Renormalization and resummation in finite temperature field theories

Resummation, i.e. reorganization of perturbative series, can result in an inconsistent perturbation theory, unless the counterterms are reorganized in an appropriate way. In this paper two methods are presented for resummation of counterterms: one is a direct method where the necessary counterterms are constructed order by order; the other is a general one, based on renormalization group arguments. Using the ${\Phi }^4$ model, we demonstrate on one hand that, in mass-independent schemes, mass resummation can be performed by gap equations renormalized prior to the substitution of the resummed mass for its argument. On the other hand it is shown that any (momentum-independent) form of mass and coupling constant resummation is compatible with renormalization, and one can explicitly construct the corresponding counterterms.