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\noindent
{\LARGE {\bf Nuclei cross sections in Extensive Air Showers}}
\vspace{.5cm}

\noindent
{\Large Tadeusz Wibig and Dorota Sobczy\'{n}ska}
\vspace{.3cm}
%\author{T. Wibig and D. Sobczy\'{n}ska}

{
\large
%\address
\noindent
{ \it Experimental Physics Dept., University of \L odz, \\
Pomorska 149/153, 90-236 \L odz, Poland}
\vspace{.5cm}

%\date{\today}


%\vspace{-.5cm}

\noindent
{\bf Abstract}
{Cross sections for proton inelastic collision with different nuclei
are described within the Glauber approximation.
The significant difference between approximate ``Glauber'' formula
and exact calculations with geometrical scaling assumption is shown
for very high energy cross section calculations.
%Experimental values of proton--proton cross sections
%obtained using extensive air shower data are based on the
%relationship of proton--proton and respective proton--air absorption
%cross sections.
According to obtained results values of proton--proton cross section reported
by the Akeno and Fly's Eye experimental groups are about 10\%
overestimated.
%The proper energy dependence of absorption cross section for collisions
%with air nuclei is of a great importance for studies of high energy
%cosmic rays using the Monte Carlo technique. The
%``simplified Glauber'' (or multiple scattering) formula used for
%energies about 10$^18$ eV can strongly bias derived physical
%conclusions.
}

\vspace{.2cm}

%\pacs{13.85.-t, 12.39.-x, 13.87.Fh, 12.40.-y}

%\section {Introduction}
The rise of the proton-proton cross section (total, inelastic) as the
interaction energy increases is an important feature of the strong
interaction picture. The growth itself is established quite well both
from theoretical and experimental point of view. However the question
how fast do cross sections rise is discussed permanently. A definite answer
is still lacking. Theoretical predictions agree well with one another and
with accelerator data in the region where data exists ($\sqrt{s} \sim 20
\div 2000$ GeV) but they differ above. Before the LHC shifts the
direct measurements limit to 10 $\sim$ 14 TeV the only existing information
can be derived from the cosmic ray extensive air shower (EAS) data.
%Two EAS experiments, Akeno \cite{akeno} and Fly's Eye \cite{FE}, gave
%estimations about proton--proton total cross section at about $\sqrt{s}
%\approx 10^4$ GeV.

The important difference between the collider and EAS proton--proton
cross section measurements is that in fact
the proton--air interactions are involved in the EAS development thus
the value which is real measured is the cross section for the interactions
with air nuclei. The value of proton--proton cross section is obtained
from it using a theory for nuclei interactions.
In most
of recent papers concerning this subject it can be found just a few sentences
like ``calculations have been made in the standard Glauber formalism''
or something very similar \cite{akeno,FE}.

The original Glauber paper \cite{glauber} has been published about forty
years ago.
% and it looks like that since that time some misunderstandings appear.
Rather complicated equations for scattering cross sections can
be simplified significantly applying some additional assumptions which
validity is limited.
%It was mentioned already about ten years after the
%initial Glauber paper in
%\cite{glaubermat} that some expressions which are most frequently identified
%with the nuclear optical model should not be used at least for light nuclei,
%but this happens still nowadays.
%In this paper
We would like to compare results of calculations with and without
these simplifications.
%It will be shown that there is quite significant difference between them.
The exact Glauber formalism will be used to evaluate
the proton--proton cross section values from the Akeno and Fly's Eye data.

%The paper is organized as follows. In the next section the detail description
%of proton--proton scattering, the basis of further nuclei cross section
%calculations, is given. In Sec. III
%the proton-nucleus cross section evaluation for both methods will be given.
%In Sec. IV the quantitative results will be presented and discussed.

