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% Reactor Measurement of $\theta_{13}$ and Its Complementarity  %
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%               K. Inoue et al.                                 %
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\documentclass[preprint,showpacs,titlepage,aps,prd,tightenlines,amsmath,byrevtex,nofootinbib]{revtex4}

\usepackage{graphicx}


\begin{document}

\preprint

\title{
Reactor Measurement of $\theta_{13}$ and Its Complementarity to 
Long-Baseline Experiments
}

\author{H.~Minakata}
\email{E-mail: minakata@phys.metro-u.ac.jp}
\author{H.~Sugiyama}
\email{E-mail: hiroaki@phys.metro-u.ac.jp}
\author{O.~Yasuda}
\email{E-mail: yasuda@phys.metro-u.ac.jp}
\affiliation{Department of Physics, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan}

\author{K.~Inoue}
\email{E-mail: inoue@awa.tohoku.ac.jp}
\author{F.~Suekane}
\email{E-mail: suekane@awa.tohoku.ac.jp}
\affiliation{Research Center for Neutrino Science, Tohoku University,
Sendai, Miyagi, 980-8578, Japan}

\date{\today}

\vglue 1.4cm
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%    Abstract
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\hfuzz=25pt
\begin{abstract}
A possibility to measure $\sin^22\theta_{13}$
using reactor neutrinos is examined in detail.
It is shown that the sensitivity 
$\sin^22\theta_{13}>0.02$ can be reached with 20 ton-year data by 
placing identical CHOOZ-like detectors at near and far distances
from a giant nuclear power plant whose total thermal energy is 
24.3 ${\text{GW}_{\text{th}}}$. 
It is emphasized that this measurement is free from
the parameter degeneracies which occur in accelerator
appearance experiments, and therefore the reactor measurement
plays a role complementary to accelerator experiments.
It is also shown that the reactor measurement may be able to
resolve the degeneracy in $\theta_{23}$
if $\sin^22\theta_{13}$ and $\cos^22\theta_{23}$
are relatively large.
\end{abstract}

\pacs{14.60.Pq,25.30.Pt,28.41.-i}
%\vskip2pc]


\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Despite the accumulating knowledges of neutrino masses and the 
lepton flavor mixing by the atmospheric \cite{SKatm}, 
the solar \cite{solar,Ahmad:2002ka}, and the accelerator \cite{K2K} neutrino 
experiments, the (1-3) sector of the Maki-Nakagawa-Sakata (MNS)
matrix \cite{MNS} is still in the dark.
%
At the moment, we only know that 
$|U_{e3}| = \sin{\theta_{13}} \equiv s_{13}$ is small, 
$s_{13}^2 \alt 0.03$, by the bound imposed by the CHOOZ reactor 
experiment \cite{CHOOZ}.
%
In this paper we assume that the light neutrino sector 
consists of only three active neutrinos. We use the 
standard notation \cite{Hagiwara:pw} for the MNS matrix with 
$\Delta m^2_{ij} \equiv m^2_i - m^2_j$ where 
$m_i$ is the mass of the $i$th eigenstate.

One of the challenging goals in an attempt to explore
the full structure of lepton flavor mixing would be measuring 
the leptonic CP or T violating phase $\delta$ in the MNS 
matrix. 
If KamLAND \cite{KamLAND} confirms the Large-Mixing-Angle (LMA)
Mikheev-Smirnov-Wolfenstein (MSW) \cite{Mikheev:1986wj,Wolfenstein:1978ue}
solution of the solar neutrino problem, 
the most favored one by the recent analyses of solar neutrino data 
\cite{Ahmad:2002ka,solaranalysis}, we will have an open route toward the goal. 
Yet, there might still exist the last impasse, namely the 
possibility of too small value of $\theta_{13}$.
%
Thus, it is emphasized more and more strongly recently that the 
crucial next step toward the goal would be the determination 
of $\theta_{13}$.


In this paper, we raise the possibility that $\bar{\nu}_{e}$ 
disappearance experiment using reactor neutrinos could be potentially
the fastest (and the cheapest) way to detect the effects 
of nonzero $\theta_{13}$.
In fact, such an experiment using the Krasnoyarsk reactor 
complex has been described earlier \cite{krasnoyarsk}, in which the 
sensitivity to $\sin^2{2\theta_{13}}$ can be as low as 
$\sim 0.01$, an order of magnitude lower than the CHOOZ experiment. 
We will also briefly outline basic features of our proposal, 
and reexamine the sensitivity to $\sin^2{2\theta_{13}}$ in this paper. 


It appears that the most popular way of measuring $\theta_{13}$ 
is the next generation long baseline (LBL) neutrino oscillation 
experiments, MINOS \cite{MINOS} and JHF phase I \cite{JHF}.
It may be followed either by conventional superbeam \cite{lowecp}
experiments, JHF phase II \cite{JHF} and possibly others 
\cite{SPL,NuMI}, or by neutrino factories 
\cite{golden,nufact}.
%
It is pointed out, however, that the measurement
of $\theta_{13}$ in LBL experiment with only neutrino channel 
(as planned in JHF phase I) would suffer from large intrinsic 
uncertainties, on top of the experimental errors, due to 
the dependence on an unknown CP phase and the 
sign of $\Delta m^2_{31}$ \cite{KMN02}.
%
Furthermore, it is noticed that the ambiguity remains in 
determination of $\theta_{13}$ and other parameters even if 
precise measurements of appearance probabilities in neutrino 
as well as antineutrino channels are carried out, the problem of 
parameter degeneracy
\cite{Burguet-Castell:2001ez,Minakata:2001qm,Barger:2001yr,DMM02,Minakata:2002qi,KMN02}.
(For a global overview 
of parameter degeneracy, see \cite{Minakata:2002qi}.) 
%
While some ideas toward a solution are proposed the problem is 
hard to solve experimentally and it is not likely to be
resolved in the near future.


We emphasize in this paper that reactor $\bar{\nu}_{e}$ 
disappearance experiment provide particularly clean environment
for the measurement of $\theta_{13}$.
%
Namely, it can be regarded as a dedicated experiment 
for determination of $\theta_{13}$: it is insensitive to 
the ambiguity due to all the 
remaining oscillation parameters 
%(except for exceptional cases, see below) 
as well as to the matter effect. 
This is in sharp contrast with the features of LBL experiments
described above. 
%
Thus, the reactor measurement of $\theta_{13}$ will provide us 
valuable information complementary to the one from LBL 
experiments and will play an important role in resolving 
the problem of parameter degeneracy.
It will be shown that reducing the systematic errors is crucial for
the reactor measurement of $\theta_{13}$ to be competitive in 
accuracy with LBL experiments.  We will present a preliminary 
analysis of its possible roles in this context.

It is then natural to think about the possibility that 
one has better control by combining the two complementary way 
of measuring $\theta_{13}$, the reactor and the accelerator 
methods. In fact, we will show in this paper that nontrivial 
relations exist between the $\theta_{13}$ measurements by both 
methods thanks to the complementary nature of these two methods, 
so that in the luckiest case one may be able to derive constraints 
on the value of the CP violating phase $\delta$, or to determine 
the neutrino mass hierarchy.



\section{Reactor experiment as a clean laboratory for 
$\theta_{13}$ measurement}


Let us examine in this section how clean the measurement of 
$\theta_{13}$ by the reactor experiments is. 
We examine possible "contamination" by other quantities, 
$\delta$, the matter effect, the sign of $\Delta m^2_{31}$, and 
the solar parameters one by one.


We first note that, due to its low neutrino energy of 
a few MeV, the reactor experiments are inherently 
disappearance experiments, which can measure only the survival 
probability $P(\bar{\nu}_{e} \rightarrow \bar{\nu}_{e})$. 
%
It is well known that the survival probability does not 
depend on the CP phase $\delta$ in arbitrary matter densities 
\cite{delta-indep}. 


