On Finiteness of 2- and 3-point Functions and the Renormalisation Group

TL;DR

This paper investigates the divergence properties of 2- and 3-point functions of composite operators in quantum field theories, showing how renormalisation group methods relate divergences to anomalous dimensions and providing explicit examples in QCD-like theories.

Contribution

It establishes the governing role of anomalous dimensions in divergences of correlation functions and extends the R^2 anomaly calculation to NNLO order.

Findings

Certain correlators are finite when all orders are summed but divergent at each perturbative order.

Finite correlation functions satisfy unsubtracted dispersion relations.

The R^2 anomaly is extended to NNLO using recent computations.

Abstract

Two and three point functions of composite operators are analysed with regard to (logarithmically) divergent contact terms. Using the renormalisation group of dimensional regularisation it is established that the divergences are governed by the anomalous dimensions of the operators and the leading UV-behaviour of the $1/\epsilon$-coefficient. Explicit examples are given by the $<G^2G^2>$-, $<\Theta \Theta>$-trace of the energy momentum tensor) and $<\bar q q \bar q q>$- correlators in QCD-like theories. The former two are convergent when all orders are taken into account but divergent at each order in perturbation theory implying that the latter and the the $\epsilon \to 0$ limit do not generally commute. Finite correlation functions obey unsubtracted dispersion relations which is of importance when they are directly related to physical observables. As a byproduct the $R^2$-anomaly is extended to NNLO ($O(\alpha^5)$) using a recent $<G^2G^2>$-computation.