Gauge invariant Ansatz for a special three-gluon vertex

TL;DR

This paper develops a gauge-invariant Ansatz for a three-gluon vertex in QCD, ensuring consistency with Ward and Slavnov-Taylor identities, crucial for accurate Schwinger-Dyson equation truncations.

Contribution

It introduces a novel, all-order valid Ansatz for the background-quantum gluon vertex, facilitating gauge-invariant truncations in QCD Schwinger-Dyson equations.

Findings

Derivation of constraints relating ghost Green's functions form factors.

Identification of non-trivial cancellations in the gluon propagator equation.

Ensures self-consistency of gauge-invariant truncation approach.

Abstract

We construct a general Ansatz for the three-particle vertex describing the interaction of one background and two quantum gluons, by simultaneously solving the Ward and Slavnov-Taylor identities it satisfies. This vertex is known to be essential for the gauge-invariant truncation of the Schwinger-Dyson equations of QCD, based on the pinch technique and the background field method. A key step in this construction is the formal derivation of a set of crucial constraints (shown to be valid to all orders), relating the various form factors of the ghost Green's functions appearing in the aforementioned Slavnov-Taylor identity. When inserted into the Schwinger-Dyson equation for the gluon propagator, this vertex gives rise to a number of highly non-trivial cancellations, which are absolutely indispensable for the self-consistency of the entire approach.