\section{Positron - Electron Annihilation into Muon - Anti-muon}

The class {\tt G4AnnihiToMuPair} simulates the electromagnetic production
of muon pairs by the annihilation of high-energy positrons with atomic 
electrons.  Details of the implementation are given below and can also be
found in Ref. \cite{AnnihiToMuPair}.

\subsection{Total Cross Section}

The annihilation of positrons and target electrons producing muon pairs in 
the final state (${\rm e}^+{\rm e}^- \to \mu^+\mu^-$) may give an
appreciable contribution to the total number of muons produced in
high-energy electromagnetic cascades.  The threshold positron energy in the 
laboratory system for this process with the target electron at rest is
\begin{equation}\label{e0}
E_{\rm th}=2m_\mu^2/m_e-m_e\approx 43.69\:{\rm GeV}\,,
\end{equation}
where $m_\mu$ and $m_e$ are the muon and electron masses, respectively.
The total cross section for the process on the electron is
\begin{equation}\label{e1}
\sigma=\frac{\pi\,r_\mu^2} 3\, \xi\left(1+\frac\xi2\right)
\sqrt{1-\xi}\,,
\end{equation}
where $r_\mu=r_e\, m_e/m_\mu$ is the classical muon radius, 
$\xi=E_{\rm th}/E$, and $E$ is the total positron energy in the laboratory 
frame.  In Eq.\,\ref{e1}, approximations are made that utilize the 
inequality $m_e^2\ll m_\mu^2$.

\begin{figure}[htbp]
\center
\includegraphics[scale=0.8]{electromagnetic/standard/AnnihiToMuPair1.eps}
\caption{Total cross section for the process 
${\rm e}^+{\rm e}^- \rightarrow \mu^+\mu^-$ as a function of the positron 
energy $E$ in the laboratory system.}
\label{plot:AnnihiToMuPair1}
\end{figure}
\noindent
The cross section as a function of the positron energy $E$ is shown in 
Fig.\ref{plot:AnnihiToMuPair1}.  It has a maximum at 
$E = 1.396 \, E_{\rm th}$ and the value at the maximum is 
$\sigma_{\max}=0.5426\,r_\mu^2 = 1.008\,\mu{\rm b}$.

\subsection{Sampling of Energies and Angles}

It is convenient to simulate the muon kinematic parameters in the
center-of-mass (c.m.) system, and then to convert into the laboratory 
frame.

The energies of all particles are the same in the c.m. frame and equal to
\begin{equation}\label{e2}
E_{\rm cm}=\sqrt{\frac12\,m_e(E+m_e)}\,.
\end{equation}
The muon momenta in the c.m. frame are 
$P_{\rm cm}=\sqrt{E_{\rm cm}^2-m_\mu^2}$.  In what follows, let the cosine 
of the angle between the c.m. momenta of the $\mu^+$ and $e^+$ be denoted as 
$x=\cos\theta_{\rm cm}$ . 

From the differential cross section it is easy to derive that, apart from
normalization, the distribution in $x$ is described by
\begin{equation}\label{e3}
f(x)\,d x=(1+\xi+x^2\,(1-\xi))\,d x\,, \quad -1\le x \le1\,.
\end{equation}
The value of this function is contained in the interval 
$(1+\xi)\le f(x)\le 2$ and the generation of $x$ is straightforward using 
the rejection technique.  Fig.\,\ref{plot:AnnihiToMuPair2} shows both 
generated and analytic distributions.
\begin{figure}[htpb]
\center
\includegraphics[scale=.8]{electromagnetic/standard/AnnihiToMuPair2.eps}
\caption{Generated histograms with $10^6$ entries each and the expected 
$\cos\theta_{\rm cm}$ distributions (dashed lines) at $E=50$ and 500\,GeV 
positron energy in the lab frame.  The asymptotic 
$1+\cos\theta_{\rm cm}^2$ distribution valid for $E \rightarrow \infty$ is 
shown as dotted line.}
\label{plot:AnnihiToMuPair2}
\end{figure}

