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) & thorn; 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 (N4LO). The computation of the threeloop 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 N4LO 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.