3D effective field theory for finite temperature scalar electrodynamics

The effective field theory approach to high temperature field theory can be used to study the phase transition in theories with spontaneously broken symmetry. I construct a sequence of two effective three-dimensional field theories which are valid on successively longer distance scales for a specific model: $N$ charged scalars coupled to a $U\left(1\right)\mathrm{}$ gauge field. The resulting effective Lagrangian can be used to investigate the phase transition, in particular the order of the phase transition as a function of $N,$ using lattice simulations.