin{equation}
-{\cal L}_m =
 \dfrac{1}{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.
\end{equation}
where $M_L$ is the $3 \times 3$ left-handed Majorana mass matrix,
$M_R$ is an $n_s \times n_s$ 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 $(M_L)^T = M_L \ , (M_R)^T = M_R$. 
Because the extension of the SM to include $\chi_R$ does not include a 
Higgs field with $T=1$, $Y=2,$  allowing a renormalizable, dimension-4 Yukawa term 
that would yield a left-handed Majorana mass, one may take $M_L=0$ at this
level (but see below for dimension-5 contributions). The
diagonalization of this mass matrix yields the neutrino masses and
the corresponding transformation relating the neutrino weak eigenstates to the mass
eigenstates.  

The same comments apply to the extended MSSM and SU(5) GUT. 
In the extended SU(5) GUT, the Dirac neutrino mass term arises most simply 
from the Yukawa couplings of the $5_R$ with a 5-dimensional Higgs
representation $H^\alpha$ 
(in terms of component fields): 
\begin{equation}
\bar \psi_{R \alpha} M_D \chi^c_L H^\alpha + h.c. 
\end{equation}
and the bare Majorana mass term $\chi_R^T M_R \chi_R + h.c.$. 

In the extended SM, MSSM, or SU(5) GUT, one could consider the addition of the
$\chi_R$ fields as {\it ad hoc}.  However, a more complete grand unification is
achieved with the (SUSY) SO(10) GUT, since all of the fermions of a
given generation fit into a single representation
 of SO(10), namely, the 16-dimensional spinor
representation $\psi_L$.  In this theory the states $\chi_R$ are not {\it
ad hoc} additions, but are guaranteed to exist.  
In terms of SU(5) representations
(recall, SO(10) $\supset$ SU(5) $\times$ U(1))
\begin{equation}
16_L = 10_L + \bar 5_L + 1_L
\end{equation}
so for each generation, in addition to the usual 15 Weyl fermions comprising
the 10$_L$ and $5_R$, (equivalently $\bar 5_L$) of SU(5), there is also an
SU(5)-singlet, $\chi^c_L$ (equivalently, $\chi_R$). 
So in SO(10) GUT, 
electroweak-singlet neutrinos are guaranteed to occur, with number equal to the
number of SM generations, inferred to be $n_s=3$.  Furthermore, 
the generic scale for the coefficients in $M_R$ is expected to be the GUT 
scale, $M_{GUT} \sim 10^{16}$ GeV. 

There is an important mechanism, which originally arose in the context of
GUT's, but is more general, that naturally predicts light neutrinos.  This is
the seesaw mechanism \cite{seesaw}.  
The basic point is that because the Majorana mass term 
$\chi_R^T C M_R \chi_R$ is an electroweak singlet, the associated Majorana 
mass matrix $M_R$ should not be related to the electroweak mass scale $v$, and 
from a top-down point of view, it should be much larger than this scale. Denote
this generically as $m_R$. 
This has the very important consequence that when we diagonalize the joint
Dirac-Majorana mass matrix above, the eigenvalues (masses) will be comprised of
two different sets: $n_s$ heavy masses, of order $m_R$, and 3 light
masses.  We illustrate this in the simplest case of a single generation and 
$n_s=1$.  Then the mass matrix is simply 
\begin{equation}
 -{\cal L}_m = \dfrac{1}{2}(\bar\nu_L \ \bar{\chi^{c}}_L) \left( \begin{array}{cc} 0 & m_D \\ m_D & m_R
\end{array} \right )\left( \begin{array}{c} \nu^{c}_R \\ 
                                            \chi_R  \end{array} \right ) + h.c.
\end{equation} 
The diagonalization yields the eigenvalues 
\begin{equation} 
\lambda =
\frac{1}{2}\Biggl [ m_R \pm \sqrt{m_R^2 + 4m_D^2} \ \Biggr ]
\end{equation} 
Since $m_D \sim h^{(D)}v$ while $m_R$ is naturally $ >> v$ and hence $m_R >> 
m_D$, we can expand to get 
\begin{equation} 
\lambda_> \simeq m_R
\end{equation} 
and 
\begin{equation} 
\lambda_< \simeq -\frac{m_D^2}{m_R}\biggl [ 1 + O\Bigl (
\frac{m_D^2}{ m_R^2} \Bigr ) \biggr ].
\label{seesaw1}
\end{equation} 
(The minus sign is not physically important.) 
The largeness of $m_R$ then naturally explains the smallness of the masses  of the known neutrinos.  
This appealing mechanism also applies in the
physical case of three generations and for $n_s \ge 2$. 

However, at a phenomenological level, without further theoretical assumptions,
there is a large range of values for the light $m_\nu$, since i) the
actual scale of $m_R$ is theory-dependent, and ii) it is, {\it a priori}, not
clear what to take for $m_D$ since the known (Dirac) masses range over 5 orders
of magnitude, from $m_e, m_u \sim $ MeV to $m_t = 174$ GeV, and this
uncertainty gets squared. 

