Analytic regularization and infrared divergences

For the infra-red cancellation mechanism to work in the context of analytic regularization it is necessary to replace the virtual photon propagator (k²)^-1 by - Gamma (z)(-k²)^-z and the photon phase-space by the continuous mass distribution 2 pi integral d⁴k theta (k_o) theta (k²)/ Gamma (1-z)(k²)^z. The cancellation for charged-particle scattering in a weak external field is shown.