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% -------------------------------------------------------------------
% Journal abbreviations
%
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\def\prl#1#2#3{ Phys. Rev. Lett. ${\bf{#1}}$: #2 (#3)}
\def\pl#1#2#3{ Phys. Lett. ${\bf{#1}}$: #2 (#3)}
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\def\epj#1#2#3{ Eur. Phys. J. ${\bf{#1}}$: #2 (#3)}
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\def\sjnp#1#2#3{ Sov. J. Nucl. Phys. ${\bf{#1}}$: #2 (#3)}
\def\mpl#1#2#3{ Mod. Phys. Lett. ${\bf{#1}}$: #2 (#3)}
\def\fortp#1#2#3{ Fortsch. Phys. ${\bf{#1}}$: #2 (#3)}
\def\cpc#1#2#3{Comput. Phys. Commun. ${\bf{#1}}$: #2 (#3)}
\def\nuovo#1#2#3{ Nuovo Cim. ${\bf{#1}}$: #2 (#3)}
\def\arnps#1#2#3{Ann. Rev. Nucl. Part. Sci. ${\bf{#1}}$: #2 (#3) }
\def\JPhysG#1#2#3{J. Phys. G ${\bf{#1}}$: #2 (#3)}
\def\JHEP#1#2#3{JHEP ${\bf{#1}}$: #2 (#3)}
\def\AnnPhys#1#2#3{Ann. Phys. (N.Y.)${\bf{#1}}$: #2 (#3)}
\def\ptp#1#2#3{ Prog. Theor. Phys. ${\bf{#1}}$: #2 (#3)}
\def\APPol#1#2#3{ Acta Phys. Pol. ${\bf{#1}}$: #2 (#3)}





\begin{document}
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\begin{flushright}
THES-TP 2002/03 \\
PM/02-55 \\
% hep-ph/xxxx \\
December  2002
\\
 \end{flushright}
\vspace{2cm}
%---------------------title ---------------------------------------
\begin{center}
{\Large\bf  The flavor distribution of
Cosmic Neutrinos\footnote{Partially
supported by EU contract   HPRN-CT-2000-00149, and the
PLATON French-Greek Collaboration project, 2002.}.}
 \vspace{1.5cm}  \\
%-----------------------------------------------------------------
{\large G.J. Gounaris$^a$ and G.  Moultaka$^b$ }\\ \vspace{0.4cm}
$^a$Department of Theoretical Physics, Aristotle University of
Thessaloniki,\\ Gr-541 24, Thessaloniki, Greece.\\
\vspace{0.2cm}
$^b$Physique
Math\'{e}matique et Th\'{e}orique,
UMR 5825\\
Universit\'{e} Montpellier II,
 F-34095 Montpellier, France.\\




\vspace{2.cm}

{\bf Abstract }
\end{center}

\noindent
For cosmic neutrinos, assuming that the neutrino mixing
angles lie  in the vicinity of their
experimentally favored values,  we derive  simple analytical
expressions for their relative flavor
fluxes on Earth, in terms of the
fluxes at the cosmic sites. This enables to disentangle clearly
the  sensitivity to the initial production fluxes as well as
 to small variations of the  mixing angles.
 Such expressions should be useful in facilitating the analysis
 of the physical properties of  cosmic neutrino
  production sites.


\vspace{1cm}
\noindent
PACS number(s):






%
%
%
\def\thefootnote{\arabic{footnote}}
\setcounter{footnote}{0}
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%

The relative flavors of energetic  Neutrinos reaching the Earth,
after they have been emitted at various cosmic sites, provide
useful information on the physical conditions there. Such sites
may consist of exploding Supernovae creating neutrinos
with energies at the ten-MeV range  \cite{Raffelt1},
or more energetic extragalactic  sites like
Gamma Ray Bursts and Active Galactic Nuclei
(AGN), that may produce neutrinos that can reach the $10^3$TeV
\cite{Waxman}, or even the $10^6$TeV scale
\cite{Halzen, Athar, Yasuda, Magnetar}. Galactic candidates
that may emit neutrinos of up to 100 TeV
have also been identified at distances of at least
$2.6 {\rm kpc}$   \cite{galactic-neutrinos, Magnetar}.


