1 | \section{Longitudinal string excitation} |
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2 | |
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3 | \subsection{Hadron--nucleon inelastic collision} |
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4 | |
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5 | \hspace{1.0em}Let us consider collision of two hadrons with their c. m. momenta |
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6 | $P_1 = \{E^{+}_1,m^2_1/E^{+}_1,{\bf 0}\}$ and $P_2 = |
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7 | \{E^{-}_2,m^2_2/E^{-}_2,{\bf 0}\}$, where the light-cone variables |
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8 | $E^{\pm}_{1,2} = E_{1,2} \pm P_{z1,2}$ are defined through hadron |
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9 | energies $E_{1,2}=\sqrt{m^2_{1,2} + P^2_{z1,2}}$, hadron longitudinal |
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10 | momenta $P_{z1,2}$ and hadron masses $m_{1,2}$, respectively. Two |
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11 | hadrons collide by two partons with momenta $p_1 = \{x^{+}E^{+}_1,0,{\bf |
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12 | 0}\}$ and $p_2 = \{0, x^{-}E^{-}_2,{\bf 0}\}$, respectively. |
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13 | |
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14 | \subsection{The diffractive string excitation} |
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15 | |
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16 | In the diffractive string excitation (the Fritiof approach \cite{FRITIOF87}) |
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17 | only momentum can be transferred: |
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18 | \begin{equation} |
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19 | \label{LSE1} |
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20 | \begin{array}{cc} |
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21 | P^{\prime}_1 = P_1 + q\\ |
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22 | P^{\prime}_2 = P_2 -q, |
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23 | \end{array} |
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24 | \end{equation} |
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25 | where |
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26 | \begin{equation} |
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27 | \label{LSE2}q=\{-q^2_t/(x^{-}E^{-}_2),q^2_t/(x^{+}E^{+}_1),\bf q_t \} |
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28 | \end{equation} |
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29 | is parton |
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30 | momentum transferred and ${\bf q_t}$ is its transverse component. |
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31 | We use the Fritiof approach to simulate the diffractive excitation of |
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32 | particles. |
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33 | |
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34 | \subsection{The string excitation by parton exchange} |
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35 | |
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36 | \hspace{1.0em}For this case the parton exchange (rearrangement) and |
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37 | the momentum exchange are allowed \cite{QGSM82},\cite{DPM94},\cite{Am86}: |
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38 | \begin{equation} |
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39 | \label{LSE3} |
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40 | \begin{array}{cc} |
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41 | P^{\prime}_1 = P_1 - p_1 + p_2 + q \\ |
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42 | P^{\prime}_2 = P_2 + p_1 - p_2 - q, |
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43 | \end{array} |
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44 | \end{equation} |
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45 | where $q= \{0,0, {\bf q_t}\}$ is parton momentum transferred, i. e. only |
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46 | its transverse components ${\bf q_t} = 0$ is taken into account. |
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47 | |
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48 | \subsection{Transverse momentum sampling} |
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49 | |
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50 | \hspace{1.0em}The transverse component of the parton momentum |
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51 | transferred is generated according to probability |
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52 | \begin{equation} |
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53 | \label{LSE4}P({\bf q_t})d{\bf q_t} = |
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54 | \sqrt{\frac{a}{\pi}} \exp{(-aq^2_t)}d{\bf q_t}, |
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55 | \end{equation} |
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56 | where parameter $a = 0.6$ GeV$^{-2}$. |
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57 | |
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58 | \subsection{Sampling x-plus and x-minus} |
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59 | |
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60 | Light cone parton quantities $x^{+}$ and $x^{-}$ are generated |
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61 | independently and according to distribution: |
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62 | \begin{equation} |
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63 | \label{LSE5} u(x) \sim x^{\alpha}(1 - x)^{\beta}, |
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64 | \end{equation} |
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65 | where $x=x^{+}$ or $x=x^{-}$. |
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66 | Parameters $\alpha =-1$ and $\beta = 0$ are chosen for the FRITIOF approach |
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67 | \cite{FRITIOF87}. In the case of the QGSM approach \cite{Am86} $\alpha = -0.5$ |
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68 | and $\beta = 1.5$ or $\beta = 2.5$. Masses of the excited strings |
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69 | should satisfy the kinematical constraints: |
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70 | \begin{equation} |
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71 | \label{LSE6} P^{\prime +}_1 P^{\prime -}_1 \geq m^2_{h1} + q^2_t |
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72 | \end{equation} |
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73 | and |
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74 | \begin{equation} |
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75 | \label{LSE7} P^{\prime +}_2 P^{\prime -}_2 \geq m^2_{h2} + q^2_t, |
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76 | \end{equation} |
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77 | where hadronic masses $m_{h1}$ and $m_{h2}$ (model parameters) are |
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78 | defined by string quark contents. Thus, the random selection of the |
