1 | \section{The high energy $\gamma$-nucleon and $\gamma$-nucleus |
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2 | interactions.} |
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3 | |
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4 | \hspace{1.0em}To simulate high energy photon interactions with nucleon and |
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5 | nucleus we use the approach\cite{PRW95}. |
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6 | We consider the following kinematic |
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7 | variables for $\gamma$-nucleon |
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8 | scattering: the Bjorken-$x$ variable defined as $x=Q^2/2m\nu$ with $Q^2$, $\nu$ |
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9 | and $m$ the photon virtuality, the photon energy and nucleon mass, |
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10 | respectively. |
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11 | The the squared total energy of the $\gamma$-nucleon system is given by |
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12 | $s=Q^2(1-x)/x + m^2$. We restrict consideration to |
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13 | the range of small $x$-values and $Q^2$ is much |
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14 | less than $s$. |
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15 | |
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16 | The Generalized Vector Dominance Model (GVDM) \cite{BSY78} |
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17 | assumes that the virtual photon |
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18 | fluctuates into intermediate $q\bar{q}$-states $V$ of mass $M$ which |
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19 | subsequently may |
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20 | interact with a nucleon $N$. |
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21 | Thus the total photon-nucleon cross section |
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22 | can be expressed by a relation \cite{PRW95}: |
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23 | \begin{equation} |
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24 | \begin{array}{c} |
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25 | \label{HEGI1}\sigma_{\gamma N}(s,Q^2)=4\pi\alpha_{em}\int_{M^2_0}^{M^2_{1}} |
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26 | dM^2D(M^2)\times \\ |
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27 | \times (\frac{M^2}{M^2+Q^2})^2(1+\epsilon\frac{Q^2}{M^2})\sigma_{VN}(s,Q^2), |
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28 | \end{array} |
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29 | \end{equation} |
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30 | where integration over $M^2$ should be performed between $M^2_0=4m^2_{\pi}$ |
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31 | and $M^2=s$. |
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32 | Here $\alpha_{em} = e^2/4\pi = 1/137$ and the density |
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33 | of $q\bar{q}$-system per |
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34 | unit mass-squared is given by |
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35 | \begin{equation} |
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36 | \label{HEGI2}D(M^2)= \frac{R_{e^{+}e^{-}}(M^2)}{12\pi^2M^2}, |
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37 | \end{equation} |
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38 | \begin{equation} |
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39 | \label{HEGI3} R_{e^{+}e^{-}}(M^2)=\frac{\sigma_{e^{+}e^{-}\rightarrow |
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40 | hadrons}(M^2)}{\sigma_{e^{+}e^{-}\rightarrow |
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41 | \mu^{+}\mu^{-}}(M^2)}\approx 3\Sigma_{f}e^2_{f}, |
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42 | \end{equation} |
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43 | where $e^2_{f}$ the squared charge of quark with flavor $f$. |
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44 | $\epsilon$ is the |
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45 | ratio between the fluxes of longitudinally |
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46 | and transversally polarized photons. |
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47 | |
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48 | Similarly the |
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49 | inelastic cross section for the scattering of a $\gamma$ with virtuality |
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50 | $Q^2$ and with a nucleus $A$ at impact parameter $B$ |
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51 | and the $\gamma$-nucleon c.m. |
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52 | energy squared $s$ is given by \cite{ERR97}: |
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53 | \begin{equation} |
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54 | \begin{array}{c} |
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55 | \label{HEGI4}\sigma_{\gamma A}(s,Q^2,B)=4\pi\alpha_{em}\int_{M^2_0}^{M^2_{1}} |
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56 | dM^2D(M^2)\times \\ |
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57 | \times (\frac{M^2}{M^2+Q^2})^2(1+\epsilon\frac{Q^2}{M^2})\sigma_{VA}(s,Q^2,B), |
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58 | \end{array} |
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59 | \end{equation} |
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60 | |
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61 | To calculate $\gamma$-nucleon or $\gamma$-nucleus inelastic cross sections |
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62 | we need model for the $M^2$-, |
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63 | $Q^2$- and $s$-dependence of the $\sigma_{VN}$ or $\sigma_{VA}$. For |
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64 | these we apply the |
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65 | Gribov-Regge approach, similarly as it was done for $h$-nucleon or $h$-nucleus |
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66 | inelastic cross sections. |
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67 | |
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68 | The |
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69 | effective cross section for the interaction of a $q\bar{q}$-system with |
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70 | squared mass $M^2$ with nucleus for the coherence length |
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71 | \begin{equation} |
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72 | \label{HEGI5} d=\frac{2\nu}{M^2+Q^2} |
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73 | \end{equation} |
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74 | exceeding the average distance between two nucleons |
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75 | can be written as follows |
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76 | \begin{equation} |
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77 | \begin{array}{c} |
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78 | \label{HEGI6}\sigma_{V A}(s,Q^2,B)=\int \prod_{i=1}^{A} |
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79 | d^3 r_i\rho_A({\bf r}_i) |
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80 | \times \\ |
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81 | \times (1 - |\prod_{i=1}^{A}[1-u(s,Q^2,M^2, b^2_i)]|^2). |
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82 | \end{array} |
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83 | \end{equation} |
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84 | Here the amplitude (eikonal) |
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85 | $u(s,Q^2,M^2, b^2_i)$ for the interaction of |
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86 | the hadronic fluctuation with $i$-th nucleon |
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87 | is given by \cite{ERR97} |
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88 | \begin{equation} |
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89 | \begin{array}{c} |
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90 | \label{HEGI7}u(s,Q^2,M^2,{\bf b}_i)= |
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91 | \frac{\sigma_{VN}(s,Q^2,M^2)} {8 \pi \lambda(s,Q^2,M^2)} \times \\ |
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92 | \times (1-i \rho \exp{[-\frac{b^2}{4\lambda(s,Q^2,M^2)}]}, |
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93 | \end{array} |
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94 | \end{equation} |
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95 | where $\rho\approx 0$ is the ratio of real and imaginary parts of scattering |
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96 | amplitude at $0$ angle. |
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97 | The amplitude parameters: the effective $q\bar{q}$-nucleon cross section |
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98 | \begin{equation} |
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99 | \label{HEGI8} \sigma_{VN}(s,Q^2,M^2)=\frac{\tilde{\sigma}_{VN}(s,Q^2)}{M^2+Q^2+C^2}, |
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100 | \end{equation} |
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101 | where $C^2=2$ \ GeV$^2$, |
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102 | and |
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103 | \begin{equation} |
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104 | \label{HEGI9}\lambda(s,Q^2,M^2)=2+\frac{m^2_{\rho}}{M^2+Q^2} + |
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105 | \alpha_{P}^{\prime} |
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106 | \ln{(\frac{s}{M^2+Q^2})}. |
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107 | \end{equation} |
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108 | The values of $\tilde{\sigma}_{VN}(s,Q^2)$ are calculated in paper |
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109 | \cite{ERR97}. |
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110 | It was shown \cite{ERR97} that $Q^2$ dependence of $\sigma_{VN}(s,Q^2)$ |
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111 | is very week at $Q^2 < m^2_{rho} + C^2$, where $m_{\rho}$ is |
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112 | $\rho$-meson mass, and we omitted this dependence. We also use |
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113 | $\sigma_{VN}(s,Q^2)$ calculated in \cite{ERR97} at $M^2=m^2_{rho}$. |
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114 | |
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115 | If coherence length is smaller that an internuclear distance integrated |
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116 | over $B$ then cross section |
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117 | $\sigma_{VA}=A\sigma_{VN}$. |
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118 | |
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