Hard thermal contributions to phase transition observables at NNLO

To construct the high-temperature effective field theory of gauge-Higgs models upto O(g6) in the gauge coupling, we integrate out hard modes to three-loop level and use thenext-to-next-to-leading order effective potential. For the Abelian Higgs model, we quantify theimpact of both higher-dimensional operators and higher-loop corrections on thermodynamicparameters relevant for gravitational-wave observables, finding that one-loop dimension-six effects typically dominate over two- and three-loop corrections to super-renormalizableparameters for the strongest transitions. We derive the three-loop scalar and Debye masses for the U(1) and SU (N) gauge-Higgs models, as well as the two-loop quartic couplings for the Abelian case, show gauge independence of physical parameters, and demonstrate that no new master integrals are required for the matching, while consistency of 4d and 3d renormalizability points to previously missing contributions in these master integrals. As a byproduct, we report a previously missing contribution to the three-loop QCD Debye mass.