[1208] | 1 | \section{Sample of collision participants |
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| 2 | in nuclear collisions.} |
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| 3 | |
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| 4 | \subsection{MC procedure to define collision participants.} |
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| 5 | \hspace{1.0em} The inelastic |
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| 6 | hadron--nucleus interactions at ultra--relativistic energies are considered |
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| 7 | as |
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| 8 | independent hadron--nucleon collisions. It was shown long |
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| 9 | time ago \cite{CK78} for the hadron--nucleus collision that such a |
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| 10 | picture can be obtained starting from the Regge--Gribov |
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| 11 | approach \cite{BT76}, when one assumes that the hadron-nucleus elastic |
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| 12 | scattering amplitude is a result of reggeon exchanges between the |
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| 13 | initial hadron and nucleons from target--nucleus. This result leads to |
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| 14 | simple and efficient MC procedure \cite{Am86} to define |
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| 15 | the interaction cross sections and the number of the nucleons |
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| 16 | participating in the inelastic hadron--nucleus collision: |
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| 17 | \begin{itemize} |
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| 18 | \item We should randomly distribute |
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| 19 | $B$ |
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| 20 | nucleons from the target-nucleus on the impact parameter plane according |
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| 21 | to the weight function |
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| 22 | $T([\vec{b}^{B}_{j}])$. This function represents |
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| 23 | probability density to find sets of the nucleon impact parameters |
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| 24 | $[\vec{b}^{B}_{j}]$, where |
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| 25 | $j=1,2,...,B$. |
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| 26 | \item For each pair of projectile hadron $i$ and target nucleon |
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| 27 | $j$ with choosen impact parameters $\vec{b}_{i}$ and |
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| 28 | $\vec{b}^{B}_{j}$ we should check whether they interact inelastically or |
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| 29 | not using the probability $p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)$, |
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| 30 | where $s_{ij}=(p_{i}+p_{j})^2$ is the squared total c.m. energy of the |
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| 31 | given pair with the $4$--momenta $p_{i}$ and $p_{j}$, respectively. |
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| 32 | \end{itemize} |
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| 33 | |
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| 34 | In the Regge--Gribov approach\cite{BT76} the probability for an inelastic |
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| 35 | collision of pair of $i$ and $j$ as a function at the squared impact |
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| 36 | parameter difference $b_{ij}^2=(\vec{ b}_i-\vec{ b}_j^B)^2 $ and $s$ |
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| 37 | is given by |
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| 38 | \begin{equation} |
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| 39 | \label{SP3} |
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| 40 | p_{ij}(\vec{ b}_i-\vec{ b}_j^B,s)= |
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| 41 | c^{-1}[1-\exp{\{-2u(b_{ij}^2,s)\}}] = |
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| 42 | \sum_{n=1}^{\infty}p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s), |
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| 43 | \end{equation} |
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| 44 | where |
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| 45 | \begin{equation} |
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| 46 | \label{SP4} |
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| 47 | p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s) |
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| 48 | =c^{-1}\exp{\{-2u(b_{ij}^2,s)\}} |
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| 49 | \frac{[2u(b_{ij}^2,s)]^{n}}{n!}. |
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| 50 | \end{equation} |
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| 51 | is the probability to find the $n$ cut Pomerons or the probability for |
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| 52 | $2n$ strings produced in an inelastic hadron-nucleon collision. These |
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| 53 | probabilities are defined in terms of the (eikonal) amplitude of |
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| 54 | hadron--nucleon elastic scattering with Pomeron exchange: |
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| 55 | \begin{equation} |
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| 56 | \label{SP5}u(b_{ij}^2,s)=\frac{z(s)}{2}\exp (-b_{ij}^2/4\lambda (s)). |
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| 57 | \end{equation} |
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| 58 | The quantities $z(s)$ and $\lambda (s)$ are expressed through the |
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| 59 | parameters of the Pomeron trajectory, $\alpha _P^{^{\prime }}=0.25$ |
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| 60 | $GeV^{-2}$ and $\alpha _P(0)=1.0808$, and the parameters of the |
