Nature of Intrinsic Breakdown of Chiral Symmetry in Terms of Physical Fields

A method for considering the intrinsic breakdown of chiral $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ symmetry is presented. The method is principally based on the knowledge of the physical masses of the particles whose fields transform linearly under the symmetry group. The exact Hamiltonian of asymptotic fields for the parity couples ($0^{+}$,$0^{-}$) and ($1^{+}$,$1^{-}$) is considered, and the nature of the resulting transformations is described. It is shown that the transformation does differ appreciably from the simple (3,$3^{*}$) form, which has been recently assumed. It is found that the representations which include subgroups of $\mathrm{SU}\left(3\right)\mathrm{}$ beyond the singlet and octet are restricted by the physical masses but cannot all simultaneously be made to vanish. It is shown that a universal condition on the fields considered exists, which requires the vanishing of the commutator between the axial charge and the divergence of the $\mathrm{SU}\left(3\right)\mathrm{}$ vector generator for those specific cases in which $\Delta I>\mathrm{}1\mathrm{}$ and/or $\Delta Y>\mathrm{}1\mathrm{}$. This condition provides mass formulas among the scalar and pseudoscalar particles and among the vector and axial-vector particles which are in good agreement with experiment.