Strongly interacting neutrinos as the highest energy cosmic rays \footnote{Talk given at Beyond the Desert '03, Castle Ringberg, 9-14 June, 2003.}
Z. Fodor\inst{1,2}
A. Ringwald\inst{3}
H. Tu\inst{3}
We show that all features of the ultrahigh energy cosmic ray spectrum from $10^{17}$~eV to $10^{21}$~eV can be described with a simple power-like injection spectrum of protons under the assumption that the neutrino-nucleon cross-section is significantly enhanced at center of mass energies above $\approx 100$~TeV. In our scenario, the cosmogenic neutrinos produced during the propagation of protons through the cosmic microwave background initiate air showers in the atmosphere, just as the protons. The total air shower spectrum induced by protons and neutrinos shows excellent agreement with the observations. A particular possibility for a large neutrino-nucleon cross-section exists within the Standard Model through electroweak instanton-induced processes. %Possible tests of this strongly interacting neutrino scenario %range from observations at cosmic ray facilities and neutrino telescopes %to QCD instanton searches at HERA.
ringberg03.tex
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\begin{document}
%
\title*{Strongly interacting neutrinos as the highest energy cosmic rays
\footnote{Talk given at Beyond the Desert '03, Castle Ringberg, 9-14 June, 2003.}}
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\toctitle{Strongly interacting neutrinos as
\protect\newline the highest energy cosmic rays}
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\titlerunning{Strongly interacting neutrinos}
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\author{Z. Fodor\inst{1,2}
\and S.D. Katz\inst{3}\thanks{On leave from Institute for Theor. 
Physics E\"otv\"os University, Budapest, Hungary.}
\and A. Ringwald\inst{3}
\and H. Tu\inst{3}}
%
\authorrunning{Z. Fodor et al.}
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% please abbreviate author list for running head
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\institute{Department of Physics, University of Wuppertal,  
Germany
\and Institute for Theoretical Physics, E\"otv\"os University, 
Budapest, Hungary
\and Deutsches Elektronen-Synchrotron DESY, 
Hamburg, Germany}

\maketitle              % typesets the title of the contribution

\vspace*{-6.5cm}
\normalsize\rightline{WUB 03-11}\rightline{ITP-BUDAPEST 603}
\rightline{DESY 03-163}\rightline{hep-ph/0310112}
\vspace{5cm}

\begin{abstract}
We show that all features of the ultrahigh energy cosmic ray spectrum
from $10^{17}$~eV to $10^{21}$~eV
can be described with a simple power-like injection spectrum of
protons under the assumption that the neutrino-nucleon cross-section
is significantly enhanced at center of mass energies above $\approx
100$~TeV. In our scenario, the cosmogenic neutrinos produced during the
propagation of protons through the cosmic microwave 
background initiate air showers in the atmosphere, just
as the protons.
The total air shower spectrum 
induced by protons and neutrinos
shows excellent agreement with the observations.
A particular possibility for a large neutrino-nucleon cross-section 
exists within the Standard Model through electroweak instanton-induced 
processes.
%Possible tests of this strongly interacting neutrino scenario 
%range from observations at cosmic ray facilities and neutrino telescopes
%to QCD instanton searches at HERA.
\end{abstract}

\section{\label{intro}Introduction}

The spectrum of cosmic rays 
extends in energy up to almost $10^{21}$~eV.
About twenty mysterious events were 
observed above 10$^{20}$~eV
by five different air shower observatories  
(AGA\-SA~\cite{Takeda:1998ps}, Fly's Eye~\cite{Bird:yi},
 Haverah Park~\cite{Lawrence:cc},
 HiRes~\cite{Abu-Zayyad:2002ta},
 and Yakutsk~\cite{Efimov91}).
%for reviews, see Ref.~\cite{Nagano:ve}).
Though some small-angle clustering in the arrival direction of the 
ultrahigh energy cosmic rays (UHECRs) is observed, 
the overall event distribution is
isotropic.
This indicates that they originate from several, isotropically
distributed sources. 

Nucleons produced at large distances
with energies above 
the Greisen-Zatse\-pin-Kuzmin (GZK) cutoff~\cite{Greisen:1966jv}
$E_{\rm GZK}$$\approx 4\cdot 10^{19}$~eV 
interact with the cosmic microwave background (CMB) and produce pions
which decay into neutrinos. This way the nucleons lose their
energy during propagation. The typical interaction
length of nucleons above $E_{{\rm GZK}}$ is around $50$~Mpc. Thus
all events above $10^{20}$~eV should originate from small distances.
However, no source within a distance of 
$50$~Mpc is known in the arrival directions of the post-GZK 
events.
The angular distribution of UHECRs above $E_{\rm GZK}$ 
does not show a correlation
with our galactic plane which also indicates that they originate
from large distances. No conventional explanation exists to the
problem how can they reach us with energies above 
$10^{20}$~eV without an apparent energy loss.
%\footnote{There is an ongoing
%debate whether the excess of events above $E_{\rm GZK}$ is 
%significant and whether the data from different collaborations are 
%consistent~\cite{Bahcall:2002wi}.
%We will comment on this later.}.

At the relevant energies, among the known particles only neutrinos can 
propagate without 
significant energy loss from cosmological distances to us. 
It is this fact which led, on the one hand, 
to scenarios invoking 
hypothetical -- beyond the Standard Model --  
strong interactions of ultrahigh energy 
cosmic 
neutrinos~\cite{Beresinsky:qj} and, on the other hand, to the Z-burst 
scenario~\cite{Fargion:1997ft}.


Interestingly, the flux of 
neutrinos coming from the pions produced during the
propagation of nucleons -- the cosmogenic neutrinos~\cite{Beresinsky:qj} 
-- shows a nice
agreement with the observed UHECR flux 
above 
$E_{{\rm GZK}}$~\cite{Yoshida:pt,Protheroe:1995ft}.
Assuming a large enough neutrino-nucleon cross-section
at these high energies, these neutrinos could initiate extensive
air showers high up in the atmosphere, like hadrons, and
explain the existence of the post-GZK events.
This large cross-section
is usually ensured by new types of TeV-scale 
interactions beyond the Standard Model,
such as 
arising through gluonic bound state 
leptons~\cite{Bordes:1997bt},
TeV-scale grand unification with leptoquarks~\cite{Domokos:2000dp}, 
or Kaluza-Klein modes from 
compactified extra 
dimensions~\cite{Domokos:1998ry} 
(see, however, Ref.~\cite{Kachelriess:2000cb}); 
for earlier and further proposals, see Refs.~\cite{Domokos:1986qy} and 
\cite{Barshay:2001eq}, respectively. 

