Probing the tensor structure of lattice three-gluon vertex in Landau gauge

TL;DR

This study investigates the tensor structure of the lattice three-gluon vertex in Landau gauge, revealing that continuum-based tensor approximations can cause significant errors, but certain kinematic configurations allow exact lattice descriptions.

Contribution

The paper identifies specific kinematic configurations where lattice three-gluon vertex tensors match continuum structures exactly, improving the accuracy of lattice gauge theory analyses.

Findings

Continuum tensor bases may lead to errors in lattice vertex representations.

Special kinematics allow exact lattice descriptions of vertex tensors.

Results are obtained for SU(2) gauge theory in 2D and 3D with various actions.

Abstract

In this paper we test an approximate method that is often used in lattice studies of the Landau gauge three-gluon vertex. The approximation consists in describing the lattice correlator with tensor bases from the continuum theory. With the help of vertex reconstruction, we show that this "continuum" approach may lead, for general kinematics, to significant errors in vertex tensor representations. Such errors are highly unwelcome, as they can lead to wrong quantitative estimates for vertex form factors and related quantities of interest, like the three-gluon running coupling. As a possible solution, we demonstrate numerically and analytically that there exist special kinematic configurations for which the vertex tensor structures can be described exactly on the lattice. For these kinematics, the dimensionless tensor elements are equal to the continuum ones, regardless of the details of the lattice implementation. We ran our simulations for an $SU(2)$ gauge theory in two and three spacetime dimensions, with Wilson and $\mathcal{O}(a^2)$ tree-level improved gauge actions. Our results and conclusions can be straightforwardly generalised to higher dimensions and, with some precautions, to other lattice correlators, like the ghost-gluon, quark-gluon and four-gluon vertices.