Testing Lorentz invariance and CPT symmetry using gamma-ray burst neutrinos

A recent work [Y. Huang and B.-Q. Ma, Commun. Phys. {\bf 1}, 62 (2018)] associated all four PeV neutrinos observed by IceCube to gamma-ray bursts (GRBs), and revealed a regularity which indicates a Lorentz violation scale $E_{\rm LV}=(6.5\pm0.4)\times10^{17}$ GeV with opposite sign factors $s=\pm 1$ between neutrinos and antineutrinos. The association of "time delay" and "time advance" events with neutrinos and antineutrinos (or vice versa) is only a hypothesis since the IceCube detector cannot tell the chirality of the neutrinos, and further experimental tests are needed to verify this hypothesis. We derive the values of the CPT-odd Lorentz violating parameters in the standard-model extension (SME) framework, and perform a threshold analysis on the electron-positron pair emission of the superluminal neutrinos (or antineutrinos). We find that different neutrino/antineutrino propagation properties, suggested by Y. Huang and B.-Q. Ma, can be described in the SME framework with both Lorentz invariance and CPT symmetry violation, but with a threshold energy constraint. A viable way on testing the CPT symmetry violation between neutrinos and antineutrinos is suggested.