In this review
we discuss strongly interacting
neutrino scenarios to solve the GZK problem, and in particular
give an example
which -- in contrast to previous proposals -- 
is based entirely on the Standard Model of particle physics. 
It exploits non-perturbative electroweak instanton-induced 
processes
for the interaction of cosmogenic neutrinos with nucleons in the atmosphere, 
which may have a sizeable cross-section above a threshold energy 
$E_{\rm th}={\mathcal O}( (4\pi m_W/\alpha_W )^2)/(2 m_p) = {\mathcal O}( 10^{18})$~eV, 
where 
$m_W$ denotes the W-boson mass and $\alpha_W$ the electroweak fine structure 
constant~\cite{Aoyama:1986ej,Morris:1993wg,Ringwald:2002sw}.  
We present  a 
detailed statistical analysis of the agreement between observations and 
predictions from strongly interacting 
neutrino scenarios.   
 
Our scenario is based on a standard power-like primary spectrum of
protons injected from sources at cosmological distances.
After propagation, these protons will have energies below $E_{{\rm GZK}}$, so
they can well describe the low energy part of the UHECR spectrum.
The cosmogenic neutrinos interact with the atmosphere and thus
give a second component to the UHECR flux, which describes
the high energy part of the spectrum. The relative normalization
of the proton and neutrino fluxes is fixed in this scenario, so
the low and high energy parts of the spectrum are explained 
simultaneously without
any extra normalization.
Details of this analysis can be found in Ref.~\cite{Fodor:2003bn}.

The structure of this review is as follows. In the next section
we give the fluxes of protons and cosmogenic neutrinos both at 
their production and at detection. In Sect. \ref{inst-spect}
the possibility of using electroweak instantons as a source
for large cross-section is discussed and the induced
air shower rate is calculated. In Sect. \ref{comparison}
we compare the predictions with observation and determine the
goodness of fit, while conclusions are given in Sect. ~\ref{conclusions}.

\section{\label{fluxes}Proton and cosmogenic neutrino fluxes}

We start with a power-like 
injection spectrum per co-moving volume  of protons with energy $E_i$, spectral index $\alpha$, 
and redshift ($z$) evolution index $n$,
\begin{equation}
j_p =j_0\,E^{-\alpha}_i\,\left(1+z\right)^n\,\theta(E_{\rm max}-E_i)\,
\theta(z-z_{\rm min})\,\theta(z_{\rm max}-z)\,.
\end{equation}
Here, $j_0$ is a normalization factor, $E_{\rm max}$ is
the maximal energy, which can be reached through 
astrophysical 
accelerating processes in a bottom-up 
scenario, and $z_{\rm min/max}$ takes into account the absence of
nearby/very early sources.
The overall normalization $j_0$ will be fixed by the observed flux, 
and our predictions are quite insensitive to the
specific choice for 
$E_{\rm max}$, $z_{\rm min}$, and $z_{\rm max}$, within their anticipated
values. 
The main sensitivity arises from the spectral parameters $\alpha$ and $n$, 
for which we determine the 1- and 2-sigma confidence regions in 
Sect.~\ref{comparison}.

The propagation of particles can be 
described~\cite{Yoshida:pt,Bahcall:1999ap}
by $P_{b|a} (r,E_i;E )$ functions, which give the expected number of 
particles of type $b$ 
above the threshold energy $E$ if one particle of type $a$ 
started at  a distance $r$ with energy $E_i$. 
With the help of these propagation functions, 
the differential flux of protons ($b=p$) 
and cosmogenic neutrinos 
($b=\nu_i, \bar\nu_i$) at earth
can be given as   
\begin{equation}
\label{flux-earth}
F_{b} ( E ) =\frac{1}{4\pi}
 \int_0^\infty {\rm d}E_i \int_0^\infty {\rm d}r 
\,(-)\frac{\partial P_{b|p}(r,E_i;E)}{\partial E}
\,j_p (r,E_i)\,.
\end{equation}
In our analysis we go, according to ${\rm d}z = (1+z)\,H(z)\,{\rm d}r/c$,  
out to distances $R_{\rm max}$ 
corresponding to $z_{\rm max} = 2$ (cf.~Ref.~\cite{Waxman:1995dg}), while 
we choose $z_{\rm min}=0.012$ in order to take into account the fact that within 
$50$~Mpc there are no astrophysical sources of UHECRs. 
We use the expression 
$H^2(z) = H_0^2\,\left[ \Omega_{M}\,(1+z)^3 
+ \Omega_{\Lambda}\right]$ 
for the relation of the Hubble expansion rate at redshift $z$ to the present one.
Uncertainties of the latter, $H_0=h$ 100 km/s/Mpc, with 
$h=(0.71\pm 0.07)\times^{1.15}_{0.95}$~\cite{Hagiwara:fs}, 
are included. 
$\Omega_{M}$ and $\Omega_{\Lambda}$, with $\Omega_M+\Omega_\Lambda =1$, are the present 
matter and vacuum energy densities in terms of the critical density. As default values we choose
$\Omega_M = 0.3$ and $\Omega_\Lambda = 0.7$, as favored today. Our results
turn out to be rather 
insensitive to the precise values of the cosmological parameters.

We calculated $P_{b|a}(r,E_i;E)$ in two steps. 
{\em i)} First, the SOPHIA
Monte-Carlo program~\cite{Mucke:1999yb} was 
used for the simulation of photohadronic processes of protons with the CMB photons. 
For $e^+e^-$ pair production we used the continuous energy loss approximation, since the
inelasticity is very small ($\approx 10^{-3}$).
We calculated
the $P_{b|a}$ functions for ``infinitesimal'' steps ($1 \mbox{ -- } 10$~kpc) as a function of 
the redshift $z$.
{\em ii)} 
We multiplied the corresponding infinitesimal probabilities  
starting at a distance $r(z)$ down to earth with $z=0$. 
%The details of the calculation of the $P_{b|a}(r,E_i;E )$ functions for
%protons, neutrinos, charged leptons, and photons will be published
%elsewhere~\cite{P:xxx}.

Since the propagation functions are of universal usage, we decided
to make the latest versions of $-\partial P_{b|a}/\partial E$ 
available for the public via the World-Wide-Web URL 
www.desy.de/\~{}uhecr \,.

