Pion and Kaon Electromagnetic Masses in Chiral Perturbation Theory

We examine the electromagnetic self-energies of the ground-state mesons in chiral perturbation theory about SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) and SU(3) \ifmmode\times\else\texttimes\fi{} SU(3), to leading order in the electromagnetic interaction. If $\ensuremath{\epsilon}$ is a measure of chiral breaking, then the leading-order corrections to the ground-state meson level splitting relative to the symmetry limit is $\ensuremath{\epsilon}\mathrm{ln}\ensuremath{\epsilon}$. For the pion electromagnetic-mass difference expanded around the chiral SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) limit, all terms of $O({{\ensuremath{\mu}}_{\ensuremath{\pi}}}^{2}\mathrm{ln}{{\ensuremath{\mu}}_{\ensuremath{\pi}}}^{2})$ are calculated and are found to give only a 12% correction. For chiral SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) one has Dashen's sum rule ${{\ensuremath{\mu}}_{{\ensuremath{\pi}}^{+}}}^{2}\ensuremath{-}{{\ensuremath{\mu}}_{{\ensuremath{\pi}}^{0}}}^{2}={{\ensuremath{\mu}}_{{K}^{+}}}^{2}\ensuremath{-}{{\ensuremath{\mu}}_{{K}^{0}}}^{2}$ in the symmetry limit. We explicitly calculate the leading correction to the symmetry limit and find that it is larger than the symmetric term, establishing the breakdown of chiral SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) perturbation theory in this application and the assumption on which the Dashen sum rule was based. We conjecture that these remarks are valid in the context of gauge models and suggest that the power-series expansions in $\ensuremath{\alpha}=\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\ensuremath{-}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$ constant can fail in some gauge models, there being terms of the form $\ensuremath{\alpha}\mathrm{ln}\ensuremath{\alpha}$.