Departures from the Eightfold Way. III. Pseudoscalar-Meson Electromagnetic Masses

The leading contributions to the ${\ensuremath{\pi}}^{+}\ensuremath{-}{\ensuremath{\pi}}^{0}$ and ${K}^{+}\ensuremath{-}{K}^{0}$ mass differences are calculated. The contributions of pseudoscalar-meson and vector-meson intermediate states are considered, and the Feynman integrations are performed assuming very general momentum dependence at the vertices. Using form factors having poles at the vector-meson masses, and the unitary-symmetric vector mixing model of Coleman and Schnitzer, we find ${m}_{\ensuremath{\pi}+}\ensuremath{-}{m}_{\ensuremath{\pi}0}=4.9$ MeV and ${{m}_{K}}^{+}\ensuremath{-}{{m}_{K}}^{0}=2.9$ MeV. It is difficult to give a reliable estimate of the errors in these calculations; we believe they are correct to within 1 MeV. The uncertainty lies partly in the determination of the $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\rho}\ensuremath{-}\ensuremath{\pi}$ coupling constant and partly in the dynamical assumptions. When the scalar-meson contribution suggested by Coleman and Glashow is included, the $\ensuremath{\pi}$-meson mass difference is unchanged and the $K$-meson mass difference becomes ${{m}_{K}}^{+}\ensuremath{-}{{m}_{K}}^{0}=\ensuremath{-}1.4$ MeV. Our numerical values have been tabulated without discussion in two earlier papers of this series.