# Thread: Theory of superluminal neutrinos

1. Theory of Superluminal Neutrinos

Here, within the atom-like structure of baryons, I calculated the neutrino speeds for the MINOS and OPERA experiments and for the supernova SN 1987A explosion.
The triplet n-p scattering length is approximately 5.4 fm. The singlet n-p effective range is approximately 2.7 fm whereas the triplet n-p effective range is approximately 1.7 fm . Assume that outside of the core of nucleons the Titius-Bode law for strong interactions r(d) = A + dB where A = 0.7 fm, B = 0.5 fm, and d = 0, 1, 2, 4 is obligatory. The diameter of the last orbit is, therefore, 2r(d = 4) = 2(A + 4B) = 5.4 fm, the radius of last orbit is r(d = 4) = A + 4B = 2.7 fm, whereas the radius of the last but one orbit is r(d = 2) = A + 2B = 1.7 fm. Assume that due to the strong interactions the spin speed of the pions on the equator of the core of baryons should be equal to the c. This means that the Schwarzschild radius for the strong interactions is 1.4 fm. From the Uncertainty Principle follows that such is also the range of the neutral pions produced in centre of the baryons. Assume that the muons, pions and W bosons (denote their mass by M) arise in the centre of the core of nucleons as the entangled gluon-ball quadrupoles. Such a quadrupole can be entangled with a neutrino (denote its mass by m) on the Schwarzschild surface. This means that the centrifugal force is directly proportional to the product 4Mm, where m<< M. Assume that one of the two pairs in a quadruple is exchanged between the second pair and the neutrino on the Schwarzschild surface for the strong interactions. When one of the two entangled pairs appears near the neutrino then the centrifugal force is directly proportional to the product 2M(2M + m). In approximation, the increase in the centrifugal force F acting on the neutrino, is
(1) F ~ 2M(2M + m)  4Mm ≈ 4MM
The force F is responsible for the radial speed of the exchanged object so for the radial speed of the neutrino also. The exchanged gluon-ball pairs can appear in the area between the surface of the core (radius is 0.7 fm) and the sphere which radius is equal to the range of the strong interactions (radius is 2.7 fm). The central value for the time of exchange t is for the Schwarzschild surface. This means that the broadening of the time of exchange t is defined in approximation by following interval (t/2, 2t).
Pions appear in the main channels of the decay of the Lambda and Sigma+ hyperons. During the decay of the hyperon Lambda, negatively charged and neutral pions appear. On the basis of this experimental data  we can assume that a neutron with a probability of x about 0.63 is composed of a positively charged core and a negative pion. Furthermore, the probability (1  x) is composed of a neutral core and a neutral pion. During the decay of the hyperon Sigma+, neutral and positively charged pions appear. On the basis of this experimental data  we can assume that the proton with a probability y about 0.51 is composed of a positively charged core and a neutral pion and the probability (1  y) is composed of a neutral core and a positive pion. We know that the nucleon-nuclear magnetic moment ratios are about +2.79 for a proton  and -1.91 for a neutron . On the basis of these experimental results, we can assume that the mass of the charged core is about H(charged) ~ 727 MeV and the relativistic charged pion is W(charged) ~ 216 MeV. Such values of the probabilities and masses leads to the experimental data for magnetic moments of nucleons. On the other hand there is the Feigenbaum constant δ = 4.669 . Assume that this constant maps the internal structure of proton in the Einstein spacetime. Assume that the core of baryons consists of two parts i.e. of the point mass in the centre of the core of baryons (in approximation Y = 424 MeV = 4·106 MeV) which is responsible for the weak interactions of baryons, and of the torus responsible for the spin and strong interactions of baryons (in approximation X = 318 MeV = 3·106 MeV). We can see that there should be in existence a quadruple symmetry for the weak interactions of baryons. The mass distance between the X + Y and the 727 MeV, i.e. about 15 MeV, is in approximation both the binding energy of the two components of the core of baryons and the volumetric binding energy per nucleon in the atomic nuclei. Outside the core of nucleons there is the relativistic charged or neutral pion. Because the charged pion has relativistic mass equal to about 216 MeV then the relativistic mass of the neutral pion should be about 208 MeV i.e. the mean value is R = 212 MeV = 2·106 MeV. Ratio of (X + Y)/R and X/Y is in approximation 14/3 = 4.667. Obtained result is close to the Feigenbaum constant. We can see that the mass of the strange quark and the mass of muon are indirectly associated with the Feigenbaum universality. On the boundary of the strong field, the strange quark-antiquark pairs (in approximation the rest mass of a pair is 2·106 MeV), due to the gluon-photon transitions, can change into muon-antimuon pairs (in approximation the rest mass of a pair is 2·106 MeV). The core of baryons carries the electric charge and spin of the baryons. This means that due to the law of conservation of charge and spin, inside the core cannot appear other electric charges. There can appear the gluon balls and gluon loops only. The gluon-ball pairs arising in the centre of the core of baryons as the gluon-ball quadrupoles can outside the core of baryons transform into the strange quark-antiquark pairs or, outside the strong field, into the muon-antimuon pairs.
Most important are the gluon-ball quadruples carrying energy equal to the Y = 424 MeV = 4·106 MeV and X + Y = 727 MeV = 4·181.7 MeV. Resultant spin, charge and resultant internal helicity of a quadruple, composed of dipoles which carry spin equal to 1, can be equal to zero. This is the quadruple symmetry. Such objects do not violate the laws of conservation of spin, charge and internal helicity of nucleons.

