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.
Highlights
Conformal invariance and its implications for correlation functions is a well-studied subject [1]
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
We expect the renormalised correlators to satisfy a Callan-Symanzik equation with beta function terms. These beta functions are for sources that couple to composite operators and not for couplings that appear in the Lagrangian of the theory, so there is no contradiction here with the fact that we are discussing CFT correlation functions
Summary
Conformal invariance and its implications for correlation functions is a well-studied subject [1]. Semilocal singularities of the triple-K integral can be removed by a renormalisation of the sources for the scalar operators In this case we find a surprising new result: that the corresponding renormalised 3-point correlators contain double logarithms of momenta.. Section 4.3.1 discusses our regularisation procedure for the divergent triple-K integral: this is most accomplished by infinitesimally shifting the dimensions of operators and of the spacetime itself These shifts give rise to corresponding shifts in the indices of the Bessel-K functions that appear in the triple-K integral, as well as in the power of the integration variable. For this additional momentum-space conformal symmetry to be present, the leading divergence of the regulated triple-K integral must be nonlocal. We present a complete worked example of holographic renormalisation for the 3-point function of a marginal operator in three dimensions
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