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\begin{document}
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%\font\fortssbx=cmssbx10 scaled \magstep2
%\hbox to \hsize{{\fortssbx University of Wisconsin - Madison}
\hfill\vtop{
\hbox{AMES-HET-02-04}
\hbox{BUHEP-02-26}
\hbox{MADPH-02-1274}
\hbox
\hbox{}}

\vspace*{.25in}
\begin{center}
{\large\bf Off--axis Beams and Detector Clusters:\\
Resolving Neutrino Parameter Degeneracies}\\[10mm]
V. Barger$^1$, D. Marfatia$^2$ and K. Whisnant$^3$\\[5mm]
\it
$^1$Department of Physics, University of Wisconsin,
Madison, WI 53706, USA\\
$^2$Department of Physics, Boston University,
Boston, MA 02215, USA\\
$^3$Department of Physics and Astronomy, Iowa State University,
Ames, IA 50011, USA

\end{center}
\thispagestyle{empty}

\begin{abstract}

\vspace*{-.35in}

\noindent

There are three parameter degeneracies inherent in the three--neutrino
analysis of long--baseline neutrino experiments.  We develop a
systematic method for determining whether or not a set of measurements
in neutrino oscillation appearance experiments with approximately
monoenergetic beams can completely resolve these ambiguities. We then
use this method to identify experimental scenarios in which the
parameter degeneracies may be efficiently resolved. Generally speaking,
with two appearance measurements degeneracies can occur over wide areas
of the $(\delta,\theta_{13})$ parameter space; with three measurements
they occur along lines in the parameter space and with four measurements
they occur only at isolated points. If two detectors are placed at the
same distance from the source but at different locations with respect to
the main axis of the beam (a detector cluster), each detector will
measure neutrinos at different energies. Then one run with neutrinos and
one run with antineutrinos will give the four independent measurements
that in principle can resolve all of the parameter degeneracies if
$\sin^22\theta_{13} \geq 0.002$. We also examine scenarios with detector
clusters using only neutrino beams. Without detector clusters, the
measurement of neutrinos and antineutrinos at a short distance and only
neutrinos at a longer distance may also work.

\end{abstract}

\newpage

\section{Introduction}

The recent solar neutrino data from the Sudbury Neutrino
Observatory~\cite{Ahmad:2001an,SNONC}, which infers different neutrino
fluxes from the charged-- and neutral--current measurements, provides
convincing evidence that electron neutrinos do in fact change flavor as
they travel from the Sun to the Earth. The neutral--current
measurement~\cite{SNONC} is also consistent with the solar neutrino flux
predicted in the Standard Solar Model~\cite{Bahcall:2000nu}. The
preponderance of solar neutrino data increasingly prefers the Large
Mixing Angle (LMA) solution to the solar neutrino puzzle, with $\delta
m^2_{21} \sim 5 \times 10^{-5}$ eV$^2$ and amplitude close to
0.8~\cite{SNONC,bmww,LMA}. This solution will be tested decisively by
the KamLAND reactor neutrino experiment~\cite{kamland}.

The atmospheric neutrino deficit also gives a strong indication that
neutrinos have mass and oscillate from one flavor to another -- the most
compelling interpretation is that $\nu_\mu$'s created in the atmosphere
oscillate to $\nu_\tau$ with almost maximal amplitude and mass-squared
difference $\delta m^2_{31} \sim 3 \times
10^{-3}$~eV$^2$~\cite{Toshito:2001dk}. The K2K
experiment~\cite{Hill:2001gu} with a baseline of 250 km has preliminary
results that are in agreement with this interpretation. Oscillations of
$\nu_\mu$ to $\nu_e$ as an explanation of the atmospheric anomaly are
ruled out by the CHOOZ~\cite{CHOOZ} and Palo Verde~\cite{paloverde}
reactor experiments, which place a bound on the $\nu_\mu \to \nu_e$
oscillation amplitude smaller than 0.1 at the 95\% C.L. in the $\delta
m^2_{31}$ region of interest. The MINOS~\cite{minos},
ICARUS~\cite{icarus} and OPERA~\cite{opera} experiments are expected to
come online in 2005 and study aspects of the oscillations at the
atmospheric scale~\cite{Barger:2001yx}. The low energy beam at MINOS
will allow a very accurate determination of the leading oscillation
parameters. ICARUS and OPERA should provide concrete evidence that
$\nu_\mu \rightarrow \nu_\tau$ oscillations are responsible for the
atmospheric neutrino deficit by identifying tau neutrino events. These
long--baseline experiments, when combined with KamLAND, should provide
further information on all of the parameters in the three--neutrino
mixing matrix, except for the $CP$--violating phase ($\delta$) and
possibly not the mixing angle associated with $\nu_\mu \to \nu_e$
oscillations in atmospheric and long--baseline experiments
($\theta_{13}$). It will take a new generation of experiments to provide
accurate measurements of these parameters.

Future measurements of $\theta_{13}$ and $\delta$ are not completely
straightforward. The parameter $\theta_{13}$ cannot be determined from a
single $\nu$ measurement since the measured oscillation rate also
depends on the value of $\delta$, which is not known. Even with both a
$\nu$ oscillation measurement and a $\bar\nu$ oscillation measurement
there are several parameter degeneracies that enter the determination of
the three--neutrino mixing matrix. These include the ambiguities
associated with (i) the parameters in the $U_{e3}$ (=$\sin \theta_{13}$
e$^{-i\delta}$) element~\cite{ambiguity,peak,bmw,minakata2,huber}, (ii)
the sign of the mass--squared difference responsible for the
oscillations of atmospheric neutrinos ($\delta
m^2_{31}$)~\cite{peak,bmw,minakata2,huber,minakata1}, and (iii) the
primary mixing angle for the oscillation of atmospheric neutrinos
($\theta_{23}$)~\cite{bmw,huber}. Each of these ambiguities can mix $CP$
violating ($CPV$) and $CP$ conserving ($CPC$) solutions. To resolve the
ambiguities requires multiple measurements; the usual prescription
involves making measurements with different beam energies, baseline
lengths, and/or beam particles (neutrinos versus antineutrinos). A
typical scenario requires measurements of neutrinos and antineutrinos,
each at two different energies~\cite{bmw}, which would entail four
separate runs, each of which would take years to complete.

In this paper we discuss alternate possibilities that capitalize on the
fact that the energy spectrum of a conventional neutrino beam varies
with the angle from the main beam axis~\cite{bnl,jhfsk,para}. The
off--axis components of a neutrino beam have very narrow spectra for a
given off--axis angle. Viewed this way, a single neutrino beam is
actually a continuum of approximately monoenergetic beams; the energy
can be tuned simply by varying the angle with respect to the main beam
axis. This has the advantage that experiments at more than one neutrino
energy may be run simultaneously using a single beam line, simply by
placing detectors at slightly different angles from the main axis of the
beam (a detector cluster). One could also locate one detector on the
beam axis and another off--axis. Furthermore, the high energy tail of
off--axis components of the beam is suppressed, reducing the
high--energy contribution to backgrounds. One disadvantage is that the
neutrino flux decreases as the off--axis distance increases, which may
give a practical limit for the choices of neutrino energies.

Detailed studies have already been made for off--axis beams with single
detectors~\cite{huber,barenboim,hagiwara}, but how to resolve the
parameter ambiguities has not been thoroughly addressed. In this paper
we present some possible scenarios for resolving parameter degeneracies
using detector clusters with off--axis neutrino beams. In particular we
examine how well parameter degeneracies may be resolved with one run of
a neutrino beam and one run of an antineutrino beam. We also discuss
other, more speculative, possibilities involving detector clusters, such
as multiple detectors and only neutrino beams. We compare these results
to the ability of more conventional experimental setups (without
detector clusters) to resolve parameter degeneracies.

The organization of our paper is as follows. In Sec.~\ref{sec:prob} we
present oscillation probability formulas based on a constant density
approximation that works well for the situations that we consider. In
Sec.~\ref{sec:determine} we determine in a general way the number and
type of measurements that are needed to resolve parameter
degeneracies. Specific detector scenarios are presented and discussed in
Sec.~\ref{sec:scenarios}. Concluding remarks are made in
Sec.~\ref{sec:discussion}.

\section{Oscillation probabilities in matter}
\label{sec:prob}

We work in the three--neutrino scenario. For oscillation studies, the
neutrino mixing matrix $U$ can be specified by 3 mixing angles
($\theta_{23}, \theta_{12}, \theta_{13}$) and a $CP$-violating phase
$\delta$. We adopt the parametrization
%
\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) \,,
\label{eq:MNS}
\end{equation}
%
where $c_{jk} \equiv \cos\theta_{jk}$ and $s_{jk} \equiv
\sin\theta_{jk}$. For Majorana neutrinos there are two additional
phases, but they do not affect oscillations. In the most general $U$,
the $\theta_{ij}$ are
restricted to the first quadrant, $0\le \theta_{ij} \le \pi/2$,
with $\delta$ in the range $0 \le \delta < 2\pi$. We assume that
$\nu_3$ is the neutrino eigenstate that is separated from the other two,
and that the sign of $\delta m^2_{31}$ can be either positive or
negative, corresponding to the case where $\nu_3$ is either above or
below, respectively, the other two mass eigenstates. The magnitude of
$\delta m^2_{31}$ determines the oscillation length of atmospheric
neutrinos, while the magnitude of $\delta m^2_{21}$ determines the
oscillation length of solar neutrinos, and thus $|\delta m^2_{21}| \ll
|\delta m^2_{31}|$. If we accept the likely conclusion that the solar
solution is LMA~\cite{SNONC,bmww,LMA}, then $\delta m^2_{21} > 0$ and we
can restrict $\theta_{12}$ to the range $[0,\pi/4]$.  It is known from
reactor neutrino data that $\theta_{13}$ is small, with
$\sin^22\theta_{13} \le 0.1$ at the  95\% C.L.~\cite{CHOOZ}. Thus a set
of parameters that unambiguously spans the space is $\delta m^2_{31}$
(magnitude and sign), $\delta m^2_{21}$,  $\sin^22\theta_{12}$,
$\sin\theta_{23}$, and $\sin^22\theta_{13}$; only the $\theta_{23}$
angle can be below or above $\pi/4$.

