Within the Standard Model (SM), neutrinos are massless, so we need a theory beyond the SM to explain their masses.

In this article, we apply the Scale-Symmetric Theory (SST) [1] to describe the most fundamental problems concerning the neutrinos, i.e. the neutrino-mass problem and the neutrino-oscillation problem.

In SST appear 5 different fundamental scales [1]: the SST tachyons that are the components of the SST Higgs field and gravitational fields, the closed string composed of K 2 = ( 0.7896685548 × 10 10 ) 2 the SST tachyons (a pair of the closed strings, i.e. the spin-1 entanglon, is responsible for the quantum entanglement; the neutrinos are built of the entanglons embedded in the Higgs field), the core of the electron-neutrino and muon-neutrino that are the very stable neutrinos (the tau-neutrino consists of three entangled stable neutrinos), the core of baryons, and the core of the Protoworld that was built of dark matter.

The properties of the tachyons and density of the Higgs field (the 6 initial parameters) combine the first and second scale and lead to the reduced Planck constant ℏ , to the speed of light c , and to the gravitational constant G . On the other hand, the saturation of interactions combines the next four scales while the nuclear strong interactions of the core of baryons lead to the atom-like structure of baryons (there is the core composed of the torus/electric-charge, the central spacetime condensate Y that is responsible for the nuclear weak interactions, and there is one or more relativistic pions on quantized orbits outside the core).

In SST, the stable neutrinos acquire their mass as follows. The very dense initial inflation field as a whole was lefthanded. Due to the rotations of the tachyons (they have infinitesimal spin) in the initial inflation field, created during the inflation, the closed strings have the internal helicities defined by the toroidal and poloidal motions. On the other hand, the entanglons (i.e. the binary systems of the closed strings) that the neutrinos consist of are embedded in the Higgs field, so the internal helicity of the closed strings and the viscosity of them and tachyons cause stable neutrinos to produce a gradient in the Higgs field, i.e. produce the gravitational field. Due to the very high energy densities in entanglons and neutrinos, the symmetry of the left-handedness and right-handedness was not broken. The left-right symmetry was broken at the end of the inflation and is encoded in the matter-antimatter asymmetry, i.e. there are many more baryons with the lefthanded internal helicity.

The stable neutrinos cannot change their flavours directly because of the tremendous non-gravitating energy frozen inside them—there is about 119 orders of magnitude more non-gravitating energy than gravitating one [1]. It explains why the zero-point energy of the “vacuum” suggested by quantum field theory does not cause a large cosmological constant.

The four stable neutrinos (i.e. the electron-neutrino, muon-neutrino and their antiparticles) differ by orientation of the spins of the entanglons on their torus/weak-charge or/and by the internal helicity. It means that their masses, when their spins do not rotate, are the same. We will show that the experimental mass distance between the electron-neutrino and muon-neutrino follows from the fact that transition between them is possible via the tau-neutrino.

Emphasize that the internal helicity defined by the toroidal-poloidal motions also defines the external helicity of the neutrinos (only of neutrinos) that is defined by spin and particle velocity. Lefthanded internal helicity of neutrinos leads to righthanded external helicity—it is showed in Figure 1.

Figure 1. Weak charge of electron-antineutrino with righthanded external helicity and lefthanded internal helicity.

Some properties of selected fermions are presented in Table 1. The electron first of all chooses the electron-antineutrino because the internal helicities and electric-weak charges of electron and electron-antineutrino are opposite. Next the electron chooses the muon-neutrino because the interactions of the opposite electric-weak charges dominate over the interactions of the internal helicities. Next the electron chooses the muon-antineutrino because their internal helicities are opposite. Such is the origin of creation of the negatively charged muon and pion. The electron does not interact with the electron-neutrino because their internal helicities and signs of electric-weak charges are the same. We also see that external helicity of neutrinos is always lefthanded whereas of antineutrinos is always righthanded. Our neutrino model is consistent with all experimental data.

Pressure in the Higgs-field region limited by the weak charge of neutrino is a little lower so the tachyons interacting with it more frequently detach from the weak charge on its inner surface. We see that the neutrino recoil in direction of motion follows from the internal helicity of the neutrinos.

Table 1. Properties of selected fermions.

It is assumed that if neutrinos have masses, the weak eigenstates, ν α , produced in a weak interaction are linear combinations of the mass eigenstates ν i

where α = e , μ , τ are the neutrino flavours, i = 1 , 2 , 3 , the n is the number of neutrino species, and U is the mixing matrix [2].

The m i is the mass of the neutrino mass eigenstate ν i .

In the Standard Model, for the neutrino mass-squared splitting we have [2]

whereas in SST are obligatory two definitions, i.e. the definition in (2) and Δ m i j 2 = ( Δ m i j ) 2 , i.e. there is a resonance.

In baryons, for the nuclear weak interactions is responsible the Y = 424.12176   MeV SST-absolute-spacetime condensate that is composed of the non-rotating-spin-1 neutrino-antineutrino pairs—they are the carriers of the elementary photons, they are the elementary dark photons. Such a condensate is located in the centre of the core of the baryons [1].

In [1], we showed that nature tries to replicate certain behaviors on different scales, so we can assume that the Y condensate consists of K 2 dark pions composed of the elementary dark photons. Then mass of the SST dark pion is

Contrary to the neutrino-antineutrino pairs (the neutrinos are entangled) in the neutral pion, in the SST dark pions the spins of the pairs do not rotate, so mass of the dark pion is about 19 orders of magnitude lower than mass of the neutral pion.

