Linear Response Theory for Symmetry Improved Two Particle Irreducible Effective Actions

TL;DR

This paper examines the linear response of an O(N) scalar quantum field theory using a symmetry improved effective action approach, revealing unphysical effects and pathologies that challenge the method's consistency.

Contribution

It identifies fundamental issues in the symmetry improved two particle irreducible effective action formalism at the linear response level, highlighting limitations and potential directions for improvement.

Findings

Goldstone boson fluctuations are over-determined and must be set to zero.

Higgs field fluctuations propagate masslessly despite correct mass in propagator.

Pathologies persist regardless of truncation or Ward identity relaxation.

Abstract

We investigate the linear response of an O(N) scalar quantum field theory subject to external perturbations using the symmetry improved two particle irreducible effective action formalism [A. Pilaftsis and D. Teresi, Nucl. Phys. B874, 594 (2013)]. Despite satisfactory equilibrium behavior, we find a number of unphysical effects at the linear response level. Goldstone boson field fluctuations are over-determined, with the only consistent solution being to set the fluctuations and their driving sources to zero, except for momentum modes where the Higgs and Goldstone self-energies obey a particular relationship. Also Higgs field fluctuations propagate masslessly, despite the Higgs propagator having the correct mass. These pathologies are independent of any truncation of the effective action and still exist even if we relax the over-determining Ward identities, so long as the constraint is formulated O(N)-covariantly. We discuss possible reasons for the apparent incompatibility of the constraints and linear response approximation and possible ways forward.