Soft symmetry improvement of two particle irreducible effective actions

TL;DR

This paper introduces soft symmetry improvement (SSI), a new method for 2PI effective actions that relaxes symmetry constraints, aiming to find practical solutions in quantum field theory with broken symmetries.

Contribution

The paper proposes SSI, a novel approach that interpolates between unimproved and symmetry-improved 2PIEA, addressing solution existence issues in finite volume and temperature settings.

Findings

SSI is IR sensitive and requires finite volume and temperature.

Three distinct limits of SSI-2PIEA are identified, with different physical properties.

Solutions may fail to exist depending on the constraint strength parameter 

Abstract

Two particle irreducible effective actions (2PIEAs) are valuable non-perturbative techniques in quantum field theory; however, finite truncations of them violate the Ward identities (WIs) of theories with spontaneously broken symmetries. The symmetry improvement (SI) method of Pilaftsis and Teresi attempts to overcome this by imposing the WIs as constraints on the solution; however the method suffers from the non-existence of solutions in linear response theory and in certain truncations in equilibrium. Motivated by this, we introduce a new method called soft symmetry improvement (SSI) which relaxes the constraint. Violations of WIs are allowed but punished in a least-squares implementation of the symmetry improvement idea. A new parameter $\xi$ controls the strength of the constraint. The method interpolates between the unimproved ($\xi \to \infty$) and SI ($\xi \to 0$) cases and the hope is that practically useful solutions can be found for finite $\xi$. We study the SSI-2PIEA for a scalar O(N) model in the Hartree-Fock approximation. We find that the method is IR sensitive: the system must be formulated in finite volume $V$ and temperature $T=\beta^{-1}$ and the $V\beta \to \infty$ limit taken carefully. Three distinct limits exist. Two are equivalent to the unimproved 2PIEA and SI-2PIEA respectively, and the third is a new limit where the WI is satisfied but the phase transition is strongly first order and solutions can fail to exist depending on $\xi$. Further, these limits are disconnected from each other; there is no smooth way to interpolate from one to another. These results suggest that any potential advantages of SSI methods, and indeed any application of (S)SI methods out of equilibrium, must occur in finite volume.