%\section {Proton--proton cross sections}
%Introducing the impact parameter formalism cross sections can be
%described using one, in general complex, function $\chi$
%in the form
%
%\begin{eqnarray}
%{\sigma}_{\rm tot}~=~2\:\int\:\left[\:1\:-\:{\rm Re}\left (
%{\rm e}^{i \chi ({b})}\right)\: \right]
%d^2 {\bf b}~~,
%\nonumber \\
%{\sigma}_{\rm el}~=~\int\:| \:1\:-\:{\rm e}^{i \chi ({b})}\:| ^2\:
%d^2 {\bf b}~~,
%\label{sigsig}
%\nonumber \\
%{\sigma}_{\rm inel}~=~\int\:1\:-\:| \:{\rm e}^{i \chi ({b})}\:| ^2\:
%d^2 {\bf b} ~~.
%\end{eqnarray}
%
%The phase shift $\chi$ is related to the scattering amplitude
%by the two dimensional Fourier transform
%
%\begin{eqnarray}
%1\:-\:{\rm e}^{i\chi({\bf b})}
%~=~{{1} \over {2\: \pi \: i}}\int\:{\rm e}
%^{-i{\bf b\:t}} S({\bf t}) d^2 {\bf t} ; \nonumber \\
%S({t}) ~=~{i \over {2\: \pi \: }}\int\:{\rm e}
%^{ i{\bf b\:t}} \left(
%1\:-\:{\rm e}^{i\chi({\bf b})} \right)
%d^2 {\bf b} .
%\label{eq2}
%\end{eqnarray}
%
%Using the optical analogy one can interpret the $
%1\:-\:{\rm e}^{i\chi({\bf b})}
%$ function as a transmission coefficient
%%opacity of the scattering potential along a straight-line path defined
%for a given impact parameter. Considering two colliding object we can
%assume (for pure scattering process)
%
%\begin{equation}
%\chi({b})~=~i\: \omega(b)~=~i\: K_{ab}\int \: d^2{\bf b'}\:
%\rho_a({\bf b})\rho_b({\bf b}\:+\:{\bf b'}) .
%\label{rho}
%\end{equation}
%
%\noindent
%where $\rho_h$ is a particle ``opaqueness''
%(the matter density integrated
%along the collision axis). To some extend the hadronic matter density
%could be identified with the charge density measured precisely at the
%leptonic scattering experiments. However for high energy hadron--hadron
%collisions the imaginary part of the phase shift $\chi$ can be
%quite considerable. The simple interpretation given in Eq.(\ref{rho})
%ought to be modified at least by introducing some dependence on the
%interaction energy ($s$).Thus the
%general phase shift $\chi$ is a two-variable complex function. Two
%possibilities of simplify the situation have been proposed in the literature.
%They are the factorization hypothesis (FH) and geometrical scaling (GS). They
%can be expressed as
%
%\begin{eqnarray}
%\chi(s,\:b) ~=~ i\: \omega(b)\:f(s) ~~~~~~{\rm (FH)} \nonumber0\\
%\chi(s,\:b) ~=~ i\: \omega \left(b\: / b_0(s)\right) ~~~~~~{\rm (GS)}
%\label{gsfh}
%\end{eqnarray}
%
%From the optical point of view, the FH means that the hadron is
%gutting blacker as the energy increase while the GS means that it is getting
%bigger.
%
%The main information about the hadron phase shift function $\chi$ comes from
%elastic scattering experiments, more precisely: from the measured differential
%elastic cross section. The value of the total cross section can be obtained
%from the imaginary part of the forward scattering amplitude using the
%optical theorem. Analysis of the elastic data above $\sqrt{s} \sim 20$ GeV
%shows that both assumptions given in Eq.(\ref{gsfh}) are not realized
%exactly
%(\cite{chouyang,amaldi}).
%However, the combination of them can reproduce the data quite
%well.
%
%In the present work the form of $\chi$ is assumed after \cite{men1}
%in the form
%
%\begin{equation}
% \chi(s,b)  ~=~(\lambda(s)+i)\: \omega(b,s)
%\label{lambdach}
%\end{equation}
%
%\noindent
%[with $\omega$ defined by Eq.(\ref{rho})]
%which follows the original GS idea \cite{buddd} and
%differs from the well-known Martin's formula where the ratio of
%real to imaginary part of the scattering amplitude depends also on the
