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

Bzowski, Adam; McFadden, Paul; Skenderis, Kostas

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.

Citation

Bzowski, A., McFadden, P., & Skenderis, K. (2016). Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies. Journal of High Energy Physics, 2016(03), Article 066. https://doi.org/10.1007/jhep03%282016%29066

• Journal Article Type: Article
• Acceptance Date: Feb 17, 2016
• Online Publication Date: Mar 11, 2016
• Publication Date: Mar 11, 2016
• Deposit Date: Oct 26, 2017
• Publicly Available Date: Oct 31, 2017
• Journal: Journal of High Energy Physics
• Print ISSN: 1126-6708
• Electronic ISSN: 1029-8479
• Publisher: Scuola Internazionale Superiore di Studi Avanzati (SISSA)
• Peer Reviewed: Peer Reviewed
• Volume: 2016
• Issue: 03
• Article Number: 066
• DOI: https://doi.org/10.1007/jhep03%282016%29066
• Public URL: https://durham-repository.worktribe.com/output/1341783
• Related Public URLs: https://arxiv.org/abs/1510.08442