Lepton Mass Matrix Models

The smallness and hierarchy in fermion parameters could be the result of selection rules due to an Abelian horizontal symmetry broken by a small parameter. When applied to the lepton sector, then for a large class of models, a number of interesting order of magnitude relations arise: with $i<j$, $m(\nu_i)/m(\nu_j)\sim\sin^2\theta_{ij}$; $m(\ell^-_i)/m(\ell^-_j)\lsim\sin\theta_{ij}$; $m(\nu_i)/m(\nu_j)\gsim m^2(\ell^-_i)/m^2(\ell^-_j)$; $m(\nu_e)\lsim m(\nu_\mu)\lsim m(\nu_\tau)$. The relations between neutrino masses and mixings may become exact if the horizontal symmetry together with holomorphy induce certain zero entries in the lepton mass matrices. A full high energy theory is likely to include scalars with flavor changing couplings and heavy leptons in vector representations; however, the masses of these particles are too heavy to be directly observed in experiment. Indirect evidence for the horizontal symmetry may arise from other sectors of the theory: non-degenerate sleptons are allowed as the symmetry aligns lepton and slepton mass matrices; light leptoquarks are allowed as the symmetry can make their couplings diagonal and chiral.