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\begin{document}
\title{Effects of neutrino oscillation on supernova neutrino:
inverted mass hierarchy}
\author{Keitaro Takahashi$^{\rm a}$ and Katsuhiko Sato$^{\rm a,b}$\\
{\it $^{\rm a}$Department of Physics, University of Tokyo,
7-3-1 Hongo, Bunkyo,}\\ {\it Tokyo 113-0033, Japan}\\
{\it $^{\rm b}$Research Center for the Early Universe, 
University of Tokyo,}\\
{\it 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan}}
      \maketitle

\baselineskip=16pt

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We study the effects of neutrino oscillation on supernova neutrino
in the case of the inverted mass hierarchy ($m_{3} \ll m_{1} < m_{2}$). 
This is an extended study of our previous study \cite{KT} where all analyses 
are performed with normal mass hierarchy ($m_{1} < m_{2} \ll m_{3}$).
Numerical analysis using a realistic supernova and presupernova model
allow us to discuss quantitatively a possibility to probe neutrino 
oscillation parameters. We show that we can break partly the degeneracy of 
the solar neutrino problem (LMA or SMA) and probe the magnitude of 
$\theta_{13}$ to some extent by the ratios of high-energy events and 
low-energy events at SuperKamiokande and SNO and the presence of the 
Earth effects. Further, if the magnitude of $\theta_{13}$ is known roughly, 
we can identify the mass hierarchy.
The cases of LOW and VO solutions are also analyzed and
we find that we can distinguish LMA and (LOW and VO) by the Earth effects
when the mass hierarchy is inverted or $\theta_{13}$ is small, 
although the value of $\Delta m^{2}_{12}$ ($= 10^{-7} {\rm eV}^{2}$ or
$10^{-9} {\rm eV}^{2}$) cannot be determined statistically significantly.

\noindent
$PACS:$ 14.60.Pq; 14.60.Lm; 96.40.Tv; 97.60.Bw; 

\noindent
$Keywords:$ Neutrino oscillation; Supernovae; neutrino mass hierarchy;

\end{abstract}

\clearpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Neutrino mixing and mass spectrum are the keys to probe new physics
beyond the standard model of particle physics. Some of the neutrino
oscillation parameters have been revealed dramatically by the observation
of the atmospheric neutrino \cite{Fukuda1999} and the solar neutrino
\cite{Fukuda2001,SNO,Bahcall2002,Barger2002Sun,Holanda2002}. 
An upper bound on $\theta_{13}$ has also been
obtained from CHOOZ experiment \cite{Apollonio1999} and a lower bound is
expected to obtained from single and double beta decay experiments
\cite{MinakataSugiyama}. KamLAND experiment is expected to probe LMA 
solution of the solar neutrino problem with high accuracy
\cite{Marfatia2001,Murayama2002}. But there still remain some ambiguities 
in the properties of neutrinos: the mass hierarchy, i.e., normal or
inverted, the solution of the solar neutrino problem and the magnitude
of $\theta_{13}$.

In such present situation, much attention have been paid to another 
neutrino source, supernova (for recent review, see \cite{Raffelt2002,Cei2002}).
This is a completely different system from sun, 
atmosphere, accelerator, and reactor in regard to neutrino energy, 
flavor of produced neutrinos, propagation length and so forth. Then 
neutrino emission from a supernova is expected to give valuable information 
that can not be obtained from neutrinos from other sources.
In fact, pioneering observations of neutrinos from SN1987A
\cite{K2_SN1987a,IMB_SN1987a} contributed significantly to our knowledge of 
the fundamental properties of neutrinos \cite{Arafune,Sato,Goldman}. 
Especially there have been many studies about the implication for the 
mass hierarchy from the observed neutrino events and the inverted
hierarchy is disfavored if $\sin^{2}{\theta_{13}}$ is rather large
($\gtrsim 10^{-4}$) \cite{Jegerlehner1996,LunardiniSmirnov,Minakata2001}.
(See, however, \cite{Barger2002}.)
Here, normal and inverted mass hierarchies are the mass pattern
$m_{3} \gg m_{2} > m_{1}$ and $m_{2} > m_{1} \gg m_{3}$, respectively.
In our notation $\Delta m^{2}_{ij} = m^{2}_{j} - m^{2}_{i}$, 
$\Delta m^{2}_{12}$ and $\Delta m^{2}_{13} \sim \Delta m^{2}_{23}$ are
the mass squared differences which are related with the solutions of the 
solar and atmospheric neutrino problems, respectively.

In this paper, we calculate numerically the effects of neutrino oscillation
on supernova neutrino, extending our previous study \cite{KT} where all 
analyses are performed with normal mass hierarchy, and
investigate the possibility to identify the mass hierarchy and to probe 
the neutrino oscillation parameters ($\theta_{12}, \theta_{13}$) by the 
observation of the neutrinos from the next galactic supernova. 
Although a global solar neutrino analysis favors LMA \cite{SNO2002},
our analysis includes SMA, LOW and VO cases as well as LMA case.
This is because it is important to check the solar neutrino analysis
independently by supernova neutrino. Furthermore, analysis with SMA 
is helpful to understand the dynamics of neutrino oscillation in supernova.
We use the original neutrino spectra from supernova based on a realistic 
supernova model and the density profile of the progenitor star based on a 
realistic presupernova model. The Earth matter effects are also discussed,
which have already been studied in the case of normal hierarchy in 
\cite{KTearth,KT3}.