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| 61 | Pomeron-hadron vertex $R_P$ and $\gamma _P$: |
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| 62 | \begin{equation} |
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| 63 | \label{SP6}z(s)=\frac{2c\gamma _P}{\lambda (s)}(s/s_0)^{\alpha _P(0)-1} |
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| 64 | \end{equation} |
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| 65 | \begin{equation} |
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| 66 | \label{SP7}\lambda (s)=R_P^2+\alpha _P^{^{\prime }}\ln (s/s_0), |
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| 67 | \end{equation} |
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| 68 | respectively, where $s_0$ is a dimensional parameter. |
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| 69 | |
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| 70 | In Eqs. (\ref{SP3},\ref{SP4}) the so--called shower enhancement |
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| 71 | coefficient $c$ is introduced to determine the contribution of |
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| 72 | diffractive dissociation\cite{BT76}. Thus, the probability for |
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| 73 | diffractive dissociation of a pair |
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| 74 | of nucleons can be computed as |
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| 75 | \begin{equation} |
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| 76 | \label{SP8}p_{ij}^d(\vec b_i-\vec b_j^B,s)=\frac{c-1}{c}[p_{ij}^{tot}(\vec |
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| 77 | b_i-\vec b_j^B,s)-p_{ij}(\vec b_i-\vec b_j^B,s)], |
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| 78 | \end{equation} |
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| 79 | where |
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| 80 | \begin{equation} |
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| 81 | \label{SP9}p_{ij}^{tot}(\vec b_i-\vec b_j^B,s)=(2/c)[1-\exp |
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| 82 | \{-u(b_{ij}^2,s)\}]. |
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| 83 | \end{equation} |
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| 84 | |
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| 85 | The Pomeron parameters are found from a global fit of the total, |
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| 86 | elastic, differential elastic and diffractive cross sections of the |
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| 87 | hadron--nucleon interaction at different energies. |
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| 88 | |
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| 89 | For the nucleon-nucleon, pion-nucleon and kaon-nucleon collisions |
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| 90 | the Pomeron vertex |
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| 91 | parameters and shower enhancement coefficients are found: |
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| 92 | $R^{2N}_{P}=3.56$ $GeV^{-2}$, $\gamma^{N}_P=3.96$ |
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| 93 | $GeV^{-2}$, $s^{N}_{0} = 3.0$ $GeV^{2}$, $c^{N}=1.4$ and |
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| 94 | $R^{2\pi}_{P} = 2.36$ $GeV^{-2}$, $\gamma^{\pi}_P = 2.17$ $GeV^{-2}$, |
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| 95 | and $R^{2K}_{P} = 1.96$ |
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| 96 | $GeV^{-2}$, $\gamma^{K} _P = 1.92$ $GeV^{-2}$, $s^{K}_{0} = 2.3$ |
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| 97 | $GeV^{2}$, $c^{\pi}=1.8$. |
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| 98 | |
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| 99 | \subsection{Separation of hadron diffraction excitation.} |
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| 100 | |
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| 101 | \hspace{1.0em}For each pair of target hadron $i$ and projectile |
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| 102 | nucleon $j$ with choosen impact |
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| 103 | parameters $\vec{b}_{i}$ and $\vec{b}^{B}_{j}$ we should check |
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| 104 | whether they interact inelastically or not using the probability |
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| 105 | \begin{equation} |
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| 106 | \label{SP14} |
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| 107 | p^{in}_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)= |
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| 108 | p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s) |
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| 109 | + p_{ij}^d(\vec b_i^A-\vec b_j^B,s). |
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| 110 | \end{equation} |
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| 111 | If interaction will be realized, then |
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| 112 | we have to consider it to be diffractive or nondiffractive with probabilities |
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| 113 | \begin{equation} |
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| 114 | \label{SP15} |
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| 115 | \frac{p_{ij}^d(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij} |
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| 116 | (\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)} |
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| 117 | \end{equation} |
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| 118 | and |
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| 119 | \begin{equation} |
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| 120 | \label{SP16} |
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| 121 | \frac{p_{ij}(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij} |
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| 122 | (\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)}. |
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| 123 | \end{equation} |
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