  For computing thermodynamics of the electroweak phase transition, we discuss
a minimal approach that reconciles both gauge invariance and thermal
resummation. Such a minimal setup consists of a two-loop dimensional reduction
to three-dimensional effective theory, a one-loop computation of the effective
potential and its expansion around the leading-order minima within the
effective theory. This approach is tractable and provides formulae for
resummation that are arguably no more complicated than those that appear in
standard techniques ubiquitous in the literature. In particular, we implement
renormalisation group improvement related to the hard thermal scale. Despite
its generic nature, we present this approach for the complex singlet extension
of the Standard Model which has interesting prospects for high energy collider
phenomenology and dark matter predictions. The presented expressions can be
used in future studies of phase transition thermodynamics and gravitational
wave production.

High Energy Physics - Phenomenology