This paper is organized as follows. In section II we summarize the
properties of supernova neutrino briefly. The method of
analysis is described and the results are shown in section III.
We discuss some ambiguities in the basis of our study and summarize
our results in section IV.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Supernova Neutrino}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Here we summarize the properties of supernova neutrino.
For details, see, for example, \cite{Suzuki}.
Almost all of the binding energy of the neutron star,
\begin{equation}
E_{\rm b} \simeq \frac{G M_{\rm NS}^{2}}{R_{\rm NS}}
\simeq 3 \times 10^{53} {\rm erg} 
\left( \frac{M_{\rm NS}}{M_{\odot}} \right)^{2}
\left( \frac{10 {\rm km}}{R_{\rm NS}} \right),
\end{equation}
is radiated away as neutrinos. Here $M_{\rm NS}$ and $R_{\rm NS}$ are
the mass and radius of the neutron star, respectively.
Neutrino emission from supernova
can be divided into three phases: the neutronization burst, the matter
accretion phase and the cooling of the proto-neutron star. 
In the neutronization burst, only $\nu_{e}$'s
are emitted and their total energy is $\sim 2 \times 10^{51} {\rm erg}$.
On the other hand, in the second and third phases neutrinos and 
antineutrinos of all flavors are emitted and the energy is approximately
equipartitioned between all flavors. 

Neutrinos are trapped in the interior of the  high-density neutron star
and they are emitted from the relatively well defined surface,
neutrinosphere. The nonelectron neutrinos, $\nu_{x} \equiv \nu_{\mu},
\nu_{\tau}, \bar{\nu}_{\mu}, \bar{\nu}_{\tau}$, have smaller opacities,
since their energies are too low for charged-current interactions.
These flavors decouple at higher densities and temperatures than
$\nu_{e}$ and $\bar{\nu}_{e}$. Therefore, they are emitted with higher
average energies. In this paper, we do not discriminate nonelectron neutrinos
(for possible differences between these, see \cite{Akhmedov2002}).
Further, since there are fewer protons than neutrons
in the neutron star, $\bar{\nu}_{e}$'s have smaller opacities and then
higher energies than $\nu_{e}$'s. To summarize, each flavor has different
average energy:
\begin{equation}
\langle E_{\nu_{e}} \rangle < \langle E_{\bar{\nu}_{e}} \rangle < 
\langle E_{\nu_{x}} \rangle.
\end{equation}
These differences are essential in this paper.

We use a realistic model of a collapse-driven supernova
by the Lawrence Livermore group\cite{Wilson1986} to calculate
the neutrino luminosities and energy spectra, as we did in \cite{KT}.
The time-integrated energy spectra and the time evolution of neutrino
fluxes are shown in Fig. \ref{fig:original}\cite{Totani1998}.
As can be seen in Fig. \ref{fig:original} (left), the average energies are
different between flavors:
\begin{equation}
\langle E_{\nu_{e}} \rangle \simeq 13 {\rm MeV}, \;\;
\langle E_{\bar{\nu}_{e}} \rangle \simeq 16 {\rm MeV}, \;\;
\langle E_{\nu_{x}} \rangle \simeq 23 {\rm MeV}.
\label{eq:nu_energy}
\end{equation}
These neutrinos, which are produced in the high dense region of the iron
core, interact with matter before emerging from the supernova. Due to
the nonzero masses and the mixing in vacuum among various neutrino flavors,
flavor conversions can occur in supernova. When the mixing angle is small,
these conversions occur mainly in the resonance layer, whose density is
\begin{equation}
\rho_{\rm res} \simeq 1.4 \times 10^{6} {\rm g/cc}
\left( \frac{\Delta m^{2}}{1 {\rm eV}^{2}} \right)
\left( \frac{10 {\rm MeV}}{E_{\nu}} \right)
\left( \frac{0.5}{Y_{e}} \right)
\cos{2 \theta},
\end{equation}
where $\Delta m^{2}$ is the mass squared difference, $\theta$ is the
mixing angle, $E_{\nu}$ is the neutrino energy, and $Y_{e}$ is the mean 
number of electrons per baryon. Since the supernova core is dense enough, 
there are two resonance points in supernova envelope. One that occurs at 
higher density is called H-resonance and another is called L-resonance.
If the mass hierarchy is normal, both resonances occur in neutrino sector.
On the other hand, if the mass hierarchy is inverted, H-resonance occurs
in antineutrino sector and L-resonance occurs in neutrino sector.
The schematic level crossing diagram for normal and inverted mass hierarchies
are shown in Fig. \ref{fig:level_crossing}.

The dynamics of conversions including large mixing case is determined by the
adiabaticity parameter $\gamma$ \cite{Friedland2001}:
\begin{equation}
\gamma \equiv 
\cos{\theta} \Delta m^{2}  \frac{\left[ \sin^{2}{2 \theta} + 
(\cos{2 \theta} - \xi)^{2} \right]^{1/2}}{\left| d\theta_{\rm m}/dr 
\right|_{\theta_{\rm m} = \pi / 4 + \theta / 2}},
\label{eq:adiabaticity}
\end{equation}
where $\theta_{\rm m}$ is mixing angle in matter and
\begin{equation}
\xi \equiv \frac{2 \sqrt{2} G_{F} n_{e} E_{\nu}}{\Delta m^{2}}
\end{equation}
is a dimensionless matter parameter.
Here $\Delta m^{2}$ and $\theta$ are
\begin{equation}
\theta_{13} \; {\rm and} \; \Delta m^{2}_{13} \; {\rm at \; H-resonance},
\end{equation}
\begin{equation}
\theta_{12} \; {\rm and} \; \Delta m^{2}_{12} \; {\rm at \; L-resonance}.
\end{equation}
When $\gamma \gg 1$, the resonance is adiabatic and 
the fluxes of the two involved flavors are completely exchanged. On the 
contrary, when $\gamma \ll 1$, the resonance is nonadiabatic and the 
conversion between flavors does not occur. For the detail of the dynamics of
the resonance in supernova, see \cite{Dighe2000}.