For the full case with three generations and $n_s > 1$, and assuming, as is
generic, that $det(M_R) \ne 0$ so that $M_R^{-1}$ exists, the set of three
 light neutrino mass eigenstates is determined by the matrix analogue of eq.
(\ref{seesaw1}): 
\begin{equation}
M_\nu = - M_D M_R^{-1} M_D^T
\label{seesaw}
\end{equation}

A different way to get neutrino masses is to interpret the SM as a low-energy
effective field theory, as is common in modern quantum field theory. Provided that their coefficients, of dimension 
$4-d_{\cal O}$ in mass units, are sufficiently small, (nonrenormalizable) operators ${\cal O}$ in the Lagrangian of mass
dimension $d_{\cal O} > 4,$ are then allowed.  In this case, the dimension-5 operator~\cite{pdg96} 
\begin{equation}
{\cal O} = \frac{1}{M_X}\sum_{a,b}h_{L,ab}
(\epsilon_{ik}\epsilon_{jm}+\epsilon_{im}\epsilon_{jk})
\Bigl [ {\cal L}^{Ti}_{a L}C {\cal L}^j_{b L} \Bigr ] \phi^k \phi^m + h.c.
\end{equation}
(where $a,b$ are flavor indices, $i,j,k,m$ are SU(2) indices) is an electroweak
singlet. Upon electroweak symmetry breaking (EWSB), this operator 
yields a left-handed Majorana mass
term 
\begin{equation}
\sum_{a,b=1}^3 \nu_{L,a}^T C (M_L)_{ab} \nu_{L,j} + h.c. 
\end{equation}
with 
\begin{equation}
(M_L)_{ab} = \frac{(h_L)_{ab}(v/\sqrt{2})^2}{M_X}
\end{equation}
Since the SM is phenomenologically very successful, one 
should have $M_X >> v$, so
again these dimension-5 operators lead naturally to light neutrinos.  The
diagonalization of the above operator determines the unitary transformation
relating the mass eigenstates to the weak eigenstates, 
\begin{equation}
\nu_{\ell_a} = \sum_{i=1}^3 U_{a i} \nu_i \ , \quad \ell_1=e, \ \ell_2=\mu, \ 
\ell_3=\tau
\end{equation}
\textit{i.e.}, 
\begin{equation}
\left( \begin{array}{c} {\nu_e} \\ {\nu_\mu} \\ {\nu_\tau} \end{array} \right) 
=  \left( \begin{array}{ccc}
              U_{e1} & U_{e2} & U_{e3}  \\
              U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\
              U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{array} \right)
\left( \begin{array}{c} \nu_1 \\
                        \nu_2 \\
                         \nu_3 \end{array} \right) 
\end{equation}

For the case of electroweak-singlet neutrinos and the resultant seesaw, because of
the splitting of the masses into a light set and a heavy set, the observed
weak eigenstates of neutrinos are again, to a very good approximation, linear
combinations of the three light mass eigenstates, so that the full $(3+n_s)
\times (3+n_s)$ mixing matrix breaks into block diagonal form involving the $3
\times 3$ $U$ matrix and an analogous $n_s \times n_s$ matrix for the heavy
sector.  In terms of the flavor vectors, this is 
\begin{equation} 
\left ( \begin{array}{c} \nu_\ell \\ 
                         \chi^c \end{array} \right ) = 
       \left( \begin{array}{cc} U & 0 \\ 0 & U_{heavy}
\end{array} \right ) \left( \begin{array}{c} \nu_i \\ \chi^c_{i,m}\end{array} 
\right )
\end{equation}

If all of the data indicating neutrino masses is accepted, including the solar
neutrino deficiency, atmospheric neutrinos, and LSND experiments, 
then light sterile (electroweak-singlet) 
neutrinos with masses of $\sim $ eV or smaller are needed.  These are usually 
considered unnatural, because electroweak-singlet neutrinos naturally have
masses $\sim m_R >> M_{ew} = v$.  

\subsection{Tests for Neutrino Masses in Decays} 

Given the focus of this report, we shall not review the well-known kinematic
tests for neutrino masses except to mention that these are of three main
types.  First there are direct tests, which search for the masses of the dominantly
coupled neutrino mass eigenstates emitted in particle and nuclear decays;
these yield the current upper bounds on these eigenstates for the three
dominantly coupled mass components in $\nu_e$, $\nu_\mu$, and $\nu_\tau$. 
Second, there are tests for rather massive neutrinos emitted, via
lepton mixing, in particle and nuclear decays.  Third, there are
searches for neutrinoless double beta decay, which would occur if there are
massive Majorana neutrinos.  The quantity on which  limits are put in searches
for neutrinoless double beta decay is $\langle m_\nu \rangle = |U_{ei}^2m(\nu_i)|$ provided that their coefficients, 
%of dimension $4-d_{\cal O}$ in mass units, 
 are sufficiently small.  Note that since $U_{ei}$ is complex, destructive interference can
occur in this sum.  At present, the upper limit on this quantity
is $\langle m_\nu \rangle \sim 0.4$ eV \cite{baudis}. 
A number of new proposals for more sensitive
experiments have been put forward, including GENIUS, EXO, MOON, and MAJORANA, 
among others, which hope to reach a sensitivity below 0.01~eV in 
$\langle m_\nu \rangle$ \cite{dbfuture}.  

