Lorentz Symmetry Violation and Very High-Energy Cross Sections

We discuss the implications of a recently proposed pattern of Lorentz symmetry violation on very high-energy cross sections. As a consequence of the breaking of local Lorentz invariance by the introduction of a fundamental length, $a$ , the kinematics is modified and the properties of final states are fundamentally different in collider-like (two incoming particles with equal, opposite momenta with respect to the vacuum rest frame) and fixed-target (one of the incoming particles at rest with respect to the vacuum rest frame) situations. In the first case, the properties of the allowed final states are similar to relativistic kinematics, as long as the relevant wave vectors are much smaller than the critical wave vector scale $a^{-1}$ . But, if one of the incoming particles is close to rest in the vacuum rest frame, energy conservation reduces the final-state phase space at very high energy and can lead to a sharp fall of cross sections starting at incoming-particle wave vectors well below the inverse of the fundamental length. Then, the Froissart bound may cease to be relevant, as total cross sections seem to become much smaller than it would be allowed by local, Lorentz-invariant, field theory. Important experimental implications of the new scenario are found for cosmic-ray astrophysics and for very high-energy cosmic rays reaching the earth.