Schwinger-Dyson Equation, Area Law and Chiral Symmetry in QCD

TL;DR

This paper derives a Schwinger-Dyson equation for the quark propagator within a specific formalism and area law model, demonstrating how modifications to the approximation and inclusion of backward trajectories are crucial for chiral symmetry breaking and Goldstone's theorem.

Contribution

It introduces a modified equal time approximation and shows the importance of backward quark trajectories for consistency with chiral symmetry and Goldstone's theorem in QCD.

Findings

Chiral symmetry breaking occurs at zero quark mass.

Backward quark trajectories are essential for model consistency.

The model aligns with Goldstone's theorem.

Abstract

A Schwinger-Dyson equation for the quark propagator is derived in the context of a Bethe-Salpeter second order formalism developped in preceding papers and of the Minimal Area Law model for the evaluation of the Wilson loop. We discuss how the equal time straight line approximation has to be modified to include correctly trajectories going backwards in time. We also show, by an appropriate selection of the solution of the SD equation, that in the limit of zero quark mass chiral symmetry breaking and a zero mass pseudoscalar meson actually occur. The inclusion of backward quark trajectories proves to be essential to make the model consistent with Goldstone's theorem.