\subsection{Models for Neutrino Masses and Mixing} 

We discuss the seesaw mechanism in further detail here. 
In the SM, a single Higgs field $\phi$ breaks the gauge symmetry and gives
masses to the fermions.  In the MSSM, it requires two $T=1/2$ Higgs fields, 
$H_1$ and $H_2$ with opposite hypercharges $Y=1$ and $Y=-1$ to do this. 
GUT theories may have more complicated Higgs sectors; typically  
different Higgs are used to break the gauge symmetry and
give masses to fermions.  For the 
Clebsch-Gordan decomposition of the representations in the fermion mass term we have
%
\begin{equation}
16 \times 16 = 10_s + 120_a + 126_s
\end{equation}
%
Hence, {\it a priori}, one considers using Higgs of dimension 10, 120, and
126.  The coupling to the 10-dimensional Higgs fields yields Yukawa 
terms of the following form (suppressing generation indices). 
\begin{equation}
\psi_L^T C \psi_L \bar \phi_{10} = (\bar d_R d_L + \bar e_R e_L)\phi_{10}
(\bar 5) + (\bar u_R u_L + \bar \nu_R \nu_L) \phi_{10}(5) 
\end{equation}
The coupling to the 126-dimensional Higgs yields a term 
\begin{equation}
\chi_R^T C \chi_R \phi_{126}(1) 
\end{equation}
together with other linear combinations of $\bar u_R u_L$, $\bar \nu_R \nu_L$, 
$\bar d_R d_L$, and $\bar e_R e_L$ times appropriate SU(5)-Higgs; these four
types of terms are also produced by the coupling to a 120-dimensional Higgs.  
Hence, in this approach, 
one expects some similarity in Yukawa matrices, and thus
Dirac mass matrices, for $T_3=+1/2$ fermions, \textit{i.e.}, the up-type quarks $u,c,t$
and the neutrinos:
\begin{equation}
M^{(u)} \sim M^{(\nu)}_D \ , \quad M^{(d)} \sim M^{(\ell)}_D
\end{equation}
However, in many string-inspired models, high-dimension Higgs representations such as
the 120- and 126-dimensional representations in SO(10), are avoided.  Instead, one
constructs the neutrino mass terms from nonrenormalizable higher-dimension 
operators.  Some reviews of models are in Ref. \cite{thy}. 

To get a rough idea of the predictions, suppose that 
$M_D$ and $M_R$ are 
diagonal and let $m_R$ denote a typical entry in $M_R$.  Denote 
$m_{u,1} = m_u$, $m_{u,2} = m_c$, $m_{u,3} = m_t$. 
Then, (neglecting physically irrelevant minus signs) 
\begin{equation}
m(\nu_i) \simeq \frac{m_{u,i}^2}{m_R}
\end{equation}
This is the quadratic seesaw. For $m(\nu_3)$, one gets 
\begin{equation}
m(\nu_3) \sim \frac{m_t^2}{m_R} \simeq \Biggl ( \frac{175 \ {\rm GeV}}
{10^{16} \ {\rm GeV}} \Biggr ) (1.75 \times 10^{11} \ {\rm eV}) 
\sim 10^{-3} \ {\rm eV} 
\end{equation}
which, given the uncertainties in the inputs, is comparable to the value 
\begin{equation}
m(\nu_3) \simeq  \sqrt{\Delta m^2_{32}} = \simeq 0.05 \ {\rm eV}
\end{equation}
inferred from the SuperK data with the assumption $\nu_\mu \to \nu_\tau$ and 
$m(\nu_2) << m(\nu_3)$.  This gives an idea of how the seesaw mechanism could
provide a neutrino mass in a region relevant to the SuperKamiokande data. 

In passing, we note that string theories allow a low string scale, perhaps as
low as 100 TeV.  These models have somewhat different phenomenological
implications for neutrinos than conventional models with a string scale
comparable to the Planck mass. 

\subsection{Lepton Mixing} 
   We proceed to consider off-diagonal structure in $M_R$, as part of the more
general topic of lepton mixing.  Neutrino mass terms naturally couple different
generations and hence violate lepton family number; the Majorana mass terms
also violate total lepton number.  Lepton mixing angles are determined by
diagonalizing the charged lepton and neutrino mass matrices, just as the quark
mixing angles in the CKM (Cabibbo-Kobayashi-Maskawa) matrix are determined by
diagonalizing the up-type and down-type quark mass matrices.  Before the
atmospheric neutrino anomaly was reported, a common expectation was that lepton
mixing angles would be small, like the known quark mixing angles.  This was one
reason why theorists favored the MSW mechanism over vacuum oscillations as
an explanation of the solar neutrino deficiency -- MSW could produce the
deficiency with small lepton mixing angles, whereas vacuum oscillations needed
nearly maximal mixing.  It was long recognized that an explanation of the
atmospheric neutrino anomaly requires maximal mixing, and while neutrino masses
are not surprising or unnatural to most theorists, the maximal mixing has been
something of a challenge for theoretical models to explain.

Denoting the lepton flavor vectors as $\ell=(e,\mu,\tau)$ and 
$\nu = (\nu_e,\nu_\mu,\nu_\tau)$, we have, for the leptonic weak charged
current, 
\begin{equation}
J^\lambda = \bar \ell_L \gamma^\lambda \nu_L
\end{equation}
The mass terms are 
\begin{equation}
 \bar\ell_L M_\ell \ell_R + \bar\nu_L M_\nu \nu^c_R + h.c. 
\end{equation}
where, as above, $M_\nu = -M_D M_R^{-1} M_D^T$ and we have used the splitting 
of the neutrino eigenvalues into a light sector and a very heavy sector. 
We diagonalize these so that, in terms of the associated unitary 
transformations, with the notation 
$\ell_m=(e_m,\mu_m,\tau_m)$ and $\nu_m=(\nu_1,\nu_2,\nu_3)$, 
for charged lepton and neutrino mass eigenstates, the 
the charged current is 
\begin{equation}
J^\lambda = \bar\nu_{mL} U_L^{(\nu)} \gamma^\lambda U_L^{(\ell) \dagger} 
\ell_{mL} = \bar\nu_{mL} U \gamma^\lambda \ell_{mL}
\end{equation}
where the lepton mixing matrix is 
\begin{equation}
U = U_L^{(\nu)} U_L^{(\ell)\dagger}
\end{equation}
%
Although many theorists expected before the SuperK results indicating that 
$\sin^2(2\theta_{23})=1$ that leptonic mixing angles would be small, like the
quark mixing angles, after being confronted with the SuperK  results, they have
constructed models that can accommodate large mixing angles.  Of course, 
$\theta_{13}$ must be small to fit experiment.  Models are able to yield either
$\sin^2(2\theta_{12}) \sim 1$ for the LMA, LOW, and just-so solutions, or 
$\sin^2(2\theta_{12}) << 1$ for the SMA solution. 

