Scattering amplitudes in high-energy limit of projectable Horava gravity

We study the high-energy limit of projectable Ho\v rava gravity using on-shell graviton scattering amplitudes. We compute the tree-level amplitudes using symbolic computer algebra and analyze their properties in the case of collisions with zero total momentum. The amplitudes grow with collision energy in the way consistent with tree-level unitarity. We discuss their angular dependence and derive the expression for the differential cross section that happens to depend only on the essential combinations of the couplings. One of our key results is that the amplitudes for arbitrary kinematics are finite when the coupling $\lambda$ in the kinetic Lagrangian is taken to infinity -- the value corresponding to candidate asymptotically free ultraviolet fixed points of the theory. We formulate a modified action which reproduces the same amplitudes and is directly applicable at $\lambda=\infty$, thereby establishing that the limit $\lambda\to\infty$ of projectable Ho\v rava gravity is regular. As an auxiliary result, we derive the generalized Ward identities for the amplitudes in non-relativistic gauge theories.