In any reactor experiment on the earth, short or long baseline, 
the matter effect is very small because the energy is quite low 
and can be ignored to a good approximation. 
It can be seen by comparing the matter and the vacuum effects 
(as the matter correction comes in only through this combination 
\cite{golden})
%
\begin{eqnarray}
\frac{aL}{|\Delta_{31}|} = 2.8 \times 10^{-4} 
\left(\frac{|\Delta m^2_{31}|}{2.5 \times10^{-3} \mbox{eV}^2} \right)^{-1}
\left(\frac{E}{4 \mbox{MeV}} \right)
\left(\frac{\rho}{2.3 \mbox{g}\cdot\mbox{cm}^{-3}} \right)
\left(\frac{Y_e}{0.5} \right)
\end{eqnarray}
where 
\begin{eqnarray}
\Delta_{ij} \equiv \frac{\Delta m^2_{ij} L}{2E}, 
\end{eqnarray}
%
with $E$ being the neutrino energy and $L$ baseline length.  
The best fit value of $|\Delta m^2_{31}|$ is given by
$|\Delta m^2_{31}|=2.5\times10^{-3}$eV$^2$ from the
Super-Kamiokande atmospheric neutrino
data \cite{shiozawa},
and throughout this paper we use this as the reference value
for $|\Delta m^2_{31}|$.
$a = \sqrt{2} G_F N_e$ denotes the index of refraction 
in matter with $G_F$ being the Fermi constant and 
$N_e$ the electron number density in the earth which is related 
to earth matter density $\rho$ as 
$N_e = Y_e\rho/m_p$ where $Y_e$ is proton fraction. 
%
Once we know that the matter effect is negligible we 
immediately recognize that the survival probability 
is independent of the sign of $\Delta m^2_{31}$.


Therefore, the vacuum probability formula applies. 
Its exact expression is given by 
%
\begin{eqnarray}
1-P(\bar{\nu}_{e} &\rightarrow& \bar{\nu}_{e}) =   
\sin^2{2 \theta_{13}}
\sin^2{\frac{\Delta_{31}}{2}} 
\nonumber \\
& & \hspace*{-15mm}
{}+ \frac{1}{\,2\,} c^2_{12}
\sin^2{2 \theta_{13}}
\sin{\Delta_{31}}\sin{\Delta_{21}} +
\left(
c^4_{13} \sin^2{2 \theta_{12}} + 
c^2_{12} \sin^2{2 \theta_{13} \cos{\Delta_{31}}}
\right)
\sin^2{\frac{\Delta_{21}}{2}}. 
\label{Pvac}
\end{eqnarray}
%
The three terms in the second line of (\ref{Pvac}) are suppressed 
relative to the main depletion term, 
the first term of the right-hand-side of (\ref{Pvac}) , by 
$\epsilon$,
$\epsilon^2/\sin^2{2 \theta_{13}}$,
$\epsilon^2$, respectively, where 
$\epsilon \equiv \Delta m^2_{21}/\Delta m^2_{31}$.
%
Assuming that 
$|\Delta m^2_{31}| = (1.6\mbox{-}3.9) \times 10^{-3}$ eV$^2$ \cite{shiozawa}, 
$\epsilon \simeq 0.1\mbox{-}0.01$ for the LMA MSW 
solar neutrino solution \cite{Ahmad:2002ka,solaranalysis}.
Then, the first and the third terms in the second line can be 
ignored, but the second term can be of order unity compared 
with the main depletion term provided that $\epsilon \simeq 0.1$. 
(Notice that we are considering the measurement of 
$\sin^2{2 \theta_{13}}$ in the range of 0.1-0.01.)
%
Therefore, assuming that $|\Delta m^2_{31}|$ is determined by LBL 
experiments in good accuracies, the reactor $\bar{\nu}_e$ 
disappearance experiments gives us a clean measurement of $\theta_{13}$
which is independent of any solar parameters except for
the case of high $\Delta m^2_{21}$ LMA solutions.


If the high $\Delta m^2_{21}$ LMA solution with 
$\Delta m^2_{21} \sim 10^{-4}$ eV$^2$ turns out to be
the right one,
we need a special care for the second term of the second line of 
(\ref{Pvac}). In this case, the determination of $\theta_{13}$ and 
the solar angle $\theta_{12}$ is inherently coupled,\footnote{
The effect of nonzero $\theta_{13}$ for measurement of 
$\theta_{12}$ at KamLAND is discussed in \cite{concha-carlos}.}
and we would need joint analysis of near-far detector complex 
(see the next section) and KamLAND. 



\section{Near-far detector complex: 
basic concepts and estimation of sensitivity}

In order to obtain a good sensitivity to $\sin^2{2 \theta_{13}}$,
selection of an optimized baseline and having the small statistical
and systematic errors are crucial.
For instance, the baseline length that gives oscillation maximum 
for reactor $\bar{\nu}_e$'s which have typical energy 4 MeV is 1.7\,km for
$\Delta m^2 \simeq 2.5\times 10^{-3}$eV$^2$.
Along with this baseline selection, if systematic error
and statistical error can be reduced to 1~\% level, which is
2.8 times better than the CHOOZ experiment \cite{CHOOZ},
one order magnitude of improvement for the $\sin^2{2 \theta_{13}}$
sensitivity at $\Delta m^2 \simeq 2.5\times 10^{-3}$eV$^2$ is possible.
In this section we demonstrate that such kind of an experiment is
potentially possible if we place a CHOOZ like detector at a
baseline 1.7\,km in 200\,m underground near the Kashiwazaki-Kariwa
nuclear power plant whose maximum energy generation is
24.3 ${\text{GW}_{\text{th}}}$.

Major part of systematic errors is caused by uncertainties of the
neutrino flux calculation, number of protons
and the detection efficiency. For instance,
in the CHOOZ experiment,
the uncertainty of the neutrino flux is 2.1~\%, that of
number of protons is 0.8~\%, and that of detection efficiency
1.5~\% as is shown in the Table \ref{table}.
The uncertainty of the neutrino flux includes ambiguity of
reactor thermal power generation, reactor fuel component,
neutrino spectra from fissions, and so on.
The uncertainty of the detection efficiency includes
systematic shift of defining the fiducial volume.
These systematic uncertainties, however, cancel out
if identical detectors are placed near the reactors and data
taken at the far and near detectors are
compared.\footnote{This is more or less the strategy taken in the
Bugey experiment \cite{bugey}.  The Krasnoyarsk group also plans in
their Kr2Det proposal \cite {krasnoyarsk} to construct two identical
50\,ton liquid scintillator detectors at 1100\,m and 150\,m from the
Krasnoyarsk reactor. They indicate that the systematic error can be
reduced down to 0.5\,\% by comparing the front and far detector.}
To estimate how good the cancellation will be, we study the case
of the Bugey experiment, which uses three identical detectors
to detect reactor neutrinos at 14/40/90m.
For the Bugey case, the uncertainty of the neutrino flux
improved from 3.5\,\% to 1.7\,\% and the error on the solid angle
remained the same (0.5\,\%$\rightarrow$0.5\,\%).
If the ratios of improvedment for the Bugey case is directly
applicable to our case, the systematic uncertainty will
improve from 2.7\,\% to 0.8\,\% as shown in the Table \ref{table}.
The ambiguity of the solid angle will be
negligibly small because the absolute baseline is much longer than
the Bugey case.
We are thinking of a case that front detectors locate at 300m from the 
reactors.