It is commonly believed that, these neutrinos are produced mainly
through the
decay of   high energy $\pi^\pm$ and K mesons, which
implies that the initial relative neutrino flavors
 at the cosmic site satisfy
$F^0_\mu/F^0_e\simeq 2$ and $F^0_\tau\simeq 0$,
\cite{Athar, Yasuda, Magnetar}. It may be useful to remember
though that our present understanding of the mechanisms for generating
high energy neutrinos is rather primitive, and sites may exist in
the Universe where the produced neutrinos have
a different initial structure \cite{Halzen, King}.
 The measurement, therefore, of the relative intensities
 of the various neutrino flavors on Earth,
 should provide useful direct information on the mechanism
 responsible for their  generation in the Cosmos.




Once  the various neutrino flavors appear at the surface
of some cosmic object,
 they  propagate oscillating through  space, following the
vacuum oscillation formalism. It will therefore be useful to have
simple  formulae giving  the observable relative
numbers of neutrino flavors  $F_e, ~F_\mu, ~F_\tau$,
in terms of the initially  produced
ones $F^0_e, ~F^0_\mu, ~F^0_\tau$ at the surface of the various
cosmic objects.
Here we present  such  formulae,  assuming only
three active neutrino flavors\footnote{See \eg
\cite{Neutrino-factory}.} which propagate oscillating
among themselves. In particular no cosmic neutrino decay is assumed
\cite{Beacom}.


For deriving  the aforementioned formula,
we  take into account the basic experimental
 characteristics of the neutrino masses and mixings.
These are summarized as follows: The recent SNO \cite{SNO}  data
combined with those of Super-Kamiokande \cite{SuperKamiokande} strongly
favor the LMA MSW   \cite{MSW} solar solution with three active
neutrinos and $\theta_{12}\simeq \pi/6$ and $|m_2^2-m_1^2|\simeq
5 \times 10^{-5} \, eV^2$ \cite{solar}. The atmospheric
neutrino \cite{atmospheric-SuperKamiokande} data imply
$\theta_{23}\simeq \pi/4$ and $|m_3^2-m_2^2|\simeq
2.5 \times 10^{-3} \, eV^2$; while the CHOOZ experiment constrains
$\theta_{13} \lsim 0.2$, \cite{CHOOZ}. Defining then in the
standard notation \cite{Neutrino-factory}
%
\bq
s_{12} \equiv \sin \theta_{12} \equiv \frac{1}{2}+\delta s_{12} ~~~,~~~
s_{23} \equiv  \sin \theta_{23}
\equiv \frac{1}{\sqrt{2}}+\delta s_{23}~~, \label{angle-range-def}
\eq
we find \cite{solar,atmospheric-SuperKamiokande,Neutrino-factory},
%
\bqa
& -0.04 \lsim \delta s_{12} \lsim 0.19 &  ~~, \label{s12-range} \\
& -0.15 \lsim \delta s_{23} \lsim 0.15 & ~~,~~~
 -0.2 \lsim s_{13} \cos\delta \lsim 0.2  ~~, \label{s23-s13-range}
\eqa
where $s_{13}\equiv \sin\theta_{13}$.


For  realistic  neutrino mass differences, and
  neutrino energies in the range
 $  E \lsim 10^6 \, {\rm TeV}$,
the vacuum  oscillation lengths $\lambda_{ij}= 4 \pi
E/|m_i^2-m_j^2|$,   always satisfy
$\lambda_{ij} \lsim 1\, {\rm pc}$,
which is much smaller than
the distances to all cosmic neutrino emitting
sites, beyond our solar system \cite{galactic-neutrinos}.
Consequently, the number of
oscillations performed  by   the cosmic
 neutrinos before arriving at the
Earth, is so  large, that
$\sin^2 (\pi L/ \lambda_{ij})$ average  to $ 1/2$,  and
the CP-violating contributions vanish.

