  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.
