  We study a QED extension that is unitary, CPT invariant and
super-renormalizable, but violates Lorentz symmetry at high energies, and
contains higher-dimension operators (LVQED). Divergent diagrams are only one-
and two-loop. We compute the one-loop renormalizations at high and low energies
and analyse the relation between them. It emerges that the power-like
divergences of the low-energy theory are multiplied by arbitrary constants,
inherited by the high-energy theory, and therefore can be set to zero at no
cost, bypassing the hierarchy problem.