Due to the 4-object symmetry and saturation of interactions [1], the number of entangled single objects can be [3]

where d = 0 , 1 , 2 , 4 , 8 , 16 , 32 , ⋯ are the Titius-Bode (TB) numbers.

For an object composed of entangled binary systems we have

Neutral pion decays to two photons so for neutrinos in neutral pion we have d ≥ 1 .

Neutral pions that contain more the neutrino-antineutrino pairs are preferred because then energies of the components are lower.

Calculate mass of a dark pion composed of N d = 32 = 4 32 the entangled non-rotating-spin neutrino-antineutrino pairs (or of N d = 32 = 2 × 4 32 the entangled non-rotating-spin neutrinos)

where m N e u t r i n o = 3.3349269504 × 10 − 67   kg is the mass of the lightest non-rotating-spin neutrino [1].

Notice that the mass of the dark pion built of 2 × 4 32 lightest neutrinos is very close to the mass of dark pion calculated in (3). It suggests that N d = 32 = 2 × 4 32 can be the upper limit for number of entangled single objects. For example, each of the two cosmological loops inside the Protoworld was composed of 2 × 4 32 the neutron black holes (NBHs) [1][3]. It follows from the relation

i.e. from the fact that nature tries to replicate certain behaviors on different scales.

Assume that range of the dark pion is equal to the de Broglie wavelength

Due to the four-object symmetry, population of dark pions with masses 4 times lower than the calculated in (9) should be higher (the dark pion decays into 4 parts)—it leads to a conclusion that the neutrinos should also disappear over distances 4×182.29 km≈729 km. The SST results, i.e. ∼182 km and ∼729 km are consistent with experimental data: KamLAND find that reactor ν e a n t i disappear over distances of about 180 km while MINOS observes the disappearance of accelerator ν μ ’s at distance of 735 km [2].

Here we showed that the dark mass of energetic neutrinos consists of the Y weak black holes [1] that are built of the dark pions. It looks as a “dark star”. In [4], it is shown that the final result of the gravitational collapse is a quantum object, an extremely compact “dark star”.

Rotating-spin neutrinos, due to viscosity of the SST absolute spacetime [1], force the more ordered motions of the non-rotating-spin neutrino-antineutrino pairs in the absolute spacetime. It decreases the local pressure in the absolute spacetime so the local number density of the neutrino-antineutrino pairs increases—there are created the Y spacetime condensates composed of the entangled dark pions that behave practically as single objects.

Total mass of the dark pions surrounding a rotating-spin neutrino is practically equal to its rotational energy E ν . But the total mass is a part of the zero-point energy, so its detection is impossible—the same concerns the photons.

From (8) and (9) results that mean range of the rotating-spin neutrinos is about L S S T = 181   km because such is the mean range of the SST dark pions surrounding them. It is a little greater distance than the reactor baseline L o = 180   km in the KamLAND experiment [5].

Emphasize that number densities, ρ N , for different energies of neutrinos, E ν , at the SST distance L S S T (so at L o as well) are different because of both the atom-like structure of baryons and creations of groups of entangled neutrons in the reactor nuclear matter. We should observe some maxima and minima in the following function

Let’s calculate the L S S T E ν e a n t i ratios for the extrema in ρ N = f ( L S S T E ν e a n t i ) .

From (4) and (5) follows that the β-decays of single neutrons and the β-decays of groups of entangled 2, 4 and 8 neutrons are most probable. Assume that created electrons in such β-decays are the virtual particles so the whole decay energy is absorbed by the groups of entangled electron-antineutrinos. There can appear the collapse of the wavefunctions that describe such groups, i.e. one of the electron-antineutrino in a group absorbs whole energy of the group. It causes that there appear electron-antineutrinos with following energies

We see that we should observe some maxima for

The maxima 35 km/MeV and 70 km/MeV are very well observed in the KamLAND data [5]. It is for pairs and quadrupoles of neutrons in nuclear matter (quadrupoles are most abundant).

The minima appear for the restorations of the negatively charged relativistic pions in neutrons, i.e. for W ( o ) , d = 1 → W ( − ) , d = 1 .

We have

When the electron is virtual then the energy absorbed by the ν e a n t i is W ( o ) , d = 1 − W ( − ) , d = 1 = − 7.1176   MeV [1]. When absorbed energy is equally distributed between e − and ν e a n t i then the energy absorbed by the ν e a n t i is − 7.1176 2 = − 3.5588   MeV .

We see that we should observe only two minima for

The minimum for 51 km/MeV is observed in the KamLAND data [5].

SST shows that the minimum ∼25 km/MeV should be deeper.

In [5], the P e e is the ν e a n t i survival probability. From a global analysis of neutrino oscillation data involving solar, accelerator, and reactor neutrinos, we have [5]

On the other hand, here we showed that there are the two dominating processes, i.e. the β-decays that increase the mean amplitude in the function ρ N = f ( L S S T E ν e a n t i ) and the reconstructions of the W ( − ) , d = 1 relativistic pions that decrease the mean amplitude. We can assume that in the first processes, the mean amplitude, A o , 1 , is the ratio of the possible maximum energy of the electron-antineutrino when the electron is a real particle to such an energy when the electron is virtual

In the second processes, similar phenomena for the electron lead to

From (16) and (17) we obtain the SST mean amplitude

The A o in SST relates to the mean survival probability P e e .