The transverse momenta of the $\mu^+$ and $\mu^-$ particles are the same,
both in the c.m. and the lab frame, and their absolute values are equal to
\begin{equation}\label{perp}
P_\perp=P_{\rm cm} \, \sin\theta_{\rm cm}=P_{\rm cm} \, \sqrt{1-x^2}\,.
\end{equation}
The energies and longitudinal components of the muon momenta in the 
lab system may be obtained by means of a Lorentz transformation.  The 
velocity and Lorentz factor of the center-of-mass in the lab frame may be
written as
\begin{equation}\label{e5}
\beta=\sqrt{\frac{E-m_e}{E+m_e}}\,,\quad \gamma\equiv\frac1{\sqrt{1-\beta^2}}=
\sqrt{\frac{E+m_e}{2 m_e}} = \frac{E_{\rm cm}}{m_e}\,.
\end{equation}
The laboratory energies and longitudinal components of the momenta of the
positive and negative muons may then be obtained:
\begin{eqnarray}
\label{e6}
E_+&=&\gamma\,(E_{\rm cm}+x \, \beta \,P_{\rm cm})\,,\quad
P_{+_\parallel}=\gamma\,(\beta E_{\rm cm} + x \, P_{\rm cm})\,, \\
\label{e7}
E_-&=&\gamma\,(E_{\rm cm}-x \, \beta \,P_{\rm cm})\,,\quad
P_{-_\parallel}=\gamma\,(\beta E_{\rm cm} -x \, P_{\rm cm})\,.
\end{eqnarray}
Finally, for the vectors of the muon momenta one obtains:
\begin{eqnarray}
\label{e8}
{\bf P}_+&=&(+P_\perp\cos\varphi ,+P_\perp \sin\varphi,P_{+_\parallel})\,,\\
\label{e9}
{\bf P}_-&=&(-P_\perp\cos\varphi,
-P_\perp\sin\varphi,P_{-_\parallel})\,,
\end{eqnarray}
where $\varphi$ is a random azimuthal angle chosen between 0 and $2\,\pi$. 
The $z$-axis is directed along the momentum of the initial positron in the
lab frame.

The maximum and minimum energies of the muons are given by
\begin{equation}\label{e10}
E_{\max}\approx\frac12\,E\left(1+\sqrt{1-\xi}\right)\,,
\end{equation}
\begin{equation}\label{e11}
E_{\min}\approx\frac12\,E\left(1-\sqrt{1-\xi}\right)=
\frac{\displaystyle
E_{\rm th}}{\displaystyle 2\left(1+\sqrt{1-\xi} \right)}\,.
\end{equation}
The fly-out polar angles of the muons are approximately
\begin{equation}\label{e12}
\theta_+\approx P_{{}\perp}/P_{+_\parallel},\quad \theta_-\approx
P_{{}\perp}/P_{-_\parallel}\,;
\end{equation}
the maximal angle $\displaystyle\theta_{\max}\approx\frac{m_e}{m_\mu}\,
\sqrt{1-\xi}\,$ is always small compared to 1.
\medskip

\subsection*{Validity}

The process described is assumed to be purely electromagnetic.  It is 
based on virtual $\gamma$ exchange, and the $Z$-boson exchange and 
$\gamma - Z$ interference processes are neglected.  The $Z$-pole corresponds 
to a positron energy of $E = M_Z^2 / 2 m_e = 8136\,{\rm TeV}$.
The validity of the current implementation is therefore restricted to 
initial positron energies of less than about 1000\,TeV.

\subsection{Status of this document}
05.02.03 created by H.Burkhardt \\
14.04.03 minor re-wording by D.H. Wright

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{AnnihiToMuPair}
H.~Burkhardt, S.~Kelner, and R.~Kokoulin, ``Production of muon pairs in 
annihilation of high-energy positrons with resting electrons,'' 
CERN-AB-2003-002 (ABP) and CLIC Note 554, January 2003.

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}

\item H.~Burkhardt, S.~Kelner, and R.~Kokoulin, ``Production of muon pairs in 
annihilation of high-energy positrons with resting electrons,'' 
CERN-AB-2003-002 (ABP) and CLIC Note 554, January 2003.

\end{enumerate}

\end{htmlonly}