The basic idea of constraining Lorentz symmetry violation by considering properties of the highest energy cosmic rays is due to Coleman and Glashow (see my longer discussion). They barely treated the case where the cosmic ray is a proton, however, because protons are complicated composite objects which makes the analysis involved. It turns out it also makes the analysis fruitful, though.

The highest energy cosmic rays are probably hadrons (see the review article on high energy cosmic rays) and the highest energy cosmic ray observed, with an energy of about 300 000 000 000 GeV, was probably a hadron and not a photon. The arrival of such a cosmic ray from an astrophysical distance tells us that some type of hadron is stable to vacuum decay, via strong or electroweak interactions, when it has such a large energy. The fact that such a cosmic ray was accelerated, presumably by ordinary electromagnetic fields, implies that they were charged particles at the time of the acceleration, which means that a charged hadron must be stable on a time scale of at least minutes with an energy of the highest energy cosmic rays observed.

In our paper, Gagnon and I show that the right way to determine the propagation speed of a particle, when different species (such as quarks, gluons, leptons, and electroweak gauge bosons) have different limiting velocities, is as a weighted average over the constituents, with the weighting determined by the momentum weighted parton distribution functions. When higher dimension operators cause dispersion relations to be nonlinear in a particle's momentum, this is accounted for in a composite particle by parton distribution functions weighted by the appropriate power of Bjorken x. We wrote a computer program to evolve the parton distribution functions, including all electroweak interactions relevant at leading order as well as QCD at NLO, to an arbitrary scale. The code is available here, and an example input file for the code is available here. With these in hand, and using initial conditions determined from the publicly available CTEQ and MRST parton distribution function generators, we determined the PDF's at the relevant scale (which turns out to be set by the particle energy in the Earth's rest frame, for a typical Lorentz violating situation). Enforcing the condition that a charged hadron be stable at the observed energy then places a number of severe constraints on possible Lorentz violating operators, as detailed in our paper. In particular, even CPT conserving, dimension 6 operators are constrained to arise at a scale at least 100 to 1000 times higher than the scale where quantum gravitational effects must be important. This makes it difficult to see how Lorentz invariance could be broken by quantum gravity without violating these constraints.