TTT in CFT: Trace Identities and the Conformal Anomaly Effective Action

TL;DR

The authors develop a covariant 1PI framework to study stress-energy correlators in 4D CFTs with curved backgrounds, focusing on the conformal anomaly. They show that anomaly-induced massless $1/k^2$ poles appear in the three-point function $raket{TTT}$ and that both the non-local anomaly action and its local conformalon form reproduce the same pole structure, matching results from momentum-space conformal Ward identities and the algebraic reconstruction method. A key result is the explicit demonstration that anomaly contributions to $raket{TTT}$ are fixed by the anomaly and are equivalent across distinct approaches, underscoring the fundamental role of the conformal anomaly in shaping higher-point stress-tensor correlators. The work highlights the physical significance of the conformalon as a propagating scalar degree of freedom associated with long-range gravitational effects and clarifies the separation between anomalous, Weyl-invariant, and local terms in the effective action.

Abstract

Stress-energy correlation functions in a general Conformal Field Theory (CFT) in four dimensions are described in a fully covariant approach, as metric variations of the quantum effective action in an arbitrary curved space background field. All Conservation, Trace and Conformal Ward Identities (CWIs), including contact terms, are completely fixed in this covariant approach. The Trace and CWIs are anomalous. Their anomalous contributions may be computed unambiguously by metric variation of the exact 1PI quantum effective action determined by the conformal anomaly of $\left\langle T^{μν}\right\rangle$ in $d = 4$ curved space. This action implies the existence of massless propagator poles in three and higher point correlators of $T^{μν}$ . The metric variations of the anomaly effective action in its local form in terms of a scalar conformalon field are carried out explicitly for the case of the correlator of three CFT stress-energy tensors, and the result is shown to coincide with the algebraic reconstruction of $\left\langle TTT\right\rangle$ from its transverse, tracefree parts, determined independently by the solution of the CWIs in d dimensional flat space in the momentum representation. This demonstrates that the specific analytic structure and massless poles predicted by the general curved space anomaly effective action are in fact a necessary feature of the exact solution of the anomalous CWIs in any $d = 4$ CFT.