\section{\label{inst-spect}Neutrino induced air shower rate}

The main assumption of our scenario is that the neutrino-nucleon
cross-section $\sigma_{\nu N}^{\rm tot}$ 
suddenly becomes much larger than $\approx 1$~mb above 
center of mass energies $\sqrt{s} \approx 100$~TeV.
In this case, the corresponding neutrino interaction length 
$\lambda_\nu \equiv m_p/\sigma_{\nu N}^{\rm tot}$, with 
$\sigma_{\nu N}^{\rm tot}=\sigma_{\nu N}^{\rm cc}+ \sigma_{\nu N}^{\rm s}$,
falls below $X_0=1031$~g/cm$^2$ -- the vertical depth 
of the atmosphere at sea level -- above the threshold 
energy $\approx 10^{19}$~eV.  
Here $\sigma_{\nu N}^{\rm cc}$ and
$\sigma_{\nu N}^{\rm s}$ denote the charged current and the new contribution
to the cross-section.
Above the neutrino threshold energy, the atmosphere becomes opaque to cosmogenic neutrinos 
and most of them will end up as 
air showers. Quantitatively, this fact can be described by   
\begin{equation}
\label{shower-rate}
F'_\nu (E) =
F_\nu (E)\,
\left[ 1 - {\rm e}^{- \frac{X(\theta )}{\lambda_\nu (E)}}\right] 
\,,
\end{equation}
which gives 
the spectrum 
of neutrino-initiated air showers, 
for an incident cosmogenic neutrino flux  
$F_\nu =\sum_i [F_{\nu_i}+F_{\bar\nu_i}]$ 
from Eq.~(\ref{flux-earth}), 
in terms of the atmospheric depth $X(\theta )$,  
with $\theta$ being the zenith angle. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURE%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\vspace{-1.4cm}
\begin{center}
\includegraphics*[width=5.5cm,clip=]{sig_i_nuN_alt.eps} \hspace*{0.5cm}
\includegraphics*[width=5.5cm,clip=]{intlength.eps}
\vspace{-0.5cm}
\caption[dum]{
{\em Left:} 
Prediction of the electroweak instanton-induced neutrino-nucleon cross-section 
$\sigma_{\nu N}^{(I)}$ (solid) in comparison with the charged current cross-section 
$\sigma_{\nu N}^{\rm cc}$ (dotted) from Ref.~\cite{Gandhi:1998ri}, as a function of 
the neutrino energy $E_\nu$ in the nucleon's rest frame. 
{\em Right:} Neutrino interaction length due to combined effects of charged current
interactions and instanton-induced processes.
\label{cross-nuN}}
\vspace{-0.5cm}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Such suddenly increasing cross-sections have been proposed
in various models involving physics beyond the Standard 
Model~\cite{Bordes:1997bt,Domokos:2000dp,Domokos:1998ry,Kachelriess:2000cb,Domokos:1986qy,Barshay:2001eq}.
In Fig.~\ref{cross-nuN} we show another example which is based entirely 
on the Standard Model exploiting non-perturbative electroweak 
instanton-induced processes~\cite{Aoyama:1986ej,Morris:1993wg,Ringwald:2002sw}.
Our quantitative analysis~\cite{Fodor:2003bn} was based on 
this cross-section, however it is expected to be insensitive to the 
exact form of it as long as it rises abruptly far above 1~mb.
Note that such a behaviour is consistent with present upper bounds
on electroweak instanton-induced cross-sections~\cite{Bezrukov:2003er}.
It should be noted that such a cross-section will lead, 
via dispersion relations, to 
lower energy deviations of Standard Model predictions for elastic 
scattering from their perturbative values. 
However, it is easily checked that, for the one shown in 
Fig.~\ref{cross-nuN} (left), 
these corrections will be unobservably small in the energy regime available at 
present accelerators~\cite{Goldberg:1998pv}.


\section{\label{comparison}Comparison with UHECR data}

%In this section, we compare the predicted air shower spectrum from Eqs.~(\ref{flux-earth}) and 
%(\ref{shower-rate}), the latter averaged over the appropriate range of 
%zenith angles $\theta$, 
The predicted air shower rate induced by protons and neutrinos
is given by
\begin{equation}
\label{flux-pred}
F_{\rm pred} (E; \alpha , n, E_{\rm max},  z_{\rm min}, z_{\rm max}, j_0 ) 
= F_{p} ( E; \ldots  ) + F'_{\nu} (E; \ldots )\,.
\end{equation}
We performed a statistical 
analysis to compare (\ref{flux-pred})
with the observations 
and presented a measure for the goodness of the scenario~\cite{Fodor:2003bn}. 
We gave the best fit to the observations and the 1- and 2-sigma
confidence regions in the ($\alpha$,$n$) plane.


UHECR collaborations usually publish their results for the
detected fluxes in a binned form. The first step of our analysis
is to convert these fluxes into event numbers in each bin. 
We use the most recent results of the HiRes and AGASA
collaborations and do our analysis separately with both data sets.
We use the energy range $10^{17.2}$~eV -- $10^{21}$~eV which is divided
into 38 equal logarithmic bins. 
In the low energy region, there are no 
published results available from AGASA and only low statistics results
from HiRes-2.
Therefore, we included the results of the predecessor 
collaborations -- Akeno~\cite{Nagano:1991jz} and Fly's Eye, 
respectively -- into the analysis. 
%With a small normalization correction,  
%it was possible to continuously 
%connect the AGASA data~\cite{Takeda:1998ps} with the Akeno ones and the 
%HiRes-1 monocular data~\cite{Abu-Zayyad:2002ta} with 
%the Fly's Eye stereo ones~\cite{Bird:yi},
%respectively (cf. Fig.~\ref{fit}). 

The goodness of the scenario is 
determined by a statistical analysis. 
We determined the compatibility of different
($\alpha$,$n$) pairs with the experimental data.
For some fixed ($\alpha$,$n$) pair, the expected number of events in 
individual bins are (${\bf \lambda}=\{\lambda_1,...,\lambda_r\}$ with
$r$ being the total number of bins (in our case 38).
The probability of getting an experimental outcome 
${\bf k}=\{k_1,...k_r\}$ (where $k_i$ are non-negative integer numbers)
is given by the probability distribution function $P({\bf k})$,
which is just the product of Poisson
distributions for the individual bins.
It is easy to include also the $\approx 30\%$ overall energy uncertainty
into the $P({\bf k})$ probability distribution. 
We denote the experimental result 
by ${\bf s}=\{s_1,...s_r\}$, where the $s_i$-s are
non-negative, integer numbers. 
%The probability of this
%outcome, $P({\bf n})$ is usually very small.
The ($\alpha$,$n$) pair is compatible with the 
experimental results if 
\begin{equation}\label{summation}
\sum_{{\bf k}|P({\bf k})>P({\bf s})}P({\bf k})<c\,.
\end{equation}
For a 1-(or 2-)sigma compatibility one takes c=0.68 
(or c=0.95), respectively. 
The best fit is found by minimizing the sum on the left hand side.

