Cosmological phase transitions at three loops: The final verdict on perturbation theory

We complete the perturbative program for equilibrium thermodynamics of cosmological first-order phase transitions of gauge-Higgs theories that map into the (three-dimensional) superrenormalizable SU⁡(2)+doublet effective theory at high temperatures. To this end, we determine their finite-temperature effective potential at next-to-next-to-next-to-next-to-leading order (N4⁢LO). The computation of the three-loop effective potential required to reach this order is also presented for U(1) gauge theories and is readily extendable to generic models in dimensionally reduced effective theories. Our N4⁢LO result is the last perturbative order before confinement renders electroweak gauge-Higgs theories nonperturbative at four loops. By contrasting our analysis with nonperturbative lattice results, we find a remarkable agreement. As a direct application for predictions of gravitational waves produced by a first-order transition, our computation provides the final fully perturbative results for the phase transition strength and speed of sound.