%momentum transfer. The discussion and some recent
%references can be found in \cite{men1}.
%Anyhow, due to the lacking both theoretical and experimental information
%about the phase shift the proportionality in Eq.(\ref{lambdach})
%can be treated as a first approximation.
%In the present work the parametrization of $\lambda$ used is given by
%
%\begin{equation}
%\lambda(s)~=~{{0.077\: \ln (s/s_0)} \over {
%1~+~0.18\: \ln (s/s_0)~+~.015\: \ln ^2 (s/s_0)}}~~~,~~s_0~=~500~{\rm GeV}^2~.
%\label{lamb}
%\end{equation}
%
%For the $\omega$ energy dependence in Eq.(\ref{gsfh})
%the GS is assumed, thus
%
%\begin{equation}
%\omega(b,s) ~=~
%\omega(\widetilde b ) ~~~~~{\rm with} ~~~~
%\widetilde b
%~=~  b \:  \left[{{\sigma_{\rm inel}(s_0)}
%\over {\sigma_{\rm inel}(s)}} \right]
%^{ \frac 1 2 } ,
%\label{gsomega}
%\end{equation}
%
%\noindent
%where $s_0$ is the
%center of mass energy for which the detail shape of $\omega$ has been
%originally determined.
%%We have used elastic $p$--$p$ scattering data
%%at $\sqrt{s} \sim 20$ GeV
%%from NA22 \cite{na22}.
%The accurate data description has been found \cite{ws1}
%``matter'' distribution
%
%\begin{equation}
%\rho_h({\bf b})~=~\int d z {{ m_h} \over {8 \pi}} {\rm e}^{-m_h {\bf r}}
%\label{pdens}
%\end{equation}
%
%with the coefficients $m_h$ [and $K_{ab}$ in Eq.(\ref{rho})]
%adjusted to the hadron--proton elastic scattering data
%at $\sqrt{s} \sim 20$ GeV.
%The energy dependence of the phase shift is thus introduced by using
%the elastic to total cross sections ratio change ($\lambda (s)$ function)
%and by the scaling of the interaction impact parameter
%($\sigma_{\rm inel} (s)$)
%which reproduces the increase of cross section as the interaction
%energy increases.
%
%The parametrization of ${\sigma_{\rm inel}(s)} $ has been made in the
%$\ln ^2 (s)$ form
%
%\begin{equation}
%{\sigma_{\rm inel}(s)} = 32.4~-~1.2 \ln (s)~+~0.21 \ln ^2 (s)
%\label{inel}
%\end{equation}
%
%\noindent
%which gives the cross section in mb when $s$ is given in GeV$^2$.
%The deviation from the Pomeron-type power-law fit
%(see e.g, \cite{blca})
%is negligible in the region where the fit can be compared with existing data.
%The advantage of using the $\ln ^2$ form is that we do not concern about
%the violation of the Froissart unitarity bound when using our formula at
%very high energies.
%
%Using Eqs.(\ref{pdens}, \ref{lamb}, and \ref{inel}) we
%can calculate the hadron--nucleon cross sections at any energy of interest
%from the point of view of extensive air shower physics.
%
%The quality of proposed parametrizations is presented in Figs.\ref{elast}
%and \ref{sigmas}.
%
%\begin{figure}
%\centerline{\psfig{file=fig1.eps,width=10cm}}
%\caption{Differential $p$--$p$ elastic cross sections obtained
%using the proposed parametrization of $\chi$ for different energies
%compared with experimental data from FNAL, ISR and SPS [12]}
%\label{elast}
%\end{figure}
%\cite{eldata}
%
%\begin{figure}
%\centerline{\psfig{file=fig2a.eps,width=9cm}
%\psfig{file=fig2b.eps,width=9cm}}
%\caption{The energy dependence of $p$--$p$
%total cross sections (a) and imaginary to real part of elastic forward
%amplitude (b)
%calculated using the proposed parametrization of $\lambda$
%compared with experimental data from [13]
%The long dashed line shows(wy\.{z}ej) Block and Cahn fit [11]
%and the short dashed line Durand and Pi [14] calculation result}
%\label{sigmas}
%\end{figure}
%\cite{totdata}
%\cite{blca}
%\cite{durpi}
%
%\section {Nucleon--nucleus cross section calculations}