Since we have 38 variables, it is practically impossible to calculate
the sum in equation (\ref{summation}) exactly. 
If we rewrite it as
\begin{equation}\label{monte-carlo}
\sum_{{\bf k}|P({\bf k})>P({\bf s})}P({\bf k})=
{\sum_{{\bf k}} P({\bf k})\, \theta[P({\bf k})-P({\bf s})]
\over \sum_{{\bf k}} P({\bf k})}\,,
\end{equation}
then we can calculate this sum approximately using
an importance sampling based Monte-Carlo technique.
We have to generate the components of ${\bf k}$ 
with Poisson distribution and take only those in the sum, for which
$P({\bf k})>P({\bf s})$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURE%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
%\vspace{-1.6cm}
\begin{center}
\includegraphics*[width=5.0cm,clip=]{E3fits.eps} \hspace*{1cm}
\includegraphics*[width=5.0cm,clip=]{regions.eps}
%\vspace{-1.5cm}
\caption[dum]{{\em Left:} Ultrahigh energy cosmic ray data 
(Akeno + AGASA on the upper panel and Fly's Eye + HiRes on the lower
panel)
and their best fits within the electroweak instanton scenario (solid) for 
$E_{\rm max}=3\cdot 10^{22}$~eV, $z_{\rm min}=0.012$, $z_{\rm max}=2$
consisting of a proton component (dotted) plus a cosmogenic 
neutrino-initiated component (dashed).  
{\em Right:} Confidence regions in the $\alpha$--$n$ plane 
for fits to the Akeno + AGASA data (2-sigma (long dashed))
and to the Fly's Eye + HiRes 
data (1-sigma (solid); 2-sigma (short-dashed)), respectively.
\label{fit}}
\vspace{-0.6cm}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Figure~\ref{fit} (left) shows our best fits for the AGASA and for 
the HiRes UHECR data. The best fit values are $\alpha=2.68(2.68)$ and 
$n=2.65(2.9)$, for AGASA(HiRes).
We can see very nice agreement with the data
within an energy range of nearly four orders of magnitude. 
The fits are insensitive to the value of $E_{\rm max}$ as far as
we choose a value above $\approx 3\cdot 10^{21}$~eV.
The shape of the curve between $10^{17}$~eV and $10^{19}$~eV is mainly
determined by the redshift evolution index $n$. At these energies the
universe is already transparent for protons created at $z\approx 0$ while
protons from sources with larger redshift accumulate in this region.
%The larger $n$ is the stronger this accumulation should be.
The peak around $4\cdot 10^{19}$~eV shows the accumulation of particles
due to the GZK effect. Neutrinos start to dominate over protons at around 
$10^{20}$~eV.

It is important to note that, if we omit the neutrino component, 
then the model is ruled out on the 3-sigma level for both experiments. 
This is due to the fact that we excluded nearby sources by setting
$z_{\rm min} \neq 0$ (see also Ref.~\cite{Kachelriess:2003yy}). 
A choice of $z_{\min}=0$ makes the HiRes data 
compatible with a proton-only scenario on the 2-sigma level 
(see also Refs.~\cite{Abu-Zayyad:2002ta,Bahcall:2002wi}). 

Figure~\ref{fit} (right) 
displays the confidence regions in the ($\alpha$,$n$)
plane
for AGASA and HiRes. The scenario is consistent on the 2-sigma level with
both experiments. For HiRes, the compatibility is even true on the 1-sigma level.
It is important to note that both experiments favor the same values for $\alpha$ and 
$n$, demonstrating their mutual compatibility on the 2-sigma level 
(see also Ref.~\cite{DeMarco:2003ig}). 
%If we ignore the energy uncertainty in the determination of the goodness 
%of the fit, they turn out to be inconsistent.

Finally, let us discuss the consistency of our scenario with the currently
available limits on deeply penetrating showers from Fly's Eye~\cite{Baltrusaitis:mt}
and AGASA~\cite{Yoshida:2001icrc}. 
Taking into account -- in distinction to Ref.~\cite{Anchordoqui:2002vb} --  
the atmospheric attenuation  of the cosmogenic neutrino flux predicted in our scenario 
and the uncertainties in the estimate of the range of depth within which the shower must originate to
trigger the array, we find  that
AGASA should have seen $1 \mbox{ -- } 10$ quasi-horizontal air 
showers ($\theta \gwig 60^{\circ}$) from the electroweak instanton-induced processes
during a running time of $1710.5$~days. 
This is consistent with AGASA's present analysis of their respective data~\cite{Yoshida:2001icrc}.
The Fly's Eye upper limit on the product of the total neutrino flux times 
neutrino-nucleon cross-section, 
$(F_{\nu}\, \sigma_{\nu N}^{\rm tot})_{\rm Fly's\ Eye}$~\cite{Baltrusaitis:mt}, 
in the energy range $10^{17 \mbox{ -- } 20}$~eV, 
can be translated, for a given predicted neutrino flux $F_\nu^{\rm pred}$, 
into an upper limit on 
$\sigma_{\nu N}^{\rm tot}<(F_{\nu}\, \sigma_{\nu N}^{\rm tot})_{\rm Fly's\ Eye}/F_\nu^{\rm pred}$, 
as long as it is smaller than 
$10$~$\mu$b~\cite{Morris:1993wg,Tyler:2000gt}. 
We find that, for our predicted cosmogenic neutrino flux, 
the right-hand-side of this inequality is larger than $10$~$\mu$b in the whole
energy range, such that the
Fly's Eye non-observation of quasi-horizontal air showers does not give any constraint. 
We therefore conclude that our prediction 
of the neutrino-nucleon cross-section, as shown in Fig.~\ref{cross-nuN} (left),
does not contradict any constraints from cosmic ray experiments 
so far, as long as the ultrahigh energy cosmic neutrino flux is at the
cosmogenic level we have predicted. 

