Probes of Lorentz Violation in Neutrino Propagation
  It has been suggested that the interactions of energetic particles with the foamy structure of space-time thought to be generated by quantum-gravitational (QG) effects might violate Lorentz invariance, so that they do not propagate at a universal speed of light. We consider the limits that may be set on a linear or quadratic violation of Lorentz invariance in the propagation of energetic neutrinos, v/c=[1 +- (E/M_\nuQG1)] or [1 +- (E/M_\nu QG2}^2], using data from supernova explosions and the OPERA long-baseline neutrino experiment. Using the SN1987a neutrino data from the Kamioka II, IMB and Baksan experiments, we set the limits M_\nuQG1 > 2.7(2.5)x10^10 GeV for subluminal (superluminal) propagation, respectively, and M_\nuQG2 >4.6(4.1)x10^4 GeV at the 95% confidence level. A future galactic supernova at a distance of 10 kpc would have sensitivity to M_\nuQG1 > 2(4)x10^11 GeV for subluminal (superluminal) propagation, respectively, and M_\nuQG2 > 2(4)x10^5 GeV. With the current CNGS extraction spill length of 10.5 micro seconds and with standard clock synchronization techniques, the sensitivity of the OPERA experiment would reach M_\nuQG1 ~ 7x10^5 GeV (M_\nuQG2 ~ 8x10^3 GeV) after 5 years of nominal running. If the time structure of the SPS RF bunches within the extracted CNGS spills could be exploited, these figures would be significantly improved to M_\nuQG1 ~ 5x10^7 GeV (M_\nuQG2 ~ 4x10^4 GeV). These results can be improved further if similar time resolution can be achieved with neutrino events occurring in the rock upstream of the OPERA detector: we find potential sensitivities to M_\nuQG1 ~ 4x10^8 GeV and M_\nuQG2 ~ 7x10^5 GeV. 
John Ellis, Nicholas Harries, Anselmo Meregaglia, Andre Rubbia,   Alexander Sakharov