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\begin{center}
{\large\bf{Effects of Neutrino Oscillation\\
on the Supernova Neutrino Spectrum }}
~\\
~\\
Keitaro Takahashi$^*$, Mariko Watanabe$^*$ and Katsuhiko Sato$^{*,**}$\\
~\\
   $^*$Department of Physics, School of Science,
    the University of Tokyo,\\
    Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan\\
   $^{**}$Research Center for the Early Universe, School of Science,
    the University of Tokyo,\\
     Hongo 7-3-1, Bunkyo-Ku, Tokyo 113-0033, Japan	\\
~\\
and
~\\
Tomonori Totani   \\
~\\
   Theory Division, National Astronomical Observatory,\\
   Mitaka, Tokyo 181-8588, Japan
\end{center}

%\date{\today}

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\begin{abstract}
The effects of three-flavor neutrino oscillation on the supernova
neutrino spectrum are studied. 
We calculate the expected event rate and energy spectra, and their 
time evolution at the Superkamiokande (SK) 
and the Sudbury Neutrino Observatory (SNO), by
using a realistic neutrino burst model based on numerical
simulations of supernova explosions.
We also employ a realistic density profile based on a presupernova model
for the calculation of neutrino conversion 
probability in supernova envelopes. These realistic models and
numerical calculations allow us to quantitatively estimate the effects 
of neutrino oscillation in a more realistic way than previous studies.
We then found that the degeneracy of the solutions of the solar
neutrino problem can be broken by the combination of the
SK and SNO detections of a future Galactic supernova. 
\end{abstract}
\vskip2pc]

\vskip 1cm 

\section{Introduction}

\label{sec:Introduction}

Neutrinos are the most mysterious particle in the 
standard model of particle physics, and the only elementary particles
showing evidence for new physics beyond the standard model.
While the evidence for the existence of neutrino
oscillations from solar \cite{SKsolar,solar} 
and atmospheric neutrino \cite{SKatm} data
is rather convincing now, the values of the mass
squared differences and mixing angles are not
firmly established. For the observed $\nu_{e}$ suppression
of solar neutrinos, for example, four solutions are still
possible: large mixing angle (LMA), small mixing angle (SMA), 
low $\Delta m^{2}$ (LOW), and vacuum oscillation (VO). For $\theta_{13}$,
the mixing angle between mass eigenstate $\nu_{1}, \nu_{3}$,
only upper bound is known from reactor experiment \cite{CHOOZ}
and combined three generation analysis \cite{Fogli}.
Also the nature of neutrino mass hierarchy (normal or
inverted) is still a matter of controversy.

There is another neutrino source: supernovae.
This is a completely different system from 
solar, atmospheric, accelerator, and reactor neutrinos 
in regard to neutrino energy and 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.

Neutrino astrophysics entered a new phase when neutrinos
from SN1987A in the Large Magellanic Cloud were detected
by the Kamiokande \cite{K2_SN1987a} and 
IMB \cite{IMB_SN1987a}. These pioneering observations
contributed significantly to our knowledge of the
fundamental properties of neutrinos 
\cite{Arafune,Minakata,Smirnov1987A,Sato,Goldman} as well as our
understanding of the mechanisms involved in a collapse-driven 
supernova. However, event numbers at the Kamiokande
and IMB, 11 and 8 events respectively, are too
small to set statistically robust constraints.

The next Galactic supernova will be
even more valuable because of the abundance of neutrino
events produced by a closer source and detected by
new neutrino detectors which are now available. 
For example, Kamiokande has been upgraded into SuperKamiokande(SK),
whose volume is about 15 times larger than the old one.
Sudbury Neutrino Observatory (SNO), which is unique in 
its use of heavy water, has already been in operation
\cite{SNO}.