To calculate the time evolution of neutrino wave functions, we use the
progenitor star density profile calculated numerically by Woosley and
Weaver\cite{Woosley1995}. The progenitor mass was set to $15 M_{\odot}$
and the metallicity was set to be the same as that of the Sun. The 
density profile is shown in Fig. \ref{fig:density}. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Method and Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we describe the method of analysis and show the results.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Conversion Probabilities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the framework of three-flavor neutrino oscillation, the time evolution
equation of the neutrino wave functions can be written as follows:
\begin{equation}
i\frac{d}{dt}\left(
	\begin{array}{ccc}\nu_e\\ \nu_{\mu}\\ \nu_{\tau}
	\end{array}\right)
= H(t)\left(
	\begin{array}{ccc}\nu_e\\ \nu_{\mu}\\ \nu_{\tau}
	\end{array}\right),
\end{equation}
\begin{equation}
H(t)\equiv 
U\left(
	\begin{array}{ccc}
		0 & 0 & 0\\
		0 & \Delta m^2_{12} /2E & 0\\
		0 & 0 & \Delta m^2_{13} /2E
	\end{array}\right)U^{-1} 
 	+\left(
	\begin{array}{ccc}
		A(t) & 0 & 0\\
		0 & 0 & 0\\
		0 & 0 & 0
	\end{array}\right),
\end{equation}
where $A(t)=\sqrt{2}G_{F}n_{e}(t)$, $G_{F}$ is Fermi constant, $n_{e}(t)$
is the electron number density, 
$\Delta m^2_{ij}$ is the mass squared differences, and $E$
is the neutrino energy. In case of antineutrino, the sign of 
$A(t)$ changes.
Here U is a unitary 3 $\times$ 3 mixing matrix in vacuum:
\begin{equation}
U  =  \left(\begin{array}{ccc}
c_{12}c_{13} & s_{12}c_{13} & s_{13}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13} & c_{12}c_{23}-s_{12}s_{23}s_{13} 
& s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13} & -c_{12}s_{23}-s_{12}c_{23}s_{13} 
& c_{23}c_{13}
\end{array}\right)\label{mixing_matrix},
\end{equation}
where $s_{ij} = \sin{\theta_{ij}}, c_{ij} = \cos{\theta_{ij}}$ 
for $i,j=1,2,3 (i<j)$.
We have here put the CP phase equal to zero in the CKM matrix.

By solving numerically these equations along the density profile in
Fig. \ref{fig:density}, we obtain conversion probabilities 
$P(\alpha \rightarrow \beta)$, i.e., probability that $\nu_{\alpha}$
at the center of the supernova becomes $\nu_{\beta}$ at the surface
of the progenitor star.

In our previous paper\cite{KT}, we assumed the normal mass hierarchy and
took four models for neutrino oscillation parameters. They are shown in 
Table \ref{table:parameter}. Here LMA-L means that $\theta_{12}$ and
$\Delta m^{2}_{12}$ are taken to be the LMA solution of the solar neutrino
problem and $\theta_{13}$ is large. As to $\theta_{23}$ and 
$\Delta m^{2}_{13}$, we take the solution of the atmospheric neutrino 
problem for all models. In this paper we take four more models 
which correspond to the inverted mass hierarchy cases of each model in 
\cite{KT}. We call eight models like normal-LMA-L, inverted-SMA-S and so on.
The nomenclature is obvious. In our notation, 
$\Delta m^{2}_{ij} = m^{2}_{j} - m^{2}_{i}$ so that $\Delta m^{2}_{13} > 0$ 
in the normal hierarchy case and $\Delta m^{2}_{13} < 0$ in the inverted 
hierarchy case. Therefore, normal-LMA-L and inverted-LMA-L are different
only in the sign of $\Delta m^{2}_{13}$.

We show in Fig. \ref{fig:conv_prob} demonstrations of conversion 
probabilities in the inverted hierarchy cases. The left figure is the 
time evolution of $P(e \rightarrow e)$ and the right figure is that of 
$P(\bar{e} \rightarrow \bar{e})$. Four lines for the same model correspond 
to the neutrino of energy, 5 MeV, 10 MeV, 40 MeV, and 70 MeV, respectively.
As can be seen, for example in the case of inv-SMA-L, the L-resonance in
neutrino sector occurs nonadiabatically at the He layer. The H-resonance
in antineutrino sector occurs completely adiabatically at the O+Ne+Mg or
O+C layer and $\bar{\nu}_{e}$ and $\bar{\nu}_{x}$ are completely exchanged.
For conversion probabilities in the case of the normal hierarchy, see 
\cite{KT}.

More likely solutions of the solar neutrino problem than SMA
are LOW and VO solutions. We also calculate conversion probabilities 
with parameter sets which correspond to the LOW and VO solutions of 
solar neutrino problem:
\begin{eqnarray}
({\rm LOW}) & \Delta m^{2}_{\odot} \approx
             10^{-7} {\rm eV}^2, \;\; &
             \sin^{2} 2 \theta_{\odot}  \approx 1.0 \\
({\rm VO}) & \Delta m^{2}_{\odot} \approx
             10^{-9}  {\rm eV}^2, \;\; &
             \sin^{2} 2 \theta_{\odot}  \approx  1.0 
\end{eqnarray}
In these cases, the magnitude of $\theta_{12}$ is almost the same
as LMA. But since $\Delta m^2_{12}$ is much smaller than LMA,
L resonance occurs less adiabatically and at further radius than LMA.
From (\ref{eq:adiabaticity}), we find that the adiabaticity of L resonance
is the largest in case of LMA and the smallest in case of SMA.
So hereafter most of the results in this paper are shown only in 
two extreme cases, LMA and SMA. We discuss at the end of section
\ref{subsection:distinction} and \ref{subsection:earth} the possibility 
of distinguishing between all four solutions of the solar neutrino problem.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Event Rates}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

After obtaining the conversion probabilities, the neutrino fluxes at the Earth
are obtained by multiplying the conversion probabilities by the original
spectra and the distance factor $1/4 \pi d^{2}$. Here we take 10 kpc
for the distance $d$ between the Earth and the supernova.
Further, by multiplying these fluxes by the cross sections of the detection
interactions, the detector volume and the detector efficiency, we obtain
the event rates at the detectors. Here we consider two detectors: 
SuperKamiokande and SNO.

