Renormalizing the Schwinger-Dyson equations in the auxiliary field formulation of $λϕ^4$ field theory

Cooper, Fred; Mihaila, Bogdan; Dawson, John F.

Renormalizing the Schwinger-Dyson equations in the auxiliary field formulation of $λϕ^4$ field theory

Authors

Cooper, Fred

Mihaila, Bogdan

Dawson, John F.

Description

In this paper we study the renormalization of the Schwinger-Dyson equations that arise in the auxiliary field formulation of the O(N) $ϕ^4$ 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 $χ$, 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 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 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's functions with the related \emph{sunset} approximation to the 2-PI equations discussed by Van Hees and Knoll, and comment on the importance of the Landau pole effect.
20 pages, 10 figures