  Current experiments do not exclude the possibility that one or more neutrinos
are very slightly superluminal or that they have a very small tachyonic mass.
Important bounds on the size of a hypothetical tachyonic neutrino mass term are
set by lepton pair Cerenkov radiation (LPCR), i.e., by the decay channel nu ->
e^+ e^- nu which proceeds via a virtual Z0 boson. Here, we use a
Lorentz-invariant dispersion relation which leads to very tight constraints on
the tachyonic mass of neutrinos; we also calculate decay and energy loss rates.
A possible cutoff seen in the IceCube neutrino spectrum for E_nu > 2 PeV, due
to the potential onset of LPCR, is discussed.
