Effective field theory approach to high-temperature thermodynamics

An effective field theory approach is developed for calculating the thermodynamic properties of a field theory at high temperature T and weak coupling g. The effective theory is the three-dimensional field theory obtained by dimensional reduction to the bosonic zero-frequency modes. The parameters of the effective theory can be calculated as a perturbation series in the running coupling constant ${\mathit{g}}^2$(T). The free energy is separated into the contributions from the momentum scales T and gT, respectively. The first term can be written as a perturbation series in ${\mathit{g}}^2$(T). If all forces are screened at the scale gT, the second term can be calculated as a perturbation series in g(T) beginning at order ${\mathit{g}}^3$. The parameters of the effective theory satisfy renormalization group equations that can be used to sum up leading logarithms of T/(gT). We apply this method to a massless scalar field with a ${\mathrm{\Phi }}^4$ interaction, calculating the free energy to order ${\mathit{g}}^6$lng and the screening mass to order ${\mathit{g}}^5$lng.