%In fact there are two different ways of
%treatment nuclei cross section calculations. Results of both of them are
%quite similar, but the foundations are quite different.
%In general it can
%be said that the Glauber method is based on scattering amplitudes the other
%one which we call hereafter the multiple scattering method works on
%probabilities.
%The multiple scattering approximation is widely presented in \cite{czyzma}
%mainly for the elastic scattering of different particles on nuclei, but the
%probabilistic formalism can be used for cross sections of other specific
%processes.


%Of course we do not want to judge which method describes reality better.
%The comparison with experimental data is possible in principle,
%however as it will be shown expected differences between these two
%approaches are comparable with other effects which has to be known
%with appropriate precision (e.g. quasi-elastic scattering).
%
%The description given below is performed for the proton--nucleus interactions,
%but the extension to the nucleus--nucleus case is straightforward.

%\subsection{Glauber approximation}
%In original Glauber approximation
For the scattering of particle on the close many particle system (nucleus),
if each interaction can be treated as a two particle one, the
overall phase shift for incoming wave is a sum of all two-particle
phase shifts.
\begin{equation}
\chi_A(b,\: \{{\bf d}\})~=~\sum_{j=1}^A \: \chi _j ({\bf b}\: -\: {\bf d}_j)
\label{chiskla}
\end{equation}
\noindent
where $\{{\bf d}\}$ is a set of nucleon positions in the nucleus
(${\bf d}_j$ is a position of the $j$th nucleon in the plane
perpendicular to the interaction axis).
The equation (\ref{chiskla}) is the essence of Ref.\cite{glauber}
and in fact it defines the Glauber approximation.
% (at least in this paper).
The scattering amplitude is thus given by
\begin{equation}
S(t)~=~{i \over {2 \pi }} \int {\rm e}^{i{\bf t b}} d^2{\bf b}
\int | \psi(\{{\bf d}\} ) | ^ 2 \: \left\{ \: 1 \: - \: {\rm e}
^{i \chi_A(b,\: \{{\bf d}\})}  \right\}
\prod _{j=1}^A d^2 {\bf d}_j~~,
\label{st1}
\end{equation}
\noindent
where $\psi$ describes the wave function of the nucleus with nucleons
distributed according to $\{{\bf d}\}$.
If one neglect position correlations of the nucleons and denotes by
$\varrho _j$ each single nucleon density and
%\begin{equation}
%|\psi(\{{\bf d}\})|^2~=~\prod_{j=1}^A \varrho_j ({\bf d}_j)
%~~~{\rm with}~~~
%\int \varrho_j ({\bf r}_j) d^3 {\bf r}~=~1 ~~.
%\end{equation}
%
%\noindent
if all interactions can be described by the same phase shift
function $\chi$ then
%
\begin{eqnarray}
S(t)=
%{{i} \over {2 \pi }} \int {\rm e}^{i{\bf t b}} d^2{\bf b}
%\int
%\prod _{j=1}^A \varrho_j ({\bf d}_j)
% \left\{ 1  -  {\rm e} ^{i \sum_{j=1}^{A}\chi({\bf b} - {\bf d}_j)} \right\}
%d^2 {\bf d}_j
%= \nonumber \\
{{i} \over {2 \pi }} \int {\rm e}^{i{\bf t b}} d^2{\bf b}
\left\{ 1  -   \int  \prod _{j=1}^A \varrho_j ({\bf d}_j)
{\rm e}^{i \chi({\bf b} - {\bf d}_j)} d^2 {\bf d}_j \right\} ~.
\label{st2}
\end{eqnarray}

