Towards the renormalization group flow of Horava gravity in $(3+1)$ dimensions

We compute the renormalization group running of the Newton constant and the parameter $\lambda$ in $(3+1)$-dimensional projectable Horava gravity. We use the background field method expanding around configurations with flat spatial metric, but non-vanishing shift. This allows us to reduce the number of interaction vertices and thereby drastically simplify the calculations. The gauge invariant $\beta$-function of $\lambda$ has two families of zeros, attractive in the infrared and ultraviolet respectively. They are candidates for the fixed points of the full renormalization group flow of the theory, once the $\beta$-functions for the rest of the couplings are added.