  The in-medium dispersion of hard partons, encoded in their so-called
asymptotic mass, receives large non-perturbative contributions from classical
gluons, i.e. soft gluons with large occupation numbers. Here, we discuss how
the analytical properties of thermal amplitudes allow for a non-perturbative
determination of the infrared classical contribution through lattice
determinations in the dimensionally-reduced effective theory of hot QCD, EQCD.
We show how these lattice determinations need to be complemented by
perturbative two-loop matching calculations between EQCD and QCD, so that the
unphysical (classical) ultraviolet behavior of EQCD is replaced by its proper
quantum QCD counterpart. We show how lattice and perturbative EQCD are in good
agreement in the UV and present an outlook on the two-loop quantum QCD
contribution.
