source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/PartonString/SampleParticipants.tex @ 1345

\section{Sample of collision participants
in nuclear collisions.}

\subsection{MC procedure to define collision participants.}
\hspace{1.0em} The inelastic
hadron--nucleus interactions at ultra--relativistic energies are considered 
as
independent hadron--nucleon  collisions.  It was shown long
time ago \cite{CK78} for the hadron--nucleus collision that such a
picture can be obtained starting from the Regge--Gribov
approach \cite{BT76}, when one assumes that the hadron-nucleus elastic
scattering amplitude is a result of reggeon exchanges between the
initial hadron and  nucleons from target--nucleus. This result leads to 
simple and efficient MC procedure \cite{Am86} to define
the interaction cross sections and the number of the nucleons
participating in the inelastic hadron--nucleus collision:
\begin{itemize}
\item We should randomly distribute 
 $B$
nucleons from the target-nucleus on the impact parameter plane according
to the weight function 
$T([\vec{b}^{B}_{j}])$. This function represents
probability density to find sets of the nucleon impact parameters
 $[\vec{b}^{B}_{j}]$, where
$j=1,2,...,B$.
\item For each pair of projectile hadron $i$ and target nucleon 
$j$ with choosen impact parameters $\vec{b}_{i}$ and
$\vec{b}^{B}_{j}$ we should check whether they interact inelastically or
not using the probability $p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)$,
where $s_{ij}=(p_{i}+p_{j})^2$ is the squared total c.m.  energy of the
given pair with the $4$--momenta $p_{i}$ and $p_{j}$, respectively.
\end{itemize}
 
In the Regge--Gribov approach\cite{BT76} the probability for an inelastic
collision of pair of $i$ and $j$ as a function at the squared impact
parameter difference $b_{ij}^2=(\vec{ b}_i-\vec{ b}_j^B)^2 $ and $s$
is given by
\begin{equation}
\label{SP3}
 p_{ij}(\vec{ b}_i-\vec{ b}_j^B,s)=
 c^{-1}[1-\exp{\{-2u(b_{ij}^2,s)\}}] = 
\sum_{n=1}^{\infty}p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s), 
\end{equation}
where
\begin{equation}
\label{SP4}
 p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s)
=c^{-1}\exp{\{-2u(b_{ij}^2,s)\}}
 \frac{[2u(b_{ij}^2,s)]^{n}}{n!}.
\end{equation}
is the probability to find the $n$ cut Pomerons or the probability for
$2n$ strings produced in an inelastic hadron-nucleon collision.  These
probabilities are defined in terms of the (eikonal) amplitude of
hadron--nucleon elastic scattering with Pomeron exchange:
\begin{equation}
\label{SP5}u(b_{ij}^2,s)=\frac{z(s)}{2}\exp (-b_{ij}^2/4\lambda (s)). 
\end{equation}
The quantities $z(s)$ and $\lambda (s)$ are expressed through the
parameters of the Pomeron trajectory, $\alpha _P^{^{\prime }}=0.25$
$GeV^{-2}$ and $\alpha _P(0)=1.0808$, and the parameters of the
Pomeron-hadron vertex $R_P$ and $\gamma _P$:
\begin{equation}
\label{SP6}z(s)=\frac{2c\gamma _P}{\lambda (s)}(s/s_0)^{\alpha _P(0)-1} 
\end{equation}
\begin{equation}
\label{SP7}\lambda (s)=R_P^2+\alpha _P^{^{\prime }}\ln (s/s_0), 
\end{equation}
respectively, where $s_0$ is a dimensional parameter.

In Eqs. (\ref{SP3},\ref{SP4}) the so--called shower enhancement
coefficient $c$ is introduced to determine the contribution of
diffractive dissociation\cite{BT76}.  Thus, the probability for 
diffractive dissociation of a pair
of nucleons can be computed as
\begin{equation}
\label{SP8}p_{ij}^d(\vec b_i-\vec b_j^B,s)=\frac{c-1}{c}[p_{ij}^{tot}(\vec
b_i-\vec b_j^B,s)-p_{ij}(\vec b_i-\vec b_j^B,s)],
\end{equation} 
where 
\begin{equation}
\label{SP9}p_{ij}^{tot}(\vec b_i-\vec b_j^B,s)=(2/c)[1-\exp
\{-u(b_{ij}^2,s)\}]. 
\end{equation}

The Pomeron parameters are found from a global fit of the total,
elastic, differential elastic and diffractive cross sections of the
hadron--nucleon interaction at different energies.

For the nucleon-nucleon, pion-nucleon and kaon-nucleon collisions 
the Pomeron vertex
parameters and shower enhancement coefficients are found:
$R^{2N}_{P}=3.56$ $GeV^{-2}$, $\gamma^{N}_P=3.96$
$GeV^{-2}$, $s^{N}_{0} = 3.0$ $GeV^{2}$, $c^{N}=1.4$ and
$R^{2\pi}_{P} = 2.36$ $GeV^{-2}$, $\gamma^{\pi}_P = 2.17$ $GeV^{-2}$,
 and $R^{2K}_{P} = 1.96$
$GeV^{-2}$, $\gamma^{K} _P = 1.92$ $GeV^{-2}$, $s^{K}_{0} = 2.3$
$GeV^{2}$, $c^{\pi}=1.8$.

\subsection{Separation of hadron diffraction excitation.}

\hspace{1.0em}For each pair of target hadron $i$ and projectile 
nucleon $j$ with choosen impact
parameters $\vec{b}_{i}$ and $\vec{b}^{B}_{j}$ we should check
whether they interact inelastically or not using the probability
\begin{equation}
\label{SP14}
p^{in}_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)=
p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)
+ p_{ij}^d(\vec b_i^A-\vec b_j^B,s).
\end{equation}
 If interaction will be realized, then 
we have to consider it to be diffractive or nondiffractive with probabilities
\begin{equation}
\label{SP15}
\frac{p_{ij}^d(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij}
(\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)}
\end{equation}
and
\begin{equation}
\label{SP16}
\frac{p_{ij}(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij}
(\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)}.
\end{equation}