The Free Energy in Scalar Electrodynamics at High Temperature

Massless scalar electrodynamics is studied at high temperature and zero chemical potential. I derive the free energy to order $\lambda^2$, $\lambda e^2$ and $e^4$ by effective field theory methods. The first step consists of the construction of an effective three-dimensional field theory that is valid on distance scales $R\gg 1/T$ and whose parameters can be written as power series in $\lambda$ and $e^2$. These coupling constants encode the contribution to physical quantities from the scale $T$. The second step is the use of the effective Lagrangian and perturbative calculations yield the contributions to the free energy from the scale $eT\sim\sqrt{\lambda}T$.