On the other hand, the scattering process
can be treated as a one collision process
with its own nuclear phase shift $\chi_{\rm opt}(b)$
\begin{equation}
S(t)~=~{{i} \over {2 \pi }} \int {\rm e}^{i{\bf t b}}
\left\{ 1 \: - \: {\rm e}^{i \chi_{\rm opt}(b)} \right\} d^2 {\bf b}~~.
\label{chiop}
\end{equation}
The comparison with Eq.(\ref{st1}) gives
\begin{eqnarray}
 {\rm e}^{i \chi_{\rm opt}(b)}~=~
\int | \psi(\{{\bf d}\})|^2\:
{\rm e}^{i \sum_{j=1}^A \: \chi _j ({\bf b}\: -\: {\bf d}_j)}
\prod _{j=1}^A d^2 {\bf d}_j
~=~\left\langle {\rm e}^{i \chi(b,\: \{{\bf d}\})} \right\rangle~~,
\label{chiopt}
\end{eqnarray}
\noindent
where the $\langle \ \rangle$ means the averaging over the all possible
configuration of nucleons $\{ {\bf d} \} $.
To go further with the calculations of $\chi_{\rm opt}$ the commonly used
assumption has to be made. If we assume that the number of scattering centers
($A$) is large (with the transparency of the nucleus as a whole constant)
then
\begin{equation}
\chi_{\rm opt}(b)
~=~i\: \int d^2 {\bf d} \rho_A({\bf d})\:
\left[ 1 - {\rm e}^{i \chi({\bf b} - {\bf d})} \right]~.
\label{exact}
\end{equation}
\noindent
where $\rho_A$ is the
distribution of scattering centers (nucleons) positions
in the nucleus ($\sum \varrho_j$).
When the individual nucleon opacity $| 1-{\rm e}^{i \chi(b)} |$ is a very
sharply peaked compared with $\rho_A$ then with the help of the
optical theorem the simple formula for the scattering amplitude can be found.
%for the scattering on nucleus is obtained
%\begin{equation}
%\chi_{\rm opt}(b)~=~{1 \over 2}\: \sigma_{pp}^{\rm tot} \left[
%\left( {{{\rm Re} S(0)} \over{ {\rm Im} S(0)}} \right) \: + \: i \right]
%\rho_A(b)~~.
%\left[ 1 \: - \: {\rm e} ^{- A T(b) } \right].
%\label{chig}
%\end{equation}
\noindent
%Function $\rho_A(b)$ can be identified with the
%the nucleus opacity.
%Substituting (\ref{chig}) to (\ref{sigsig})
The proton nucleus inelastic cross section is thus
\begin{equation}
\sigma_{pA}^{\rm inel}~=~
\int d^2 {\bf b}
\left[ 1 - {\rm e}^{- \sigma_{pp}^{\rm tot} \rho_A(b)} \right]
~=~
\int d^2 {\bf b}
\left\{ 1 -
\left[ 1-
\sigma_{pp}^{\rm tot} {\rho_A \over A} \right] ^A \right\}
\label{ginel}
\end{equation}
\noindent
%where the last equality holds in the large $A$ limit [
%Eq.(\ref{ginel}) can not be used for $A=1$ to compare results for
%$\sigma_{pp}^{\rm inel}$]
This result is often but not quite correctly called
``the Glauber approximation''.
% while, as it has been shown,
%the original Glauber assumption given in Eq.(\ref{chiskla})
%has to be supported by small nucleon sizes and large value of A.
%
%\subsection {Multiple scattering approach}
%In multiple scattering approach
%the $\sigma_{\rm ine}$ given in Eq.(\ref{sigsig}) can be interpreted in the
%probabilistic way by identifying the
%$[1-|\exp(i\chi)|^2]$
%term as a
%probability of inelastic scattering at impact parameter $b$. This can be
%extended over the interaction with nucleus in a straightforward way.
%If we denote this probability by
%
%\begin{equation}
%P(b)~=~1-|{\rm e}^{i \chi(b)}|^2
%\end{equation}
%
%\noindent
%and nucleons in a nucleus $A$ are distributed according to $\rho_A$
%then the probability of inelastic interaction with one of the nucleons
%is
%
%\begin{equation}
%\overline{P}_A(b)~=~
%\int d^2 {\bf d} \ {{\rho_A({\bf b})} \over A} \ P({\bf b} - {\bf d}).
%\end{equation}
%
%The inelastic cross section with the whole nucleus is then
%
%\begin{equation}
%\sigma_{pA}^{\rm inel}~=~
%\int d^2 {\bf b} \left\{
%1-
%\left[ 1-
%\overline{P({\bf b})}\right]
%^A
%\right\}~~.
%\end{equation}
%
%In the multiple scattering picture the point nucleon approximation can be
%also introduced simplifying the cross section formula. If one put
%$P(b)~=~\delta^2(b)\:\sigma_{pp}^{\rm inel}$ then
%
%\begin{equation}
%\overline{P}_A(b)~=~
%{{\rho_A({\bf d})} \over A} \sigma_{pp}^{\rm inel},
%\end{equation}
%
%\noindent
%what leads to
%
%\begin{equation}
%\sigma_{pA}^{\rm inel}~=~
%\int d^2 {\bf b} \left\{ 1 - \left[ 1-
%\sigma_{pp}^{\rm inel} {{\rho_A ({\bf b})} \over A}\right]^A \right\}
%\label{minel}
%\end{equation}
%
%The above equation has a very similar form to the
%Eq.(\ref{ginel}) but the difference is also quite clear.
%
%It is interesting to note that Eq.(\ref{minel})
%is often called also ``Glauber approximation'' (see e.g.,
%\cite{gs2}).

