1 | \section[Gamma Conversion into a Muon - Anti-mu Pair] |
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2 | {Gamma Conversion into a Muon - Anti-muon Pair} |
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3 | |
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4 | The class {\tt G4GammaConversionToMuons} simulates the process of gamma |
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5 | conversion into muon pairs. Given the photon energy and $Z$ and $A$ of the |
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6 | material in which the photon converts, the probability for the conversions |
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7 | to take place is calculated according to a parameterized total cross section. |
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8 | Next, the sharing of the photon energy between the $\mu^+$ and $\mu^-$ is |
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9 | determined. Finally, the directions of the muons are generated. Details of |
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10 | the implementation are given below and can be also found in\,\cite{MuPgen}. |
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11 | |
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12 | \subsection{Cross Section and Energy Sharing} |
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13 | In the field of the nucleus, muon pair production on atomic electrons, |
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14 | $\gamma+e\to e+\mu^+ +\mu^-$, has a threshold of |
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15 | $2m_\mu(m_\mu+m_e)/m_e\approx 43.9\;{\rm GeV}$ . |
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16 | %hbu check in MuonBkg.C in fact 43.905 GeV |
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17 | Up to several hundred GeV this process has a much lower cross section than |
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18 | the corresponding process on the nucleus. At higher energies, the cross |
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19 | section on atomic electrons represents a correction of $\sim 1/Z$ to the |
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20 | total cross section. |
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21 | |
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22 | For the approximately elastic scattering considered here, momentum, but no |
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23 | energy, is transferred to the nucleon. The photon energy is fully shared |
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24 | by the two muons according to |
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25 | \begin{equation} |
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26 | E_\gamma = E_\mu^+ + E_\mu^- |
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27 | \end{equation} |
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28 | or in terms of energy fractions |
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29 | \[ x_+ = \frac{E_\mu^+}{E_\gamma}, \qquad x_- = \frac{E_\mu^-}{E_\gamma}, \qquad x_+ + x_- = 1\;. |
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30 | \] |
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31 | The differential cross section for electromagnetic pair creation of muons in |
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32 | terms of the energy fractions of the muons is |
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33 | \begin{equation} |
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34 | \frac{d\sigma}{d x_+} = 4 \, \alpha \, Z^2 \, r_c^2 |
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35 | \left(1-\frac43\,x_+x_-\right)\log(W)\;, |
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36 | \label{eq:dSigxPlus} |
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37 | \end{equation} |
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38 | where $Z$ is the charge of the nucleus, $r_c$ is the classical radius of the |
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39 | particles which are pair produced (here muons) and |
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40 | \begin{equation} |
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41 | W = W_\infty \; |
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42 | \frac{1+(D_n\sqrt{e}-2)\,\delta\,/m_\mu}{1+B \, Z^{-1/3}\, \sqrt e \,\delta\,/m_e} |
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43 | \label{eq:W} |
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44 | \end{equation} |
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45 | where |
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46 | \[ |
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47 | W_\infty = \frac{B \, Z^{-1/3}}{D_n} \, \frac{m_\mu}{m_e} \qquad |
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48 | \delta = \frac{m_\mu^2}{2\,E_\gamma\,x_+x_-} \qquad |
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49 | \sqrt{e}=1.6487\dots . |
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50 | \] |
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51 | \begin{eqnarray} |
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52 | \mbox{For hydrogen} \qquad B=202.4 \qquad D_n=&1.49 & \nonumber\\ |
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53 | \mbox{and for all other nuclei} \qquad B=183 \qquad D_n=&1.54 \, A^{0.27}. & |
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54 | \end{eqnarray} |
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55 | These formulae are obtained from the differential cross section for muon |
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56 | bremsstrahlung \cite{Kelner:1995hu} by means of crossing relations. The |
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57 | formulae take into account the screening of the field of the nucleus by the |
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58 | atomic electrons in the Thomas-Fermi model, as well as the finite size of |
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59 | the nucleus, which is essential for the problem under consideration. |
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60 | % |
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61 | The above parameterization gives good results for $E_\gamma \gg m_\mu$. The |
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62 | fact that it is approximate close to threshold is of little practical |
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63 | importance. Close to threshold, the cross section is small and the few low |
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64 | energy muons produced will not travel very far. The cross section |
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65 | calculated from Eq.\,(\ref{eq:dSigxPlus}) is positive for |
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66 | $E_\gamma > 4 m_\mu$ and |
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67 | \begin{equation} |
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68 | x_{\rm min} \leq x \leq x_{\rm max} \quad {\rm with} \quad |
