Universal relations for two- and three-point functions of the CFT energy-momentum tensor and the Weyl anomaly

Abstract

We establish universal relations, valid for a generic conformal field theory in even dimension $d\geqslant6$, between the coefficients $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ of the energy-momentum tensor three-point function and the coefficients of the quadratic and cubic terms in the Weyl tensor in the Weyl anomaly. Alternatively, we can replace the three-point function coefficients with the two-point function coefficient $C_T$ and the two energy flux parameters $t_2$ and $t_4$. Our derivation combines well-known holographic results for $C_T$ with more recent holographic results for $t_2$ and $t_4$, along with a newly obtained Chern--Gauss--Bonnet formula for compact Einstein manifolds. This theorem isolates the Weyl-squared and Weyl-cubed contributions in the relation between the Euler density and the $Q$-curvature, allowing us to identify the relevant quadratic and cubic terms unambiguously. We also provide two genuine CFT derivations for $C_T$ based on the renormalization-group running of the $TT$-correlator with respect to the arbitrary but necessary mass scale $\mu$. We revisit several examples to illustrate and validate the general result. These include free scalars, GJMS operators, and holographic duals of higher-derivative gravities.