Cubic Weyl gravity does not renormalize dimension-eight matter–curvature operators: a spin-two projection and a conformal zero

The cubic Weyl operator C3 and its parity-odd partner are the only pure-gravity operators at dimension six. We show that at one loop they generate no counterterm for the dimension- eight operators that couple Weyl tensors to the Higgs, by two different mechanisms. With two Weyl tensors, at the leading gravitational power, this holds for any matter state X in C2X and every monomial of Standard Model couplings. The reason is a projection: the off-shell C3 source with two on-shell gravitons is transverse and traceless, by linearized diffeomorphism and Weyl invariance and in any dimension, so it couples only to the spin-two part of the matter stress tensor, which is not renormalized, and the one spin-zero channel that could have carried a new divergence, the curvature coupling ξRH†H, is annihilated by the same projector. We verify this for X = (H†H)2 at order λ2 in the Higgs quartic with the on-shell unitarity method of Baratella et al., where the mixing first requires an internal graviton and a tree amplitude with the same external states already exists, so that the running of its couplings must be subtracted. The seven two-particle cuts are individually nonzero and configuration dependent, and their sum equals the inherited running with nofree parameter, to a relative residual of 10−15. With three Weyl tensors the trace of the source turns on and the spin-zero channel opens. We compute that entry, (H†H)C3, from its four cuts. There the curvature coupling is redundant, because the Weyl rescaling that removes it generates the target operator, so only one combination of ξ and C(H†H)C3 is physical. The cut is proportional to the trace of the matter stress tensor, hence to the distance from conformal coupling, and the invariant com-bination runs multiplicatively with no source. The zero sits at conformal coupling because the only channel that can feed this entry is itself redundant.