SuperKamiokande is a water Cherenkov detector with 32,000 ton pure
water based at Kamioka in Japan. The relevant interactions of
neutrinos with water are as follows:
\begin{eqnarray}
\bar{\nu_e} +p & \rightarrow & n + e^+ \quad(\rm{CC}) 
\label{interaction:anuep}\\
\nu_e + e^- & \rightarrow & \nu_e + e^- \quad(\rm{CC \quad and \quad NC})\\
\bar{\nu_e} + e^- & \rightarrow & \bar{\nu_e} + e^- 
\quad(\rm{CC \quad and \quad NC})\\
\nu_x + e^- & \rightarrow & \nu_x + e^- \quad(\rm{NC})\\
\nu_e + {\rm O} & \rightarrow & {\rm F} + e^- \quad(\rm{CC})\\
\bar{\nu_e} + {\rm O} & \rightarrow & {\rm N} + e^+ \quad(\rm{CC})
\end{eqnarray}
where CC and NC stand for charged current and neutral current
interactions, respectively. 

SuperKamiokande is not in operation now because of the unfortunate accident
\cite{SKhomepage}. However it is expected to be repaired and will work
by the end of 2002 with lower performance. Since the number of 
photomultiplier tubes reduce to the half of the number before the accident,
the energy threshold will become larger (5 $\rightarrow \sim$ 8 MeV) and the 
energy resolution will become $\sqrt{2}$ times worse.
But these lower performance does not matter in this work because
the number of events between 5 MeV and 8 MeV is much smaller than the
total event number and the energy resolution is not important in our
analysis as you will see later. The angular resolution will also become
worse but it plays no role in our analysis.

For the cross sections of the detection interactions, we refer to 
\cite{Totsuka1992}. We assume the efficiency is 100 $\%$ above 8 MeV.
In the detection interactions at SuperKamiokande, $\bar{\nu}_{e}p$ CC
interaction has the largest contribution ($\sim 95 \%$) to the
total events. Hence the energy spectrum detected at SK is almost the
same as the spectrum of $\bar{\nu}_{e}p$ CC interaction only.
Fig. \ref{fig:SK} show the time-integrated energy spectra (left) and the time
evolution of the number of neutrino events (right). In these figures,
only $\bar{\nu}_{e}p$ CC interaction is taken into account. Event numbers of
each interaction are shown in Table \ref{table:event_SK}.
In this table, the contribution from neutronization burst phase is also shown.
Here the neutronization phase means the period from 41msec to
48msec after the bounce.

Sudbury Neutrino Observatory(SNO) is a water $\check{\rm{C}}$herenkov
detector based at Sudbury, Ontario. SNO is unique in its use of 1000
tons of heavy water, by which both the charged-current and
neutral-current interactions can be detected. The interactions of
neutrinos with heavy water are as follows,
\begin{eqnarray}
\nu_e + d & \rightarrow & p + p + e^-\quad(\rm{CC})\label{eq:SNO_CC_nue}\\
\bar{\nu_e} + d & \rightarrow & n + n + e^+\quad(\rm{CC})
	\label{eq:SNO_CC_anue}\\
\nu_x + d & \rightarrow & n + p + \nu_x\quad(\rm{NC})\\
\bar{\nu_x} + d & \rightarrow & n + p + \bar{\nu_x}\quad(\rm{NC})
\end{eqnarray}
We take only (\ref{eq:SNO_CC_nue}) and (\ref{eq:SNO_CC_anue}) into account
since the numbers of the NC events are not affected by the neutrino
oscillation. (\ref{eq:SNO_CC_nue}) and (\ref{eq:SNO_CC_anue}) are detected 
when the electron or the positron emit $\check{\rm{C}}$herenkov light and
can be clearly distinguished from CC events. It is also possible to
distinguish between (\ref{eq:SNO_CC_nue}) and (\ref{eq:SNO_CC_anue})
by the detection of the neutron-capture signal. However the capture
efficiency is only $24 \%$ now. In order to enhance the neutron capture
efficiency ${}^{35}$Cl will be added to the D${}_{2}$O  in the form of
NaCl for the second ``salt'' running phase\cite{Virtue2001}.
This will raise the neutron capture efficiency to approximately $83 \%$.

For the cross sections of (\ref{eq:SNO_CC_nue}) and (\ref{eq:SNO_CC_anue}),
we refer to \cite{Ying1989}. The detection efficiency is set to be one
because we have no information about it.
Fig. \ref{fig:SNO} and \ref{fig:SNO_anti} show the time-integrated energy 
spectra (left) and the time evolution of the number of neutrino events (right)
for (\ref{eq:SNO_CC_nue}) and (\ref{eq:SNO_CC_anue}), respectively. 
Event numbers of each interaction are shown in Table \ref{table:event_SNO}.
In this table, the contribution from neutronization burst phase is also shown.
Here the neutronization phase means the period from 41msec to
48msec after the bounce.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Distinction between Models\label{subsection:distinction}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In general neutrino oscillation makes the $\nu_{e}$ and $\bar{\nu}_{e}$
spectra harder, since the original average energies of 
$\nu_{e}$ and $\bar{\nu}_{e}$ are smaller than those of $\nu_{x}$.
In other words neutrino oscillation produces high energy
$\nu_{e}$ and $\bar{\nu}_{e}$ from $\nu_{x}$. As a result, the high-energy
events increase and the low-energy events decrease. The boundary between
high energy and low energy is around 25 MeV. Note that how much
these increase and decrease are depends on
the neutrino oscillation parameters, as can be seen in Fig. \ref{fig:SK},
\ref{fig:SNO} and \ref{fig:SNO_anti}. This feature can be used as a criterion
of the magnitude of the neutrino oscillation effects. We define
the following ratios of high-energy to low-energy events at both
detectors:
\begin{equation}
R_{\rm SK} \equiv \frac{\makebox{number of events at } 25<E<70\rm{MeV}}
{\makebox{number of events at } 5<E<25\rm{MeV}},
\end{equation}
\begin{equation}
R_{\rm SNO} \equiv \frac{\makebox{number of events at } 25<E<70\rm{MeV}}
{\makebox{number of events at } 5<E<25\rm{MeV}}.
\end{equation}
The plots of $R_{\rm SK}$ vs $R_{\rm SNO}$ are shown in Fig. \ref{fig:ratio}.
Here ``both'' means both normal and inverted hierarchy cases.
In the left figure, we consider only $\nu_{e} d$ CC events at SNO for
$R_{\rm SNO}$ assuming $\nu_{e} d$ CC event and $\bar{\nu}_{e} d$ CC event
can be distinguished completely. On the other hand, in the right figure
we assume that $\nu_{e} d$ CC event and $\bar{\nu}_{e} d$ CC event
can not be distinguished at all and we sum $\nu_{e} d$ CC events 
and $\bar{\nu}_{e} d$ CC events for $R_{\rm SNO}$. The error bars
represent the statistical errors.