Expanding the standard vacuum oscillation formulae   to first
order in $\delta s_{12}, ~\delta s_{23}$ and $s_{13}$, we get
for  the induced
neutrino relative flavors  on Earth,
%
\bqa
 F_e &=& \frac{3+7 F^0_e}{16}
 -\,\frac{\delta s_{12}}{2} (3 F^0_e-1)
 +\delta s_{123} (F^0_\tau -F^0_\mu)  ~~ , \nonumber  \\
 F_\mu &=& \frac{13-7 F^0_e}{32}
 +\,\frac{\delta s_{12}}{4} (3 F^0_e-1)
 +\delta s_{123}   (F^0_\mu -F^0_e)  ~~ , \nonumber \\
 F_\tau &=& \frac{13-7 F^0_e}{32}
 +\,\frac{\delta s_{12}}{4} (3 F^0_e-1)
 +\delta s_{123}   (F^0_e -F^0_\tau)  ~~ , \label{relative-flavors}
\eqa
%
where
%
\bq
\delta s_{123} \equiv \frac{\sqrt{3}}{8}
 (\sqrt{6} \delta s_{23}-s_{13} \cos\delta ) ~~, \label{a123-def}
\eq
and $F^0_e,~  F^0_\mu, ~ F^0_\tau$ are
the initial neutrino relative flavors
 at the cosmic site.
In writing (\ref{relative-flavors}) we took into account
the unitarity relation
\bq
F_e +F_\mu + F_\tau = F^0_e +F^0_\mu + F^0_\tau =1 ~~,
\label{normalization}
\eq
where   the right hand side  is just a normalization.
Equation (\ref{relative-flavors})
is our basic result. It depends  on two
mixing angle parameters only;  namely $\delta s_{12}$ which is
experimentally constrained by (\ref{s12-range}), and the
combination $\delta s_{123}$
for which (\ref{s23-s13-range}) implies
%
\bq
-0.12 \lsim \delta s_{123} \lsim +0.12 ~~ .\label{a123-range}
\eq
%


We next turn to the discussion of three
interesting specific cases.

\vspace{0.5cm}
Because of unitarity, if the initial relative flavors
satisfy $F^0_e=F^0_\mu=F^0_\tau$, then the final ones  also obey
$F_e=F_\mu=F_\tau=1/3$, irrespective of the neutrino mixing angles.
This is the situation expected \eg for  neutrinos
(or antineutrinos)
generated in supernovae explosions \cite{Raffelt1}.
We call it "Supernova-type case",
allowing it  to cover also the possibility of TeV   neutrino
sources which somehow produce  equal neutrino fluxes
for all neutrino and antineutrino flavors.



\vspace{0.5cm}
High energy neutrinos are commonly assumed to be generated
in connection to the high energy cosmic rays, through some
beam-dumb process producing  unstable mesons (mainly $\pi^\pm$),
which subsequently decay while crossing regions of space with rather small
energy density. In  such regions it is then  expected that
$F^0_e=1/3$, $F^0_\mu=2/3$, $F^0_\tau=0$  \cite{Athar, Yasuda,
Magnetar, Halzen, canonical-F0}, which leads to
%
%
\bqa
&& F_e = \frac{1}{3}(1 ~-~2\,  \delta s_{123})
 ~~ , \nonumber \\
&&  F_\mu=F_\tau  = \frac{1}{3}(1 ~+~\,  \delta s_{123})
~~ .  \label{canonical-relative-flavors}
\eqa
This situation we call "canonical case". The
result (\ref{canonical-relative-flavors})
agrees with the conclusion of \cite{Yasuda, Yasuda1}
that for bimaximal neutrino mixing with very small  $s_{13}$, the
relative neutrino flavor  fluxes are
$F_e \simeq F_\mu\simeq F_\tau\simeq 1/3$.
Our formalism  goes beyond this though, since it predicts  that the
 arriving neutrino fluxes are  independent of
$\delta s_{12}$, and that they only depend
on the specific combination of
$\delta s_{23}$ and $s_{13}\cos \delta$ entering (\ref{a123-def}).


