Renormalization in self-consistent approximation schemes at finite temperature: Theory

Within finite temperature field theory, we show that truncated nonperturbative self-consistent Dyson resummation schemes can be renormalized with local counterterms defined at the vacuum level. The requirements are that the underlying theory is renormalizable and that the self-consistent scheme follows Baym's Phi-derivable concept. The scheme generates both the renormalized self-consistent equations of motion and the closed equations for the infinite set of counterterms. At the same time the corresponding two-particle irreducible generating functional and the thermodynamical potential can be renormalized, consistent with the equations of motion. This guarantees that the standard Phi-derivable properties such as thermodynamic consistency and exact conservation laws hold also for the renormalized approximation schemes. The proof uses the techniques of Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization to cope with the explicit and the hidden overlapping vacuum divergences.