Renormalization in Self-Consistent Approximations schemes at Finite Temperature I: Theory

Within finite temperature field theory, we show that truncated non-perturbative self-consistent Dyson resummation schemes can be renormalized with local counter-terms defined at the vacuum level. The requirements are that the underlying theory is renormalizable and that the self-consistent scheme follows Baym’'s $\mathrm{\Phi }$-derivable concept. The scheme generates both, the renormalized self-consistent equations of motion and the closed equations for the infinite set of counter terms. At the same time the corresponding 2PI-generating functional and the thermodynamical potential can be renormalized, in consistency with the equations of motion. This guarantees the standard $\mathrm{\Phi }$-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold. The proof uses the techniques of BPHZ-renormalization to cope with the explicit and the hidden overlapping vacuum divergences.