To determine oscillation probabilities we use the constant density
approximation and expand in terms of the small parameters $\theta_{13}$
and $\delta m^2_{21}$~\cite{cervera,freund}, which has been shown to
reproduce well the exact oscillation probabilities for $E_\nu >
0.5$~GeV, $\theta_{13}$ not too large, and $L < 4000$~km~\cite{bmw}.
Up to second order in $\alpha$ and $\theta_{13}$, the oscillation
probabilities for $\delta m^2_{31} > 0$ and $\delta m^2_{21} > 0$ are
%
\begin{eqnarray}
P(\nu_\mu \to \nu_e) =
|xf + y g e^{i(\Delta+\delta)}|^2 =
x^2 f^2 + 2 x y f g (\cos\delta\cos\Delta - \sin\delta\sin\Delta)
+ y^2 g^2 \,,
\label{eq:P}\\
\bar P(\bar\nu_\mu \to \bar\nu_e) =
|xf + y g e^{i(\Delta-\delta)}|^2 =
x^2 \bar f^2 + 2 x y \bar f g (\cos\delta\cos\Delta
+ \sin\delta\sin\Delta) + y^2 g^2 \,,
\label{eq:Pbar}
\end{eqnarray}
%
respectively, where
%
\begin{eqnarray}
x &\equiv& \sin\theta_{23} \sin 2\theta_{13} \,,
\label{eq:x}\\
y &\equiv& \alpha \cos\theta_{23} \sin 2\theta_{12} \,,
\label{eq:y}\\
f, \bar f &\equiv& \sin((1\mp\hat A)\Delta)/(1\mp\hat A) \,,
\label{eq:f}\\
g &\equiv& \sin(\hat A\Delta)/\hat A \,,
\label{eq:g}
\end{eqnarray}
%
and
%
\begin{eqnarray}
\Delta &\equiv& |\delta m_{31}^2| L/4E_\nu
= 1.27 |\delta m_{31}^2/{\rm eV^2}| (L/{\rm km})/ (E_\nu/{\rm GeV}) \,,
\label{eq:D}\\
\hat A &\equiv& |A/\delta m_{31}^2| \,,
\label{eq:Ahat}\\
\alpha &\equiv& |\delta m^2_{21}/\delta m^2_{31}| \,.
\label{eq:alpha}
\end{eqnarray}
%
In Eq.~(\ref{eq:Ahat}), $A/2E_\nu$ is the  amplitude for coherent forward
charged-current $\nu_e$ scattering on electrons, with
%
\begin{equation}
A = 2\sqrt 2\, G_F \, N_e \, E_\nu = 1.52\times10^{-4}\,{\rm eV^2}
Y_e \, \rho\,({\rm g/cm^3}) E_\nu\,(\rm GeV) \;,
\label{eq:A}
\end{equation}
%
and $N_e$ is the electron number density, which is the product of
the electron fraction $Y_e(x)$ and matter density $\rho(x)$. In the
Earth's crust and mantle the average matter density is typically
3--5~g/cm$^3$ and $Y_e\simeq 0.5$. In all of our calculations we use the
average $N_e$ along the neutrino path, assuming the Preliminary
Reference Earth Model~\cite{PREM}.

The coefficients $f$ and $\bar f$ differ due to matter effects ($\hat A
\ne 0$). The values of $f$, $\bar f$, and $g$ (scaled by
$\sqrt{E_\nu}/L$ to account for the dependence of neutrino cross section
with energy and the flux with distance) are shown versus $\Delta$ in
Fig.~\ref{fig:ffbg} for $L = 300$~km (JHF to Kamioka), $730$~km
(Fermilab to Soudan or CERN to Gran Sasso), $1290$~km (Fermilab to
Homestake), $1770$~km (Fermilab to Carlsbad), and $2900$~km (Fermilab to
SLAC); the scaling factor is chosen this way since the oscillation
probabilities are quadratic functions of $f$, $\bar f$, and $g$.

To obtain the probabilities for $\delta m^2_{31} < 0$, the
transformations $\hat A \to - \hat A$, $y \to -y$ and $\Delta \to
-\Delta$ (implying $f \leftrightarrow -\bar f$ and $g \to -g$) can be
applied to the probabilities in Eqs.~(\ref{eq:P}) and (\ref{eq:Pbar}) to
give
%
\begin{eqnarray}
P(\nu_\mu \to \nu_e) =
x^2 \bar f^2 - 2 x y \bar f g (\cos\delta\cos\Delta
+ \sin\delta\sin\Delta) + y^2 g^2\,,
\label{eq:P2}\\
\bar P(\bar\nu_\mu \to \bar\nu_e) =
x^2 f^2 - 2 x y f g (\cos\delta\cos\Delta
- \sin\delta\sin\Delta) + y^2 g^2 \,.
\label{eq:Pbar2}
\end{eqnarray}
%

In practice, neutrino beams are not monoenergetic, even for a narrow
band beam or an off--axis component of a beam. The number of appearance
events is $N(\nu_\mu \to \nu_e) =$ \mbox{$\int P(\nu_\mu \to \nu_e)$}
$\Phi \sigma \, dE_\nu$, where $\Phi$ is the energy--dependent neutrino
flux, $\sigma$ the interaction cross section, and $P$ the oscillation
probability from Eq.~(\ref{eq:P}). Then for $\delta m^2_{31} > 0$, $N$
has the approximate form
%
\begin{equation}
N(\nu_\mu \to \nu_e) =  A_1 x^2 + A_2 x\cos\delta + A_3 x\sin\delta
+A_4 \,,
\label{eq:N}
\end{equation}
%
where
%
\begin{eqnarray}
A_1 &=& \int f^2 \Phi \sigma \, dE_\nu \,,
\nonumber\\ 
A_2 &=& \int 2yfg\cos\Delta \Phi \sigma \, dE_\nu \,,
\nonumber\\ 
A_3 &=& - \int 2yfg\sin\Delta \Phi \sigma \, dE_\nu \,,
\nonumber\\ 
A_4 &=& \int y^2 g^2 \Phi \sigma \, dE_\nu \,.
\label{eq:A_n}
\end{eqnarray}
%
There are similar expressions for $\bar N(\bar\nu_\mu \to
\bar\nu_e)$, and for $\delta m^2_{31} < 0$. In all cases, whether one is
describing the number of events at a single energy or total events
integrated over an energy spectrum, the result can be written
approximately as a linear combination of $x^2$, $x\cos\delta$, and
$x\sin\delta$. In fact the probability for constant density can be
written as $A\cos\delta + B\sin\delta + C$ without any
approximation~\cite{kimura}. It is this generic property that we exploit
in this paper, so many of the qualitative aspects of our analysis should
hold for neutrino beams that are not monoenergetic if only the total
number of events is being used.

\section{Determining the oscillation parameters}
\label{sec:determine}

\subsection{Parameter degeneracies with two measurements}
\label{sec:two}

It is expected that $\delta m^2_{31}$ and $\sin^22\theta_{23}$ will be
well--measured in $\nu_\mu$ survival experiments; if the solar
solution is LMA, as it now seems to be~\cite{SNONC,bmww,LMA}, then
$\delta m^2_{21}$ and $\sin^22\theta_{12}$ will be well--determined by
KamLAND~\cite{kamland}. Thus it is the parameters $\theta_{13}$ and
$\delta$ that will be primarily measured in appearance experiments,
modulo uncertainties in the other parameters.

A single measurement is not sufficient to determine
$\theta_{13}$ since the oscillation probability depends on both
$\theta_{13}$ and $\delta$ (see, e.g., Eqs.~(\ref{eq:P}) and
(\ref{eq:Pbar})). In fact the inferred measurement of
$\sin^22\theta_{13}$ could be uncertain by an order of magnitude with
only one measurement~\cite{bmw}.
The usual approach for determining $\theta_{13}$ and $\delta$ is a
measurement of both $P$ and $\bar P$ at one $L$ and $E_\nu$. Then
$\theta_{13}$ and $\delta$ can be determined from Eqs.~(\ref{eq:P}) and
(\ref{eq:Pbar}). Unfortunately, there are usually two solutions with
distinct $\theta_{13}$ and $\delta$ (the well--known $(\delta,
\theta_{13})$ ambiguity~\cite{ambiguity,peak,bmw,minakata2,huber}). The
existence of this ambiguity can be understood as follows. If
$\theta_{13}$ is held fixed, then as $\delta$ is varied an ellipse is
traced out in ($P,\bar P$) space; the eccentricity of the ellipse is
determined by the oscillation argument
$\Delta$~\cite{bmw,minakata1}. Ellipses for two different values of
$\theta_{13}$ can intersect, so that a single point in $(P,\bar P)$
space can be obtained by two different sets of parameters $(\delta,
\theta_{13})$. For a given $(P,\bar P)$ there are also often two
solutions for $\delta m^2_{31} < 0$ from Eqs.~(\ref{eq:P2}) and
(\ref{eq:Pbar2}) (the sgn($\delta m^2_{31}$) ambiguity). Furthermore,
since only $\sin^22\theta_{23}$ is known from atmospheric neutrino
experiments (or from measuring $\nu_\mu$ survival), if $\theta_{23} \ne
\pi/4$, each solution discussed above also has two possible values of
$\theta_{23}$ (each with a different $\theta_{13}$ and $\delta$), one
with $\theta_{23} < \pi/4$ and one with $\theta_{23} > \pi/4$ (the
$(\theta_{23}, \pi/2-\theta_{23})$ ambiguity). Thus, in general, there
can be as much as an 
eight--fold degeneracy of solutions from a measurement of $P$ and $\bar
P$ at one $L$ and $E_\nu$~\cite{bmw}, which has already required two
experimental runs (one with neutrinos and one with
antineutrinos). Further measurements at different oscillation arguments
$\Delta$ are needed to eliminate all of the ambiguities, requiring
additional runs.

A judicious choice of the $L$ and $E_\nu$ can help reduce the effect of
the ambiguities. For example, the $(\delta, \theta_{13})$ ambiguity can
be reduced to a $(\delta, \pi - \delta)$ ambiguity by choosing $L/E_\nu
= (2n -1) (410 {\rm~km/GeV}) (3\times10^{-3}$~eV$^2/ |\delta m^2_{31}|$,
where $n$ is an integer (which implies $\Delta = (n - {1\over2})\pi$,
the approximate location of the appearance oscillation peak).  The
parameter $\theta_{13}$ is removed from the ambiguity (and therefore in
principle determined) since the ellipse in $(P,\bar P)$ space collapses
to a line and the lines for different $\theta_{13}$ no longer
intersect~\cite{peak,bmw,minakata2}. Furthermore, if $L$ is sufficiently
large, the large matter effects separate the ellipses for $\delta
m^2_{31} > 0$ and $\delta m^2_{31} < 0$, so that they do not intersect,
removing the sgn($\delta m^2_{31}$) ambiguity~\cite{peak,bmw}. However,
measuring ($P,\bar P$) at a single $L$ and $E_\nu$ will still leave a
$(\delta, \pi-\delta)$ ambiguity, and may have a $\theta_{23}$ ambiguity
if $\theta_{23} \ne \pi/4$ (with a possible associated $CPV/CPC$
confusion), so that additional measurements are needed to completely
determine the parameters and resolve all of the ambiguities.

