Constraining 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 can, in principle, allows one to place constraints on Lorentz invariance violation (LIV) in the neutrino sector. After first arguing that at least one of the IceCube events was of extragalactic origin, I derive an upper limit for {\it the difference} between putative superluminal neutrino and electron velocities of $\le \sim 5.6 \times 10^{-19}$ 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 in the Crab Nebula flare of September, 2010. The inference that the $>$ 1 GeV $\gamma$-rays from synchrotron emission in the flare were produced by electrons of energy up to $\sim 5.1$ PeV indicates the non-occurrence of vacuum \'{C}erenkov radiation by these electrons. This implies a new, strong constraint on superluminal electron velocities $\delta_e \le \sim 5 \times 10^{-21}$. It immediately follows that one then obtains an upper limit on the superluminal neutrino velocity {\it alone} of $\delta_{\nu} \le \sim 5.6 \times 10^{-19}$, many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are {\it subluminal} the constraint on $|\delta_e| \le \sim 8 \times 10^{-17}$, obtained from the Crab Nebula $\gamma$-ray spectrum, places a weaker constraint on superluminal neutrino velocity of $\delta_{\nu} \le \sim 8 \times 10^{-17}$.