Neutrino masses, the μ-term and mathcal{ PSL}₂(7)

Using an $SO(10)$-inspired form for the Dirac neutrino mass, we map the neutrino data to right-handed neutrino Majorana mass-matrix, $\mathcal M$, and investigate a special form with \emph{seesaw} tribimaximal mixing; it predicts a normal hierarchy, and the values of the light neutrino masses. It may be generated by mapping the top quark hierarchy onto the vacuum values of familon fields transforming under the family group $\mathcal{ PSL}_2(7)$. We next investigate the hypothesis that these familons play a dual role, generating a hierarchy in the supersymmetric $\mathbf \mu$-mass matrix of Higgses carrying family quantum numbers. A special $\mathcal{ PSL}_2(7)$ invariant coupling produces a $\mu$-matrix with a hierarchy of thirteen orders of magnitude. Only one Higgs field (per hypercharge sector) is light enough (with a $\mu$-mass $\sim 10-100$ GeV) to be destabilized by SUSY soft breaking at the TeV scale, and upon spontaneous symmetry breaking, gives \emph{tree-level} masses for the heaviest family.