Renormalization group flow of projectable Ho\v{r}ava gravity in (3+1) dimensions

We report a comprehensive numerical study of the renormalization group flow of marginal couplings in $(3+1)$-dimensional projectable Ho\v{r}ava gravity. First, we classify all fixed points of the flow and analyze their stability matrices. We find that some of the stability matrices possess complex eigenvalues and discuss why this does not contradict unitarity. Next, we scan over the renormalization group trajectories emanating from all asymptotically free fixed points. We identify a unique fixed point giving rise to a set of trajectories spanning the whole range of the kinetic coupling $\lambda$ compatible with unitarity. This includes the region $0<\lambda-1\ll 1$ assumed in previous phenomenological applications. The respective trajectories closely follow a single universal trajectory, differing only by the running of the gravitational coupling. The latter exhibits non-monotonic behavior along the flow, vanishing both in the ultraviolet and the infrared limits. The requirement that the theory remains weakly coupled along the renormalization group trajectory implies a natural hierarchy between the scale of Lorentz invariance violation and a much larger value of the Planck mass inferred from low-energy interactions.