There have been some studies on future supernova neutrino
detection taking neutrino oscillation into account. 
Dighe and Smirnov \cite{DigheSmirnov} estimated qualitatively the
effects of neutrino oscillation in a collapse-driven
supernova on the neutronization peak, the distortion of
energy spectra, and the Earth matter effects. They 
concluded that it is possible to identify the solar neutrino
solution and to probe the mixing angle $\theta_{13}$.
Dutta et al. \cite{Dutta} showed numerically that the events 
involving oxygen targets increase dramatically when there 
is neutrino mixing.

In this paper, we calculate numerically the effects of 
three flavor oscillation on the supernova neutrino spectra,
taking into account the constraints on the neutrino mixing
and masses imposed by solutions consistent with the
solar and atmospheric neutrino problems.
For the original neutrino flux and density profile of
the supernova, which were not considered accurately in previous
studies, we use ones which are
based on the realistic numerical presupernova and supernova models.
We then calculate number of events expected to be detected
at SuperKamiokande and SNO.
Finally we propose a method to discriminate quantitatively
the solutions of the solar neutrino problem. We do not
include the Earth matter effects in our calculation, and
this point will be discussed when we conclude the paper.

This paper is organized as follows. In section II we describe
the features of supernova neutrinos. In section III we
calculate dynamics of neutrino conversion on their way
out to the surface of the star. Then, in section IV, we obtain
neutrino energy spectra at detectors and time evolution of
neutrino number luminosity. Features of results of section IV
are discussed in section V and a measure of neutrino oscillation
is proposed. Finally in section VI, we summarize our results. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Supernova neutrinos
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Supernova neutrinos \label{section:Supernova}}

Supernova neutrino emission process can be divided into two
distinct phases \cite{Supernova}; the neutronization burst 
and nearly thermal neutrino emission. Almost all of the binding energy,
\begin{equation}
E_{b} = 1.5 \sim 4.5 \times 10^{53} {\rm erg},
\end{equation}
is radiated away as neutrinos, while a small fraction of which 
($\sim 2 \times 10^{51}$ erg) are
emitted during the first phase. While only $\nu_{e}$ is
emitted during the first phase, neutrinos and antineutrinos
of all types are emitted during the second phase with rougly the same
luminosity. The average
energies are different between flavors:
\begin{eqnarray}
\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}, 
\end{eqnarray}
where $\nu_{x}$ means either of
$\nu_{\mu}$, $\nu_{\tau}$, and their antineutrinos.
In this paper we use a realistic model of a collapse-driven supernova
by the Lawrence Livermore group \cite{Wilson} to calculate the
neutrino luminosity and energy spectrum.
Time-integrated energy spectra and time evolution of neutrino flux
are are shown in Fig. \ref{figure:original_spectra}
and Fig. \ref{figure:original_time}. (See
Totani et al. \cite{Totani} for detail.)

These neutrinos, which are produced in the high dense region
of the iron core, interact with matter before emerging from
the supernova. The presence of non-zero masses and mixing
in vacuum among various neutrino flavors results in strong
matter dependent effects, including conversion from one flavor
to another. In supernova, the conversions occur mainly in the
resonance layers. The resonance matter density can be written as
\begin{equation}
\rho_{res} \sim 1.4 \times 10^{6} (\frac{\Delta m^{2}}{1 {\rm eV^{2}}})
(\frac{10 {\rm MeV}}{E})(\frac{0.5}{Y_{e}}) \cos{\theta} \;{\rm g/cc},
\end{equation}
where$\Delta m^{2}$ is the mass squared difference, $E$ is the 
neutrino energy, and $Y_{e}$ is the mean number of electrons
per baryon.
In normal mass hierarchy scheme ($m_{3} > m_{2} > m_{1}$), 
the system has two resonances in neutrino sector: 
one at higher density(H-resonance)
and the other at lower density(L-resonance).
On the other hand, antineutrino sector has no resonance.

