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].

The annihilation of positrons and target electrons producing muon pairs in
the final state (
) 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

where is the classical muon radius, , and is the total positron energy in the laboratory frame. In Eq.8.51, approximations are made that utilize the inequality .

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

From the differential cross section it is easy to derive that, apart from
normalization, the distribution in is described by

The transverse momenta of the and particles are the same,
both in the c.m. and the lab frame, and their absolute values are equal to

The laboratory energies and longitudinal components of the momenta of the positive and negative muons may then be obtained:

Finally, for the vectors of the muon momenta one obtains:

where is a random azimuthal angle chosen between 0 and . The -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

The fly-out polar angles of the muons are approximately

the maximal angle is always small compared to 1.

The process described is assumed to be purely electromagnetic. It is based on virtual exchange, and the -boson exchange and interference processes are neglected. The -pole corresponds to a positron energy of . The validity of the current implementation is therefore restricted to initial positron energies of less than about 1000TeV.

14.04.03 minor re-wording by D.H. Wright

- 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.