\section{\label{conclusions}Summary and conclusions}

We have shown that a simple scenario with a single power-like 
injection spectrum of protons can describe all 
the features of the UHECR spectrum 
in the energy range $10^{17 \mbox{ -- } 21}$~eV. 
In our scenario, the  injected protons
produce neutrinos during their propagation and these
neutrinos are assumed to have large enough cross-section to 
produce air showers high up in the atmosphere. As an example
we discussed the possibility that Standard Model electroweak instanton-induced
processes may give a cross section which is suitable for this scenario.
The model has few parameters from which only two -- the power index $\alpha$ and the redshift  
evolution index $n$ -- has a strong effect on the final shape of the spectrum.
We found that for certain values of $\alpha$ and $n$ this scenario is 
compatible 
with the available observational data from the AGASA and HiRes experiments 
(combined with their predecessor  experiments, 
Fly's Eye and Akeno, respectively) on the 2-sigma level (also 1-sigma for
HiRes). 
The ultrahigh energy neutrino component can be experimentally tested by 
studying the zenith angle dependence of the events in the range 
$10^{18 \mbox{ -- } 20}$~eV and possible correlations with distant
astrophysical sources~\cite{Tinyakov:2001nr}
at cosmic ray facilities such as the Pierre Auger 
Observatory~\cite{Zavrtanik:2000zi}, 
and by looking for enhanced rates for throughgoing muons
at neutrino telescopes such as AMANDA~\cite{Andres:2001ty}.  
As laboratory tests, one may search for a model-independent enhancement 
in (quasi-)elastic lepton-nucleon scattering~\cite{Goldberg:1998pv}
or for signatures of QCD instanton-induced processes in deep-inelastic
scattering~\cite{Ringwald:1994kr}, e.g. at HERA.

%Finally, let us emphasize that the same fit results are valid
%for all strongly interacting neutrino models if the neutrino-nucleon 
%cross-section has a similar threshold-like behavior as in Fig.~\ref{cross-nuN}. 
%The instanton scenario, however, has the advantage that it is based solely
%on the Standard Model and can be falsified in the near future by 
%a negative outcome of QCD instanton searches at HERA.


\section*{Acknowledgments}
This work was partially supported by Hungarian
Science Foundation grants No. OTKA-T034980/T037615.


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\end{thebibliography}

\end{document}

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hep-ph0310112.extracted_bib
\begin{thebibliography}{99}

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%http://www-akeno.icrr.u-tokyo.ac.jp/AGASA/
%\,; 
%date: 24$^{\rm th}$ February 2003.

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D.~J.~Bird {\it et al.},  %[HIRES Collaboration],
%``Evidence For Correlated Changes In The Spectrum And Composition Of Cosmic Rays At Extremely High-Energies,''
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%%CITATION = PRLTA,71,3401;%%
%
%\cite{Bird:wp}
%\bibitem{Bird:wp}
%D.~J.~Bird {\it et al.}  [HIRES Collaboration],
%``The Cosmic Ray Energy Spectrum Observed By The Fly's Eye,''
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%%CITATION = ASJOA,424,491;%%
%
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%\bibitem{Bird:1994uy}
%D.~J.~Bird {\it et al.},
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%Astrophys.\ J.\  
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%%CITATION = ASJOA,441,144;%%

%\cite{Lawrence:cc}
\bibitem{Lawrence:cc}
M. A.~Lawrence, R. J.~Reid, A. A.~Watson,
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%
%\cite{Ave:2000nd}
%\bibitem{Ave:2000nd}
M.~Ave, {\em et al.},
%``New constraints from Haverah Park data on the photon and iron fluxes of  UHE cosmic rays,''
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%\cite{Abu-Zayyad:2002ta}
\bibitem{Abu-Zayyad:2002ta}
T.~Abu-Zayyad {\it et al.}  %[High Resolution Fly's Eye Collaboration],
[HiRes Collaboration], 
%``Measurement of the flux of ultrahigh energy cosmic rays from monocular  
%observations by the High Resolution Fly's Eye experiment,''
%arXiv:
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%%CITATION = ASTRO-PH 0208243;%%
%
%\cite{Abu-Zayyad:2002sf}
%\bibitem{Abu-Zayyad:2002sf}
%T.~Abu-Zayyad {\it et al.}  [High Resolution Fly's Eye Collaboration],
%``Measurement of the spectrum of UHE cosmic rays by the FADC detector of  the HiRes experiment,''
%arXiv:
astro-ph/0208301.
%%CITATION = ASTRO-PH 0208301;%%


\bibitem%Efimov et al.(1991)]
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K.~Greisen,
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%%CITATION = PRLTA,16,748;%%
%
%\bibitem{Zatsepin:1966jv}
G.~T.~Zatsepin and V.~A.~Kuzmin,
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%%CITATION = JTPLA,4,78;%%


%\cite{Beresinsky:qj}
\bibitem{Beresinsky:qj}
V.~S.~Beresinsky and G.~T.~Zatsepin,
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%``Ultrahigh energy neutrino scattering onto relic light neutrinos in  galactic halo 
%as a possible source of highest energy extragalactic  cosmic rays,''
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%
%\cite{Weiler:1997sh}
%\bibitem{Weiler:1997sh}
T.~J.~Weiler,
%``Cosmic ray neutrino annihilation on relic neutrinos revisited:  
%A mechanism for generating air showers above the  Greisen-Zatsepin-Kuzmin cut-off,''
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%
%\cite{Yoshida:1998it}
%\bibitem{Yoshida:1998it}
S.~Yoshida, G.~Sigl and S.~J.~Lee,
%``Extremely high energy neutrinos, neutrino hot dark matter, and the  highest energy cosmic rays,''
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%[arXiv:hep-ph/9808324].
%%CITATION = HEP-PH 9808324;%%
%\cite{Fodor:2001qy}
%\bibitem{Fodor:2001qy}
Z.~Fodor, S.~D.~Katz and A.~Ringwald,
%``Determination of absolute neutrino masses from Z-bursts,''
Phys.\ Rev.\ Lett.\  {\bf 88} (2002) 171101; 
%[arXiv:hep-ph/0105064].
%%CITATION = HEP-PH 0105064;%%
%
%\bibitem{Fodor:2002hy}
%Z.~Fodor, S.~D.~Katz and A.~Ringwald,
%``Relic neutrino masses and the highest energy cosmic rays,''
JHEP {\bf 0206} (2002) 046;
%[arXiv:hep-ph/0203198].
%%CITATION = HEP-PH 0203198;%%
%\cite{Kalashev:2001sh}
%\bibitem{Kalashev:2001sh}
O.~E.~Kalashev {\em et al.},
%``Ultra-high energy cosmic rays from neutrino emitting acceleration  sources?,''
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%[arXiv:hep-ph/0112351].
%%CITATION = HEP-PH 0112351;%%

