Scenarios for electroweak baryogenesis require an understanding of the effective potential at finite temperature near a first-order electroweak phase transition. Working in Landau gauge, we present a calculation of the dominant two-loop corrections to the ringimproved one-loop potential in the formal limit g 4 ≪ λ ≪ g 2 , where λ is the Higgs self-coupling and g is the electroweak coupling. The limit λ ≪ g 2 ensures that the phase transition is significantly first-order, and the limit g 4 ≪ λ allows us to use high-temperature expansions. We find corrections from 20 to 40% at Higgs 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 super-daisy approximations do not correctly compute these corrections.

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S. Ramanujan introduced a technique, known as Ramanujan's Master Theorem, which provides an analytic expression for the Mellin transform of a function. The main identity of this theorem involves the extrapolation of the sequence of coefficients of the integrand, defined originally as a function on N to C. The history and proof of this result are reviewed. Applications to the evaluation of a variety of definite integrals is presented.