Absolute values of three neutrino masses from atmospheric mixing and an ansatz for the mixing-matrix elements

Using data from atmospheric neutrino mixing, and a simple functional form for mixing angles, the absolute values of three neutrino masses are calculated: m3 = 5.37 × 10 −2 eV, m2 = 1.94 × 10 −2 eV, m1 = 1.46 × 10 −2 eV. The quantities relevant for solar neutrino mixing are calculated: (m22 −m 2 1) = 1.63 × 10 eV, with non-maximal mixing tan θsol = 0.56. The analysis gives a suggestion of a dynamical origin for the empirical, large CP-violating phase associated with an intrinsically, very small mixing angle in the quark sector. With the experimental determination [1, 2, 3] of two differences between squared masses for neutrinos, and nearly maximal mixing [1, 4, 5] for atmospheric neutrinos, the matter of of the absolute value of the neutrino masses is of importance for many physical processes [6]. The purpose of this note is to use the atmospheric measurements [1] ∆m atm ∼= 2.5 × 10 eV, tan θatm ∼= 1, together with a simple ansatz for the functional form of the main elements of the lepton mixing matrix which is motivated by certain analogies to the quark sector, to determine three absolute values for neutrino masses: m3 ∼= 5.37 × 10 −2 eV, m2 ∼= 1.94 × 10 −2 eV, m1 ∼= 1.46 × 10 −2 eV. These masses are hierarchal, although not as markedly so as the masses of the three, charge (-1/3) quarks. A further essential result following from our calculations is that the second sizable mixing angle, responsible for the disappearence of solar, electron neutrinos is not maximal (i. e. 45◦), rather tan θsol ∼= 0.56. A comparison of the pattern of lepton mixing to that in the quark sector shows that in both cases, there appears to be a single, intrinsically, very small mixing angle (< 10 of either of the other two mixing angles). In the quark mixing matrix, this very small angle is associated with the KobayashiMaskawa [7], CP-violating phase δ13, which as presently determined, may well be large [8] (> ∼ 45 ◦). This can suggest that there may exist some kind of milli-weak interaction which itself grossly violate CP invariance. This interaction dynamically gives rise to a sizable CP-violating phase in the quark CKM matrix. This matrix then phenomenologically describes the “corrected” weak interactions with a third, very small mixing angle associated with the milli-weak strength, and a large CPviolating phase associated with the gross violation of the symmetry. The mixing of three neutrino mass states, m1 < m2 < m3, is described by the