All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory

TL;DR

This paper rigorously proves the perturbative renormalisability of Euclidean Yang-Mills theories, providing detailed bounds on correlation functions of gauge-invariant operators using renormalisation group techniques.

Contribution

It offers a complete, mathematically rigorous proof of renormalisability and bounds for correlation functions in Yang-Mills theory, including control of BRST invariance.

Findings

Established all-order bounds for correlation functions

Proved renormalisability of Euclidean Yang-Mills theories

Controlled BRST invariance of gauge-invariant operators

Abstract

We give a complete, self-contained, and mathematically rigorous proof that Euclidean Yang-Mills theories are perturbatively renormalisable, in the sense that all correlation functions of arbitrary composite local operators fulfil suitable Ward identities. Our proof treats rigorously both all ultraviolet and infrared problems of the theory and provides, in the end, detailed analytical bounds on the correlation functions of an arbitrary number of composite local operators. These bounds are formulated in terms of certain weighted spanning trees extending between the insertion points of these operators. Our proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, combined with estimation techniques based on tree structures. Compared with previous mathematical treatments of massless theories without local gauge invariance [R. Guida and Ch. Kopper, arXiv:1103.5692; J. Holland, S. Hollands, and Ch. Kopper, Commun. Math. Phys. 342 (2016) 385] our constructions require several technical advances; in particular, we need to fully control the BRST invariance of our correlation functions.