Planck-scale deformation of Lorentz symmetry as a solution to the ultrahigh energy cosmic ray and the TeV-photon paradoxes

One of the most puzzling current experimental physics paradoxes is the arrival on Earth of ultrahigh energy cosmic rays (UHECRs) with energies above the Greisen-Zatsepin-Kuzmin threshold $(5\mathrm{}\times {10}^{19}\hspace{0.5em}\mathrm{eV}).\mathrm{}$ Photopion production by cosmic microwave background radiation photons should reduce the energy of these protons below this level. The recent observation of 20 TeV photons from Mk 501 (a BL Lac object at a distance of 150 Mpc) is another somewhat similar paradox. These high energy photons should have disappeared due to pair production with IR background photons. A common feature of these two paradoxes is that they can both be seen as “threshold anomalies”: energies corresponding to an expected threshold (pion production or pair creation) are reached but the threshold is not observed. Several (relatively speculative) models have been proposed for the UHECR paradox. No solution has yet been proposed for the $\mathrm{TeV}-\gamma$ paradox. Remarkably, the single drastic assumption of the violation of ordinary Lorentz invariance would resolve both paradoxes. We present here a formalism for the systematic description of the type of Lorentz-invariance deformation (LID) that could be induced by the nontrivial short-distance structure of space-time, and we show that this formalism is well suited for comparison of experimental data with LID predictions. We use the UHECR and $\mathrm{TeV}-\gamma$ data, as well as upper bounds on time-of-flight differences between photons of different energies, to constrain the parameter space of the LID. A model with only two free parameters, an energy scale and a dimensionless parameter characterizing the functional dependence on the energy scale, is shown to be sufficient to solve both the UHECR and the $\mathrm{TeV}-\gamma$ threshold anomalies while satisfying the time-of-flight bounds. The allowed region of the two-parameter space is relatively small, but, remarkably, it fits perfectly the expectations of the quantum-gravity-motivated space-time models known to support such deformations of Lorentz invariance: an integer value of the dimensionless parameter and a characteristic energy scale constrained to a narrow interval in the neighborhood of the Planck scale.