Constraining the effective action by a method of external sources

Björn Garbrecht,Peter Millington

doi:10.1016/j.nuclphysb.2016.02.022

Björn Garbrecht, Peter Millington

Open Access

https://doi.org/10.1016/j.nuclphysb.2016.02.022

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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.

Highlights

The use of effective-action techniques has become ubiquitous across theoretical physics, both in the relativistic regime of high-energy processes and the non-relativistic setting of condensed matter systems
We have described an alternative method of evaluating the effective action, where, in contrast to the usual approach, the physical limit is obtained in the presence of non-vanishing external sources in vacuum
We have illustrated the utility of this general approach by means of three concrete examples: (i) By forcing the system to follow its extremal quantum trajectory, we are able to recover the CJT effective action

Summary

Introduction

The use of effective-action techniques has become ubiquitous across theoretical physics, both in the relativistic regime of high-energy processes and the non-relativistic setting of condensed matter systems. [24, 25]) and the symmetry-improved CJT effective action [26] of Pilaftsis and Teresi (PT), in which the Ward identities are imposed through the method of Lagrange multipliers The latter variant of the effective action has the advantage that, aside from ensuring the masslessness of the Goldstone boson in the SSB phase and the correct second-order phase transition [27], it yields the correct decay thresholds for both the Higgs and Goldstone modes. It remains an open question as to the correct method by which to determine the tunneling rate when the global minimum of the potential is generated radiatively [73] This latter issue is relevant to the Coleman-Weinberg scenario of SSB [74], as well as symmetry restoration at finite temperature [75,76,77].

Method of external sources

CJT 2PI effective action

CV 2PPI effective action

Global symmetries

Conclusions