The dynamics of conversions in each resonance is determined by the
adiabaticity parameter $\gamma$,
\begin{equation}
\gamma \equiv \frac{\Delta m^{2}}{2E} 
              \frac{\sin^{2}2\theta}{\cos2\theta}
              \frac{n_{e}}{dn_{e}/dr},
\end{equation}
where $\theta$ is mixing angle, and $n_{e}$ is the electron
number density. The flip probability $P_{f}$, the probability that
a neutrino in one matter eigenstate jumps to the other
matter eigenstate, is,
\begin{equation}
P_{f} = \exp(-\frac{\pi}{2} \gamma),
\end{equation}
as given by the Landau-Zener formula \cite{Landau}. Adiabatic resonance
corresponds to $\gamma \gg 1$. Note that adiabaticity of
resonance depends on the mixing angle and the squared mass difference, 
that is \cite{DigheSmirnov},
\begin{eqnarray}
{\rm H \; resonance} \longrightarrow \theta_{13}, \Delta m^{2}_{13}, \\
{\rm L \; resonance} \longrightarrow \theta_{12}, \Delta m^{2}_{12}.
\end{eqnarray}
The neutrino spectra observed at the detectors can be dramatically
different from the original spectra according to the adiabaticity
of these two resonances.
If neutrino oscillation occurs, for example, between $\nu_{e}$ and
$\nu_{x}$, observed energy spectrum will be a mixture
of original $\nu_{e}$ and $\nu_{x}$ spectra and the average
energy of $\nu_{e}$ will be higher than the original $\nu_{e}$
average energy.

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





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Conversion Probability
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Numerical Calculation of Conversion Probabilities}

\subsection{Time Evolution of Conversion Probabilities in the Framework
of Three Flavor Neutrinos 
	\label{section:time_ev_calc}}

In the framework of three-flavor neutrino oscillation, the time
evolution equation of the neutrino wave functions can be written as
follows:

%%%%%%%%%  differential equation to solve  %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{eqnarray}
H(t) & \equiv &
U\left(
	\begin{array}{ccc}
		0 & 0 & 0\\
		0 & \Delta m^2_{21} /2E & 0\\
		0 & 0 & \Delta m^2_{31} /2E
	\end{array}\right)U^{-1} \nonumber \\
    & &	+\left(
	\begin{array}{ccc}
		A(t) & 0 & 0\\
		0 & 0 & 0\\
		0 & 0 & 0
	\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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:
%%%%%%%%% 3*3 unitary matrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.

In $H(t)$, the first term is the origin of vacuum oscillation, and
the second term A(t), which is the only time-dependent term in H(t), is
the origin of MSW effect.

By solving numerically the above differential equations from the center of
supernova to the outside of supernova, we obtain conversion 
probabilities $P(\nu_{\alpha \rightarrow \beta})$, i.e., probabilities
that a neutrino of flavor $\alpha$ produced at the center of supernova
is observed as a neutrino of flavor $\beta$.

We assume normal mass hierarchy and use the sets of mixing
parameters shown in table \ref{table:parameter}.
Here $\theta_{12}$ and $\Delta m_{12}^{2}$ correspond to the solutions of
solar neutrino problem and $\theta_{23}$ and $\Delta m_{13}^{2}$ correspond
to the solution of atmospheric neutrino. The value of $\theta_{13}$ is taken to
be consistent with current upper bound from reactor experiment \cite{CHOOZ}.
These models are named after their values of mixing angle:
LMA-L means that $\theta_{12}$ is set to be LMA of solar neutrino problem
and $\theta_{13}$ is large.

Fig.\ref{figure:LMA_ad_demo} - \ref{figure:SMA_nonad_demo} 
show the time evolution of coversion probability. 
$P(e \rightarrow e)$ and $P(e \rightarrow x)$ means the probability
that $\nu_{e}$ produced at the center of supernova become $\nu_{e}$,
$\nu_{x} (\nu_{\mu},\nu_{\tau}) $, respectively. 
Since we set $\sin^{2} 2 \theta_{23} = 1$, probabilities to become
$\nu_{\mu}$ and $\nu_{\tau}$ are the same. Four lines correspond
to a neutrino of energy, 5MeV, 10MeV, 40MeV, 70MeV, respectively.


