Patterns of gauge symmetry in the background field method

TL;DR

This paper demonstrates, through explicit calculations, that the strong Slavnov-Taylor identities hold for the background three-gluon vertex in Yang-Mills theories, revealing intricate cancellations and the role of background Ward identities.

Contribution

It provides the first explicit proof that the strong Slavnov-Taylor identities apply to the background three-gluon vertex without truncations or assumptions.

Findings

Strong Slavnov-Taylor identities hold for the background three-gluon vertex.

Cancellations occur in the Schwinger-Dyson equations without explicit integrations.

Background Ward identities relate derivatives of propagators to zero-momentum insertions.

Abstract

The correlation functions of Yang-Mills theories formulated in the background field method satisfy linear Slavnov-Taylor identities, which are naive generalizations of simple tree level relations, with no deformations originating from the ghost sector of the theory. In recent years, a stronger version of these identities has been found to hold at the level of the background gluon self-energy, whose transversality is enforced separately for each special block of diagrams contributing to the gluon Schwinger-Dyson equation. In the present work we demonstrate by means of explicit calculations that the same distinct realization of the Slavnov-Taylor identity persists in the case of the background three-gluon vertex. The analysis is carried out at the level of the exact Schwinger-Dyson equation for this vertex, with no truncations or simplifying assumptions. The demonstration entails the contraction of individual vertex diagrams by the relevant momentum, which activates Slavnov-Taylor identities of vertices and multi-particle kernels nested inside these graphs; the final result emerges by virtue of a multitude of extensive cancellations, without the need of performing explicit integrations. In addition, we point out that background Ward identities amount to replacing derivatives of propagators by zero-momentum background-gluon insertions, in exact analogy to standard properties of Abelian gauge theories. Finally, certain potential applications of these results are briefly discussed.