We present a one loop calculation in the context of Hořava-Lifshitz gravity. Due to the complexity of the calculation in the full theory we focus here on the study of a toy model, namely the conformal reduction of the z = 2 projectable theory in 2 + 1 dimensions. For this value of the dimension there are no gravitons, hence the conformal mode is the only physical degree of freedom, and thus we expect our toy model to lead to qualitatively correct answers regarding the perturbative renormalization of the full theory. We find that Newton's constant (dimensionless in Hořava-Lifshitz gravity) is asymptotically free. However, the DeWitt supermetric approaches its Weyl invariant form with the same speed and the effective interaction coupling remains constant along the flow. In other words, the would-be asymptotic freedom associated to the running Newton's constant is exactly balanced by the strong coupling of the scalar mode as the Weyl invariant limit is approached. We conclude that in such model the UV limit is singular at one loop order, and we argue that a similar phenomenon can be expected in the full theory, even in higher dimensions.

The scaling properties of quantum gravity are discussed by employing a class of proper-time regulators in the functional flow equation for the conformal factor within the formalism of the background field method. Renormalization group trajectories obtained by projecting the flow on a flat topology are more stable than those obtained from a projection on a spherical topology. In the latter case the ultraviolet flow can be characterized by a Hopf bifurcation with an ultraviolet attractive limiting cycle. Although the possibility of determining the infrared flow for an extended theory space can be severely hampered due to the conformal factor instability, we present a robust numerical approach to study the flow structure around the non-Gaussian fixed point as an inverse problem. In particular, we show that the possibility of having a spontaneous breaking of the diffeomorphism invariance can be realized with nonlocal functionals of the volume operator.

We study the Brans-Dicke theory with arbitrary potential within a functional renormalization group framework. Motivated by the asymptotic safety scenario of quantum gravity and by the well-known relation between f(R) gravity and Brans-Dicke theory at the classical level, we concentrate our analysis on the fixed-point equation for the potential in four dimensions and with Brans-Dicke parameter ω = 0. For two different choices of gauge, we study the resulting equations by examining both the local and global properties of the solutions, by means of analytical and numerical methods. As a result of our analysis we do not find any nontrivial fixed point in one gauge, but we find a continuum of fixed points in the other one. We interpret such an inconsistency as a result of the restriction to ω = 0, and thus we suggest that it indicates a failure of the equivalence between f(R) gravity and Brans-Dicke theory at the quantum level.