Nonperturbative study of the four gluon vertex

TL;DR

This paper investigates the nonperturbative structure of the four-gluon vertex in SU(3) gauge theory, revealing a zero-crossing and divergence in the form factor due to ghost masslessness, with implications for lattice simulations.

Contribution

It provides a novel nonperturbative analysis of the four-gluon vertex, identifying specific tensor structures and their infrared behavior using an approximation scheme with dressed propagators.

Findings

The form factor associated with a tensor orthogonal to the tree-level vertex exhibits zero-crossing and logarithmic divergence.

The tree-level tensor form factor remains finite and close to its naive value across all momenta.

The chosen kinematic configuration is suitable for lattice measurements, isolating the genuine vertex contributions.

Abstract

In this paper we study the nonperturbative structure of the SU(3) four-gluon vertex in the Landau gauge, concentrating on contributions quadratic in the metric. We employ an approximation scheme where "one-loop" diagrams are computed using fully dressed gluon and ghost propagators, and tree-level vertices. When a suitable kinematical configuration depending on a single momentum scale $p$ is chosen, only two structures emerge: the tree-level four-gluon vertex, and a tensor orthogonal to it. A detailed numerical analysis reveals that the form factor associated with this latter tensor displays a change of sign (zero-crossing) in the deep infrared, and finally diverges logarithmically. The origin of this characteristic behavior is proven to be entirely due to the masslessness of the ghost propagators forming the corresponding ghost-loop diagram, in close analogy to a similar effect established for the three-gluon vertex. However, in the case at hand, and under the approximations employed, this particular divergence does not affect the form factor proportional to the tree-level tensor, which remains finite in the entire range of momenta, and deviates moderately from its naive tree-level value. It turns out that the kinematic configuration chosen is ideal for carrying out lattice simulations, because it eliminates from the connected Green's function all one-particle reducible contributions, projecting out the genuine one-particle irreducible vertex. Motivated by this possibility, we discuss in detail how a hypothetical lattice measurement of this quantity would compare to the results presented here, and the potential interference from an additional tensorial structure, allowed by Bose symmetry, but not encountered within our scheme.