Proton-Neutron Mass Difference

Incorporating the methods of the preceding paper we examine the electromagnetic contributions to the masses of the nucleons in terms of the graviton-nucleon vertex. Using the methods of sidewise dispersion relations, we consider this vertex as a function of the mass of one of the external nucleons, which enables us to express the contributions to δM=Mₚ-Mₙ in terms of integrals over scattering amplitudes. Applying the hypothesis of threshold dominance, we find that a simple explanation for the observed mass difference δM=-1.3 MeV emerges. In the low-energy region to which we restrict our attention, the only contributing states are the intermediate Nγ and Nπ states. For the photon contributions we find the usual unsuccessful result δM^γ~+0.5 MeV as a consequence of the dominance of the Coulomb over the magnetic energy in the threshold region. However, in calculating the contribution of the nucleon mass shift back on itself from the Nπ states of energy W, there is a term under the dispersion integral proportional to the difference of nucleon pole terms ${({W}²{-}{M}ₙ)}^{{-}1}{-}{({W}²{-}{{M}ₚ}²)}^{{-}1}{}2M{δ}M{({W}²{-}{M}²)}^{{-}2}$, implying a large contribution due to the enhancement at threshold ${W}²{→}{M}²$. Including this contribution we have ${δ}M=+0.5 MeV+1.3{δ}M$ or ${δ}M{}{-}1.7$ MeV, which suggests how the nucleon mass difference emerges in spite of the sign of the photon contribution.