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\title[Breaking 8--Fold Neutrino Parameter Degeneracies]
{Breaking 8--Fold Neutrino Parameter Degeneracies}

\author{K. Whisnant}

\address{Dept. of Physics and Astronomy, Iowa State Univ.,
Ames, IA 50011, USA}

\begin{abstract}

Parameter degeneracies are a generic problem in the three--neutrino
analysis of long--baseline neutrino appearance measurements.
Degeneracies can lead to different inferred values for the neutrino
mixing angle $\theta_{13}$, and they often mix $CP$ violating and
$CP$ conserving solutions. Appropriate experimental strategies can
reduce or eliminate degeneracies and/or the $CPV/CPC$ confusion.

\end{abstract}

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\section{Introduction}

With the strong evidence for neutrino oscillations from
solar~\cite{solar} and atmospheric~\cite{atmos} neutrino experiments,
the focus is now changing from establishing the existence of neutrino
oscillations to obtaining a complete understanding of the parameters
in the neutrino mass matrix.  Solar neutrino data suggest $|\delta
m^2_{31}| \sim 3\times10^{-3}$~eV$^2$ and $\sin^22\theta_{23} \simeq
1.0$\ (in the standard three--neutrino description), while atmospheric
neutrino data provide give $\delta m^2_{21} \sim
5\times10^{-5}$~eV$^2$ and $\sin^22\theta_{12} \simeq 0.8$. Reactor
experiments~\cite{reactor} put an upper bound on $\sin^22\theta_{13}$
of about 0.1. In the future, KamLAND~\cite{KamLAND} will likely
provide a more precise measurement of the solar neutrino parameters
and $\nu_\mu \to \nu_\mu$ survival in long--baseline experiments will
provide a more precise measurement of the atmospheric neutrino
parameters (to $\le 10$\%).

The heretofore unknown neutrino parameters are the $CP$ phase $\delta$
that can appear in three--neutrino oscillations, the mixing angle
$\theta_{13}$, and the sign of $\delta m^2_{31}$, which determines the
ordering of the neutrino mass eigenstates. In this talk I discuss the
difficulties in making a unique determination of these neutrino
parameters, and how experiments might be designed to overcome them.

\section{Neutrino parameter degeneracies}

The standard method for determining the parameters $\theta_{13}$ and
$\delta$ is to measure the rates for $\nu_\mu \to \nu_e$ and
$\bar\nu_\mu \to \bar\nu_e$ in a long--baseline neutrino experiment.
For our analysis we use approximate expressions for the the
oscillation probabilities similar to those derived in Ref.~\cite{approx}.
For $\delta m^2_{31}>0$
%
\begin{eqnarray}
P(\nu_\mu \to \nu_e) &=& A x^2
+ B x y(\cos\delta\cos\Delta - \sin\delta\sin\Delta) + C y^2 \,,
\label{eq:P}\\
\bar P(\bar\nu_\mu \to \bar\nu_e) &=& \bar A x^2
+ \bar B x y(\cos\delta\cos\Delta + \sin\delta\sin\Delta) + C y^2 \,,
\label{eq:Pbar}
\end{eqnarray}
%
where $x \equiv \sin\theta_{23} \sin2\theta_{13}$, $y \equiv
|\delta m^2_{21}/\delta m^2_{31}|\cos\theta_{23}\sin2\theta_{12}$,
and $\Delta = 1.27 |\delta m^2_{31}/{\rm eV}^2|
(L/{\rm km})/(E_\nu/{\rm GeV})$ is the oscillation argument.
The coefficients $A$, $B$, $C$, etc., depend on $|\delta m^2_{31}|$,
$\Delta$, and the density of matter. Matter effects lead to a
suppression of $\bar A$ ($\bar B$) relative to $A$ ($B$). The
probabilities for $\delta m^2_{31}<0$ are given by
making the exchanges $A \leftrightarrow \bar A$, $B \leftrightarrow
- \bar B$, and $\delta \to - \delta$. More details, including a
discussion of their range of validity, are given in Ref.~\cite{bmw}.
These approximations have a similar form to those for low
energy~\cite{arafune} and the exact expressions for constant
density~\cite{kimura}.