Hereafter we take 2~\% and 0.8~\% as the reference values for
the total systematic error.
Let us examine the physics potential of such a reactor experiment 
assuming these reference values for the systematic error.
%
We take, for concreteness, the Kashiwazaki-Kariwa reactor of 
24.3 ${\text{GW}_{\text{th}}}$ thermal power and assume its operation with 80\,\% 
efficiency. A liquid scintillator detector is located at 1.7\,km 
away from the reactor and assumed to detect $\bar{\nu}_e$ by 
delayed coincidence with 70\,\% detection efficiency.
The $\bar{\nu}_e$'s of 1-8\,MeV visible energy,
$E_{\text{visi}} = E_{\bar{\nu}_e} - 0.8\,{\text{MeV}}$, are used
and the number of events are counted in 14 bins of 0.5\,MeV\@.
Without oscillation,
20\,ton-year measurement yields 40,000 $\bar{\nu}_e$ events
which is naively comparable to a 0.5\,\% statistical error.
First, let us calculate how much we could constrain
$\sin^2{2\theta_{13}}$.
Unlike the analysis in \cite{bugey} which
uses the ratio of the numbers of events
at the near and the far detectors,
we use the difference of the numbers of events
$N_i(L_2)-(L_1/L_2)^2N_i(L_1)$, because
the analysis of the ratio becomes complicated.
(See, e.g., \cite{Fogli:1995xu}.)
In the actual case two near detectors are necessary. 
However, in this analysis both the near detectors are considered as one
detector for simplicity. 
The definition of $\Delta\chi^2$, which
stands for the deviation from the best fit point 
(non-oscillation point) 
is given by
\begin{eqnarray}
%&&
\hspace*{-20mm}
\displaystyle
&{\ }&\Delta\chi^2(\sin^2{2\theta_{13}}, \Delta m^2_{31})\nonumber\\
&\equiv&
 \sum_{i=1}^{14} \frac{\left\{
\left[
N_{i(0)}(L_2)- \left({L_1 \over L_2}\right)^2N_{i(0)}(L_1)\right]
-\left[N_i(L_2) - \left({L_1 \over L_2}\right)^2N_i(L_1)\right]
\right\}^2}
{{N_i}_{(0)}(L_2) + \left({L_1 \over L_2}\right)^4{N_i}_{(0)}(L_1)
+\sigma^2_{\text{sys}}{N_i}_{(0)}^2(L_2)}, 
\label{chi2}
\end{eqnarray}
%\\[3mm]
%&&
%\hspace*{-20mm}
\begin{eqnarray}
 N_i(L_j) \equiv N_i(\sin^2{2\theta_{13}}, \Delta m^2_{31}; L_j) ,\ \
 {N_i}_{(0)}(L_j) \equiv N_i(0, 0; L_j) ,
\nonumber
\end{eqnarray}
where $\sigma_{\text{sys}}$ is the relative systematic error and
$N_i(\sin^2{2\theta_{13}}, \Delta m^2_{31})$ denotes the theoretical
number of $\bar{\nu}_e$ events within the $i$th energy bin.  In
principle both the systematic errors $\sigma^{\text{abs}}_{\text{sys}}$
(absolute normalization) and $\sigma_{\text{sys}}$ (relative normalization)
appear in the denominator of 
(\ref{chi2}), but by taking the difference, we have
$(1+\sigma^{\text{abs}}_{\text{sys}})[(1+\sigma_{\text{sys}})
N_i(L_2)-(L_1/L_2)^2N_i(L_1)]
-[N_i(L_2)-(L_1/L_2)^2N_i(L_1)]=\sigma_{\text{sys}} N_i(L_2)
+\sigma^{\text{abs}}_{\text{sys}}
[N_i(L_2)-(L_1/L_2)^2N_i(L_1)]$ which indicates
the systematic error is dominated by the relative
error $\sigma_{\text{sys}}$, as the second term $[N_i(L_2)-(L_1/L_2)^2N_i(L_1)]$
is supposed to be small.  In fact we have explicitly verified numerically
that the presence of
$(\sigma^{\text{abs}}_{\text{sys}})^2
[N_i(L_2)-(L_1/L_2)^2N_i(L_1)]^2$ in the denominator of 
(\ref{chi2}) does not affect
any result.  The 90\,\%~CL\ exclusion limits,
which corresponds to $\Delta \chi^2 = 2.7$, are presented
in Fig.~\ref{exclude} for $\sigma_{\text{sys}} = 2.0\,\%$
with detector size 5\,ton-year
and $\sigma_{\text{sys}} = 0.8\,\%$ with detector size 20\,ton-year, respectively.
The CHOOZ result \cite{CHOOZ} is also depicted in Fig.~\ref{exclude}.
%Fig.~\ref{exclude} 
The figure shows that it is possible to measure $\sin^2{2\theta_{13}}$
down to 0.02 at the maximum sensitivity with respect to $\Delta m^2_{31}$
and to 0.04 for larger $\Delta m^2_{31}$
by a 20\,ton-year measurement, provided the quoted value of
the systematic errors is realized.

 Next, let us examine how precisely we could measure
$\sin^2{2\theta_{13}}$.
 The definition of $\Delta\chi^2$ is
\begin{eqnarray}
\hspace*{-20mm}
\displaystyle
&{\ }&\Delta\chi^2(\sin^2{2\theta_{13}}, \Delta m^2_{31})\nonumber\\
&\equiv&
 \sum_{i=1}^{14} \frac{\left\{
\left[
N_{i({\text{best}})}(L_2)- \left({L_1 \over L_2}\right)^2N_{i({\text{best}})}(L_1)\right]
-\left[N_i(L_2) - \left({L_1 \over L_2}\right)^2N_i(L_1)\right]
\right\}^2}
{{N_i}_{({\text{best}})}(L_2) + \left({L_1 \over L_2}\right)^4{N_i}_{({\text{best}})}(L_1)
+\sigma^2_{\text{sys}}{N_i}_{({\text{best}})}^2(L_2)}
\nonumber
\end{eqnarray}
where ${N_i}_{({\text{best}})}$ denotes $N_i$
for the set of the best fit parameters
($\sin^2{2\theta^{({\text{best}})}_{13}}$, $|\Delta m^{2({\text{best}})}_{31}|$)
given artificially.
The 90\,\% CL\ allowed regions,
whose bounds correspond to $\Delta \chi^2$ = 4.6,
are presented in Fig.~\ref{allow} for
$\sin^2{2\theta^{({\text{best}})}_{13}}$ = $0.04,\cdots,0.08$
(in the case of 5~ton-year measurement with $\sigma_{\text{sys}}$=2.0\%)
and $\sin^2{2\theta^{({\text{best}})}_{13}}$ = $0.02,\cdots,0.08$
(in the case of 20~ton-year measurement with $\sigma_{\text{sys}}$=0.8\%),
respectively.
Assuming that the value of $|\Delta m^2_{31}|$ is
known to a precision of $10^{-4}$eV$^2$ by the JHF
experiment, we can read off the error at 90\%CL in $\sin^2{2\theta_{13}}$
from Fig.~\ref{allow}, which suggests that
the error is almost independent of the central
value $\sin^2{2\theta^{({\text{best}})}_{13}}$.  Thus we have
\begin{eqnarray}
 \sin^2{2\theta_{13}} &=& \sin^2{2\theta^{({\text{best}})}_{13}}\pm0.034
\qquad(\mbox{\rm at 90\%CL,~d.o.f.=2})
\nonumber\\
&{\ }&\mbox{\rm for}~~\sin^2{2\theta^{({\text{best}})}_{13}}\agt0.04
\nonumber
\end{eqnarray}
in the case of
$\sigma_{\text{sys}}$=2.0\% with 5~ton-year measurement, and
\begin{eqnarray}
 \sin^2{2\theta_{13}} &=& \sin^2{2\theta^{({\text{best}})}_{13}}\pm0.015
\qquad(\mbox{\rm at 90\%CL,~d.o.f.=2})
\nonumber\\
&{\ }&\mbox{\rm for}~~\sin^2{2\theta^{({\text{best}})}_{13}}\agt0.02
\nonumber
\end{eqnarray}
in the case of
$\sigma_{\text{sys}}$=0.8\% with 20~ton-year measurement.

If the JHF experiment measures the value of
$|\Delta m^2_{31}|$ to the accuracy of $10^{-4}$eV$^2$ \cite{JHF},
the analysis of the allowed region of the reactor experiment reduces
to one dimensional one ($\Delta \chi^2$ = 2.7) and the error becomes
\begin{eqnarray}
 \sin^2{2\theta_{13}} &=& \sin^2{2\theta^{({\text{best}})}_{13}}\pm0.012
\qquad(\mbox{\rm at 90\%CL,~d.o.f.=1})
\nonumber\\
&{\ }&\mbox{\rm for}~~\sin^2{2\theta^{({\text{best}})}_{13}}\agt0.013
\nonumber
\end{eqnarray}
for 20~ton-year measurement with $\sigma_{\text{sys}}$=0.8\%.



