  OPERA has claimed the discovery of superluminal propagation of neutrinos. We
analyze the consistency of this claim with previous tests of special
relativity. We find that reconciling the OPERA measurement with information
from SN1987a and from neutrino oscillations requires stringent conditions. The
superluminal limit velocity of neutrinos must be nearly flavor independent,
must decrease steeply in the low-energy domain, and its energy dependence must
depart from a simple power law. We construct illustrative models that satisfy
these conditions, by introducing Lorentz violation in a sector with light
sterile neutrinos. We point out that, quite generically, electroweak quantum
corrections transfer the information of superluminal neutrino properties into
Lorentz violations in the electron and muon sector, in apparent conflict with
experimental data.