\subsection{Relevant Near- and Mid-Term Experiments} 

There are currently intense efforts to confirm and extend the evidence for
neutrino oscillations in all of the various sectors - solar, atmospheric, and
accelerator.  Some of these experiments are now running.  In addition to
SuperKamiokande and Soudan-2, these include the Sudbury Neutrino Observatory,
SNO, and the K2K long baseline experiment between KEK and Kamioka.  Others are
in the development and testing phases, such as BOONE, MINOS, the CERN-Gran Sasso
(GNGS) program, KAMLAND, and Borexino \cite{anl}.  Among the long baseline neutrino
oscillation experiments, the approximate distances are $L \simeq 250$ km for
K2K, 730 km for both MINOS, from Fermilab to Soudan, and the proposed CNGS experiments.  K2K is a $\nu_\mu$ disappearance experiment with a
conventional neutrino beam having a mean energy of about 1.4 GeV, going from
KEK  to the SuperK detector; it has a near detector for beam
calibration.  It has obtained results consistent with the SuperK experiment,
and has reported that its data disagree by $2\sigma$ with the no-oscillation
hypothesis \cite{k2k}.  MINOS is another conventional neutrino beam experiment
that takes a beam from Fermilab to a detector in the Soudan mine in
Minnesota.  It too uses a near detector for beam flux measurements and has
opted for a low-energy configuration, with the flux peaking at about 3 GeV.
This experiment expects to start taking data in early 2004 and, after some
years of running, to obtain higher statistics than the K2K experiment and to
achieve a sensitivity down to roughly  the level $\Delta m^2_{32} \sim
10^{-3} $eV$^2$.  The CNGS program will come on later, around
2005.  It will involve taking a higher energy neutrino beam from CERN to the
Gran Sasso deep underground laboratory in Italy.  This program will emphasize
detection of the $\tau$'s produced by the $\nu_\tau$'s that result from the
inferred neutrino oscillation transition $\nu_\mu \to \nu_\tau$.  The OPERA
experiment will do this using emulsions \cite{opera}, while the ICARUS proposal
uses a liquid argon chamber \cite{icanoe}.  Moreover, at Fermilab, the
MiniBOONE experiment plans to run in the next few years and to confirm or
refute the LSND claim after a few years of running.  

There are also several relevant solar neutrino experiments.  The SNO experiment
is currently running and should report their first results in spring 2001.
These will involve measurement of the solar neutrino flux and energy
distribution using the charged current reaction on heavy water, $\nu_e + d \to e
+ p + p$.  Subsequently, they will measure the neutral current reaction $\nu_e
+ d \to \nu_e + n + p$.  The KamLAND experiment in Japan expects to begin
taking data in late 2001.  This is a reactor antineutrino experiment using
baselines of order 100-250~km and will search for $\bar\nu_e$ disappearance.
On a similar time scale, the Borexino experiment in Gran Sasso expects to turn
on and hopes to measure the $^7$Be neutrinos from the sun.  These experiments
should help to decide which of the various solutions to the solar neutrino
problem is preferred, and hence the corresponding values of $\Delta m^2_{21}$
and $\sin^2(2\theta_{12})$.

This, then, is the program of relevant experiments during the period
2001-2010.  By the end of this period, we may expect that much will have been learned
about neutrino masses and mixing.  However, there will remain several
important quantities that will not be well measured and which can be measured by a
Neutrino Factory. 

\subsection{Oscillation Experiments at a Neutrino Factory } 

Although a Neutrino Factory based on a muon storage ring will turn on several
years after this near-term period in which K2K, MINOS, and the CNGS
experiments will run, it has a valuable role to play, given the
very high-intensity neutrino beams of fixed flavor-pure content, including, in
particular, $\nu_e$ and $\bar\nu_e$ beams as well as the conventional $\nu_\mu$
and $\bar\nu_\mu$ beams.  The potential of the
neutrino beams from a muon storage ring is that, in contrast to a conventional
neutrino beam, which, say, from $\pi^+$ decay, is primarily $\nu_\mu$ with some
admixture of $\nu_e$'s and other flavors from $K$ decays, the neutrino beams
from the muon storage ring would be extremely high purity: $\mu^-$ beams
would yield 50 \% $\nu_\mu$ and 50 \% $\bar\nu_e$, and viceversa for the charge
conjugate case of $\mu^+$ beams.  Furthermore, these could be produced with
extremely high intensities; we shall take the BNL design value of $\approx 10^{20}\mu$ decays per Snowmass year, $10^7$~s. 

