Schwinger-Dyson equations and the quark-antiquark static potential

TL;DR

This paper compares the Schwinger-Dyson equations approach and the flux tube model in lattice QCD to understand quark confinement, highlighting their successes, challenges, and how they relate to each other.

Contribution

It provides an analysis of how the Schwinger-Dyson equations and flux tube models offer different insights into quark confinement and discusses the challenges in connecting these perspectives.

Findings

Schwinger-Dyson equations suggest an inverse quartic potential in momentum space.

Lattice QCD confirms a linear potential consistent with flux tube confinement.

Matching the two approaches remains an open problem.

Abstract

In lattice QCD, a confining potential for a static quark-antiquark pair can be computed with the Wilson loop technique. This potential, dominated by a linear potential at moderate distances, is consistent with the confinement with a flux tube, an extended and scalar system also directly observable in lattice QCD. Quantized flux tubes have also been observed in another class of confinement, the magnetic confinement in type II superconductors. On the other hand the solution of Schwinger Dyson Equations, say with the Landau gauge fixing and the truncation of the series of Feynman diagrams, already at the rainbow level for the self energy and at the ladder level for the Bethe Salpeter equation, provides a signal of a possible inverse quartic potential in momentum space derived from one gluon and one ghost exchange, consistent with confinement. Here we address the successes, difficulties and open problems of the matching of these two different perspectives of confinement, the Schwinger-Dyson perspective versus the flux tube perspective.