Non-perturbative renormalization of Φ-derivable approximations in theories with fermions

We show how to renormalize Φ-derivable approximations in a theory with a fermionic field coupled to a self-interacting scalar field through a Yukawa interaction. The non-perturbative renormalization concerns the self-interaction coupling of the scalar field which is renormalized via a set of nested Bethe–Salpeter equations for the scalar and fermionic four-point functions. We use this information to construct explicit finite equations of motion in the symmetric phase. We work in the context of quantum field theory in equilibrium and show that renormalization can be carried out without introducing temperature dependent counterterms.