The types of neutrino oscillations that can be explored with the neutrino
factory based on a muon storage ring are listed below for the case of $\mu^-$
decaying into  $\nu_\mu e^- \bar\nu_e$: 

\begin{enumerate}

\item

$\nu_\mu \to \nu_\mu$, $\nu_\mu \to \mu^-$ (survival) 

\item

$\nu_\mu \to \nu_e$, $\nu_e \to e^-$ (appearance)

\item 

$\nu_\mu \to \nu_\tau$, $\nu_\tau \to \tau^-$; $\tau^- \to (e^-, \mu^-)...$
 (appearance$^*$)

\item

$\bar\nu_e \to \bar\nu_e$, $\bar\nu_e \to e^-$ (survival) 

\item 

$\bar\nu_e \to \bar\nu_\mu$, $\bar\nu_\mu \to \mu^+$ (appearance)

\item 

$\bar\nu_e \to \bar\nu_\tau$, $\bar\nu_\tau \to \tau^+$; $\tau^+ \to 
(e^+, \mu^+)...$ (appearance$^*$)

\end{enumerate}
where the $*$ on the term appearance refers to the greater difficulty in
experimentally inferring the production of the $\tau$ particle. 
It is clear from the list of processes above that, since the beam 
contains both neutrinos and antineutrinos, the only way to determine 
the identity of the parent neutrino is to determine the identity of the final-state charged lepton and measure its sign.  One aspect of the experiments will
involve the measurement of $\nu_\mu \to \nu_\mu$ as a disappearance
experiment.  A unique aspect for the Neutrino Factory will be the measurement
of the oscillation $\bar\nu_e \to \bar\nu_\mu$, giving a wrong-sign $\mu^+$. 
Of greater difficulty would be the measurement of the transition 
$\bar\nu_e \to \bar\nu_\tau$, giving a $\tau^+$ which will decay part of the
time to $\mu^+$.  These physics goals mean that a detector must have excellent
capability to identify muons and measure their charge sign. 
The oscillation $\nu_\mu \to \nu_e$ would be difficult to
observe, since it would be difficult to identify an electron shower from a
hadron shower.  From the above formulas for oscillations, we can see that,
given the knowledge of $|\Delta m^2_{32}|$ and $\sin^2(2\theta_{23})$ available
 by the time a Neutrino Factory is built, the measurement of the 
$\bar\nu_e \to \bar\nu_\mu$ transition yields the value of $\theta_{13}$. 

To get a rough idea of how the sensitivity of 
an oscillation experiment would scale with energy and baseline length, 
recall that the event rate in the absence of oscillations is 
simply the neutrino flux times the cross section.  
First of all, neutrino cross sections in the region above
about 10 GeV (and slightly higher for $\tau$ production) grow linearly with
the neutrino energy.  Secondly, the beam divergence is
a function of the initial muon storage ring energy; 
this divergence yields a flux, as a
function of $\theta_d$, the angle of deviation from the forward direction, that
goes like $1/\theta_d^2 \sim E^2$.  Combining this with the linear $E$
dependence of the neutrino cross section 
and the overall $1/L^2$ dependence of the flux far from the
production region, one finds that the event rate goes like 
\begin{equation} 
\frac{dN}{dt} \sim \frac{E^3}{L^2}
\label{eventrate}
\end{equation}
Estimated event rates have been given in the Fermilab Neutrino Factory Working 
 Group Report~\cite{INTRO:ref9},~\cite{INTRO:ref10}. For
a stored muon energy of 20 GeV, as considered in this report, and a distance of 
$L=2900$ to the WIPP Carlsbad site in New Mexico, these event rates amount to 
several thousand events per kton of detector per year, \textit{i.e.}, they are
satisfactory for the physics program. This is also true for the other
pathlengths under consideration, namely $L=2500$ km from BNL to Homestake and
$L=1700$ km to Soudan.  A usual racetrack design would only allow a single
pathlength $L$, but a bowtie design could allow two different pathlengths
(\textit{e.g.},~\cite{zp}).  