Note that $\nu_{e}$ flux and $\bar{\nu}_{e}$ flux have essentially
different information about the neutrino oscillation parameters.
For example, inverted-LMA-L and inverted-LMA-S are distinguishable
from $\bar{\nu}_{e}$ events but are not from $\nu_{e}$ events.
So it is more effective to distinguish between models if $\nu_{e} d$
CC events and $\bar{\nu}_{e} d$ CC events at SNO can be distinguished 
perfectly. This can be clearly seen in Fig. \ref{fig:ratio}. In the left figure
it is easier to distinguish between normal-LMA-L and (normal-LMA-S and
inverted-LMA-S) and between inverted-LMA-L and inverted SMA-L than
in the right figure. But even in the left figure, it may be difficult
to distinguish between normal-LMA-L and (normal-LMA-S and
inverted-LMA-S) considering some ambiguities discussed in the next
section. But as we discuss in the next subsection, these degeneracy
can be broken by the Earth effects.

As for the cases of LOW and VO, L resonance becomes less adiabatic
than LMA, since $\Delta m_{12}^{2}$ is much smaller. In fact,
$\Delta m_{12}^{2}$ is still large enough for L resonance to be 
adiabatic in case of LOW. On the other hand, in case of VO, 
L resonance is slightly nonadiabatic. This can be seen in 
Fig. \ref{fig:ratio}. Unfortunately, the difference between LMA
and VO is not large enough to be distinguished statistically.
But as we discuss in the next subsection, these degeneracy
can also be broken partly by the Earth effects.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Earth Effects\label{subsection:earth}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Neutrinos go through the Earth in general before they reach the detector.
The matter effects inside the Earth can change the neutrino spectra again.
The numerical process is the same as described before except that
the density profile of the Earth is required. We use a realistic Earth
density profile \cite{Dziewonski1981}.

We found that the Earth effects are seen significantly in the $\nu_{e}$ 
spectrum of the model normal-LMA-L, normal-LMA-S and inverted-LMA-S and
in the $\bar{\nu}_{e}$ spectrum of the model normal-LMA-S, inverted-LMA-L
and inverted-LMA-S. The distorted spectra due to the Earth effects are shown
in Fig. \ref{fig:Earth}. The left figure is the $\nu_{e}$ spectra at SNO
and the right figure is the $\bar{\nu}_{e}$ spectra at SuperKamiokande.
Neutrino spectra in the case that the nadir angle is $30^{\circ}$ and 
$70^{\circ}$ are shown with that in absence of the Earth effects.
As can be seen, some dips appear due to the Earth effects, which
was discussed in detail in \cite{KTearth}.
These dips are the signal of the Earth effects. Although it appears that
dips in the spectra at SuperKamiokande is too small to identify,
the large event number at SuperKamiokande will make it possible \cite{KT3}.
By observing the dips in spectra due to the Earth effects, we will be able 
to distinguish between models that may not be able to be distinguished 
clearly by Fig. \ref{fig:ratio}:
between normal-LMA-L and (normal-LMA-S and inverted-LMA-S), and between
inverted-LMA-L and inverted-SMA-L. Combined with the discussion of the
previous subsection, if the mass hierarchy is known (normal or inverted), 
we can distinguish between all four models. If it is known, instead, 
that $\theta_{13}$ is enough large for the H-resonance to be adiabatic, 
we can distinguish between the normal and inverted mass hierarchy. 
Degeneracies exist only between normal-LMA-S and inverted-LMA-S, 
and between normal-SMA-S and inverted-SMA-S. This degeneracies comes from
the fact that the mass hierarchy determines which sector (neutrino or 
anti-neutrino) the H-resonance occurs in and that the H-resonance results
in no observable effects when $\theta_{13}$ is small enough.

In cases of LOW and VO, there is no Earth effect in both channel.
Hence we can distinguish LMA and (LOW and VO) by the Earth effects
when the mass hierarchy is inverted or $\theta_{13}$ is small, 
although the value of $\Delta m^{2}_{12}$ ($= 10^{-7} {\rm eV}^{2}$ or
$10^{-9} {\rm eV}^{2}$) cannot be determined.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion and Conclusion\label{section:discussion}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are some ambiguities besides the statistical errors.
The first is the direction of the supernova. If the supernova can
be observed optically, the direction can be known with enough accuracy.
But if the supernova is at the Galactic Center, it might be hidden by 
the large amount of gas and could not seen optically. Pointing by the
electron scattering events of the supernova neutrino is studied
by several authors \cite{Beacom1999,Ando2001} and the accuracy is
expected to be $\sim 7^{\circ}$. More detailed analyses of the Earth 
effects considering the locations of the detectors were studied in 
\cite{KT3,Lunardini2001}.

