Two- and three-point functions of composite operators are analyzed with regard to (logarithmically) divergent contact terms. Using the renormalization group of dimensional regularization it is established that the divergences are governed by the anomalous dimensions of the operators and the leading UV behavior of the $1/\epsilon$ coefficient. Explicit examples are given by the $⟨G^2{G}^2⟩$, $⟨\mathrm{\Theta }\mathrm{\Theta }⟩$ (trace of the energy momentum tensor) and $⟨\overline{q}q\overline{q}q⟩$ correlators in QCD-like theories. The former two are convergent when the $1/\epsilon$ poles are resummed but divergent at fixed order implying that perturbation theory and 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 by-product the $R^2$ term of the trace anomaly is extended to next-to-next-to-leading order [$\mathcal{O}(a_s^5)$], in the minimal subtraction scheme, using a recent $⟨G^2{G}^2⟩$ computation.

• Received 27 January 2017

DOI:https://doi.org/10.1103/PhysRevD.95.065027