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 [1]. 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 [2] 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 [2] 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 [2] and -1.91 for a neutron [2]. 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 [3]. 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.