Testing the equivalence principle and Lorentz invariance with PeV neutrinos from blazar flares

It was recently proposed that a giant flare of the blazar PKS B1424-418 at redshift $z=1.522$ is in association with a PeV-energy neutrino event detected by IceCube. Based on this association we here suggest that the flight time difference between the PeV neutrino and gamma-ray photons from blazar flares can be used to constrain the violations of equivalence principle (EP) and the Lorentz invariance for neutrinos. From the calculated Shapiro delay due to clusters or superclusters in the nearby universe, we find that violation of the equivalence principle for neutrinos and photons is constrained to an accuracy of at least $10^{-5}$, which is two orders of magnitude tighter than the constraint placed by MeV neutrinos from supernova 1987A. Lorentz invariance violation (LIV) arises in various quantum-gravity theories, which predicts an energy-dependent velocity of propagation in vacuum for photons and neutrinos. We find that the association of the PeV neutrino with the gamma-ray outburst set limits on the energy scale of possible LIV to $> 0.01 E_{pl}$ for linear LIV models and $>6\times 10^{-8} E_{pl}$ for quadratic order LIV models, where $E_{pl}$ is the Planck energy scale. These are the most stringent constraints on neutrino LIV for subluminal neutrinos.