Recoil Momentum Distribution in Electron Pair Production

Electron pair production by a very energetic photon in the field of a particle of arbitrary mass (in particular in the fields of an electron and a nucleus) is studied following the work of Borsellino. The distribution of recoil momenta $q$ is calculated for $q$ of order of the electron mass and it is shown that the recoil distribution is independent of the mass of the recoil particle if appropriate variables are used. It is also explicitly shown that the mass of the recoil particle does not make any difference in the recoil distribution for very small $q$ (of order $q_{min}$). The total cross section must therefore be independent of the mass of the recoil particle in the high-energy limit, as previously stated by Borsellino. The Wheeler-Lamb result for pair production in the field of a bound electron is also justified. The results also describe the electromagnetic production of any fermion pair if certain restrictions are satisfied.