Weak coupling phase structure of the Abelian Higgs model at finite temperature

Using the 1-loop reduced 3D action of the Abelian Higgs-model we discuss the order of its finite temperature phase transition. A two-variable saddle point approximation is proposed for the evaluation of the effective potential. The strength of the first order case scales like ∼e3−6. Analytic asymptotic weak coupling and numerical small coupling solutions are compared with special emphasis on the cancellation of divergences.