The arguments above can be generalized to show that {\it any} two
measurements of neutrino and/or antineutrino appearance probabilities,
even if they are not at the same $L$ and $E_\nu$, are subject to the
same degeneracy problem. Each of the neutrino or antineutrino
probabilities in an appearance experiment (e.g., Eqs.~(\ref{eq:P}) and
(\ref{eq:Pbar}), or (\ref{eq:P2}) and (\ref{eq:Pbar2})) is a linear
combination of the parameters $x^2$, $x\cos\delta$, and
$x\sin\delta$. The coefficients of these parameters involve $f$, $\bar
f$, $g$, and $\Delta$, which depend only on $L$, $E_\nu$, $\delta
m^2_{31}$, and $y$. It is not hard to show that for two such
measurements of $P$ and/or $\bar P$, if $\theta_{13}$ is fixed and
$\delta$ is varied, an ellipse is traced out in the two--dimensional
probability space. Ellipses with somewhat different values of
$\theta_{13}$ will overlap, so that there is always a $(\delta,
\theta_{13})$ ambiguity for any two appearance measurements. The only
exception is the special situation where the ellipse collapses to a
straight line, in which case lines for different $\theta_{13}$ no longer
overlap, but there is still an ambiguity involving $\delta$ since the
two halves of the collapsed ellipse then overlap (e.g., the
$(\delta,\pi-\delta)$ ambiguity for ($P,\bar P)$ when $\Delta =
\pi/2$). Similarly, there is a $\theta_{23}$ ambiguity if $\theta_{23}
\ne \pi/4$, and there may be a sgn($\delta m^2_{31}$) ambiguity if the
matter effect is not large enough to separate the ellipses for
$\delta m^2_{31} > 0$ and $\delta m^2_{31} < 0$. Thus two measurements
will {\it always} have an ambiguity involving $\delta$ (which may or may
not also involve $\theta_{13}$, depending on the value of the
oscillation argument $\Delta$), and may also have ambiguities involving
sgn($\delta m^2_{31}$) and $\theta_{23}$. Ambiguities involving
sgn($\delta m^2_{31}$) and $\theta_{23}$ will have a $CPV/CPC$
confusion; the ambiguity that involves $\delta$ alone will have a
$CPV/CPC$ confusion unless $\Delta = n\pi/2$.

We emphasize that the degeneracies discussed here are exact, i.e., there
are different sets of parameters that give {\it identical}
predictions. Thus these ambiguities are present even in the limit of no
experimental uncertainties. When such uncertainties are taken into
account, nearly degenerate solutions may also be indistinguishable,
which would require further (or improved) measurements. Therefore the
conditions needed to remove the exact degeneracies are necessary but not
sufficient for removing degeneracies in an actual
experiment. Furthermore, parameter degeneracies occur nearly everywhere
in the $(\delta,\theta_{13})$ plane, since in most cases any given
ellipse (for fixed $\theta_{13}$) overlaps an adjacent ellipse (with
different $\theta_{13}$).

\subsection{Parameter degeneracies with three measurements}
\label{sec:three}

Since two measurements will generally have parameter degeneracies, the
natural question is whether a third measurement sufficient to remove all
of them. To extend the degeneracy analysis to more than two appearance
measurements, what is needed is a straightforward and systematic method
for determining whether or not a given set of such measurements resolves
all of the degeneracies over the entire region of interest in the
parameters $\delta$ and $\theta_{13}$. The method we present here uses
the approximate expressions for the probabilities given in
Sec.~\ref{sec:prob}.

We begin by analyzing the case of three appearance measurements. From
the approximate probability expressions we have three linear equations
involving $x^2$, $x\cos\delta$, and $x\sin\delta$. As long as the three
equations are linearly independent (which can be realized if no two
measurements simultaneously have the same $L$, $E_\nu$, and type of
neutrino) these are easily solved, giving a unique solution for $x$ and
$\delta$. However, in principle four possible solutions may still exist,
corresponding to the sgn($\delta m^2_{31}$) and $\theta_{23}$
ambiguities (see Table~\ref{tab:degen}); each such solution gives
identical predictions for the three probabilities being measured, but
can have different values for $\theta_{13}$ and/or $\delta$.

\begin{table}[t]
\squeezetable
\caption[]{Four classes of degenerate solutions which could potentially
remain after three appearance measurements.}
\label{tab:degen}
\begin{tabular}{ccc}
Case & sgn($\delta m^2_{31}$) & $\theta_{23}$\\
\hline\hline
I & $+$ & $<\pi/4$\\
II & $-$ & $<\pi/4$\\
III & $+$ & $>\pi/4$\\
IV & $-$ & $>\pi/4$
\end{tabular}
\end{table}

To determine if the remaining degeneracies have been resolved by the
third measurement, we use the fact that the variables $x^2$,
$x\cos\delta$, and $x\sin\delta$ are not independent, and in the
``true'' solution they {\it must} obey the relation
%
\begin{equation}
x^2 = (x\cos\delta)^2 + (x\sin\delta)^2 \,.
\label{eq:x2}
\end{equation}
%
The question is then whether the ``fake'' solutions can also satisfy the
constraint in Eq.~(\ref{eq:x2}). We have found that for many values of
$\delta$ and $\theta_{13}$, the fake solutions do not obey
Eq.~(\ref{eq:x2}) (thus removing the degeneracy). However, for some
particular values of $\delta$ and $\theta_{13}$ there exist fake
solutions that also obey Eq.~(\ref{eq:x2}), so that degeneracy may
remain even after three measurements; the values of $\delta$ and
$\theta_{13}$ for which degenerate solutions still exist form lines in
the $(\delta,\theta_{13})$ plane. It is easy to understand why this is
so: with two measurements, most points in the $(\delta,\theta_{13})$
plane have degeneracies, but the condition in Eq.~(\ref{eq:x2}) imposes
an additional constraint so that for three measurements the dimension of
the degenerate space is reduced from two (the plane) to one (lines).
Since the forms of $P$ and $\bar P$ are the same (i.e., a linear
combination of $x^2$, $x\cos\delta$, and $x\sin\delta$), {\it any}
combination of three neutrino and/or antineutrino appearance
measurements will yield lines in $(\delta,\theta_{13})$ space where the
fake degenerate solutions are not eliminated by three measurements.

For example, consider neutrino measurements made at $L = 730$~km and
$E_\nu = 2.66$, $1.77$, and $1.34$~GeV ($\Delta = \pi/3$, $\pi/2$, and
$2\pi/3$, respectively); Fig.~\ref{fig:degen1}a shows the lines in
$(\delta,\theta_{13})$ space where a fake solution remains
degenerate with the true solution (taken to be Case~I in the
figure). Figure~\ref{fig:degen1}b shows similar curves for $L = 730$~km
and $E_\nu = 3.54$, $1.77$, and $0.89$~GeV ($\Delta = \pi/4$, $\pi/2$,
and $\pi$, respectively). In Fig.~\ref{fig:degen1} we assume
%
\begin{eqnarray}
|\delta m^2_{31}| = 3\times10^{-3}{\rm~eV}^2 \,,
\sin^22\theta_{23} = 0.90 \,,
\nonumber\\
\delta m^2_{21} = 5\times10^{-5}{\rm~eV}^2 \,,
\sin^22\theta_{12} = 0.80 \,,
\label{eq:standard}
\end{eqnarray}
%
which allows the possibility of a $\theta_{23}$ ambiguity as well as the
($\delta,\theta_{13}$) and sgn($\delta m^2_{31}$) ambiguities; this will
be the standard parameter set used in this paper unless stated
otherwise. The key to determining if three measurements are sufficient
to remove degeneracies in a particular experiment is whether or not the
experiment is sensitive to the values of $\delta$ and $\theta_{13}$
where the lines of degeneracies occur. We see that in the two examples
in Fig.~\ref{fig:degen1}, degeneracies remain for a wide range of
$\delta$ when $\sin^22\theta_{13} > 0.01$. Thus either of these two sets
of measurements are not sufficient to ensure there are no degeneracies
in the next generation of long--baseline neutrino experiments, which
should be sensitive down to $\sin^22\theta_{13} = 0.01$ or
below~\cite{huber,jhfsk,barenboim,superbeams}.

\subsection{Parameter degeneracies with four measurements}
\label{sec:four}

When three measurements are not sufficient to completely resolve the
degeneracies, a fourth measurement must be made. A straightforward way
to determine whether or not the fourth measurement has removed the
remaining degeneracies is as follows. Take all possible combinations of
three measurements within the four (there are four such combinations)
and perform the above analysis on each combination of three
measurements.  Each three--measurement subgroup may have values of
$\delta$ and $\theta_{13}$ for which degeneracies remain (if not, then
that subgroup alone is sufficient to remove the degeneracies, and a
fourth measurement is not needed). If the degenerate lines in
$(\delta,\theta_{13})$ space for the four subgroups do not intersect at
a common point, then in principle the four measurements have resolved
all the degeneracies; if not, then some degeneracies remain.

For example, in the three measurements used to obtain
Fig.~\ref{fig:degen1}a there were degeneracies for a range of $\delta$
when $\sin^22\theta_{13} > 0.01$.  After a fourth measurement with $L =
730$~km and $E_\nu = 5.31$~GeV ($\Delta = \pi/6$) is added to the
measurements used in Fig.~\ref{fig:degen1}a, the lines of degeneracy for
the four subgroups of three measurements are shown in
Fig.~\ref{fig:degen2}. There is a point near $\sin^22\theta_{13} =
0.0085$ and $\delta = 0.19\pi$ where the lines of degeneracy all
intersect when comparing I and IV, so that these two cases remain
degenerate after these four measurements (see Fig.~\ref{fig:degen2}c),
Furthermore, there is also a point near $\sin^22\theta_{13} =
0.0015$ and $\delta = 0.49\pi$ where the four lines nearly intersect at
the same point when comparing I and II, so that these two cases would be
degenerate once experimental uncertainties are considered.

It is not hard to understand why some degenerate points remain in
$(\delta,\theta_{13})$ space after four measurements. The fourth data
point adds one additional constraint on the parameters, which then
reduces the dimension of the space that is degenerate from one
(lines) to zero (points). For an experiment to be assured of resolving
the parameter degeneracies, these degenerate points must lie at
$\sin^22\theta_{13}$ below the experimentally accessible region.

