Lorentz Violation in the Photon Sector and Ultra-High Energy Cosmic Rays

Lorentz symmetry breaking at very high energies may lead to photon dispersion relations of the form $\omega^2=k^2+\xi_n k^2(k/M_{\rm Pl})^n$ with new terms suppressed by a power $n$ of the Planck mass $M_{\rm Pl}$. We show that first and second order terms of size $\left|\xi_1\right|\gtrsim10^{-14}$ and $\xi_2\lesssim-10^{-6}$, respectively, would lead to a photon component in cosmic rays above $10^{19}\,$eV that should already have been detected. This suggests that LI breaking suppressed up to second order in the Planck scale are unlikely to be phenomenologically viable for photons