Effective potential and first-order phase transitions: Beyond leading order

Scenarios for electroweak baryogenesis require an understanding of the effective potential at finite temperature near a first-order electroweak phase transition. Working in the Landau gauge, we present a calculation of the dominant two-loop corrections to the ring-improved one-loop potential in the formal limit $g^4\ll \lambda \ll g^2$, where $\lambda$ is the Higgs self-coupling and $g$ is the electroweak coupling. The limit $\lambda \ll g^2$ ensures that the phase transition is significantly first order, and the limit $g^4\ll \lambda$ allows us to use high-temperature expansions. We find corrections from 20% to 40% at Higgs-boson masses relevant to the bound computed for baryogenesis in the minimal standard model. Though our numerical results seem to still rule out minimal standard model baryogenesis, this conclusion is not airtight because the loop expansion is only marginal when corrections are as big as 40%. We also discuss why superdaisy approximations do not correctly compute these corrections.