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

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