The causal structure of the quark propagator

TL;DR

This paper investigates the causal properties of the quark propagator using spectral Dyson-Schwinger equations, revealing additional complex poles that can violate causality and discussing how to avoid these in full QCD.

Contribution

It introduces a spectral approach to analyze the quark propagator's causal structure, identifying conditions for the presence of complex poles and their implications for causality.

Findings

Identifies a critical vertex strength for the quark propagator's Källén-Lehmann representation.

Finds additional complex conjugate poles at specific energy sums, affecting causality.

Discusses how full QCD may avoid causality violations through tensor structure contributions.

Abstract

We study the causal structure of the quark propagator with the spectral DSE. The spectral gap equation is solved with the input of the spectral representation of the gluon and a causal STI-construction for the quark-gluon vertex. The latter includes a potential infrared enhancement of the vertex strength of the classical tensor structure that accommodates for the physical strength of chiral symmetry breaking. We find a critical vertex strength, below which the quark has a K\"all\'en-Lehmann representation. While the nature of the first singularity does not change above the critical strength, we find that the quark propagator features at least two additional pairs of complex conjugate poles that are located approximately at the sum of quark pole mass and peak position of the quark-gluon coupling. These additional poles lead to violations of causality, if they persist in $S$-matrix elements. While the vertex strength of the classical tensor structure in full QCD is below the critical one, that of commonly used vertex models, which rely solely on the classical vertex structure, is typically above it. Finally, we discuss how these additional poles could be avoided in full QCD, where part of chiral symmetry breaking is generated by the other tensor structures in the quark-gluon vertex.