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

Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
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.
1-75
Bzowski, Adam
89bef061-563e-4b14-ac4d-0418692e9992
McFadden, Paul
4e7762ff-9b96-4516-b333-be01784fdbae
Skenderis, Kostas
09f32871-ffb1-4f4a-83bc-df05f4d17a09
Bzowski, Adam
89bef061-563e-4b14-ac4d-0418692e9992
McFadden, Paul
4e7762ff-9b96-4516-b333-be01784fdbae
Skenderis, Kostas
09f32871-ffb1-4f4a-83bc-df05f4d17a09

Bzowski, Adam, McFadden, Paul and Skenderis, Kostas (2016) Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies. Journal of High Energy Physics, 2016 (66), 1-75. (doi:10.1007/JHEP03(2016)066).

Record type: Article

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.

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More information

Accepted/In Press date: 17 February 2016
Published date: 11 March 2016
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 391636
URI: https://eprints.soton.ac.uk/id/eprint/391636
PURE UUID: 96c16ab4-2410-4a1d-ab18-5c44b5a2ae31
ORCID for Kostas Skenderis: ORCID iD orcid.org/0000-0003-4509-5472

Catalogue record

Date deposited: 24 May 2016 13:55
Last modified: 15 Aug 2019 00:37

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