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79 | values $x^{+}$ and $x^{-}$ is limited by above constraints. |
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80 | |
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81 | \subsection{The diffractive string excitation} |
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82 | |
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83 | \hspace{1.0em}In the diffractive string excitation (the |
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84 | FRITIOF approach \cite{FRITIOF87}) for each |
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85 | inelastic hadron--nucleon collision we have to select |
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86 | randomly the transverse momentum transferred ${\bf q_t}$ (in accordance |
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87 | with the probability given by Eq. ($\ref{LSE4}$)) and select randomly |
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88 | the values of $x^{\pm}$ (in accordance with distribution defined by |
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89 | Eq. ($\ref{LSE5}$)). Then we have to calculate the parton momentum |
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90 | transferred $q$ using Eq. ($\ref{LSE2}$) and update scattered hadron |
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91 | and nucleon or scatterred nucleon and nucleon momenta using |
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92 | Eq. ($\ref{LSE3}$). For each collision we have to check the constraints |
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93 | ($\ref{LSE6}$) and ($\ref{LSE7}$), which can be written more |
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94 | explicitly: |
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95 | \begin{equation} |
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96 | \label{LSE8} [E_1^{+} -\frac{q^2_t}{x^{-}E^{-}_2}][\frac{m_1^2}{E^{+}_1} + |
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97 | \frac{q^2_t}{x^{+}E^{+}_1}]\geq m^2_{h1} + q^2_t |
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98 | \end{equation} |
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99 | and |
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100 | \begin{equation} |
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101 | \label{LSE9} [E_2^{-} +\frac{q^2_t}{x^{-}E^{-}_2}][\frac{m_2^2}{E^{-}_2} - |
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102 | \frac{q^2_t}{x^{+}E^{+}_1}]\geq m^2_{h1} + q^2_t. |
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103 | \end{equation} |
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104 | |
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105 | \subsection{The string excitation by parton rearrangement} |
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106 | |
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107 | \hspace{1.0em}In this approach \cite{Am86} strings (as result of parton |
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108 | rearrangement) should be spanned not only between valence quarks of |
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109 | colliding hadrons, but also between valence and sea quarks and between |
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110 | sea quarks. The each participant hadron or nucleon should be splitted |
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111 | into set of partons: valence quark and antiquark for meson or valence |
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112 | quark (antiquark) and diquark (antidiquark) for baryon (antibaryon) and |
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113 | additionaly the $(n-1)$ sea quark-antiquark pairs (their flavours are |
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114 | selected according to probability ratios $ u:d:s = 1:1:0.35$), if hadron |
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115 | or nucleon is participating in the $n$ inelastic collisions. Thus for |
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116 | each participant hadron or nucleon we have to generate a set of light |
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117 | cone variables $x_{2n}$, where $x_{2n}=x^{+}_{2n}$ or |
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118 | $x_{2n}=x^{-}_{2n}$ according to distribution: |
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119 | \begin{equation} |
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120 | \label{LS10} f^{h}(x_1,x_2,...,x_{2n})=f_{0}\prod_{i=1}^{2n}u^h_{q_i}(x_i) |
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121 | \delta{(1-\sum_{i=1}^{2n}x_i)}, |
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122 | \end{equation} |
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123 | where $f_0$ is the normalization constant. |
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124 | Here, the quark structure functions $u_{q_i}^h(x_i)$ for valence quark |
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125 | (antiquark) $q_v$, |
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126 | sea quark and antiquark $q_s$ and valence diquark (antidiquark) $qq$ are: |
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127 | \begin{equation} |
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128 | \label{LS11} |
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129 | u^h_{q_v}(x_v)=x_v^{\alpha_v},\ u^h_{q_s}(x_s)=x_s^{\alpha_s},\ u^h_{qq}(x_{qq}) |
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130 | =x_{qq}^{\beta_{qq}}, |
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131 | \end{equation} |
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132 | where $\alpha_v = -0.5$ and $\alpha_s = -0.5$ \cite{QGSM82} |
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133 | for the non-strange quarks (antiquarks) and $\alpha_v = |
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134 | 0$ and $\alpha_s = 0$ for strange quarks (antiquarks), $\beta_{uu} = |
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135 | 1.5$ and $\beta_{ud} = 2.5$ for proton (antiproton) and $\beta_{dd} = |
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136 | 1.5$ and $\beta_{ud} = 2.5$ for neutron (antineutron). Usualy $x_i$ are |
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137 | selected between $x^{min}_i \leq x_i \leq 1$, where model parameter |
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138 | $x^{min}$ is a function of initial energy, to prevent from production of |
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139 | strings with low masses (less than hadron masses), when whole selection |
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140 | procedure should be repeated. Then the transverse momenta of partons |
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141 | ${\bf q_{it}}$ are generated according to the Gaussian probability |
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142 | Eq. ($\ref{LSE4}$) with $a = 1/4\Lambda(s)$ and under the constraint: $\sum_{i=1}^{2n}{\bf |
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143 | q_{it}}=0$. The partons are considered as the off-shell partons, |
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144 | i. e. $m^2_i \neq 0$. |
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