One could estimate that at a time when the neutrino factory turns on, $|\Delta
m^2_{32}|$ and $\sin^2(2\theta_{23})$ would be known at perhaps the 10\% level
(1~$\sigma$) from MINOS~\cite{superbeams} (we emphasize that future projections
such as this are obviously uncertain and note that JHF anticipates better
accuracy; see below).  The Neutrino Factory should improve the precision on
those two parameters, and can contribute to three important measurements:
\begin{itemize}
\item measurement of $\theta_{13}$, as discussed above
\item measurement of the sign
of $\Delta m^2_{32}$ using matter effects
\item possibly a measurement of
CP violation in the leptonic sector, if $\sin^2(2\theta_{13})$,
$\sin^2(2\theta_{21})$, and $\Delta m^2_{21}$ are sufficiently large
\end{itemize}  
It is
estimated that a Neutrino Factory with the BNL design parameters could achieve
a sensitivity down to $\sin^2 2\theta_{13}) \sim 3 \times 10^{-4}$ or better,
assuming a 50~kton water Cherenkov detector at $L=2900$~km, after three years
of running~\cite{INTRO:ref10,superbeams}.  To measure the sign of $\Delta m^2_{32}$,
one uses the fact that matter effects reverse sign when one switches from
neutrinos to antineutrinos, and carries out this switch in the charges of the
stored $\mu^\pm$.  We elaborate on this next.
\subsection{Matter Effects} 
With the advent of the muon storage ring, the distances at which detectors
can be placed are large enough that, for the first time, matter effects can be exploited in
accelerator-based oscillation experiments.  Simply put, matter effects are the
matter-induced oscillations that neutrinos undergo along their flight path
through the Earth from the source to the detector.  Given the typical density
of the earth, matter effects are important for the neutrino energy range $E
\sim O(10 \textrm{GeV})$ and $\Delta m^2_{32} \sim 10^{-3}$ eV$^2,$ values relevant for
the long baseline experiments.  After the initial discussion of matter-induced
resonant neutrino oscillations in \cite{wolf}, an early study of these effects,
including three generations, was carried out in \cite{barger80}.  The
sensitivity of an atmospheric neutrino experiment to small $\Delta m^2$ due to
the long baselines, and the necessity of taking into account matter effects, was
discussed \textit{e.g.}, in \cite{snowmass}.  After Ref. \cite{ms}, many analyses were
performed in the 1980s of the effects of resonant neutrino oscillations on the
solar neutrino flux. Matter effects in the Earth were studied, \textit{e.g.},
\cite{kp88} and \cite{baltz}, which also discussed the effect on atmospheric
neutrinos.  Recent papers on matter effects relevant to atmospheric neutrinos
include \cite{petcov,akh}. Early studies of matter
effects on long baseline neutrino oscillation experiments were carried out in
\cite{bernpark}.  More recent analyses relevant to neutrino factories include
\cite{geer,dgh}, \cite{bargergeer}-\cite{cpv}.  In recent papers~\cite{lb}, calculations were presented of the matter effect
for parameters relevant to possible long baseline neutrino experiments
envisioned for the Neutrino Factory.  In
particular, these authors compared the results obtained with constant density
along the neutrino path with results obtained by incorporating the actual
density profiles.  They studied the dependence of the oscillation signal on
both $\frac{E}{\Delta m^2_{32}}$ and on the angles in the leptonic mixing matrix, and commented on the influence of $\Delta m^2_{21}.$

In the constant-density approximation, one has 
%
\begin{equation}
P(\nu_\mu \to \nu_e) = \sin^2(2\theta_{13}^m)\sin^2 \theta_{23}
\sin^2(\omega_{32} L)
\label{pnumunue}
\end{equation}
%
where
%
\begin{equation}
\sin^2(2\theta_{13}^m) = \frac{\sin^2(2\theta_{13})}{
\sin^2(2\theta_{13}) + \Bigl [ \cos(2\theta_{13}) - \frac{2\sqrt{2}
G_FN_e E}{\Delta m^2_{32}} \Bigr ]^2 }
\label{sin2m}
\end{equation}
% 
and 
%
\beq
\omega_{32}^2 = \biggl [ \frac{\Delta m^2_{32}}{4E} \sin(2\theta_{13}) 
\biggr ]^2 + \biggl [ \frac{\Delta m^2_{32}}{4E} \cos(2\theta_{13}) - 
\frac{G_F N_e}{\sqrt{2}} \biggr ]^2 
\label{omega}
\eeq
% 
where $N_e$ is the electron number density in the medium. For antineutrinos,
one reverses the sign of the matter term $\propto G_F N_e$.  The resonance
condition is that
%
\beq
\frac{\Delta m_{32}^2}{2E}\cos(2\theta_{13}) = \sqrt{2} G_F N_e 
\label{rescon}
\eeq
%
\textit{i.e.}, $E \simeq 15$ GeV for $\Delta m^2_{32} = 3 \times 10^{-3}$ eV$^2$, $\rho
= 3$ g/cm$^2$, and $Z/A \simeq 0.5$.  Thus, if $\Delta m^2_{32} > 0$, this
resonance enhances the $\nu_e \to \nu_\mu$ transition, whereas if 
$\Delta m^2_{32} < 0$, it enhances the $\bar\nu_e \to \bar\nu_\mu$ transition.
By comparing these (using first a stored $\mu^+$ beam and then a stored $\mu^-$
beam) one can thus determine the sign of $\Delta m^2_{32}$ as well as the value
of $\sin^2(2\theta_{13})$.  A rough estimate is that this could be done to the
level $\sin^2(2\theta_{13}) \sim 10^{-3}.$ 