\section{The problem of ($\theta_{13}$, $\theta_{23}$, $\delta$) 
parameter degeneracy}


We explore in this and the following sections the possible 
significance of reactor measurement of $\theta_{13}$ in 
the context of the problem of parameter degeneracy. 
We show that reactor measurement of $\theta_{13}$ can resolve 
the degeneracy at least partly if the measurement is 
sufficiently accurate.

Toward the goal we first explain what is the problem of 
parameter degeneracy in long-baseline neutrino oscillation experiments. 
It is a notorious problem: measurement of 
the disappearance and the appearance oscillation probabilities 
$\nu_{\mu}(\bar{\nu}_{\mu}) \rightarrow \nu_{e}(\bar{\nu}_{e})$ 
in both neutrino and antineutrino channels, 
no matter how they are accurate, does not allow unique determination 
of $\theta_{13}$, $\theta_{23}$, and $\delta$. 
%
The problem was first recognized in the form of intrinsic 
degeneracy between the two set of solutions of ($\theta_{13}$, $\delta$) 
\cite{Burguet-Castell:2001ez}. 
It was noticed that the degeneracy is further duplicated  
provided that the two neutrino mass patterns, 
the normal ($\Delta m^2_{31} > 0$) or 
the inverted ($\Delta m^2_{31} < 0$) hierarchies, are allowed 
\cite{Minakata:2001qm}, and/or 
that $\theta_{23}$ is not maximal \cite{Barger:2001yr}.
In general, there exist maximal eight-fold degeneracy 
in determination of ($\theta_{23}$, $\theta_{13}$, $\delta$). 


To illuminate the point, let us first restrict our 
treatment to relatively short-baseline experiment such as 
CERN-Frejus project \cite{SPL}. 
In this case, one can use the vacuum oscillation approximation 
for the disappearance and the appearance probabilities. 
They read  
%
\begin{eqnarray}
1- P(\nu_{\mu} \rightarrow \nu_{\mu}) 
&=& \sin^2{2 \theta_{23}}
\sin^2{\frac{\Delta_{31}}{2}} 
\nonumber \\
&-&
\left(
\frac{1}{\,2\,} c^2_{12} \sin^2{2 \theta_{23}} 
- 
s_{13} s^2_{23} \sin{2 \theta_{23}} 
\sin{2 \theta_{12}} \cos{\delta}
\right)
\sin{\Delta_{21}}\sin{\Delta_{31}} 
\nonumber \\
&+&
O(\epsilon^2) + O(s^2_{13}),  
\label{P_mumu} 
\end{eqnarray}
%
\begin{eqnarray}
P[\nu_{\mu}(\bar{\nu}_{\mu}) \rightarrow \nu_{e} (\bar{\nu}_{e})] 
=  s^2_{23} \sin^2{2 \theta_{13}} 
\sin^2{\frac{\Delta_{31}}{2}} &+&
\frac{1}{\,2\,} J_r 
\sin{\Delta_{21}} \sin{\Delta_{31}} 
\cos{\delta}
\nonumber \\
&\mp& 
J_r 
\sin{\Delta_{21}} \sin^2{\frac{\Delta_{31}}{2}} 
\sin{\delta} + 
O(\epsilon s^2_{13}), 
\label{P_mue}
\end{eqnarray}
%
where $\epsilon \equiv \Delta m^2_{21}/\Delta m^2_{31}$,
$J_r \equiv \sin{2 \theta_{23}}\sin{2 \theta_{12}} c^2_{13}s_{13}$. 
The sign $\pm$ in (\ref{P_mue}) correspond to neutrino and 
antineutrino channels, respectively.
%
In fact, one can show that the matter effect comes in 
with the factor of $s^2_{13} aL /\Delta_{31}$ \cite{AKS}. 

By the disappearance measurement, for example at JHF, 
$\sin^2{2 \theta_{23}}$ and $\Delta m^2_{31}$
will be determined with accuracies of 1 \% level for both 
quantities \cite{JHF}.\footnote{
%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%
Usually one thinks of determining not $\Delta m^2_{31}$ but 
$\Delta m^2_{32}$ by the disappearance measurement. But, 
it does not appear possible to resolve difference between 
these two quantities because one has to achieve resolution of 
order $\epsilon$ for the reconstructed neutrino energy.
}
%
Then, we have two solutions for $\theta_{23}$ 
($\theta_{23}$ and $\pi/2 - \theta_{23}$) 
if $\theta_{23}$ is not maximal. For example,  
if $\sin^2{2 \theta_{23}} = 0.95$, which is perfectly allowed by 
the most recent atmospheric neutrino data \cite {shiozawa}, 
then $s^2_{23}$ can be either 0.39 or 0.61. 
Since the dominant term in the appearance probability 
depends upon $s^2_{23}$ not $\sin^2{2 \theta_{23}}$, it leads to 
$\pm20$ \% difference in the number of appearance events.
%
On the other hand, appearance measurement will allow us to 
determine $\sin^2{2 \theta_{13}}$, but up to the ambiguity 
of the above two solutions of $s^2_{23}$. 

Let us discuss the simplest possible case, the LOW or the vacuum 
(VAC) oscillation solution of the solar neutrino problem. 
(See e.g., \cite{Barenboim:2002nv} for a recent discussion.)
In this case, one can safely ignore terms of order $\epsilon$
in (\ref{P_mumu}) and (\ref{P_mue}). Then we are left with 
only the first terms in the right-hand-side of these equations, 
the one-mass scale dominant vacuum oscillation probabilities.
%
Now let us define the symbols 
$x=\sin^2{2 \theta_{13}}$ and $y=s^2_{23}$. 
Then, (\ref{P_mumu}) and (\ref{P_mue}) take the forms  
$y=y_1$ or $y_2$ (corresponding to two solutions of $s^2_{23}$)
and $xy=constant$, respectively, for given values of the 
probabilities. It is then obvious that there are 
two crossing point of these curves. 
This is the simplest version of the ($\theta_{13}, \theta_{23}$) 
degeneracy problem. 
%
%The feature as well as more elaborate ones, on which we discuss 
%immediately below, are pictorially displayed in Fig.\ref{degen1}.


We next discuss what happens if $\epsilon$ is not 
negligible though small: the case of LMA solar neutrino solution. 
In this case, the appearance curve, $xy=constant$, split into 
two curves (though they are in fact connected at their maximum 
value of $s^2_{23}$) because of the two degenerate solution 
of $\delta$ that is allowed for given values of 
$P(\nu_{\mu} \rightarrow \nu_{e})$ and 
$P(\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{e})$.
Then, we have, in general, four crossing points for a given value of 
$s^2_{23}$, the four-fold degeneracy.
%
Simultaneously, the two $y= constant$ lines are slightly 
tilted and the splitting between two curves becomes larger 
at larger $\sin^2{2 \theta_{13}}$, though the effect is 
too tiny to be clearly seen. 

If the baseline distance is longer, the earth matter effect 
comes in and further splits each appearance contour 
into two, depending upon the sign of $\Delta m^2_{31}$. 
Then, we have four appearance contours and it is clear that we have 
eight solutions, as displayed in Fig.\ref{degen1}.
%
This is a simple pictorial representation of the maximal 
eight-fold parameter degeneracy \cite {Barger:2001yr}.

