An Explanation for the Violation of Lepton Universality in Beauty-Quark Decays: The Binary Isotope Mixture of Beauty-Quarks

This paper purposes an explanation for the recent evidence for the violation of lepton universality in beauty-quark decays at CERN’s Large Hadron Collider. A beauty meson (B + ) transforms into a strange meson (K + ) with the emission of either electron-positron (e + e − ) or muon-antimuon (µ + µ − ). The ratio (R K ) of branching fractions for B + → K + µ + µ − and B + → K + e + e − decays is measured to be R K = 0.846 instead of 1 in the violation of lepton universality in the Standard Model. This paper proposes that the violation is derived from the binary isotope mixture of two beauty-quarks, b 7 (4979 MeV mass) and b 8 (143,258 MeV mass) whose masses are calculated from the periodic table of elementary particles. b 7 is the observable B, while b 8 is the hidden B to preserve the generation number symmetry between the three lepton family generations and the three quark family generations in the Standard Model. The preservation of the generation number symmetry forbids b 8 to decay into K + µ + µ − . In the transition state involving the virtual particles (γ, W± and Z˚) before the decay, b 7 and b 8 emerge to form the binary isotope mixture from B. The rates of emergence as the rates of diffuse in Graham’s law of diffusion are proportional to inverse square root of mass. The rate ratio between b 8 /b 7 is (4979/143,258) 1/2 = 0.1864. Since b 7 decays into K + , e + e − , and µ + µ − , while b 8 decays into K + , e + e − , and forbidden µ + µ − , the calculated ratio (R K ) of branching fractions for B + → K + µ + µ − and B + → K + e + e − is 0.5/(0.1864 × 0.5+ 0.5) = 0.843 in excellent agreement with the observed 0.846. The agreement between the calculated R K and the observed R K confirms the validity of the periodic table of elementary particles which provides the answers for the dominance of matter over antimatter, dark-matter, and the mass hierarchy of elementary particles. Particles, Binary Isotope Mixture of Beauty-Quarks, Ratio of Branching Fractions, Dark Matter, Dominance of Matter over Antimatter


Introduction
For about sixty years, the Standard Model (SM) of particle physics has provided the model for various properties and interactions of fundamental particles, and has been confirmed by numerous experiments. In lepton universality of the SM, the different charged leptons, the electron, muon and tau, have identical electroweak interaction strengths. Lepton universality has been confirmed in a wide range of particle decays. A recent measurement of beauty-quark decays based on proton-proton collision data collected with the LHCb detector at CERN's Large Hadron Collider shows the evidence for the breaking of lepton universality in beauty-quark decays [1]. A beauty meson (B + ) transforms into a strange meson (K + ) with the emission of leptons (ℓ + ℓ − ) which is either electron-positron (e + e − ) or muon-antimuon (µ + µ − ). Lepton universality of the SM predicts that the ratio (R K ) of branching fractions for B + → K + µ + µ − and B + → K + e + e − decays is 1. The observed R K is 0.846 instead of 1 in the violation of lepton universality in the SM.
The measurement has a significance of 3.1 standard deviations. This paper proposes that the violation is derived from the binary isotope mixture of two beauty-quarks, b 7 (4979 MeV mass) and b 8 (143,258 MeV mass) whose masses are calculated from the periodic table of elementary particles [7]. b 7 is the observable B, while b 8 is the hidden B to preserve the generation number symmetry between the three lepton family generations and the three quark family generations in the SM. The preservation of the generation number symmetry forbids b 8 to decay into K + µ + µ − . In the transition state involving the virtual particles (γ, W± and Z˚) before the decay, b 7 and b 8 emerge to form the binary isotope mixture from B. The rates of emergence as the rates of diffuse in Graham's law of diffusion are proportional to inverse square root of mass. The rate ratio between b 8 /b 7 is (4979/143,258) 1/2 = 0.1864. Since b 7 decays into K + , e + e − , and µ + µ − , while b 8 decays into K + , e + e − , and forbidden µ + µ − , the calculated ratio (R K ) of branching fractions for B + → K + µ + µ − and B + → K + e + e − is 0.5/(0.1864 × 0.5+ 0.5) = 0.843 in excellent agreement with the observed 0.846. The agreement between the calculated R K and the observed R K confirms the validity of the periodic table of elementary particles. Section 2 describes the periodic table of elementary particles and the calculation of the masses of leptons and quarks including the masses of b 7 and b 8 . Section 3 describes the beauty-quark decay and the calculation of R K for the branching fractions of B + → K + µ + µ − and B + → K + e + e − . Section 4 explains dark matter and the dominance of matter over antimatter.

