We consider a classical fermion and a classical scalar, propagating on two different kinds of 4-dimensional diffeomorphism breaking gravity backgrounds, and we derive the one-loop effective dispersion relation for matter, after integrating out gravit
ons. One gravity model involves quadratic divergences at one-loop, as in Einstein gravity, and the other model is the $z=3$ non-projectable Horava-Lifshitz gravity, which involves logarithmic divergences only. Although these two models behave differently in the UV, the IR phenomenology for matter fields is comparable: {it(i)} for generic values for the parameters, both models identify $10^{10}$ GeV as the typical characteristic scale above which they are not consistent with current upper bounds on Lorentz symmetry violation; {it(ii)} on the other hand there is always, for both models, a fine-tuning of parameters which allows the cancellation of the indicator for Lorentz symmetry violation.
We show how a mass mixing matrix can be generated dynamically, for two massless fermion flavours coupled to a Lorentz invariance violating (LIV) gauge field. The LIV features play the role of a regulator for the gap equations, and the non-analytic de
pendence of the dynamical masses, as functions of the gauge coupling, allows to consider the limit where the LIV gauge field eventually decouples from the fermions. Lorentz invariance is then recovered, to describe the oscillation between two free fermion flavours, and we check that the finite dynamical masses are the only effects of the original LIV theory.
