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

Adam Bzowski, Paul McFadden, Kostas Skenderis

TL;DR

The paper develops a complete momentum-space renormalisation framework for scalar 3-point functions in conformal field theories, showing that conformal Ward identities fix these correlators to triple-K integrals, with divergences categorized as ultralocal, semilocal, or nonlocal. Renormalisation proceeds via dimensional regularisation, counterterms, and scheme choices, yielding new conformal anomalies and beta functions for composite sources, and revealing rich analytic structures including single and double logarithms of momenta. In cases with nonlocal leading divergences, the triple-K representation itself is singular and the renormalised correlator is given by the leading nonlocal piece, often displaying dual conformal symmetry tied to massless Feynman integrals. The work is complemented by explicit free-field and AdS/CFT realizations, establishing a broad, scheme-sensitive but physically consistent picture of 3-point function renormalisation at conformal fixed points. These results illuminate how anomalies, beta functions, and dual conformal invariance interplay in momentum-space CFT data and have implications for holography and higher-point functions.

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.

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

TL;DR

The paper develops a complete momentum-space renormalisation framework for scalar 3-point functions in conformal field theories, showing that conformal Ward identities fix these correlators to triple-K integrals, with divergences categorized as ultralocal, semilocal, or nonlocal. Renormalisation proceeds via dimensional regularisation, counterterms, and scheme choices, yielding new conformal anomalies and beta functions for composite sources, and revealing rich analytic structures including single and double logarithms of momenta. In cases with nonlocal leading divergences, the triple-K representation itself is singular and the renormalised correlator is given by the leading nonlocal piece, often displaying dual conformal symmetry tied to massless Feynman integrals. The work is complemented by explicit free-field and AdS/CFT realizations, establishing a broad, scheme-sensitive but physically consistent picture of 3-point function renormalisation at conformal fixed points. These results illuminate how anomalies, beta functions, and dual conformal invariance interplay in momentum-space CFT data and have implications for holography and higher-point functions.

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.

Paper Structure

This paper contains 47 sections, 345 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conformal Ward identities
  3. 2-point functions
  4. 3-point functions
  5. Ward identities
  6. General solution
  7. Renormalisation
  8. $\bm{(---)}$-condition: $\Delta_1 + \Delta_2 + \Delta_3 = 2d +2 k_{---}$.
  9. $\bm{(--+)}$-condition: $\Delta_1 + \Delta_2 - \Delta_3 = d +2 k_{--+}$.
  10. $\bm{(-++)}$-condition: $\Delta_1 - \Delta_2 - \Delta_3 = 2 k_{-++}$.
  11. $\bm{(+++)}$-condition: $\Delta_1 + \Delta_2 + \Delta_3 = d-2 k_{+++}$.
  12. Regularisation
  13. Changing the regularisation scheme
  14. Ward identities
  15. Triple-$K$ integrals with $1/\epsilon$ singularity
  16. ...and 32 more sections

Figures (4)

  • Figure 1: Feynman graphs representing $\langle\mathcal{O}_{[3]}\mathcal{O}_{[3]}\rangle$ and $\langle \mathcal{O}_{[4]} \mathcal{O}_{[3]} \mathcal{O}_{[3]} \rangle$ for a free scalar $\Phi$ with $\mathcal{O}_{[3]} =\, :\! \Phi^3 \!:$ and $\mathcal{O}_{[4]} =\, :\! \Phi^4 \!:$.
  • Figure 2: The leading Feynman diagrams contributing to the renormalisation of the $\phi^3$ vertex.
  • Figure 3: Feynman graphs representing $\langle\mathcal{O}_{[3]}\mathcal{O}_{[3]}\rangle$ and $\langle \mathcal{O}_{[3]} \mathcal{O}_{[3]} \mathcal{O}_{[3]} \rangle$ for a free scalar $\Phi$ with $\mathcal{O}_{[3]} = \,:\! \Phi^6 \!:$.
  • Figure 4: (a) Witten diagram for the evaluation of the scalar field $\Phi$ at a point $x$ in the bulk; (b) taking $x$ to a point $x_3$ on the boundary, the diagram now represents a 3-point function.