Model for Lepton, Nucleon, and Pion Interactions

An $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times Y_L$ gauge theory is studied. The purely leptonic part is the Weinberg theory. The purely hadronic part is a form of the SU(2) × SU(2) $\sigma$ model. It is shown, following the suggestion of Weinberg, that with a cross-coupling term the spontaneous breaking of the lepton symmetry induces a spontaneous breaking of the SU(2) × SU(2) hadron symmetry, resulting in a partially-conserved-axial-vector-current condition for axial-vector diver-gences and pions which are Goldstone-Nambu particles in an appropriate limit.