Energy momentum tensor correlators in ϕ4 theory II: The spin-two sector

We extend the computation of the $C_T$ charge of the 2-point function of the Energy-Momentum Tensor to 4-loops. We show that $C_T$ decomposes into two sectors, the conformal sector, which encodes the value of the central charge at fixed points and an RG-sector that contains logarithmic and constant corrections proportional to the β-function. This latter constitutes the main new result of this work and is inaccessible via CFT methods alone. Furthermore, we demonstrate that $C_T$ satisfies an eigenvalue-like equation analogous to that of the spin-0 charge, as discussed in part I, though with a different in general eigenvalue. Finally we present three possible applications.