Nonlinear Realizations of SU(3)×SU(3) and the Symmetry-Breaking Meson Lagrangian

We show how an octet of pseudoscalar mesons which transform nonlinearly under SU(3) × SU(3) can be converted to the linear realizations ($3,\overline{3}$) and ($\overline{3},3\mathrm{}$). Our results are manifestly covariant with respect to redefinitions of the meson field. We use them to construct an effective meson Lagrangian in which all the symmetry breaking occurs in the mass term and belongs to a single representation {$(p,\overline{p})+(\overline{p},p)$}. From an analysis of $\pi -\pi$ scattering lengths, we find it unlikely that the symmetry-breaking interaction belongs either to the self-adjoint sequence (8, 8), (27, 27),..., or to the triangular one $p=6,10,15,\dots$. There are, however, many other representations, including {$(3,\overline{3})+(\overline{3},3)$}, which cannot be ruled out. We also examine weak currents and find that the $f_{-}$ form factor in $K_{l3\mathrm{}}$ decay gives rise to a serious problem in the symmetry breaking.