Helicity-changing Decays of Cosmological Relic Neutrinos

In this paper, we examine the possibility that massive neutrinos are unstable due to their invisible decays $\nu^{}_i \to \nu^{}_j + \phi$, where $\nu^{}_i$ and $\nu^{}_j$ (for $i, j = 1, 2, 3$) are any two of neutrino mass eigenstates with masses $m^{}_i > m^{}_j$ and $\phi$ is a massless Nambu-Goldstone boson, and explore the implications for the detection of cosmological relic neutrinos in the present Universe. First, we carry out a complete calculation of neutrino decay rates in the general case where the individual helicities of parent and daughter neutrinos are specified. Then, the invisible decays of cosmological relic neutrinos are studied and their impact on the capture rates on the beta-decaying nuclei (e.g., $\nu^{}_e + {^3{\rm H}} \to {^3{\rm He}} + e^-$) is analyzed. The invisible decays of massive neutrinos could substantially change the capture rates in the PTOLEMY-like experiments when compared to the case of stable neutrinos. In particular, we find that the helicity-changing decays of Dirac neutrinos play an important role whereas those of Majorana neutrinos have no practical effects. However, if a substantial fraction of heavier neutrinos decay into the lightest one, the detection of relic neutrinos will require a much higher energy resolution and thus be even more challenging.