General treatment of second-class currents in field theory

The general problem of introducing second-class currents into renormalizable field theory is considered. (a) A suitable definition of first- and second-class currents is given for arbitrary integer isospin. (b) The general construction of these currents from fermion and boson fields in renormalizable theories is given. Second-class currents constructed from fermion fields require the existence of two distinct isomultiplets of fermion fields with all of the same quantum numbers except isospin. (c) It is shown that these second-class currents can emerge from gauge theories of the weak interactions. (d) However, the requirement that these currents contribute to $\beta$ decay places severe restrictions on the field theory of the strong interactions. Only two classes of theories allow nonzero $\beta$-decay matrix elements: those which include strongly coupled scalar or pseudoscalar fields, and gauge theories for which the weak and strong gauge groups do not commute. In the latter case the physical hadrons cannot be singlets if the gauge group is not broken. Other topics considered include second-class neutral currents, the divergences of second-class currents, mechanisms such as isospin breaking that might mimic the effects of second-class currents, the phenomenology of first- and second-class currents of arbitrary isospin, the role of time-reversal violation in the weak interactions, and the role of anomalous phases associated with $P$, $C$, and $T$ transformations.