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69 | x_{\rm min} = \frac12 - \sqrt{\frac{1}{4}-\frac{m_\mu}{E_\gamma}} \qquad |
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70 | x_{\rm max} = \frac12 + \sqrt{\frac{1}{4}-\frac{m_\mu}{E_\gamma}}\;, |
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71 | \end{equation} |
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72 | except for very asymmetric pair-production, close to threshold, which can |
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73 | easily be taken care of by explicitly setting $\sigma = 0$ whenever |
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74 | $\sigma < 0$. |
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75 | |
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76 | Note that the differential cross section is symmetric in $x_+$ and $x_-$ and |
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77 | that |
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78 | \[ x_+ x_- = x - x^2 \] |
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79 | where $x$ stands for either $x_+$ or $x_-$. By defining a constant |
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80 | \begin{equation} |
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81 | \sigma_0 = 4 \, \alpha \, Z^2 \, r_c^2 \log(W_\infty)\; |
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82 | \label{eq:sigma0} |
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83 | \end{equation} |
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84 | the differential cross section Eq.\,(\ref{eq:dSigxPlus}) can be rewritten |
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85 | as a normalized and symmetric as function of $x$: |
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86 | \begin{equation} |
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87 | \frac{1}{\sigma_0} \, \frac{d\sigma}{dx} = \left[ 1-\frac43 \,(x - x^2) \right] |
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88 | \, \frac{\log W}{\log W_\infty}\;. |
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89 | \label{eq:dSigdx} |
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90 | \end{equation} |
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91 | This is shown in Fig.\,\ref{plot:dsigdx} for several elements and a wide |
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92 | range of photon energies. The asymptotic differential cross section for |
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93 | $E_\gamma \rightarrow \infty$ |
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94 | \[ \frac{1}{\sigma_0} \, \frac{d\sigma_\infty}{dx} = 1-\frac43 \,(x - x^2) \] |
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95 | is also shown. |
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96 | \begin{figure}[htpb] |
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97 | \center\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/dsigdx.eps} |
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98 | \caption{Normalized differential cross section for pair production as a |
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99 | function of $x$, the energy fraction of the photon energy carried by one of |
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100 | the leptons in the pair. The function is shown for three different |
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101 | elements, hydrogen, beryllium and lead, and for a wide range of photon |
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102 | energies.} |
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103 | \label{plot:dsigdx} |
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104 | \end{figure} |
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105 | |
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106 | \subsection{Parameterization of the Total Cross Section} |
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107 | The total cross section is obtained by integration of the differential |
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108 | cross section Eq.\,(\ref{eq:dSigxPlus}), that is |
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109 | \begin{equation} |
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110 | \sigma_{\rm tot}(E_\gamma) = \int_{x_{\rm min}}^{x_{\rm max}} \frac{d\sigma}{d x_+} \, d x_+ |
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111 | = 4 \, \alpha \, Z^2 \, r_c^2 \, \int_{x_{\rm min}}^{x_{\rm max}}\left(1-\frac43\,x_+x_-\right)\log(W) \, d x_+ \;. |
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112 | \label{eq:sigmatot} |
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113 | \end{equation} |
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114 | $W$ is a function of ($x_+, E_\gamma$) and ($Z, A$) of the element |
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115 | (see Eq.\,(\ref{eq:W})). Numerical values of $W$ are given in |
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116 | Table\,\ref{tab:W}. |
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117 | |
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118 | \begin{table}[htbp]\center |
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119 | \caption{Numerical values of $W$ for $x_+=0.5$ for different elements.} |
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120 | \label{tab:W}\vskip 1mm |
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121 | \begin{tabular}{|c|c|c|c|c|} \hline |
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122 | $E_\gamma$ & W for H & W for Be & W for Cu & W for Pb \\ |
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123 | GeV & & & & \\ \hline |
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124 | 1 & 2.11 & 1.594 & 1.3505 & 5.212 \\ |
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125 | 10 & 19.4 & 10.85 & 6.803 & 43.53 \\ |
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126 | 100 & 191.5 & 102.3 & 60.10 & 332.7 \\ |
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127 | 1000 & 1803 & 919.3 & 493.3 & 1476.1 \\ |
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128 | 10000 & 11427 & 4671 & 1824 & 1028.1 \\ |
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129 | $\infty$ & 28087 & 8549 & 2607 & 1339.8 \\ \hline |
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130 | \end{tabular} |
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131 | \end{table} |
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132 | % |
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133 | Values of the total cross section obtained by numerical integration are |
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134 | listed in Table\,\ref{tab:sigmatot} for four different elements. Units are |
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135 | in $\mu{\rm barn}\,$, where $1\,\mu{\rm barn} = 10^{-34}\,{\rm m}^2\,$. |
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136 | % |
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137 | \begin{table}[htbp]\center |
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138 | \caption{Numerical values for the total cross section} |
