On the Finite Temperature Phase Transition of Scalar QED

We analyse the finite temperature phase transition of scalar QED in euclidean spacetime. Analytical solutions of approximations to a Wilsonian renormalisation group equation are discussed. Special emphasis is put on the discussion of the $3d$ running of the Abelian charge and its effects on thermodynamical quantities. An upper bound for the range of first-order phase transitions is given. Our results are compared to those of resummed perturbation theory.