We note that to determine the parameter degeneracies between, say,
Case~I and Case~II with more than two measurements using a purely
numerical approach, for each point in ($\delta,\theta_{13}$) space for
Case~I, one would have to search the entire ($\delta,\theta_{13}$) space
for Case~II to check for an identical set of probabilities. With our
analytic approach, one need only search the parameter space once, since
the existence of parameter degeneracies is determined algebraically.

%Finally, we note that in many of the cases we consider in this paper, the
%degeneracies involving only the $\theta_{23}$ ambiguity and not the
%sgn($\delta m^2_{31}$) ambiguity (e.g., between I and III in
%Fig.~\ref{fig:degen2}b) tend to be at lower $\theta_{13}$ when compared
%to degeneracies involving sgn($\delta m^2_{31}$) (e.g., between I and II
%in Fig.~\ref{fig:degen2}a and between I and IV in
%Fig.~\ref{fig:degen2}c). Therefore, practically speaking, the
%sgn($\delta m^2_{31}$) ambiguity is usually the most serious (i.e., it
%occurs at larger $\theta_{13}$), although the $\theta_{23}$ ambiguity
%can occur in conjunction with it, e.g., I versus IV.

%For $\sin^22\theta_{13} > 0.01$, these four lines
%do not have a common intersection point, so the fourth measurement has
%removed the degeneracy for an experiment that is sensitive only to
%$\sin^22\theta_{13} > 0.01$.  However, since these four measurements
%could not ensure 
%the ambiguity is removed between I and IV if the experiment was
%sensitive to $\sin^22\theta_{13} = 0.0085$ or less.  Furthermore, when
%comparing I and IV there is a region near $\sin^22\theta_{13} =
%0.04$ and $\delta = 90^\circ$ where the four lines nearly intersect at
%the same point (see Fig.~\ref{fig:degen2}c), so experimental
%uncertainties could make it difficult to remove this degeneracy; a more
%detailed analysis including the experimental errors would be necessary
%in this case. Of course, the regions that remain degenerate after these
%four measurements occupy small areas of the parameter space, which may
%or may not be near the physical parameters, but if one wanted to
%completely avoid the possibility of degeneracies, a different set of
%measurements would have to be considered.

Figure~\ref{fig:degen3}a shows some typical parameter degeneracies
between Cases~I and II if only one measurement is made for $\nu$ and
$\bar\nu$ at $L=300$~km and $\Delta = \pi/2$ (corresponding to $E_\nu =
0.73$~GeV), assuming the same oscillation parameters as
Eq.~(\ref{eq:standard}) except $\theta_{23} = \pi/4$. We note that these
degeneracies can mix CPV and CPC solutions, especially when
$\theta_{13}$ is not small; for some sets of parameters, $CPC$ can give
identical results to maximal $CPV$ with the opposite sgn($\delta
m^2_{31}$). To illustrate how additional measurements reduce the
possibilities for degeneracies, Fig.~\ref{fig:degen3}b shows the
parameter regions in Case~I that have degeneracy with Case~II, for two,
three, and four measurements; the degenerate space is reduced from an
area to lines to points. The corresponding degenerate parameters for
Case~II in Fig.~\ref{fig:degen3}b occur at the same $\theta_{13}$ with
$\delta \to \delta \pm \pi$; it is not hard to show that this
symmetrical situation is an artifact of choosing the same energy for
$\nu$ and $\bar\nu$, and would not necessarily be true if the $\nu$ and
$\bar\nu$ energies were different.

In summary, two measurements necessarily have parameter degeneracies
over a broad range of $(\delta,\theta_{13})$ space; adding a third
measurement reduces the degeneracy to lines in the parameter space, and
adding a fourth measurement reduces it further to isolated points. This
behavior is to be expected since each additional measurement adds one
more constraint, which has the effect of reducing the
dimension of the degenerate subspace by one. Adding a fifth measurement
with a linearly independent combination of $x^2$, $x\cos\delta$ and
$x\sin\delta$ should then remove the degeneracy completely for all
values of $\delta$ and $\theta_{13}$. For example, in
Fig.~\ref{fig:degen2}, if a fifth measurement at $L = 730$~km is made
at, say, $\Delta = 5\pi/6$, then no point in the parameter space remains
degenerate. Of course, practically speaking, less than five measurements
may be sufficient to eliminate the possibility of parameter degeneracies
in any given experiment if the degeneracies occur in a region of
parameter space in which the experiment is not sensitive. In the next
section we explore some specific cases with less than five measurements.

%\begin{table}[t]
%\squeezetable
%\caption[]{Nature of degeneracies in $(\theta_{13},\delta)$ plane for a
%given number of linearly independent appearance measurements.}
%\label{tab:number}
%\begin{tabular}{lc}
%\# of $\nu$/$\bar\nu$ & Region with\\
%measurements & degeneracies\\
%\hline\hline
%2 & $(\theta_{13},\delta)$ plane\\
%3 & lines\\
%4 & isolated points\\
%5 & none
%\end{tabular}
%\end{table}

\section{Detector scenarios for resolving parameter degeneracies}
\label{sec:scenarios}

In the previous section we presented a general method for determining if
a set of more than two measurements are sufficient to remove all of the
parameter degeneracies. In this section we discuss specific detector
scenarios using this method. We emphasize scenarios where at least some
of the measurements can be made at the same $L$ but different $E_\nu$,
since these are well--suited to off--axis beams (where different
energies can be obtained by having a detector at different locations
with respect to the beam axis). If a detector cluster could be used, the
time required to make all of the necessary measurements could be
reduced. We also compare the detector cluster scenarios to scenarios
in which a detector cluster is not used.

\subsection{Two $\nu$ and two $\bar\nu$ measurements at one $L$}
\label{sec:2nu2nubar}

The standard means for the study of $CP$ violation is to measure both
$\nu_\mu \to \nu_e$ and $\bar\nu_\mu \to \bar\nu_e$ at the same $L$ and
$E_\nu$. As we have seen, there could be as much as an eight--fold
degeneracy in the parameters with just these two measurements, and it
may be impossible to establish $CPV$ from $\nu$ and $\bar\nu$
measurements at a single $L$ and $E_\nu$. The simplest way one could
imagine obtaining another set of independent measurements is to measure
$\nu_\mu \to \nu_e$ and/or $\bar\nu_\mu \to \bar\nu_e$ at the same $L$
but with different $E_\nu$; the detector and beamline remain the same,
and only the beam energy and/or primary particle in the beam ($\nu_\mu$
or $\bar\nu_\mu$) would need to be changed from one run to the
next. With a single detector this would take four separate
runs. However, with two detectors at different locations with respect to
the beam axis (so that each receives neutrinos of different energy)
these four measurements could be taken with only two runs (one with
$\nu$ and one with $\bar\nu$).

Figure~\ref{fig:2nu2nubar} shows the points in $(\delta,\theta_{13})$
space where degeneracies remain due to the sgn($\delta m^2_{31}$) and
$\theta_{23}$ ambiguities after measurements at two energies for both
neutrinos and antineutrinos, assuming that $\bar\nu$ energies are the
same as the $\nu$ energies and that the measurements are all made at
approximately the same $L$, for different choices of $L$. We examined
scenarios where one energy was chosen so that $\Delta_1 = \pi/2$, and
the second energy was chosen so that $\Delta_2 = \pi/6$, $\pi/4$,
$\pi/3$, $2\pi/3$, or $\pi$; some representative examples are shown in
the figure. We chose $\Delta_1 = \pi/2$ because the probabilities are
large there (see $f$ and $\bar f$ in Fig.~\ref{fig:ffbg}), the
$\cos\delta$ term in the probabilities vanishes (thereby isolating the
$\sin\delta$ term), and the survival channel $\nu_\mu \to \nu_\mu$ is
near an oscillation minimum. All possible degeneracies are included in
Fig.~\ref{fig:2nu2nubar} (e.g., I vs. II, II vs. I, I vs. III, etc.,
where the first case is the true solution), and different types of
degeneracies are not distinguished. In all cases there are some points
in the parameter space where degeneracies are still possible. We found
that if $\Delta_2 < \pi/2$, the degeneracies tended to be at smaller
values of $\theta_{13}$ than for $\Delta_2 > \pi/2$; the probabilities
are also higher there (see $f$ and $\bar f$ in Fig.~\ref{fig:ffbg}).

For off--axis beams, both the neutrino flux and energy fall off with
increasing off--axis angle (see Eqs.~1 and 2 of Ref.~\cite{para});
after eliminating the angular dependence one obtains for the energy
dependence of the flux
%
\begin{equation}
\Phi \propto E_\nu^2 \,,
\label{eq:phi}
\end{equation}
%
so that $\Phi \propto \Delta^{-2}$ for measurements at the same $L$,
Therefore if two off--axis measurements had very different $\Delta$
values, one would have a much reduced flux compared to the other. Hence
it may be preferable if the ratio between the two values of $\Delta$
is not too large, since then the fluxes would be more equal. For
example, with $\Delta_1 = \pi/2$, $\Delta_2 = \pi/4$ and $\pi/3$ give
very similar parameter degeneracies, but choosing $\Delta_2 = \pi/3$
would make the fluxes of the two measurements closer; even then, the
ratio of the fluxes would be 4:9. Of course, the detector further off
axis could be made larger to help equalize the event rate. The detectors
should not be too close in off--axis angle, either, since then the
$\Delta$ values would be very similar and the expressions for the
probabilities would no longer be linearly independent after experimental
uncertainties are taken into account.

Similar to the discussion of Fig.~\ref{fig:degen3}b, since the $\bar\nu$
energies are the same as the $\nu$ energies, the degenerate points in
Fig.~\ref{fig:2nu2nubar} come in symmetrical pairs, each with the same
$\theta_{13}$ and $\delta$ differing by $\pi$. It can be shown that the
degenerate parameters for Case~I (III) when it is degenerate with
Case~II (IV) exhibit this symmetry property with the parameters for II
(IV) when it is degenerate with I (III). Therefore for degeneracies
between I and II, or between III and IV, $CPC$ and $CPV$ solutions are
not mixed, since $\delta$ and $\delta \pm \pi$ are either both $CPC$ or
$CPV$ (although of course sgn($\delta m^2_{31}$) is not determined). On
the other hand, the symmetry also exists between the parameters for I
when it is degenerate with IV (III) and the parameters for II when it is
degenerate with III (IV), and between the parameters for IV (III) when
it is degenerate with I and the parameters for III (IV) when it is
degenerate with II; in all of these situations, the degenerate points
that exhibit the symmetry property do not occur between two cases
that are degenerate with each other, so there is still a $CPC/CPV$
confusion for these degeneracies.

