Perturbative $S$-matrix unitarity ($S^{\dagger}S=1$) in $R_{\mu \nu} ^2$ gravity

We show that in the quadratic curvature theory of gravity, or simply $R_{\mu \nu} ^2$ gravity, the tree-level unitariy bound (tree unitarity) is violated in the UV region but an analog for $S$-matrix unitarity ($SS^{\dagger} = 1$) is satisfied. This theory is renormalizable, and hence the failure of tree unitarity is a counter example of Llewellyn Smith's conjecture on the relation between them. We have recently proposed a new conjecture that $S$-matrix unitarity gives the same conditions as renormalizability. We verify that $S$-matrix unitarity holds in the matter-graviton scattering at tree level in the $R_{\mu \nu} ^2$ gravity, demonstrating our new conjecture.