As can be seen, for example, H resonance occurs adiabatically at 
$r = 0.02 \sim 0.05 R_{\odot}$ (O+Ne+Mg or O+C layer), since 
$\theta_{13}$ is large in model LMA-L.
Final conversion probabilities are independent of neutrino energy
because of the adiabaticity of resonance.
On the other hand, in the lower of Fig \ref{figure:SMA_ad_demo}
while H resonance is adiabatic, L resonance (He layer) is 
nonadiabatic because $\theta_{12}$ is small. Consequently, final 
conversion probabilities depend on energy. More detailed study
on dynamics of conversion probability in supernovae is done
by Watanabe \cite{Watanabe}.

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 
             (0.5 \sim 2) \times 10^{-7} {\rm eV}^2 \\
             \sin^{2} 2 \theta_{\odot}  \approx  0.9 \sim 1.0 \\
({\rm VO})   \Delta m^{2}_{\odot}  \approx 
             (0.6 \sim 6) \times 10^{-10}  {\rm eV}^2 \\
             \sin^{2} 2 \theta_{\odot}  \approx  0.8 \sim 1.0 
\end{eqnarray}
In these cases, whose parameters have nearly the same
values as LMA except for $\Delta m^2_{12}$, the final conversion
probabilities in vacuum take nearly the same values as in the case of
LMA. The difference in $\Delta m^2_{12}$ is reflected in the
radius at which L resonance occurs. Larger $\Delta m^2_{12}$
results in lager radius.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Event Rates
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Expected Event Rates in SuperKamiokande and SNO}

In this section, expected event rates at SuperKamiokande and SNO are
studied assuming a future galactic supernova at a distance d = 10kpc.
To obtain the event rates, we use the original neutrino flux mentioned
in section II and
the probabilities of the flavor conversion calculated 
in the previous section. We performed smoothing of 
the electron/positron energy spectra with a dispersion of 1 MeV,
taking into account rough energy resolutions of detectors.
In fact, the energy resolutions of detectors depend on energy itself.
But the rough estimation of energy resolution is sufficient, 
since our purpose is to see overall shapes of the spectra.
We also assume that the time delay of neutrinos due to non-zero
mass is negligible compared with the time scale considered here
($>$ msec). This assumption is secure unless the neutrino mass
hierarchy is degenerate at around $\gtrsim$ 3 MeV \cite{Totani2}.

Since the original neutrino spectra and the conversion probabilities
are the same for $\nu_{\mu}$ and $\nu_{\tau}$, the event rates
are also the same for $\nu_{\mu}$ and $\nu_{\tau}$.


\subsection{Event Rates at SuperKamiokande \label{section:SK}}

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:
%%%%%%%%%%%   interaction at SK  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 + O & \rightarrow & F + e^- \quad(\rm{CC})\\
\bar{\nu_e} + O & \rightarrow & N + e^+ \quad(\rm{CC})\label{interaction:nuebarO}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where CC and NC stand for charged current and neutral current
interactions, respectively. 
The lower limit of detection is $\sim$ 5MeV, and the energy resolution is
$\sim$ 15\% for an electron with energy 10MeV. 
For the cross sections of these interactions, we refer to \cite{SKcross}.
The appropriate detection efficiency curve is also taken into account
\cite{SKefficiency}. The efficiency is 100\% above 5MeV and 50\%
at 4.5MeV. In these interactions,
the $\bar{\nu_e}p$ CC interaction [Eq.(\ref{interaction:anuep})] has
the largest contribution to the detected events at SK. Hence the energy
spectrum detected at SK (including all the reactions) is almost the
same as the spectrum derived from the interaction Eq.(\ref{interaction:anuep}) 
only.

