  High-energy particles traversing the Quark-Gluon plasma experience modified
(massive) dispersion, although their vacuum mass is negligible compared to the
kinetic energy. Due to poor convergence of the perturbative series in the
regime of soft loop momenta, a more precise determination of this effective
mass is needed. This paper continues our investigation on the factorisation
between strongly-coupled infrared classical and perturbative ultraviolet
behavior. The former has been studied non-perturbatively within EQCD by
determining a non-local operator on the lattice. By computing the
temperature-scale contribution to the same operator in 4D QCD at
next-to-leading order (NLO), we remove the ultraviolet divergence of the EQCD
calculation with an opposite infrared divergence from the hard thermal scale.
The result is a consistent, regulator-independent determination of the
classical contribution where the emergence of new divergences signals
sensitivities to new regions of phase space. We address the numerical impact of
the classical and NLO thermal corrections on the convergence of the factorised
approach and on the partial applicability of our results to calculations of
transport coefficients.