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139 | \label{tab:sigmatot}\vskip 1mm |
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140 | \begin{tabular}{|c|c|c|c|c|} \hline |
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141 | $E_\gamma$ & $\sigma_{\rm tot}$, H & $\sigma_{\rm tot}$, Be & $\sigma_{\rm tot}$, Cu & $\sigma_{\rm tot}$, Pb \\ |
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142 | GeV & $\mu{\rm barn}\,$ & $\mu{\rm barn}\,$ & $\mu{\rm barn}\,$ & $\mu{\rm barn}\,$ \\ \hline |
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143 | 1 & 0.01559 & 0.1515 & 5.047 & 30.22 \\ |
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144 | 10 & 0.09720 & 1.209 & 49.56 & 334.6 \\ |
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145 | 100 & 0.1921 & 2.660 & 121.7 & 886.4 \\ |
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146 | 1000 & 0.2873 & 4.155 & 197.6 & 1476 \\ |
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147 | 10000 & 0.3715 & 5.392 & 253.7 & 1880 \\ |
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148 | $\infty$ & 0.4319 & 6.108 & 279.0 & 2042 \\ \hline |
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149 | \end{tabular} |
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150 | \end{table} |
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151 | % |
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152 | \begin{figure}[htpb] |
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153 | \center\includegraphics[scale=.7]{electromagnetic/standard/MuPgen/SigTot.eps} |
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154 | \caption{Total cross section for the Bethe-Heitler process |
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155 | $\gamma \rightarrow \mu^+\mu^-$ as a function of the photon energy |
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156 | $E_\gamma$ in hydrogen and lead, normalized to the asymptotic cross section |
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157 | $\sigma_\infty$.} |
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158 | \label{plot:SigTot} |
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159 | \end{figure} |
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160 | |
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161 | \noindent |
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162 | Well above threshold, the total cross section rises about linearly in |
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163 | $\log(E_\gamma)$ with the slope |
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164 | \begin{equation} |
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165 | W_M = \frac{1}{4\, D_n \, \sqrt e \,m_\mu} |
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166 | \end{equation} |
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167 | until it saturates due to screening at $\sigma_\infty$. |
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168 | Fig.\,\ref{plot:SigTot} shows the normalized cross section where |
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169 | \begin{equation} |
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170 | \sigma_\infty = \frac79 \, \sigma_0 \qquad {\rm and} \qquad \sigma_0 = 4 \, \alpha \, Z^2 \, r_c^2 \, \log(W_\infty)\;. |
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171 | \end{equation} |
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172 | Numerical values of $W_M$ are listed in Table\,\ref{tab:WM}. |
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173 | |
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174 | \begin{table}[htbp]\center |
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175 | \caption{Numerical values of $W_M$.} |
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176 | \label{tab:WM}\vskip 1mm |
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177 | \begin{tabular}{|c|c|} \hline |
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178 | Element & $W_M$ \\ |
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179 | & 1/GeV \\ \hline |
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180 | H & 0.963169 \\ |
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181 | Be & 0.514712 \\ |
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182 | Cu & 0.303763 \\ |
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183 | Pb & 0.220771 \\ |
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184 | \hline |
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185 | \end{tabular} |
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186 | \end{table} |
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187 | |
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188 | |
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189 | The total cross section can be parameterized as |
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190 | \begin{equation} |
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191 | \sigma_{\rm par} = \frac{28 \, \alpha \, Z^2 \, r_c^2}{9} \; \log(1 + W_M C_f E_g)\;, |
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192 | \label{eq:sigpar} |
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193 | \end{equation} |
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194 | with |
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195 | \begin{equation} |
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196 | E_g = \left(1-\frac{4 m_\mu}{E_\gamma}\right)^{t} |
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197 | \left(W_{\rm sat}^{s} + E_\gamma^{s} \right)^{1/s}\;. |
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198 | \end{equation} |
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199 | and |
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200 | \[ |
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201 | W_{\rm sat} = \frac{W_\infty}{W_M} = B \, Z^{-1/3} \, \frac{4\,\sqrt e \,m_\mu^2 }{m_e}\;. |
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202 | \] |
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203 | The threshold behavior in the cross section was found to be well |
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204 | approximated by $t = 1.479 + 0.00799 D_n $ and the saturation by |
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205 | $ s = -0.88 $. The agreement at lower energies is improved using an |
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206 | empirical correction factor, applied to the slope $W_M$, of the form |
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207 | \[ |
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208 | C_f = \left[ 1 + 0.04 \log \left(1+\frac{E_c}{E_\gamma}\right)\right]\;, |
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209 | \] |
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210 | where |
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211 | \[ E_c = \left[ -18.+\frac{4347.}{B \, Z^{-1/3}}\right] \;{\rm GeV}\;. |
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212 | \] |
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213 | A comparison of the parameterized cross section with the numerical |
