The uninvited guest in mixed derivative Ho\v{r}ava Gravity

We revisit the mixed derivative extension of Ho\v{r}ava gravity which was designed to address the naturalness problems of the standard theory in the presence of matter couplings. We consider the minimal theory with mixed derivative terms that contain two spatial and two temporal derivatives. Including all terms compatible with the (modified) scaling rules and the foliation preserving diffeomorphisms, we calculate the dispersion relations of propagating modes. We find that the theory contains four propagating degrees of freedom, as opposed to three in the standard Ho\v{r}ava gravity. The new degree of freedom is another scalar graviton and it is unstable at low energies. Our result brings tension to the Lorentz violation suppression mechanism that relies on separation of scales.