%\cite{Yoshida:pt}
\bibitem{Yoshida:pt}
S.~Yoshida and M.~Teshima,
%``Energy Spectrum Of Ultrahigh-Energy Cosmic Rays With Extragalactic Origin,''
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%%CITATION = PTPKA,89,833;%%
%
%\cite{Protheroe:1995ft}
\bibitem{Protheroe:1995ft}
R.~J.~Protheroe and P.~A.~Johnson,
%``Propagation of ultrahigh-energy protons over cosmological distances and implications for topological defect models,''
Astropart.\ Phys.\  {\bf 4} (1996) 253;
%[arXiv:astro-ph/9506119].
%%CITATION = ASTRO-PH 9506119;%%
%
%\cite{Yoshida:1996ie}
%\bibitem{Yoshida:1996ie}
S.~Yoshida {\em et al.},
%``Extremely high energy neutrinos and their detection,''
Astrophys.\ J.\  {\bf 479} (1997) 547;
%[arXiv:astro-ph/9608186].
%%CITATION = ASTRO-PH 9608186;%%
%
%\cite{Engel:2001hd}
%\bibitem{Engel:2001hd}
R.~Engel and T.~Stanev,
%``Neutrinos from propagation of ultra-high energy protons,''
Phys.\ Rev.\ D {\bf 64} (2001) 093010;
%[arXiv:astro-ph/0101216].
%%CITATION = ASTRO-PH 0101216;%%
%\cite{Kalashev:2002kx}
%\bibitem{Kalashev:2002kx}
O.~E.~Kalashev {\em et al.}, 
%``Ultra-high energy neutrino fluxes and their constraints,''
Phys.\ Rev.\ D {\bf 66} (2002) 063004;
%[arXiv:hep-ph/0205050].
%%CITATION = HEP-PH 0205050;%%
%\cite{Fodor:2003ph}
%\bibitem{Fodor:2003ph}
Z.~Fodor, S.~D.~Katz, A.~Ringwald and H.~Tu,
%``Bounds on the cosmogenic neutrino flux,''
hep-ph/0309171;
%%CITATION = HEP-PH 0309171;%%
%\cite{Semikoz:2003wv}
%\bibitem{Semikoz:2003wv}
D.~Semikoz and G.~Sigl,
%``Ultra-high energy neutrino fluxes: New constraints and implications,''
hep-ph/0309328.
%%CITATION = HEP-PH 0309328;%%

\bibitem{Bordes:1997bt}
J.~Bordes {\em et al.},  
%, H.~M.~Chan, J.~Faridani, J.~Pfaudler and S.~T.~Tsou,
%``Possible explanation for E > 10**20-eV air showers with flavor-changing  neutral currents,''
%arXiv:
hep-ph/9705463;
%%CITATION = HEP-PH 9705463;%%
%
%\cite{Bordes:1997rx}
%\bibitem{Bordes:1997rx}
%J.~Bordes, H.~M.~Chan, J.~Faridani, J.~Pfaudler and S.~T.~Tsou,
%``Possible test for the suggestion that air showers with E > 10**20-eV  are due to strongly interacting neutrinos,''
Astropart.\ Phys.\  {\bf 8} (1998) 135.
%[arXiv:astro-ph/9707031].
%%CITATION = ASTRO-PH 9707031;%%



\bibitem{Domokos:2000dp}
G.~Domokos, S.~Kovesi-Domokos and P.~T.~Mikulski,
%``Ultrahigh energy neutrinos and the highest energy cosmic rays,''
%arXiv:
hep-ph/0006328.
%%CITATION = HEP-PH 0006328;%%

%ar24/02
\bibitem{Domokos:1998ry}
G.~Domokos and S.~Kovesi-Domokos,
%``Strongly interacting neutrinos and the highest energy cosmic rays,''
Phys.\ Rev.\ Lett.\  {\bf 82} (1999) 1366;
%[arXiv:hep-ph/9812260].
%%CITATION = HEP-PH 9812260;%%
%
%\bibitem{Nussinov:1998jt}
S.~Nussinov and R.~Shrock,
%``Some remarks on theories with large compact dimensions and TeV-scale  quantum gravity,''
Phys.\ Rev.\ D {\bf 59} (1999) 105002; 
%[arXiv:hep-ph/9811323].
%%CITATION = HEP-PH 9811323;%%
%
%\bibitem{Jain:2000pu}
P.~Jain {\em et al.},
%``Extra dimensions and strong neutrino nucleon interactions above  10**19-eV: Breaking the GZK barrier,''
Phys.\ Lett.\ B {\bf 484} (2000) 267.
%[arXiv:hep-ph/0001031].
%%CITATION = HEP-PH 0001031;%%

%\cite{Kachelriess:2000cb}
\bibitem{Kachelriess:2000cb}
M.~Kachelriess and M.~Pl\"umacher,
%``Ultrahigh energy neutrino interactions and weak-scale string theories,''
Phys.\ Rev.\ D {\bf 62} (2000) 103006;
%[arXiv:astro-ph/0005309].
%%CITATION = ASTRO-PH 0005309;%%
%
%\cite{Anchordoqui:2000uh}
%\bibitem{Anchordoqui:2000uh}
L.~Anchordoqui {\em et al.},
%``Extensive air showers with TeV-scale quantum gravity,''
Phys.\ Rev.\ D {\bf 63} (2001) 124009.
%[arXiv:hep-ph/0011097].
%%CITATION = HEP-PH 0011097;%%

%\cite{Domokos:1986qy}
\bibitem{Domokos:1986qy}
G.~Domokos and S.~Nussinov,
%``Do Neutrinos Interact Strongly?,''
Phys.\ Lett.\ B {\bf 187} (1987) 372.
%%CITATION = PHLTA,B187,372;%%