Generally, the simplest neutral pion consists of 4 energetic neutrinos each carrying energy equal to E ν i n − p i o n = 33.744   MeV . We can use this energy to define the deviations from the mean amplitude because the β-decays and the reconstructions of the W ( − ) , d = 1 relativistic pions concern the pion inside the neutron

The total amplitude is defined as follows

The A t o t a l relates to the total survival probability in the KamLAND experiment.

To compare the SST results with the KamLAND data we can calculate the amplitudes for the three clear extrema, i.e. for ± E ν e a n t i = 5.17   MeV , − 3.56   MeV and 2.59 MeV (they relate to L S S T E ν e a n t i ≈ 35 km MeV , 51 km MeV and 70 km MeV )

The SST results and the KamLAND data are presented in Figure 2.

We see that our results are consistent with the experimental data [5][6].

Figure 2. Amplitude versus electron-antineutrino energy from SST.

According to SST, the tremendous non-gravitating energy inside the four lightest neutrinos (i.e. the electron-neutrino, muon-neutrino and their antiparticles) causes that they are the very stable particles [1], i.e. they cannot change their flavours directly.

The charged pions can decay as follows

There is a possibility that the three neutrinos that appear in the decays of charged pions will be entangled and will carry half-integral spin. We claim that such objects composed of three entangled stable neutrinos are the tau-neutrinos

where γ c a r r i e r is the neutrino-antineutrino pair that is a carrier of the elementary photon (they are the dark photons that are a part of the zero-point energy)—when the spin-1 rotates then it is the elementary photon. We see that there is possible the decoupling of a carrier of photon from the tau-neutrino and tau-antineutrino. It means that, for example, the electron-neutrino can transform into the tau-neutrino and next, due to the neutrino-photon decoupling, the tau-neutrino can transform into the muon-neutrino, and so on. Emphasize that such transformations are not some neutrino oscillations inside the neutrinos. We should change the nomenclature: here instead the neutrino oscillations we have the neutrino-photon decoupling (or coupling). Of course, there are some oscillations of the cores of neutrinos, for example, they lead to the Higgs potential that increases the effective radius of the neutrinos (the SST quantum gravity) but in such processes, the neutrinos do not change their flavour [1]. There are also the two characteristic neutrino oscillations that lead to the Δ m 32 2 and Δ m 21 2 .

Notice that the carriers of the elementary photons/gluons must contain neutrino and antineutrino and must have opposite internal helicities. It leads to four different carriers that as a whole, due to the spin-1 rotation, can be externally lefthanded or righthanded, i.e. we have 8 different states that are activated in fields with internal helicity—we see that in the nuclear strong fields which contrary to the electromagnetic fields have internal helicity, we have 8 different spin-1 Types of gluons. The 8 Types of gluons force 3 colors of quarks: there is the left-helicity (LH) color, right-helicity (RH) color, and spin-rotation (SR) color—they are the red, green and blue colors of quarks. In anticolors, charges are converted to their opposite charges.

Let’s calculate the maximum mass, M ν , r o t a t i o n , of cosmological neutrino that follows from its spin rotation—it is in the form of the Y spacetime condensates, so in the form of the dark-pions as well

where r n e u t r i n o = 1.118455577 × 10 − 35   m is the radius of the core of the stable neutrino [1]. The effective neutrino radius that follows from the Higgs potential is ∼3510.2 times higher [1].

The mass of non-rotating-spin lightest neutrino is [1]

Masses of the cosmological neutrinos are defined by the following interval

In SST, the arithmetic mean is for simultaneous interactions of a mass (then effective coupling constant is a sum of coupling constants) while for not simultaneous ones we must apply the geometric mean because then effective coupling constant is a product of successive coupling constants [1]. The masses of neutrinos concerning their rotational energies decrease asymptotically to the minimum mass, so the real mean geometric mass of the lightest cosmological neutrino is (it is the real mass of the electron-neutrino and muon-neutrino; mass of tau-neutrino is three times higher)

From (31) follows that the sum of the real masses of the three cosmological neutrinos with different flavours is

In neutrino experiments, we do not measure their masses that follow from (31) but the masses emitted by them during their interactions. SST shows that in the interacting neutrinos there are produced the counterparts of the objects that appear in the interacting cores of baryons [1]. The effective mass of the interacting core of baryons is the sum of the mass of the created spacetime condensate Y * = 424.39406   MeV and the mass of the torus/electric-charge X ± = 318.29555   MeV [1]. On the other hand, in a stable neutrino there is a spacetime condensate and the torus/weak-charge both composed of the entanglons. The mass Y * + X ± is a counterpart of the mass m ν e r e a l = 0.05745   eV .