The values of $\delta$ at which degeneracies occur in
Fig.~\ref{fig:2nu2nubar} are insensitive to the assumed values of
$\delta m^2_{21}$ and $\sin^22\theta_{12}$. However, we found that the
values of $\theta_{13}$ at which degeneracies occur vary approximately
as
%
\begin{equation}
\sin^22\theta_{13}^{degen} = (\sin^22\theta_{13}^{degen})_0
\left({\delta m^2_{21}\over5\times10^{-5}{\rm~eV}^2}\right)^2
\left({\sin^22\theta_{12}\over 0.80}\right) \,,
\label{eq:degen}
\end{equation}
%
where the degenerate parameter is $(\sin^22\theta_{13}^{degen})_0$ when
$\delta m^2_{21} = 5\times10^{-5}$~eV$^2$ and $\sin^22\theta_{12} =
0.80$. Thus the values of $\theta_{13}$ at which degeneracies occur is
somewhat sensitive to $\sin^22\theta_{12}$ (which varies from 0.67 to
0.92 at the 2$\sigma$ level~\cite{bmww}) and very sensitive to $\delta
m^2_{21}$ (which varies from $3\times10^{-5}$ to
$2\times10^{-4}$~eV$^2$ at the 2$\sigma$ level~\cite{bmww}).
We also found that a variation in $\delta m^2_{31}$ of 10\% (the
expected precision of MINOS, ICARUS, and OPERA) caused
$\sin^22\theta_{13}$ of the degenerate points to vary by 30\% or less;
smaller $\delta m^2_{31}$ moved the degeneracies to higher
$\sin^22\theta_{13}$; the largest variations occurred
at lower $\sin^22\theta_{13}$. The corresponding variation in
$\delta$ was much less, of order a few percent.

In Fig.~\ref{fig:2nu2nubar}, all of the examples at shorter $L$ have
degeneracies for $\sin^22\theta_{13} \simeq 0.01$ or larger, which is
within the expected range of future long--baseline experiments with
superbeams. However, at $L=300$~km with $\Delta = \pi$, the degeneracy
for larger $\theta_{13}$ occurs for $\sin^22\theta_{13} > 0.1$ (around
0.13), above the current upper bound. The only degeneracies with
$\sin^22\theta_{13} < 0.1$ in the $L=300$~km case occur at much smaller
$\theta_{13}$ ($\sin^22\theta_{13} \simeq 0.0013$; see
Fig.~\ref{fig:2nu2nubar}c), just below the region accessible at the
3$\sigma$ level in the SuperJHF to Hyper--K experiment~\cite{jhfsk}.
Thus $\nu$ and $\bar\nu$ measurements with both $\Delta = \pi/2$ and
$\pi$ at $L=300$~km do not have degeneracy problems over the entire
search region. However, the values of $|f|$ and $|\bar f|$ are
significantly smaller for $\Delta = \pi$ (by more than a factor of 10
compared to $\Delta = \pi/2$; see Fig.~\ref{fig:ffbg}), so that the
probabilities are also significantly smaller there, leading to
significantly reduced statistics that will limit the
$\sin^22\theta_{13}$ reach of such an experiment. Furthermore, for
slightly smaller values of $\delta m^2_{21}$ or $\sin^22\theta_{12}$ the
degeneracy at $\sin^22\theta_{13} = 0.13$ will be pushed below 0.10,
into the search region.

Since the matter effect increases with increasing $L$ and $\theta_{13}$,
one would expect that at larger $L$ degenerate solutions with opposite
sgn($\delta m^2_{31}$) can be differentiated for smaller values of
$\theta_{13}$. This property is evident in Fig.~\ref{fig:2nu2nubar};
e.g., for $L=2900$~km degeneracies occur only at $\sin^22\theta_{13}
\le 0.002$--$0.003$. If such an experiment were sensitive only to
$\sin^22\theta_{13}$ above this value, it would not have a degeneracy
problem. The $\sin^22\theta_{13}$ sensitivity depends on the total
flux (which varies with $L$ and off--axis angle), so a detailed
calculation must be made to determine the existence of
degeneracies in any given experimental situation.

There may also be approximate degeneracies that cannot be differentiated
due to statistical and systematic uncertainties. Here, too, one also
needs to include details of a particular experimental situation to
determine if approximate degeneracies are a problem; such an analysis is
beyond the scope of this paper. However, to give an indication of the
effect of approximate degeneracies, in Fig.~\ref{fig:approx} we show
regions in ($\delta,\theta_{13}$) space where the differences between
$(x\cos\delta)^2 + (x\sin\delta)^2$ and $x^2$ for the false solution,
when added in quadrature, are less than 10\%. We see that in
Figs.~\ref{fig:approx}a and \ref{fig:approx}b the approximate regions
extend to $\sin^22\theta_{13}$ that are nearly twice as large as the
maximum values for the exact degeneracies. Furthermore, in
Fig.~\ref{fig:approx}c there are no exact degeneracies in the region
shown, but approximate degeneracies extend to $\sin^22\theta_{13}$ as
large as 0.007. Therefore it is clear that approximate degeneracies will
occur at somewhat higher $\theta_{13}$ than the exact degeneracies.

\subsection{Four $\nu$ measurements at one $L$}
\label{sec:4nu}

Antineutrino measurements have two drawbacks compared to neutrino
measurements: the $\bar\nu$ cross section is approximately half as
large, and $\bar\nu$ beam fluxes are less. Thus to achieve comparable
statistics the running time for an antineutrino measurement must be more
than twice as long as a similar neutrino measurement (apart from the
possible enhancement of antineutrino oscillation probabilities at long
$L$ if $\delta m^2_{31} < 0$). If it were possible to remove the
degeneracies with neutrino measurements alone, the total time to
complete all of the necessary measurements would be substantially
reduced. With a single detector, this would require more than two
separate runs (since, as we have shown, any two measurements alone will
have degeneracies), each at a different $E_\nu$. With two detectors at
different locations with respect to the beam axis, only two runs would
give four independent measurements, and only the on--axis beam energy or
the detector off-axis angles would have to be changed between runs
(although the latter might prove to be unfeasible). Another interesting
possibility is to have a four-detector cluster, so that only a single
neutrino run would be needed to eliminate all degeneracies. Although
this many detectors might seem to be economically unfeasible, the
running time is shorter and the beam would not have to be reconfigured,
which would offset increased detector costs. In this section we examine
the ability of four $\nu$ runs to resolve degeneracies.

In Fig.~\ref{fig:4nu} we show the points in the $(\delta,\theta_{13})$
plane that have degeneracy with four neutrino measurements at the same
$L$, for several choices of $L$. We see that in all cases there are
points in the parameter space that have degeneracies even after four
measurements are taken. Similar to the case with two $\nu$ and two
$\bar\nu$ measurements, the degenerate solutions are pushed to lower
$\theta_{13}$ as $L$ is increased, due to the larger matter effect,
which helps distinguish sgn($\delta m^2_{31}$). For $L=2900$~km, in the
best cases we found that the largest $\sin^22\theta_{13}$ with a
degeneracy is 0.004--0.006. A comparison of Figs.~\ref{fig:2nu2nubar}
and \ref{fig:4nu} shows that two $\nu$ and two $\bar\nu$ measurements at
the same $L$ generally has degeneracies occurring at lower
$\sin^22\theta_{13}$ than four $\nu$ measurements at one $L$.

For a small off--axis angle $\theta$, the neutrino energy is given
approximately by~\cite{para}
%
\begin{equation}
E_\nu \simeq {0.43 E_\pi\over 1+(E_\pi\theta/m_\pi)^2} \,,
\label{eq:Enu}
\end{equation}
%
where $E_\pi$ is the pion energy in the lab frame and $m_\pi$ is the
pion mass. Four different
measurements can be made simultaneously if four detectors are placed at
different off--axis angles, and it is always possible to choose $E_\pi$
and the angles to select any desired combination of $\Delta_i$. This is
not the case if only two detectors are used and two runs are made with
different $E_\pi$ (assuming the detectors stay at the same off--axis
angles in both runs). From Eq.~(\ref{eq:Enu}), the condition that a set
of four $\Delta_i$ can be obtained in a two--detector, two--run scenario
is
%
\begin{equation}
\Delta_1(\Delta_4-\Delta_1) \ge \Delta_2(\Delta_3-\Delta_2) \,,
\label{eq:condition}
\end{equation}
%
where $\Delta_4 >\Delta_3 >\Delta_2 >\Delta_1$. In practice,
Eq.~(\ref{eq:condition}) is not hard to satisfy, especially if
$\Delta_4$ is not too small and the difference between $\Delta_3$ and
$\Delta_2$ is not too large. All of our examples in Fig.~\ref{fig:4nu}
can be obtained in two runs with two detectors.

One drawback of making only neutrino measurements is that this is not a
direct measurement of $CPV$ (such as would be the case of having both
$\nu$ and $\bar\nu$ measurements). Furthermore, if $\delta m^2_{31} <
0$, then all of the measurements would have a suppression due to matter
effects, especially at longer $L$. A detailed calculation including
experimental uncertainties would have to be made to see if the improved
resolution of degeneracies at longer $L$ can overcome the rate loss if
$\delta m^2_{31} < 0$.

\subsection{Measurements at more than one $L$}
\label{sec:differentL}

In the previous examples all of the detectors were assumed to be at the
same distance. As we have seen, measurements at longer $L$ push
the degeneracies to lower $\sin^22\theta_{13}$. Here we consider having
two $\nu$ measurements made at one distance (simultaneously with a
detector cluster) and a third $\nu$ measurement at another distance. We
examine two possibilities: one with the detector cluster at the shorter
distance, the other with the detector cluster at the longer distance.