To draw Fig.\ref{degen1}, we have calculated disappearance and 
appearance contours by using the approximate formula derived by 
Cervera {\it et al.} \cite{golden}. We take the baseline distance 
and neutrino energy as $L=295$ km and $E=400$ MeV with possible 
relevance to JHF project \cite{JHF}. The earth matter density is 
taken to be $\rho = 2.3~ \mbox{g}\cdot\mbox{cm}^{-3}$ based on 
the estimate given in \cite{KSmpl99}.
%
We assume, for definiteness, that a long-baseline 
disappearance measurement has resulted in 
$\sin^2{2 \theta_{23}} = 0.92$ and 
$\Delta m^2_{31} = 2.5 \times 10^{-3}$ eV$^2$.
For the LMA solar neutrino parameters we take 
$\tan^2\theta_{12}=0.38$ and $\Delta m^2_{21}=6.9\times10^{-5}$eV$^2$ 
\cite{Fukuda:2002pe}. 
%
The qualitative features of the figure remain unchanged even if 
we employ the parameters obtained by other analyses.



\section{Resolving the parameter degeneracy by reactor measurement 
of $\theta_{13}$} 


Now we discuss how reactor experiments can contribute to 
resolve the parameter degeneracy. To make our discussion as 
concrete as possible we use the particular long-baseline experiment, 
the JHF experiment \cite{JHF}, to illuminate the complementary 
role played by reactor and long-baseline experiments.
%
It is likely that the experiment will be carried out at around 
the first oscillation maximum ($\Delta_{31}=\pi$)
for a number of reasons:
the dip in energy spectrum in disappearance channel is deepest, 
the number of appearance events are maximal \cite {JHF}, 
and the degeneracy in $\delta$ is reduced to two-fold 
($\delta$, $\pi - \delta$) for each mass hierarchy 
\cite{KMN02,Barger:2001yr}.\footnote{
%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%
In order to have this reduction, one has to actually tune the 
energy to vanishing $\cos{\delta}$ term 
(the thinnest CP trajectory), not $\Delta_{31}=\pi$, 
after averaging over the energy spectrum \cite{KMN02}.
}
%
With the distance $L=295$ km, the oscillation maximum is 
at around $E=600$ MeV. We take the same mixing parameters as 
those used in Fig.\ref{degen1}.


\subsection{Illustration of how reactor measurement helps resolve the  
($\theta_{13}$, $\theta_{23}$) degeneracy}

Let us first give an illustrative example showing 
how reactor experiments could help resolve the  
($\theta_{13}$, $\theta_{23}$) degeneracy. 
%
In Fig.\ref{degen2} the two solutions of $s^2_{23}$ for 
$\sin^22\theta_{23}=0.92$ as well as the curves determined 
by appearance measurements of 
$P(\nu_{\mu} \rightarrow \nu_{e})$ and 
$P(\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{e})$ 
are drawn by the thick and the thin 
dotted lines for positive and the negative $\Delta m^2_{31}$, 
respectively. The values of disappearance and appearance 
probabilities are chosen arbitrarily for illustrative purpose 
and are given in the caption of Fig.\ref{degen2}.
%
We cannot resolve the two curves with different $\delta$ 
for each sign of $\Delta m^2_{31}$ because they completely overlap 
with each other at the oscillation maximum. 
%
Furthermore, the two overlapped curves of positive and the negative 
$\Delta m^2_{31}$ also approximately overlap because of 
the small matter effect due to short baseline of JHF, as indicated 
in Fig.\ref{degen2}.


Thus, the degeneracy in the set ($\theta_{13}$, $\theta_{23}$) 
is effectively two-fold, and in this particular example, 
the reactor experiment described in section III will be able 
to resolve the degeneracy.
%
Notice that once the $\theta_{23}$ degeneracy is lifted 
one can easily obtain other four solutions of 
($\delta$, $\theta_{23}$) 
(though they are still degenerate) because the relationship 
between them is given analytically in a completely general setting 
\cite{Minakata:2002qi}.

Considering the possibility that the long-baseline experiment
run with the neutrino mode only in its first phase, we present 
in Fig.\ref{degen2} the regions on 
$\sin^2{2\theta_{13}}$-$s^2_{23}$ plane that are allowed by varying 
completely arbitrary CP phase $\delta$ for a given 
$P(\nu_{\mu} \rightarrow \nu_{e})$. 
%
We observe that there is a large intrinsic uncertainty in 
$\theta_{13}$ determination, the problem addressed in \cite{KMN02}.
The two regions corresponding to positive and negative 
$\Delta m^2_{31}$ heavily overlap due to small matter effect. 




\subsection{Resolving power of the ($\theta_{13}$, $\theta_{23}$)
degeneracy by reactor measurement}


Let us make a semi-quantitative estimate of how powerful the 
reactor method is for resolving the ($\theta_{13}$, $\theta_{23}$)
degeneracy.\footnote{The possibility of resolving the
($\theta_{13}$, $\theta_{23}$) by a reactor experiment was
qualitatively mentioned in \cite{Barenboim:2002nv}.}\footnote{ 
%%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%
An alternative way to resolve the ambiguity is to look at 
$\nu_e\rightarrow\nu_\tau$ channel 
%in either superbeam experiments or in neutrino factories, 
because the main oscillation 
term in the probability $P(\nu_e\rightarrow\nu_\tau)$ 
depends upon $c^2_{13}$. 
%
Unfortunately, this idea does not appear to be explored in detail 
while it is briefly mentioned in \cite{Barger:2001yr,DMM02}.}
%%%%%%%%%%%%%%%%%%%%
The degeneracy persists even for the LOW or the VAC solutions 
of the solar neutrino problem, as we have seen in the previous 
section.



We consider, for simplicity, the special case $\Delta_{31}=\pi$, 
i.e., energy tuned at the first oscillation maximum, since the 
general case is complicated to work out. 
In this case the other solution 
$\sin^22\theta'_{13}$ is given by \cite{Barger:2001yr}
\begin{eqnarray}
\sin^22\theta_{13}^{\prime} &=&
\sin^22\theta_{13} \tan^2\theta_{23}
+ \left({\Delta m^2_{21} \over \Delta m^2_{31}} \right)^2
{\tan^2\left(aL/2\right) \over \left(aL/\pi\right)^2}
\nonumber\\
&\times&\left[1-\left(aL/\pi\right)^2 \right]
\sin^22\theta_{12} \left(1-\tan^2\theta_{23}\right) 
%\label{eqn:
\label{eqn:th23}
\end{eqnarray}
%

We already know from the Super-Kamiokande atmospheric neutrino
data \cite{shiozawa}
that $0.92\le\sin^22\theta_{23}\le1.0$ at 90\%CL which implies
$|1-\tan^2\theta_{23}|\le0.22$.
Together with $|\Delta m^2_{21}/\Delta m^2_{31}|<0.1$
for the solar solution and $|aL/2|\ll1$ for the JHF setup,
we can ignore the second term
to get a rough estimate of the difference between
$\sin^22\theta_{13}^{\prime}$ and $\sin^22\theta_{13}$
in the region of $\sin^22\theta_{13}(\agt0.03)$
which the reactor experiment can reach.
In fact in Fig.\ref{delth13}(b) the value of
$|\delta(\sin^22\theta_{13})|/(\sin^22\theta_{13})_{\text{average}}$
is calculated numerically as a function of $\sin^22\theta_{23}$, where
$\delta(\sin^22\theta_{13})\equiv\sin^22\theta'_{13}-\sin^22\theta_{13}$,
$(\sin^22\theta_{13})_{\text{average}}\equiv
(\sin^22\theta'_{13}+\sin^22\theta_{13})/2$.
For the best fit value of the two mass squared differences
$\Delta m^2_{21}$ and $\Delta m^2_{31}$, for which
$\epsilon\equiv\Delta m^2_{21}/\Delta m^2_{31}=0.028$,
there is little difference between the case with
$\sin^22\theta_{13}=0.03$ and the one with $\sin^22\theta_{13}=0.09$,
and they are all approximated by the first term in (\ref{eqn:th23}).
The solar mixing angle is taken as
$\tan^2\theta_{12}=0.38$ \cite{Fukuda:2002pe}.
For the purpose of comparison, we present in Fig.\ref{delth13}(a)
the sensitivity which is expected to be achieved in the
reactor experiment described in section III.

