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\begin{document}

\BookTitle{\itshape The 28th International Cosmic Ray Conference}
\CopyRight{\copyright 2003 by Universal Academy Press, Inc.}
%\tableofcontents
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\chapter{%   %%%%%%%%% <===== TITLE of the contribution
%%%%%%%%%%% The first letter of each word should be capital letter.
Propagation of Extremely High Energy Leptons in the Earth}

\author{%
%
% You can include as many co-authors as you wish, unless
% the title/author information fits within 1 page.
%
Shigeru Yoshida\\
{\it Department of Physics, Chiba University, 
Chiba 263-8522, Japan}
}%% end of author


\section*{Abstract}

        It has been claimed that underground neutrino telescopes 
such as AMANDA and IceCube have detection capability
of not only PeV neutrinos but EeV-ZeV neutrinos which are possibly
generated by the GZK mechanism associated with the Highest Energy
Cosmic Rays. The accurate calculations on propagation of
neutrinos and produced muons and taus in the Earth is inevitable 
for realistic evaluation of such Extremely High Energy (EHE) neutrino 
detection possibilities, however. The relevant interactions involving 
EHE neutrinos and charged leptons include charged-current interactions
contributing to charged lepton disappearance and the photo-nuclear interactions,
both of which has not been considered or only evaluated by the cross sections
without accounting newly available data of the photon-nucleon collisions.
Here we present the results of the detailed numerical calculations 
concerning particle propagations at EHE energies (EeV or greater).
We use the GZK neutrino flux as a benchmark and show the resultant
energy spectra of EHE neutrinos after their propagation in the Earth
as well as the generated secondary leptons. The expected event rate
above 0.1 EeV by a km cube detector is $\sim$ 1 event/year.

\section{Introduction}

It is well known that there exist extremely high energy (EHE)
particles in the Universe with energies up to $\sim 10^{20}$ eV.
These EHE cosmic rays may be originated and/or producing
neutrinos by the various mechanism.
Underground neutrino telescopes being operated and/or planned
to be built are expected to detect such EHE neutrinos~[1].
In their travel in the earth to the detection volume
in a telescope, EHE neutrinos collide with nuclei in the rock
due to enhancement of the cross section
at EHE range and produce secondary leptons like $\mu$ and $\tau$.
The expected mean free path is 
$\sim 600 (\rho_{rock}/2.65 {\rm g\ cm^{-3}})^{-1} (\sigma_{\nu}/10^{-32} {\rm cm^2})^{-1}$ km which is a far shorter
than a typical path length of the propagation in the earth. Moreover,
the decay lifetime is long enough at EHEs for the produced
$\mu$ and $\tau$ to survive and possibly reach the detection volume
directly. The resultant fluxes and spectrum of EHE particles are rather
complex and the accurate understanding of the EHE neutrino and lepton
propagation in the earth is inevitable for EHE neutrino search 
by the underground neutrino telescopes. Here we numerically calculate
the intensity and the energy distribution of EHE neutrinos and
their secondary produced $\mu$'s and $\tau$'s in the propagation in the earth.
The results are shown for the GZK Neutrinos~[2], 
generated from the decay of photopions produced by EHE cosmic ray
protons colliding with the cosmic thermal background photons,
since the GZK neutrino model is appropriate for a benchmark as 
the flux prediction is on the solid theoretical foundation.

\section{The Relevant Interactions and the Calculation Method}

An EHE neutrino is a subject to charged-current (CC) and neutral-current (NC)
interactions with nucleon.
As there is no direct measurement of the relevant interactions in EHE range,
the predictions of the $\nu N$ cross sections rely on
incompletely tested assumptions about the behavior of parton distributions
at very small values of the momentum fraction $x$.
Since we do not have further clues to investigate EHE neutrino interactions
in our hands, we limit our present analysis within the range of the standard 
particle physics and use the cross section estimated by
the CTEQ version 5 parton distribution~[3] functions.

$\mu$'s and $\tau$'s are the main products of the $\nu N$ interactions.
Electrons and hadrons are also produced, but they initiate shower cascades
immediately in the rock and would not run over the long distance
and their contribution to the overall EHE particle flux after
the propagation is negligible. In EHE range, $\mu$ and $\tau$ are
further interacting with nuclei (and atomic electrons) via
pair creation, Bremsstrahlung, and the photopion production.
The photopion production cross section has the largest uncertainty
at EHEs due to the same reasons for the $\nu N$ interactions.
In this calculation is used the estimation based on 
the deep-inelastic scattering formalism with the ALLM parameterization 
of the structure function, which has been considered to be most reliable
prediction.  Furthermore, the week interaction, $l^{\pm}N\to \nu X$,
to cause muon and tauon disappearances, and the heavier lepton pair production
such as $\mu^+\mu^-$ are also taken into account in the present calculation,
which leads to a visible contribution to the particle fluxes at EHEs.

