  If, as recently reported by the Super-Kamiokande collaboration, the neutrinos
are massive, the heaviest one would not be stable and, though chargeless, could
in particular decay into a lighter neutrino and a photon by quantum loop
effects. The corresponding rate is computed in the standard model with massive
Dirac neutrinos as a function of the neutrino masses and mixing angles. The
lifetime of the decaying neutrino is estimated to be approximately 10^44 years
for a mass 5 10^{-2} eV. If kinematically possible, the decay of a heavy
neutrino into a lighter one plus an e+ e- pair occurs at tree level and its
one-loop radiative corrections get enhanced by a large logarithm of the
electron mass acting as an infrared cutoff. It then largely dominates the
photonic mode by several orders of magnitude, corresponding to a lifetime
approximately equal to 10^{-2} year for a mass 1.1 MeV.
