Quantum corrections to Lorentz invariance violating theories: Fine-tuning problem

It is of general agreement that a quantum gravity theory will most probably mean a breakdown of the standard structure of space-time at the Planck-scale. This has motivated the study of Planck-scale Lorentz Invariance Violating (LIV) theories and the search for its observational signals. Yet, it has been recently shown that, in a simple scalar-spinor Yukawa theory, radiative corrections to tree-level Planck-scale LIV theories can induce large Lorentz violations at low energies, in strong contradiction with experiment, unless an unnatural fine-tuning mechanism is present. In this Letter, we show the calculation of the electron self-energy in the framework given by the Myers-Pospelov model for a Lorentz invariance violating QED. We find a contribution that depends on the preferred frame four-velocity which is not Planck-scale suppressed, showing that this model suffers from the same disease. Comparison with Hughes-Drever experiments requires a fine-tuning of 21 orders of magnitude for this model not to disagree with experiment.