Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies

TL;DR

This paper thoroughly analyzes the renormalization of scalar 3-point functions in conformal field theories, addressing divergences, anomalies, and beta functions, and illustrating the concepts with examples from free fields and AdS/CFT.

Contribution

It provides a comprehensive framework for renormalizing 3-point functions in CFTs, including handling various divergences and deriving associated anomalies and beta functions.

Findings

Classification of divergences into ultralocal, semilocal, and nonlocal.

Renormalization procedures lead to conformal anomalies and beta functions.

Enhanced symmetry under dual conformal transformations in certain cases.

Abstract

We present a comprehensive discussion of renormalisation of 3-point functions of scalar operators in conformal field theories in general dimension. We have previously shown that conformal symmetry uniquely determines the momentum-space 3-point functions in terms of certain integrals involving a product of three Bessel functions (triple-K integrals). The triple-K integrals diverge when the dimensions of operators satisfy certain relations and we discuss how to obtain renormalised 3-point functions in all cases. There are three different types of divergences: ultralocal, semilocal and nonlocal, and a given divergent triple-K integral may have any combination of them. Ultralocal divergences may be removed using local counterterms and this results in new conformal anomalies. Semilocal divergences may be removed by renormalising the sources, and this results in CFT correlators that satisfy Callan-Symanzik equations with beta functions. In the case of non-local divergences, it is the triple-K representation that is singular, not the 3-point function. Here, the CFT correlator is the coefficient of the leading nonlocal singularity, which satisfies all the expected conformal Ward identities. Such correlators exhibit enhanced symmetry: they are also invariant under dual conformal transformations where the momenta play the role of coordinates. When both anomalies and beta functions are present the correlators exhibit novel analytic structure containing products of logarithms of momenta. We illustrate our discussion with numerous examples, including free field realisations and AdS/CFT computations.