High-energy properties of the graviton scattering in quadratic gravity

We obtain the matter-graviton scattering amplitude in the gravitational theory of quadratic curvature, which has $R_{\mu \nu }^2$ term in the action. Unitarity bound is not satisfied because of the existence of negative norm states, while an analog of unitarity bound for $S$-matrix unitarity holds due to the cancelation among the positive norm states and negative norm ones in the unitarity summation in the optical theorem. The violation of unitarity bound is a counter example of Llewellyn Smith’s conjecture on the relation between tree-level unitarity and renormalizability. We have recently proposed a new conjecture that an analog of the unitarity bound for $S$-matrix unitarity gives the equivalent conditions to those for renormalizability. We show that the gravitational theory of quadratic curvature is a nontrivial example consistent with our conjecture.