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A recent proposal to associate 60~TeV to 2~PeV IceCube neutrino events withgamma-ray bursts~(GRBs) indicates the Lorentz violation of cosmic neutrinos andleads further to the $CPT$ symmetry violation between neutrinos andantineutrinos. Here we find that another 12 northern hemisphere track eventspossibly correlated with GRBs from three-year IceCube data satisfy the sameregularity at a lower energy scale around 1~TeV. The combined fitting indicatesa Lorentz violation scale ${E}_{\rm LV}=(6.4\pm 1.5)\times10^{17}~{ \rm GeV}$and an intrinsic time difference ${\Delta {t}_{\rm in}=(-2.8\pm0.7)\times10^2~{\rm s}}$, from which we find an earlier emission of neutrinosthan photons at the GRB source. We also suggest analyzing neutrino eventsdetected a few minutes before the GRB trigger time to test the $CPT$ violationof ultrahigh-energy neutrinos.

Consistent Lorentz violation features from near-TeV IceCube neutrinos