%\cite{Barshay:2001eq}
\bibitem{Barshay:2001eq}
S.~Barshay and G.~Kreyerhoff,
%``Stronger neutrino interactions at extremely high energies and the muon  anomalous magnetic moment,''
Eur.\ Phys.\ J.\ C {\bf 23} (2002) 191;
%[arXiv:hep-ph/0106047].
%%CITATION = HEP-PH 0106047;%%
%
%\cite{Barshay:2002hu}
%\bibitem{Barshay:2002hu}
%S.~Barshay and G.~Kreyerhoff,
%``Effect of possible stronger neutrino interaction at E(nu/mu) approx.  
%10**11-GeV upon the extraction of sin**2(theta(W)) from the
%neutral-current cross-section at NuTeV,''
Phys.\ Lett.\ B {\bf 535} (2002) 201;
%[arXiv:hep-ph/0203054].
%%CITATION = HEP-PH 0203054;%%
%
%\cite{Anchordoqui:2002it}
%\bibitem{Anchordoqui:2002it}
L.~A.~Anchordoqui, J.~L.~Feng and H.~Goldberg,
%``p-branes and the GZK paradox,''
Phys.\ Lett.\ B {\bf 535} (2002) 302.
%[arXiv:hep-ph/0202124].
%%CITATION = HEP-PH 0202124;%%

%\cite{Aoyama:1986ej}
\bibitem{Aoyama:1986ej}
H.~Aoyama and H.~Goldberg,
%``Anomalous Baryon Number Nonconservation In P P Collisions At 40-Tev,''
Phys.\ Lett.\ B {\bf 188} (1987) 506;
%%CITATION = PHLTA,B188,506;%%
%
%\cite{Ringwald:1989ee}
%\bibitem{Ringwald:1989ee}
A.~Ringwald,
%``High-Energy Breakdown Of Perturbation Theory In The Electroweak Instanton Sector,''
Nucl.\ Phys.\ B {\bf 330} (1990) 1; 
%%CITATION = NUPHA,B330,1;%%
%
%\cite{Espinosa:qn}
%\bibitem{Espinosa:qn}
O.~Espinosa,
%``High-Energy Behavior Of Baryon And Lepton Number Violating Scattering 
%Amplitudes And Breakdown Of Unitarity In The Standard Model,''
Nucl.\ Phys.\ B {\bf 343} (1990) 310; 
%%CITATION = NUPHA,B343,310;%%
%
%\cite{Khoze:1991mx}
%\bibitem{Khoze:1991mx}
V.~V.~Khoze and A.~Ringwald,
%``Nonperturbative contribution to total cross-sections in nonAbelian gauge theories,''
Phys.\ Lett.\ B {\bf 259} (1991) 106. 
%%CITATION = PHLTA,B259,106;%%
%

%\cite{Morris:1993wg}
\bibitem{Morris:1993wg}
D.~A.~Morris and A.~Ringwald,
%``Cosmic ray signatures of multi - W processes,''
Astropart.\ Phys.\  {\bf 2} (1994) 43.
%[arXiv:hep-ph/9308269].
%%CITATION = HEP-PH 9308269;%%


%\cite{Ringwald:2002sw}
\bibitem{Ringwald:2002sw}
A.~Ringwald,
%``Electroweak instantons / sphalerons at VLHC?,''
Phys.\ Lett.\ B {\bf 555} (2003) 227.
%[arXiv:hep-ph/0212099].
%%CITATION = HEP-PH 0212099;%%

%\cite{Fodor:2003bn}
\bibitem{Fodor:2003bn}
Z.~Fodor, S.~D.~Katz, A.~Ringwald and H.~Tu,
%``Electroweak instantons as a solution to the ultrahigh energy cosmic ray  puzzle,''
Phys.\ Lett.\ B {\bf 561} (2003) 191.
%[arXiv:hep-ph/0303080].
%%CITATION = HEP-PH 0303080;%%

%\cite{Bahcall:1999ap}
\bibitem{Bahcall:1999ap}
J.~N.~Bahcall and E.~Waxman,
%``Ultra-high energy cosmic rays may come from clustered sources,''
Astrophys.\ J.\  {\bf 542} (2000) 543; 
%[arXiv:hep-ph/9912326].
%%CITATION = HEP-PH 9912326;%%
%
%\cite{Fodor:2000yi}
%\bibitem{Fodor:2000yi}
Z.~Fodor and S.~D.~Katz,
%``Ultra high energy cosmic rays from compact sources,''
Phys.\ Rev.\ D {\bf 63} (2001) 023002.
%[arXiv:hep-ph/0007158].
%%CITATION = HEP-PH 0007158;%%

%\cite{Waxman:1995dg}
\bibitem{Waxman:1995dg}
E.~Waxman,
%``Cosmological origin for cosmic rays above 10**19-eV,''
Astrophys.\ J.\  {\bf 452} (1995) L1.
%[arXiv:astro-ph/9508037].
%%CITATION = ASTRO-PH 9508037;%%

%\cite{Hagiwara:fs}
\bibitem{Hagiwara:fs}
K.~Hagiwara {\it et al.}  [Particle Data Group],
%``Review Of Particle Physics,''
Phys.\ Rev.\ D {\bf 66} (2002) 010001.
%%CITATION = PHRVA,D66,010001;%%

%\cite{Mucke:1999yb}
\bibitem{Mucke:1999yb}
A.~M\"ucke {\em et al.}, 
%, R.~Engel, J.~P.~Rachen, R.~J.~Protheroe and T.~Stanev,
%``Monte Carlo simulations of photohadronic processes in astrophysics,''
Comput.\ Phys.\ Commun.\  {\bf 124} (2000) 290.
%[arXiv:astro-ph/9903478].
%%CITATION = ASTRO-PH 9903478;%%

%\cite{P:xxx}
%\bibitem{P:xxx}
%Z.~Fodor, S.~D.~Katz and A.~Ringwald, in preparation.