\section{CP Violation}
CP violation is measured by the (rephasing-invariant) Jarlskog product
%
\beqs
J & =& Im(U_{ai}U_{bi}^* U_{aj}^* U_{bj}) \cr\cr
& & = 2^{-3}
\sin(2\theta_{12})\sin(2\theta_{13})\cos(\theta_{13})\sin(2\theta_{23})\sin
\delta 
\eeqs
%
Leptonic CP violation also requires that each of the leptons in each charge
sector be nondegenerate with any other leptons in this sector; this is,
course, true of the charged lepton sector and, for the neutrinos, this requires
$\Delta m^2_{ij} \ne 0$ for each such pair $ij$.  In the quark sector, $J$ is 
known to be small; $J_{CKM} \sim O(10^{-5})$.  
A promising asymmetry to measure is $P(\nu_e \to \nu_\mu)-P(\bar\nu_e - 
\bar\nu_\mu)$.  As an illustration, in the absence of matter effects, 
%
\begin{eqnarray}
P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu) & = & -4J(\sin 2\phi_{32}+
\sin 2\phi_{21} + \sin 2\phi_{13}) \cr\cr
& = & -16J \sin \phi_{32} \sin \phi_{31} \sin \phi_{21} 
\label{pnuenumudif}
\end{eqnarray}
%
where
%
\begin{eqnarray}
\frac{P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu)}{
P(\nu_e \to \nu_\mu) + P(\bar\nu_e \to \bar\nu_\mu)} =
-\frac{\sin(2\theta_{12})\cot(\theta_{23})\sin\delta \sin \phi_{21}}{
\sin \theta_{13}}
\end{eqnarray}
%
In order for the CP violation in Eq.~\ref{pnuenumudif} to be large enough to 
measure, it is necessary that
$\theta_{12}$, $\theta_{13}$, and $\Delta m^2_{sol} = \Delta m^2_{21}$ not be 
too small. From atmospheric neutrino data, we have $\theta_{23}\simeq
\pi/4$ and $\theta_{13} << 1$.  If LMA describes solar neutrino data, then
$\sin^2(2\theta_{12}) \simeq 0.8$, so $J \simeq 0.1\sin(2\theta_{13})\sin
\delta$.  Say $\sin^2(2\theta_{13})=0.04$; then $J$ could be $>> J_{CKM}$.
Furthermore, for the upper part of the LMA, $\Delta m^2_{sol} \sim 4 \times
10^{-5}$ eV$^2$, so the CP violating effects might be observable. In the 
absence of matter, one would measure the asymmetry
%
\beq
\frac{P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu)}{
P(\nu_e \to \nu_\mu) + P(\bar\nu_e \to \bar\nu_\mu)} =
-\frac{\sin(2\theta_{12})\cot(\theta_{23})\sin\delta \sin(2\phi_{21})}{
4\sin(\theta_{13})\sin^2(\phi_{32}) }
\eeq
%
However, in order to optimize this, because of the smallness of $\Delta
m^2_{21}$ even for the LMA, one must go to large pathlengths $L$, and here
matter effects are important.  These make leptonic CP violation challenging to
measure, because, even in the absence of any intrinsic CP violation, these
matter effects render the rates for $\nu_e \to \nu_\mu$ and $\bar\nu_e \to
\bar\nu_\mu$ unequal since the matter interaction is opposite in sign for $\nu$
and $\bar\nu$.  One must therefore subtract out the matter effects in order to
try to isolate the intrinsic CP violation.  Alternatively, one might think of
comparing $\nu_e \to \nu_\mu$ with the time-reversed reaction $\nu_\mu \to
\nu_e$.  Although this would be equivalent if CPT is valid, as we assume, and
although uniform matter effects are the same here, the detector response is
quite different and, in particular, it is quite difficult to identify $e^\pm$.
Results from SNO and KamLAND testing the LMA will help further planning.

\subsection{Detector Considerations} 

We have commented on the requisite properties of detectors.  These should be
quite massive, O(10-100)~kton.  Possibilities include magnetized steel
calorimetors, water Cherenkov detectors, and liquid-argon chambers. A description of the type of detector presently envisioned for the Neutrino Factory is given in Chapter~\ref{detector-chapter}.


\subsection{Experiments with a High-Intensity Conventional Neutrino Beam} 
One possibility for the staging of the construction of the neutrino factory is
to start with an intense, $\sim 1$ MW proton driver with an associated program
of neutrino physics using a conventional $\nu_\mu$ neutrino beam from pion
decays.  Comparisons of the capabilities of a neutrino factory with those of
neutrino oscillation experiments with a very high luminosity conventional
neutrino beam are discussed in \cite{richter}-\cite{bargersuperbeam}.  The JHF
proposal estimates that its planned long baseline $\nu_\mu \to \nu_e$
oscillation experiment to SuperK could reach a level of $\sin^2( 2\theta_{13})$
of roughly $10^{-2}$~\cite{jhf}, and perhaps somewhat better, depending on the type of beam, the running
time, and the value of $|\Delta m^2_{32}|$.  The recent Fermilab study reached
similar conclusions~\cite{superbeams}.  The JHF plans also consider the
possibility of an upgrade to 4~MW and the construction of a much larger far
detector, namely a 1~Mton water Cherenkov detector called HyperKamiokande.
Long baseline experiments of this type also intend to carry out $\nu_\mu \to
\nu_\mu$ disappearance measurements that will yield much more precise
determinations of $\sin^2 2\theta_{23}$ and $|\Delta m^2_{32}|$ than are
currently available from the atmospheric data.  At Fermilab these plans are
being considered in conjunction with plans to construct a more intense proton
source~\cite{brighter}.  Recently also there have been studies of a number of
possible future options, including a 2100~km long baseline experiment using a
conventional neutrino beam from JHF to a detector located in the Beijing area~\cite{j2b}, an experiment taking a very low energy neutrino beam from CERN to a
detector in Frejus~\cite{cernsuperbeam}, and long baseline experiments with a
600 kton water Cherenkov detector called UNO (Ultra Underground Nucleon Decay
and Neutrino Detector)~\cite{unosuperbeam}.   
\subsection{Uses of Intense Low-Energy Muon Beams} 
The front end of a neutrino factory would be a source of intense low-energy
$\mu^\pm$ beams.  There is a rich program of physics that could be explored
with these beams.  Plans are already underway to do this at JHF, using their 3
GeV proton source~\cite{nag}, and at CERN~\cite{ellis1},~\cite{ellis2}.  One of the main areas
would be searches for lepton family number violating (LFV) decays, such as $\mu
\to e \gamma$ and $\mu \to e e \bar e$.  A review of the current status of
experimental searches for such decays is~\cite{kuno}.  The generalization of
the standard model to include massive neutrinos and lepton mixing does give
rise to these decays, but with branching ratios many orders of magnitude below
feasible levels of observation~\cite{meg}.  Models of dynamical electroweak
symmetry breaking such as technicolor generically predict large flavor-changing
neutral current processes, including these LFV decays.  This statement also
applies to many types of supersymmetric models~\cite{susylfv}.  Let us comment
on the possible improvements for various decays:

