[Paper Review] Gauge Invariant Quark Propagator in the Instanton Background

Marcus Hütter|arXiv (Cornell University)|Feb 21, 1995

Quantum Chromodynamics and Particle Interactions7 references3 citations

TL;DR

This paper presents a gauge-invariant quark propagator in the background of a single instanton by employing a path-ordered exponential to ensure gauge invariance. Using a gauge choice motivated by this construction, the authors derive a finite quark condensate without requiring an infrared cutoff or phenomenological instanton models, offering a non-perturbative framework for studying chiral symmetry breaking in QCD.

ABSTRACT

After a general discussion on the choice of gauge, we compare the quark propagator in the background of one instanton in regular and singular gauge with a gauge invariant propagator obtained by inserting a path-ordered gluon exponential. Using a gauge motivated by this analysis, we were able to obtain a finite result for the quark condensate without introducing an infrared cutoff nor invoking some instanton model.

Motivation & Objective

• To resolve the gauge dependence of quark propagators in instanton backgrounds, which compromises physical observables.
• To construct a gauge-invariant formulation of the quark propagator using path-ordered gluon exponentials.
• To compute the quark condensate in a non-perturbative, gauge-invariant manner without introducing ad hoc regulators.
• To eliminate reliance on phenomenological instanton models or infrared cutoffs in evaluating the condensate.
• To provide a consistent framework for studying chiral symmetry breaking in QCD via instanton-induced effects.

Proposed method

• Introduces a gauge-invariant quark propagator by inserting a path-ordered exponential of the gauge field along a path from the quark's position to a reference point.
• Compares the standard quark propagator in regular and singular gauges with the gauge-invariant version to assess gauge dependence.
• Derives a gauge choice motivated by the structure of the path-ordered exponential, ensuring consistency and finiteness.
• Performs the calculation of the quark condensate in this new gauge, avoiding divergences without explicit regularization.
• Uses the instanton solution as the background gauge field, focusing on one-instanton configurations.
• Applies standard QCD techniques in Euclidean space, with careful treatment of zero modes and fermionic determinants.

Experimental results

Research questions

• RQ1 How can a gauge-invariant quark propagator be consistently defined in the presence of an instanton background?
• RQ2 What is the impact of gauge choice on the quark condensate in instanton backgrounds?
• RQ3 Can the quark condensate be computed without introducing an infrared cutoff or phenomenological instanton models?
• RQ4 Does the path-ordered exponential construction yield a finite and physically meaningful result for the condensate?
• RQ5 What gauge choice emerges naturally from the requirement of gauge invariance in the instanton background?

Key findings

• The gauge-invariant quark propagator is constructed using a path-ordered exponential, ensuring invariance under local gauge transformations.
• The quark condensate is found to be finite without the need for an infrared cutoff or phenomenological instanton models.
• A specific gauge choice emerges naturally from the gauge-invariant construction, enabling finite results in the calculation.
• The method avoids the ambiguities of standard perturbative approaches in non-perturbative QCD contexts.
• The result supports the idea that instantons can generate a finite quark condensate through non-perturbative dynamics.
• The approach provides a consistent framework for studying chiral symmetry breaking in QCD using a single instanton background.