Vacuum Cherenkov radiation and photon triple-splitting in a   Lorentz-noninvariant extension of quantum electrodynamics
  We consider a CPT-noninvariant scalar model and a modified version of quantum electrodynamics with an additional photonic Chern-Simons-like term in the action. In both cases, the Lorentz violation traces back to a spacelike background vector. The effects of the modified field equations and dispersion relations on the kinematics and dynamics of decay processes are discussed, first for the simple scalar model and then for modified quantum electrodynamics. The decay widths for electron Cherenkov radiation in modified quantum electrodynamics and for photon triple-splitting in the corresponding low-energy effective theory are obtained to lowest order in the electromagnetic coupling constant. A conjecture for the high-energy limit of the photon-triple-splitting decay width at tree level is also presented. 
C. Kaufhold, F.R. Klinkhamer