Limiting superluminal electron and neutrino velocities using the 2010 Crab Nebula flare and the IceCube PeV neutrino events

The observation of two PeV-scale neutrino events reported by Ice Cube allows one to place constraints on Lorentz invariance violation (LIV) in the neutrino sector. After first arguing that at least one of the PeV IceCube events was of extragalactic origin, I derive an upper limit for the difference between putative superluminal neutrino and electron velocities of ⩽∼5.6×10-19 in units where c=1, confirming that the observed PeV neutrinos could have reached Earth from extragalactic sources. I further derive a new constraint on the superluminal electron velocity, obtained from the observation of synchrotron radiation from the Crab Nebula flare of September, 2010. The inference that the >1GeV γ-rays from synchrotron emission in the flare were produced by electrons of energy up to ∼5.1PeV indicates the non-occurrence of vacuum Ćerenkov radiation by these electrons. This implies a new, strong constraint on superluminal electron velocities δe⩽∼5×10-21. It immediately follows that one then obtains an upper limit on the superluminal neutrino velocity alone of δν⩽∼5.6×10-19, many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are subluminal the constraint on |δe|⩽∼8×10-17, obtained from the Crab Nebula γ-ray spectrum, places a weaker constraint on superluminal neutrino velocity of δν⩽∼8×10-17.