Our results are strongly dependent on the original neutrino spectra.
If the differences of the average energies of each flavor are smaller,
it is harder to distinguish between the models. Estimation of uncertainties of
the numerical supernova model is hard because there is currently
no successful simulation other than that by the Lawrence Livermore group
\cite{Wilson1986}. But there are some studies about the temperatures 
of the produced neutrinos. According to them, they are typically
\cite{Janka1992},
\begin{equation}
\langle E_{\nu_{e}} \rangle = 10 - 12 {\rm MeV}, \;\;
\langle E_{\bar{\nu}_{e}} \rangle = 14 - 17 {\rm MeV}, \;\;
\langle E_{\nu_{x}} \rangle = 24 - 27 {\rm MeV}.
\label{eq:typical_nu_energy}
\end{equation}
The neutrino average energies of our original neutrino fluxes 
(\ref{eq:nu_energy}) seem close to these typical values. But
a recent study of $\nu_{x}$ spectra formation including energy exchange
by nucleon recoil \cite{Raffelt2001} predicts $\nu_{x}$ temperature 
10-20\% smaller than typical value (\ref{eq:typical_nu_energy}). Analysis 
taking these uncertainties into account will be performed in our future work.
Furthermore, supernova neutrino spectra depend on the mass of 
the progenitor star. Although the dependence is shown to be rather 
small \cite{MayleD,Mayle1987}, analysis taking this dependence into account
will also be performed in our future work.

We studied the effects of neutrino oscillation on supernova neutrino
in the case of the inverted mass hierarchy as well as the normal
mass hierarchy which we have already analyzed before \cite{KT}.
Numerical analysis using a realistic supernova and presupernova model
allowed us to discuss quantitatively a possibility to probe neutrino 
oscillation parameters. We showed that we can distinguish between
most of the 8+1 models considered here. Degeneracies exist only 
between normal-LMA-S and inverted-LMA-S, and between normal-SMA-S and
inverted-SMA-S. 

When the solution of the solar neutrino problem is LMA, as implied by recent
studies, information about $\theta_{13}$ is obtained, which is rather
hard to be obtained from other experiments (but see, \cite{MinakataSugiyama}).
Also if $\theta_{13}$ is large, the mass hierarchy can be determined.

As to the other solutions of the solar neutrino problem, LOW and VO,
which were not included in our previous study \cite{KT},
we can distinguish LMA and (LOW and VO) by the Earth effects
when the mass hierarchy is inverted or $\theta_{13}$ is small, 
although the value of $\Delta m^{2}_{12}$ ($= 10^{-7} {\rm eV}^{2}$ or
$10^{-9} {\rm eV}^{2}$) cannot be determined.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This work was supported in part by Grants-in-Aid for Scientific Research 
provided by the Ministry of Education, Science and Culture of Japan through 
Research Grant No.4.







\begin{thebibliography}{99}

\bibitem{KT}
K. Takahashi, M. Watanabe, K. Sato and T. Totani, Phys. Rev. D
{\bf 64} (2001) 093004, .

\bibitem{Fukuda1999}
Y. Fukuda et al., Phys. Rev. Lett. 82 (1999) 2644.

\bibitem{Fukuda2001}
S. Fukuda et al., Phys. Rev. Lett. {\bf 86} (2001) 5656.

\bibitem{SNO}
SNO Collaboration, Phys.Rev.Lett. 87 (2001) 071301.

\bibitem{Bahcall2002}
J. N. Bahcall, M. C. Gonzalez-Garcia and C. Pena-Garay, .

\bibitem{Barger2002Sun}
V. Barger, D. Marfatia, K. Whisnant and B. P. Wood, .

\bibitem{Holanda2002}
P. C. de Holanda and A. Yu. Smirnov, .

\bibitem{Apollonio1999}
M. Apollonio et al., Phys. Lett. B {\bf 466} (1999) 415.

\bibitem{MinakataSugiyama}
H. Minakata and H. Sugiyama,
Phys. Lett. B {\bf 526} (2002) 335, .

\bibitem{Marfatia2001}
V. Barger, Danny Marfatia and Benjamin P. Wood,
Phys. Lett. B {\bf 498} (2001) 53, .

\bibitem{Murayama2002}
H. Murayama and A. Pierce, Phys. Rev. {\bf D 65} (2002) 013012. 

\bibitem{Raffelt2002}
G. G. Raffelt, .

\bibitem{Cei2002}
F. Cei, .

\bibitem{K2_SN1987a} 
K. Hirata et al., {Phys. Rev. Lett. {\bf 58}, 1490 (1987)}.

\bibitem{IMB_SN1987a} R. M. Bionta et al.,
	{Phys. Rev. Lett. {\bf 58}, 1494 (1987)}.

\bibitem{Arafune} J. Arafune and M. Fukugita, 
        {Phys. Rev. Lett. {\bf59}, 367 (1987)}.

\bibitem{Sato}K. Sato and H. Suzuki, 
        {Phys. Rev. Lett. {\bf 58}, 2722 (1987)}.

\bibitem{Goldman}I. Goldman et al., 
        {Phys. Rev. Lett. {\bf 60}, 1789 (1988)}.

\bibitem{Jegerlehner1996}
B. Jegerlehner, F. Neubig and G. Raffelt, Phys. Rev. D {\bf 54} (1996) 1194.

\bibitem{LunardiniSmirnov}
C. Lunardini and A. Yu. Smirnov, Phys. Rev. D {\bf 63} (2001) 073009.

\bibitem{Minakata2001}
H. Minakata and H. Nunokawa, Phys. Lett. B {\bf 504} 301 (2001).

\bibitem{Barger2002}
V. Barger, D. Marfatia and B. P. Wood, .

\bibitem{SNO2002}
SNO Collaboration, .