Very simple calculations are in next post.  2.

3. Calculations
Assume that due to the properties of the Einstein spacetime, mass of neutrino cannot change. This means that the neutrinos can be the superluminal particles but they are the non-relativistic particles so we can apply to them the Newton’s second law. The Newton’s second law we can write for neutrinos as follows

(2) m(v – c) = Ft
The strange quark-antiquark pairs and next the muon-antimuon pairs arise in the centre of the core of baryons as the gluon-ball quadrupoles i.e. as the quadrupoles of pure energy. This means that such objects are also the non-relativistic objects so we can apply to them the Newton’s second law. The force acts on the carriers of gluons i.e. on the Einstein spacetime components. They are the neutrino-antineutrino pairs i.e. the weak dipoles carrying spin equal to 1 so they are the non-relativistic particles also. Their speed is equal to the c. From the formulae (1) and (2) we obtain that the increase in the radial speed of neutrinos which appear in the beta decays is
(3a) v – c ~ 4[m(neutron) – (m(proton) + m(electron)]^2 = 4NN
The increase in the radial speed of the neutrinos appearing in the weak decays of the exchanged gluon-ball pairs is
(3b) v – 0 ~ 4MM,
where M is mass of gluon ball which decays due to the weak interactions. Energy of such gluon balls can be equal to the mass of muons or to the one fourth of the mass of the core of baryons or to the mass of the W bosons. Due to the weak interactions of the neutrinos with the gluon balls, the neutrinos appearing in the beta decays and the neutrinos appearing in the decays of the gluon balls must have the same resultant speed. From formulae (3a) and (3b) we obtain
(4) (v– c)/v = (N/M)^2.
Since v≈ c then in approximation is
(5) (v– c)/c = (N/M)^2,
or
(6) v= {1 + (N/M)^2}c.
To the gluon balls we can apply the theory of stars. The theory of stars leads to conclusion that lifetime T is inversely proportional to four powers of mass, i.e. T ~ 1/m^4, so we can rewrite the formula (5) as follows
We can see that we can calculate the neutrino superluminal speeds both from masses of particles (formula (5)) or from their lifetimes (formula (7)). Both methods lead to the experimental data.
From the Uncertainty Principle and the invariance of the neutrino mass follows that the square of the change in neutrino speed is inversely proportional to the time of exchange t. On the other hand, from formula (3a) and the relation T ~ 1/m^4 follows that similar relation is for the lifetime T. This means that the interval for the broadening of the time of exchange t, i.e. the (t/2, 2t) described in the Introduction leads to following conclusions. To obtain the maximum neutrino speed, we must multiply the central value for an increase in neutrino speed in relation to the c, i.e. the (v – c)/c, by sqrt(2) whereas to obtain the minimum speed we must divide the central value by sqrt(2). The theoretical results are the central values whereas in the round brackets we will write the increases in speed for the maximum neutrino speed.
For lower energies, such as in the MINOS experiment , there are mostly the neutrinos from the decays of neutrons and gluon-ball pairs carrying energy equal to the mass of the muon-antimuon pairs. The ratio of the lifetime of neutron to lifetime of muon is smallest (882/2.20·10^-6 = 4·10^8 ) so the obtained neutrino speed is the upper limit. From formula (7) follows that for the more precise MINOS experiment, for the neutrino speed we should obtain 1.000050(21)c i.e. the maximum neutrino speed should be 1.000071c.
