Spontaneous Breakdown of Chiral Symmetry with Linear Realizations for Asymptotic Fields

A method for considering the spontaneous breakdown of chiral symmetry without recourse to a particular Lagrangian model is presented. The physical or asymptotic fields are assigned to linear representations of chiral ${\mathrm{SU}}_3\times {\mathrm{SU}}_3$ together with appropriate $c$-number addition to the scalar fields. This $c$-number addition manifests the spontaneous breakdown of the symmetry. Expansion of the interpolating fields in terms of asymptotic fields is introduced, which, together with the spontaneously broken chiral symmetry and the arbitrariness in the choice of interpolating fields, produces sum rules relating leptonic, semileptonic, and strong-coupling constants of $0^{\pm }$ and $1^{\pm }$ mesons. Results similar to those of current algebra are obtained with some notable differences. Among the most important is the appearance of the soft-meson amplitude as a consequence of our mechanism of spontaneous breakdown. We are thereby led, in an exceedingly simple way, to generalized soft-meson sum rules. In particular, new results are given for the semileptonic decays of the $K$ meson, and generalized relations among strong-coupling constants and leptonic decays of $1^{\pm }$ mesons are derived.