Fully coupled functional equations for the quark sector of QCD

TL;DR

This paper develops a comprehensive, self-consistent framework for studying the quark sector in 2+1 flavor QCD using coupled Schwinger-Dyson equations, integrating inputs from lattice and other functional methods, and accurately determining vertex form factors and the quark propagator.

Contribution

It introduces a self-consistent approach to solve coupled quark propagator and vertex equations in QCD, incorporating external inputs from lattice and functional methods, and identifies dominant vertex form factors.

Findings

Quark propagator form factors agree with lattice and functional methods.

Identified three dominant quark-gluon vertex form factors.

Chiral condensate in the chiral limit is $(245 ext{ MeV})^3$.

Abstract

We present a comprehensive study of the quark sector of $2+1$ flavour QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with $2+1$ dynamical quark flavours, obtained from studies with Schwinger-Dyson equations, the functional renormalisation group approach, and large volume lattice simulations. The appropriate renormalisation procedure required in order to self-consistently accommodate external inputs stemming from other functional approaches or the lattice is discussed in detail, and the value of the gauge coupling is accurately determined at two vastly separated renormalisation group scales. Our analysis establishes a clear hierarchy among the vertex form factors. We identify only three dominant ones, in agreement with previous results. The components of the quark propagator obtained from our approach are in excellent agreement with the results from Schwinger-Dyson equations, the functional renormalisation group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as $(245\,\textrm{MeV})^3$. The present approach has a wide range of applications, including the self-consistent computation of bound-state properties and finite temperature and density physics, which are briefly discussed.