Fig. \ref{figure:SK_spe} and Fig. \ref{figure:SK_time} show energy 
spectrum and time evolution of number luminosity of positrons and 
electrons expected to be detected at SuperKamiokande, respectively.
Fig. \ref{figure:SK_time_big} is a zoom-up of
Fig. \ref{figure:SK_time} near the neutronization burst.

Total event numbers for all the models are shown in 
Table \ref{table:event_SK}. 
In this table, the numbers of events for each interaction and
contribution from neutronization burst phase are also shown.
Here the neutronization phase means the period from 41msec to
48msec after the bounce.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Event Rates in SNO}

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,
%%%%%%%%%%%   interaction at SNO  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The two interactions written in Eqs.(\ref{eq:SNO_CC_nue}) and
(\ref{eq:SNO_CC_anue}) are detected when electrons emit
$\check{\rm{C}}$herenkov light. These reactions produce electrons and
positrons whose energies sensitive to the neutrino energy, and hence the energy
spectra of electrons and positrons give us the information on the
original neutrino flux. In this work, we mainly take into account
these two charged current interactions. For the cross sections,
we refer to \cite{SNOcross}. The efficiency of detection is 
set to be one, because we have no information about it.

Two neutral current interactions, which produce neutrons, are detected
by observing the photons emitted at the neutron absorption. Photons give
energy to electrons, then the $\check{\rm{C}}$herenkov light from the
electrons is detected. Moreover, there is a possibility to
distinguish the two CC interactions by detecting neutrons because
the detection of the neutron and the positron at the same time indicates 
the interaction in Eq.(\ref{eq:SNO_CC_anue}).

Fig.\ref{figure:SNO_spe} and Fig.\ref{figure:SNO_time} show energy 
spectrum and time evolution of number luminosity of positrons and 
electrons, produced by the two CC interactions 
expected at SNO, respectively.
Fig.\ref{figure:SNO_time_big} is a zoom-up of 
Fig.\ref{figure:SNO_time} near the neutronization burst.

Total event numbers for all the models are shown in the following
Table \ref{table:event_SNO}. 
In this table, the numbers of events of each interaction and
contribution from neutronization burst phase are also shown.
Here the neutronization phase means the period from 41msec to
48msec after the bounce.

The SNO detector has also 7,000 tons of light water which can
be used to detect neutrinos. This can be considered to be a
miniature of SuperKamiokande (32,000 tons of light water).
Then the number of events detected by light water at SNO
is 7/32 of that at SuperKamiokande.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Discussions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}


\subsection{Features of Energy Spectra and Neutronization}

As can be seen in Fig.\ref{figure:SK_spe} and \ref{figure:SNO_spe}, 
when there is neutrino oscillation, 
neutrino spectra are harder than those
in absence of neutrino oscillation. 
This is because average energies
of $\nu_{e}$ and $\bar{\nu}_{e}$ are smaller than those of $\nu_{x}$
and neutrino oscillation produces high energy $\nu_{e}$ and $\bar{\nu}_{e}$
which was originally $\nu_{x}$. This feature can be used as
a criterion of magnitude of neutrino oscillation, which will be
discussed in the next section.

It is worth noting that number of events during neutronization burst 
phase is highly suppressed in model LMA-L and SMA-L.
This is because, due to large value of $\theta_{13}$ in these two models, 
H resonance occurs adiabatically and $\nu_{e}$ produced
at the center of supernova is detected as $\nu_{x}$ which
has small cross section. But the number of events during
neutronization burst will be too small to extract statistically
significant information (see Table II and III).

