Constraining the effective action by a method of external sources

Abstract

We propose a novel method of evaluating the effective action, wherein the physical one- and two-point functions are obtained in the limit of non-vanishing external sources. We illustrate the self-consistency of this method by recovering the usual 2PI effective action due to Cornwall, Jackiw and Tomboulis, differing only by the fact that the saddle-point evaluation of the path integral is performed along the extremal quantum, rather than classical, path. As such, this approach is of particular relevance to situations where the dominant quantum and classical paths are non-perturbatively far away from one-another. A pertinent example is the decay of false vacua in radiatively-generated potentials, as may occur for the electroweak vacuum of the Standard Model. In addition, we describe how the external sources may instead be chosen so as to yield the two-particle-point-irreducible (2PPI) effective action of Coppens and Verschelde. Finally, in the spirit of the symmetry-improved effective action of Pilaftsis and Teresi, we give an example of how the external sources can be used to preserve global symmetries in truncations of the 2PI effective action. Specifically, in the context of an O(2) model with spontaneous symmetry breaking, we show that this approach allows the Hartree-Fock approximation to be re-organized, such that the Goldstone boson remains massless algebraically in the symmetry-broken phase and we obtain the correct second-order thermal phase transition.Comment: 31 pages; typos corrected; to match published versio

Alternative flow equation for the functional renormalization group

“…In this letter, we derive an alternative flow equation from the 2PI effective action by means of the method of external sources due to Garbrecht and Millington [28]. We show that the resulting flow equation differs to that derived from the average 1PI effective action, suggesting there exists an ambiguity in the "correct" choice of exact flow equation.…”

“…In the approach of Ref. [28], and in the case of a single saddle point, the stationarity condition in Eq. (6), combined with the variation in Eq.…”

“…The latter feature is particularly relevant in the case of false vacuum decay in theories with radiatively generated spontaneous symmetry breaking (see Refs. [28,31,32]), for instance, via the Coleman-Weinberg mechanism [33]. Alternatively, we can constrain the two-point source to be local, i.e.…”

Fluctuations about the Fubini-Lipatov instanton for false vacuum decay in classically scale invariant models

“…[10]), we encounter a significant limitation of this method. Such a situation occurs for models of radiative symmetry breaking, where the true vacuum only appears through loop corrections [11] or, as in the case of interest for the present work, for approximately scale-invariant models, where the scale of the radius of the nucleating bubbles is not known before consistently accounting for quantum effects.…”

Visualising quantum effective action calculations in zero dimensions

“…[22], see also Ref. [23]), this treatment of the sources allows a self-consistent calculation of the tunnelling rate [19,24].…”