Sample of collision participants in nuclear collisions.

MC procedure to define collision participants.

The inelastic hadron-nucleus interactions at ultra-relativistic energies are considered as independent hadron-nucleon collisions. It was shown long time ago [5] for the hadron-nucleus collision that such a picture can be obtained starting from the Regge-Gribov approach [6], 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 [7] to define the interaction cross sections and the number of the nucleons participating in the inelastic hadron-nucleus collision:

• 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 $j=1,2,...,B$.
• 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.

In the Regge-Gribov approach[6] 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

$$p_{ij}(\vec{ b}_i-\vec{ b}_j^B,s)= c^{-1}[1-\exp{\{-2u(b_{... ... \sum_{n=1}^{\infty}p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s),$$ (22.8)

where

$$p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s) =c^{-1}\exp{\{-2u(b_{ij}^2,s)\}} \frac{[2u(b_{ij}^2,s)]^{n}}{n!}.$$ (22.9)

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:

$$u(b_{ij}^2,s)=\frac{z(s)}{2}\exp (-b_{ij}^2/4\lambda (s)).$$ (22.10)

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 $\gamma _P$:

$$z(s)=\frac{2c\gamma _P}{\lambda (s)}(s/s_0)^{\alpha _P(0)-1}$$ (22.11)

$$\lambda (s)=R_P^2+\alpha _P^{^{\prime }}\ln (s/s_0),$$ (22.12)

respectively, where $s_0$ is a dimensional parameter.

In Eqs. (22.8,22.9) the so-called shower enhancement coefficient $c$ is introduced to determine the contribution of diffractive dissociation[6]. Thus, the probability for diffractive dissociation of a pair of nucleons can be computed as

$$p_{ij}^d(\vec b_i-\vec b_j^B,s)=\frac{c-1}{c}[p_{ij}^{tot}(\vec b_i-\vec b_j^B,s)-p_{ij}(\vec b_i-\vec b_j^B,s)],$$ (22.13)

where

$$p_{ij}^{tot}(\vec b_i-\vec b_j^B,s)=(2/c)[1-\exp \{-u(b_{ij}^2,s)\}].$$ (22.14)

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 $R^{2K}_{P} = 1.96$ $GeV^{-2}$, $\gamma^{K} _P = 1.92$ $GeV^{-2}$, $s^{K}_{0} = 2.3$ $GeV^{2}$, $c^{\pi}=1.8$.

Separation of hadron diffraction excitation.

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

$$p^{in}_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)= p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s) + p_{ij}^d(\vec b_i^A-\vec b_j^B,s).$$ (22.15)

If interaction will be realized, then we have to consider it to be diffractive or nondiffractive with probabilities

$$\frac{p_{ij}^d(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij} (\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)}$$ (22.16)

and

$$\frac{p_{ij}(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij} (\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)}.$$ (22.17)