Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics

Recent work has shown that dispersion relations with Planck scale Lorentz violation can produce observable effects at energies many orders of magnitude below the Planck energy M. This opens a window on physics that may reveal quantum gravity phenomena. It has already constrained the possibility of Planck scale Lorentz violation, which is suggested by some approaches to quantum gravity. In this work we carry out a systematic analysis of reaction thresholds, allowing unequal deformation parameters for different particle dispersion relations. The thresholds are found to have some unusual properties compared with standard ones, such as asymmetric momenta for pair creation and upper thresholds. The results are used together with high energy observational data to determine combined constraints. We focus on the case of photons and electrons, using vacuum Čerenkov, photon decay, and photon annihilation processes to determine order unity constraints on the parameters controlling $O(E/M)\mathrm{}$ Lorentz violation. Interesting constraints for protons (with photons or pions) are obtained even at $O\left({\mathrm{}(E/M)\mathrm{}}^2\right),$ using the absence of vacuum Čerenkov and the observed GZK cutoff for ultrahigh energy cosmic rays. A strong Čerenkov limit using atmospheric PeV neutrinos is possible for $O(E/M)\mathrm{}$ deformations provided the rate is high enough. If detected, ultrahigh energy cosmological neutrinos might yield limits at or even beyond $O\left({\mathrm{}(E/M)\mathrm{}}^2\right).$