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214 | integration of the exact cross section shows that the accuracy of the |
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215 | parametrization is better than 2\%, as seen in Fig.\,\ref{plot:SigApRat}. |
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216 | \begin{figure}[htpb] |
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217 | \center\includegraphics[scale=.8333]{electromagnetic/standard/MuPgen/SigApRat.eps} |
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218 | \caption{Ratio of numerically integrated and parametrized total cross |
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219 | sections as a function of $E_\gamma$ for hydrogen, beryllium, copper and |
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220 | lead.} |
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221 | \label{plot:SigApRat} |
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222 | \end{figure} |
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223 | |
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224 | \subsection{Multi-differential Cross Section and Angular Variables} |
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225 | The angular distributions are based on the multi-differential cross section |
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226 | for lepton pair production in the field of the Coulomb center |
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227 | \[ |
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228 | \frac{d\sigma}{dx_+ \, du_+ \, du_-\,d\varphi} = \frac{4\,Z^2\alpha^3}{\pi}\,\frac{m_\mu^2}{q^4}\,u_+\,u_- |
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229 | \] |
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230 | \[ |
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231 | \left\{ \frac{u_+^2+u_-^2}{(1+u_+^2)\,(1+u_-^2)} -2x_+x_- \right. |
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232 | \] |
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233 | \begin{equation} |
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234 | \left. |
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235 | \left[\frac{u_+^2}{(1+u_+^2)^2}+\frac{u_-^2}{(1+u_-^2)^2}\right] |
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236 | -\frac{2u_+u_-(1-2x_+x_-)\,\cos\varphi}{(1+u_+^2)\,(1+u_-^2)} |
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237 | \right\} \,. \\ |
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238 | \label{eq:MultiDiff} |
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239 | \end{equation} |
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240 | Here |
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241 | \begin{equation} |
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242 | u_\pm = \gamma_\pm \theta_\pm \quad , \qquad \gamma_\pm = \frac{E_\mu^\pm}{m_\mu} |
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243 | \quad,\qquad q^2=q_{\parallel}^2+q_{\perp}^2\quad, %new |
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244 | \label{eq:uthetagamma} |
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245 | \end{equation} |
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246 | where |
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247 | \begin{eqnarray} |
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248 | q_{\parallel}^2=q_{\min}^2\,(1+x_-u_+^2+x_+u_-^2)^2\,, \nonumber \\ |
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249 | q_{\perp}^2=m_\mu^2\left[(u_+-u_-)^2+2\,u_+u_-(1-\cos\varphi) |
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250 | \right]\,. \, |
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251 | \label{eq:q2} |
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252 | \end{eqnarray} |
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253 | $q^2$ is the square of the momentum ${\bf q}$ transferred to the target |
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254 | and $q_{\parallel}^2$ and $q_{\perp}^2$ are the squares of the components |
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255 | of the vector ${\bf q}$, which are parallel and perpendicular to the |
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256 | initial photon momentum, respectively. |
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257 | The minimum momentum transfer is |
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258 | $q_{\min}=m_\mu^2/(2E_\gamma \, x_+x_-)$.\\ |
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259 | |
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260 | The muon vectors have the components |
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261 | \begin{equation} |
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262 | \begin{array}{rcl}\displaystyle |
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263 | {\bf p}_+&=&p_+\,(\;\;\;\sin\theta_+\cos(\varphi_0+\varphi/2)\,,\,\;\;\;\sin\theta_+\sin(\varphi_0+\varphi/2)\,,\,\cos\theta_+)\,,\\ |
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264 | {\bf p}_-&=&p_-\,( -\sin\theta_-\cos(\varphi_0-\varphi/2)\,,\, -\sin\theta_-\sin(\varphi_0-\varphi/2)\,,\,\cos\theta_-)\,, |
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265 | \end{array} |
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266 | \label{eq:pvec} |
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267 | \end{equation} |
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268 | where $p_{\pm}=\sqrt{E_{\pm}^2-m_\mu^2}$. |
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269 | The initial photon direction is taken as the $z$-axis. |
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270 | The cross section of Eq.\,(\ref{eq:MultiDiff}) does not depend on |
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271 | $\varphi_0$. Because of azimuthal symmetry, $\varphi_0$ can simply be |
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272 | sampled at random in the interval $(0,\,2\,\pi)$. |
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273 | |
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274 | Eq.\,(\ref{eq:MultiDiff}) is too complicated for efficient Monte Carlo |
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275 | generation. To simplify, the cross section is rewritten to be symmetric |
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276 | in $u_+$, $u_-$ using a new variable $u$ and small parameters $\xi,\beta$, |
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277 | where $u_\pm=u \pm \xi/2$ and $\beta = u \,\varphi$. When higher powers |
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278 | in small parameters are dropped, the differential cross section in terms |
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279 | of $u,\xi,\beta$ becomes |
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280 | \begin{eqnarray} |
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281 | \label{mupgen.a} |
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282 | \frac{d\sigma}{dx_+ \, d\xi \, d\beta\, u du} & = & \frac{4\,Z^2\alpha^3}{\pi} |
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283 | \frac{m_\mu^2}{\left(q_{\parallel}^2+m_\mu^2(\xi^2+\beta^2)\right)^2} \\ |
