Is Compton scattering in strong magnetic fields really infrared   divergent?
  The infrared behavior of QED changes drastically in the presence of a strong magnetic field: the electron self-energy and the vertex function are infrared {\em finite}, in contrast with field-free QED, while new infrared divergences appear that are absent in free space. One famous example of the latter is the infrared catastrophe of magnetic Compton scattering, where the cross section for scattering of photons from electrons which undergo a transition to the Landau ground state {\em diverges} as the frequency of the incoming photon goes to zero. We examine this divergence in more detail and prove that the singularity of the cross section is {\em removed} as soon as proper account is taken of all quantum electrodynamical processes that become indistinguishable from Compton scattering in the limit of vanishing frequency of the incident photon. 
M. Kachelriess, D. Berg, and G. Wunner