Quark-gluon vertex in the complex plane

TL;DR

This work investigates the nonperturbative quark-gluon vertex in the complex plane, focusing on the soft-gluon limit, revealing its analytic structure and singularities relevant for meson studies.

Contribution

It introduces a novel method to analyze the quark-gluon vertex in the complex plane, extending the understanding of its analytic properties beyond previous approaches.

Findings

Determined the domain of reliable vertex form factors in the complex plane.

Identified the first singularity in the vertex integrals affecting analytic continuation.

Extended the analysis to arbitrary gluon momenta and discussed implications for the quark propagator.

Abstract

In the present work we explore for the first time the general structure and properties of the nonperturbative quark-gluon vertex in the complex plane. Specifically, we focus on the transversely-projected quark-gluon vertex that emerges from a recently developed symmetry-preserving approach for the study of meson properties beyond the rainbow-ladder approximation. The analysis focuses on the so-called "soft-gluon" limit, which reduces the momentum-dependence of the corresponding vertex form factors to a single momentum variable. The complexification of this variable inside the defining integrals furnishes unambiguously all eight vertex form factors within a concrete domain of the complex variable, delimited by a characteristic parabola. The extent of this reliable domain is determined by the appearance of the first singularity in the integrands of the vertex integrals, where the standard Wick rotation must be duly supplemented by additional crucial contributions. This primary analytic region may be extended considerably by resorting to standard extrapolation methods, which remain valid up until the appearance of complex structures associated with the onset of physical processes. The generalization of the method to arbitrary gluon momenta, and its relevance for the determination of the quark propagator in the complex plane, are briefly discussed.