source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/CrossSection/GlauberModel.tex @ 1345

\subsection{Glauber model at high energies.}

\hspace{1.0em}We can use Glauber approach \cite{Glauber55} to calculate 
the total, elastic and differential elastic hadron-nucleus and nucleus-
nucleus cross sections at high (more than hundreds of MeV) energies.

\subsubsection{The hadron--nucleus and nucleus--nucleus total and 
elastic cross sections.}
\hspace{1.0em}
The knowledge of the nuclear elastic scattering amplitude $F(\vec{q},s)$, 
where $s$ is the total hadron-nucleon or nucleon-nucleon c.m. energy squared
and $\vec{q}$ is the momentum transfer vector,  
gives us a possibility to calculate  the total cross section (the optical
 theorem)
\begin{equation}
\label{GM1} \sigma_{tot}(s) = \frac{4\pi}{k} Im F(0,s),
\end{equation}
where $k$ is a hadron or nucleon projectile momentum in the target nucleus 
rest frame.
Using this amplitude we are also able to calculate the differential elastic 
cross section 
\begin{equation}
\label{GM2} \frac{d\sigma_{el}(s)}{d\Omega} = |F(\vec{q},s)|^2
\end{equation}
or 
\begin{equation}
\label{GM3} \frac{d\sigma_{el}(s)}{dt} =\frac{\pi}{k^2} |F(\vec{q},s)|^2
\end{equation}
and total elastic cross section
\begin{equation}
\label{GM4} \sigma_{el}(s)= \int d\Omega|F(\vec{q},s)|^2 = 
\frac{1}{k^2}\int dq|F(\vec{q},s)|^2.
\end{equation}

The elastic scattering amplitude can be expressed through the profile 
function
\begin{equation}
\label{GM5}\Gamma(\vec{B},s)= 1-S(\vec{B},s) 
\end{equation}
as the following
\begin{equation}
\label{GM6} F(\vec{q},s)=\frac{ik}{2\pi}\int
d^2\vec{B}\exp{[i\vec{q}\Gamma(\vec{B},s)]},
\end{equation}
where $S(\vec{B},s)$ is the $S$-matrix and $\vec{B}$ 
is the impact parameter vector perpendicular to the 
incident momentum $\vec{k}$.

The total and elastic cross sections can be obtained from
  the profile function $\Gamma(\vec{B},s)$:
\begin{equation}
\label{GM7} \sigma_{tot}(s)= 2\int d^2\vec{B}Re\Gamma(\vec{B},s)
\end{equation}
and 
\begin{equation}
\label{GM8} \sigma_{el}(s)= \int d^2\vec{B}|\Gamma(\vec{B},s)|^2.
\end{equation}

Thus to calculate the total, elastic and differential cross sections we need 
to know $S$-matrix $S(\vec{B},s)$.

\subsubsection{The hadron--nucleus and nucleus--nucleus 
$S$-matrix.}
\hspace{1.0em}

Let us consider the nucleus-nucleus scattering at
given impact parameter $\vec{B}$ and at given squared total
c.m. nucleon--nucleon energy $s$. 
 
In Glauber approach \cite{Glauber55} an
elastic nucleus--nucleus interaction is a result of the 
interactions between nucleons from the projectile and target nuclei.
Thus, the $S$-scattering matrix $S^{AB}(\vec B,s)$ for 
nucleus $A$ on nucleus $B$
collision in the impact parameter representation can 
be expressed as follows:
\begin{equation}
\label{GM9}S^{AB}( \vec
B,s)=<\prod\limits_{i=1}^A\prod\limits_{j=1}^B
[1 - \Gamma_{ij}(\vec{B}+ \vec b_i^A-\vec b_j^B,s)]> 
\end{equation}
where $<...>$ means integration over the sets $\{\vec b_i^A\}$ and
$\{\vec b_j^B\}$ with weight functions $T _A$$(\{\vec b^A\})$ and
$T_B$$(\{\vec b^B\})$.  These functions 
\begin{equation}
\label{GM10} T_{A,B}(\vec b_i^{A,B})=\int
\rho ((\vec b_i^{A,B}z_i)dz_i
\end{equation}
 can be obtained from the nucleon
densities $\rho ((\vec b_i^{A,B},z_i)$. The nucleon profile function is 
\begin{equation}
\label{GM11} \Gamma_{ij}(\vec{B}+ \vec b_i^A-\vec b_j^B,s) = 
\frac{\sigma_{ij}(s)}{4\pi \beta_{ij}(s)} \exp{[-\frac{(\vec{B} + 
\vec b_i^A-\vec b_j^B)^2}{2\beta_{ij}(s)}]}.
\end{equation}
The last equation can be obtained in the case of nucleon-nucleon 
amplitude parametrization:
\begin{equation}
\label{GM12} f_{ij}(q,s) = \frac{ik \sigma_{ij}(s)}{4\pi}
\exp{[-\frac{1}{2}\beta_{ij}(s) q^2]}.
\end{equation}
The equation $(\ref{GM9})$ is a result of the assumptions that
the $AB$-scattering phase is sum of
the nucleon--nucleon scattering phases and no correlations between nucleons 
inside a nucleus are taken into account.

The hadron-nucleus $S$-matrix is calculated in similar way using 
Eq. $(\ref{GM9})$ for $i = 1$ and $\vec b_i = 0$. In this case 
we need to use the corresponding parameter $\sigma_{hN}(s)$ 
and $\beta_{hN}(s)$
in nucleon profile function.

 As we will show below
the hadron-nucleon and nucleon--nucleon elastic scattering amplitudes at high 
energies
 can be expressed
through the reggeon-nucleon vertex parameters and the parameters of the
reggeon trajectory\cite{BT76}.