Generalized current algebra including first- and second-class currents

The total weak $\Delta S=0\mathrm{}$ hadronic current is usually split into $\mathrm{CP}$-odd and $\mathrm{CP}$-even, or first- and second-class currents. It is postulated that these currents obey an (SU(2) ⊗ SU(2)) ⊗ (SU(2) ⊗ SU(2)) algebra. This implies a set of Adler-Weisberger relations which, in particular, indicate the coupling of weak leptonic $\eta \to \pi l\nu$ decay as being of order one. It is possible to consider the strong interactions approximately invariant under the group generated by all these currents by introducing an additional set of scalar Goldstone bosons as well as $C$-odd pseudoscalar mesons. Since the latter seem not to exist, one has an alternative realization by $\mathrm{CP}$ doublets. As a special example, a generalized $\sigma$ model is discussed. Further consequences are partial conservation of the second-class vector current and Goldberger-Treiman-type equations. Finally, the extension of the full algebra to leptons is discussed, which suggests the interesting fact that electrons and muons are $\mathrm{CP}$ partners of each other and the separate conservation of electron and muon numbers is a consequence of $\mathrm{CP}$ conservation.