Asymptotic freedom in (3+1)-dimensional projectable Hořava gravity

We derive the full set of beta functions for the marginal essential couplings of projectable Hořava gravity in (3+1)-dimensional spacetime. To this end we compute the divergent part of the one-loop effective action in static background with an arbitrary spatial metric. The computation is done in several steps: reduction of the problem to three dimensions, extraction of an operator square root from the spatial part of the fluctuation operator, and evaluation of its trace using the method of universal functional traces. This provides us with the renormalization of couplings in the potential part of the action which we combine with the results for the kinetic part obtained previously. In the next step, we investigate the renormalization group flow of the theory. The flow possesses a number of asymptotically free fixed points. We find a family of trajectories connecting one of these fixed points in the ultraviolet to the region of the parameter space where the kinetic term of the theory acquires the general relativistic form. The gravitational coupling exhibits nonmonotonic behavior along the flow, vanishing both in the ultraviolet and the infrared.