% % % You can include as many co-authors as you wish
unless % the title/author information fits within 1 page. % M. Selvi$^1$ on behalf of the LVD Collaboration and F. Vissani
{LNGS - INFN.} %
Assergi
Italy.
009420-2.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ``The 28th International Cosmic Ray Conference"
% icrc2003.tex
%
%_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
%
%  Universal Academy Press, Inc.
%  BR-Hongo-5 Bldg., 6-16-2 Hongo Bunkyo-ku, Tokyo 113-0033, Japan
%  Tel: +81-3-3813-7232
%  Fax: +81-3-3813-5932
%  E-mail address for the proceedings: icrc@uap.co.jp
%  URL: http://www.uap.co.jp
%
%_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
%

\documentclass[12pt]{book}

\usepackage[dvips]{graphicx,color}
\usepackage{makeidx,tsukuba}
% graphicx
% is available from
% CTAN: one of CTAN; http://ring.etl.go.jp/pub/text/CTAN/

\makeauthorindex
\makeindex
\def\thefootnote{\fnsymbol{footnote}}

\begin{document}

\BookTitle{\itshape The 28th International Cosmic Ray Conference}
\CopyRight{\copyright 2003 by Universal Academy Press, Inc.}
%\tableofcontents
\pagenumbering{arabic}

\chapter{Study of the Effect of Neutrino Oscillation on the SuperNova Neutrino Signal with the LVD Detector}

\author{%
%
% You can include as many co-authors as you wish, unless
% the title/author information fits within 1 page.
%
M. Selvi$^1$ on behalf of the LVD Collaboration and F. Vissani, {\it LNGS - INFN.}
%, Assergi, Italy.}
\\{\it (1) Bologna University and INFN.} 
}
%%Makoto Sasaki,$^1$ Akiko Kawachi,$^1$ and Yutaka Matsubara$^2$ \\
%%{\it (1) Institute for Cosmic Ray Research, University of Tokyo,
%%Kashiwa, Chiba 277-8582, Japan\\
%%(2) Solar-Terrestrial Environment Laboratory,  Nagoya University, Nagoya,
%%Aichi 464-8602, Japan} \\
%% end of author


\section*{Abstract}
%\vspace{-0.5cm}
%In this paper we present some updates, with respect to previous analysis [2], in the study of the impact of $\nu$ oscillations in the expected $\nu$ signal from a supernova core collapse in the LVD detector at LNGS. The updates regard the re-calculation of the expected number of inverse beta decay events after recent cross section refinements, the study of inverted mass hierarchy case, the calculation of the expected number of $\nu$-Fe interactions and the Earth matter effects.
We present an update of our previous study [2] on how $\nu$ oscillations affect the signal from a supernova core collapse observed in the LVD detector at LNGS. In this paper we use a recent, more precise determination of the cross section [8] to calculate the expected number of inverse beta decay events, we introduce in the simulation also the $\nu$-{\rm Fe} interactions, we include the Earth matter effects and, finally, we study also the inverted mass hierarchy case.


