Paper detail

All-order existence of and recursion relations for the operator product expansion in Yang-Mills theory

We prove the existence of the operator product expansion (OPE) in Euclidean Yang-Mills theories as a short-distance expansion, to all orders in perturbation theory. We furthermore show that the Ward identities of the underlying gauge theory are reflected in the OPE; especially, the OPE of an arbitrary number of gauge-invariant composite operators only involves gauge-invariant composite operators. Moreover, we derive recursion relations which allow to construct the OPE coefficients, the quantum BRST differential and the quantum antibracket order by order in perturbation theory, starting from the known free-theory objects. These relations are completely finite from the start, and do not need any further renormalisation as is usually the case in other approaches. Our results underline the importance of the OPE as a general structure underlying quantum field theories. The proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, and generalise similar results recently obtained for scalar field theories [J. Holland and S. Hollands, Commun. Math. Phys. 336 (2015) 1555; J. Holland and S. Hollands, J. Math. Phys. 56 (2015) 122303]. Combining their results with our recursion formula, we also obtain associativity of the OPE coefficients.