Operator product expansion algebra

Abstract

We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean $\varphi^{4}$-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of arXiv:1105.3375, that the 3-point OPE, $< O_{A_1} O_{A_2} O_{A_3} > = \sum_{C} \cal{C}_{A_1 A_2 A_3}^{C} <O_{C}>$, usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity $\cal{C}_{A_1 A_2 A_3}^{B}=\sum_{C}\cal{C}_{A_1 A_2}^{C} \cal{C}_{C A_3}^{B}$ is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such `consistency conditions' and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.