\bibitem{KTearth}
K. Takahashi, M. Watanabe and K. Sato, Phys. Lett. B {\bf 510} (2001) 189,
.

\bibitem{KT3}
K. Takahashi and K. Sato, Phys. Rev. D, in press, .

\bibitem{Suzuki} H. Suzuki: Supernova Neutrino
in {\em Physics and Astrophysics of Neutrino}, edited by M. Fukugita 
and A. Suzuki (Springer-Verlag, Tokyo, 1994).

\bibitem{Akhmedov2002}
E. Kh. Akhmedov, C. Lunardini and A. Yu. Smirnov, .

\bibitem{Wilson1986}J. R. Wilson, R. Mayle, S. Woosley, T. Weaver,
{Ann. NY Acad. Sci. {\bf 470}, 267 (1986)}. 

\bibitem{Totani1998}T. Totani, K. Sato, H. E. Dalhed and J. R. Wilson, 
{Astrophys. J. {\bf 496}, 216 (1998)}.

\bibitem{Friedland2001}
A. Friedland, Phys. Rev. D {\bf 64} (2001) 013008.

\bibitem{Dighe2000}A. S. Dighe and A. Yu. Smirnov, 
{Phys. Rev. D {\bf 62}, 033007 (2000)}.

\bibitem{Woosley1995}S. E. Woosley and T. A. Weaver, 
{ApJ. Suppl. {\bf 101}, 181 (1995)}.

\bibitem{SKhomepage}
See, http://www-sk.icrr.u-tokyo.ac.jp/index.html.

\bibitem{Totsuka1992}
Y. Totsuka, {Rep. Prog. Phys. {\bf 55}, 377 (1992)};
K. Nakamura, T. Kajita and A. Suzuki, {\em Kamiokande},
in {\em Physics and Astrophysics of Neutrino}, edited by M.Fukugita 
and A.Suzuki (Springer-Verlag, Tokyo, 1994).

\bibitem{Virtue2001}
C. J. Virtue (SNO Collaboration), .

\bibitem{Ying1989}
S. Ying, W. C. Haxton and E. M. Henley, {Phys. Rev. D {\bf 40}, 3211 (1989)}.

\bibitem{Dziewonski1981}
A. M. Dziewonski and D. L. Anderson, Phys. Earth. Planet Inter. {\bf 25}
(1981) 297.

\bibitem{Beacom1999} 
J. F. Beacom and P. Vogel,
Phys.Rev. {\bf D 60} (1999) 033007.

\bibitem{Ando2001} 
S. Ando and K. Sato, .

\bibitem{Lunardini2001}
C. Lunardini and A. Yu. Smirnov, .

\bibitem{Janka1992}
H. -T. Janka, in: F. Giovannelli and G. Mannocchi (eds.),
Proc. Vulcano Workshop 1992 {\it Frontier Objects in Astrophysics
and Particle Physics}, Conf. Proc. Vol. 40 (Soc. Ital. Fis).

\bibitem{Raffelt2001}
G. G. Raffelt, Astrophys. J. {\bf 561} (2001) 890.

\bibitem{MayleD} 
R. Mayle, Ph. D. Thesis, University of California (1987).

\bibitem{Mayle1987}               
R. Mayle, J. R. Wilson, and D. N. Schramm, Astrophys. J. {\bf 318} (1987) 288.
\end{thebibliography}




\clearpage


\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{original_spe.eps}
\epsfxsize=3.5in
\epsffile{original_time.eps}
\end{center}
\caption{The original neutrino energy spectra and the time evolution of 
the original neutrino number luminosity \cite{Totani1998}.
\label{fig:original}}
\end{figure}



\begin{figure}
\begin{center}
\epsfxsize=7in
\epsffile{resonance1.eps}
\end{center}
\caption{Schematic level crossing diagram for normal (left) and 
inverted (right) mass hierarchies. The circles shows the resonance points.
\label{fig:level_crossing}}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{density.eps}
\end{center}
\caption{Density profile of the presupernova star model used in the
paper\cite{Woosley1995}. The progenitor mass is set to be $15 M_{\odot}$.
\label{fig:density}}
\end{figure}




\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{conv_prob.eps}
\epsfxsize=3.5in
\epsffile{conv_prob_anti.eps}
\end{center}
\caption{Time evolution of conversion probabilities $P(e \rightarrow e)$ 
and $P(\bar{e} \rightarrow \bar{e})$ in the inverted hierarchy cases.
Four lines of the same marking correspond to neutrino energy, 5MeV, 
10MeV, 40MeV, and 70MeV, respectively. For the corresponding figures
in the case of the normal mass hierarchy, see \cite{KT}.
\label{fig:conv_prob}}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{SK_spectrum.eps}
\epsfxsize=3.5in
\epsffile{SK_time.eps}
\end{center}
\caption{The time-integrated energy spectra (left) and the time
evolution of the number of neutrino events (right) at SuperKamiokande. 
In these figures, only $\bar{\nu}_{e}p$ CC interaction is taken into account.
The ``both-'' means both normal and inverted hierarchies.
\label{fig:SK}}
\end{figure}



\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{SNO_spectrum.eps}
\epsfxsize=3.5in
\epsffile{SNO_time.eps}
\end{center}
\caption{The time-integrated energy spectra (left) and the time
evolution of the number of neutrino events (right) at SNO. In these figures,
only $\nu_{e}d$ CC interaction is taken into account.
The ``both-'' means both normal and inverted hierarchies.
\label{fig:SNO}}
\end{figure}




