# Computation of the Deuteron Mass and Force Unification via the Rotating Lepton Model

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## Abstract

**:**

## 1. Introduction

## 2. The Rotating Lepton Model (RLM)

_{o}is the neutrino rest mass, and γ is the Lorentz factor (1 − v

^{2}/c

^{2})

^{−1/2}, with Newton’s gravitational law using the gravitational masses of the neutrinos. According to special relativity [2,3,4], γm

_{o}is equal to the relativistic mass of a particle of rest mass m

_{o}, while γ

^{3}m

_{o}is equal to its inertial mass [2,3,4]. According to the equivalence principle, the latter is equal to the gravitational mass, m

_{g}, i.e.,

_{g}, is defined from Newton’s gravitational law, i.e., from

^{2}[4,5,6,7,8,9,10], i.e., a value within the range of the heaviest neutrino mass, 0.048 ± 0.01 eV/c

^{2}[4,14,15], and exhibits bistability, i.e., there are two γ values satisfying Equation (6) for any r value above $\left(2.96\right)\mathrm{G}{\mathrm{m}}_{\mathrm{o}}/{\mathrm{c}}^{2}(=1.71\times {10}^{-64}\mathrm{m})$ [4]. The high γ branch corresponds to highly relativistic speeds; as shown in Figure 2, the model solution is found by the intersection of Equation (6) with the de Broglie equation

^{2}to

^{2}[13] and ${\mathrm{m}}_{\mathrm{o}}=$ 0.0437250 eV/c

^{2}[4,11,12], where the m

_{o}value lies again within the range of the heaviest neutrino mass, (0.048 ± 0.01 eV/c

^{2}) [4,11,12], and where ${\mathrm{m}}_{\mathrm{Pl}}{(=\hslash \mathrm{c}/\mathrm{G})}^{1/2}=1.220890\times {10}^{28}\text{}{\mathrm{eV}/\mathrm{c}}^{2}$.

## 3. The Deuteron Model

^{2}, which is 2.225 MeV/c

^{2}smaller than the sum of the proton (938.272 MeV/c

^{2}) and neutron (939.565 MeV/c

^{2}) masses, respectively. The difference is commonly called the interaction energy.

_{o}, eigenstate, i.e., of mass 3 in the normal hierarchy [4,5,14,15] (Figure 1). These neutrinos in their gravitationally confined relativistic state have γ values as high as 10

^{10}; hence, their relativistic mass is of the order of 0.3 GeV/c

^{2}, i.e., in the range of quarks, thus acting as u and/or d quarks. Therefore, it follows that the deuteron can be modeled as a composite particle comprising six neutrinos, three from its proton and three from its neutron initial constituents.

## 4. Mathematical Modeling and Mass Computation

^{2}[1,11], located at the center of the structure, to obtain

_{o}value of 0.0437250 eV/c

^{2}found to yield exactly the neutron mass via Equation (9), Equations (19) and (20) give m

_{d}= 1876.368 MeV/c

^{2}. Exact agreement with the experimental value of m

_{d}= 1875.613 MeV is obtained for the value ${\mathrm{m}}_{\mathrm{o},\mathrm{d}}=0.0436986\text{}\mathrm{eV}/{\mathrm{c}}^{2}$. This value differs by less than 0.06% from the m

_{o}value, denoted ${\mathrm{m}}_{\mathrm{o},\mathrm{n}}=0.0437250\text{}\mathrm{eV}/{\mathrm{c}}^{2}$, which gives exact agreement with the neutron mass m

_{n}(Equation (9)). Furthermore, if we use ${\mathrm{m}}_{\mathrm{o},\mathrm{n}}^{}$ or ${\mathrm{m}}_{\mathrm{o},\mathrm{d}}^{}$ to compute both the neutron and the deuteron masses from Equations (9), (19), and (20), the computed values differ by less than 0.05%.

## 5. Conclusions and Future Work

_{d}value (1876.37 MeV/c

^{2}) and the experimental value (1875.612 MeV/c

^{2}), without use of any adjustable parameters, shows that, similarly to the strong force [4,5,10], the residual strong force or nuclear force can also be modeled quantitatively as relativistic gravity. This force is computed via Equation (5), which accounts for the special relativistic equation for gravitational mass ${\mathrm{m}}_{\mathrm{g}}(={\mathsf{\gamma}}^{3}{\mathrm{m}}_{\mathrm{o}})$ [2,3] in conjunction with Newton’s universal gravitational law.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Glossary/Nomenclature/Abbreviations

QCD | Quantum chromodynamics |

RLM | Rotating lepton model |

m_{1}, m_{2}, m_{3} | Rest neutrino eigenmasses |

m_{n} | Neutron mass |

m_{e} | Electron mass |

m_{μ} | Muon mass |

m_{Pl} | ${(\hslash \mathrm{c}/\mathrm{G})}^{1/2}=1.221\times {10}^{28}\mathrm{eV}/{\mathrm{c}}^{2},$ Planck mass |

γ | Lorentz factor = ${(1-{\mathrm{v}}^{2}/{\mathrm{c}}^{2})}^{-1/2}$ |

m_{o} | Rest mass |

v | Particle speed |

r | Rotational radius |

G | Gravitational constant, 6.6743 × 10^{−11} m^{3}/(kgs^{2}) |

m_{g} | Gravitational mass |

m_{i} | Inertial mass |

c | Speed of light in vacuum |

$\hslash $ | Reduced Planck’s constant |

eV/c^{2} | Mass unit, 1.783 × 10^{−36} kg |

$\overline{\mathsf{\gamma}}$ | Mean γ value |

F_{1} | Centripetal force in the equatorial rotation |

F_{2} | Centripetal force in the top–bottom rotation |

m_{d} | Total deuteron mass |

m_{d,r} | Mass of the rotating components of the deuteron |

m_{o,d} | Neutrino rest mass in the deuteron |

m_{o,n} | Neutrino rest mass in the neutron |

1 | Equatorial rotation |

2 | Top–bottom rotation |

d | Deuteron |

Quantum chromodynamics (QCD) | The strong force theory of the standard model (SM). |

