Renormalization of Φ-derivable approximations in scalar field theories

We discuss the renormalization of Φ-derivable approximations for scalar field theories. In such approximations, the self-energy is obtained as the solution of a self-consistent equation which effectively resums infinite subsets of diagrams of perturbation theory. We show that a consistent renormalization can be carried out, and we provide an explicit construction of the counterterms needed to eliminate the subdivergences. These counterterms are calculated from the solution of an auxiliary gap equation which determines the dominant part of the self-energy at large momentum. This auxiliary gap equation may be chosen as the gap equation of the massless theory at zero temperature. We verify explicitly that, as expected, the counterterms determined at zero temperature are sufficient to eliminate the divergences which occur in finite temperature calculations.