Schwinger-Dyson truncations in the all-soft limit: a case study

TL;DR

This paper investigates a specific Schwinger-Dyson equation in SU(3) Yang-Mills theory using the background field method, deriving exact results in the all-soft limit and assessing truncation errors with lattice data.

Contribution

It provides an exact form of a background gluon-ghost vertex in the all-soft limit and evaluates the accuracy of truncation schemes using lattice data.

Findings

Exact vertex form in the all-soft limit derived from Ward identity.

Quantitative assessment of truncation errors in Schwinger-Dyson equations.

Validation of background field method relations in the Schwinger-Dyson framework.

Abstract

We study a special Schwinger-Dyson equation in the context of a pure SU(3) Yang-Mills theory, formulated in the background field method. Specifically, we consider the corresponding equation for the vertex that governs the interaction of two background gluons with a ghost-antighost pair. By virtue of the background gauge invariance, this vertex satisfies a naive Slavnov-Taylor identity, which is not deformed by the ghost sector of the theory. In the all-soft limit, where all momenta vanish, the form of this vertex may be obtained exactly from the corresponding Ward identity. This special result is subsequently reproduced at the level of the Schwinger-Dyson equation, by making extensive use of Taylor's theorem and exploiting a plethora of key relations, particular to the background field method. This information permits the determination of the error associated with two distinct truncation schemes, where the potential advantage from employing lattice data for the ghost dressing function is quantitatively assessed.