The LPM effect in sequential bremsstrahlung: incorporation of "instantaneous'' interactions for QCD

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 (formation lengths) of two consecutive splitting processes overlap, avoiding soft-emission approximations. Previous work made a ``nearly-complete'' calculation of the effect of overlapping formation times on gluonic splittings such as $g \to gg \to ggg$ (with simplifying assumptions such as an infinite QCD medium and the large-$N_c$ limit). In this paper, we extend those previous rate calculations from nearly-complete to complete by including processes involving the exchange of longitudinally-polarized gluons. In the context of Lightcone Pertubation Theory, used earlier for the ``nearly-complete'' calculation, such exchanges are instantaneous in lightcone time and have their own diagrammatic representation.