Renormalizing the Schwinger-Dyson equations in the auxiliary field formulation of λφ4 field theory

In this paper we study the renormalization of the Schwinger-Dyson equations that arise in the auxiliary field formulation of the O(N) φ4 field theory. The auxiliary field formulation allows a simple interpretation of the large-N expansion as a loop expansion of the generating functional in the auxiliary field χ, once the effective action is obtained by integrating over the φ fields. Our all orders result is then used to obtain finite renormalized Schwinger-Dyson (SD) equations based on truncation expansions which utilize the two-particle irreducible (2-PI) generating function formalism. We first do an all orders renormalization of the two- and three-point function equations in the vacuum sector. This result is then used to obtain explicitly finite and renormalization constant independent self-consistent SD equations valid to order 1/N, in both 2+1 and 3+1 dimensions. We compare the results for the real and imaginary parts of the renormalized Green's functions with the related sunset approximation to the 2-PI equations discussed by Van Hees and Knoll, and comment on the importance of the Landau pole effect