\section{Supernova neutrino signal}
At the end of its burning phase a massive star ($M \ge 8 M_\odot$) explodes into a supernova (SN), originating a neutron star which cools emitting about 99~\% of
the liberated gravitational binding energy
%, $E_b \simeq 3 \cdot 10^{53}$ erg 
in neutrinos.
The time-integrated spectra can be well approximated by the
pinched Fermi--Dirac distribution,
 with an effective degeneracy parameter
$\eta$, which is assumed, for simplicity, to be null.
%$i.e.$,$$
%%\begin{eqnarray}
%F^0_\alpha(E,T_\alpha,\eta_\alpha,L_\alpha,D) =
%\frac{L_\alpha}{4\pi D^2 T^4_\alpha F_3(\eta_\alpha)} \frac{E^2}{e^{E/
%T_{\alpha}-\eta_\alpha}+1}$$ ,
%%\label{eq3}
%%\end{eqnarray}
%where $D$ is the distance to the supernova, $E$ is the energy of the neutrinos, $L_\alpha$ is the
%luminosity of the flavor $\nu_\alpha$, and $T_\alpha$ represents the
%effective temperature of the $\nu_\alpha$ gas inside the neutrinosphere.
%In this work, we assume for simplicity that $\eta=0$ for all 
%neutrino flavors. 
%This choice results in the relation between mean energy and neutrinosphere temperature $\langle E_\alpha \rangle \simeq 3.15 ~T_\alpha$.
%Due to the different trapping processes, the different neutrino
%flavors originate in layers of the supernova with different
%temperatures. The electron (anti)neutrino flavor is kept in thermal
%equilibrium by $\beta$ processes up to a certain radius usually referred to as
%the ``neutrinosphere'', beyond which the neutrinos stream off
%freely. However, the practical absence of muons and taus in the
%supernova core implies that the other two neutrino flavors, here
%collectively denoted by $\nu_x$ ($\nu_\mu, \nu_\tau, \bar\nu_\mu,
%\bar\nu_\tau$), interact primarily by less efficient neutral-current processes
%Therefore, their spectra are determined at deeper, $i.e.$ hotter,
%regions. In addition, since the content of neutrons is larger
%than that of protons, $\nu_e$'s escape from outer regions than
%$\bar{\nu}_e$'s. 
%This rough picture leads to 
Although the hierarchy
$\langle E_{\nu_e}\rangle < \langle E_{\bar\nu_e}\rangle < \langle
 E_{\nu_x}\rangle $ remains valid
(here $\nu_x$ refers to both $\nu_\mu$ and $\nu_\tau$),
%Typical values of the average energies of the time-integrated neutrino
%spectra obtained in simulations
%are $\langle E_{{\nu}_e} \rangle \sim 12$ MeV, $\langle E_{\bar{\nu}_e} \rangle
%\sim 15$ MeV, and $\langle E_{{\nu}_x} \rangle \sim 24$ MeV
%\cite{Janka:1992jk,Totani:1998vj}.
%However, 
recent studies with an improved treatment of $\nu$ transport,
microphysics, the inclusion of the nucleon bremsstrahlung, and the
energy transfer by recoils, find somewhat smaller differences between the
$\bar\nu_e$ and $\nu_x$ spectra [5].

In the following, we assume a future galactic SN explosion at a typical
distance of $D = 10$~kpc, with a binding energy of $E_b = 3 \cdot
10^{53}$~erg and a total energy of $E^{\rm tot}_{\nu_e} = E^{\rm
tot}_{\bar\nu_e} = E^{\rm tot}_{\nu_x}$.
%that is we assume energy equipartion among flavours, even if it is known that equipartition is valid within a factor of two or so.
%where $\xi$
%parameterizes a possible deviation from energy
%equipartition~\cite{Mezzacappa:2000jb}. 
We also assume that the fluxes of
$\nu_\mu$, $\nu_\tau$, $\bar\nu_\mu$, and $\bar\nu_\tau$ are identical, we
fix 
%$\tau_E \equiv 
%$\langle
%E_{\nu_x} \rangle/ \langle E_{\bar{\nu}_e} \rangle = 1.5$, 
$T_{\nu_x} / T_{\bar{\nu}_e} = 1.5$, 
%% $\xi_E \equiv 
%$\langle E_{\nu_e} \rangle/ \langle E_{\bar{\nu}_e} \rangle = 0.8$
$T_{\nu_e} / T_{\bar{\nu}_e}  = 0.8$
and $T_{\bar{\nu}_e} 
%\in \{4,~7\}~ 
=5~{\rm MeV}$ [5].

\section{LVD detector and neutrino reactions}

\begin{figure}[t]
  \begin{minipage}{.48\columnwidth}
    \begin{center}
      \includegraphics[height=17pc]{figsig.eps}
    \end{center}
    \vspace{-1.cm}
    \caption{Cross sections of the main neutrino interactions in LVD.}
    \label{fi:sig}
  \end{minipage}
  \hspace{1pc} %%%%% space between two figures
  \begin{minipage}{.48\columnwidth}
    \begin{center}
      \includegraphics[height=17pc]{figosc.eps}
    \end{center}
    \vspace{-1.cm}
    \caption{Effect of neutrino oscillations in the signal detected in LVD.}
    \label{fi:osc}
  \end{minipage}
\end{figure}