The neutrino-photon coupling-decoupling concerns the transformation of the electron-neutrino into muon-neutrino, or vice versa (it concerns the antineutrinos as well) via the tau-neutrino. In the core of baryons, a photon can create the X + X − pair, so for some counterpart we obtain

The second solution that results from (2) is as follows. The electron-neutrino can emit a counterpart to Y * (it leads to m 3 = m ν e r e a l and m 2 = 0.0328286   eV ) or to X ± (it leads to m 3 = m ν e r e a l and m 2 = 0.0246214   eV ). The first scenario leads to Δ m 32 2 = m 3 2 − m 2 2 = 2.2228 × 10 − 3   eV 2 whereas the second one to Δ m 32 2 = m 3 2 − m 2 2 = 2.6943 × 10 − 3   eV 2 . The mean value is Δ m 32 , m e a n 2 = 2.459 × 10 − 3   eV 2 that is very close to the result in (33)—the mean value of the three results is Δ m 32 , m e a n ∗ 2 = 2.45 × 10 − 3   eV 2 . We can say that there is a resonance between the SST results.

Interacting core of baryons can also produce a quanta with an energy that is a sum of the mass distance Y * − X ± and the weak mass of the π + π − pair (the coupling constant is α w ( p ) = 0.0187229 [1]), so for some counterpart in a stable neutrino we obtain

The second solution that results from (2) is as follows. The electron-neutrino can emit a counterpart to Y * − X ± (it leads to m 2 = 0.0082071   eV ) or an imaginary counterpart to i E ν i n − p i o n = i 33.7442   MeV (it leads to i m 1 = i 0.0026102   eV ). It leads to Δ m 21 2 = m 2 2 + m 1 2 = 7.417 × 10 − 5   eV 2 . This second SST result is very close to the result in (34)—the mean value is Δ m 21 , m e a n 2 = 7.42 × 10 − 5   eV 2 . We can say that there is a resonance between the SST results.

Notice that decoupling of the carrier of the elementary photon from the tau-neutrino causes that the tau-neutrino behaves as the electron-neutrino or muon-neutrino, so then we have only two different flavours of neutrinos

Mixing angles are the angles between two sets of orthogonal basis vectors. The choice of angles (parameterization) is based on convention, i.e. is not unique.

In the μ + μ − pair, there are 4 stable neutrinos (the 4-fermion symmetry) so we have

The real total mass of the 3 neutrino flavours is 5 times higher than mass of a stable neutrino and there are 5 stable neutrinos (see (32)). Moreover, in the π + μ − pair, there are 5 stable neutrinos so we have

In the π + π − pair, there are 6 stable neutrinos, so we have

Emphasize that the neutrino mixing angles can change their values because of the nuclear weak interactions of the π + π − pairs that appear in (36)-(39). It means that we should take into account such interactions when we interpret experimental results. For example, the weak mass of the charged pion is α w ( p ) π ± = 2.61316   MeV , where α w ( p ) = 0.0187229 is the coupling constant for the nuclear weak interactions [1]. Then we obtain Θ 12 / ∘ = 33.708539 and sin 2 Θ 12 = 0.3080 .

In such a way we should interpret the neutrino mixing angles.

The SST results in (36)-(39) are consistent with experimental data [6].

When we also take into account the mixing angle in (39) then Formula (1) has not a physical meaning (then for mass of the muon-neutrino we obtain value about 50% too high), so it suggests that probability of the angle Θ 23 / ∘ = 49.6335 is very low.

We assume that in the neutrino oscillations there is obligatory the CP conservation

Notice that cos ( − 180 ∘ ) = − 1 .

In SST, the Pontecorvo-Maki-Nakagava-Sakata (PMNS) matrix is unitary.

Table 2. The PMNS-SST matrix for neutrinos.

From (36), (37), (38) and (40) we obtain the PMNS-SST mixing matrix for neutrinos—see Table 2.

To obtain the real masses of neutrinos we must assume that the masses m 1 , m 2 and m 3 , i.e. the masses of the neutrino mass eigenstates ν i that appear in Formula (1), are as follows

The m 1 relates to two stable neutrinos as it is in the muon μ ± .

The m 2 is the virtual mass of the weak charge, i.e. it is a counterpart of − X ± .

The m 3 relates to three stable neutrinos as it is in the charged pion π ± .

We see that the masses m 1 , m 2 and m 3 are not associated with the Δ m 32 2 and Δ m 21 2 .

From the PMNS-SST mixing matrix for neutrinos and from (41)-(43), within the PMNS formalism we can calculate the real masses of neutrinos

The results in (44)-(46) are very close to the SST results in (31).

For the weak correction for the π + π − pair we obtain

From (47)-(49) we obtain the second PMNS-SST mixing matrix for neutrinos (see Table 3).

Table 3. The second PMNS-SST matrix.

From the second PMNS-SST mixing matrix for neutrinos and from (41)-(43), we obtain the second solution that is very close to the first solution but the results are a little worse

We showed that production of the electron-antineutrinos in reactors concern the decays of the W ( − ) , d = 1 charged relativistic pions inside the neutron nuclear strong field. It means that number of such weak decays can be changed because of a change in coupling constant for the nuclear strong interactions. The ratio α s N N , π α s N N , π + α s represents the transition from the nuclear strong interactions via the pions (the strong coupling constant is α s N N , π = 14.40076 ) and via the fundamental gluon loops (at low energies the strong coupling constant is α s = 1 ) to the strong interactions only via pions outside the strong field [1]. We see that when the β-decays of the negative relativistic pions take place outside the neutron strong field then number of such decays is lower

It solves the reactor electron-antineutrino anomaly. Our result is consistent with the experimental data [7].

