Non-perturbative comparison of QCD effective charges

TL;DR

This paper compares two non-perturbative definitions of the QCD effective charge, showing they are nearly identical across all momenta and exactly coincide in the infrared, revealing insights into QCD dynamics.

Contribution

It demonstrates the near equivalence of two different non-perturbative QCD effective charges and derives a key identity linking their components in the Landau gauge.

Findings

The two effective charges almost coincide across all momenta.

They exactly match in the deep infrared, both freezing at a finite value.

The infrared finiteness of the gluon propagator is crucial for the effective charge behavior.

Abstract

We study the non-perturbative behavior of two versions of the QCD effective charge, one obtained from the pinch technique gluon self-energy, and one from the ghost-gluon vertex. Despite their distinct theoretical origin, due to a fundamental identity relating various of the ingredients appearing in their respective definitions, the two effective charges are almost identical in the entire range of physical momenta, and coincide exactly in the deep infrared, where they freeze at a common finite value. Specifically, the dressing function of the ghost propagator is related to the two form factors in the Lorentz decomposition of a certain Green's function, appearing in a variety of field-theoretic contexts. The central identity, which is valid only in the Landau gauge, is derived from the Schwinger-Dyson equations governing the dynamics of the aforementioned quantities. The renormalization procedure that preserves the validity of the identity is carried out, and various relevant kinematic limits and physically motivated approximations are studied in detail. A crucial ingredient in this analysis is the infrared finiteness of the gluon propagator, which is inextricably connected with the aforementioned freezing of the effective charges. Some important issues related to the consistent definition of the effective charge in the presence of such a gluon propagator are resolved. We finally present a detailed numerical study of a special set of Schwinger-Dyson equations, whose solutions determine the non-perturbative dynamics of the quantities composing the two effective charges.