Nonlinear effects in quantum electrodynamics

Abstract

The fourth-rank vacuum polarization tensor, which is related to the lowest-order nonlinear interaction between four electromagnetic fields in quantum electrodynamics, is exactly calculated in terms of rational, logarithm and dilogarithm functions when two of the four electromagnetic fields describe photons off the mass shell. This task has been accomplished by a not exccedingly laborious effort with the aid of double dispersion relations which proved to be a very convenient tool for the treatment of these problems (in particular, photon-photon scattering). From the explicit expression of the polarization tensor we have easily obtained the exact amplitudes for photon-photon scattering, photon splitting and photon coalescence into photons on nuclei. Moreover we give the real and imaginary part of Delbruck scattering in the form of threefold integrals over the momentum transferred to the nucleus by one of the two virtual photons. This can be compared with the fivefold and sixfold integrals for the imaginary and real part, respectively, available in the existing literature on Delbruck scattering. Finally we give an explicit expression for the differential cross-section of Delbruck scattering in the limit of low energies.