Rotating lepton model (RLM) | It has the same goals as QCD, but has no adjustable parameters. |

Deuteron | The nucleus of the deuterium, comprising a proton and a neutron. |

Quarks | Building blocks of hadrons, identified as relativistic rotating neutrinos in the RLM. |

Strong force | Force binding the constituents of hadrons. |

Relativistic gravity | Newtonian gravity, also accounting for special relativity. |

Residual strong force | The force binding nucleons with other nucleons. |

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**Figure 1.**Comparison of computed via the RLM (horizontal dotted lines) [12] and experimental [14,15] neutrino eigenmasses; m

_{n}and m

_{μ}are the neutron and muon masses, respectively. The m

_{3}, m

_{2}, and m

_{1}expressions were retrieved from [12]. These equations imply that, interestingly, m

_{3}/m

_{2}= m

_{2}/m

_{1}≈ 6.3 ≈ 2π.

**Figure 2.**Plot of the relativistic equation for circular motion of three gravitating neutrinos each of rest mass m

_{o}(Equation (6)) and of the de Broglie equation for each of these particles (Equation (7)), which accounts for its dual wave–particle nature, showing that the intersection of these two curves defines the neutron mass (939.565 MeV/c

^{2}) and radius (0.630 fm) in quantitative agreement with experiment [16]. The heaviest neutrino mass value used (m

_{o}= 0.0437·eV/c

^{2}) is computed from the equation ${\mathrm{m}}_{\mathrm{o}}={\mathrm{m}}_{\mathrm{n}}^{3/2}/{3}^{1/8}{\mathrm{m}}_{\mathrm{Pl}}^{1/2}$ derived from Equations (5) and (7), and it lies within the current experimental uncertainty limits of the heaviest neutrino mass measured at the Super-Kamiokande [14,15]. The neutron mass (939.565 MeV/c

^{2}) is a factor of γ (=7.163 × 10

^{9}) larger than the rest mass of the three neutrinos (0.131 eV/c

^{2}) according to Equation (9).

**Figure 3.**Schematic of the RLM structure of the triangular octahedron model (

**a**) for the deuteron, drawn using the de Broglie–Compton wavelength spheres of the rotating neutrinos (

**b**,

**c**) each of radius $\mathrm{r}=0.631\text{}\mathrm{f}\mathrm{m}=\hslash /\overline{\mathsf{\gamma}}{\mathrm{m}}_{\mathrm{o}}\mathrm{c}$ and showing the seven rotational axes, of which four are equatorial type (

**b**) and three are top–bottom type (

**c**).

**Figure 4.**Centripetal force components (marked by red arrows) of the forces F

_{1}and F

_{2}for the equatorial (

**a**–

**c**) and top–bottom (

**d**,

**e**) neutrino rotations, respectively. Equatorial forces: ${\mathrm{F}}_{1}=2{\mathrm{F}}_{\mathrm{a}}+2{\mathrm{F}}_{\mathrm{b}}+{\mathrm{F}}_{\mathrm{c}};$ ${\mathrm{F}}_{\mathrm{a}}=(1/2\sqrt{3})(\mathrm{G}{\mathrm{m}}^{2}/{\mathrm{r}}^{2});$ ${\mathrm{F}}_{\mathrm{b}}=(1/6\sqrt{3})(\mathrm{G}{\mathrm{m}}^{2}/{\mathrm{r}}^{2});$ ${\mathrm{F}}_{\mathrm{c}}=(1/3\sqrt{6})(\mathrm{G}{\mathrm{m}}^{2}/{\mathrm{r}}^{2})$. Top–bottom forces: ${\mathrm{F}}_{2}=4{\mathrm{F}}_{\mathrm{d}}+{\mathrm{F}}_{\mathrm{e}};\text{}{\mathrm{F}}_{\mathrm{d}}=(\sqrt{2}/6)(\mathrm{G}{\mathrm{m}}^{2}/{\mathrm{r}}^{2});$ ${\mathrm{F}}_{\mathrm{e}}=(1/6)(\mathrm{G}{\mathrm{m}}^{2}/{\mathrm{r}}^{2})$.

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**MDPI and ACS Style**

Vayenas, C.G.; Grigoriou, D.; Tsousis, D.; Parisis, K.; Aifantis, E.C.
Computation of the Deuteron Mass and Force Unification via the Rotating Lepton Model. *Axioms* **2022**, *11*, 657.
https://doi.org/10.3390/axioms11110657

**AMA Style**

Vayenas CG, Grigoriou D, Tsousis D, Parisis K, Aifantis EC.
Computation of the Deuteron Mass and Force Unification via the Rotating Lepton Model. *Axioms*. 2022; 11(11):657.
https://doi.org/10.3390/axioms11110657

**Chicago/Turabian Style**

Vayenas, Constantinos G., Dimitrios Grigoriou, Dionysios Tsousis, Konstantinos Parisis, and Elias C. Aifantis.
2022. "Computation of the Deuteron Mass and Force Unification via the Rotating Lepton Model" *Axioms* 11, no. 11: 657.
https://doi.org/10.3390/axioms11110657