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284 | & & \left\{\xi^2\left[\frac1{(1+u^2)^2}-2\,x_+x_-\,\frac{(1-u^2)^2}{(1+u^2)^4}\right]+ |
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285 | \frac{\beta^2(1-2x_+x_-)}{(1+u^2)^2}\right\}\,, \nonumber |
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286 | \label{eq:MultiDiff2} |
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287 | \end{eqnarray} |
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288 | where, in this approximation, |
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289 | $$ |
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290 | q_{\parallel}^2=q_{\min}^2\,(1+u^2)^2\,.\, |
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291 | $$ |
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292 | For Monte Carlo generation, it is convenient to replace ($\xi,\beta$) by |
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293 | the polar coordinates ($\rho,\psi$) with $\xi=\rho\,\cos\psi$ and |
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294 | $\beta=\rho\,\sin\psi$. Integrating Eq.~\ref{mupgen.a} over $\psi$ and |
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295 | using symbolically $du^2$ where $du^2 = 2 u \, du$ yields |
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296 | \begin{equation} |
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297 | \label{mupgen.b} |
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298 | \frac{d\sigma}{dx_+\,d\rho\,du^2} =\frac{4Z^2\alpha^3}{m_\mu^2}\,\frac{\rho^3}{(q_{\parallel}^2/m_\mu^2+\rho^2)^2} |
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299 | \,\,\left\{\frac{1-x_+x_-}{(1+u^2)^2}-\frac{x_+x_-(1-u^2)^2}{(1+u^2)^4}\right\}.\, |
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300 | \end{equation} |
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301 | Integration with logarithmic accuracy over $\rho$ gives |
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302 | \begin{equation}\label{q4} |
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303 | \int\!\frac{\rho^3\,d\rho}{(q_{\parallel}^2/m_\mu^2+\rho^2)^2} |
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304 | \approx \int\limits_{q_{\parallel}/m_\mu}^1\!\frac{d\rho}{\rho} |
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305 | =\log\left(\frac{m_\mu}{q_{\parallel}}\right)\,. |
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306 | \end{equation} |
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307 | Within the logarithmic accuracy, $\log(m_\mu/q_{\parallel})$ can be |
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308 | replaced by $\log(m_\mu/q_{\min})$, so that |
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309 | \begin{equation} |
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310 | \frac{d\sigma}{dx_+\,du^2}=\frac{4\,Z^2\alpha^3}{m_\mu^2}\, |
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311 | \left\{\frac{1-x_+x_-}{(1+u^2)^2}-\frac{x_+x_-(1-u^2)^2}{(1+u^2)^4}\right\}\, |
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312 | \log\left(\frac{m_\mu}{q_{\min}}\right)\,. |
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313 | \end{equation} |
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314 | Making the substitution $u^2 = 1/t -1$, $du^2 = -dt \, / t^2$ gives |
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315 | \begin{equation} |
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316 | \frac{d\sigma}{dx_+\,dt}=\frac{4\,Z^2\alpha^3}{m_\mu^2}\, |
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317 | \left[1-2\,x_+x_-+4\,x_+x_-t\,(1-t)\right]\, |
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318 | \log\left(\frac{m_\mu}{q_{\min}}\right) . \, |
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319 | \label{eq:sigmadxdt} |
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320 | \end{equation} |
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321 | Atomic screening and the finite nuclear radius may be taken into account by |
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322 | multiplying the differential cross section determined by |
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323 | Eq.\,(\ref{eq:MultiDiff2}) with the factor |
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324 | \begin{equation}\label{q5} |
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325 | \left(F_a(q)-F_n(q)\,\right)^2\,, |
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326 | \end{equation} |
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327 | where $F_a$ and $F_n$ are atomic and nuclear form factors. |
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328 | Please note that after integrating Eq.~\ref{mupgen.b} over $\rho$, the |
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329 | $q$-dependence is lost. |
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330 | |
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331 | \subsection{Procedure for the Generation of Muon - Anti-muon Pairs} |
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332 | |
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333 | Given the photon energy $E_\gamma$ and $Z$ and $A$ of the material in which |
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334 | the $\gamma$ converts, the probability for the conversions to take place is |
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335 | calculated according to the parametrized total cross section |
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336 | Eq.\,(\ref{eq:sigpar}). The next step, determining how the photon energy |
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337 | is shared between the $\mu^+$ and $\mu^-$, is done by generating $x_+$ |
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338 | according to Eq.\,(\ref{eq:dSigxPlus}). The directions of the muons are |
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339 | then generated via the auxilliary variables $t,\,\rho,\,\psi$. In more |
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340 | detail, the final state is generated by the following five steps, in which |
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341 | $R_{1,2,3,4,...}$ are random numbers with a flat distribution in the |
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342 | interval [0,1]. The generation proceeds as follows. |
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343 | \\ \\ |
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344 | {\bf 1)} Sampling of the positive muon energy $E_\mu^+ = x_+ \, E_\gamma$. \\ |
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345 | This is done using the rejection technique. |
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346 | $x_+$ is first sampled from a flat distribution within kinematic limits |
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347 | using |
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348 | \[ x_+ = x_{\rm min} + R_1 (x_{\rm max} - x_{\rm min}) \] |
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349 | and then brought to the shape of Eq.\,(\ref{eq:dSigxPlus}) by keeping all |
