Integral equation for gauge invariant quark two-point Green's function in QCD

TL;DR

This paper derives an integral equation for gauge invariant quark two-point Green's functions in QCD, involving Wilson loops and functional derivatives, offering a new approach to study quark confinement.

Contribution

It introduces a novel integral equation for gauge invariant Green's functions using path-ordered phase factors and Wilson loops, expanding the theoretical framework in QCD.

Findings

Derived an integral equation involving Wilson loops and functional derivatives.

Established functional relations between Green's functions with different path segments.

Proposed an approximate resolution using the lowest-order kernel with two functional derivatives.

Abstract

Gauge invariant quark two-point Green's functions defined with path-ordered gluon field phase factors along skew-polygonal lines joining the quark to the antiquark are considered. Functional relations between Green's functions with different numbers of path segments are established. An integral equation is obtained for the Green's function defined with a phase factor along a single straight line. The equation implicates an infinite series of two-point Green's functions, having an increasing number of path segments; the related kernels involve Wilson loops with contours corresponding to the skew-polygonal lines of the accompanying Green's function and with functional derivatives along the sides of the contours. The series can be viewed as an expansion in terms of the global number of the functional derivatives of the Wilson loops. The lowest-order kernel, which involves a Wilson loop with two functional derivatives, provides the framework for an approximate resolution of the equation.