The Periodic Table of Elementary Particles
The periodic table of elementary particles for baryonic matter and dark matter [2]- [8] is based on the seven principal mass dimensions (d's) for stable baryonic matter leptons (electron and neutrinos), gauge bosons (all forces), gravity, and dark matter (five sterile dark matter neutrinos) and the seven auxiliary mass dimensions (a's) for unstable leptons (muon and tau) and quarks (d, u, s, c, b, and t) as in Figure 1 and Table 1.  The periodic table of elementary particles provides the answers for the dominance of matter over antimatter [8], dark-matter [5], and the mass hierarchy of elementary particles [2]- [8]. The masses of all leptons, quarks, gauge bosons, the Higgs boson, gravity, and dark matter can be calculated by the periodic table of elementary particles [2]- [8]. Since this paper deals with mostly beauty quark, only the masses of leptons and quarks are calculated.
The mass of mass dimensional fermion and the mass of mass dimensional boson are related to each other with three simple formulas as the follows.
where d is the mass dimension number, F is fermion, and B is boson. Each dimension has its own α d , and all α d 's except α 7 (α w ) of the seventh dimension (weak interaction) are equal to α, the fine structure constant of electromagnetism.
The lepton mass formula and the quark mass formula are derived from the incorporation of basic gluon (g* = B 6 = M F6 /α = M e /α = 70 MeV from Equation (1)) from Table 1 to electron. The incorporation of basic gluon as flux quanta follows the composite fermion theory for the FQHE (fractional quantum Hall effect) [9] [10]. In the composite fermion model for FQHE, the formation of composite fermion is through the attachment of an even number of magnetic flux quanta to electron, while the formation of composite boson is through the attachment of an odd number of magnetic flux quanta to electron. In the same way, the formation of composite fermion is through the attachment of an even number of basic gluons to electron, while the formation of composite boson is through the attachment of an odd number of basic gluons to electron. The formation of composite boson is equal to the formation of composite di-leptons, so the formation of composite lepton is through the attachment of one half of an odd number of basic gluons to electron. As a result, the muon (µ) mass formula is as follows.
which is in excellent agreement with the observed 105.6584 MeV [11] for the mass of muon. The masses of leptons follow the Barut lepton mass formula [12] as follows.
where a = 0, 1, and 2 are for e, μ 7 , and τ 7 , respectively. The calculated mass of τ 7 is 1786.2 MeV in good agreement with the observed mass as 1776.82 MeV. According to Barut, the second term, which is in excellent agreement with observed 1776.82 MeV, and means that during this dipole-interaction in a circular orbit for τ, an electron with total mass of 17M e is lost. 17M e is shown as the observed 17 MeV for 34M e in the light boson (17 ee̅ ) [13] [14]. Quark has fractional charge (±1/3 or ±2/3), 3-color gluons (red, green, and blue) for 3g*, and both the principal mass dimensions and axillary mass dimensions, so similar to Equation (4), d and u in the principal mass dimension involves e/3 or 2e/3 and 3g* as follows.
where a = 1, 2, 3, 4, and 5 for u 7 /d 7 , s 7 , c 7 , b 7 , and t 7 , respectively. The quark mass at a = 5 for the auxiliary mass dimension at d = 7 is the maximum mass below the mass of B 7 , so the next auxiliary mass dimension has  (4), the mass of μ 8 is as follows.
Since at d = 7, there are 3-color basic gluons, at d = 8, 3-color basic gluons are not needed, and only one basic gluon (g* 7 ) at d = 7 is used. Similar to Equations (7) and (9), the quark mass formulas for the principal and auxiliary mass dimensions are as follows.
The quark mass formula at d = 8 is the combination of Equations (10) and (11) as follows.
Combining Equations (10) and (14), the quark mass formula is as follows.   Table 1. To be symmetrical to the absent µ 8 , b 8 quark is also absent. In other words, b 8 quark is hidden. As a result, the observable b 7 and the hidden b 8 form the binary isotope mixture. A beauty meson (B + ) transforms into a strange meson (K + ) with the emission of leptons (ℓ + ℓ − ) which is either electron-positron (e + e − ) or muon-antimuon (µ + µ − ). The decay process includes a transition state where the decay process is mediated by virtual particles that can have a physical mass larger than the mass difference between the initial-and final-state particles. In the SM, these virtual particles include the electroweak-force carriers, the γ, W± and Z˚ bosons, and the top quark as Figure 2.
In the transition state before the decay, from B + , the binary isotope mixture of beauty quarks emerges along with the virtual particles (γ, W± and Z˚) that can have a physical mass larger than the mass difference between the initial-and final-state particles. In the transition state, the binary isotope mixture is virtual. The rates of emergence as the rates of diffuse in Graham's law of diffusion are proportional to inverse square root of mass.
The calculated ratio (R K ) of branching fractions for B + → K + µ + µ − and B + → K + e + e − is 0.843 in excellent agreement with the observed 0.846. The agreement between the calculated R K and the observed R K confirms the validity of the periodic table of elementary particles. On molecular level, the emergence of the binary isotope mixture of b 7 and the hidden b 8 from B + is similar to the diffusion of the binary isotope mixture of  gaseous molecules. One of the well-known examples of the binary isotope mixture is the binary isotope mixture of 3 He/ 4 He from the degassing of the Earth's mantle through magmatism that results in the irreversible loss of helium to space, and high 3 He/ 4 He ratios observed in oceanic basalts have been considered the main evidence for a "primordial" undegassed deep mantle reservoir [15]. The initial ratio of the degassing rates from Graham's law between 3 He/ 4 He is as follows.
( ) ( ) As a result, the initial rate ratio of degassing is 1.15 which is confirmed by the observation. Another example of the binary isotope mixture is the neon isotope mixture of 20 Ne/ 22 Ne [16]. The isotopic diffusivity ratio for neon ( 20 Ne/ 22 Ne) in silicate glasses appears to equal the inverse square-root of the isotopic masses as 1.05.

