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 $\delta M=M_p-M_n$ in terms of integrals over scattering amplitudes. Applying the hypothesis of threshold dominance, we find that a simple explanation for the observed mass difference $\delta M=-1.3\mathrm{}$ MeV emerges. In the low-energy region to which we restrict our attention, the only contributing states are the intermediate $N\gamma$ and $N\pi$ states. For the photon contributions we find the usual unsuccessful result $\delta M^{\gamma }\sim +0.5\mathrm{}$ 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 $N\pi$ states of energy $W$, there is a term under the dispersion integral proportional to the difference of nucleon pole terms ${(W^2-M_n)}^{-1\mathrm{}}-{(W^2-{M_p}^2)}^{-1\mathrm{}}\simeq 2M\delta M{(W^2-M^2)}^{-2\mathrm{}}$, implying a large contribution due to the enhancement at threshold $W^2\to M^2$. Including this contribution we have $\delta M=+0.5\mathrm{} \mathrm{MeV}+1.3\mathrm{}\delta M$ or $\delta M\simeq -1.7\mathrm{}$ MeV, which suggests how the nucleon mass difference emerges in spite of the sign of the photon contribution.