\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{SNO_anti_spectrum.eps}
\epsfxsize=3.5in
\epsffile{SNO_anti_time.eps}
\end{center}
\caption{The time-integrated energy spectra (left) and the time
evolution of the number of neutrino events (right) at SNO. In these figures,
only $\bar{\nu}_{e}d$ CC interaction is taken into account.
The ``both-'' means both normal and inverted hierarchies.
\label{fig:SNO_anti}}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{ratio.eps}
\epsfxsize=3.5in
\epsffile{ratio_sum.eps}
\end{center}
\caption{Plots of $R_{\rm SK}$ vs $R_{\rm SNO}$.
In the left figure, only $\nu_{e} d$ CC events at SNO are considered for
$R_{\rm SNO}$. In the right figure we sum $\nu_{e} d$ CC events 
and $\bar{\nu}_{e} d$ CC events for $R_{\rm SNO}$. The error bars
represent the statistical errors and some of them are slightly shifted
to be easy to see. Here LOW and VO means $\theta_{12}$ is
the same as LMA and $\Delta m^{2}_{12} = 10^{-7},10^{-9} {\rm eV}^{2}$,
respectively.
The ``both-'' means both normal and inverted hierarchies.
\label{fig:ratio}}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3.5in
\epsffile{SNO_angle2.eps}
\epsfxsize=3.5in
\epsffile{SK_angle2.eps}
\end{center}
\caption{Distorted spectra due to the Earth effects. The left figure 
is the $\nu_{e}$ spectra at SNO and the right figure is the 
$\bar{\nu}_{e}$ spectra at SuperKamiokande. Neutrino spectra in the case that
the nadir angle is $30^{\circ}$ and $70^{\circ}$ are shown with that
in absence of the Earth effects.
The ``both-'' means both normal and inverted hierarchies.
\label{fig:Earth}}
\end{figure}



\clearpage

\small

\begin{table*}
\caption{Sets of mixing parameter for calculation
	\label{table:parameter}}
\begin{center}
\begin{tabular}{ccccccc}
model  & $\sin^{2} 2 \theta_{12}$ & $\sin^{2} 2 \theta_{23}$ & $\sin^{2} 2 \theta_{13}$ 
& $\Delta m_{12}^{2}({\rm eV}^{2})$  & $|\Delta m_{13}^{2}| ({\rm eV}^{2})$ 
& $\nu_{\odot}$ problem  \\ \hline 
LMA-L &  0.87  & 1.0 & 0.043 & $7.0 \times 10^{-5}$ & $3.2 \times 10^{-3}$ & LMA \\ 
LMA-S &  0.87  & 1.0 & $1.0 \times 10^{-6}$ & $7.0 \times 10^{-5}$ & $3.2 \times 10^{-3}$ 
& LMA \\   
SMA-L &  $5.0 \times 10^{-3}$  & 1.0 & 0.043  & $6.0 \times 10^{-6}$ & $3.2 \times 10^{-3}$ 
& SMA \\ 
SMA-S &  $5.0 \times 10^{-3}$  & 1.0 & $1.0 \times 10^{-6}$ & $6.0 \times 10^{-6}$ & $3.2 \times 10^{-3}$
& SMA \\ 
\end{tabular}
\end{center}
\end{table*}

\begin{table}
\caption{Number of events at SuperKamiokande
	\label{table:event_SK}}
\begin{center}
\begin{tabular}{c|cccc|cccc|c}
hierarchy & normal & & & & inverted & & & & \\
model & LMA-L & LMA-S & SMA-L & SMA-S & LMA-L & LMA-S & SMA-L & SMA-S 
& no osc\\ \hline
$\bar{\nu_e}p$ & 9459 & 9427 & 8101 & 7967 & 12269 & 9441 & 12270 & 7982  & 8036 \\
$\nu_ee^{-}$ & 186 & 171 & 189 & 131 & 171 & 171 & 131 & 130 &  132\\
$\bar{\nu_e}e^{-}$& 46 & 46 & 41 & 42 & 56 & 46 & 56 & 41 & 42\\
$\nu_{\mu}e^{-}$ & 25 & 26 & 25 & 30 & 27 & 26 & 29 & 30 & 30 \\
$\bar{\nu_{\mu}}e^{-}$ & 24 & 23 & 24 & 24 & 12 & 23 & 12 & 24 & 24 \\
$\nu_{\tau}e^{-}$ & 25 & 26 & 25 & 30 & 26 & 26 & 30 & 30 & 30 \\
$\bar{\nu_{\tau}}e^{-}$& 24 & 23 & 24 & 24 & 12 & 23 & 12 & 24 & 24 \\
$O\nu_e$ & 297 & 214 & 297 & 108 & 215 & 214 & 109 & 106 & 31 \\
$O\bar{\nu_e}$ & 160 & 158 & 95 & 92 & 296 & 159 & 296 & 93 & 92 \\
\hline
total & 10245 & 10114 & 8822 & 8447 & 13084 & 10129 & 12945 & 8460 & 8441  \\ 
\begin{tabular}{c}neutronization \\ burst \end{tabular}
& 15.7 & 16.7 & 9.0 & 10.1 & 20.1 & 16.7 & 20.3 & 10.1 & 12.4
\end{tabular}
\end{center}
\end{table}


\begin{table}
\caption{Number of events (CC) at SNO
	\label{table:event_SNO}}
\begin{center}
\begin{tabular}{c|cccc|cccc|c}
hierarchy & normal & & & & inverted & & & & \\
model & LMA-L & LMA-S & SMA-L & SMA-S & LMA-L & LMA-S & SMA-L & SMA-S
& no osc \\ \hline
$\nu_e d {\rm (CC)}$ & 237 & 185 & 237 & 111 & 185 & 185 & 112 & 130 & 68 \\ 
$\bar{\nu_e}d {\rm (CC)}$ & 118 & 117 & 84 & 82 & 190 & 118 & 190 & 82 & 82 \\
\hline 
total & 355 & 302 & 321 & 193 & 375 & 303 & 302 & 212 & 150 \\
\begin{tabular}{c}neutronization \\ burst \end{tabular}
& 0.6 & 1.1 & 0.5 & 1.1 & 1.1 & 1.1 & 1.1 & 1.1 & 2.1
\end{tabular}
\end{center}
\end{table}


\end{document}
