Lorentz violation and ultrahigh-energy photons

The propagation of photons, electrons and positrons at ultrahigh energies above $\sim {10}^{19}\mathrm{eV}$ can be changed considerably if the dispersion relations of these particles are modified by terms suppressed by powers of the Planck scale. We recently pointed out that the current nonobservation of photons in the ultrahigh-energy cosmic ray flux at such energies can put strong constraints on such modified dispersion relations. In the present work we generalize these constraints to all three Lorentz invariance-breaking parameters that can occur in the dispersion relations for photons, electrons and positrons at first- and second-order suppression with the Planck scale. We also show how the excluded regions in these three-dimensional parameter ranges would be extended if ultrahigh-energy photons were detected in the future.

Figure 1

Case $n=1$. Region excluded by present upper limits on the UHE photon flux (${10}^{19}\mathrm{eV}\lesssim \omega \lesssim {10}^{20}\mathrm{eV}$).

Figure 2

Case $n=1$. Region excluded if a ${10}^{19}\mathrm{eV}$ photon were detected (shaded region). The allowed region has the shape of a double cone centered at the origin and opening towards the positive and negative ${\xi }_1$ directions.

Figure 3

Case $n=1$, ${\eta }_1^{+}={\eta }_1^{-}$. Combined constraint using both the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (blue shaded, checkered region) and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (yellow shaded region).

Figure 4

Case $n=1$, ${\eta }_1^{+}={\eta }_1^{-}$. Combined constraint using the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (gray plus blue shaded, checkered regions), in the energy range between ${10}^{19}$ and $5\times {10}^{19}\mathrm{eV}$ (blue region), and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (yellow shaded region).

Figure 5

Case $n=2$. Region excluded by present upper limits on the UHE photon flux (${10}^{19}\mathrm{eV}\lesssim \omega \lesssim {10}^{20}\mathrm{eV}$).

Figure 6

Case $n=2$. Region excluded if a ${10}^{19}\mathrm{eV}$ photon were detected.

Figure 7

Case $n=2$. Combined constraint using both the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (blue shaded, checkered region) and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (uncheckered region).

Figure 8

Case $n=2$, ${\eta }_2^{+}={\eta }_2^{-}$. Combined constraint using both the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (blue shaded, checkered region) and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (yellow shaded region).

Figure 9

Case $n=2$, ${\eta }_2^{+}={\eta }_2^{-}$. Combined constraint using the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (gray plus blue regions), in the energy range between ${10}^{19}$ and $5\times {10}^{19}\mathrm{eV}$ (blue shaded, checkered region), and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (yellow shaded region).

Figure 10

Case $n=2$, ${\eta }_2^{+}=-{\eta }_2^{-}$. Combined constraint using both the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (blue shaded, checkered region) and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (yellow shaded region).

Figure 11

Case $n=2$, ${\eta }_2^{-}=0$. Combined constraint using both the current upper limits on the photon fraction in the energy range between ${10}^{19}$ and ${10}^{20}\mathrm{eV}$ (blue shaded, checkered region) and assuming that a ${10}^{19}\mathrm{eV}$ photon were detected (yellow shaded region).