%\subsection{Comparison with the exact Glauber formula}

%Both approaches discussed above lead to slightly different formulas
%for inelastic proton--nucleus cross section. To see how big the difference
%is we have calculated respective cross sections using exactly the same
%procedures, nucleus shapes and proton--proton cross sections.
%It is interesting to compare results for different nucleus masses and
%incoming proton energies.
%
%The energy dependence is in the discussed approximation introduced only
%via the proton-proton cross sections change. Respective formulae
%have been given in Sect. II.
%
%The distributions of the nucleon
%position in the nuclei of the form given in the FRITIOF
%interaction model \cite{frit} have been used
%
%\begin{equation}
%\varrho_j~\sim~\left\{ \matrix{ {\textstyle
%\left\{\: {1+\exp
%\left[ \left( r- r_0 A^{1/3} \right)/C \right] \:
% } \right\} }
%^{-1}
%& \ \ \ \  {\rm for} & A > 16 \cr
%{\textstyle
%\left[ 1+ {{A-4} \over 6} \left( r \over d \right) ^2 \right]
%\exp \left( - {r \over d}^2 \right)} & \ & A \leq 16
%}
%\right.
%\end{equation}
%
%\noindent
%with the parameters given there, except for $A=4$ where we have used
%the so-called ``parabolic Fermi distribution'' \cite{gs2} and for
%lighter ($A=2,3$) nuclei the simple Gaussian was used. The minimum
%allowable distance between two nucleons was introduced (0.8 fm) which
%modify the ``initial'' (uncorrelated) distribution rather strongly
%specially for very light nuclei, so the detail shape of the light
%nuclei uncorrelated density distribution is not essential for our
%subject.
%
%The validity of the nucleus description used is shown in Fig.\ref{paxsect}
%where the
%proton--nucleus inelastic cross section data measured at low energies
%are compared with our calculations. The interaction energy is about
%the same that it was used as the reference energy $s_0$ for the
%proton ``hadronic matter density distribution'' ($\omega$) estimation,
%so the energy dependence of all proton--proton cross section parameters
%are not involved here.
%
%\begin{figure}
%\centerline{\psfig{file=fig3.eps,width=10cm}}
%\caption{
%Inelastic cross sections measured at the laboratory proton
%momentum of $\sim$ 200 GeV/$c$ compared with our calculation results
%as a function of target atomic mass. Data from Ref.[18].}
%\label{paxsect}
%\end{figure}
%\cite{pa}
%
%The comparison between the simplified Glauber [Eq.(\ref{ginel})] and
%multiple scattering [Eq.(\ref{minel})] results is given in
%Fig.~\ref{xsecc1} for proton--nucleus as well as for nucleus--nucleus
%interactions. We have chosen here the ``air nucleus'' as a target and
%five different projectiles which represent main components of primary
%cosmic ray mass spectrum. The difference is not very significant. The energy
%dependence is similar and the systematic shift is rather constant as it
%can be expected. It should be remembered that for light nuclei (specially
%for protons) Eq.(\ref{ginel}) loose his physical grounds.
%
%\begin{figure}
%\centerline{
%\psfig{file=fig4.eps,width=12cm}}
%\caption{Cross section of collisions
%of different nuclei with the ``air nucleus'' calculated
%with simplified Glauber (solid lines)
%and multiple scattering (dashed lines) approximations for different
%interaction energies.}
%\label{xsecc1}
%\end{figure}
%