We thus find from (\ref{canonical-relative-flavors}) and the
constraint (\ref{a123-range}), that all relative neutrino fluxes
on Earth should satisfy
%
\bqa
&& 0.25 ~ \lsim F_e ~\lsim ~ 0.41 ~~ , \nonumber \\
&& 0.29 ~ \lsim F_\mu=F_\tau ~\lsim ~ 0.37 ~~ ,
\label{Fe-Fnu-Ftau-range-can}
\eqa
while for $\nu_\mu/\nu_e$ number ratio we get
\bq
0.6 ~ \lsim ~ \frac{F_\mu}{F_e}\simeq (1+3 ~ \delta s_{123})
 ~\lsim  ~1.4 ~~. \label{Fmu-Fe-ratio-range-can}
\eq

It is worthwhile to note that the ranges in
(\ref{Fe-Fnu-Ftau-range-can}, \ref{Fmu-Fe-ratio-range-can}), which
have been derived   analytically, are
very similar to those derived in the
numerical analysis of \cite{Yasuda, Yasuda1}.
A virtue of the present derivation, is that  the
effect of a future reduction of the experimental uncertainties on the
mixing angles, can be straightforwardly read
from (\ref{canonical-relative-flavors}).





As an example, we  note that if  it turns out that \eg
$\delta s_{123}= 0.1$ (compare (\ref{a123-range})), then
(\ref{canonical-relative-flavors}) would imply $F_e=0.27$, and
$F_\mu=F_\tau= 0.37$; which, in a future
sufficiently large  neutrino telescope,  might be possible to discriminate
from the "Supernova-type" case predicting $F_e=F_\mu=F_\tau=0.33$.





\vspace{0.5cm}
As a next rather  exotic  situation we consider the case
 $F^0_e=1$, $F^0_\mu=F^0_\tau=0$. In this  case
%
\bqa
F_e &=&\frac{5}{8} \, - \delta s_{12}~~, \nonumber \\
F_\mu &=&\frac{3}{16} +\frac{\delta s_{12}}{2}
\, - \delta s_{123}  ~~, \nonumber \\
F_\tau &=& \frac{3}{16} +\frac{\delta s_{12}}{2}
\, + \delta s_{123}  ~~ , \label{Fe-1-relative-flavours}
\eqa
%
where, in contrast to the previous situation, the relative neutrino
fluxes have  some sensitivity to $\delta s_{12}$ also. Using
(\ref{s12-range}, \ref{a123-range}) we then find
%
\bqa
&& 0.44 ~ \lsim F_e ~\lsim ~ 0.67 ~~ , \nonumber \\
&& - 0.24 ~ \lsim F_\mu -F_\tau ~\lsim ~ 0.24 ~~ ,
\label{Fe-Fnu-Ftau-range-exotic}
\eqa
in which the presentation has been chosen so that
the uncertainties induced by  $\delta s_{12}$ and $\delta
s_{123}$, are separated.


\vspace{0.5cm}
In the present paper we have assumed just three neutrino flavors
that propagate in space oscillating among themselves, without the presence
of any sterile neutrinos or neutrino decay
processes\footnote{The above formulae
can of course straightforwardly be extended to cases including
sterile neutrinos.}.
Assuming then that the deviations  of the neutrino mixing angles
from their "canonical values" $s_{12}=1/2$,
$s_{23}=1/\sqrt{2}$ and $s_{13}=0$ are small, we
expressed  the observable neutrino fluxes on Earth,
in terms of the original ones at the cosmic sites,
keeping only linear terms in the aforementioned angle-deviations.


The simplicity of these expressions  should
render them  useful in the analysis of future Neutrino
Astronomy data. In particular they may help us  in performing
cosmic scans using
 the physical properties of the neutrino fluxes
arriving  from various directions of the Universe.
They may thus facilitate the analysis of the physical properties
of some intriguing cosmic objects.


\vspace{1cm}
\noindent
\underline{Acknowledgement}\\
One of us (GJG) would like to thank C. Quigg for inspiring
discussions.










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\end{thebibliography}




\end{document}