The Large Volume Detector (LVD) in the INFN Gran Sasso National Laboratory,
Italy, consists of an array of 840  
liquid scintillator (LS) counters, 1.5 m$^3$ each. These are 
interleaved by streamer tubes, and arranged in a 
%compact and 
modular 
geometry; a detailed description is in [1].
The active scintillator mass is $M=1000$ t. 
There are two subsets of counters: the
external ones ($43 \%$),
operated at energy threshold ${\cal E}_h\simeq 7$ {\rm MeV},
and inner ones ($57 \%$), better shielded from rock radioactivity and 
operated at ${\cal E}_h\simeq 4$ {\rm MeV}. 
In order to tag the delayed $\gamma$ pulse due to
$n$-capture, all counters are equipped with an additional discrimination
channel, set at a lower
threshold, ${\cal E}_l\simeq 1$ {\rm MeV}.
%Relevant features of the detector are: $(i)$ good event localization; $(ii)$
%accurate absolute and relative timing:
%$\Delta\mathrm{t}_{\rm abs}=1\, \mu \rm{s}$, 
%$\Delta\mathrm{t}_{\rm rel}=12.5\, \rm{ns}$;
%$(iii)$ energy resolution: 
%$\sigma_{E}/{E} = 0.07 + 0.23\cdot ({E} /\rm{MeV})^{-0.5}$.

In the following we will focus on $\nu$ reactions with free protons and iron nuclei:
% We don't consider, in this paper, the $\nu$ interactions with carbon nuclei
%The observable neutrino reactions that occur in the liquid scintillator (LS) are:\\
%%\begin{itemize}
%\item
 {\em (1)}\
$\bar\nu_e p,e^+ n$, observed through a prompt signal from ${e}^+$ above
threshold ${\cal E}_h$ (detectable energy $E_d \simeq E_{\bar\nu_{e}}-1.8$ {\rm MeV}
$+ 2 m_e c^2 $), followed by the signal from the ${n p,d} \gamma$ capture
($E_{\gamma} = 2.2$ {\rm MeV}), above ${\cal E}_l$ and with a mean delay
$\Delta t \simeq 180~\mu \mathrm{s}$.  The cross section for this reaction has been recently recalculated [8] with a better treatment of the $10-100~{\rm MeV}$ region, i.e. the SN neutrino energy.
%crucial for the supernova $\nu$ detection. 
The cross section behaviour with energy is shown in figure \ref{fi:sig} The efficiency for the prompt signal is $\epsilon_{\bar\nu_e\, p,e^+\, n} = 95\% $. The total number of free protons in the LS is $9.34~\cdot 10^{31}$.