The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a unitary matrix that concerns the flavour-changing weak interactions of quarks so there also should appear the energetic neutrinos with the energy equal to E ν i n − p i o n = 33.744   MeV , energy equal to n − p = 1.29333   MeV , and neutrinos from the W ( − ) , d = 1 → W ( o ) , d = 1 transition: their maximum energy is Δ W d = 1 = W ( − ) , d = 1 − W ( o ) , d = 1 = 7.11764   MeV .

Mass of the bottom quark follows from the peripheral interactions of the baryons [1]. The first mixing angle we define as the ratio of the mass of the bottom quark, m b = 4190.33   MeV , to the mass of the source of the nuclear strong interactions and electromagnetic interactions ( X ± = 318.29555   MeV [1])

The second mixing angle is the ratio of the mass of the fundamental gluon loop m F G L = 67.54441313   MeV [1] to X ±

The third mixing angle is the ratio of the mass of the boson that creates the Titius-Bode states for the strong-electroweak interactions, M T B = 750.2957682   MeV [1], to X ±

We see that the mass X ± normalizes the quark mixing angles.

In such a way we should interpret the quark mixing angles.

The SST results in (54)-(56) are consistent with experimental data [6].

We assume that in the quark weak interactions there is obligatory the CP conservation

Notice that cos ( − 0 ∘ ) = 1 .

What is the difference between the δ C P = 180 ∘ in the PMNS matrix and the δ C P = 0 ∘ in the CKM matrix? In SST, a charge (i.e. weak, electromagnetic, dark-matter charge) is a torus with an internal helicity and with all spins of its components directed towards the inside of the torus or all of them directed towards the outside of the torus. From Table 1 follows that flavour of a stable neutrino we can change by changing only weak charge or only internal helicity, i.e. δ C P = 180 ∘ . To change flavour of a quark in a V i j , we must change simultaneously both electric charge (the spins must be inverted) and internal helicity, i.e. δ C P = 180 ∘ + 180 ∘ = 360 ∘ = 0 ∘ .

Why the stable neutrinos have the 4 different states that leads to δ C P = 180 ∘ whereas the quarks and electrically charged leptons have only 2 different states (i.e. particle and its antiparticle) that leads to δ C P = 0 ∘ ? From SST follows that the neutrinos are embedded in the Higgs field (Hf) whereas the quarks and electrically charged leptons are embedded in the SST absolute spacetime (SST-As) which density is about 4 × 10⁴² higher than the Higgs field [1]. It means that viscosity of the SST-As is much, much higher than Hf. It also causes that at low energy, gravitational interactions are much weaker than electromagnetic interactions. Then on the tori of the quarks and electrically charged leptons, there appears a spin-helicity (internal) coupling. Such a coupling causes that when the spins of a torus components are directed towards the outside of the torus then there is preferred the right internal helicity whereas when they are directed towards the inside then there is preferred the left internal helicity—in such a way behaves the ball that is kicked in such a way that it starts spinning.

Emphasize that in SST, the matter-antimatter symmetry was broken at the end of the SST inflation because of the external left helicity of the initial inflation field, i.e. there between the angular velocity and velocity was the angle equal to 180˚. The matter-antimatter asymmetry does not concern the PMNS matrix or CKM matrix. It could be possible when the total internal helicity of the SST spacetime (i.e. of the Hf plus SST-As) was not equal to zero. But it is zero because the excessive left-handedness is encoded in the baryons. Emphasize that the nuclear strong field has helicity not equal to zero.

From (54)-(57) we obtain the CKM-SST mixing matrix for quarks—see Table 4.

Table 4. The CKM-SST mixing matrix for quarks.

The SST result for V t d is about 33% lower than the Standard Model (SM) result. Current experimental results are too imprecise to determine which model is correct.

The up-down quark-mixing element of the CKM matrix is V u d = 0.97367 ( 32 ) [8]. We see that the SST result, V u d , S S T = 0.97371 , is very close to the central value of the result presented in [8]. On the other hand, the element V u d depends on both Θ 12 and Θ 13 —it suggests that the SST definitions of the Θ 12 and Θ 13 can be correct. Notice that the SST definitions of Θ 12 and Θ 13 contain the mass of the bottom quark and the mass of the fundamental gluon loop, respectively, and they are normalized via the mass of the electric charge inside the core of baryons.

The fit result for the magnitude of V t b is (it is from [6]: 12. CKM Quark-Mixing Matrix): V t b = 0.999118 − 0.000034 + 0.000029 —this result is consistent with the SST result. On the other hand, the element V t b depends on both Θ 23 and Θ 13 .

The up-bottom quark-mixing element of the CKM matrix from B → π l ν is V u b = 0.00370 ± 0.00010 ( exp ) ± 0.00012 ( t h ) [9]—it depends only on Θ 13 and it is consistent with the SST result.

We see that the V u d from [8], V t b from [6] and V u b from [9] plus the condition of unitarity of the matrix lead to the CKM-SST matrix.

In the right vector in (1) (it is a vector of the mass eigenstates of the down-type quarks), there appear the masses of the down quark, strange quark and bottom quark. In SST, the mass of the s -quark is equal to half the mass of the relativistic neutral pion in the d = 2 state that is responsible for the structures of the hyperons: m s = 87.85483   MeV , whereas the mass of the b-quark is m b = 4190.33   MeV [1].

The masses of the three involved quarks in the right vector in (1) and the CKM-SST mixing matrix for quarks lead to the masses in the left vector in (1) (there are the weak-interaction doublet partners of down-type quarks)

In (58), there appear the β-decay and the weak decay of the neutral pion.

