Neutron-Proton Mass Difference According to the Bound-State Model

The proton must be heavier than the neutron according to any correct calculation treating them as pionnucleon bound states whose masses are shifted because of one-photon-exchange corrections to the binding forces, and which takes into account no particles other than pions and nucleons. The reason is simply that only the neutron can contain two charged constituents, $p$ and ${\pi }^{-}$, and that both the electric and the magnetic forces between these are attractive, thus binding the neutron more tightly. Dashen's previous calculation of this effect was based on an unreliable variant of the Dashen-Frautschi method for eliminating infrared divergences; the sources of the mistakes in that calculation are pointed out. On the basis of the pion-nucleon bound-state picture, we give a simple and physically well-based estimate of the Coulomb contribution to the mass splitting in terms of the pion and nucleon form factors; as compared with experiment it has the right order of magnitude but necessarily the wrong sign. The magnetic energy is more difficult to estimate, apart from its sign; but it is probably much smaller. We conclude that a consistent calculation, if it is to be successful, must include other baryons and mesons. As a by-product we obtain a simple dynamical interpretation of the fact that the neutron's charge form factor is very small.