Polarization of Electrons in Muon Decay with General Parity-Nonconserving Interactions

The polarization of electrons in muon decay is studied in the case of the four-component neutrino theory with general parity-nonconserving interactions. The spectrum for each of the two states of longitudinal polarization is described by a set of three parameters of which only two are independent. By observing the electron polarization in arbitrary directions, one can determine ten parameters of which six are related to the two longitudinal polarizations and the rest to the transverse part. It is seen that the prediction of the two-component neutrino theory about the electron polarization in muon decay is quite specific and can be distinguished from that of the general case, if the latter involves the interactions of the $S$, $T$, $P$ types in charge-retention order. However, from the study of the polarization of electrons, it is impossible to distinguish the two-component theory from the four-component theory which involves only the vector-type interactions. It is noticed that the $\mathrm{TCP}$ theorem determines a detailed correspondence between the polarization dependence of the spectra for the ${\mu }^{+}$ and ${\mu }^{-}$ decays. This provides us with an opportunity to determine experimentally whether weak interactions actually obey the $\mathrm{TCP}$ theorem or not.