Dark Matter and the Dominance of Matter over Antimatter
It is speculated that since the SM is unable to explain cosmological observations of the dominance of matter over antimatter, the apparent dark-matter content of the Universe, or explain the patterns seen in the interaction strengths of the particles, the new physics derived from the violation of lepton universality can explain the SM's shortcomings [1]. In the Section 2, the periodic table of elementary particles explains the patterns seen in the interaction strengths of the particles for leptons and quarks. In this Section, dark matter and the dominance of matter  Table 2.
The lowest energy gauge boson (B 5 ) at d = 5 is the Coulomb field for electromagnetism. The second gauge lowest boson (B 6 ) at d = 6 is basic gluon (g* = 70 MeV ≈ one half of pion) is the strong force as the nuclear force in the pion theory [19] where pions mediate the strong interaction at long enough distances (longer than the nucleon radius) or low enough energies. B 6 is denoted as basic gluon, g*. At short enough distances (shorter than the nucleon radius) or high enough energies, gluons emerge to confine fractional charge quarks. Fractional  isolated fractional charge quark is allowed, and only collective integer charge quark composites are allowed. In general, collective fractional charges are confined by the short-distance confinement force field where the sum of the collective fractional charges is integer [20]. As a result, fractional charges are confined and collective. The confinement force field includes gluons for collective fractional charge quarks in hadrons and the magnetic flux quanta for collective fractional charge quasiparticles in the fractional quantum Hall effect (FQHE) [21] [22] [23].
The third lowest boson (B 7 ) at d = 7 is Z L for the left-handed weak interaction among leptons and quarks. Massive weak bosons produce short-distance interaction. B 8 at d = 8 is Z R for the right-handed weak interaction among dark matter neutrinos as dark matter neutrino oscillation. The symmetry between Z R and Z L provides the neutrino oscillation for both baryonic matter neutrinos [24] and dark matter neutrinos. B 9 as the gauge boson represents dark matter repulsive force. The condensed baryonic gas at the critical surface density (derived from the acceleration constant a 0 in MOND [25] [26]) induces the creation tensor for dark matter repulsive force to transform dark matter in the region into repulsive dark matter repulsing one another, corresponding to the Farnes' repulsive dark matter [8] [27].
Before the emergence of dark matter repulsive force, dark matter in the CMB was not repulsive. B 10 at d = 10 is for the gauge boson for particle-antiparticle asymmetry to provide the slight excess of particle in particle-antiparticle at the Big Bang, while B 8 has particle-antiparticle symmetry. (B 9 emerged long after the Big Bang.) As a result, the excess of particle is α 4 (2.8 × 10 −9 ) per particle-antiparticle (photon) for the ratio between B 8 and B 10 . Since baryonic matter is 1/6 of dark matter and baryonic matter [5] [8] [17], the baryonic matter excess is 4.7 × 10 −10 which is in a good agreement with 6 × 10 −10 for the ratio of the numbers between baryonic matter and photons in the Big Bang nucleosynthesis [28].

Summary
In summary, this paper purposes an explanation for the recent evidence for the into K + , e + e − , and µ + µ − , while b 8 decays into K + , e + e − , and forbidden µ + µ − , the calculated ratio (R K ) of branching fractions for B + → K + µ + µ − and B + → K + e + e − is 0.5/(0.1864 × 0.5 + 0.5) = 0.843 in excellent agreement with the observed 0.846.
The agreement between the calculated R K and the observed R K confirms the validity of the periodic table of elementary particles which provides the answers for the dominance of matter over antimatter, dark-matter, and the mass hierarchy of elementary particles.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.