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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 x U(1)_Y gauge symmetry. The overall neutrino mass matrix M turns out to be a symmetric (p+q) x (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 x p light Majorana neutrino mass matrix M_νand the q x q heavy Majorana neutrino mass matrix M_R. In other words, the numbers of massive left- and 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.