  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. Of recent
interest is 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). In previous papers, we have developed
methods for computing such corrections without making soft-gluon
approximations. However, our methods require consistent treatment of canceling
ultraviolet (UV) divergences associated with coincident emission times, even
for processes with tree-level amplitudes. In this paper, we show how to use
dimensional regularization to properly handle the UV contributions. We also
present a simple diagnostic test that any consistent UV regularization method
for this problem needs to pass.
