  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 prefered's 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.
