Two-body decays in deformed relativity

Deformed relativistic kinematics is a framework which captures effects, that are expected from particles and fields propagating on a quantum spacetime, effectively. They are formulated in terms of a modified dispersion relation and a modified momentum conservation equation. In this work we use Finsler geometry to formulate deformed relativistic kinematics in terms of particle velocities. The relation between the Finsler geometric velocity dependent formulation and the original momentum dependent formulation allows us to construct deformed Lorentz transformations between arbitrary frames. Moreover, we find the corresponding compatible momentum conservation equation to first order in the Planck scale deformation of special relativity based on the κ-Poincaré algebra in the bicrossproduct basis. We find that the deformed Lorentz transformations, as well as the deformed time dilation factor, contain terms that scale with the energy of the particle under consideration to the fourth power. We derive how the distributions of decay products are affected when the deformed relativity principle is satisfied and find, for the case of a pion decaying into a neutrino and a muon, that the ratio of expected neutrinos to muons with a certain energy is just slightly modified when compared to the predictions based on special relativity. We also discuss the phenomenological consequences of this framework for cosmic-ray showers in the atmosphere.