Calculating extremely high-energy bremsstrahlung in matter

Ultra-relativistic electrons initiate electromagnetic showers in ordinary matter that evolve through bremsstrahlung and pair production. At very high energy, the quantum mechanical duration of bremsstrahlung becomes longer than the mean free time to elastically scatter from the medium, leading to a significant suppression known at the Landau-Pomeranchuk-Migdal (LPM) effect. For some ranges of bremsstrahlung photon and initial electron energies $(k_γ,E)$, the duration becomes so long that it will also overlap with subsequent pair production by the bremsstrahlung photon, disrupting LPM suppression and drastically changing LPM predictions. We have previously calculated this change for extremely high energies ($k_γ\gg 2$ TeV or more, depending on the medium), for which the electron mass and medium-induced photon mass could be ignored. In this paper, we extend that analysis to lower (but still ultra-relativistic) energy by accounting for those masses, leading to a rich map of behavior in different regions of $(k_γ,E)$.