Superluminal neutrinos in long baseline experiments and SN1987a

Precise tests of Lorentz invariance in neutrinos can be performed using long baseline experiments such as MINOS and OPERA or neutrinos from astrophysical sources. The MINOS collaboration reported a measurement of the muonic neutrino velocities that hints to super-luminal propagation, very recently confirmed at 6 sigma by OPERA. We consider a general parametrisation which goes beyond the usual linear or quadratic violation considered in quantum-gravitational models. We also propose a toy model showing why Lorentz violation can be specific to the neutrino sector and give rise to a generic energy behaviour E^alpha, where alpha is not necessarily an integer number. Supernova bounds and the preferred MINOS and OPERA regions show a tension, due to the absence of shape distortion in the neutrino bunch in the far detector of MINOS. The energy independence of the effect has also been pointed out by the OPERA results.