Behaviour of propagator and quark confinement

TL;DR

This paper investigates the behavior of the quark propagator in the timelike region using Schwinger-Dyson equations, revealing that quark confinement is associated with a complex mass function that prevents real poles.

Contribution

It introduces a method to calculate the quark propagator in timelike momenta via Temporal Euclidean space and analyzes the impact of different gluon propagator assumptions.

Findings

The quark mass function becomes complex below the perturbative threshold.

Infrared mass value is approximately equal to 5555555 555 555 555 555 with phase 5555555.

Timelike dynamical chiral symmetry breaking is linked to the Euclidean gap equation.

Abstract

The propagator of confined quarks is calculated for timelike momenta by transforming Minkowski Greens functions to the Temporal Euclidean space. Based on the framework of the Schwinger-Dyson equations the QCD quark propagator is obtained in two approximations which differ by assuming behaviour of gluon propagator. In both studied cases we get universal result for the light quarks: The quark mass function becomes complex bellow expected perturbative threshold, the obtained absolute value of the infrared mass is $M\simeq \Lambda_{QCD} $ with the infrared phase $\simeq {\pi\over 2}$. Permanent confinement of quarks is maintained by generation of the complex mass function which prevents a real pole in the propagator. We will show that timelike dynamical Chiral Symmetry Breaking (CSB) solution is approximately, but non-trivially determined by the solution of gap equation in the standard Euclidean space.