The LPM effect in sequential bremsstrahlung: nearly complete results 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 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. Previous work has computed overlap effects for double splitting $g \to gg \to ggg$. To make use of those results, one also needs calculations of related virtual loop corrections to single splitting $g \to gg$ in order to cancel severe (power-law) infrared (IR) divergences. This paper provides calculations of nearly all such processes involving gluons and discusses how to organize the results to demonstrate the cancellation. In the soft emission limit, our results reproduce the known double-log behavior of earlier authors who worked in leading-log approximation. We also present a first (albeit numerical and not yet analytic) investigation of sub-leading, single IR logarithms. Ultraviolet divergences appearing in our calculations correctly renormalize the coupling $\alpha_{\rm s}$ in the usual LPM result for leading-order $g \to gg$.