Simple way to calculate neutrino masses

The formula of neutrino masses was received on the basis of simple physical assumptions, and neutrino masses of 3 types were calculated for the moment of their birth

It was shown in [1] that when an elementary particle is emitting Higgs virtual bosons in the form of spherical waves, this particle creates its own confining potential, as a result of the impulse 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 νе, νμ, and ντ [1,3] for the moment of their birth in the decay and the other processes.

It was shown in [4-7] that neutrinos have a complex internal structure as a result of virtual transitions ${\nu }_{\ell }$ ↔ $\ell$ ^- +W⁺, ${\tilde{\nu }}_{\ell }$ ↔ $\ell$ ⁺ +W^-, where the subscript $\ell$ means e, μ or τ, W - intermediate vector bosons, carriers of weak interaction with mass M_w = 80.4 GeV/c² [8]. Taking into account such virtual transitions, in [4-7] it was found that the square of the electromagnetic neutrino radius is:

2( ${\nu }_{\ell }$ )>=(3GF/8 $\sqrt{2}$ π²ћc)[(5/3)Lnα+(8/3)Ln(M_w/ $m_{\ell }$ )+η] (1)

where GF = 8.95 x 10^-44 Mev cm3 is the constant of weak interaction, α =e²/ћc $\cong$ 1/137, the numerical constant η $\cong$ 1 $÷$ 2. For the mean value η = 1.5, taking into account m_ес² = 0.511 Mev, m_μс²= 105.66 MeV and m_τc² = 1777 MeV, it follows from (1) that the characteristic values ​​of the squares of the neutrino radii are:

2(ν_e)> $\cong$ 3.10^-33cm² , 2(ν_μ)> $\cong$ 1.3.10^-33 cm², 2(ν_τ )> $\cong$ 4.2.10^-34 cm² (2)

Naturally, we must refer these values 2(ν_ℓ)> to the moment of birth of the corresponding neutrinos, while they do not change their status in their move.

To determine the neutrino masses in [1,3], the following assumptions were made:

• Although neutrinos do not have an electric charge, they apparently have a small electrostatic energy, due to that the spatial distribution of opposite small electric charges,created by virtual pairs ( $\ell$ , W) in neutrino structure, is different. In this case, the neutrino's electrostatic energy has the value U( ${\nu }_{\ell }$ $\delta$ )=( ${\nu }_{\ell }$ )e2/r , where r is the electromagnetic radius of the neutrino, $\delta$ ( ${\nu }_{\ell }$ ) is an unknown small dimensionless parameter related to the charge distribution in the structure of .
• The virtual rest energy of the neutrino consists of a confining potential W_S= s4πr2 and an electrostatic energy:

E = σ4πr² + $\delta$ ( ${\nu }_{\ell }$ )e²/r (3)

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

The energy constant σ was determined earlier in [2] using the neutral pion mass mo = 134.963 MeV / c² on the basis of 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 spherical lepton shell with “surface tension” σ and radius R. The number and type of these quanta are determined from the decay scheme of these elementary particles: 2- for the muon , 3- for the pion and 21 minimal quanta- for the kaon. As shown in [1,2], this assumption implies that the virtual rest energy of a neutral pion is written as

E = σ4πR² + 4.5ћπc/R (4)

where the value ћπc/R is the energy of one quantum of virtual neutrino in the resonator.

After minimizing the virtual energy E by R , we obtain the equation

m0 c² = 3πσ^1/3(4.5ћс)^2/3 (5)

and the value of σ is

σ =4 $\times$ 3^-7π^-3(m_oc²)³/(ћc)² = 3.72410 $\times$ ²³ Mev/cm² (6)

The neutrino mass could be found by finding the minimum of the virtual energy (3), but since the value of $\delta$ ( ${\nu }_{\ell }$ ) is not known, we should use equation

m( ${\nu }_{\ell }$ )c²= 12σπ r_m² = f r_m² (7)

which is obtained by minimizing the virtual energy (3), where the coefficient f = 12σπ ≅ 1.404.10²⁵ MeV / cm² , r_m is the value of r corresponding to the minimum of the rest energy (3).

Substituting the values ​​of 2( ${\nu }_{\ell }$ )> from (2) into formula (7) instead of r_m² , we find:

m(ν_e)c² $\cong$ 4.310 ^-2eV, m(νμ)c² $\cong$ 210 ^-2eV, m(ντ )c² $\cong$ 6.10^-3eV (8)

Similar values ​​were found for the base neutrino masses (ν1, ν2, ν3) in [9] on the basis of the experimental results of the Super-Kamiokande neutrino Laboratory [10] in the case of supposition of inverse neutrino masses hierarchy :

m1c² = 0.049 eV, m^{2c2 = 0.050 eV, m3c2 = 0.0087 eV (9)}

Formula (7) for neutrino masses with allowance for (1) can be transformed to the form:

m( ${\nu }_{\ell }$ ) = 3-5 $\sqrt{2}$ π^-4 F [(5/3) Lnα + (8/3) Ln (M_w / $m_{\ell }$ ) + η] m0 (10)

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

Using the alternative formula for neutrino charge radius [11]

2( ${\nu }_{\ell }$ )> =(GF/4 $\sqrt{2}$ π²ћc)[3+ 4Ln(M_w/ $m_{\ell }$ )] (11)

we will get masses of the similar order but somewhat larger :

m(ν_e)c² $\cong$ 5.810 ^-2eV, m(ν_μ)c² $\cong$ 3.410 ^-2eV , m(ν_τ )c² $\cong$ 2.1 $\times$ 10 ^-2eV (12)

Knowing the neutrino masses (8), we find the values of $\delta$ ( ${\nu }_{\ell }$ ):

$\delta$ ( ${\nu }_e$ ) $\cong$ 1.10 $\times$ 10^-11 , $\delta$ ( ${\nu }_{\mu }$ ) $\cong$ 3.17 $\times$ 10^-12 , $\delta$ ( ${\nu }_{\tau }$ ) $\cong$ 5.6 $\times$ 10^-13 (13)

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