For higher energies, such as in the OPERA experiment , there are mostly the neutrinos from the weak decays of the neutrons and gluon-ball pairs carrying energy equal to the half of the mass of the core of baryons. Mass of one gluon ball is 181.7 MeV. This means that lifetime of such gluon ball which decays due to the weak interactions at once into 3 neutrinos and electron, is 8.74 times shorter than lifetime of muon. This leads to conclusion that the neutrino speed is 1.0000169(70)c i.e. the maximum speed is 1.0000239c so the time-distance between the fronts of the neutrino and photon beams is 58.4 ns.
For highest energies, such as in the explosions of the neutron cores of supernovae, dominate the neutrinos from the decays of the neutrons and gluon-ball pairs carrying energy equal to the mass of the W boson-antiboson pairs. The distance of mass between the point mass and the torus in the core of baryons is equal to the mass of muon whereas the mass of the point mass, which is responsible for the weak interactions of baryons, is 4 times greater than the muon. The quadruple symmetry shows that creation of systems containing 4 elements is preferred. This means that the lifetime of the muon is characteristic also for the point mass (i.e. 424 MeV = 4·105.7 MeV – each one of the four muons lives 2.2·10^-6 s ). This leads to conclusion that lifetime of the W bosons (mass = 80,400 MeV ) which decay due to the weak interactions is T(lifetime-of-W-boson) = 2.2·10^-6 s/(80,400/424)^4 = 1.7·10^-15 s.
This leads to following neutrino speed 1.0000000014(6)c i.e. maximum speed is 1.000000002c (i.e. (1 + 2·10^-9)c). This result is consistent with the observational facts . The time-distance Δt between the fronts of the neutrino and photon beams, measured on the Earth for the SN 1987A, should be
Δt = 168,000 ly · 365 days · 24 hours · 2·10^-9 = 3 hours.
If before the explosion, the mass of the SN 1987A was close but greater than four masses of the Type Ia supernovae, i.e. greater than 5.6 times the mass of the sun, then due to the quadruple symmetry, during the gravitational collapse, there could arise the system containing 4 the Type Ia supernovae. After simultaneous explosion of the 4 supernovae, we should not observe there a remnant i.e. neutron core. Due to gravitational collapse a supernova transforms into neutron star. Next, there are the beta decays of the neutrons and nuclear fusions of the nucleons. These two processes appear simultaneously. This means that neutrinos and photons appear on surface and inside neutron star simultaneously. When mass of a neutron star is equal to mass of the Type Ia supernova then neutrinos and photons appear simultaneously in whole volume of the star. We can see that a supernova which has mass in approximation 5.6 times the mass of the sun practically should not have some plasma layer around the four neutron stars. This means that during the explosion of such quadrupole of neutron stars there should not be a time-distance between the fronts of the neutrino and photon beams. The observed on the Earth the 3-hours delay must be due to the superluminal speed of neutrinos.