The precision of $\sin^22\theta_{23}$ which is expected
to be achieved at the JHF experiment \cite{JHF} is approximately
1\% for $0.92\le\sin^22\theta_{23}\le1.0$ (See Fig.~11 in \cite{JHF}).
Therefore, if the central value of $\sin^22\theta_{23}$
from the JHF measurement of $\nu_\mu\rightarrow\nu_\mu$ turns out
to be 1.0 then $|1-\tan^2\theta_{23}|\alt0.2$
implies about 20\% error in $\sin^22\theta_{13}$.
In this case the precision of the reactor experiment
of $\sin^22\theta_{13}$ is comparable or poorer than that
of JHF, irrespective of the value of $\sin^22\theta_{13}$.
However, if the central value of $\sin^22\theta_{23}$
turns out to be smaller than 1.0 ($\alt$0.98)
and if $\sin^22\theta_{13}$
is relatively large ($\agt$0.06) then the reactor experiment
may enable us to eliminate a fake solution $\sin^22\theta_{13}^{\prime}$
and determine whether $\theta_{23}$ is smaller or larger
than $\pi/4$. (See Fig. \ref{delth13} (a) and (b).)
On the other hand,
if the ratio $\epsilon\equiv\Delta m^2_{21}/\Delta m^2_{31}$
is much larger than that at the best fit point, then
the second term in (\ref{eqn:th23}) is not negligible.
In Fig.\ref{delth13}(b)
$|\delta(\sin^22\theta_{13})|/(\sin^22\theta_{13})_{\text{average}}$
is plotted in the case of $\epsilon=0.12$ with
$\sin^22\theta_{13}=0.03, 0.06, 0.09$ and we observe that
the suppression in the first term in (\ref{eqn:th23})
is compensated by the second term for $\sin^22\theta_{13}=0.03$,
i.e., the degeneracy is small and therefore
resolving the degeneracy is difficult in this case.
Estimation of the significance of the fake solution
requires detailed calculations of the accelerator
experiment which include the statistical and systematic
errors as well as the correlations of errors and
the parameter degeneracies, and it will be worked out
in future communication.



\section{More about reactor vs. long-baseline experiments}


The foregoing discussion in the previous section implicitly assumes 
that the sensitivities of reactor and LBL experiments with both 
$\nu$ and $\bar{\nu}$ channels are good enough to detect effects 
of nonzero $\theta_{13}$.
However, it need not be true, in particular, in coming 10 years.
%
To further illuminate complementary roles played by reactor and LBL 
experiments, we examine their possible mutual relationship 
including the cases that there is signal in the former but 
no signal in the latter experiments, or vice versa. 

For ease of understanding by the readers,  
we restrict our presentation in this section to very intuitive 
level by using figures. It is, of course, possible to make it more 
precise by deriving inequalities based on analytic approximate 
formula \cite {golden}.


If a reactor experiment sees positive evidence for disappearance 
in $\bar{\nu}_e \rightarrow \bar{\nu}_e$ (case of Reactor Affirmative), 
it would be possible 
to determine $\theta_{13}$ up to certain experimental errors. 
%
In this case, the appearance probability in LBL experiment 
must fall into the region 
$P(\nu)_{\pm}^{\text{min}} \leq P(\nu)_{\pm} \leq  P(\nu)_{\pm}^{\text{max}}$ 
if the mass hierarchy is known, and by
$P(\nu)_{-}^{\text{min}} \leq P(\nu)_{\pm} \leq  P(\nu)_{+}^{\text{max}}$
otherwise. The similar inequalities are present also for 
antineutrino appearance channel.


In Fig.\ref{envelope} we present allowed regions on a plane spanned by 
$P(\nu_{\mu} \rightarrow \nu_{e})$ and 
$P(\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{e})$ by taking the three 
reference values $\sin^22\theta_{13}=0.09,~0.06,~0.03$ 
(labeled as a, b, c). 
They are inside the sensitivity region of the reactor experiment 
discussed in section III. 
In fact, we have used the one dimensional $\chi^2$
analysis (i.e., the only parameter is $\sin^22\theta_{13}$) to 
obtain the allowed regions in Fig.\ref{envelope} 
since the disappearance experiment of JHF is supposed to give
us quite an accurate value for $|\Delta m^2_{31}|$. 
In doing this we have used the same systematic error of 0.8 \% and 
the statistical errors corresponding to 20\,ton-year measurement 
by the detector considered in section III.
%
For $\sin^22\theta_{13}< 0.012$, the particular reactor experiment 
would fail (case of Reactor Negative) but the allowed region 
can be obtained by the same procedure, and presented in 
Fig.\ref{envelope}, the region labeled as d.
%
We use the same LMA parameters as used earlier for Fig.\ref{degen1} 
and Fig.\ref{degen2}. 



We discuss four cases depending upon the two possibilities of 
positive and negative evidences (denoted as Affirmative, and Negative)
in each disappearance and appearance search in reactor and 
long-baseline accelerator experiments, respectively. 
However, it is convenient to organize our discussion by classifying 
them into two categories, (Reactor Affirmative), and (Reactor Negative).


\subsection{Reactor Affirmative}


We have two alternative cases, LBL appearance 
search Affirmative, or Negative.

\noindent
{\bf LBL Affirmative}: 

Implications of positive evidence in appearance search in LBL 
experiments differ depending upon which region 
the observed appearance probability $P(\nu)$ falls in:

\noindent
(1) $P_{-}^{\text{min}} \leq P(\nu) \leq P_{+}^{\text{min}}$, or 
(2) $P_{-}^{\text{max}} \leq P(\nu) \leq P_{+}^{\text{max}}$: 

These case correspond to the two intervals which are given by
the projection on the $P$ axis of all the shadowed regions
(a, b or c)
minus the projection on the $P$ axis of the darker
shadowed region (a, b or c) in Fig.\ref{envelope}.
It is remarkable that in these cases the sign of 
$\Delta m^2_{31}$ is determined. 
If it is in the former (latter) region, the sign is 
negative (positive) which implies the inverted (normal) 
hierarchy of neutrino masses.
%


\noindent
(3) $P_{+}^{\text{min}} \leq P(\nu) \leq P_{-}^{\text{max}}$: 

This case corresponds to the interval which is given by
the projection on the $P$ axis of the darker
shadowed region (a, b or c) in Fig.\ref{envelope}.
In this case, the sign of $\Delta m^2_{31}$ cannot be determined. 
The CP phase  $\delta$ is determined up to the two-fold ambiguity 
$\delta \leftrightarrow \pi - \delta$ 
for each sign of $\Delta m^2_{31}$, and 
in general there is four degenerate solutions (apart from 
$\theta_{23} \leftrightarrow \pi/2 - \theta_{23}$ 
degeneracy).

It may be worth noting that if the reactor determination of 
$\theta_{13}$ is accurate enough, it could be advantageous 
for LBL appearance experiments to run only in neutrino mode 
(where the cross section is larger by a factor of 2-3)
to determine $\delta$, and possibly the sign of $\Delta m^2_{31}$ 
depending upon which region $P(\nu)$ falls in. 



\noindent
{\bf LBL Negative}:

In principle, it is possible to have no appearance event even 
though the reactor sees evidence for disappearance.
This case corresponds to the left edge of the region c in Fig.\ref{envelope},
i.e., the allowed region with $\sin^22\theta_{13}\simeq0.02$
whose projection on the $P$ axis falls below $P=0.003$.
%
In order for this case to occur the sensitivity limits 
$P(\nu)_{\text{limit}}$ of the LBL experiment must satisfy, 
assuming our ignorance to the sign of $\Delta m^2_{31}$,
$P_{-}^{\text{min}} <  P(\nu)_{\text{limit}}$.
%
If it occurs that 
$P_{-}^{\text{min}} < P(\nu)_{\text{limit}} < P_{+}^{\text{min}}$,  
then the sign of $\Delta m^2_{31}$ is determined to be minus.


