Renormalised 3-point functions of stress tensors and conserved currents in CFT

TL;DR

This work develops a complete momentum-space framework for renormalising tensorial 3-point functions in CFTs, solving conformal Ward identities with triple-$K$ integrals and regulating divergences in a conformally invariant manner. By carefully separating transverse-traceless components, reconstructing full correlators from Ward identities, and introducing covariant counterterms, the authors obtain finite, renormalised correlators for <JJJ>, <TJJ>, and <TTT> across dimensions, with explicit results in $d=3$ and $d=4$. A central result is the dimension-dependent degeneracy linking to the Euler (type A) anomaly in four dimensions, where the Euler contribution appears as a square of the chiral anomaly and is decoupled from the scale dependence governed by the Weyl-squared coefficient. The framework yields both scheme-dependent and scheme-independent constants, enabling extraction of universal CFT data such as the OPE coefficient $C_1$ and, in 4d, the Euler coefficient $a$, and opens pathways to nonlocal effective actions, anomaly matching, and potential $a$-theorem perspectives. Overall, the momentum-space approach provides a universal, highly systematic method to understand tensorial CFT correlators and their anomalies with broad implications for holography, RG flows, and conformal bootstrap.

Abstract

We present a complete momentum-space prescription for the renormalisation of tensorial correlators in conformal field theories. Our discussion covers all 3-point functions of stress tensors and conserved currents in arbitrary spacetime dimensions. In dimensions three and four, we give explicit results for the renormalised correlators, the anomalous Ward identities they obey, and the conformal anomalies. For the stress tensor 3-point function in four dimensions, we identify the specific evanescent tensorial structure responsible for the type A Euler anomaly, and show this anomaly has the form of a double copy of the chiral anomaly.