In (59), there can appear the transition from the heaviest hyperon Ω − that contains three strange quarks to the lightest hyperon Λ that contains one strange quark.

In (59) and (60), there appear the weak decays of the relativistic charged pion W ( − ) , d = 1 .

The experimental mass of the strange quark is m s , e x p = 93.5 ( 8 )   MeV [5]. It differs from the SST results. It is because in the atom-like structure of baryons, in the d = 2 state there appears symmetrical decay of the virtual mass/energy

M T B 2 = 375.147884   MeV [1]. Such a mass can also decay to a pair of the s s a n t i pairs, i.e. to a quadrupole

From (61) we have

The SST mass m s * = 93.787   MeV can mimic the experimental value m s , e x p = 93.5 ( 8 )   MeV .

The mass of top quark calculated within the SST is m t = 171.85   GeV [1] whereas the experimental value is

. It follows from the fact that at high energies such a quark can absorb its electromagnetic mass, i.e. α e m , h i g h m t , where α e m , h i g h = 1 127.54251 is the fine-structure constant at high energy [1]. Then the SST mean value is

This SST result is consistent with experimental value [6].

The Wolfenstein parameterization of the CKM matrix leads to the unitarity triangle with three angles/phases [6]

Experimental data lead to [6]: sin 2 β = 0.709 ± 0.011 , i.e. β / ∘ = 22.58 ± 0.35 , α / ∘ = 84.1 − 3.8 + 4.5 and δ / ∘ = 65.7 ± 3.0 . We see that the sum of the central values is 172.4 ∘ < 180 ∘ . We claim that in SST, the three angles are associated with the components of the core of baryons, i.e. with X ± and Y , and with the muon μ ± plus Δ π = π ± − π o produced by the core (they are the objects from decay of the charged pion). The Θ 12 , Θ 13 and Θ 23 in the CKM-SST are normalized via X ± but the CKM matrix is associated with the d -type quarks (they appear in the right vector in (1)), so we can define three angles normalized by mass of the d -quark that in SST is equal to half the mass distance between the two states of neutron: m d = 4.890481   MeV [1]

We see that the SST angles are consistent with the experimental values presented in [6] and are very close to the experimental central values.

The sum of the three SST angles is

When in (67), instead the μ ± + Δ π there is the charged pion π ± then we obtain β π , S S T / ∘ = π ± m d = 28.5392 and then the sum is

so we can say about the SST unitarity triangle for π ± , Y , and X ± normalized via m d .

We should not confuse the phases in (64)-(66) or in (67)-(69) with the CP phase δ C P / ∘ = 0 in (57).

Within SST we can define the three mixing angles in such a way to obtain from them the unitarity triangle

The relativistic charged pion, the X ± and Y are the three fundamental objects in the neutron while the 2 m d is the mass distance between the two states of the neutron [1]—we see that the SST mixing angles are closely related to the internal structure of the neutron which undergoes β-decay.

For scalars, pseudoscalars and electrically charged fermions we have

From (72)-(75) we obtain the SST unitary mixing matrix (see Table 5).

The SST mixing matrix and Formula (1) lead to

*the doublet scalars s ′ s ≡ π + π − and Y ;

*the doublet pseudoscalars p ′ p ≡ − π o (virtual) and π o ;

*the doublet fermions f ′ f ≡ X ± and E ν i n − p i o n .

Table 5. The SST mixing matrix.

The calculated from (1) masses of the weak eigenstates differ from the real values less than 1 part in 1 α w ( p ) ≈ 53 parts, where α w ( p ) is the coupling constant for the nuclear weak interactions [1].

From Table 5 we obtain that probability of the Y → π + π − transition is P s ′ s = V s ′ s 2 = 0.137 . From Table 5, it follows that there is preferred the decay of Y into X + μ − .

Notice that instead of saying about the scalar, pseudoscalar and fermion in the mass eigenstate vector ( Y , π o , E ν i n − p i o n ) we can say about the oscillations of the Y scalar because the diameter → circle oscillations lead to Y π ≈ 135.00   MeV ≈ π o while the next 4-object-symmetry decays lead to Y 4 π ≈ 33.75   MeV ≈ E ν i n − p i o n . It means that there instead the vector ( Y , π o , E ν i n − p i o n ) we have the vector composed of the scalars ( Y , Y π , Y 4 π ).

The Cabibbo angle, Θ c , was introduced to preserve the universality of the weak interaction (for example, the β -decay Fermi constant of the strange Λ baryon was a few times smaller). The Cabibbo angle in the CKM-SST matrix is (see Table 4)

The Cabibbo angle leads to the relative probability that strange and down quarks decay into up quark.

By analogy to the Cabibbo angle we can define the Cabibbo-SST angle for the SST matrix. From Table 5 we have

This angle leads to the relative probability that neutral pion (pseudoscalar) and scalar Y decay into the π + π − pair.

Notice that there appear some angle resonances

Emphasize that we should not confuse the angle δ / ∘ = γ / ∘ = 65.7 ± 3.0 with the CP phase δ C P / ∘ = 0 that appears in (57) and (75).

From (77) and (79) we obtain

where m b and m d are the masses of the bottom and down quarks, respectively, so the CKM-SST and SST matrices are combined via masses of such two quarks.

