Neutrinos may acquire small Dirac or Majorana masses by new low-energy physics in terms of the chiral gravitational anomaly, as proposed by Dvali and Funcke (2016). This model predicts fast neutrino decays, $ u_ito u_j+phi$ and $ u_itobar{ u}_j+phi$, where the gravi-majorons $phi$ are pseudoscalar Nambu-Goldstone bosons. The final-state neutrino and antineutrino distributions differ depending on the Dirac or Majorana mass of the initial state. This opens a channel for distinguishing these cases, for example in the spectrum of high-energy astrophysical neutrinos. In particular, we put bounds on the neutrino lifetimes in the Majorana case, ${tau_2}/{m_2}> 1.1times 10^{-3}(6.7times 10^{-4})~{rm s/eV}$ and ${tau_3}/{m_3}> 2.2times 10^{-5}(1.3times 10^{-4})~{rm s/eV}$ at 90% CL for hierarchical (degenerate) masses, using data from experiments searching for antineutrino appearance from the Sun.
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