Massive and Massless Neutrinos on Unbalanced Seesaws

The observation of neutrino oscillations requires new physics beyond the standard model (SM). A SM-like gauge theory with p lepton families can be extended by introducing q heavy right-handed Majorana neutrinos but preserving its SU(2)L×U(1)Y gauge symmetry. The overall neutrino mass matrix M turns out to be a symmetric (p+q)×(p+q) matrix. Given p > q, the rank of M is in general equal to 2q, corresponding to 2q non-zero mass eigenvalues. The existence of (p − q) massless left-handed Majorana neutrinos is an exact consequence of the model, independent of the usual approximation made in deriving the Type-I seesaw relation between the effective p×p light Majorana neutrino mass matrix Mν and the q×q heavy Majorana neutrino mass matrix MR. In other words, the numbers of massive leftand right-handed neutrinos are fairly matched. A good example to illustrate this “seesaw fair play rule” is the minimal seesaw model with p = 3 and q = 2, in which one massless neutrino sits on the unbalanced seesaw.