It is possible that He and H layers of progenitor star are missing
when supernova burst occurs, and density decreases abruptly
outside the O+C layer. Then L resonance would occur nonadiabatically
to some extent even in case of LMA, and differences between
LMA and SMA would become smaller.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Comparison of the Energy Spectra at SK and SNO
	\label{section:comparison}}


As mentioned in the previous subsection, neutrino oscillation
makes $\nu_{e}$ and $\bar{\nu}_{e}$ spectra harder. Therefore,
the ratio of high-energy events to low-energy events will be a
good measure of neutrino oscillation effects.
We calculated the following ratio of events at both detectors:

\begin{equation}
	R_{SK} \equiv \frac{\makebox{number of events at } 30<E<70\rm{MeV}}
		{\makebox{number of events at } 5<E<20\rm{MeV}}
\end{equation}
\begin{equation}
	R_{SNO} \equiv \frac{\makebox{number of events at } 25<E<70\rm{MeV}}
		{\makebox{number of events at } 5<E<20\rm{MeV}}
\end{equation}


The plots of $R_{SK}$ vs. $R_{SNO}$ are shown in Fig.\ref{figure:ratio}.
The errorbars include only statistical errors. At first glance,
it seems to be possible to distinguish all the models including
the no oscillation case. But there are other ambiguities besides
statistical errors. 

One is the mass of the progenitor star. Supernovae with different progenitor
masses may result in different original neutrino spectra and neutrino
oscillation effects. Studies on this point are now in progress.
But dependence of shape of neutrino spectra on progenitor mass 
is not so large \cite{mass} and we would be able to distinguish the models. 
The difference among the following three groups will still be clear:
(1)LMA-L and LMA-S, (2)SMA-L, and (3)SMA-S and no oscillation.

Another ambiguity is the direction of supernova. Depending on
the direction, neutrinos from supernova may travel through the Earth
before they reach the detectors. In this case, we have
to take the Earth matter effect into account. This effect
has already been studied by our previous work\cite{Takahashi}. 
In this work we concluded that we can
differentiate LMA-L from LMA-S, by observing the Earth matter effects.




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

\section{Summary}

We studied quantitatively the effects of the three-flavor oscillation
on supernova neutrinos, by using more realistic 
neutrino profiles and presupernova density profiles than previous studies.
Our calculations are based on a realistic
numerical supernova model calculated by the Lawrence Livermore group
\cite{Wilson} and a realistic model of density profile of a presupernova
star by Woosley and Weaver\cite{Woosley}. 
First we calculate time evolution of conversion probabilities.
Then energy spectra and time evolution of number luminosity 
are obtained assuming a supernova at 10kpc. Neutronization burst
is highly suppressed in models LMA-L and SMA-L.
In case that there is neutrino mixing, energy spectra are
harder than in case of no oscillation. 
By comparing ratios of high-energy events to low-energy events at
SuperKamiokande and SNO, we found that we will be able to 
distinguish the solutions of solar neutrino problem and probe $\theta_{13}$.


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

\acknowledgments

We would like to thank S. E. Woosley for giving the
numerical data of the progenitor star, and J. R. Wilson and H. E. Dalhed 
for the neutrino emission data from the supernova explosion.  
We also want to acknowledge T.Kajita, Y.Totsuka , Y.Suzuki and Y.Fukuda for 
recent results of neutrino oscillation analysis at SuperKamiokande.
This work was supported in part by Grant-in-Aid for Scientific
Research provided by the Ministry of Education, Science and Culture
of Japan through Research Grant No.07CE2002 3.

