Renormalised CFT 3-point functions of scalars, currents and stress tensors

TL;DR

The paper develops a nonperturbative momentum-space framework to renormalise mixed 3-point functions of scalars, conserved currents, and stress tensors in general dimension via triple-K integrals and conformal Ward identities. It identifies and classifies anomalies (type B) and beta functions arising from (---) and (+--) singularities, providing explicit results in d=3 and d=4 for operators with Δ=d and Δ=d-2. The work reveals universal tensor structures that persist across operator dimensions and demonstrates how anomalous dilatation and special conformal Ward identities emerge from counterterm structures and Weyl covariance. These results illuminate the renormalisation of mixed correlators, clarify the role of contact terms in conformal perturbation theory, and offer tools for applications in cosmology and holography. The analysis also lays groundwork for future momentum-space explorations of higher-point functions and conformal manifolds.

Abstract

We discuss the renormalisation of mixed 3-point functions involving tensorial and scalar operators in conformal field theories of general dimension. In previous work we analysed correlators of either purely scalar or purely tensorial operators, in each case finding new features and new complications: for scalar correlators, renormalisation leads to beta functions, novel conformal anomalies of type B, and unexpected analytic structure in momentum space; for correlators of stress tensors and/or conserved currents, beta functions vanish but anomalies of both type B and type A (associated with a $0/0$ structure) are present. Mixed correlators combine all these features: beta functions and anomalies of type B, plus the possibility of new type A anomalies. Following a non-perturbative and general momentum-space analysis, we present explicit results in dimensions $d=3,4$ for all renormalised 3-point functions of stress tensors, conserved currents and scalars of dimensions $Δ=d$ and $Δ=d-2$. We identify all anomalies and beta functions, and explain the form of the anomalous conformal Ward identities. In $d=3$, we find a $0/0$ structure but the corresponding type A anomaly turns out to be trivial. In addition, the correlators of two currents and a scalar, and of two stress tensors and a scalar, both feature universal tensor structures that are independent of the scalar dimension and vanish for opposite helicities.