The factor f = α w ( p ) α e m = 2.56571 means that there is a transition from the electromagnetic interactions to the nuclear weak interactions and that then involved mass increases f times [1]. When we use the SST matrix then the mass eigenstate vector ( b , s , d ), i.e. the d -type quarks arranged from heaviest to lightest, leads to the following weak eigenstate vector

where H ± = 727.4392   MeV is the mass of the charged core of baryons [1]. The mass distances between doublet partner and corresponding to it the right side in (1) is smaller than 1 part in 140 parts.

There is obligatory the following equation that relates the mass of the bottom quark to mass of the electric charge in the core of baryons

We define the mass distance between the charged core of baryons and neutral one [1] as the mass distance between the down and up quarks

We define the mass distance between the relativistic charged pion in the d = 1 state and neutral one [1] as the sum of masses of the down and up quarks

From (85) and (86) we obtain

The transition between the two states of neutron [1] creates the quark field that is a part of the zero-point energy—there appear the quarks with positive mass and ‘holes’ in spacetime with negative mass. When we neglect the virtual spacetime holes then we obtain

The transition between the two states of proton leads to [1]

When we neglect the atom-like structure of baryons, as it is in the quark model, then from (89) and (90) we obtain the electric charges of the up and down quarks

We also have

where n 1 , n 2 , p 1 and p 2 are the mass states of the nucleons.

Our conclusions are as follows.

*In interactions/collisions of nucleons, we can measure the mass distances between the characteristic masses of the nucleon components that follow from the atom-like structure of baryons.

*The new description via quarks replicates the total electric charges of the nucleons—such an adaptation symmetry forces quarks to have fractional electric charges, which is achieved through appropriate radii of the tori/electric-charges of the quarks. The elementary torus/electric-charge in the core of baryons has the equatorial radius equal to A = 0.6974425   fm [1], so the electric charge equal to + 2 3 e has the equatorial radius equal to 2 A 3 whereas the charge − 1 3 e has the equatorial radius equal to A 3 . The electric tori interact with photons so their mass/energy is not determined by torus equatorial radius.

*It is impossible to describe properties of hadrons only via quarks, gluons and electroweak interactions—we need also the atom-like structure of baryons.

*Due to (93) and (94), there can appear a virtual field composed of the quark-antiquark pairs that is a part of the zero-point energy.

*The proton is the only baryon that is the stable particle because of its lowest mass and because it is the strong black hole [1]. In SST, the lower-mass state of the proton consists of three objects ( W ( o ) , d = 1 , Y , X + ) so it leads to the electric-charge eigenstate vector k i = ( 0 , 0 , + 1 ) . The quark model of proton ( u , u , d ) leads to the following electric-charge eigenstate vector k i = ( + 2 3 , + 2 3 , − 1 3 ) . On the other hand, the Kasner metric is an exact solution to Einstein’s equations of gravitation in an anisotropic space free from the matter when the following Kasner conditions for the 4D spacetime are satisfied [10]

The two extreme solutions that follow from (95) and (96) are ( 0 , 0 , + 1 ) and ( + 2 3 , + 2 3 , − 1 3 ) , so they overlap with the SST/quark electric-charge eigenstate vectors for proton.

To obtain within SST the correct masses of nucleons, we must assume that the quark field is a result of the mass/electric-charge oscillations of the nucleons and that the created quark field is associated with the zero-point energy. Moreover, the oscillations of both the virtual electroweak field [1] and the virtual quark field of nucleons cause that information about the internal structure of the nucleons leaks outside the nucleons—it is very important in cosmology [11] and in the SST chaos theory.

We can interpret the Kasner conditions as an real-virtual oscillation between the two extreme states of the electric-charge eigenstate vectors defined by both the real SST-model and the virtual quark-model, i.e. the exchanged elementary real electric charge in the SST model forces creation of the three virtual fractional electric charges in the quark model, and vice versa.

We showed that in SST, the Kasner conditions concern electric charges (not masses) and concern the zero-point energy field so we can say about the Kasner anisotropic space “free from the matter”. In collisions/interactions of the nucleons, the quark field can behave as a real field.

We see that we can partially unify the general relativity (GR) and the standard model (SM) via the Kasner conditions.

In [1] we calculated masses of the three heaviest quarks whereas in this article we calculated masses of the three lightest quarks and the second solution for t -quark. The SST results and the experimental data from [6] are collected in Table 6.

SST shows that the base of the QGP is a superfluid composed of the cores of baryons. In the plasma core, there is destroyed the atom-like structure of baryons outside the cores of baryons. On the other hand, only mass of the t -quark follows from the structure of the baryonic core [1]. It means that the other quarks are produced only in the plasma corona, i.e. outside the plasma core. In this article we showed that there are 8 Types of gluons.

Table 6. Masses of quarks from SST and [6].

In the Standard Model of particle physics, we have three distinct neutrino flavours. But several anomalous observations suggest that there should be an additional neutrino state [12]. Here we show that the anomalies can be explained within presented here the SST model of neutrinos.

