Pseudo-Goldstone pion masses in a gauge model

Abstract

A ${\mathrm{SU}}_{L}(2)\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$ gauge model of electromagnetic and weak interactions is presented, which has three pseudo-Goldstone bosons identified as pions. We can include strong interactions in the pion masses without generating mass counterterms, if the strong gauge symmetry is U(1) or "color" SU(3). At the one-loop level (without strong interactions) the ${\ensuremath{\pi}}^{0}$ remains massless, while the ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$ pick up masses. From our analysis, the one-loop calculation should be interpreted as the electromagnetic and weak contribution to the pion mass difference. Our result is not too large in magnitude as compared with the results found in other models. The pion mass difference including strong interactions has the same form as in the Weinberg model where pions are not pseudo-Goldstone bosons. We also exhibit formulas for the pion masses including strong interactions, which however we can not evaluate.