%\item 
%\noindent {\em (2)}\ $\nu_e\,^{12}\mathrm{C},^{12}\!\mathrm{N}\ e^-$,
%observed through two signals: the
%prompt one due to the $e^-$ above ${\cal E}_h$ (detectable energy
%$E_d \simeq E_{\nu_e}-17.8$ MeV) followed by the signal, above ${\cal E}_h$, from the
%$\beta^+$ decay of $^{12}\!\mathrm{N}$ (mean life time $\tau = 15.9$ ms).\\
%%\item 
%\noindent {\em (3)}\ $\bar\nu_e\, ^{12}\rm{C},^{12}\!\rm{B}\ e^+$, observed
%through two signals: the prompt one due to the $e^+$ (detectable energy
%$E_d \simeq E_{\bar\nu_e}-13.9\, \mathrm{MeV} + 2m_e c^2$) followed by the signal
%from the $\beta^-$ decay of $^{12}\!{\rm B}$ (mean life time $\tau= 29.4$ ms).
%As for reaction {\em (2)}, the second signal is detected above the threshold
%${\cal E}_h$.\\
%%\item 
%\noindent {\em (4)}\ $\stackrel{\scriptscriptstyle (-)}{\nu}_{\!\ell}\,$
%${}^{12}{\rm C},\stackrel{\scriptscriptstyle (-)}{\nu}_{\!\ell}\,$
%${}^{12}{\rm C}^*$
%($\ell=e,\mu,\tau$), whose signature is the monochromatic photon from
%carbon de-excitation ($E_{\gamma}=15.1$ MeV), above ${\cal E}_h.$ Cross sections for reactions $(2),~(3)$ and $(4)$ are taken from \cite{CrossC} and shown in figure \ref{fi:cross}.\\
%%\item 
%\noindent {\em (5)}\ $\stackrel{\scriptscriptstyle (-)}{\nu}_{\!\ell}\,$
%$e^-,\stackrel{\scriptscriptstyle (-)}{\nu}_{\!\ell}\, e^-$, which yields a single
%signal, above ${\cal E}_h$, due to the recoil electron.
%%\end{itemize}
The LVD detector presents an iron support structure made basically by two components: the tank (mean thickness: $0.4 ~ cm$) which contains the LS and the portatank (mean thickness: $1.5 ~ cm$) which hosts a cluster of 8 tanks. Indeed, the higher energy part of the $\nu$ flux could be detected also with the $\nu (\bar\nu)  {\rm Fe}$ interaction, which results in an electron (positron) that could exit iron and release energy in the LS. The considered  reactions are:\\
\noindent {\em (2)}\ $\nu_e\,^{56}\mathrm{Fe},^{56}\!\mathrm{Co}\ e^-$. The binding energy difference between the ground levels is $E_b^{\rm{Co}} - E_b^{\rm{Fe}} = 4.566~{\rm MeV}$; moreover the first $\rm{Co}$ allowed state is at $3.589~{\rm MeV}$. Indeed, in this work we considered  $E_{e^-} = E_{\nu_e} - 8.15~ {\rm MeV}$. 
%The beta decay products of $^{56}\!\mathrm{Co}$ are not taken into account because they give some $MeV$ gammas that are trapped in the iron. 
A full simulation of the LVD support structure and LS geometry has been developed in order to get the efficiency for an electron, generated randomly in the iron structure, to reach the LS with energy higher than ${\cal E}_h$. It is greater than $20\%$ for $E_\nu > 30~{\rm MeV}$ and grows up to $70\%$ for $E_\nu > 100~{\rm MeV}$ . On average, the electron energy detectable in LS is $E_d \simeq 0.45 \times E_\nu$. The total number of iron nuclei is $7.63~\cdot10^{30}$.

\noindent {\em (3)}\ $\bar\nu_e\,^{56}\mathrm{Fe},^{56}\!\mathrm{Mn}\ e^+$,
the energy threshold is very similar to reaction {\em (2)} and the same considerations could be done.
The cross section for reactions {\em (2),(3)} are taken respectively from [6,9] and plotted in figure \ref{fi:sig}

%The number of all the possible targets present in the LVD detector is listed in table \ref{ta:targ}

%\begin{table}[h]
% \caption{Number of targets in the LVD detector.}
% \label{ta:targ}
%\begin{center}
%\begin{tabular}{l|c|c|c}
%\hline
%Target Type & Contained in &       Mass       & Number of targets      \\
%\hline
%Free protons         & Liquid Scintillator     & $1000~t$ & $9.34~10^{31}$   \\
%Electrons     & LS & $1000~t$     & $3.47~10^{32}$  \\
%C Nuclei     & LS &    $1000~t$  & $4.23~10^{31}$      \\
%Fe Nuclei     & Support Structure & $710~t$     & $7.63~10^{30}$   \\
%\hline
%\end{tabular}
%\end{center}
%\end{table}







\section{Neutrino oscillation and MSW effect in the SN and in the Earth}
In the study of SN neutrinos,
$\nu_{\mu}$ and $\nu_{\tau}$ are indistinguishable, both in the star and in
the detector because of the corresponding charged lepton production threshold; consequently, in the frame of three-flavor oscillations,
the relevant parameters are just
$(\Delta m^2_{{\rm sol}}, U_{e2}^2)$ and 
$(\Delta m^{2}_{\rm atm}, U_{e3}^2)$. 
We will adopt the following numerical values:
$\Delta m^2_{{\rm sol}}=7 \cdot 10^{-5}{\rm eV}^2$, 
$\Delta m^{2}_{\rm atm}=2.5 \cdot 10^{-3} {\rm eV}^2$, 
$U_{e2}^2=0.33;$ the selected solar parameters 
$(\Delta m^2_{{\rm sol}}, U_{e2}^2)$ describe the
LMA-I solution, as it results from a global analysis including solar, CHOOZ and KamLAND $\nu$ data [4].