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350 | $x_+$ which satisfy |
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351 | \[ \left(1-\frac43\,x_+x_-\right)\frac{\log(W)}{\log(W_{\rm max})} < R_2 \,. \] |
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352 | Here $W_{\rm max}= W(x_+=1/2)$ is the maximum value of $W$, obtained for |
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353 | symmetric pair production at $x_+=1/2$. About 60\% of the events are kept |
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354 | in this step. Results of a Monte Carlo generation of $x_+$ are illustrated |
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355 | in Fig.\,\ref{plot:xPlusGen}. The shape of the histograms agrees with the |
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356 | differential cross section illustrated in Fig.\,\ref{plot:dsigdx}. |
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357 | \begin{figure}[htpb] |
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358 | \center\includegraphics[scale=.7]{electromagnetic/standard/MuPgen/xPlusGen.eps} |
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359 | \caption{Histogram of generated $x_+$ distributions for beryllium at three |
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360 | different photon energies. The total number of entries at each energy is |
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361 | $10^6$.} |
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362 | \label{plot:xPlusGen} |
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363 | \end{figure} |
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364 | \\ \\ |
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365 | {\bf 2)} Generate $t ( = \frac{1}{\gamma^2 \theta^2 + 1} )$ . \\ |
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366 | The distribution in $t$ is obtained from Eq.(\ref{eq:sigmadxdt}) as |
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367 | \begin{equation}\label{t} |
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368 | f_1(t)\,dt=\frac{1-2\,x_+x_-+4\,x_+x_-t\,(1-t)} |
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369 | {1+C_1/t^2}\,dt\,,\quad 0<t\le 1\,. |
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370 | \end{equation} |
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371 | with form factors taken into account by |
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372 | \begin{equation}\label{C_1} |
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373 | C_1= |
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374 | \frac{(0.35\,A^{0.27})^2}{x_+x_-\,E_\gamma/m_\mu }\,. |
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375 | \end{equation} |
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376 | In the interval considered, the function $f_1(t)$ will always be bounded |
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377 | from above by |
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378 | \[ |
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379 | \max [f_1(t)]=\frac{1-x_+x_-}{1+C_1}\;.\, |
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380 | \] |
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381 | For small $x_+$ and large $E_\gamma$, $f_1(t)$ approaches unity, as shown |
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382 | in Fig.\,\ref{f1t.eps}. |
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383 | \begin{figure}[htpb] |
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384 | \includegraphics[scale=.65]{electromagnetic/standard/MuPgen/f1t_10.eps}\hfill\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/f1t_1000.eps} |
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385 | \caption{The function $f_1(t)$ at $E_\gamma = 10\,{\rm GeV}$ (left) and |
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386 | $E_\gamma = 1\,{\rm TeV}$ (right) in beryllium for different values of |
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387 | $x_+$.} |
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388 | \label{f1t.eps} |
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389 | \end{figure} |
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390 | |
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391 | \begin{figure}[htpb] |
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392 | % \begin{minipage}[t]{7.5cm} |
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393 | % \includegraphics[scale=.62]{electromagnetic/standard/MuPgen/f1tgen.eps} |
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394 | \center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/f1tgen.eps} |
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395 | \caption{Histograms of generated $t$ distributions for |
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396 | $E_\gamma = 10\,{\rm GeV}$ (solid line) and |
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397 | $E_\gamma = 100\,{\rm GeV}$ (dashed line) with $10^6$ events each.} |
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398 | \label{f1t_gen.eps} |
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399 | % \end{minipage} |
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400 | \end{figure} |
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401 | % \hfill |
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402 | \begin{figure}[htpb] |
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403 | % \begin{minipage}[t]{7.5cm} |
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404 | % { \center\includegraphics[scale=.6]{electromagnetic/standard/MuPgen/PsiGen.eps} |
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405 | \center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/PsiGen.eps} |
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406 | \caption{Histograms of generated $\psi$ distributions for beryllium at |
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407 | four different photon energies.} |
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408 | \label{plot:PsiGen.eps} |
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409 | % } |
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410 | % \end{minipage} |
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411 | \end{figure} |
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412 | |
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413 | \noindent |
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414 | The Monte Carlo generation is done using the rejection technique. About |
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415 | 70\% of the generated numbers are kept in this step. Generated |
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416 | $t$-distributions are shown in Fig.\,\ref{f1t_gen.eps}. |
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417 | \\ \\ |
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418 | {\bf 3)} Generate $\psi$ by the rejection technique using $t$ generated in |
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419 | the previous step for the frequency distribution |
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420 | \begin{equation}\label{q3} |