In all cases we assume the first measurement is made with a $\nu$ beam
at $L_1=300$~km and $\Delta_1=\pi/2$ (the approximate values for the
proposed JHF to Super--K experiment). If the second $\nu$ measurement is
also at $L_2=300$~km with $\Delta_2 \ne \Delta_1$, and a third $\nu$
measurement at $L_3 > L_2$, we found that the degeneracies are less severe
(i.e., they occur at smaller $\sin^22\theta_{13}$) when $\Delta_3 =
\Delta_2 < \pi/2$; some of the better examples we found are shown in
Table~\ref{tab:2+1}. For the longer distance we used the $L$ values
considered in Fig.~\ref{fig:4nu}, plus 2100~km (nominally JHF to
Beijing). Having $\Delta_2 < \pi/2$ and $\Delta_3 = \pi/2$ also gave
good results.  In the cases we tested with the second and third $\nu$
measurements both at longer $L$, the best results were achieved if one
of the measurements at the longer distance had $\Delta=\pi/2$; some of
the better examples we found are shown in Table~\ref{tab:1+2}.

\begin{table}[t]
\squeezetable
\caption[]{Maximum $\sin^22\theta_{13}$ that has a degeneracy when the
first $\nu$ measurement is made at $L_1 = 300$~km and $\Delta_1 =
\pi/2$, a second at $L_2 = 300$~km with $\Delta_2 \ne \Delta_1$ , and a
third at $L_3 > L_2$, for different choices of the second and third
measurements. The other neutrino parameters are the same as in
Eq.~(\ref{eq:standard}).}
\label{tab:2+1}
\begin{tabular}{cc||ccccc}
& & \multicolumn{5}{c}{$L_3$}\\
$\Delta_2 (\nu)$ & $\Delta_3 (\nu)$
& 730~km & 1290~km & 1770~km & 2100~km & 2900~km\\
\hline\hline
$\pi/4$ & $\pi/4$ & .0012 & .0013 & .0013 & .0013 & .0013\\
        & $\pi/2$ & .0028 & .0033 & .0038 & .0042 & .0060\\
\hline
$\pi/3$ & $\pi/3$ & .0011 & .0012 & .0012 & .0013 & .0013\\
        & $\pi/2$ & .0021 & .0025 & .0030 & .0034 & .0047
\end{tabular}
\end{table}

\begin{table}[t]
\squeezetable
\caption[]{Maximum $\sin^22\theta_{13}$ that has a degeneracy when the
first $\nu$ measurement is made at $L_1 = 300$~km and $\Delta_1 =
\pi/2$, and two other $\nu$ measurements are made at $L_2 = L_3 > L_1$,
for different choices of $\Delta$ for the second and third measurements.
The other neutrino parameters are the same as in Eq.~(\ref{eq:standard}).}
\label{tab:1+2}
\begin{tabular}{cc||ccccc}
& & \multicolumn{5}{c}{$L_2=L_3$}\\
$\Delta_2 (\nu)$ & $\Delta_3 (\nu)$
& 730~km & 1290~km & 1770~km & 2100~km & 2900~km\\
\hline\hline
$\pi/2$ & $\pi/4$ & .0027 & .0030 & .0033 & .0038 & .0047\\
$\pi/2$ & $\pi/3$ & .0021 & .0022 & .0025 & .0027 & .0035\\
$\pi/2$ & $2\pi/3$ & .0030 & .0042 & .0067 & .0094 & .032
\end{tabular}
\end{table}

In both Tables~\ref{tab:2+1} and \ref{tab:1+2}, experimental $L$ and
$E_\nu$ choices can be made such that degeneracies occur only at
$\sin^22\theta_{13} = 0.003$ or less (assuming standard values for
$\sin^22\theta_{12}$ and $\delta m^2_{12}$). Thus, although there are
only three measurements, with appropriate choices of $L$ and $\Delta$,
degeneracies are less of a problem (i.e., they occur at smaller
$\theta_{13}$) than for the scenarios where all measurements are done at
the same $L$. Of course, for only three measurements the regions where
degeneracies occur are lines in $(\delta,\theta_{13})$ space, rather
than isolated points, so they are more likely to occur in experiments
with $\sin^22\theta_{13}$ sensitivity of order 0.003 or
less. Furthermore, as with the scenario with four $\nu$ measurements at
one $L$, a determination of $CPV$ is indirect (depending on the
three--neutrino parametrization) and probabilities would be suppressed
if $\delta m^2_{31} < 0$.

\subsection{Scenarios without a detector cluster}
\label{sec:nocluster}

All of our previous examples included some measurements at more than one
energy, but at the same $L$, which is particularly suited to detector
cluster experiments. Here we examine scenarios which do not use a
detector cluster.

First we consider a $\nu$ and $\bar\nu$ measurement at one $L$ and
$E_\nu$, and another $\nu$ measurement at the same $L$ and different
$E_\nu$. This is similar to the case of two $\nu$ and two $\bar\nu$
measurements all at the same $L$ discussed in Sec.~\ref{sec:2nu2nubar},
except that the second $\bar\nu$ measurement is missing, and the two
$\nu$ measurements are done in separate runs with different $\nu$
energies. Since there are only three measurements, the parameter
degeneracies are lines in $(\delta,\theta_{13})$ space. We examined
cases where $\nu$ and $\bar\nu$ measurements were done at $\Delta =
\pi/2$ and another $\nu$ measurement was done at $\Delta = \pi/6$,
$\pi/4$, $\pi/3$, $2\pi/3$, $3\pi/4$, or $\pi$, for $L = 300$, $730$,
$1290$, $1770$, $2100$, or $2900$~km. We found that for $L \le
1290$~km, in the best cases degeneracies can occur for
$\sin^22\theta_{13}$ as high as 0.01 (0.10 for $L = 300$~km). For the
best cases with $L \ge 1770$~km degeneracies can occur for
$\sin^22\theta_{13}$ as high as 0.005-0.008, which is a factor of two or
more higher than for two $\nu$ and two $\bar\nu$ energies at the same
$L$. Thus making a second $\bar\nu$ measurement not only reduces the
set of degenerate points from lines to points, but it also pushes the
degeneracies to lower $\sin^22\theta_{13}$.

Next we consider a $\nu$ and $\bar\nu$ measurement at one $L$ and
$E_\nu$, and another $\nu$ measurement at a different $L$. The neutrino
energies for the two $\nu$ measurements may or may not be the same. We
examine two possibilities, where the $\bar\nu$ measurement is done at
either the shorter or longer $L$. In all cases we assume the first
measurement is made with a $\nu$ beam at $L_1=300$~km and
$\Delta_1=\pi/2$. If the $\bar\nu$ measurement is also at $L_2=300$~km
with $\Delta_2 = \pi/2$, and a $\nu$ measurement at $L_3 > 300$~km, we
found that for $\Delta_3 \simeq \pi/3$ ($\Delta_3 \simeq 2\pi/3$)
the degeneracies are less severe at larger (intermediate) $L_3$; see
Table~\ref{tab:nubarL1}.  Another possibility is to have the $\bar\nu$
measurement done at the longer $L$. Typical results are shown in
Table~\ref{tab:nubarL2}.  Generally speaking the measurements in
Tables~\ref{tab:nubarL1} and \ref{tab:nubarL2} have degeneracies for
higher $\sin^22\theta_{13}$ than with three $\nu$ measurements, one at a
different $L$ (Tables~\ref{tab:2+1} and \ref{tab:1+2}).

\begin{table}[t]
\squeezetable
\caption[]{Maximum $\sin^22\theta_{13}$ that has a degeneracy when $\nu$
and $\bar\nu$ measurements are made at $L_1 = L_2 = 300$~km and
$\Delta_1 = \Delta_2 = \pi/2$ and another $\nu$ measurement is made at
$L_3 > 300$~km, for different choices of $L_3$ and $\Delta_3$. The other
neutrino parameters are the same as in Eq.~(\ref{eq:standard}).}
\label{tab:nubarL1}
\begin{tabular}{c||ccccc}
& \multicolumn{5}{c}{$L_3$}\\
$\Delta_3 (\nu)$ & 730~km & 1290~km & 1770~km & 2100~km & 2900~km\\
\hline\hline
$\pi/3$  & .089 & .018 & .008 & .006 & .004\\
$2\pi/3$ & .035 & .010 & .016 & .025 & .071
\end{tabular}
\end{table}

\begin{table}[t]
\squeezetable
\caption[]{Maximum $\sin^22\theta_{13}$ that has a degeneracy when a
$\nu$ measurement is made at $L_1=300$~km and $\Delta_1=\pi/2$, and
$\nu$ and $\bar\nu$ measurements are made at $L_2 = L_3 > 300$~km,
for different choices of the second and third measurements. The other
neutrino parameters are the same as in Eq.~(\ref{eq:standard}).}
\label{tab:nubarL2}
\begin{tabular}{cc||ccccc}
& & \multicolumn{5}{c}{$L_2=L_3$}\\
$\Delta_2$ ($\nu$) & $\Delta_3$ ($\bar\nu$)
& 730~km & 1290~km & 1770~km & 2100~km & 2900~km\\
\hline\hline
$\pi/3$ & $\pi/2$   & .050 & .014 & .005 & .006 & .010\\
$2\pi/3$ & $\pi/2$  & .038 & .010 & .016 & .022 & .067\\
\hline
$\pi/4$ & $\pi/4$   & .056 & .013 & .007 & .005 & .008\\
$\pi/3$ & $\pi/3$   & .038 & .008 & .005 & .006 & .013\\
$2\pi/3$ & $2\pi/3$ & .033 & .011 & .011 & .015 & .063

\end{tabular}
\end{table}

\section{Summary and discussion}
\label{sec:discussion}

We summarize the important points of our paper as follows:

\begin{enumerate}

\item[(i)] For any two appearance measurements there may be as much as
an eight--fold parameter degeneracy (resulting from simultaneous
$(\delta,\theta_{13})$, sgn($\delta m^2_{31}$) and $(\theta_{23},
\pi/2-\theta_{23})$ ambiguities) for most points in the
($\delta,\theta_{13}$) plane. Making a third appearance measurement
resolves the $(\delta,\theta_{13})$ ambiguity and reduces the regions
where the remaining degeneracies occur to lines in
$(\delta,\theta_{13})$ space. Making a fourth appearance measurement
reduces these degeneracies to isolated points in the
$(\delta,\theta_{13})$ plane; a fifth measurement then in principle
removes all remaining degeneracies.

\item[(ii)] Two $\nu$ and two $\bar\nu$ measurements at the same $L$ can
be made so that parameter degeneracies only occur at isolated points in
$(\delta, \theta_{13})$ space at $\sin^22\theta_{13} \leq 0.01$--$0.02$
($0.002$--$0.003$) for $L= 300~(2900)$~km. If the $\bar\nu$ energies are
the same as the $\nu$ energies, there will be no $CPC/CPV$ confusion for
degeneracies between Cases~I and II (defined in Table~\ref{tab:degen}),
or between Cases~III and IV, although sgn($\delta m^2_{31}$) is not
determined. This scenario could be completed in two runs (one with a
$\nu$ beam and one with a $\bar\nu$ beam) with two detectors at
different positions with respect to the beam axis (a two--detector
cluster).

