We derive the one-loop effective 3-dimensional Lagrangian that describes the high-temperature limit of the electroweak theory, to order \( \mathcal{O} \) (g6) in coupling constants g, including corrections due to Matsubara modes of both fermionic and bosonic degrees of freedom. We clarify certain aspects of the gauge-independence of physical parameters. We also extend the calculation to the Standard Model effective field theory, paving the way, in particular, for a precise study of the electroweak phase transition within this framework.
