Nonlinear effective Lagrangian for pseudoscalar mesons in broken SU(3)×SU(3)

A symmetry-breaking term with derivative coupling $\mathrm{Tr}\left[\right(a^{\prime }+b^{\prime }{\lambda }_8+c^{\prime }{\lambda }_3)M{\partial }_{\mu }M{\partial }_{\mu }{M}^{†}+\mathrm{H}.\mathrm{c}.\mathrm{}]$ originating from the $(3,\overline{3})+(\overline{3},3)$ quark mass term is added to the minimal nonlinear meson Lagrangian for a unified description of SU(3)×SU(3) breaking. These effects arise from a rescaling of the meson fields which gives automatically the $K_{l3\mathrm{}}$ Callan-Treiman relation. SU(3) breaking for $\frac{f_K}{f_{\pi }}$ and $\frac{f_{{\eta }_8}}{f_{\pi }}$ is understood to come from the current quark mass $m_s$. Using this result, we find $\Gamma (\eta \to 2\gamma )=289\mathrm{}\pm 20$ eV for ${\theta }_P=10.5\mathrm{}°$, in good agreement with the measured value 324±50 eV. It is also shown that the $K\to 3\pi$ slope parameters and relative decay rates are unaffected by this symmetry breaking.