Summary and references are in next post.  4. Summary
The calculated neutrino speed for the MINOS experiment is 1.000050(21)c. The maximum neutrino speed is 1.000071c. The calculated time-distance between the fronts of the neutrino and photon beams for the OPERA experiment is 58.4 ns whereas the neutrino speed is 1.0000169(70)c i.e. maximum neutrino speed is 1.0000239c. The calculated time-distance between the fronts of the neutrino and photon beams, observed on the Earth, for the supernova SN 1987A is 3 hours whereas the neutrino speed is 1.0000000014(6)c.
Neutrino speed depends on internal structure of baryons and phenomena responsible for creation of particles which decay due to the weak interactions. The MINOS and OPERA experiments and the data concerning the supernova SN 1987A suggest that there is in existence an atom-like structure of baryons. In MINOS dominated neutrinos from decays of neutrons caused by gluon-ball pairs which energy is two times greater than mass of muon, in OPERA caused by gluon-ball pairs which energy is two times smaller than the mass of the core of baryons whereas in the supernova SN 1987A explosion, by gluon-ball pairs which energy is two times greater than the mass of the W bosons.

This is the end but applying the Everlasting Theory

http://www.cosmology-particles.pl

we can calculate the neutrino speed for the MINOS experiment also in different third way. Neutrons and muons decay due to the weak interactions. From formula (51) (see my book) follows that coupling constant for weak interactions is in proportion to square of exchanged mass whereas theory of stars leads to T ~ 1/m^4. This means that square root from lifetime is inversely proportional to coupling constant so applying also formula (57) (see my book) we can rewrite formula (7) as follows
(8) X = α_weak-for-beta-decay/α_weak-for-muon = c/(v– c) = 19,685.3
From this formula we obtain
(9) v = c(X + 1)/X = 1.0000508c.
Due to the weak interactions, the mass of the electron-positron pair can increase the Xtimes whereas the resultant mass due to the quadrupole symmetry can increase the four times. The final mass is 80,473 MeV and it is the mass of the W boson.
We can see that due to the quadruple symmetry there are the 4 basic quadrupoles leading to the superluminal neutrinos. Their masses are as follows. The 4N, in approximation the mass of the point mass in the centre of the core of baryons 424 MeV and mass of the core of baryons 727 MeV, and the mass of the W boson 80,473 MeV.
There is the relativity of lifetimes for entangled particles. To free a neutron from an atomic nucleus is needed the mean energy equal to the volumetric binding energy 14.952 MeV (see the description concerning formula (183)). On the other hand we know that the binding energy of the mass X and Y is 14.98 MeV. This energy is close to the volumetric binding energy so the free neutrinos can frequently be entangled with the volumetric binding energy i.e. the bound neutrons can simultaneously interact with energy two times higher than the volumetric binding energy. From relation T ~ 1/m^4 and formula (95) follows that lifetime of neutron entangled with the volumetric binding energy 14.97 MeV is 888 s. Similarly the distance of mass between the two charged states of the core of baryons is 2.67 MeV. When a muon is entangled with such energy then its lifetime is 2.21·10^-6 s.

References

 K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010)
 Feigenbaum Mitchell, Universal Behaviour in Nonlinear Systems, “Los Alamos Science” 1 (1981)
 P. Adamson et al. (MINOS Collaboration) (2007). "Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam". Physical Review D 76 (7). arXiv:0706.0437.
 OPERA Collaboration, T. Adam et al. (2011), “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam”, arXiv:1109.4897 [hep-ex].
 K. Hirata et al., Phys. Rev. Lett. 58 (1987) 1490;
R. M. Bionta et al., Phys. Rev. Lett. 58 (1987) 1494;
M. J. Longo, Phys. Rev. D 36 (1987) 3276.  Bookmarks
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