Renormalizing the Schwinger-Dyson Equations

In this paper we study the renormalization of the Schwinger-Dyson (S-D) equations that arise in the auxiliary field formulation of the O(N) lambda phi^4 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 chi, once the effective action is obtained by integrating over the phi fields. Our all orders result is then used to obtain finite renormalized Schwinger-Dyson equations based on truncation expansions that utilize the two-particle irreducible (2-PI) generating function formalism. We first perform 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 S-D 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 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.