The operator product expansion converges in massless $\varphi_4^4$-theory

TL;DR

This paper proves that the operator product expansion (OPE) converges in massless 4 theory, extending previous results from massive theories using new techniques to handle exceptional momentum singularities.

Contribution

It establishes the convergence of the OPE in massless 4 theory, employing novel methods to manage momentum singularities with flow equations.

Findings

OPE converges in massless 4 theory at all loop orders.

New techniques are developed to control momentum singularities.

Bounds are expressed via weight factors associated with tree graphs.

Abstract

It has been shown recently that the mathematical status of the operator product expansion (OPE) is better than was expected before: namely considering massive Euclidean $\varphi^4$-theory in the perturbative loop expansion, the OPE converges at any loop order when considering (as is usually done) composite operator insertions into correlation functions. In the present paper we prove the same result for the massless theory. While the short-distance properties of massive and massless theories may be expected to be similar on physical grounds, the proof in the massless case requires entirely new techniques. In our inductive construction we have to control with sufficient precision the exceptional momentum singularities of the massless correlation functions. In fact the bounds we state are expressed in terms of weight factors associated to certain tree graphs. Our proof is again based on the flow equations of the renormalisation group.