Spontaneous Breakings of Chiral Symmetries. II. Mass Relations and Particle Mixings

A detailed analysis of SU(3) × SU(3) symmetry breaking by a (3, ¯3) + (¯3, 3) linear term is given, with no reference to a Lagrangian model and by carefully taking into account the SU(3) nonivariance of the vacuum. Sum rules are derived and a detailed discussion of the $\kappa$ mass and the $\eta -{\eta }^{\prime }$ mixing angle is given. By carefully handling the delicate problem of approximations, no inconsistency is found with the experimental data, in contrast with results that have appeared in the literature. The over-all picture is instead in very good agreement with experiments, thus showing that SU(3) × SU(3) symmetry breaking is a very precise framework for discussion of elementary-particle symmetries. Corrections to SU(3) results, arising from the SU(3) noninvariance of the vacuum, are discussed. it is found that, in general, neglecting ${\lambda }_8\equiv 〈0\left|u_8\right|0\mathrm{}〉$ is a dangerous approximation. In particular, the value of the $\eta -{\eta }^{\prime }$ mixing angle is shown to depend very strongly on ${\lambda }_8$, a point which has not been realized in the literature. A generalized Gell-Mann-Okubo mass formula taking into account the SU(3) noninvariance of the vacuum is derived and it is found to be in extremely good agreement with experiment, thus showing that SU(3) × SU(3) symmetry breaking may be even more precise than SU(3) symmetry breaking. The high degree of accuracy of the sum rules resolves the ambiguity in the identification of the ninth pseudoscalar meson and provides a very precise determination of the $\eta -{\eta }^{\prime }$ mixing angle.