The operator product expansion converges in perturbative field theory

TL;DR

This paper proves that the Wilson operator product expansion in Euclidean φ^4 quantum field theory converges at finite distances, not just asymptotically, using renormalization group flow equations to estimate the remainder.

Contribution

It demonstrates the convergence of the OPE in four-dimensional Euclidean φ^4 theory, a significant advancement over the previous belief of asymptotic behavior, by employing renormalization group methods.

Findings

OPE converges at finite distances in Euclidean φ^4 theory.

Remainder estimates decrease with higher operator dimensions.

Gradient expansion of the effective action is also convergent.

Abstract

We show, within the framework of the Euclidean $\phi^4$-quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further "spectator fields". The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the "gradient expansion" of the effective action is convergent.