The standard method for determining $\theta_{13}$ and $\delta$ is to
measure $P$ and $\bar P$ at one $L$ and $E_\nu$. If $\theta_{13}$ is
held fixed and $\delta$ is varied from $0$ to $2\pi$, an ellipse is
traced out in the ($P,\bar P$) plane~\cite{minakata} (it is assumed
that the other oscillation parameters are well--measured from other
experiments, as mentioned above). The measured values of $P$ and $\bar
P$ then in principle should determine $\theta_{13}$ and $\delta$.
However, there are three possible two--fold degeneracies that can
occur:\cite{bmw}

\begin{enumerate}

\item The ellipses for two different values of $\theta_{13}$ can
overlap, so that the same $P$ and $\bar P$ are described by two
different sets of parameters ($\delta,\theta_{13}$) and
($\delta^\prime,\theta_{13}^\prime$) (the ($\delta, \theta_{13}$)
ambiguity). The values of $\theta_{13}$ and $\theta_{13}^\prime$ can
be very different, and if $\delta$ is $CP$ violating, then
$\delta^\prime$ can be $CP$ conserving ($CPV/CPC$ confusion). In some
cases solutions with maximal $CP$ violation can give the same
predictions for $P$ and $\bar P$ as a solution with $CP$ conserved.

\item Ellipses for $\delta m^2_{31}>0$ are shifted towards the $P$ axis,
while those for $\delta m^2_{31}<0$ are shifted towards the $\bar P$
axis, due to different matter effects. These ellipses can overlap, so
that for many values of $P$ and $\bar P$, the parameters ($\delta,
\theta_{13}$) for $\delta m^2_{31}>0$ and a different set of
parameters ($\delta^\prime, \theta_{13}^\prime$) for $\delta
m^2_{31}<0$ can both reproduce the data (the sgn($\delta m^2_{31}$)
ambiguity). Here $\theta_{13}$ and $\theta_{13}^\prime$ are not as
far apart as they can be for the ($\delta, \theta_{13}$) ambiguity,
but a large $CPV/CPC$ confusion is possible.

\item Since only $\sin^22\theta_{23}$ is measured in atmospheric and
$\nu_\mu \to \nu_\mu$ experiments, there are two possible values for
$\theta_{23}$, one with $\theta_{23} < \pi/4$ and one with
$\theta_{23} > \pi/4$~\cite{lisi}. The ellipses for these two
solutions can overlap, once again with different solutions for
$\theta_{13}$ and $\delta$ (the ($\theta_{23}, \pi/2-\theta_{23}$)
ambiguity). As with the ($\delta, \theta_{13}$) ambiguity, the
uncertainties in $\theta_{13}$ can be large and a large $CPV/CPC$
confusion is also possible. This ambiguity vanishes for $\theta_{23} =
\pi/4$, the value preferred by atmospheric data.

\end{enumerate}

\noindent Each of these two--fold ambiguities may exist for a given
data set, so that it is possible to have an eight--fold ambiguity in
some cases, i.e., eight distinct sets of the parameters ($\delta,
\theta_{13}$) that describe the data.

\section{Strategies for breaking parameter degeneracies}

Appropriate experimental strategies may be used to either reduce or
eliminate the effects of parameter degeneracies. The simplest is to
choose your baseline so that the $\cos\delta$ terms in the oscillation
probabilities vanish. For a narrow spectrum this can be achieved by
choosing $L/\langle E_\nu \rangle$ (where $\langle E_\nu \rangle$ is
the average energy) such that $\Delta = (n - 1/2)\pi$, at the peak of
the oscillation in vacuum (which also is near the peak after matter is
taken into account). Then the ellipses in ($P,\bar P$) space collapse
to lines, and the line for one value of $\theta_{13}$ no longer
overlaps the lines for other $\theta_{13}$ (for the same sgn($\delta
m^2_{31}$) and $\theta_{23}$), thereby removing $\theta_{13}$ from the
($\delta, \theta_{13}$) ambiguity. Since only $\sin\delta$ is being
measured, there is a residual ($\delta, \pi - \delta$) ambiguity;
however, $CPV$ and $CPC$ solutions are no longer mixed for the same
sgn($\delta m^2_{31}$) and $\theta_{23}$. There still can be large
$CPV/CPC$ confusion due to the sgn($\delta m^2_{31}$) ambiguity,
unless $\delta \simeq 3\pi/2$ ($\pi/2$) for $\delta m^2_{31} > 0$
($<0$), in which case the sgn($\delta m^2_{31}$) ambiguity is avoided.
On a peak the $CPV/CPC$ confusion due to the ($\theta_{23}, \pi/2 -
\theta_{23}$) ambiguity is small, although the uncertainty in
$\sin^22\theta_{13}$ can be as much as a factor of two for
$\sin^22\theta_{23} = 0.90$~\cite{bmw}.

Another tactic is to go to longer baseline $L$ to increase the matter
effect, thereby splitting apart the predictions for $\delta m^2_{31} >
0$ and $\delta m^2_{31} < 0$. We have found that predictions for the
two signs of $\delta m^2_{31}$ do not overlap for $\sin^22\theta_{13}
> 0.01 (0.005) [0.002]$ for $L = 1290 (1770) [2900]$~km, assuming the
values of the solar and atmospheric parameters given above~\cite{bmw}.
We have also found that the first peak of the oscillation (i.e.,
$\Delta = \pi/2$) is best for resolving the sgn($\delta m^2_{31}$)
ambiguity, since higher peaks tend to have more overlap between
$\delta m^2_{31} > 0$ and $\delta m^2_{31} < 0$~\cite{bmw}.

