Time-Like Momenta in Quantum Electrodynamics

We have considered electron pair production by high-energy muons in a Coulomb potential as a test of quantum electrodynamics in the time-like region. If triple coincidence configurations are selected whereby the total momentum of the three final leptons is in the incident muon direction and the electron-positron pair is detected symmetrically with respect to the muon scattering plane, then the differential cross section is obtained in a simple analytical form. With these constraints it is seen that for a given effective mass of the pair, the contributions to the cross section involving timelike photon propagators become dominant as the muon scattering angle increases, although the cross section decreases slowly. If the constraints are relaxed, the cross section increases by orders of magnitude but the above features of the cross section persist. Thus, for a 10-BeV/c incident muon, nuclear charge $Z=10\mathrm{}$, and effective pair mass 100 MeV, the cross section is nominally ${10}^{-32\mathrm{}}$ ${\mathrm{cm}}^2$/${\mathrm{}\left(\mathrm{MeV}\right)\mathrm{}}^2$ ${\mathrm{}\left(\mathrm{sr}\right)\mathrm{}}^3$, the time-like contribution gives more than 80% of the cross section, and the momentum transfer to the nucleus is less than 50 MeV/c over ranges of polar angles $\Delta \theta =20\mathrm{}$ mrad, azimuthal angles $\Delta \phi =40\mathrm{}°$ and momenta $\Delta P=1\mathrm{}$ BeV/c. The characteristic behavior of the cross section is presented in graphical form and a simple method of computing the differential cross section (without involving large cancellations of gauge-variant quantities) is discussed.