The important point of this paper is to show how the point-nucleon
approximation changes the results.
Formulas given in Eq.
%(\ref{minel}) and
(\ref{ginel}) are in fact in
agreement with the factorization hypothesis for individual nucleon--nucleon
$\chi$ function % [Eq.(\ref{gsfh})].
%\begin{eqnarray}
$\chi(s,\:b) ~=~ i\: \omega(b)\:f(s)$. % ~~~~~~{\rm (FH)} \nonumber \\
%\chi(s,\:b) ~=~ i\: \omega \left(b\: / b_0(s)\right) ~~~~~~{\rm (GS)}
%\label{gsfh}
%\end{eqnarray}
According to it nucleons are getting
blacker
as the interaction energy increases.
However, for some time (see, e.g., Ref.\cite{amaldi}) it is known that
this is not the case. Experimental data
% Our analysis presented in Sect. II
strongly favour the geometrical scaling hypothesis
$\chi(s,\:b) ~=~ i\: \omega \left(b\: / b_0(s)\right)$% ~~~~~~{\rm (GS)}
 (see, e.g., Ref. \cite{buddd})
which treats nucleons
as getting bigger. Nucleus profiles obtained using exact Glauber formula
(Eq.(\ref{exact})] differ from the
$A \sigma_{pp}^{\rm inel} {\rho_A \over A}$ suggested by
Eq.(\ref{ginel}). The difference can be seen clearly
in Fig.~1.

%\begin{figure}
%\centerline{
%\psfig{file=nxfig1.eps,width=10cm}}
%\caption{Nitrogen nucleus profile functions ($\chi_{p-{\rm N}}$)
%obtained using the
%exact Glauber formula with geometrical scaling (solid lines)
%and factorization hypothesis
%(simplified Glauber) (dashed lines) for
%different interaction energy (per nucleus)}
%\label{ddis}
%\end{figure}

The significant change of the nucleus size have to influence the value
of inelastic cross section. Fig.~2 shows the
change of inelastic cross section of proton--nucleus with the
interaction energy calculated using geometrically scaled and
factorized nucleus profiles.
%\begin{figure}
%\centerline{\psfig{file=nxfig2.eps,width=10cm}}
%\caption{
%Cross section of collisions
%of different nuclei with the ``air nucleus'' calculated
%using exact Glauber formula with geometrical scaling
%(solid lines) and simplified formulas (dashed lines)
%as a function of interaction energy.}
%\label{finxsec}
%\end{figure}
As it can be seen the difference at very high energies is remarkable.

%\section {Proton--proton cross section from cosmic ray data.}

The results presented above indicate the importance of
reexamination of the proton--proton cross section estimation based
on proton--air data measured in EAS experiment.

The conversion from proton--air to proton--proton cross section
presented in Fig.~3 is obtained using the exact Glauber
formalism.
%\begin{figure}
%\centerline{\psfig{file=nxfig3.eps,width=10cm}}
%\caption{Relationship between inelastic proton--air cross section
%and the value of proton--proton cross section (inelastic - solid curve
%and total - dashed one)
%calculated using exact Glauber formula with geometrical scaling.
%Solid horizontal line represent the value measured in Fly's Eye [3]
%experiment (dashed area shows 1$\sigma$ bounds).}
%\label{pppair}
%\end{figure}
The original Fly's Eye estimation of proton--proton total cross section
given in Ref.\cite{FE} is 120 mb while according to results given in
Fig.~3 it is equal to 110 mb. The same procedure should be applied
to the Akeno data.
% and also all their proton--proton cross sections appear to be 10\% overestimated.

%The point plotted in Fig.\ref{sigmas}(a) are taken from the original works.
%They agree quite well with two phenomenological descriptions plotted
%there. The approximation of the cross section rise used in the present work
%[Eq.(\ref{inel})] agrees with the Block and Cahn and with the Durand and Pi
%parametrizations in the direct measurements energy region, but goes about
%10\% below the original EAS proton--proton data points.

In Fig.~4 the calculated proton--air cross section energy
dependence is given. The solid line represents result
obtained using exact Glauber formula [Eq.(\ref{chiopt})] and proton--proton
phase shift $\chi$ function described in Ref.\cite{xsecprep}.
%the Sect.II (with proton--proton cross section rise defined by Eq.(\ref{inel}).
The outcome of the
``simplified Glauber'' approach [Eq.(\ref{ginel})] is given for a comparison
by the long dashed line.