For a normal mass hierarchy (NH) scheme,  
$\nu$ (not $\bar \nu$) cross two resonance layers:
one at higher density (H), which corresponds to $\Delta m^{2}_{\rm atm}, U_{e3}^2$, and
the other at lower density (L), corresponding to 
$\Delta m^{2}_{{\rm sol}}, U_{ e2}^2$.
For inverted mass hierarchy (IH), transitions at
the higher density layer occur in the $\bar \nu$
sector, while at the lower density layer they occur 
in the $\nu$ sector. 

Given the energy range of SN $\nu$ (up to $\sim 100~{\rm MeV}$)
%($5\ {\rm MeV} \le E_{\nu} \le 100$ MeV),
and considering a star density profile $\rho \propto 1/r^3$, 
the adiabaticity condition is always satisfied at the L resonance
for any LMA solution, while at the H resonance, this 
depends on the value of $U_{e3}^2$. 
When $U_{e3}^2 \geq 5 \cdot 10^{-4}$ 
the conversion is completely adiabatic, meaning that the flip probability between two adiacent mass eigenstates is null ($P_h=0$).
%; conversely, when $U_{e3}^2 \le 5 \cdot 10^{-4}$ 
%the conversion is completely non adiabatic, meaning that the flip probability is $P_h=1$. We used in the calculation 
%$U_{e3}^2=10^{-2}$ which is just behind the corner of the CHOOZ upper limit.
In the adiabatic case and NH, the $\bar \nu_e$ produced in the SN core arrive at Earth as $\nu_1$, and they have a high ($U_{e1}^2 \simeq cos^2 \theta_{12} \simeq 0.7$) probability to be detected as $\bar\nu_e$. 
On the other hand, the original $\bar \nu_x$ arrive at Earth as $\nu_2$ and $\nu_3$ and are detected as $\bar \nu_e$ with probability $U_{e2}^2 \simeq sin^2 \theta_{12}$. Given the higher energy spectrum of $\bar \nu_x$ this configuration results in a larger number of interactions, with respect to the no-oscillation case, due to the increasing cross sections with energy. In the adiabatic-IH case the detected $\bar \nu_e$ completely come from the original $\bar \nu_x$ flux in the star and the number of interaction is still greater, as shown in figure \ref{fi:osc}
The oscillations scheme can be summarized as:
%$F_e = P_h \sin^2 \theta_{12} F_e^0 + (1-P_h \sin^2 \theta_{12}) F_x^0$\\
%$F_{\bar e} = \cos^2 \theta_{12} F_{\bar e}^0 + \sin^2 \theta_{12} F_{\bar x}^0$ for normal hierarchy and\\
%$F_e = \sin^2 \theta_{12} F_e^0 + \cos^2 \theta_{12} F_x^0$\\
%$F_{\bar e} = P_h \cos^2 \theta_{12} F_{\bar e}^0 + (1 - P_h \cos^2 \theta_{12}) F_{\bar x}^0$ for inverted hierarchy, 
 $F_e = P_h U_{e2}^2 F_e^0 + (1-P_h U_{e2}^2) F_x^0$ and  $F_{\bar e} = U_{e1}^2 F_{\bar e}^0 + U_{e2}^2 F_{\bar x}^0$ for normal hierarchy; 
$F_e = U_{e2}^2 F_e^0 + U_{e1}^2 F_x^0$ and $F_{\bar e} = P_h U_{e1}^2 F_{\bar e}^0 + (1 - P_h U_{e1}^2) F_{\bar x}^0$ for inverted hierarchy, 
where $F_{any}^0$ are the original neutrino fluxes in the star and $F_{any}$ are the observed $\nu$ fluxes. One can notice that, in the antineutrino channel, the non adiabatic ($P_h=1$), IH case, is equivalent to the NH case (which does not depend on adiabaticity). 
%the and the non adiabatic ones (both NH and IH) give the same observable $\bar \nu_e$ fluxes.
%The increase in the number of $\nu$ interactions and in the detected $\nu$ mean energy for different oscillation schemes is shown in figure \ref{fi:osc} 
In figure \ref{fi:iron} is shown the contribution of $(\nu_e+\bar \nu_e)$ {\rm Fe} interactions in the total number of events. For the chosen SN and oscillation parameters they are about $18\%$ of the signal.
%%%, indeed they must be considered in the estimation.

