  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 $g \to gg$.
