The 3d effective field theory of the high temperature abelian Higgs model

We study the 3d effective theory of the high T Abelian Higgs model with N real scalars and gauge and scalar couplings denoted g and $\lambda$, respectively. We find for the three cases a) $6g^2/\lambda =0(1)$, b) $6g^2/\lambda=0(N)$ and c) $6g^2/\lambda=O(N^{2\over3})$ the following results: a) The O(N)+O(1) potential admits only a second order phase transition, b) The O(1) potential admits a first order transition and c) the O(N^{1\over3}) +O(1) result gives a first order transition whose strength diminshes with N.