Currently there is an experiment planned~\cite{jhfsk} to go from the
Japan Hadron Facility to the Super--Kamiokande detector ($L = 295$~km
and $\langle E_\nu \rangle \simeq 0.7$~GeV). This lies approximately
on the first peak of the oscillation, but the $L$ is not long enough
to resolve the sgn($\delta m^2_{31}$) ambiguity except for extreme
values of $\delta$. Therefore it is interesting to know what
additional measurements are needed to resolve the remaining ($\delta,
\pi - \delta$), sgn($\delta m^2_{31}$) and ($\theta_{23}, \pi/2 -
\theta_{23}$) ambiguities.

In general, for two measurements, ambiguities occur across wide areas
of the ($\delta, \theta_{13}$) plane. We have shown~\cite{bmw2} that,
in principle, (i) a third measurement can resolve the ($\delta, \pi -
\delta$) ambiguity, and the remaining ambiguities occur only along
lines in the ($\delta, \theta_{13}$) plane, (ii) a fourth measurement
may still have sgn($\delta m^2_{31}$) and ($\theta_{23}, \pi/2 -
\theta_{23}$) ambiguities, but they only occur at isolated points in
the ($\delta, \theta_{13}$) plane, and (iii) a fifth measurement can
remove all of the ambiguities. All of the degeneracies do not
necessarily have to be completely removed in a given experiment, since
they are only a problem if they occur in the region of parameter space
accessible to the experiment. We have found that making the
measurements at longer $L$ pushes the degeneracies to lower values of
$\theta_{13}$; furthermore, making measurements at different $L$ tends
to push the degeneracies to lower $\theta_{13}$ than if all of the
measurements are at the same $L$~\cite{bmw2}. For example, ignoring
experimental uncertainties, the degeneracies for two $\nu$ and two
$\bar\nu$ measurements at the same $L$ lie below $\sin^22\theta_{13} =
0.01 (0.002)$ for $L = 300 (2900)$~km; with only three measurements at
different $L$, they can be pushed to $\sin^2 = 0.001$ or lower. For
comparison, the sensitivity of the SuperJHF to Hyper--Kamiokande
experiment~\cite{jhfsk} is expected to be around $\sin^22\theta_{13} =
0.002$.

In summary, sitting on an oscillation peak reduces the ($\delta,
\theta_{13}$) ambiguity to a ($\delta, \pi - \delta$) ambiguity with
no $CPV/CPC$ confusion, which a third measurement can then
resolve. Going to long enough $L$ can resolve the sgn($\delta
m^2_{31}$) ambiguity, and perhaps the ($\theta_{23}, \pi/2 -
\theta_{23}$) ambiguity, especially if more than two measurements are
made. Even if the ($\theta_{23}, \pi/2 - \theta_{23}$) ambiguity
cannot be resolved, the resulting uncertainty in $\delta$ is small,
although $\theta_{13}$ will not be as well determined.

I gratefully acknowledge the collaboration of V. Barger and
D. Marfatia.

\section*{References}

\begin{thebibliography}{99}

\bibitem{solar}
See talk by T. Kajita, these proceedings.

\bibitem{atmos}
See talk by S. Biller, these proceedings.

\bibitem{reactor}
See plenary talk by D. Markoff, these proceedings.

\bibitem{KamLAND}
See parallel talk by D. Markoff, these proceedings.

\bibitem{approx}
A. Cervera {\it et al.} 2000 \NP {\bf B 579} 17; M. Freund 2001 \PR
{\bf D 64} 053003.

\bibitem{bmw}
V. Barger, D. Marfatia and K. Whisnant 2002 \PR {\bf D 65} 073023;
hep-ph/0108090.

\bibitem{arafune}
J. Arafune, M. Koike and J. Sato 1997 \PR {\bf D 56} 3093.

\bibitem{kimura}
K. Kimura, A. Takamura and H. Yokomakura 2002 \PL {\bf B 537} 86;
hep-ph/0205295.

\bibitem{minakata}
H. Minakata and H. Nunokawa 2001 JHEP {\bf 0110} 001.

\bibitem{lisi}
G. L. Fogli and E. Lisi 1996 \PR {\bf D 54} 3667.

\bibitem{bmw2}
V. Barger, D. Marfatia and K. Whisnant 2002 \PR {\bf D 66} 053007.

\bibitem{jhfsk}
Y. Itow {\it et al.}, hep-ex/0106019.

\end{thebibliography}

\end{document}
