Constraint on super-luminal neutrinos from vacuum Cerenkov processes

We examine the Cerenkov-like emission of $e^+ e^-$ from muon super-luminal muon neutrinos assuming a quadratic energy dependence of the neutrino velocity arising from Lorentz violating interactions. We find that with the OPERA result for the neutrino-photon velocity difference, the decay length for the process $\nu_\mu\rightarrow \nu_\mu e^+ e^-$ is 17,039 km which is much larger than the OPERA neutrinos path length of 730 km. We also calculate the pion rate for super-luminal outgoing neutrinos, and we find that the deviation of the pion decay length from the standard Lorentz conserving case at the OPERA neutrino energy is 2%. We conclude that if the muon-neutrino velocity has a quadratic energy dependence, then OPERA result is consistent with non-observation of forbidden neutrino decays and large deviations from the standard pion decay lifetime.