\item[(iii)] Four $\nu$ measurements at the same $L$ can be made so that
parameter degeneracies only occur at isolated points in $(\delta,
\theta_{13})$ space at $\sin^22\theta_{13} < 0.04~(0.004)$ for $L =
300~(2900)$~km. Such a scenario could be implemented in two runs (two
$\nu$ beams with different on--axis energies) with a two--detector
cluster, or in one run (a single $\nu$ beam) with a four--detector
cluster. It would have the advantage that the $\nu$ cross section and
flux would be larger than for $\bar\nu$; disadvantages include the fact
that it is an indirect measurement of $CPV$ and it would have rates
suppressed by matter effects at longer $L$ if $\delta m^2_{31} < 0$.

\item[(iv)] Two $\nu$ measurements at one $L$ and a third at another $L$
can be made so that parameter degeneracies only occur at
$\sin^22\theta_{13} \leq 0.001$--$0.003$. Since there are three
measurements, the degeneracies occur along lines in the
($\delta,\theta_{13}$) plane. Such a scenario could be implemented in
two runs, with a two--detector cluster at one $L$ and a single detector
at the other $L$. This scenario uses the matter effect at longer $L$ to
help push degeneracies to lower $\theta_{13}$, and, since it uses
only $\nu$ beams, has similar advantages and disadvantages as (iii).

\item[(v)] Both a $\nu$ oscillation measurement and a $\bar\nu$
oscillation measurement at one $L$ and a $\nu$ measurement at another
$L$ can be made so that parameter degeneracies occur along lines in the
($\delta,\theta_{13}$) plane for $\sin^22\theta_{13} \leq
0.005$--$0.010$. Similar results can be obtained for two separate $\nu$
measurements and one $\bar\nu$ measurement at the same $L$. These
scenarios do not use a detector cluster.

\item[(vi)] All of the examples we show assume $\delta m^2_{21} =
5\times10^{-5}$~eV$^2$ and $\sin^22\theta_{12} = 0.80$. We found that
the maximum $\theta_{13}$ that may have parameter degeneracies varied
strongly with $\delta m^2_{21}$, and less so with $\sin^22\theta_{12}$
(see Eq.~\ref{eq:degen}); the corresponding $\delta$ values were
unaffected. There was also a dependence on $\delta m^2_{31}$, but if
$\delta m^2_{31}$ is known to 10\% its uncertainty does not greatly
affect our degeneracy analysis.

\end{enumerate}

A summary of the requirements for the scenarios discussed in
Sec.~\ref{sec:scenarios} is given in Table~\ref{tab:scenarios}; also
shown is the best--case (i.e., lowest) maximum value of
$\sin^22\theta_{13}$ that we found for which a parameter degeneracy
could occur, assuming the standard parameter set of
Eq.~(\ref{eq:standard}). Generally speaking, we found that measurements
at more than one $L$ did better at resolving parameter degeneracies than
if all measurements were done at a single $L$; for example, a third
$\nu$ measurement at a different $L$ than the first two measurements
appears to be better than a third and a fourth at the same $L$. Also,
scenarios with detector clusters resolved degeneracies as well as or
better than those without a detector cluster, and required fewer
runs. Most scenarios resolved parameter degeneracies the best at longer
$L$ (1770~km or more), except for the scenario with a two--detector
cluster at one $L$ and a single detector at another $L$, which actually
did better when the second distance was not as long (1290~km or less).

\begin{table}[t]
%\squeezetable
\caption[]{A summary of detector scenarios discussed in this paper. The
right--most column indicates the best--case (i.e., lowest) maximum value
of $\sin^22\theta_{13}$ that we found for which a parameter degeneracy
could occur, assuming the parameters of Eq.~(\ref{eq:standard}).}
\label{tab:scenarios}
\begin{tabular}{l||c|c|c|c|c}
          &       &           &       &              & max. $\sin^22\theta_{13}$\\
\# of     & \# of & \# of     & \# of & \# of        & without\\
detectors & beams & beamlines & runs  & measurements & degeneracy\\
\hline\hline
2 at one $L$ & 2 ($\nu$ and $\bar\nu$) & 1 & 2 & 4 & .002\\
2 at one $L$ & 2 ($\nu$ at $E_1$ and $E_2$) & 1 & 2 & 4 & .004\\
4 at one $L$ & 1 ($\nu$) & 1 & 1 & 4 & .004\\
2 at $L_1$, 1 at $L_2$ & 1 ($\nu$) & 2 & 2 & 3 & .001\\
\hline
%          &       &           &       &              & max. $\sin^22\theta_{13}$\\
%\# of     & \# of & \# of     & \# of & \# of        & without\\
%detectors & beams & beamlines & runs  & measurements & degeneracy\\
%\hline
1 at one $L$ & 3 ($\nu$ at $E_1$ and $E_2$, $\bar\nu$) & 1 & 3 & 3 & .005\\
1 at $L_1$, 1 at $L_2$ & 2 ($\nu$ and $\bar\nu$) or 3 (2 $\nu$
and 1 $\bar\nu$) & 2 & 3 & 3 & .004\\
\end{tabular}
\end{table}

If $\theta_{23} \simeq \pi/4$ (the value favored by atmospheric neutrino
experiments), then the $\theta_{23}$ ambiguity vanishes and the only two
cases to consider after three measurements are I and II (the sgn($\delta
m^2_{31}$) ambiguity). We have also performed the degeneracy analyses
for $\sin^22\theta_{23} = 1$, and found that all of the relevant
degeneracies are at least a factor of two lower in $\sin^22\theta_{13}$
than shown in Table~\ref{tab:scenarios} for the shorter distances ($L
\le 1290$~km), and in most cases lower than that for the longer
distances. Therefore if it is known that $\theta_{23}$ is close to
$\pi/4$ (for example, from measurements at MINOS or ICARUS), the
parameter degeneracy problem is greatly reduced.

Some of the scenarios we discussed involve measurements with neutrinos
only (not antineutrinos). Although it is possible to extract the
neutrino mixing angles and $CP$ phase from such measurements, the result
relies on the three--neutrino mixing assumption. A more robust
determination of $CP$ violation which would not be as model dependent
would include a measurement involving antineutrinos; then the
measurement of an asymmetry between neutrino and antineutrino
oscillation probabilities (after correcting for the $CP$ violation
induced by matter effects) would be direct and definitive.

%Of course such an additional measurement would make these
%scenarios less attractive.

The detector cluster scenarios are especially well--suited for detector
designs that emphasize large, cheaply--built detectors. We note that the
detectors in a cluster would not necessarily have to be the same size;
detectors at larger off--axis angles could be made larger to compensate
for the reduced flux off--axis.

We emphasize that our analysis only considered exact degeneracies for
monoenergetic beams. Experimental uncertainties will expand the regions
with degeneracies, especially at small $\sin^22\theta_{13}$ (where
probabilities and hence events rates are lower) and at longer $L$ (where
beam fluxes fall off). On the other hand, neutrino beams are not
precisely monoenergetic, and energy spectrum information can help
resolve the degeneracies, even for relatively narrow spectrum
beams~\cite{huber}; the survival channel $\nu_\mu \to \nu_\mu$ can in
principle also provide additional information, although the effects of
$\theta_{13}$ and $\delta$ are not at leading order. A definitive
analysis would have to include all of these factors to determine if
parameter degeneracies are a problem in a particular experiment. The
results of this paper can serve as a guideline for which detector/beam
scenarios are likely to encounter difficulties with parameter
degeneracies.


\section*{Acknowledgments}

This research was supported in part by the U.S.~Department of Energy
under Grants No.~DE-FG02-95ER40896, No.~DE-FG02-01ER41155 and 
No.~DE-FG02-91ER40676, and in
part by the University of Wisconsin Research Committee with funds
granted by the Wisconsin Alumni Research Foundation.

\clearpage

\begin{thebibliography}{99}

\bibitem{Ahmad:2001an}
Q.~R.~Ahmad {\it et al.}  [SNO Collaboration],
%``Measurement of the charged of current interactions produced by B-8  solar
%neutrinos at the Sudbury Neutrino Observatory,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 071301 (2001)
.
%%CITATION = ;%%

\bibitem{SNONC}
Q.~R.~Ahmad {\it et al.}  [SNO Collaboration],
arXiv:
%%CITATION = ;%%
 arXiv:.
%%CITATION = ;%%

\bibitem{Bahcall:2000nu}
J.~N.~Bahcall, M.~H.~Pinsonneault and S.~Basu,
%``Solar models: Current epoch and time dependences, neutrinos, and
%helioseismological properties,''
Astrophys.\ J.\  {\bf 555}, 990 (2001)
.
%%CITATION = ;%%

\bibitem{bmww}
V. Barger, D. Marfatia, K. Whisnant, and B.P.~Wood,
arXiv:.
%%CITATION = ;%%

\bibitem{LMA}
A. Bandyopadhyay, S. Choubey, S. Goswami and D.P. Roy,
arXiv:;
%%CITATION = ;%%
J.~N.~Bahcall, M.~C.~Gonzalez-Garcia and C.~Pena-Garay,
arXiv:;
%%CITATION = ;%%
P.~Aliani, V.~Antonelli, R.~Ferrari, M.~Picariello, and E.~Torrente--Lujan,
arXiv:;
%%CITATION = ;%%
P.C.~de~Holanda and A.~Yu.~Smirnov,
arXiv:.
%%CITATION = ;%%

\bibitem{kamland}
P.~Alivisatos {\it et al.},
%``KamLAND: A liquid scintillator anti-neutrino detector at the  Kamioka
%site,''
STANFORD-HEP-98-03;
V.~Barger, D.~Marfatia and B.~P.~Wood,
%``Resolving the solar neutrino problem with KamLAND,''
Phys.\ Lett.\ B {\bf 498}, 53 (2001)
.

\bibitem{Toshito:2001dk}
T.~Toshito  [Super-Kamiokande Collaboration],
%``Super-Kamiokande atmospheric neutrino results,''
arXiv:.
%%CITATION = ;%%

\bibitem{Hill:2001gu}
J.~E.~Hill  [K2K Collaboration],
%``Results from the K2K long-baseline neutrino oscillation experiment,''
arXiv:.
%%CITATION = ;%%

\bibitem{CHOOZ}
M.~Apollonio {\it et al.}  [CHOOZ Collaboration],
%``Limits on neutrino oscillations from the CHOOZ experiment,''
Phys.\ Lett.\ B {\bf 466}, 415 (1999)
.
%%CITATION = ;%%

\bibitem{paloverde}
F.~Boehm {\it et al.},
%``Final results from the Palo Verde neutrino oscillation experiment,''
Phys.\ Rev.\ D {\bf 64}, 112001 (2001)
.
%%CITATION = ;%%

\bibitem{minos}
MINOS Collaboration, Fermilab Report No. NuMI-L-375 (1998).