Some neutrino-flavour change corresponds to a mass-squared splitting of Δ m 41 2 ≳ 10 − 2   eV 2 [12], which is much greater than the measured Δ m 32 2 and Δ m 21 2 in (33) and (34), respectively. On the other hand, the mass distance between the tau-neutrino and electron-neutrino (the ν e ⇆ ν μ oscillations are possible only via the tau-neutrino) leads to

The second solution that results from (2) is as follows. The sum of the rest masses of neutrinos in the muon is m 4 = m ν e a n t i r e a l + m ν μ r e a l = 0.1149   eV . The decay of the muon leads to m 1 = 0   eV in the electron, so we have Δ m 41 2 = m 4 2 − m 1 2 = 1.320 × 10 − 2   eV 2 . It is the mean value because this result and result in (98) are the same, i.e. Δ m 41 , m e a n 2 = 1.32 × 10 − 2   eV 2 that is consistent with the data in [12].

The interplay between oscillation of intrinsic ν μ and ν e components in a beam leads to Δ m 41 2 = 1.2 or 1.4   eV 2 [12]. Its origin is as follows.

The derived relationship between the center-of-mass energy s N N and the characteristic masses created in the SST nuclear matter, M , is as follows [13]

We obtain such a formula for the ratio v c in the formula for the relativistic mass equal to 1 1 + a e , where a e = 0.0011596522 is the anomalous magnetic moment of the electron. We obtain it by calculating the relativistic mass of the electron when the wavefunction of the bare electron propagates with the maximum speed c . Then we have [1]: m e , b a r e m e = v c = 1 1 + a e , where m e is the rest mass of the electron while m e , b a r e is the rest mass of the bare electron. Next from the Einstein’s formula for the relativistic mass we obtain that the relativistic mass of the electron is C M times higher than its rest mass.

We can apply the factor C M also to the neutrinos. Then we obtain

The second solution that results from (2) is as follows: m 4 = M ν e = 1.1940   eV and m 1 = m ν e r e a l = 0.057450   eV . It leads to Δ m 41 2 = m 4 2 − m 1 2 = 1.422   eV 2 . The mean value of this result and the result in (102) is Δ m 41 , m e a n 2 = 1.36   eV 2 that is consistent with the data in [12].

The value in (102), i.e. Δ m 41 2 = 1.29   eV 2 appears in [14].

Let’s consider a neutrino counterpart to 4 Y * = 1697.576   MeV for 4 different entangled stable neutrinos that mass increased C M times (see (99)). Then we have

The value in (103) and (105) are consistent with the results presented in [15]: m 4 = 2.70 ± 0.22   eV and Δ m 41 2 = 7.3 ± 1.17   eV 2 , respectively.

SST shows that the neutrinos are the Dirac fermions.

Here within SST, we showed that due to the cloud of the dark pions surrounding the energetic neutrinos, they should disappear over distances of ∼182 km and ∼729 km.

Due to the spin rotation, masses of neutrinos can change from ~3 × 10⁻⁶⁷ kg to masses close to the Planck mass but they are a part of the zero-point energy.

We described two oscillations of neutrinos that lead to Δ m 32 , m e a n 2 = 2.45 × 10 − 3   eV 2 and Δ m 21 , m e a n 2 = 7.42 × 10 − 5   eV 2 . We showed that they follow from the SST structure and weak interactions of the core of the stable neutrinos. Emphasize that they are not associated with the mass eigenstates. We also described three Standard-Model anomalies that follow from following mass-squared splitting equal to Δ m 41 , m e a n 2 = 1.32 × 10 − 2   eV 2 , Δ m 41 , m e a n 2 = 1.36   eV 2 , and Δ m 41 2 = 7.395   eV 2 —SST shows that sterile neutrinos are not needed.

The cores of neutrinos pulsate that lead to the Higgs potential that increases the effective radius of the neutrinos (the SST quantum gravity), but in such processes, the neutrinos do not change their flavour [1]—such pulsations activate the neutrino oscillations.

We explained the dependence of the survival probability on the electron-antineutrino energy in the KamLAND data via the atom-like structure of baryons. There are two dominant phenomena, i.e. the β-decays and the reconstructions of the W ( − ) , d = 1 charged relativistic pions.

We showed that the experimental results for masses of cosmological neutrinos lead to the radius of the neutrino core and vice versa.

We showed that there can be the neutrino-photon decoupling (or coupling). In such processes, the flavours of neutrinos can change.

We also solved the reactor electron-antineutrino anomaly that follows from a change in the strong coupling.

We used the SST to show how we should correctly interpret the PMNS formalism. The mixing angles relate to two symmetries, i.e. to the four-fermion symmetry and to the saturation-of-interaction symmetry for the three neutrinos with different flavours. On the other hand, the internal structure of the stable neutrinos leads to the CP phase δ C P / ∘ = 180 . But we can show that there exists an alternative set of mixing angles (the choice of angles is based on convention). We have: Θ 13 / ∘ = 2 π ± E ν i n − p i o n = 8.272 , Θ 12 / ∘ = Λ E ν i n − p i o n = 33.063 , Θ 23 / ∘ = μ B , c h , u p p e r , T = 0 E ν i n − p i o n = 41.99 , and Θ 23 / ∘ = Ω − E ν i n − p i o n = 49.56 , where Λ = 1115.683 ( 6 )   MeV and Ω − = 1672.45 ( 29 )   MeV are the masses of the lightest and the heaviest hyperons [6], and μ B , c h , u p p e r , T = 0 = 1417   MeV is the upper limit for the characteristic baryonic chemical potential for emission of the relativistic pion W ( − ) , d = 1 by neutron at T = 0 [13]. Such a set of mixing angles practically does not differ from the original set.