Lepton Private Higgs and the discrete group Σ(81)

We use the discrete group Σ(81)≡(Z3×Z3×Z3)⋊Z3 to explore a particular region of parameter space in the Private Higgs model. In doing so we suggest a relation among the off-diagonal entries of the neutrino mass matrix and a possible explanation for the muon magnetic moment anomaly, aμexp−aμSM∼10−9. We predict three new nearly degenerate Higgs doublets with masses of order ∼500GeV to ∼1TeV, and three nearly degenerate SM-singlet TeV-scale neutrinos. The largest scale in the model is ∼10TeV, so there is no severe hierarchy problem. Appendix A is devoted to the group theory of Σ(81).