In-medium loop corrections and longitudinally polarized gauge bosons in high-energy showers.

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths of two consecutive splitting processes overlap (which is important for understanding corrections to standard treatments of the LPM effect in QCD), avoiding soft-emission approximations. In this particular paper, we show (i) how the "instantaneous" interactions of Light-Cone Perturbation Theory must be included in the calculation to account for effects of longitudinally-polarized gauge bosons in intermediate states, and (ii) how to compute virtual corrections to LPM emission rates, which will be necessary in order to make infrared-safe calculations of the characteristics of in-medium QCD showering of high-energy partons. In order to develop these topics in as simple a context as possible, we will focus in the current paper not on QCD but on large-$N_f$ QED, where $N_f$ is the number of electron flavors.