Asymptotic Freedom in Hořava-Lifshitz Gravity

We use the Wetterich equation for foliated spacetimes to study the renormalization group flow of projectable Hořava-Lifshitz gravity coupled to $n$ Lifshitz scalars. Using novel results for anisotropic heat kernels, the matter-induced beta functions for the gravitational couplings are computed explicitly. The renormalization group flow exhibits an UV attractive anisotropic Gaussian fixed point where Newton’s constant vanishes and the extra scalar mode decouples. This fixed point ensures that the theory is asymptotically free in the large–$n$ expansion, indicating that projectable Hořava-Lifshitz gravity is perturbatively renormalizable. Notably, the fundamental fixed point action does not obey detailed balance.

Figure 1

$z$ dependence of the fixed point arising from (13). The dashed horizontal line represents the general relativistic case, while the thick vertical line represents the line of Gaussian fixed points. In the shaded area, the extra scalar mode of HL gravity is classically unstable [10].

Figure 2

Scale-dependence of the wave-function renormalizations $G_k$ and ${\lambda }_k$ and the marginal couplings ${\tilde{g}}_i$ in $d=z=3$. The AGFP (19), (21) acts as an UV attractor of the RG flow.