Do quantified flavor neutrino masses exist?

Mass-squared differences and neutrino mixing angles can be determined from neutrino oscillations. Three neutrino masses m_i (i = 1, 2, 3) corresponding to three mass eigenstates can all be calculated, when one additional relation is available. The so-called geometric mean mass relation may serve as such.

Alternatively, when the neutrino mass m_1 is known, the other two masses can be calculated from measured mass-squared differences. As an example, a mass m_1 of 1.530 meV/c^2 has previously been obtained by combination of the magnetic moment of a massive Dirac neutrino deduced in the context of the electroweak interaction at the one-loop level and the so-called Wilson-Blackett law. Curiously, about the same result for m_1 is found from the geometric mean mass relation.

Furthermore, the existence of neutrino masses mα (α = e, μ, τ) corresponding to the flavor eigenstates will be conjectured. They are defined by the product of the unitary PMNS mixing matrix and the column vector with masses mi as components. For a zero Dirac CP phase δ, the masses m_α appear to be approximately quantized, where the ratio of the three masses is equal to: ⅿ_e : m_μ : m_τ = 1.00 : 3.02 : 2.03. The mass set mα appears to possess a higher total energy than the mass set m_i.

Finally, the toroidal model of neutrinos is applied to all neutrino masses. It appears that neutrinos with masses m_i (i = 2, 3) and m_α (α = e, μ, τ) can all be described by the same spindle torus model. In addition, expressions for the radii and angular momenta of all neutrinos are deduced. The obtained parameters all depend on simple functions of the toroidal factor N.