\begin{figure}[t]
  \begin{minipage}{.48\columnwidth}
    \begin{center}
      \includegraphics[height=17pc]{figiron.eps}
    \end{center}
    \vspace{-1.cm}
    \caption{Impact of iron interactions in the global neutrino signal in LVD.}
    \label{fi:iron}
  \end{minipage}
  \hspace{1pc} %%%%% space between two figures
  \begin{minipage}{.48\columnwidth}
    \begin{center}
      \includegraphics[height=17pc]{figmswe.eps}
    \end{center}
    \vspace{-1.cm}
    \caption{Effect of the Earth matter in the $\bar\nu_e p,e^+ n$ signal in LVD.}
    \label{fi:mswe}
  \end{minipage}
\end{figure}


If we consider the effect of Earth in the neutrino path to the detector, we must replace, in the detected flux estimation, $U_{ei}^2$ with $P_{ei}~ (i=1,2)$, the probability for the mass eigenstate $\nu_i$ to be detected as $\nu_e$ after path in the Earth [7], which depends on the solar oscillation parameters and on the travelled density profile through the Earth.
We developed a complete 3-flavour calculation, describing the earth interior as made of 12 equal density steps, following the PREM matter density profile. For each constant density step we compute the exact propagator of the evolution matrix and we get the global amplitude matrix by multiplying the propagators of the traversed density layers, following the strategy of [3].
%In order to measure the Earth matter effect at least two detector at opposite sites in the world must get the signal, and one of them must be shielded by the Earth.
In figure \ref{fi:mswe} the effect of Earth matter in the SN neutrino signal is shown for a nadir angle $\theta_n=50^\circ$, which corresponds to neutrinos passing through the mantle only. Earth matter effect are more relevant in the $\nu$ than in the $\bar \nu$ channel, so the effect in reaction {\em (1)} is quite weak (it also depends on the rather high $\Delta m^2_{{\rm sol}}$), but it could be detected if compared with a high statistic sample (i.e. SuperKamiokande) or if a larger number of events is available, i.e. a closer SN.
% or the use of the SK detector as reference.
% or the SuperKamiokande detector as the reference one ($\sim 20$ times higher signal). 
 

%
%Therefore, the SN neutrino signal could feel the effect of $U_{e3}^2$
%(and could also help to discriminate the type of mass hierarchy)
%\cite{Panta}, \cite{Dutta}, \cite{Dighe}.\\
%\\ \noindent $\star$ We included the active mass of the detector 
%and the energy thresholds, as described in Sect.2.
%We used the following values of detection efficiencies 
%above threshold:

%$\epsilon_{n\, p,d\, \gamma_{2.2}}=50\% $;
%$\epsilon_{\nu_e\, {\rm C},{\rm N}\, e^-}= 85\% $;
%$\epsilon_{\bar\nu_e\, {\rm C},{\rm B}\, e^+}= 70\% $;
%$\epsilon_{\nu_\ell\, \rm{C},\nu_\ell\, \rm{C}}= 55\% $ 
%\cite{ANT91}.
%\\ \noindent $\star$ 
%In the oscillation case, 
%we used two extreme values for $U_{e3}^2$:
%$U_{e3}^2=10^{-2}$ and $U_{e3}^2=10^{-6}$,
%and the above mentioned mixing
%parameters.
% (normal mass hierarchy, LMA solution).


%%%\section{Results}


%%%\section{Conclusions}


\section{References}



%\re
%4.\ Ikegami H.\ et al.\ 1987, in Black Hole, ed.\ Wheeler (Universal Academy
%Press, Tokyo)

%\begin{thebibliography}{99}

%\bibitem{Raffelt1}

%\bibitem{LVD}
\re
1. M. Aglietta et al., {\em Il Nuovo Cimento A} {\bf 105} (1992), 1793.

%\bibitem{taup01}
\re
2. M. Aglietta et al., {\em Nucl. Phys. B Proc. Suppl.} {\bf 110} (2002), 410-413.

