How a Non-hierarchical Neutrino Mass Matrix Can Arise

One puzzle of neutrino masses and mixings is that they do not exhibit the kind of strong “hierarchy” that is found for the quarks and charged leptons. Neutrino mass ratios and mixing angles are not small. A possible reason for this is proposed here. It is based on the fact that typical realistic grand unified models contain particles with unification-scale masses which, when integrated out, can yield a neutrino mass matrix that is not of the standard seesaw form. In the standard (or type I) see-saw mechanism [1], the observed lefthanded neutrinos derive their masses from their Dirac couplings to righthanded neutrinos. These right-handed neutrinos have Majorana masses that in grand unified theories (GUTs) are expected to be of order the unified scale MGUT . When the right-handed neutrinos are integrated out, an effective d = 5 operator is induced that produces a mass matrix for the left-handed neutrinos given by the well-known “see-saw formula”: Mν = −MNM −1 R M T N . (1) Here MN is the Dirac mass matrix of the neutrinos, whose elements are less than or of order the Weak scale. MR is the Majorana mass matrix of the right-handed neutrinos. And Mν is the effective mass matrix of the light left-handed neutrinos.