Hořava gravity with mixed derivative terms: power counting renormalizability with lower order dispersions

It has been argued that Hořava gravity needs to be extended to include terms that mix spatial and time derivatives in order to avoid unacceptable violations of Lorentz invariance in the matter sector. In an earlier paper we have shown that including such mixed derivative terms generically leads to 4th instead of 6th order dispersion relations and this could be (naïvely) interpreted as a threat to renormalizability. We have also argued that power counting renormalizability is not actually compromised, but instead the simplest power counting renormalizable model is not unitary. In this paper we consider the Lifshitz scalar as a toy theory and we generalize our analysis to include higher order operators. We show that models which are power counting renormalizable and unitary do exist. Our results suggest the existence of a new class of theories that can be thought of as Hořava gravity with mixed derivative terms.