%\cite{Gandhi:1998ri}
\bibitem{Gandhi:1998ri}
R.~Gandhi, C.~Quigg, M.~H.~Reno and I.~Sarcevic,
%``Neutrino interactions at ultrahigh energies,''
Phys.\ Rev.\ D {\bf 58} (1998) 093009.
%[hep-ph/9807264].
%%CITATION = HEP-PH 9807264;%%

%\cite{Bezrukov:2003er}
\bibitem{Bezrukov:2003er}
F.~Bezrukov {\em et al.},
%``Semiclassical study of baryon and lepton number violation in  high-energy electroweak collisions,''
Phys.\ Rev.\ D {\bf 68} (2003) 036005;
%[arXiv:hep-ph/0304180]
%%CITATION = HEP-PH 0304180;%%
%\cite{Ringwald:2003ns}
%\bibitem{Ringwald:2003ns}
A.~Ringwald,
%``An upper bound on the total cross-section for electroweak baryon number  violation,''
hep-ph/0307034;
%%CITATION = HEP-PH 0307034;%%
%\cite{Han:2003ru}
%\bibitem{Han:2003ru}
T.~Han and D.~Hooper,
%``Effects of electroweak instantons in high-energy neutrino telescopes,''
hep-ph/0307120.
%%CITATION = HEP-PH 0307120;%%


%\cite{Goldberg:1998pv}
\bibitem{Goldberg:1998pv}
H.~Goldberg and T.~J.~Weiler,
%``Lower energy consequences of an anomalous high-energy neutrino  cross-section,''
Phys.\ Rev.\ D {\bf 59} (1999) 113005.
%[arXiv:hep-ph/9810533].
%%CITATION = HEP-PH 9810533;%%

%\cite{Nagano:1991jz}
\bibitem{Nagano:1991jz}
M.~Nagano {\it et al.},
%``Energy spectrum of primary cosmic rays above 10**17-eV determined from the extensive air shower experiment at Akeno,''
J.\ Phys.\ G {\bf 18} (1992) 423.
%%CITATION = JPHGB,G18,423;%%

%\cite{Kachelriess:2003yy}
\bibitem{Kachelriess:2003yy}
M.~Kachelriess, D.~V.~Semikoz and M.~A.~Tortola,
%``New hadrons as ultra-high energy cosmic rays,''
Phys.\ Rev.\ D {\bf 68} (2003) 043005.
%[arXiv:hep-ph/0302161].
%%CITATION = HEP-PH 0302161;%%

%\cite{Bahcall:2002wi}
\bibitem{Bahcall:2002wi}
J.~N.~Bahcall and E.~Waxman,
%``Has the GZK cutoff been discovered?,''
Phys.\ Lett.\ B {\bf 556} (2003) 1.
%[arXiv:hep-ph/0206217].
%%CITATION = HEP-PH 0206217;%%


%\cite{DeMarco:2003ig}
\bibitem{DeMarco:2003ig}
D.~De Marco, P.~Blasi and A.~V.~Olinto,
%``On the statistical significance of the GZK feature in the spectrum of  ultra high energy cosmic rays,''
Astropart.\ Phys.\  {\bf 20} (2003) 53.
%[arXiv:astro-ph/0301497].
%%CITATION = ASTRO-PH 0301497;%%

%\cite{Baltrusaitis:mt}
\bibitem{Baltrusaitis:mt}
R.~M.~Baltrusaitis {\it et al.},
%``Limits On Deeply Penetrating Particles In The > 10**17-Ev Cosmic Ray Flux,''
Phys.\ Rev.\ D {\bf 31} (1985) 2192.
%%CITATION = PHRVA,D31,2192;%%


%\cite{Yoshida:2001icrc}
\bibitem{Yoshida:2001icrc}
S.~Yoshida {\it et al.} [AGASA Collaboration], in
{\it Proc. 27th International Cosmic Ray Conference}, 
Hamburg, Germany, 2001, Vol. 3, p. 1142.

%\cite{Anchordoqui:2002vb}
\bibitem{Anchordoqui:2002vb}
L.~A.~Anchordoqui {\em et al.},
%, J.~L.~Feng, H.~Goldberg and A.~D.~Shapere,
%``Neutrino bounds on astrophysical sources and new physics,''
Phys.\ Rev.\ D {\bf 66} (2002) 103002.
%[arXiv:hep-ph/0207139].
%%CITATION = HEP-PH 0207139;%%

%\cite{Tyler:2000gt}
\bibitem{Tyler:2000gt}
C.~Tyler, A.~V.~Olinto and G.~Sigl,
%``Cosmic neutrinos and new physics beyond the electroweak scale,''
Phys.\ Rev.\ D {\bf 63} (2001) 055001;
%[arXiv:hep-ph/0002257].
%%CITATION = HEP-PH 0002257;%%    
%\cite{Ringwald:2001vk}
%\bibitem{Ringwald:2001vk}
A.~Ringwald and H.~Tu,
%``Collider versus cosmic ray sensitivity to black hole production,''
Phys.\ Lett.\ B {\bf 525} (2002) 135.
%[arXiv:hep-ph/0111042].
%%CITATION = HEP-PH 0111042;%%

%\cite{Tinyakov:2001nr}
\bibitem{Tinyakov:2001nr}
P.~G.~Tinyakov and I.~I.~Tkachev,
%``BL Lacertae are sources of the observed ultra-high energy cosmic rays,''
JETP Lett.\  {\bf 74} (2001) 445
[Pisma Zh.\ Eksp.\ Teor.\ Fiz.\  {\bf 74} (2001) 499];
%[arXiv:astro-ph/0102476].
%%CITATION = ASTRO-PH 0102476;%%
%\cite{Tinyakov:2001ir}
%\bibitem{Tinyakov:2001ir}
%P.~G.~Tinyakov and I.~I.~Tkachev,
%``Tracing protons through the galactic magnetic field: A clue for charge composition of ultrahigh-energy cosmic rays,''
Astropart.\ Phys.\  {\bf 18} (2002) 165.
%[arXiv:astro-ph/0111305].
%%CITATION = ASTRO-PH 0111305;%%


%\cite{Zavrtanik:2000zi}
\bibitem{Zavrtanik:2000zi}
D.~Zavrtanik  [AUGER Collaboration],
%``The Pierre Auger Observatory,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 85} (2000) 324.
%%CITATION = NUPHZ,85,324;%%

%\cite{Andres:2001ty}
\bibitem{Andres:2001ty}
E.~Andres {\it et al.} [AMANDA Collaboration],
%``Observation of high-energy neutrinos using Cerenkov detectors embedded deep in Antarctic ice,''
Nature {\bf 410} (2001) 441.
%%CITATION = NATUA,410,441;%%

%\cite{Ringwald:1994kr}
\bibitem{Ringwald:1994kr}
A.~Ringwald and F.~Schrempp,
%``Towards the phenomenology of QCD instanton induced particle production at HERA,''
hep-ph/9411217; 
%%CITATION = HEP-PH 9411217;%%
%\cite{Ringwald:2000gt}
%\bibitem{Ringwald:2000gt}
%A.~Ringwald and F.~Schrempp,
%``Zooming-in on instantons at HERA,''
Phys.\ Lett.\ B {\bf 503} (2001) 331.
%[arXiv:hep-ph/0012241].
%%CITATION = HEP-PH 0012241;%%



\end{thebibliography}