Action principle for OPE

TL;DR

This paper introduces an action principle for the operator product expansion in quantum field theories, providing a regulator-free way to understand how OPE coefficients evolve under deformations, especially in conformal theories.

Contribution

It formulates a novel, regulator-free action principle for OPEs that defines RG flow and conformal data evolution in general quantum field theories.

Findings

Provides a natural definition of RG flow for OPE coefficients

Derives coupled dynamical equations for conformal data

Applicable to general Euclidean quantum field theories

Abstract

We formulate an "action principle" for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings "In memoriam Rudolf Haag" in September 2016 and the "Wolfhart Zimmermann memorial symposium" in May 2017.