# Nonperturbative Ball-Chiu construction of the three-gluon vertex

## TL;DR

This paper develops a nonperturbative method to derive the longitudinal three-gluon vertex consistent with Slavnov-Taylor identities, revealing features like zero-crossings and comparing well with lattice and Schwinger-Dyson results.

## Contribution

It introduces a nonperturbative Ball-Chiu construction for the three-gluon vertex, accounting for gluon mass effects and infrared finiteness, advancing understanding of nonperturbative QCD vertices.

## Key findings

- Vertex form factors show suppression below 1 GeV
- Characteristic zero-crossing near 100-200 MeV
- Good agreement with lattice and Schwinger-Dyson data

## Abstract

We present the detailed derivation of the longitudinal part of the three-gluon vertex from the Slavnov-Taylor identities that it satisfies, by means of a nonperturbative implementation of the Ball-Chiu construction; the latter, in its original form, involves the inverse gluon propagator, the ghost dressing function, and certain form factors of the ghost-gluon kernel. The main conceptual subtlety that renders this endeavor nontrivial is the infrared finiteness of the gluon propagator, and the resulting need to separate the vertex into two pieces, one that is intimately connected with the emergence of a gluonic mass scale, and one that satisfies the original set of Slavnov-Taylor identities, but with the inverse gluon propagator replaced by its "kinetic" term. The longitudinal form factors obtained by this construction are presented for arbitrary Euclidean momenta, as well as special kinematic configurations, parametrized by a single momentum. A particularly preeminent feature of the components comprising the tree-level vertex is their considerable suppression for momenta below 1 GeV, and the appearance of the characteristic "zero-crossing" in the vicinity of 100-200 MeV. Special combinations of the form factors derived with this method are compared with the results of recent large-volume lattice simulations as well as Schwinger-Dyson equations, and good overall agreement is found. A variety of issues related to the distribution of the pole terms responsible for the gluon mass generation are discussed in detail, and their impact on the structure of the transverse parts is elucidated. In addition, a brief account of several theoretical and phenomenological possibilities involving these newly acquired results is presented.