Universal relation between $C_{T}$ and the CFT Weyl anomaly

We establish a universal relation between the coefficient $C_T$ of the energy momentum tensor two point function and the coefficient $c$ multiplying the term quadratic in the Weyl tensor in the Weyl anomaly of a generic even dimensional conformal field theory. Our first derivation combines long known holographic results for $C_T$ and for the Weyl anomaly in Einstein bulk gravity with a recently obtained Chern Gauss Bonnet formula for compact Einstein manifolds. This theorem isolates the Weyl squared contribution in the relation between the Euler density and the $Q$ curvature, allowing us to identify the relevant quadratic term unambiguously. We then provide a genuine CFT derivation based on the renormalization group running of the TT correlator with respect to the arbitrary but necessary mass scale $\mu$. Several known examples are revisited to illustrate and validate the general result.