The calculation of the intensity and energy distribution of propagating
particles in the earth is carried out with numerically resolving
the transport equations as
%=====================================================
{\small
\begin{eqnarray}
{dJ_{\nu}\over dX} &=& -N_A\sigma_{\nu N, CC+NC}J_{\nu}+\
{m_l\over c\rho\tau^d_l}\int dE_l {1\over E_l} {dn^d_l\over dE_{\nu}}J_{l}\
+ N_A \int dE^{'}_{\nu} {d\sigma_{\nu N, NC}\over dE_{\nu}}J_{\nu}\nonumber\\
&& + N_A \int dE^{'}_{l} {d\sigma_{l N}\over dE_{\nu}}J_l\nonumber\\
{dJ_{l}\over dX} &=& -N_A\sigma_{l N}J_{l}-{m_l\over c\rho\tau^d_l E_l}J_{l}\
+ N_A \int dE^{'}_{\nu} {d\sigma_{\nu N, CC}\over dE_{l}}J_{\nu}\
+ N_A \int dE^{'}_{l} {d\sigma_{l N}\over dE_{l}}J_l\nonumber\\
&& +{m_l\over c\rho\tau^d_l}\int dE^{'}_l 
{1\over E^{'}_l} {dn^d_l\over dE_l}J_{l},\nonumber
\end{eqnarray}
}
%=====================================================
where $J_{l}=dN_l/dE_l$ and $J_{\nu}=dN_{\nu}/dE_{\nu}$ are differential
fluxes of charged leptons and neutrinos, respectively, $N_A$ is the Avogadro's number,
$\rho$ is the local density of the medium (rock) in the propagation path,
$\sigma$ is the relevant interaction cross sections, $dn^d_l/dE$
is the energy distribution of the decay products which is derived from
the decay rate per unit energy,
$m_l$ and $\tau^d_l$ are mass and the decay life time of the lepton $l$,
respectively. We numerically calculated these equations by building the matrices
describing the particle propagation over infinitesimal distance as
described in~[4-5].

\section{Results}

%=======================================================================
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=0.7\textwidth,clip=true]{nuMuNuTauToTau_89Deg10EeV.eps}
  \end{center}
  \vspace{-1.5pc}
  \caption{The energy distributions of $\nu_{\tau}$ and 
$\tau$ emerging from the ground for the trajectory with nadir angle
of 89 degree which corresponds to path length of 223 km. 
Primary input spectrum is monochromatic energy distribution of $10^{10}$ GeV
of $\nu_{\mu}$ and $\nu_{\tau}$.}
\end{figure}
%=======================================================================

Fig.1 shows an example of the energy distribution of propagating particles
when spectra of primary EHE neutrinos is monochromatic. The secondary produced
neutrinos, muons, and tauons are clearly seen. It should be remarked that
$\nu_{\tau}$ and $\tau$ from primary $\nu_{\mu}$ are originated in
the heavy lepton production such as $\mu\to\tau^{-}\tau^{+}$ which
becomes sizable in EHE range. These energy distributions as well as
its relative intensities of individual particles give the overall
fluxes and the energy spectra of $\nu,\mu$, and $\tau$ for a given
model of EHE neutrino productions. 

%=======================================================================
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=0.9\textwidth,clip=true]{gzk_4_4.eps}
  \end{center}
  \vspace{-1.5pc}
  \caption{The intensity of the GZK neutrinos and secondary produced
muons and tauons propagating in the earth. Left panel shows the downward going
case while the right panel shows the upward going case.}
\end{figure}
%=======================================================================

Fig.2 shows the fluxes of the GZK neutrinos and the secondary produced
$\mu$ and $\tau$ in the earth. We used the primary flux given in~[5-6].
The downward going case represents when neutrinos enter into 
the earth vertically and propagate through 1500 m under the ground.
If a neutrino telescope is located at 1500 m below the surface,
such neutrinos and leptons blasts an instrumented volume.
In the upward going case the fluxes emerging from the ground
strongly depend on nadir angle of the trajectory as the slant depth of propagation path
varies significantly depending on the geometry.
It is seen that EHE neutrinos are absorbed when
nadir angle of trajectory is 70 degree due to the increasing cross sections
at EHE. 

The secondary produced $\mu$ and $\tau$ during EHE neutrino propagation
in the earth have intensities with $\sim 2.5$ orders of magnitude
lower than neutrinos. $\mu$ and even $\tau$ in EHE region
would not decay but rather interact with nuclei or survive 
to complete their propagation in the earth. Therefore
{\sf these EHE $\mu$ and $\tau$ can be a prime target in EHE neutrino
search}. Below $\sim 10^8$ GeV, $\tau$ becomes more likely to decay
and regenerate $\nu_{\tau}$. Consequently the $\nu_{\tau}$ flux
are higher than the others as clearly seen in the case of 
the downward going trajectories.

Down-going events are of advantage to detection of EHE neutrino induced $\mu$ and $\tau$'s 
due to the wider solid angle factor, if the atmospheric muon background
can be completely excluded. Our results indicates 
$N_{\mu,\tau}(E\geq 0.1EeV)\sim 4\times 10^{-20}$ cm$^{-2}$ sec$^{-1}$ sr$^{-1}$,
which gives $\sim$ 1 event/year in a km cube scale of neutrino telescope
such as IceCube. More study involving the event/background and detector simulation
is necessary but the GZK neutrino signature is certainly 
detectable by a km cube telescope.

\section{References}

\re
1. Alvarez-Mu\~niz J., Halzen F.\ 2001, Phys. Rev. D 63, 037302
\re
2.\ Hill C. T., Schramm D. N.\ 1983, Phys. Lett.B 131, 247
\re
3.\ Lai H.\ et al.\ 1997, Phys. Rev.D 55, 1280
\re
4.\ Protheroe R. J., Johnson P.\ 1996, Astropart. Phys. 4, 253
\re
5.\ Yoshida S., Teshima M.\ 1993, Prog. Theor. Phys. 89, 833
\re
6.\ Yoshida S.\ et al.\ 1997, ApJ 479, 547

\endofpaper
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