1 | |
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2 | \section{Positron - Electron Annihilation into Muon - Anti-muon} |
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3 | |
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4 | The class {\tt G4AnnihiToMuPair} simulates the electromagnetic production |
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5 | of muon pairs by the annihilation of high-energy positrons with atomic |
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6 | electrons. Details of the implementation are given below and can also be |
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7 | found in Ref. \cite{AnnihiToMuPair}. |
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8 | |
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9 | \subsection{Total Cross Section} |
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10 | |
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11 | The annihilation of positrons and target electrons producing muon pairs in |
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12 | the final state (${\rm e}^+{\rm e}^- \to \mu^+\mu^-$) may give an |
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13 | appreciable contribution to the total number of muons produced in |
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14 | high-energy electromagnetic cascades. The threshold positron energy in the |
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15 | laboratory system for this process with the target electron at rest is |
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16 | \begin{equation}\label{e0} |
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17 | E_{\rm th}=2m_\mu^2/m_e-m_e\approx 43.69\:{\rm GeV}\,, |
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18 | \end{equation} |
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19 | where $m_\mu$ and $m_e$ are the muon and electron masses, respectively. |
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20 | The total cross section for the process on the electron is |
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21 | \begin{equation}\label{e1} |
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22 | \sigma=\frac{\pi\,r_\mu^2} 3\, \xi\left(1+\frac\xi2\right) |
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23 | \sqrt{1-\xi}\,, |
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24 | \end{equation} |
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25 | where $r_\mu=r_e\, m_e/m_\mu$ is the classical muon radius, |
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26 | $\xi=E_{\rm th}/E$, and $E$ is the total positron energy in the laboratory |
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27 | frame. In Eq.\,\ref{e1}, approximations are made that utilize the |
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28 | inequality $m_e^2\ll m_\mu^2$. |
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29 | |
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30 | \begin{figure}[htbp] |
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31 | \center |
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32 | \includegraphics[scale=0.8]{electromagnetic/standard/AnnihiToMuPair1.eps} |
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33 | \caption{Total cross section for the process |
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34 | ${\rm e}^+{\rm e}^- \rightarrow \mu^+\mu^-$ as a function of the positron |
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35 | energy $E$ in the laboratory system.} |
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36 | \label{plot:AnnihiToMuPair1} |
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37 | \end{figure} |
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38 | \noindent |
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39 | The cross section as a function of the positron energy $E$ is shown in |
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40 | Fig.\ref{plot:AnnihiToMuPair1}. It has a maximum at |
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41 | $E = 1.396 \, E_{\rm th}$ and the value at the maximum is |
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42 | $\sigma_{\max}=0.5426\,r_\mu^2 = 1.008\,\mu{\rm b}$. |
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43 | |
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44 | \subsection{Sampling of Energies and Angles} |
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45 | |
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46 | It is convenient to simulate the muon kinematic parameters in the |
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47 | center-of-mass (c.m.) system, and then to convert into the laboratory |
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48 | frame. |
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49 | |
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50 | The energies of all particles are the same in the c.m. frame and equal to |
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51 | \begin{equation}\label{e2} |
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52 | E_{\rm cm}=\sqrt{\frac12\,m_e(E+m_e)}\,. |
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53 | \end{equation} |
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54 | The muon momenta in the c.m. frame are |
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55 | $P_{\rm cm}=\sqrt{E_{\rm cm}^2-m_\mu^2}$. In what follows, let the cosine |
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56 | of the angle between the c.m. momenta of the $\mu^+$ and $e^+$ be denoted as |
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57 | $x=\cos\theta_{\rm cm}$ . |
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58 | |
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59 | From the differential cross section it is easy to derive that, apart from |
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60 | normalization, the distribution in $x$ is described by |
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61 | \begin{equation}\label{e3} |
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62 | f(x)\,d x=(1+\xi+x^2\,(1-\xi))\,d x\,, \quad -1\le x \le1\,. |
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63 | \end{equation} |
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64 | The value of this function is contained in the interval |
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65 | $(1+\xi)\le f(x)\le 2$ and the generation of $x$ is straightforward using |
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66 | the rejection technique. Fig.\,\ref{plot:AnnihiToMuPair2} shows both |
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67 | generated and analytic distributions. |
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68 | \begin{figure}[htpb] |
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69 | \center |
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70 | \includegraphics[scale=.8]{electromagnetic/standard/AnnihiToMuPair2.eps} |
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71 | \caption{Generated histograms with $10^6$ entries each and the expected |
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72 | $\cos\theta_{\rm cm}$ distributions (dashed lines) at $E=50$ and 500\,GeV |
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73 | positron energy in the lab frame. The asymptotic |
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74 | $1+\cos\theta_{\rm cm}^2$ distribution valid for $E \rightarrow \infty$ is |
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75 | shown as dotted line.} |
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76 | \label{plot:AnnihiToMuPair2} |
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77 | \end{figure} |
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78 | |
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79 | The transverse momenta of the $\mu^+$ and $\mu^-$ particles are the same, |
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80 | both in the c.m. and the lab frame, and their absolute values are equal to |
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81 | \begin{equation}\label{perp} |
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82 | P_\perp=P_{\rm cm} \, \sin\theta_{\rm cm}=P_{\rm cm} \, \sqrt{1-x^2}\,. |