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






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\end{thebibliography}



\clearpage


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{original_spe.eps}
\end{center}
\caption{ Original energy spectra of neutrinos [18].
Solid, dashed, and long-dashed lines correspond to $\nu_{e}$,
$\bar{\nu}_{e}$, and $\nu_{x}$, respectively.
\label{figure:original_spectra}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{original_time.eps}
\end{center}
\caption{ Time evolution of the original neutrino number
luminosity [18].
Solid, dashed, and long-dashed lines correspond to $\nu_{e}$,
$\bar{\nu}_{e}$, and $\nu_{x}$, respectively.
\label{figure:original_time}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{density.eps}
\end{center}
\caption{ Density profile of the presupernova star model used in the
paper [20]. The progenitor mass is set to be $15 M_{\odot}$.
%\cite{Woosley}
\label{figure:density}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{LMA_ad_demo.eps}
\end{center}
\caption{  Time evolution of conversion probability for model LMA-L.
In the upper figure, solid and dashed lines show $P(e \rightarrow e)$ and 
$P(e \rightarrow x)$, respectively. In the lower figure, solid and dashed 
lines show $P(x \rightarrow e)$ and 
$P(x \rightarrow x)$, respectively. Four lines of the same marking
correspond to neutrino energy, 5MeV, 10MeV, 40MeV, and 70MeV,
respectively.
\label{figure:LMA_ad_demo}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{LMA_nonad_demo.eps}
\end{center}
\caption{  Time evolution of conversion probability for model LMA-S.
In the upper figure, solid and dashed lines show $P(e \rightarrow e)$ and 
$P(e \rightarrow x)$, respectively. In the lower figure, solid and dashed 
lines show $P(x \rightarrow e)$ and 
$P(x \rightarrow x)$, respectively. Four lines of the same marking
correspond to neutrino energy, 5MeV, 10MeV, 40MeV, and 70MeV,
respectively.
\label{figure:LMA_nonad_demo}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SMA_ad_demo.eps}
\end{center}
\caption{ Time evolution of conversion probability for model SMA-L.
In the upper figure, solid and dashed lines show $P(e \rightarrow e)$ and 
$P(e \rightarrow x)$, respectively. In the lower figure, solid and dashed 
lines show $P(x \rightarrow e)$ and 
$P(x \rightarrow x)$, respectively. Four lines of the same marking
correspond to neutrino energy, 5MeV, 10MeV, 40MeV, and 70MeV,
respectively.
\label{figure:SMA_ad_demo}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SMA_nonad_demo.eps}
\end{center}
\caption{ Time evolution of conversion probability for model SMA-S.
In the upper figure, solid and dashed lines show $P(e \rightarrow e)$ and 
$P(e \rightarrow x)$, respectively. In the lower figure, solid and dashed 
lines show $P(x \rightarrow e)$ and 
$P(x \rightarrow x)$, respectively. Four lines of the same marking
correspond to neutrino energy, 5MeV, 10MeV, 40MeV, and 70MeV,
respectively.
\label{figure:SMA_nonad_demo}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SK_spe.eps}
\end{center}
\caption{ Energy spectrum of positrons and 
electrons expected to be detected at SuperKamiokande.
Solid, dashed, long-dashed, dash-dot-dash, and dotted lines
correspond to no oscillation, model LMA-L, LMA-S, SMA-L, 
and SMA-S, respectively.
\label{figure:SK_spe}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SK_time.eps}
\end{center}
\caption{ Time evolution of number luminosity of positrons and 
electrons expected to be detected at SuperKamiokande.
Solid, dashed, long-dashed, dash-dot-dash, and dotted lines
correspond to no oscillation, model LMA-L, LMA-S, SMA-L, 
and SMA-S, respectively.
\label{figure:SK_time}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SK_time_big.eps}
\end{center}
\caption{ A zoom-up of Fig.\ref{figure:SK_time} near the 
neutronization burst. 
Solid, dashed, long-dashed, dash-dot-dash, and dotted lines
correspond to no oscillation, model LMA-L, LMA-S, 
SMA-L, and SMA-S, respectively.
\label{figure:SK_time_big}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SNO_spe.eps}
\end{center}
\caption{ Energy spectrum of positrons and 
electrons expected to be detected at SNO taking only CC events into account.
Solid, dashed, long-dashed, dash-dot-dash, and dotted lines
correspond to no oscillation, model LMA-L, LMA-S, SMA-L, 
and SMA-S, respectively.
\label{figure:SNO_spe}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SNO_time.eps}
\end{center}
\caption{ Time evolution of number luminosity of positrons and 
electrons expected to be detected at SNO taking only CC events into account.
Solid, dashed, long-dashed, dash-dot-dash, and dotted lines
correspond to no oscillation, model LMA-L, LMA-S, SMA-L, 
and SMA-S, respectively.
\label{figure:SNO_time}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{SNO_time_big.eps}
\end{center}
\caption{ A zoom-up of Fig.\ref{figure:SNO_time} near the 
neutronization burst. 
Solid, dashed, long-dashed, dash-dot-dash, and dotted lines
correspond to no oscillation, model LMA-L, LMA-S, SMA-L, 
and SMA-S, respectively.
\label{figure:SNO_time_big}
}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3.2in
~\epsffile{ratio.eps}
\end{center}
\caption{ The plot of $R_{SK}$ vs. $R_{SNO}$ for all the models. 
The error-bars represent the
statistical errors. \label{figure:ratio}
}
\end{figure}