\begin{itemize}
\item $\mu \to e \gamma$.  A series of experiments of progressively better
sensitivity at SIN, TRIUMF, and LASL have been performed to search for this
decay.  In 1988, the Crystal Box experiment at LASL achieved the limit $B(\mu^+
\to e^+ \gamma) < 4.9 \times 10^{-11}$ \cite{bolton88}.  This was improved by a
factor of 4 by the MEGA experiment at LASL, to $B(\mu^+ \to e^+ \gamma) < 1.2
\times 10^{-11}$ \cite{mega}.  The MEGA experiment took advantage of a stopping
$\mu^+$ rate of about $10^8$ $\mu$/sec. A proposal has been approved
\cite{psimeg} for a $\mu \to e \gamma$ search at PSI with a single event
sensitivity of about $10^{-14}$.  With the increase in the stopping $\mu$ decay
rate to $10^{13}$ or more that would be achieved at a low-energy muon facility
as part of the neutrino factory, one might envision that it could be possible,
if requisite improvements in background suppression and detector technology
could be made, to get to a single event sensitivity of $10^{-15}$ or better.

\item 

$\mu^+ \to e^+e^+e^-$.  The current upper limit on this decay was set by the
SINDRUM experiment in 1988~\cite{meee}: $B(\mu^+ \to e^+e^+e^-) < 1.0 \times
10^{-12}$.  As is the case with $\mu \to e \gamma$, if the necessary background
reduction can be achieved and detectors can be designed to take the much
greater rates, then with the much higher stopping muon rates at the front end
of a neutrino factory, one might be able to reach a sensitivity of $10^{-15}$
or better in this search.

\item 

$\mu N \to e N$.  The current upper limit on muon to electron conversion in the
field of a nucleus was set by a PSI experiment~\cite{psi_meco}: $\sigma(\mu^- +
Ti \to e^+ + Ca)/\sigma(\mu^- + Ti \to \nu_\mu + Sc) < 1.7 \times 10^{-12}$.
Upgrades of this experiment at PSI hope to reach a sensitivity of $\sim
10^{-13}$.  The MECO~\cite{meco} experiment at Brookhaven plans to search for $\mu + Al \to
e + Al$ conversion down to a sensitivity of order $10^{-16}- 10^{-17}$.  This is predicated upon obtaining a stopped muon rate of
$10^{11}$ per sec.  With the increase in this rate at a neutrino factory to
$10^{13}-10^{14}$ per sec, again if backgrounds can be controlled, one might
envision an improvement in the sensitivity of a muon to electron conversion
experiment down to the level of perhaps $10^{-18}$.

\end{itemize}

There are also many other interesting experiments that could be pursued.  The
Brookhaven muon $g-2$ experiment has reported a 2.6 $\sigma$ discrepancy
between the measured value of the anomalous magnetic moment of $\mu^+$ and the
theoretical prediction~\cite{e821,ml}.  Further $\mu^+$ data and, in
addition, $\mu^-$ data, will be analyzed in the near future.  The projected
sensitivity of this experiment in $a_\mu$ is about $0.4 \times 10^{-9}$.  The
current rate of stopping $\mu$'s at BNL is about $10^8$ per sec.  With the
increase rate at a neutrino factory, one could perform a higher-statistics
version of this experiment.  This is particular interest in view of the
discrepancy that has been reported between the measured value of the anomalous
magnetic moment and the theoretical prediction.  

At Brookhaven, a proposal~\cite{yannis} has been submitted for an experiment making use of
the existing muon storage ring to search for a muon electric dipole moment
(EDM) down to the level of $10^{-22}$ e-cm in a first stage, with an upgrade
having a sensitivity of $10^{-24}$ e-cm.  A more intense source
of $\mu^\pm$ would also enable one to push this sensitivity down, perhaps to
$10^{-25}$ e-cm or better.  


\subsection{Conclusions}

Neutrino masses and mixing are generic theoretical expectations.
The seesaw mechanism naturally yields light neutrinos, although its detailed
predictions are model-dependent and may require a lower mass scale than the GUT
mass scale.  One of the most interesting findings from the atmospheric data has
been the maximal mixing in the relevant channel, which at present is favored to
be $\nu_\mu \to \nu_\tau$.  Even after the near-term program of experiments by
K2K, MINOS, CNGS, and MiniBOONE, a high-intensity
Neutrino Factory at BNL with $10^{20}$ $\mu$ decays per Snowmass year
and a stored $\mu^\pm$ energy of 20~GeV, coupled with a long-baseline neutrino
oscillation experiment, say with $L=2900$ km to the WIPP facility in Carlsbad,
would make a valuable contribution to the physics of neutrino masses and lepton
mixing. In particular, the Neutrino Factory should be able to improve the accuracy of
the measurement of $\sin^2(2\theta_{23})$ and $\Delta m^2_{32}$ and to measure 
$\sin^2(2\theta_{13})$ and the sign of $\Delta m^2_{32}$.  It might also be
able to measure leptonic CP violation.  