%\begin{figure}
%\centerline{\psfig{file=nxfig4.eps,width=10cm}}
%\caption{Inelastic proton--air cross sections
%calculated using exact Glauber formula with geometrical scaling
%as a function of interaction energy (per nucleus).
%Experimental points are from Akeno and Fly's Eye experiments
%(squares and the circle, respectively). Results of
%calculations with the ``simplified Glauber'' formula with the
%same proton--proton cross section energy dependence are given by the
%long dashed line. The default CORSIKA proton--air
%cross section is shown for a comparison by the short dashed line.}
%\label{pair}
%\end{figure}

%As it is seen the proposed cross section rise gives very good agreement
%with EAS measurements at about $\sqrt{s} \sim 10^4$ GeV.

%\section{Summary}
Concluding, we have shown that
the geometrical scaling hypothesis with the exact Glauber formalism
gives the value of proton--proton total cross section
at about 30 TeV about 10\% smaller than that reported in original
Fly's Eye and Akeno papers. Results presented in this work
have been obtained using the following cross section energy dependence
\begin{equation}
{\sigma_{\rm inel}(s)} = 32.4~-~1.2 \ln (s)~+~0.21 \ln ^2 (s)~~,
\end{equation}
\noindent
which fits quite well accelerator measurements as well as the EAS
points diminish by $\sim$ 10\%.
%The fit for $\sigma_{pp}^{\rm inel}$
%given in Eq.(\ref{inel}) leads to $\sigma_{pp}^{\rm tot}$ shown in
%Fig.\ref{sigmas}(a) by the solid line and correctly reproduces
%$\sigma_{p-air}$ cross section measured by EAS experiments.

The rise of the $\sigma_{A-air}$ predicted by
the geometrical scaling hypothesis with the exact Glauber formalism
is significantly faster than the one which can be obtained using
simplified formulas (Fig.~2).
This can change the physical conclusions based on
Monte Carlo simulations of the EAS development at very high energies.

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\end{thebibliography}



%\twocolumn
%\newpage

\vspace{.3cm}


%\begin{figure}
\centerline{
\hspace{.8cm}
\psfig{file=nxfig1.eps,width=8.5cm}
\hspace{-1.cm}
%}
%\label{ddis}
%\end{figure}
%
%\begin{figure}
%\centerline{
\psfig{file=nxfig2.eps,width=8.5cm}}
%\label{finxsec}
%\end{figure}
\vspace{.5cm}

%\begin{figure}
\centerline{
\hspace{.8cm}
\psfig{file=nxfig3.eps,width=8.5cm}%}
\hspace{-1.cm}
%\label{pppair}
%\end{figure}
%
%\begin{figure}
%\centerline{
\psfig{file=nxfig4.eps,width=8.5cm}}
%\label{pair}
%\end{figure}
%\vspace{.2cm}
\vspace{.8cm}

%FIGURE CAPTIONS:
%\footnotesize

\vspace{.3cm}
%\caption
Fig.1.
{Nitrogen nucleus profile ( phase shift: $\chi_{p-{\rm N}}$)
obtained using the
exact Glauber formula with geometrical scaling (solid lines)
and factorization hypothesis
(simplified Glauber) (dashed lines) for
different interaction energy (per nucleon--nucleon collision)}
\vspace{.3cm}

Fig.2.
Cross section of collisions
of different nuclei with the ``air nucleus'' calculated
using exact Glauber formula with geometrical scaling
(solid lines) and simplified formulas (dashed lines)
as a function of interaction energy.
\vspace{.3cm}

%\caption
{Fig.3.
Relationship between inelastic proton--air cross section
and the value of proton--proton cross section (inelastic - solid curve
and total - dashed one)
calculated using exact Glauber formula with geometrical scaling.
Solid horizontal line represent the value measured in Fly's Eye [2]
experiment (dashed area shows 1$\sigma$ bounds).}
\vspace{.3cm}

%\caption
{Fig.4. Inelastic proton--air cross sections
calculated using exact Glauber formula with geometrical scaling
as a function of interaction energy (per nucleus).
Experimental points are from Akeno and Fly's Eye experiments
(squares and the circle, respectively). Results of
calculations with the ``simplified Glauber'' formula with the
same proton--proton cross section energy dependence are given by the
long dashed line. The default CORSIKA \cite{corsika} proton--air
cross section is shown for a comparison by the short dashed line.}


\end{document}

