Renormalizing the Schwinger-Dyson equations in the auxiliary field formulation ofλϕ4field theory

Abstract

In this paper we study the renormalization of the Schwinger-Dyson equations that arise in the auxiliary field formulation of the O($N$) ${\ensuremath{\phi}}^{4}$ 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 $\ensuremath{\chi}$, once the effective action is obtained by integrating over the $\ensuremath{\phi}$ 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.