Stringent constraint on neutrino Lorentz-invariance violation from the two IceCube PeV neutrinos

It has been speculated that Lorentz-invariance violation (LIV) might be generated by quantum-gravity (QG) effects. As a consequence, particles may not travel at the universal speed of light. In particular, superluminal extragalactic neutrinos would rapidly lose energy via the bremssthralung of electron-positron pairs (nu -> nu e+ e-), damping their initial energy into electromagnetic cascades, a figure constrained by Fermi-LAT data. We show that the two cascade neutrino events with energies around 1 PeV recently detected by IceCube -if attributed to extragalactic diffuse events, as it appears likely- can place the strongest bound on LIV in the neutrino sector, namely delta =(v^2-1) < O(10^(-18)), corresponding to a QG scale M_QG ~ 10^5 M_Pl (M_QG >~ 10^(-4) M_Pl) for a linear (quadratic) LIV, at least for models inducing superluminal neutrino effects (delta > 0).