Another point to add to this thread -- it's kind of a cool idea to think there might be different energy levels for the proton (or neutron). I gather that the idea is that the constituent particles of the proton (currently believed to be quarks) can be in different states of angular momentum (in contrast to intrinsic spin, which presumably is conserved), and together perhaps provide some kind of shell model, comparable to the electron shell model of the atom and the nuclear shell model of the nucleus. In this case there would be a ground state and then different excited states for the proton as a whole.
If a shell-model approach is suitable, perhaps most protons would be in the ground state and then there would be brief periods where some of them are nudged into an excited state, and perhaps a few that are in a longer-lasting metastable state. These states would relax and give off a photon through an immediate or a proximate interaction of some kind. If a quantum system with relaxed and excited states is involved, I doubt that a Gaussian distribution would describe the energies (masses) across the population. Eric