In the case of the JHF experiment in its phase I 
$P(\nu)_{\text{limit}}$ is estimated to be $3 \times 10^{-3}$ 
\cite{JHF}.\footnote{
%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%
The sensitivity limit of $\sin^2{2 \theta_{13}}$ quoted in 
\cite{JHF}, $\sin^2{2 \theta_{13}} \leq 6 \times 10^{-3}$, 
obtained by using one-mass scale approximation ($\epsilon \ll 1$)
may be translated into this limit for $P(\nu)$.
}
Then, by using the mixing parameters typical to 
the LMA solution, the case of LBL Negative can only occur 
if $\sin^2{2 \theta_{13}} \alt 0.01$.



\subsection{Reactor Negative}


If the reactor experiment does not see disappearance of $\bar{\nu}_e$
one obtains the bound $\theta_{13} \leq \theta_{13}^{\text{RL}}$.
This corresponds to the region d in Fig.\ref{envelope} for the 
experiment described in section III.
We have again two alternative cases, LBL appearance 
search Affirmative, or Negative.

\noindent
{\bf LBL Affirmative}:

If a LBL experiment measures the oscillation probability $P(\nu)$.
%
Then, for a given value of $P(\nu)$ the allowed region of 
$\sin{2 \theta_{13}}$ is given by 
%
$\sin{2 \theta_{\pm}^{\text{min}}} \leq
\sin{2 \theta_{13}} \leq
\sin{2 \theta_{\pm}^{\text{max}}}$
%
if the sign of $\Delta m^2_{31}$ is known, and by 
%
$\sin{2 \theta_{+}^{\text{min}}} \leq
\sin{2 \theta_{13}} \leq
\sin{2 \theta_{-}^{\text{max}}}$
otherwise.
%
We denote below the maximum and the minimum values of $\theta_{13}$ 
collectively as $\theta_{\text{max}}$ and $\theta_{\text{min}}$, respectively.
%
In Fig.\ref{degen2}, the region bounded by 
$\sin{2 \theta_{+}^{\text{min}}}$ and $\sin{2 \theta_{+}^{\text{max}}}$ 
($\sin{2 \theta_{-}^{\text{min}}}$ and $\sin{2 \theta_{-}^{\text{max}}}$) 
are indicated as a region bounded by the thick (thin) solid line 
for a given value of $s^2_{23}$.

Then, there are three possibilities which we discuss one by one:

\noindent
(i) $\theta_{13}^{\text{RL}} \geq \theta_{\text{max}}$: 
In this case no additional information is obtained by nonobservation 
of disappearance of $\bar{\nu}_e$ in reactor experiment. 


\noindent
(ii) $\theta_{\text{min}} \leq \theta_{13}^{\text{RL}} \leq \theta_{\text{max}}$: 
In this case we have a nontrivial constraint 
$\theta_{\text{min}} \leq \theta_{13} \leq \theta_{13}^{\text{RL}}$.


\noindent
(iii) $\theta_{13}^{\text{RL}} \leq \theta_{\text{min}}$: 
This case can only occur if reactor sensitivity is good enough, 
for example very roughly speaking $\sin^2{2 \theta_{13}} \leq 0.01$ 
in JHF.



\noindent
{\bf LBL Negative}:

In this case, we obtain the upper bound on $\theta_{13}$, which 
however depends on the assumed values of $\delta$ and the sign of 
$\Delta m^2_{31}$. A $\delta$-independent bound can also be derived:
$\theta_{13} \leq$ min[$\theta^{\text{RL}}, \theta_{\text{min}}$].




\section{Discussion and Conclusions}

We have explored in detail the possibility to measure $\sin^22\theta_{13}$
using reactor neutrinos,
We stressed that this measurement is free from
the parameter degeneracies from which accelerator
appearance experiments suffer, and that the reactor measurement
is complementary to accelerator experiments.
We have shown that sensitivity to $\sin^22\theta_{13}\agt0.02$ (0.04)
is obtained with a 24.3 ${\text{GW}_{\text{th}}}$ reactor and
data size of 20\,(5)\,ton-year
by placing identical detectors at near and far distances,
assuming that the relative systematic error is
0.8\% (2.0\%), respectively.
In particular,
if the relative systematic error is 0.8\%, the
error in $\sin^22\theta_{13}$ is 0.013 which is
smaller than the uncertainty due to the combined
(intrinsic and hierarchical) parameter degeneracies
expected in accelerator experiments.
We also have shown that the reactor measurement can
resolve the degeneracy in $\theta_{23}\leftrightarrow\pi/2-\theta_{23}$
and determine whether $\theta_{23}$ is smaller or larger
than $\pi/4$ if $\sin^22\theta_{13}$ and $\cos^22\theta_{23}$
are relatively large.
We have taken 2.0\% and 0.8\% as the reference values for
the relative systematic error.
2.0\% is exactly the same figure as the Bugey experiment
while 0.8\% is what we naively expect in the case we have
identical detectors which are exactly the same as the CHOOZ detector. 
It is also technically possible to dig a 200m depth shaft hole with
diameter wide enough to place a CHOOZ-size detector in. 
Therefore the discussions in this paper are realistic.
We hope the present
paper stimulates interest of the community in reactor measurements
of $\theta_{13}$.

%%%%%%%%%%%%%%%% acknowledgments %%%%%%%%%%%%%%%%
\begin{acknowledgements}
We thank Yoshihisa Obayashi for correspondence. 
HM thanks Andre de Gouvea for discussions, and 
Theoretical Physics Department of Fermilab for hospitality.
This work was supported by the Grant-in-Aid for Scientific Research
in Priority Areas No. 12047222 and No. 13640295, Japan Ministry
of Education, Culture, Sports, Science, and Technology.
\end{acknowledgements}



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\end{thebibliography}


\newpage
%\vglue 4.5cm
\hglue -1.8cm
%\tightenlines
\begin{table}
\vglue 4.5cm
\hglue -1.8cm
\begin{tabular}{|l|c|c|}
\hline
Bugey & absolute normalization & relative normalization\\
\hline
flux& 2.8\% & 0.0\% \\
number of protons& 1.9\% & 0.6\% \\ 
solid angle& 0.5\% & 0.5\% \\ 
detection efficiency& 3.5\% & 1.7\% \\ 
\hline
total& 4.9\% & 2.0\% \\ 
\hline 
\end{tabular}
%\vglue -2.63cm
\vglue 1.63cm
\hglue -0.2cm
%\tightenlines
\begin{tabular}{|l|c|c|}
\hline
CHOOZ--like & absolute normalization & relative normalization (expected)\\
\hline
flux& 2.1\% & 0.0\% \\
number of protons& 0.8\% & 0.3\% \\ 
detection efficiency& 1.5\% & 0.7\% \\ 
\hline
total& 2.7\% & 0.8\% \\ 
\hline 
\end{tabular}
\vglue 2.5cm
\label{tab:error}
\caption{Systematic errors in the Bugey and the CHOOZ--like experiments.
Relative errors in the CHOOZ--like experiment are expectation.}
\label{table}
\end{table}


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\includegraphics[scale=0.7]{exclude.eps}
%\vglue 1.0cm
\caption{
Shown are the 90\%~CL\ exclusion limits
for measurements with ($\sigma_{\text{sys}}$=2\%, 5 t$\cdot$yr),
($\sigma_{\text{sys}}$=2\%, $\infty$ t$\cdot$yr),
($\sigma_{\text{sys}}$=0.8\%, 20 t$\cdot$yr),
($\sigma_{\text{sys}}$=0.8\%, $\infty$ t$\cdot$yr),
respectively.
 The solid line is the CHOOZ result, and the 90\%CL interval
$1.6\times10^{-3}$eV$^2\le\Delta m^2_{31}\le 3.9\times10^{-3}$eV$^2$
of the Super-Kamiokande atmospheric neutrino data
is shown as a shaded strip.
}
\label{exclude}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
%\vglue 4.3cm
\vglue -0.5cm
%\hglue -0.5cm
\includegraphics[scale=0.6]{allow1.eps}
\includegraphics[scale=0.6]{allow2.eps}
\vglue 1.5cm
\caption{
 Shown are the allowed regions at 90\%CL
with ((a): $\sigma_{\text{sys}}$=2\%, 5 t$\cdot$yr) and 
((b): $\sigma_{\text{sys}}$=0.8\%, 20 t$\cdot$yr), respectively
for $\sin^2{2\theta^{({\text{best}})}_{13}}$=$0.04,\cdots,0.08$ ((a)),
$0.02,\cdots,0.08$ ((b)).
 The reference value of $|\Delta m^{2({\text{best}})}_{31}|$ is
$2.5\times10^{-3}\,{\text{eV}}^2$.
}
\label{allow}
\end{figure}

