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Subsections

Positron - Electron Annihilation into Muon - Anti-muon

The class 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. [1].

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{displaymath}
E_{\rm th}=2m_\mu^2/m_e-m_e\approx 43.69\:{\rm GeV}\,,
\end{displaymath} (8.50)

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{displaymath}
\sigma=\frac{\pi\,r_\mu^2} 3\, \xi\left(1+\frac\xi2\right)
\sqrt{1-\xi}\,,
\end{displaymath} (8.51)

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.8.51, approximations are made that utilize the inequality $m_e^2\ll m_\mu^2$.

Figure 8.1: 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.
\includegraphics[scale=0.8]{electromagnetic/standard/AnnihiToMuPair1.eps}
The cross section as a function of the positron energy $E$ is shown in Fig.8.1. It has a maximum at $E = 1.396 \, E_{\rm th}$ and the value at the maximum is $\sigma_{\max}=0.5426\,r_\mu^2 = 1.008\,\mu{\rm b}$.

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{displaymath}
E_{\rm cm}=\sqrt{\frac12\,m_e(E+m_e)}\,.
\end{displaymath} (8.52)

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{displaymath}
f(x)\,d x=(1+\xi+x^2\,(1-\xi))\,d x\,, \quad -1\le x \le1\,.
\end{displaymath} (8.53)

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.8.2 shows both generated and analytic distributions.
Figure 8.2: Generated histograms with $10^6$ entries each and the expected $\cos\theta_{\rm cm}$ distributions (dashed lines) at $E=50$ and 500GeV 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.
\includegraphics[scale=.8]{electromagnetic/standard/AnnihiToMuPair2.eps}

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{displaymath}
P_\perp=P_{\rm cm} \, \sin\theta_{\rm cm}=P_{\rm cm} \, \sqrt{1-x^2}\,.
\end{displaymath} (8.54)

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{displaymath}
\beta=\sqrt{\frac{E-m_e}{E+m_e}}\,,\quad \gamma\equiv\frac1{...
...ta^2}}=
\sqrt{\frac{E+m_e}{2 m_e}} = \frac{E_{\rm cm}}{m_e}\,.
\end{displaymath} (8.55)

The laboratory energies and longitudinal components of the momenta of the positive and negative muons may then be obtained:
$\displaystyle E_+$ $\textstyle =$ $\displaystyle \gamma\,(E_{\rm cm}+x \, \beta \,P_{\rm cm})\,,\quad
P_{+_\parallel}=\gamma\,(\beta E_{\rm cm} + x \, P_{\rm cm})\,,$ (8.56)
$\displaystyle E_-$ $\textstyle =$ $\displaystyle \gamma\,(E_{\rm cm}-x \, \beta \,P_{\rm cm})\,,\quad
P_{-_\parallel}=\gamma\,(\beta E_{\rm cm} -x \, P_{\rm cm})\,.$ (8.57)

Finally, for the vectors of the muon momenta one obtains:
$\displaystyle {\bf P}_+$ $\textstyle =$ $\displaystyle (+P_\perp\cos\varphi ,+P_\perp \sin\varphi,P_{+_\parallel})\,,$ (8.58)
$\displaystyle {\bf P}_-$ $\textstyle =$ $\displaystyle (-P_\perp\cos\varphi,
-P_\perp\sin\varphi,P_{-_\parallel})\,,$ (8.59)

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{displaymath}
E_{\max}\approx\frac12\,E\left(1+\sqrt{1-\xi}\right)\,,
\end{displaymath} (8.60)


\begin{displaymath}
E_{\min}\approx\frac12\,E\left(1-\sqrt{1-\xi}\right)=
\frac{...
...le
E_{\rm th}}{\displaystyle 2\left(1+\sqrt{1-\xi} \right)}\,.
\end{displaymath} (8.61)

The fly-out polar angles of the muons are approximately
\begin{displaymath}
\theta_+\approx P_{{}\perp}/P_{+_\parallel},\quad \theta_-\approx
P_{{}\perp}/P_{-_\parallel}\,;
\end{displaymath} (8.62)

the maximal angle $\displaystyle\theta_{\max}\approx\frac{m_e}{m_\mu}\,
\sqrt{1-\xi}\,$ is always small compared to 1.


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 1000TeV.

Status of this document

05.02.03 created by H.Burkhardt
14.04.03 minor re-wording by D.H. Wright

Bibliography

  1. 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.


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Next: Synchrotron Radiation Up: Electron Incident Previous: Positron - Electron Annihilation   Contents