Lorentz violation in Hořava-Lifshitz-type theories

We show that coupling the standard model to a Lorentz symmetry-violating sector may coexist with viable phenomenology provided that the interaction between the two is mediated by higher-dimensional operators. In particular, if the new sector acquires anisotropic-scaling behavior above a “Hořava-Lifshitz” energy scale ${\Lambda }_{\mathrm{HL}}$ and couples to the standard model through interactions suppressed by $M_{\mathrm{pl}}$, the transmission of the Lorentz violation into the standard model is protected by the ratio ${\Lambda }_{\mathrm{HL}}^2/M_{\mathrm{pl}}^2$. A wide-scale separation ${\Lambda }_{\mathrm{HL}}\ll M_{\mathrm{pl}}$ can then make Lorentz-violating terms in the standard model sector within experimental bounds without fine-tuning. We first illustrate our point with a toy example of Lifshitz-type neutral fermion coupled to photon via the magnetic moment operator, and then implement similar proposal for the Hořava-Lifshitz gravity coupled to conventional Lorentz-symmetric matter fields. We find that most radiatively induced Lorentz violation can be controlled by a large-scale separation, but the existence of instantaneously propagating non-Lifshitz modes in gravity can cause a certain class of diagrams to remain quadratically divergent above ${\Lambda }_{\mathrm{HL}}$. Such problematic quadratic divergence however can be removed by extending the action with terms of higher Lifshitz-dimension, resulting in a completely consistent setup that can cope with the stringent tests of Lorentz invariance.