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Recursive construction of operator product expansion coefficients. (English) Zbl 1429.81042
Summary: We derive a novel formula for the derivative of operator product expansion (OPE) coefficients with respect to a coupling constant. The formula involves just the OPE coefficients themselves but 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 requires no further UV-renormalization, i.e., it is completely well-defined from the start. This feature is a result of a cancellation of UV- and IR-divergences between various terms in our identity. Our proof, and an analysis of the features of the identity, is given for the example of massive, Euclidean $$\varphi^4$$ theory in 4 dimensional Euclidean space. It relies on the renormalization group flow equation method and is valid to arbitrary, but finite orders in perturbation theory. The final formula, however, makes neither explicit reference to the renormalization group flow, nor to perturbation theory, and we conjecture that it also holds non-perturbatively. Our identity can be applied constructively because it gives a novel 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.

##### MSC:
 81T08 Constructive quantum field theory 81T17 Renormalization group methods applied to problems in quantum field theory
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