%\bibitem{Akhmedov}
\re
3. E. Kh. Akhmedov, {{\tt hep-ph/0001264}}.

%\bibitem{prem}
%\re
%A.M.Dziewonski et al., {\em Phys. Earth Planet. Interior} {\bf 25} (1981), 207.

%\bibitem{Fogli}
\re
4. G. L. Fogli et al.,  {{\tt hep-ph/0212127}}.

%\bibitem{Janka}
\re
5. M.~T. Keil, G.~G. Raffelt, and H.-T. Janka, {{\tt astro-ph/0208035}}.
% {\it Monte carlo study of supernova neutrino spectra formation}, 
%, and references therein.

%\bibitem{Kolbe1} 
\re
6. E. Kolbe, K. Langanke, {{\tt nucl-th/0003060}}.

%\bibitem{Lunard}
\re
7. C. Lunardini, A. Yu. Smirnov,  {{\tt hep-ph/0106149}}.

%\bibitem{Raffelt:2001ai}
%\re
%G.G. Raffelt,
%\newblock Astrophys. J. 561 (2001) 890.

%\re
%G.~G. Raffelt,   {\em Nucl. Phys. B Proc. Suppl.}
%  {\bf 110} (2002) 254--267
%, {{\tt hep-ph/0201099}}, and references therein.

%\bibitem{Buras:2002wt}
%\re
%R. Buras et~al.,
%\newblock astro-ph/0205006.

%\bibitem{VisStrum} 
\re
8. A. Strumia, F. Vissani, {{\tt astro-ph/0302055}}.
%\bibitem{CrossC} 

%\re
%M. Fukugita, Phys. Lett. B 212:139 , (1988)


\re
%\bibitem{Kolbe2} 
9. J. Toivanen et al., {\em Nuclear Physics A} {\bf 694} (2001), 395-408.


%\end{thebibliography}

\endofpaper
\end{document}




hep-ph0307287.extracted_bib
\begin{thebibliography}{99}

%\bibitem{Raffelt1}

%\bibitem{LVD}
\re
1. M. Aglietta et al., {\em Il Nuovo Cimento A} {\bf 105} (1992), 1793.

%\bibitem{taup01}
\re
2. M. Aglietta et al., {\em Nucl. Phys. B Proc. Suppl.} {\bf 110} (2002), 410-413.

%\bibitem{Akhmedov}
\re
3. E. Kh. Akhmedov, {{\tt hep-ph/0001264}}.

%\bibitem{prem}
%\re
%A.M.Dziewonski et al., {\em Phys. Earth Planet. Interior} {\bf 25} (1981), 207.

%\bibitem{Fogli}
\re
4. G. L. Fogli et al.,  {{\tt hep-ph/0212127}}.

%\bibitem{Janka}
\re
5. M.~T. Keil, G.~G. Raffelt, and H.-T. Janka, {{\tt astro-ph/0208035}}.
% {\it Monte carlo study of supernova neutrino spectra formation}, 
%, and references therein.

%\bibitem{Kolbe1} 
\re
6. E. Kolbe, K. Langanke, {{\tt nucl-th/0003060}}.

%\bibitem{Lunard}
\re
7. C. Lunardini, A. Yu. Smirnov,  {{\tt hep-ph/0106149}}.

%\bibitem{Raffelt:2001ai}
%\re
%G.G. Raffelt,
%\newblock Astrophys. J. 561 (2001) 890.

%\re
%G.~G. Raffelt,   {\em Nucl. Phys. B Proc. Suppl.}
%  {\bf 110} (2002) 254--267
%, {{\tt hep-ph/0201099}}, and references therein.

%\bibitem{Buras:2002wt}
%\re
%R. Buras et~al.,
%\newblock astro-ph/0205006.

%\bibitem{VisStrum} 
\re
8. A. Strumia, F. Vissani, {{\tt astro-ph/0302055}}.
%\bibitem{CrossC} 

%\re
%M. Fukugita, Phys. Lett. B 212:139 , (1988)


\re
%\bibitem{Kolbe2} 
9. J. Toivanen et al., {\em Nuclear Physics A} {\bf 694} (2001), 395-408.


%\end{thebibliography}