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421 | f_2(\psi) =\Big[1-2\,x_+x_-+4\,x_+x_-t\,(1-t)\,(1+\cos(2\psi))\Big]\;, \qquad 0\le\psi\le2\pi\,. |
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422 | \end{equation} |
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423 | The maximum of $f_2(\psi)$ is |
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424 | \begin{equation} |
---|
425 | \max [f_2(\psi)]=1-2\,x_+x_-\left[1-4\,t\,(1-t)\right]\,.\, |
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426 | \end{equation} |
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427 | Generated distributions in $\psi$ are shown in Fig.\,\ref{plot:PsiGen.eps}. |
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428 | \\ \\ |
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429 | {\bf 4)} Generate $\rho$. \\ |
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430 | %old The frequency distribution to generate is |
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431 | %old \begin{equation} |
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432 | %old f_2(\rho^2) = \frac{1}{\rho^2 + \kappa^2} \qquad 0 \leq \rho^2 \leq 1 |
---|
433 | %old \end{equation} |
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434 | %old where $\kappa^2=1/W$ with the value of $W$ calculated in step 1. % according to Eq.\,\ref{eq:W}. |
---|
435 | %old The distribution can be generated by direct transformation. |
---|
436 | %old By integration |
---|
437 | %old \[ |
---|
438 | %old F_2(x) = \int_0^x f_2(\rho^2) \, d \rho^2 = \log\left(1 + \frac{x}{\kappa^2} \right) |
---|
439 | %old \] |
---|
440 | %old with the inverse |
---|
441 | %old \[ |
---|
442 | %old F_2^{-1}(x) = \kappa^2 \, \left(e^x -1\right)\;. |
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443 | %old \] |
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444 | %old This is mapped by linear transform $a + b x$ to the range |
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445 | %old $(0,1)$ of the standard random generator. |
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446 | %old From |
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447 | %old \[ x=0 \quad : \qquad F_2^{-1}(a + b x) = \kappa^2 \, \left(e^a -1\right) = 0 \qquad \Rightarrow \qquad a=0 \] |
---|
448 | %old and |
---|
449 | %old \[ x=1 \quad : \qquad F_2^{-1}(a + b x) = \kappa^2 \, \left(e^b -1\right) = 1 \qquad \Rightarrow \qquad |
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450 | %old b=\log\left( 1+1/\kappa^2\right) = \log\left( 1+W^2\right) \; .\] |
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451 | %old The direct transformation $F_2^{-1}(a + b \, R)$, applied to the flat random distribution $R$ to generate $\rho^2$ |
---|
452 | %old is therefore |
---|
453 | %old \begin{equation} |
---|
454 | %old \rho^2 = \kappa^2 \left(e^{R \log\left( 1+1/\kappa^2\right)} -1\right) |
---|
455 | %old = \kappa^2 \, \left[ \left( 1+1/\kappa^2\right)^R -1 \right] |
---|
456 | %old = \frac{\left( 1+W^2\right)^R -1 }{W^2} \;. |
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457 | %old \end{equation} |
---|
458 | %old Generated distributions in $\rho^2$ are shown in Fig.\,\ref{plot:rho2.eps}. |
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459 | %old The distribution is, with growing photon energy, increasingly peaked at 0. |
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460 | %old Note that $\rho$ does not depend on $t$. One could do step 3 before 2 or after step 4. |
---|
461 | The distribution in $\rho$ has the form |
---|
462 | \begin{equation}\label{rho1} |
---|
463 | f_3(\rho)\,d\rho=\frac{\rho^3\,d\rho}{\rho^4+C_2}\,,\quad |
---|
464 | 0\le\rho\le \rho_{\rm max}\,,\, |
---|
465 | \end{equation} |
---|
466 | where |
---|
467 | \begin{equation}\label{rhomax} |
---|
468 | \rho_{\rm max}^2=\frac{1.9}{A^{0.27}}\,\left(\frac{1}{t}-1\right), \, |
---|
469 | \end{equation} |
---|
470 | and |
---|
471 | \begin{equation}\label{C2} |
---|
472 | C_2=\frac4{\sqrt{x_+x_-}}\left[\left(\frac{m_\mu}{2E_\gamma x_+x_-\,t} |
---|
473 | \right)^2+\left(\frac{m_e}{183 \, Z^{-1/3} \, m_\mu}\right)^2 |
---|
474 | \right]^2\,. |
---|
475 | \end{equation} |
---|
476 | The $\rho$ distribution is obtained by a direct transformation applied to |
---|
477 | uniform random numbers $R_i$ according to |
---|
478 | \begin{equation}\label{rho} |
---|
479 | \rho=\left[C_2(\exp(\beta\,R_i)-1)\right]^{1/4}\,, |
---|
480 | \end{equation} |
---|
481 | where |
---|
482 | \begin{equation}\label{beta} |
---|
483 | \beta=\log\left(\frac{C_2+\rho_{\rm max}^4}{C_2}\right)\,. |
---|
484 | \end{equation} |
---|
485 | Generated distributions of $\rho$ are shown in Fig.\,\ref{plot:rho.eps} |
---|
486 | |
---|
487 | \begin{figure}[htpb] |
---|
488 | % \begin{minipage}[t]{7.5cm} |
---|
489 | % \includegraphics[scale=.6]{electromagnetic/standard/MuPgen/rho.eps} |
---|
490 | \center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/rho.eps} |
---|
491 | \caption{Histograms of generated $\rho$ distributions for beryllium at |
---|
492 | two different photon energies. The total number of entries at each energy |
---|
493 | is $10^6$.} |
---|
494 | \label{plot:rho.eps} |
---|
495 | % \end{minipage} |
---|
496 | \end{figure} |
---|
497 | % \hfill |
---|
498 | \begin{figure}[htpb] |
---|
499 | % \begin{minipage}[t]{7.5cm} |
---|
500 | % \includegraphics[scale=.6]{electromagnetic/standard/MuPgen/thetaPlus.eps} |
---|
501 | \center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/thetaPlus.eps} |
---|
502 | \caption{Histograms of generated $\theta_+$ distributions at different photon energies.} |
---|
503 | \label{plot:thetaPlus} |
---|
504 | % \end{minipage} |
---|
505 | \end{figure} |
---|
506 | |
---|
507 | \noindent |
---|
508 | {\bf 5)} Calculate $\theta_+,\theta_-$ and $\varphi$ from $t, \rho,\psi$ with |
---|
509 | \begin{equation} |
---|
510 | \gamma_\pm = \frac{E_\mu^\pm}{m_\mu} \qquad {\rm and} \qquad |
---|
511 | u=\sqrt{\frac1t-1}\,. |
---|
512 | \label{eq:gammau} |
---|
513 | \end{equation} |
---|
514 | according to |
---|
515 | \begin{equation}\label{s2} |
---|
516 | \theta_+= |
---|
517 | \frac{1}{\gamma_+}\,\left(u +\frac{\rho}{2}\,\cos\psi\right)\,,\quad \theta_-= |
---|
518 | \frac{1}{\gamma_-}\,\left(u -\frac{\rho}{2}\,\cos\psi\right)\, \quad {\rm and} |
---|
519 | \quad \varphi=\frac{\rho}{u} \, \sin\psi\, . \, |
---|
520 | \end{equation} |
---|
521 | The muon vectors can now be constructed from Eq.\,(\ref{eq:pvec}), where |
---|
522 | $\varphi_0$ is chosen randomly between 0 and $2\pi$. |
---|
523 | Fig.\,\ref{plot:thetaPlus} shows distributions of $\theta_+$ at different |
---|
524 | photon energies (in beryllium). The spectra peak around $1/\gamma$ as |
---|
525 | expected. |
---|
526 | |
---|
527 | The most probable values are $\theta_+\sim m_\mu/E_\mu^+ = 1 / \gamma_+$. In the small angle |
---|
528 | approximation used here, the values of $\theta_+$ and $\theta_-$ |
---|
529 | can in principle be any positive value from 0 to $\infty$. |
---|
530 | In the simulation, this may lead (with a very small probability, of the |
---|
531 | order of $m_\mu/E_\gamma$) to unphysical events in which $\theta_+$ or |
