Fine-tuning problems in quantum field theory and Lorentz invariance: A   scalar-fermion model with a physical momentum cutoff
  We study the consistency of having Lorentz invariance as a low energy approximation within the quantum field theory framework. A model with a scalar and a fermion field is used to show how a Lorentz invariance violating high momentum scale, a physical cutoff rendering the quantum field theory finite, can be made compatible with a suppression of Lorentz invariance violations at low momenta. The fine tuning required to get this suppression and to have a light scalar particle in the spectrum is determined at one loop. 
J.L. Cortes and J. Lopez-Sarrion
hep-th gr-qc