The Emergence of a universal limiting speed

We display several examples of how fields with different limiting velocities (the ”speed of light”)at a high energy scale can nevertheless have a common limiting velocity at low energies due to theeffects of interactions. We evaluate the interplay of the velocities through the self-energy diagramsand use the renormalization group to evolve the system to low energy. The differences normallyvanish only logarithmically, so that an exponentially large energy trajectory is required in order tosatisfy experimental constraints. However, we also display a model in which the running is power-law, which could be more phenomenologically useful. The largest velocity difference should be insystem with the weakest interaction, which suggests that the study of the speed of gravitationalwaves would be the most stringent test of this phenomenon