Relation Between Nonlinear and Linear Realizations of SU(3)×SU(3): Theory and Applications

Abstract

In an earlier paper we took the most general, nonlinear form for the action [${K}_{a},{\ensuremath{\pi}}_{b}$] of chiral operators upon an octet of pseudoscalar-meson fields, and we developed functions ${z}_{\ensuremath{\alpha}}$ and ${\overline{z}}_{\ensuremath{\beta}}$ ($\ensuremath{\alpha},\ensuremath{\beta}=0, 1, 2,\dots{}, 8$) which span the (3, \ifmmode\bar\else\textasciimacron\fi{}3) and (\ifmmode\bar\else\textasciimacron\fi{}3, 3) representations of chiral SU(3) symmetry. Here we use the nonlinear properties of ${z}_{\ensuremath{\alpha}}$ to show that the 3\ifmmode\times\else\texttimes\fi{}3 matrix ($\ensuremath{\Sigma}{0}^{8}{\ensuremath{\lambda}}_{\ensuremath{\alpha}}{z}_{\ensuremath{\alpha}}$) is proportional to a unitary, unimodular matrix exp ($i\ensuremath{\Sigma}{1}^{8}{\ensuremath{\omega}}_{k}{\ensuremath{\lambda}}_{k}$). We then find that the matrix $\ensuremath{\Lambda}$ which converts an arbitrary nonlinear field into a linear realization of SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) is a pure chiral transformation with ${\ensuremath{\omega}}_{k}$ as its parameters, $\ensuremath{\Lambda}=\mathrm{exp}(i\ensuremath{\Sigma}{1}^{8}{\ensuremath{\omega}}_{k}{X}_{k})$. This result enables us to demonstrate the equivalence of different approaches to the theory of nonlinear realizations, and to construct a model for meson-baryon scattering. In the model, chiral symmetry is broken by a mass term which transforms as an admixture of singlet and octet members of the ($m,\overline{m}$) representation of SU(3) \ifmmode\times\else\texttimes\fi{} SU(3). There are thirty parameters in the most general symmetry-breaking Lagrangian, but this number can be reduced to eight with reasonable assumptions. Unfortunately this is still too large a number for us to learn anything definitive about the manner of chiral symmetry breaking.