Renormalized broken-symmetry Schwinger-Dyson equations and the two-particle irreducible $1/N$ expansion for the $O(N)$ model

We derive the renormalized Schwinger-Dyson equations for the one- and two-point functions in the auxiliary field formulation of Coleman, Jackiw, and Politzer [S. Coleman, R. Jackiw, and H. D. Politzer, Phys. Rev. D 10, 2491 (1974).] for $\lambda {\varphi }^4$ field theory, to order $1/N$, in the 2PI-$1/N$ expansion. We show that the renormalization of the broken-symmetry theory depends only on the counter terms of the symmetric theory with $\varphi =0$, as discussed in our previous paper [F. Cooper, B. Mihaila, and J. F.Dawson, Phys. Rev. D 70, 105008 (2004).]. We find that the 2PI-$1/N$ expansion violates the Goldstone theorem at order $1/N$. In using the $O(4)$ model as a low energy effective field theory of pions to study the time evolution of disoriented chiral condensates one has to explicitly break the $O(4)$ symmetry to give the physical pions a nonzero mass. In this effective theory, we expect that the additional small contribution to the pion mass due to the violation of the Goldstone theorem in the 2PI-$1/N$ equations to be unimportant for an adequate description of the phenomenology.

Figure 1

Graphs included in the 2PI effective action ${\Gamma }_2[G]$.