Nonperturbative renormalization for 2PI effective action techniques

Abstract

Nonperturbative approximation schemes based on two-particle irreducible (2PI) effective actions provide an important means for our current understanding of (non-)equilibrium quantum field theory. A remarkable property is their renormalizability, since these approximations involve selective summations to infinite perturbative orders. In this paper we show how to renormalize all n-point functions of the theory, which are given by derivatives of the 2PI-resummed effective action Γ[φ] for scalar fields φ. This provides a complete description in terms of the generating functional for renormalized proper vertices, which extends previous prescriptions in the literature on the renormalization for 2PI effective actions. The importance of the 2PI-resummed generating functional for proper vertices stems from the fact that the latter respect all symmetry properties of the theory and, in particular, Goldstone's theorem in the phase with spontaneous symmetry breaking. This is important in view of the application of these techniques to gauge theories, where Ward identities play a crucial role. *

Symmetry-improved CJT effective action

“…for terms proportional to φ 2 , ∆, φ 4 , φ 2 ∆ and ∆ 2 . To be specific, the different CTs are required to cancel the UV divergences of the two-point correlation functions [32], 5) and the respective four-point functions,…”

Renormalisability of the 2PI-Hartree approximation of multicomponent scalar models in the broken symmetry phase

“…Our aim in the present paper is to clarify fully its renormalisability by providing a simple and transparent explicit construction of the counterterms for a rather general class of scalar field theories along the method proposed in [10,21].…”

“…A compact analysis of the model at the Hartree level of truncation appeared in the literature already as part of a more complete discussion of the 2PI renormalisability in Ref. [10]. The treatment was shown renormalisable by constructing the necessary counterterms iteratively.…”

“…The counterterms are introduced by the method of [10,21] generalising the case of the O(N) model presented in section 2. One has…”

“…It is used extensively both in equilibrium [2,3,4] and out-of-equilibrium [5,6,7,8] non-perturbative investigations of phase transition phenomena. Its non-perturbative renormalisability was demonstrated as particular case of the general proof of renormalisability of the physical quantities computed in various 2PI approximations [9,1,10]. These proofs are rather involved especially in the broken symmetry phase.…”