\twocolumn[
\hsize\textwidth\columnwidth\hsize
\begin{table*}
\caption{Sets of mixing paremeter for calculation
	\label{table:parameter}}
\begin{center}
\begin{tabular}{ccccccccc}
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 &  H resonance & L resonance \\ \hline 
LMA-L &  0.87  & 1.0 & 0.043 & $7.0 \times 10^{-5}$ & $3.2 \times 10^{-3}$ & LMA & adiabatic & adiabatic \\ 
LMA-S &  0.87  & 1.0 & $1.0 \times 10^{-6}$ & $7.0 \times 10^{-5}$ & $3.2 \times 10^{-3}$ 
& LMA & nonadiabatic & adiabatic \\   
SMA-L &  $5.0 \times 10^{-3}$  & 1.0 & 0.043  & $6.0 \times 10^{-6}$ & $3.2 \times 10^{-3}$ 
& SMA & adiabatic & nonadiabatic \\ 
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 & nonadiabatic & nonadiabatic \\ 
\end{tabular}
\end{center}
\end{table*}
]


\begin{table}
\caption{Number of events at SuperKamiokande
	\label{table:event_SK}}
\begin{center}
\begin{tabular}{cccccc}
	model & LMA-L & LMA-S & SMA-L & SMA-S 
              & no osc\\
	\hline
	$\bar{\nu_e}p$ & 9459 & 9427 & 8101 & 7967 & 8036 \\
	$\nu_ee^{-}$ & 186 & 115 & 189 & 131 & 132 \\
	$\bar{\nu_e}e^{-}$& 46 & 46 & 41 & 42 & 42 \\
	$\nu_{\mu}e^{-}$ & 25 & 26 & 25 & 30 & 30 \\
	$\bar{\nu_{\mu}}e^{-}$ & 24 & 23 & 24 & 24 & 24 \\
	$\nu_{\tau}e^{-}$ & 25 & 26 & 25 & 30 & 30 \\
	$\bar{\nu_{\tau}}e^{-}$& 24 & 23 & 24 & 24 & 24 \\
	$O\nu_e$ & 297 & 214 & 297 & 108 & 31 \\
	$O\bar{\nu_e}$ & 160 & 158 & 95 & 92 & 92 \\
	\hline
        total & 10245 & 10114 & 8822 & 8447 & 8441 \\ 
        \begin{tabular}{c}neutronization \\ burst \end{tabular}
         & 15.7 & 16.7 & 9.0 & 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}{cccccc}
	model & LMA-L & LMA-S & SMA-L & SMA-S & no osc \\
	\hline
	$\nu_e d {\rm (CC)}$ & 237 & 185 & 237 & 111 & 68 \\ 
	$\bar{\nu_e}d {\rm (CC)}$ & 118 & 117 & 84 & 82 & 82 \\
	\hline 
        total & 355 & 302 & 321 & 193 & 150 \\
        \begin{tabular}{c}neutronization \\ burst \end{tabular}
              & 0.6 & 1.1 & 0.5 & 1.1 & 2.1
\end{tabular}
\end{center}
\end{table}




\end{document}