\bibitem{icarus}
A. Rubbia for the ICARUS Collaboration,
talk given at Skandinavian NeutrinO Workshop (SNOW), Uppsala, Sweden,
February (2001), which are available at
http://pc\-no\-meth4.\-cern.\-ch/\-pu\-blicpdf.html.  

\bibitem{opera}
OPERA Collaboration,
CERN/SPSC 2000-028, SPSC/P318,  LNGS P25/2000, July, 2000.

\bibitem{Barger:2001yx}
V.~Barger, A.~M.~Gago, D.~Marfatia, W.~J.~Teves, B.~P.~Wood and
R.~Z.~Funchal,
%``Neutrino oscillation parameters from MINOS, ICARUS and OPERA combined,''
arXiv:.
%%CITATION = ;%%

\bibitem{ambiguity}
J.~Burguet-Castell, M.~B.~Gavela, J.~J.~Gomez-Cadenas, P.~Hernandez and
O.~Mena,
%``On the measurement of leptonic CP violation,''
Nucl.\ Phys.\ B {\bf 608}, 301 (2001)
.
%%CITATION = ;%%

\bibitem{peak}
V.~Barger, D.~Marfatia and K.~Whisnant,
%``Neutrino superbeam scenarios at the peak,''
arXiv:.
%%CITATION = ;%%

\bibitem{bmw}
V. Barger, D. Marfatia, and K. Whisnant,
Phys.\ Rev.\ D {\bf 65}, 073023
(2002) .
%%CITATION = ;%%

\bibitem{minakata2}
T. Kajita, H.~Minakata, and H.~Nunokawa,
arXiv:.
%%CITATION = ;%%

\bibitem{huber}
P. Huber, M. Lindner, and W. Winter, arXiv:.
%%CITATION = ;%%

\bibitem{minakata1}
H.~Minakata and H.~Nunokawa,
%``Exploring neutrino mixing with low energy superbeams,''
JHEP {\bf 0110}, 001 (2001)
.
%%CITATION = ;%%

\bibitem{bnl}
D.~Beavis {\it et al.}, E889 Collaboration, BNL preprint BNL-52459, April 1995.

\bibitem{jhfsk}
Y.~Itow {\it et al.},
%``The JHF-Kamioka neutrino project,''
arXiv:.
%%CITATION = ;%%

\bibitem{para}
A. Para and M. Szleper, arXiv:.
%%CITATION = ;%%

\bibitem{barenboim}
G.~Barenboim, A.~de Gouvea, M.~Szleper, and M.~Velasco,
arXiv:.
%%CITATION = ;%%

\bibitem{hagiwara}
M.~Aoki, K.~Hagiwara, Y.~Hayoto, T.~Kobayashi, T.~Nakaya, K.~Nishikawa,
and N.~Okamura, arXiv:.
%%CITATION = ;%%

\bibitem{cervera}
A. Cervera {\it et al.} Nucl.\ Phys.\ B {\bf 579}, 17 (2000)
[Erratum-ibid.\ B {\bf 593}, 731 (2000)]
.
%%CITATION = ;%%

\bibitem{freund}
M. Freund, Phys.\ Rev.\ D {\bf 64}, 053003 (2001)
.
%%CITATION = ;%%

\bibitem{PREM}
A. Dziewonski and D. Anderson, Phys. Earth Planet. Inter. {\bf 25}, 297 (1981).

\bibitem{kimura}
K. Kimura, A. Takamura, and H. Yokomakura,
arXiv:.
%%CITATION = ;%%

\bibitem{superbeams}
V. Barger, S. Geer, R. Raja, and K. Whisnant,
Phys. Rev. D {\bf 63}, 113011(2001)
.
%%CITATION = ;%%

\end{thebibliography}

\clearpage

% 1
\begin{figure}
\centering\leavevmode
\psfig{file=lblvsE2.f.ps,width=2in}
\psfig{file=lblvsE2.fbar.ps,width=2in}
\psfig{file=lblvsE2.g.ps,width=2in}
\medskip

\caption{Value of $\sqrt{E_\nu}/L$ times the coefficients (a) $f$,
(b) $\bar f$, and (c) $g$ versus $\Delta$ for several values of
$L$, assuming $\delta m^2_{31} = 3\times10^{-3}$~eV$^2$, and where
$E_\nu$ is in GeV.}
\label{fig:ffbg}
\end{figure}
%

% 2
\begin{figure}
\centering\leavevmode
\psfig{file=lbloa7con.2.123.ps,width=2.1in}
\psfig{file=lbloa7con.13.123.ps,width=2.1in}
\medskip
\caption{Lines in ($\sin^22\theta_{13},\delta$) space for Case~I where
three neutrino measurements are not sufficient to eliminate the
ambiguity between I and II (solid curves), I and III (dashed), and I and
IV (dotted), assuming $L = 730$~km and that the other parameters have
the values given in Eq.~(\ref{eq:standard}). The values of $E_\nu$ are
chosen so that the oscillation arguments are (a) $\Delta = \pi/3$,
$\pi/2$, and $2\pi/3$, ($E_\nu = 2.66$, $1.77$ and $1.33$~GeV,
respectively), and (b) $\Delta = \pi/4$, $\pi/2$, and $\pi$ ($E_\nu =
3.54$, $1.77$ and $0.89$~GeV, respectively).}
\label{fig:degen1}
\end{figure}
%

% 3
\begin{figure}
\centering\leavevmode
\psfig{file=lbloa7con.2.I.II.ps,width=2.1in}
\psfig{file=lbloa7con.2.I.III.ps,width=2.1in}
\psfig{file=lbloa7con.2.I.IV.ps,width=2.1in}
\medskip

\caption{Lines in ($\sin^22\theta_{13},\delta$) space for Case~I where
subgroups of three neutrino measurements are not sufficient to eliminate
the parameter degeneracies between (a) I and II, (b) I and III, and (c)
I and IV, assuming $L = 730$~km and that the other parameters are the
same as in Eq.~(\ref{eq:standard}). The values of $E_\nu$ for the four
measurements are chosen so that the oscillation arguments are $\Delta =
\pi/6$, $\pi/3$, $\pi/2$, and $2\pi/3$, ($E_\nu = 5.31$, $2.66$, $1.77$
and $1.33$~GeV, respectively), and the four subgroups are labeled as
follows: $E_\nu = 5.31$~GeV omitted (solid), $E_\nu = 2.66$~GeV omitted
(dashed), $E_\nu = 1.77$~GeV omitted (dotted), $E_\nu = 1.33$~GeV
omitted (dash--dotted).}
\label{fig:degen2}
\end{figure}
%

% 4
\begin{figure}
\centering\leavevmode
\psfig{file=lbldegenpts.300.ps,width=2.1in}
\psfig{file=lbldegen.300.ps,width=2.1in}
\medskip

\caption{(a) Sample points in ($\delta,\theta_{13}$) space that
have parameter degeneracies between Cases~I (boxes) and II (crosses),
defined in Table~\ref{tab:degen}, after $\nu$ and $\bar\nu$ measurements
at $L=300$~km and $\Delta = \pi/2$ (corresponding to $E_\nu =
0.73$~GeV), assuming the other parameters are the same as in
Eq.~(\ref{eq:standard}), except for $\theta_{23} = \pi/4$. The sets of
parameters that are degenerate with each other are linked by a line. For
each pair of linked points there is another pair with $\delta \to \pi -
\delta$ (not shown).
(b) Region in Case~I parameter space that have degeneracies with Case~II
after
(i) the two measurements in (a) (area between the solid curves),
(ii) an additional $\nu$ measurement at $L = 300$~km and $\Delta = \pi/3$
(along dashed curves), and
(iii) a final $\bar\nu$ measurement at $L = 300$~km and $\Delta = \pi/3$
(boxes). The corresponding regions for Case~II are found by taking
$\delta \to \delta \pm \pi$. We note that after only the first two
measurements there is also a $(\delta,\pi-\delta)$ ambiguity everywhere
in the plane, which is removed by the third measurement.}
\label{fig:degen3}
\end{figure}
%

% 5
\begin{figure}
\centering\leavevmode
\psfig{file=lbloa8pts.2.ps,width=2.1in}
\psfig{file=lbloa8pts.3.ps,width=2.1in}
\psfig{file=lbloa8pts.5.ps,width=2.1in}
\medskip

\caption{Points in ($\sin^22\theta_{13},\delta$) space where two $\nu$
and two $\bar\nu$ measurements are not sufficient to eliminate all of
the parameter degeneracies, for several values of $L$, assuming the
other parameters are the same as in Eq.~(\ref{eq:standard}). In each case
measurements assumed to be made at two energies such that $\Delta =
\pi/2$ and (a) $\Delta = \pi/3$, (b) $\Delta = 2\pi/3$, and (c) $\Delta
= \pi$.}
\label{fig:2nu2nubar}
\end{figure}
%

% 6
\begin{figure}
\centering\leavevmode
\psfig{file=lbloa9pts.2.I.II.300.ps,width=2.1in}
\psfig{file=lbloa9pts.2.III.IV.300.ps,width=2.1in}
\psfig{file=lbloa9pts.2.III.II.300.ps,width=2.1in}
\medskip

\caption{Regions in ($\sin^22\theta_{13},\delta$) space where two $\nu$
and two $\bar\nu$ measurements at $L=300$~km with $\Delta = \pi/2$ and
$\Delta = \pi/3$ still have approximate parameter degeneracies, assuming
the other parameters are the same as in Eq.~(\ref{eq:standard}). The
three degeneracies shown are (a) I vs. II (I is the true solution
and II is the false solution). (b) III vs. IV, and (c) III vs. II.
The condition defining the approximate region is given in the
text. Exact parameter degeneracies are indicated with pluses.}
\label{fig:approx}
\end{figure}
%

% 7
\begin{figure}
\centering\leavevmode
\psfig{file=lbloa8pts.10.ps,width=2.1in}
\psfig{file=lbloa8pts.11.ps,width=2.1in}
\psfig{file=lbloa8pts.12.ps,width=2.1in}
\medskip

\caption{Points in ($\sin^22\theta_{13},\delta$) space where four $\nu$
measurements are not sufficient to eliminate all of
the parameter degeneracies, for several values of $L$, assuming the
other parameters are the same as in Eq.~(\ref{eq:standard}). In each
case measurements are made at four energies such that
(a) $\Delta = \pi/4$, $\pi/3$, $\pi/2$, and $3\pi/4$,
(b) $\Delta = \pi/6$, $\pi/3$, $\pi/2$, and $2\pi/3$, and
(c) $\Delta = \pi/6$, $\pi/3$, $\pi/2$, and $\pi$.}
\label{fig:4nu}
\end{figure}
%

\end{document}

