Lifshitz scalar fields: One loop renormalization in curved backgrounds

We consider an interacting Lifshitz field with $z=3$ in a curved spacetime. We analyze the renormalizability of the theory for interactions of the form λφⁿ, with arbitrary even n. We compute the running of the coupling constants both in the ultraviolet and infrared regimes. We show that the Lorentz-violating terms generate couplings to the spacetime metric that are not invariant under general coordinate transformations. These couplings are not suppressed by the scale of Lorentz violation and therefore survive at low energies. We point out that in these theories, unless the effective mass of the field is many orders of magnitude below the scale of Lorentz violation, the coupling to the four-dimensional Ricci scalar ξ⁽⁴⁾Rφ² does not receive large quantum corrections ξ1. We argue that quantum corrections involving spatial derivatives of the lapse function (which appear naturally in the so-called healthy extension of the Ho {r} {r}{}ava-Lifshitz theory of gravity) are not generated unless they are already present in the bare Lagrangian.