Second-class currents and their matrix elements in field theory

Second-class currents constructed from fermion fields are considered in the context of renormalizable field theory. It is shown that such currents can only be constructed if there exist two distinct isomultiplets of fermion fields that have all of the same quantum numbers except isospin. These currents can easily be generated by gauge theories of the weak interactions. However, the requirement that these currents actually contribute to $\beta$ decay places severe constraints on the field theory of the strong interactions. The only theories for which the $\beta$-decay matrix elements are nonzero are (a) those which include fundamental strongly interacting scalar or pseudoscalar fields and (b) gauge theories for which the weak and strong gauge groups do not commute. In the latter case the physical hadron states must not be singlets under the strong gauge group unless the gauge symmetry is broken.