Why are the tau neutrino and mu neutrino stable? For example, why can't we have this decay?
$$\nu_\tau \rightarrow \nu_e+\gamma$$
This doesn't seem to be forbidden by conservation of energy-momentum, charge, angular momentum, or lepton number.
[EDIT] WP has some relevant info here: https://en.wikipedia.org/wiki/Lepton_number#Violations_of_the_lepton_number_conservation_laws
Lepton flavor is only approximately conserved, and is notably not conserved in neutrino oscillation. However, total lepton number is still conserved in the Standard Model.
This leaves me unclear on whether the mu and tau neutrinos are in fact unstable in the standard model, but just with a very long lifetime.
There is also some discussion here: Rothstein, "What Do We Know About the Tau Neutrino?," https://arxiv.org/abs/hep-ph/9506443 . In eq. (8) he discusses the process I asked about, along with $\nu_\tau\rightarrow3\nu_e$ and decay into an electron neutrino plus a Higgs. Not being a particle physicist, I'm not having much luck extracting the fundamentals from this paper. It sounds like perhaps the standard model allows these decays, but with a very long lifetime, while modifications of the standard model might allow them to go a lot faster...?