Pseudo-Goldstone pions in SU3models

Motivated by the masslessness of the pseudoscalar nonet in the ${\mathrm{U}}_3$ generalization of the Bars-Lane model, we develop general principles for pseudo-Goldstone-Boson models of this type, and show (i) that the Bars-Lane model is indeed the most economical that can be constructed and (ii) that the unlocking problem for models involving S${\mathrm{U}}_3$ or higher groups is quite general, and unavoidable. As for the mysterious disappearance of the pion mass, we show by one-loop calculation that the mass becomes nonzero, irrespective of the gauge coupling, i.e., that there is no higher symmetry, and also indicate a quick method of determining which fields will be massless in lowest order. A serious problem arises when the $\eta$ and $X^0$ are shown to remain massless with the $\pi$, while the $K$ acquires a mass, once the vacuum symmetry is broken to S${\mathrm{U}}_2$ ⊗ ${\mathrm{U}}_1$; although it is possible to generate a mass for the $X^0$, it appears very unlikely that the $\eta -\pi$ degeneracy in lowest order can be removed in a realistic model based on these lines.