One-loop renormalization in a toy model of Hořava-Lifshitz gravity

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.