The Dirac nature of neutrinos in the standard model with Σ(81) symmetry

Given the current lack of conclusive evidence on whether neutrinos are Dirac or Majorana particles, we proceed by adopting a Dirac neutrino framework. We then propose a Standard Model (SM) expansion with Σ(81) symmetry. This framework successfully accounts for the mass hierarchy of charged leptons, the smallness and hierarchy of neutrino masses, and the observed neutrino mixing. Our analysis yields the following anticipated regions for the physical quantities. The lightest neutrino mass is m ℓ ∈ (0.00, 55.83) meV for normal ordering (NO), and m ℓ ∈ (3.05, 53.71) meV for inverted ordering (IO). The sum of neutrino mass is mν ∈ (58.90, 187.00) meV for NO, and mν ∈ (103.00, 200.00) meV for IO. The eective neutrino mass in the nuclear beta decay is m β ∈ (8.85, 56.50) meV for NO, and m β ∈ (49.00, 72.70) meV for IO. Furthermore, the estimated Yukawa-like couplings in the lepton sector naturally explain the observed lepton mass hierarchies.