Superluminal Neutrinos at OPERA Confront Pion Decay Kinematics

Violation of Lorentz invariance (VLI) has been suggested as an explanation of the superluminal velocities of muon neutrinos reported by OPERA. In this note we show that the amount of VLI required to explain this result poses severe difficulties with the kinematics of the pion decay, extending its lifetime and reducing the momentum carried away by the neutrinos. We show that the OPERA experiment limits $\alpha = (v_\nu - c)/c < 4 \times 10^{-6}$. We then take recourse to cosmic ray data on the spectrum of muons and neutrinos generated in the earth's atmosphere to provide a stronger bound on VLI: $(v-c)/c < 10^{-12}$.