Seesaw options for three neutrinos

The seesaw mechanism for three neutrinos is discussed, clarifying the situation where the seesaw texture results in three approximately zero mass eigenvalues. The true underlying mechanism is shown to be just the inverse (or linear) seesaw, which explains why there could be large mixing. However, these zeroes cannot occur naturally, unless there is a conserved symmetry, i.e. lepton number L, either global or gauged, which is softly or spontaneously broken at the TeV scale. We discuss in particular the case where the three heavy singlet neutrinos have L=3,-2,-1.