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

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.

Automated summary

In this study, we investigate a special Schwinger-Dyson equation in the context of a pure SU(3) Yang-Mills theory using the background field method. We focus on the vertex that describes the interaction between two background gluons and a ghost-antighost pair. By exploiting the background gauge invariance, we find that this vertex satisfies a simple Slavnov-Taylor identity that is unaffected by the ghost sector. In the limit where all momenta vanish, we obtain the exact form of this vertex from the corresponding Ward identity. Moreover, we demonstrate that this special result can be reproduced using the Schwinger-Dyson equation, taking advantage of Taylor's theorem and the specific relations in the background field method. This information allows us to determine the truncation error associated with two different truncation schemes and assess the potential benefit of using lattice data for the ghost dressing function.