Recursive Construction of Operator Product Expansion Coefficients

Abstract

We derive a novel formula for the derivative of operator product expansion (OPE) coefficients with respect to a coupling constant. The formula only involves the OPE coefficients themselves, and no further input, and is in this sense self-consistent. Furthermore, unlike other formal identities of this general nature in quantum field theory (such as the formal expression for the Lagrangian perturbation of a correlation function), our formula is completely well-defined from the start, i.e. requires no further UV-renormalization. This feature is a result of a cancelation of UV-divergences between various terms in our identity. Our proof, and an analysis of the features, of our identity is given for the example of massive, Euclidean $\varphi^4$ theory in 4 dimensional Euclidean space, and relies heavily on the framework of the renormalization group flow equations. It is valid to arbitrary, but finite orders in perturbation theory. The final formula, however, makes no explicit reference to the renormalization group flow, nor to perturbation theory, and we conjecture that it also holds non-perturbatively. The identity can be applied constructively because it gives recursive algorithm for the computation of OPE coefficients to arbitrary (finite) perturbation order in terms of the zeroth order coefficients corresponding to the underlying free field theory, which in turn are trivial to obtain. We briefly illustrate the relation of this method to more standard methods for computing the OPE in some simple examples.