---|
532 | $\theta_-$ is greater than $\pi$. To avoid this, a limiting angle |
---|
533 | $\theta_{\rm cut}=\pi$ is introduced, and the angular sampling repeated, |
---|
534 | whenever $\max(\theta_+,\,\theta_-)>\theta_{\rm cut}$ \, . |
---|
535 | |
---|
536 | \begin{figure}[htpb] |
---|
537 | \center\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/Fig1.eps} |
---|
538 | \caption{Angular distribution of positive (or negative) muons. |
---|
539 | %old $E_\gamma=10\:{\rm GeV}$, $x_+=0.3$; iron. |
---|
540 | The solid curve represents |
---|
541 | the results of the exact calculations. The histogram is the simulated |
---|
542 | distribution. The angular distribution for pairs created in the field |
---|
543 | of the Coulomb centre (point-like target) is shown by the dashed curve |
---|
544 | for comparison.} |
---|
545 | \label{plot:Fig1} |
---|
546 | \end{figure} |
---|
547 | |
---|
548 | \begin{figure}[htpb] |
---|
549 | % \begin{minipage}[t]{7.5cm} |
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550 | % { \center\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/Fig2.eps} |
---|
551 | \center\includegraphics[scale=0.8]{electromagnetic/standard/MuPgen/Fig2.eps} |
---|
552 | \caption{Angular distribution in logarithmic scale. The curve corresponds |
---|
553 | to the exact calculations and the histogram is the simulated |
---|
554 | distribution.} |
---|
555 | \label{plot:Fig2} |
---|
556 | % } |
---|
557 | % \end{minipage} |
---|
558 | \end{figure} |
---|
559 | % \hfill |
---|
560 | \begin{figure}[htpb] |
---|
561 | % \begin{minipage}[t]{7.5cm} |
---|
562 | % \includegraphics[scale=.65]{electromagnetic/standard/MuPgen/Fig3.eps} |
---|
563 | \center\includegraphics[scale=0.8]{electromagnetic/standard/MuPgen/Fig3.eps} |
---|
564 | \caption{Distribution of the difference of transverse momenta of positive |
---|
565 | and negative muons (with logarithmic x-scale).} |
---|
566 | \label{plot:Fig3} |
---|
567 | % \end{minipage} |
---|
568 | \end{figure} |
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569 | |
---|
570 | Figs.\,\ref{plot:Fig1},\ref{plot:Fig2} and \ref{plot:Fig3} show |
---|
571 | distributions of the simulated angular characteristics of muon pairs in |
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572 | comparison with results of exact calculations. The latter were obtained |
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573 | by means of numerical integration of the squared matrix elements with |
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574 | respective nuclear and atomic form factors. All these calculations were |
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575 | made for iron, with $E_\gamma=10\,{\rm GeV}$ and $x_+=0.3$. As seen from |
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576 | Fig.\,\ref{plot:Fig1}, wide angle pairs (at low values of the argument in |
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577 | the figure) are suppressed in comparison with the Coulomb center |
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578 | approximation. This is due to the influence of the finite nuclear size |
---|
579 | which is comparable to the inverse mass of the muon. Typical angles of |
---|
580 | particle emission are of the order of |
---|
581 | $1/\gamma_\pm=m_\mu/E_\mu^\pm$ (Fig.\,\ref{plot:Fig2}). Fig.\,\ref{plot:Fig3} |
---|
582 | illustrates the influence of the momentum transferred to the target on the |
---|
583 | angular characteristics of the produced pair. In the frame of the often |
---|
584 | used model which neglects target recoil, the pair particles would be |
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585 | symmetric in transverse momenta, and coplanar with the initial photon. |
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586 | |
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587 | |
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588 | \subsection{Status of this document} |
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589 | 28.05.02 created by H.Burkhardt. \\ |
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590 | 01.12.02 re-worded by D.H. Wright \\ |
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591 | |
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592 | \begin{latexonly} |
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593 | |
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594 | \begin{thebibliography}{99} |
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595 | |
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596 | \bibitem{MuPgen} |
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597 | H.~Burkhardt, S.~Kelner, and R.~Kokoulin, ``Monte Carlo Generator for Muon |
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598 | Pair Production''. CERN-SL-2002-016 (AP) and CLIC Note 511, May 2002. |
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599 | |
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600 | \bibitem{Kelner:1995hu} |
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601 | S.~R. Kelner, R.~P. Kokoulin, and A.~A. Petrukhin, ``About cross section |
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602 | for high energy muon bremsstrahlung,''. Moscow Phys. Eng. Inst. 024-95, 1995. |
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603 | 31pp. |
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604 | |
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605 | \end{thebibliography} |
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606 | |
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607 | \end{latexonly} |
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608 | |
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609 | \begin{htmlonly} |
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610 | |
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611 | \subsection{Bibliography} |
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612 | |
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613 | \begin{enumerate} |
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614 | \item H.~Burkhardt, S.~Kelner, and R.~Kokoulin, |
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615 | ``Monte Carlo Generator for Muon |
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616 | Pair Production''. CERN-SL-2002-016 (AP) and CLIC Note 511, May 2002. |
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617 | |
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618 | \item S.~R. Kelner, R.~P. Kokoulin, and A.~A. Petrukhin, ``About cross section |
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619 | for high energy muon bremsstrahlung,''. Moscow Phys. Eng. Inst. 024-95, 1995. |
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620 | 31pp. |
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621 | \end{enumerate} |
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622 | |
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623 | \end{htmlonly} |
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624 | |
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625 | |
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