Calculation of Neutrino Masses for the Moment of their Birth

In the article, the author presented his original explanation for the origin of small masses of mysterious neutrino particles. A formula for neutrino masses was obtained for the moment of their birth, using results for the neutrino charge radius, received by several theoretical groups, and the masses of three types are calculated. The paper assumes that the main contribution to the neutrino mass (2/3) comes from their small electrostatic energy.

It was shown in [1] that when an elementary particle is emitting Higgs virtual bosons in the for_m of spherical waves, this particle creates its confining potential (as a result of the effect of momentum recoil), due to which the mass of the particle is stabilized during its lifetime. Allowance for the confining potential allows, in particular, to calculate the mass ratio for elementary particles e, μ, π⁰, π^±, K⁰, K^± [2] and calculate the neutrino masses of three types ν_е, ν_μ, and ν_τ [1] for the moment of their birth in the decay and the other processes.

It was shown in [3-6] that neutrinos have a complex internal structure as a result of virtual transitions ν_ℓ↔ℓ-+W⁺, ν̃_ℓ↔ℓ++W^-, where the subscript ℓ means e, μ or τ, W – inter_mediate vector bosons, carriers of weak interaction with mass M_w = 80.4 GeV/c².

Taking into account such virtual transitions, in [3-6] it was found that the square of the electromagnetic neutrino radius is:

r²(ν_ℓ)>=(3GF/8π²2^1/2ћc)[(5/3)Lnα+(8/3)Ln(M_w/m_ℓ)+η (1A)

here η = 1-2, GF = 8.95 x 10-44 MeV cm₃ is the constant of weak interaction, α = e²/ћc ≈1/137.

For the mean value η = 1.5, taking into account m_ес² = 0.511 MeV, m_μс²= 105.66 MeV and mτc² = 1777 MeV, it follows from (1A) that the characteristic values of the squares of the neutrino radii are:

r²(νe)> ≈ 3 x 10^-33cm^-2, r²(ν_μ)> ≈1.3 x 10^-33 cm^-2, r²(ν_τ )> ≈ 4.2 x 10^-34 cm^-2 (2A)

Using the alternative formula for charge neutrino radius [7], we have:

r²(ν_ℓ)>=(3GF/8π²2^1/2ћc)[(8/3)Ln(M_w/m_ℓ) + 2] (1B)

and the characteristic values of the squares of the neutrino radii are:

r²(νe)> ≈ 4.1 x 10^-33 cm^-2, r²(ν_μ)> ≈ 2.3 x 10^-33cm^-2, r²(ν_τ )> ≈ 1.4 x 10^-33 cm^-2 (2B)

To deter_mine the neutrino masses in [1], the following assumptions were made:

1. Although neutrinos do not have an electric charge, they have a small electrostatic energy due to the spatial distribution of opposite small electric charges created by virtual pairs (ℓ, W) is different. In this case, the neutrino's electrostatic energy has the value U(ν_ℓ) = δ (ν_ℓ)e²/-r, where r is the electromagnetic radius of the neutrino, δ(ν_ℓ) is an unknown small dimensionless parameter related to the charge distribution in the structure of ν_ℓ.

2. The virtual rest energy of the neutrino consists of a confining potential WS = σ4πr² and an electrostatic energy:

E = σ4πr² + δ(ν_ℓ)e²/-r (3)

3. The quantity σ is the same for all leptons (i.e. neutrinos, e, µ, τ), and pions and kaons.

The energy constant σ was deter_mined earlier in [2] using the neutral pion mass m_o = 134.963 MeV/c²

based on the initial model assumption that the muon, pion, and kaon elementary particles in the stopped state can be represented as resonators for quanta of virtual neutrinos, excited inside the ”elastic” lepton shell:

σ = 4 x 3-7π-3(moc²)3/(ћc)2 (=3.724 x 10²³ Mev/cm^-2) (4)

The neutrino mass could be found by finding the minimum of the virtual energy (3), but since the value of δ (ν_ℓ) is not known, we should use the equation which is obtained by minimizing the virtual energy (3):

m(ν_ℓ)c² = 12σπr_m² = f r_m² (5)

where the coefficient

f = 12σπ = 1.404 x 10²⁵ MeV/cm^-2 (6)

r_m is the value of r, corresponding to the minimum of the rest energy (3).

Substituting the values of r²(ν_ℓ)> from (2A) and (2B) into formula (5) instead of r_m², we find respectively:

m(νe)c² ≈ 4.3 x 10^-2eV, m(ν_μ)c² ≈ 1.9 x 10^-2eV, m(ν_τ )c² ≈ 6 x 10^-3eV (7A)

and m(ν_e)c² ≈ 5.7 x 10^-2eV, m(ν_μ)c² ≈ 3.3 x 10^-2eV, m(ν_τ )c² ≈ 2 x 10^-2eV (7B)

Similar values were found for the base neutrino masses (m₁, m₂, m₃) in [8] based on the experimental results of the Super-Kamiokande neutrino laboratory [9] in the case of supposition of inverse neutrino masses hierarchy:

m₁c² = 0.049 eV, m₂c² = 0.050 eV, m₃c² = 0.0087 eV (8)

For_mula (5) for neutrino masses with allowance for (1A) and (1B) can be transfor_med to the for_m (9A) and (9B) respectively:

m(ν_ℓ) = 3^-5 2^1/2 π^-4 F [(5/3) Lnα + (8/3) Ln (M_w /m_ℓ) + η] m0 (9A)

m(ν_ℓ) = 3^-5 2^1/2 π^-4 F [(8/3) Ln (M_w/m_ℓ) + 2] m₀ (9B)

where the dimensionless small value F = GF (m₀c²)²/(ħc)³ = 2.116 x 10^-7.

Knowing the neutrino masses, we find the values of δ (ν_ℓ):

δ(νe) ≈ 1.10 x 10^-11, δ(ν_μ) ≈ 3.17 x 10^-12, δ(ν_τ) ≈ 5.6 x 10^-13 (10A)

or

δ(νe) ≈ 1.7 x 10^-11, δ(ν_μ) ≈ 7.34 x 10^-12, δ(ν_τ) ≈ 3.5 x 10^-12 (10B)

It should be noted that the sum of neutrino masses for cases (7A), (7B), and (8) is:

S(A)= 0.068 eV/c² , S(B) = 0.11 eV/c² , S(8) = m₁ + m₂ + m₃ = 0.1077 eV/c² (11)

and as can be seen from (11) case (B) is preferable to case (A).

Using 2 experimental equations | (m₂)2 – (m₁)2 | ≅ 7.59 x 10-5 eV2

and |(m₃)2 – (m₂)2|  2.43 x 10-3 eV2 we can add the third equation m₁ + m₂ + m₃ = m(ν_e) + m(ν_μ) + m(ν_τ) = 0.11 eV and resolve this system of 3 equations for m₁, m₂, m₃ using numerical method and a computer. The results will be close to that of Hajdukovich.

The mass values ​​calculated in the article for m(ν_e), m(ν_μ), m(ν_τ) (for the moment of their birth) turned out to be of the same order, as the basic neutrino masses m₁ , m₂, m₃ obtained along with the assumption of an inverse hierarchy of neutrino masses. It should be noted that the results of calculating the neutrino masses using the Bernabeu, et al. formula, turned out to be more satisfactory. Since for the sum m(ν_e) + m(ν_μ) + m(ν_τ) = 0.11 ev/c² is close to the sum m₁ + m₂ + m₃ = 0.1077 ev/c² ± errors, obtained from the article of Hajdukovich.

Supplementary-Information

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