Calculation of cross sections in Lorentz-violating theories

We revisit, starting from the first principles, the calculation of scattering cross sections in theories with violation of the Lorentz invariance. Our approach takes into account all information about the Lagrangian of the theory and is applicable to the case of Lorentz violation by higher-dimension operators. As an illustration, we consider spinor electrodynamics with Lorentz violating operators of dimension four and six that preserve CPT, parity, and rotational invariance in a preferred frame. We derive the Feynman rules and formulas for the sums over polarizations in this model and use them to compute the probabilities of several astrophysically relevant processes. We calculate the rates of photon decay and vacuum Cherenkov radiation along with the cross sections of electron-positron pair production on background radiation and in the Coulomb field. The latter process is essential for detection of photon-induced air showers in the atmosphere.