Associativity of the operator product expansion

Abstract

We consider a recursive scheme for defining the coefficients in the operator product expansion (OPE) of an arbitrary number of composite operators in the context of perturbative, Euclidean quantum field theory in four dimensions. Our iterative scheme is consistent with previous definitions of OPE coefficients via the flow equation method, or methods based on Feynman diagrams. It allows us to prove that a strong version of the "associativity" condition holds for the OPE to arbitrary orders in perturbation theory. Such a condition was previously proposed in an axiomatic setting in [1] and has interesting conceptual consequences: 1) One can characterise perturbations of quantum field theories abstractly in a sort of "Hochschild-like" cohomology setting, 2) one can prove a "coherence theorem" analogous to that in an ordinary algebra: The OPE coefficients for a product of two composite operators uniquely determine those for $n$ composite operators. We concretely prove our main results for the Euclidean $\varphi^4_4$ quantum field theory, covering also the massless case. Our methods are rather general, however, and would also apply to other, more involved, theories such as Yang-Mills theories.