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83 | \end{equation} |
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84 | The energies and longitudinal components of the muon momenta in the |
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85 | lab system may be obtained by means of a Lorentz transformation. The |
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86 | velocity and Lorentz factor of the center-of-mass in the lab frame may be |
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87 | written as |
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88 | \begin{equation}\label{e5} |
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89 | \beta=\sqrt{\frac{E-m_e}{E+m_e}}\,,\quad \gamma\equiv\frac1{\sqrt{1-\beta^2}}= |
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90 | \sqrt{\frac{E+m_e}{2 m_e}} = \frac{E_{\rm cm}}{m_e}\,. |
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91 | \end{equation} |
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92 | The laboratory energies and longitudinal components of the momenta of the |
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93 | positive and negative muons may then be obtained: |
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94 | \begin{eqnarray} |
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95 | \label{e6} |
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96 | E_+&=&\gamma\,(E_{\rm cm}+x \, \beta \,P_{\rm cm})\,,\quad |
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97 | P_{+_\parallel}=\gamma\,(\beta E_{\rm cm} + x \, P_{\rm cm})\,, \\ |
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98 | \label{e7} |
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99 | E_-&=&\gamma\,(E_{\rm cm}-x \, \beta \,P_{\rm cm})\,,\quad |
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100 | P_{-_\parallel}=\gamma\,(\beta E_{\rm cm} -x \, P_{\rm cm})\,. |
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101 | \end{eqnarray} |
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102 | Finally, for the vectors of the muon momenta one obtains: |
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103 | \begin{eqnarray} |
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104 | \label{e8} |
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105 | {\bf P}_+&=&(+P_\perp\cos\varphi ,+P_\perp \sin\varphi,P_{+_\parallel})\,,\\ |
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106 | \label{e9} |
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107 | {\bf P}_-&=&(-P_\perp\cos\varphi, |
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108 | -P_\perp\sin\varphi,P_{-_\parallel})\,, |
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109 | \end{eqnarray} |
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110 | where $\varphi$ is a random azimuthal angle chosen between 0 and $2\,\pi$. |
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111 | The $z$-axis is directed along the momentum of the initial positron in the |
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112 | lab frame. |
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113 | |
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114 | The maximum and minimum energies of the muons are given by |
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115 | \begin{equation}\label{e10} |
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116 | E_{\max}\approx\frac12\,E\left(1+\sqrt{1-\xi}\right)\,, |
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117 | \end{equation} |
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118 | \begin{equation}\label{e11} |
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119 | E_{\min}\approx\frac12\,E\left(1-\sqrt{1-\xi}\right)= |
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120 | \frac{\displaystyle |
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121 | E_{\rm th}}{\displaystyle 2\left(1+\sqrt{1-\xi} \right)}\,. |
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122 | \end{equation} |
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123 | The fly-out polar angles of the muons are approximately |
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124 | \begin{equation}\label{e12} |
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125 | \theta_+\approx P_{{}\perp}/P_{+_\parallel},\quad \theta_-\approx |
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126 | P_{{}\perp}/P_{-_\parallel}\,; |
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127 | \end{equation} |
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128 | the maximal angle $\displaystyle\theta_{\max}\approx\frac{m_e}{m_\mu}\, |
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129 | \sqrt{1-\xi}\,$ is always small compared to 1. |
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130 | \medskip |
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131 | |
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132 | \subsection*{Validity} |
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133 | |
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134 | The process described is assumed to be purely electromagnetic. It is |
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135 | based on virtual $\gamma$ exchange, and the $Z$-boson exchange and |
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136 | $\gamma - Z$ interference processes are neglected. The $Z$-pole corresponds |
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137 | to a positron energy of $E = M_Z^2 / 2 m_e = 8136\,{\rm TeV}$. |
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138 | The validity of the current implementation is therefore restricted to |
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139 | initial positron energies of less than about 1000\,TeV. |
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140 | |
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141 | \subsection{Status of this document} |
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142 | 05.02.03 created by H.Burkhardt \\ |
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143 | 14.04.03 minor re-wording by D.H. Wright |
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144 | |
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145 | \begin{latexonly} |
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146 | |
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147 | \begin{thebibliography}{99} |
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148 | |
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149 | \bibitem{AnnihiToMuPair} |
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150 | H.~Burkhardt, S.~Kelner, and R.~Kokoulin, ``Production of muon pairs in |
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151 | annihilation of high-energy positrons with resting electrons,'' |
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152 | CERN-AB-2003-002 (ABP) and CLIC Note 554, January 2003. |
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153 | |
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154 | \end{thebibliography} |
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155 | |
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156 | \end{latexonly} |
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157 | |
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158 | \begin{htmlonly} |
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159 | |
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160 | \subsection{Bibliography} |
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161 | |
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162 | \begin{enumerate} |
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163 | |
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164 | \item H.~Burkhardt, S.~Kelner, and R.~Kokoulin, ``Production of muon pairs in |
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165 | annihilation of high-energy positrons with resting electrons,'' |
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166 | CERN-AB-2003-002 (ABP) and CLIC Note 554, January 2003. |
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167 | |
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168 | \end{enumerate} |
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169 | |
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170 | \end{htmlonly} |
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