Loss of solution in the symmetry improved Phi-derivable expansion scheme

TL;DR

This paper investigates the loss of solutions in a symmetry-improved Phi-derivable approximation for an O(2) scalar model, highlighting issues with IR regulators and the implications for maintaining Goldstone's theorem.

Contribution

It demonstrates the conditions under which solutions are lost in the symmetry-improved Phi-derivable scheme and discusses the impact of UV regulators and potential remedies.

Findings

Solutions are lost with smooth UV regulators as IR scale decreases.

Infrared regular solutions are artifacts of non-analytic UV regulators.

Loss of solutions occurs at both zero and finite temperature.

Abstract

We consider the two-loop Phi-derivable approximation for the O(2)-symmetric scalar model, augmented by the symmetry improvement introduced in [A. Pilaftsis and D. Teresi, Nucl. Phys. B874, 594 (2013)], which enforces Goldstone's theorem in the broken phase. Although the corresponding equations admit a solution in the presence of a large enough infrared (IR) regulating scale, we argue that, for smooth ultraviolet (UV) regulators, the solution is lost when the IR scale becomes small enough. Infrared regular solutions exist for certain non-analytic UV regulators, but we argue that these solutions are artifacts which should disappear when the sensitivity to the UV regulator is removed by a renormalization procedure. The loss of solution is observed both at zero and at finite temperature, although it is simpler to identify in the latter case. We also comment on possible ways to cure this problem.