\newpage 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vglue -4.0cm
\hglue -2.2cm
\includegraphics[scale=0.6]{degene4-3.eps}
\vglue 3.0cm
\caption{
The contours are depicted on the
$\sin^2{2\theta_{13}}$-$s^2_{23}$
plane that are determined by given values of the appearance 
probabilities $P(\nu_\mu\rightarrow\nu_e)=0.01$ and 
$P(\bar{\nu}_\mu\rightarrow\bar{\nu}_e)=0.015$.
%
The thick and the thin dotted lines correspond to positive 
and negative $\Delta m^2_{31}$, respectively.
%
The present bound on $\sin^2\theta_{23}$
($0.36<\sin^2\theta_{23}<0.64$) from the atmospheric neutrino data is
denoted by the dotted dash lines.
There are four solutions for each $s^2_{23}$ that are allowed for 
a given value of $\sin^2{2\theta_{23}}$ to be measured in LBL 
experiments.  $\sin^2{2\theta_{23}} = 0.92$ is assumed in this figure
and there are eight solutions which are denoted by blobs.
The oscillation parameters are taken as follows:
$\Delta m^2_{31}=2.5\times10^{-3}$eV$^2$,
$\Delta m^2_{21}=6.9\times10^{-5}$eV$^2$,
$\tan^2\theta_{12}=0.38$.  The Earth density
is taken to be $\rho$=2.3 g/cm$^3$.
}
\label{degen1}
\end{figure}

\newpage 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vglue -4.0cm
\hglue -2.2cm
\includegraphics[scale=0.6]{degene56.eps}
\vglue 3.0cm
\caption{
The allowed region on the $\sin^2{2\theta_{13}}$-$s^2_{23}$
plane with a given value of $P(\nu_\mu\rightarrow\nu_e)$ (in this case
$P(\nu_\mu\rightarrow\nu_e)=0.025$) where the CP
phase $\delta$ varies from $0$ to $2\pi$.  The shadowed region bounded by a
thick line is for the normal hierarchy ($\Delta m^2_{31}>0$) while the
one bounded by a thin line is for the inverted hierarchy ($\Delta
m^2_{31}<0$).
Furthermore, if the value of
$P(\bar{\nu}_\mu\rightarrow\bar{\nu}_e)$ is measured
(in this case
$P(\bar{\nu}_\mu\rightarrow\bar{\nu}_e)=0.035$), then the
shadowed region shrinks to two lines (because of the intrinsic
degeneracy there are two solutions).  Because the reference values for
the parameters gives oscillation maximum $\Delta_{31}=\pi$, these
two lines become identical (denoted as a dotted line).  Also since the
matter effect is small at the JHF experiment, the dotted lines for
the normal (the thick dotted one) and the inverted (the thin dotted one)
hierarchies largely overlap.  The thick dotted line terminates at the
boundary of the allowed region of the case with $\Delta m^2_{31}>0$.
The bound $0.36<\sin^2\theta_{23}<0.64$ is depicted by the dotted dash
lines as in Fig.\ref{degen1}.  The oscillation parameters and the
Earth density are the same as those in Fig.\ref{degen1}.
}
\label{degen2}
\end{figure}

\vglue 0.5cm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vglue 3.0cm\hglue -8.8cm
\includegraphics[scale=0.6]{delth13.eps}
\vglue -8.6cm\hglue 8.4cm
\includegraphics[scale=0.6]{tanth23.eps}
\vglue 1.0cm
\caption{
(a) The normalized error at 90\%CL in the reactor
measurements of $\theta_{13}$ for $\sigma_{\text{sys}}$=2\%, 5 t$\cdot$yr
(d.o.f.=2, $|\delta(\sin^22\theta_{13})|\le0.034$), for $\sigma_{\text{sys}}$=0.8\%,
20 t$\cdot$yr (d.o.f.=2, $|\delta(\sin^22\theta_{13})|\le0.015$),
and for $\sigma_{\text{sys}}$=0.8\%, 20 t$\cdot$yr
(d.o.f.=1, $|\delta(\sin^22\theta_{13})|\le0.012$), respectively.
Without the precise information on the value of $|\Delta m^2_{31}|$
from the JHF experiment, the degrees of freedom of the analysis is 2,
but once the the value of $|\Delta m^2_{31}|$ is known from JHF
the degrees of freedom becomes 1.\\
(b) The plot of
$|\delta(\sin^22\theta_{13})|/(\sin^22\theta_{13})_{\text{average}}$
as a function of $\sin^22\theta_{23}$, where
$\delta(\sin^22\theta_{13})\equiv\sin^22\theta'_{13}-\sin^22\theta_{13}$,
$(\sin^22\theta_{13})_{\text{average}}\equiv
(\sin^22\theta'_{13}+\sin^22\theta_{13})/2$,
$\epsilon\equiv\Delta m^2_{21}/\Delta m^2_{31}$
($\epsilon=6.9\times10^{-5}$eV$^2/2.5\times10^{-3}$eV$^2=0.028$
is for the best fit and an extreme case with
$\epsilon=1.9\times10^{-4}$eV$^2/1.6\times10^{-3}$eV$^2=0.12$,
which is allowed at 90\%CL (atmospheric) or 95\%CL
(solar), is also shown for illustration),
and $\sin^22\theta'_{13}$ stands for
a fake solution due to the ($\theta_{13}$, $\theta_{23}$)
degeneracy.
$\sin^22\theta_{23}\ge0.92$ has to be satisfied
due to the constraint from the Super-Kamiokande
atmospheric neutrino data.  In the best fit case with $\epsilon=0.028$,
the contribution from the second term in (\ref{eqn:th23}) is
small and there is little difference between the case with
$\sin^22\theta_{13}=0.03$ and the one with $\sin^22\theta_{13}=0.09$.
If the value of
$\cos^22\theta_{23}$ is large enough, the value of
$|\delta(\sin^22\theta_{13})|/(\sin^22\theta_{13})_{\text{average}}$
increases and lies outside of the normalized error of the reactor
experiment, so the reactor result may resolve the $\theta_{23}$ ambiguity.
}
\label{delth13}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\hglue 2.6cm
\includegraphics[scale=0.8]{env.eps}
\vglue 2.8cm 
\caption
{
Allowed region on the $P$-$\bar{P}$ plane
($P\equiv P(\nu_\mu\rightarrow\nu_e)$,
$\bar{P}\equiv P(\bar{\nu}_\mu\rightarrow\bar{\nu}_e)$)
after an affirmative (a negative) result
is obtained.  The cases a, b, c, d correspond to
~a: $\sin^22\theta_{13}=0.09\pm0.012$,
b: $\sin^22\theta_{13}=0.06\pm0.012$,
c: $\sin^22\theta_{13}=0.03\pm0.012$,
d: $\sin^22\theta_{13}<0.012$, respectively.
The regions bounded by the dashed lines and
dotted lines are for
the normal hierarchy ($\Delta m^2_{31}>0$)
and the inverted hierarchy ($\Delta m^2_{31}<0$),
respectively.
The thickest dotted or dashed lines are for
$\sin^2\theta_{23}=0.5$, the second thickest lines
are for $\sin^2\theta_{23}=0.36$ and the thinnest lines
are for $\sin^2\theta_{23}=0.64$.
}
\label{envelope}
\end{figure}



\end{document}

