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The particle physics Standard Model involves three charge 0 neutrinos, three charge e leptons, three charge (2/3)
*e* quarks, and three charge
−(1/3)
*e* quarks, where
*e* is electron charge. However, the Standard Model cannot explain why there are three generations of particles in each charge state and makes no predictions relating to quark and lepton masses. This analysis, treating Standard Model particles as spheres with radii 1/4 the particle Compton wavelength, explains three, and only three fermions are in each charge state and relates first generation quark masses to the electron mass.

The particle physics Standard Model cannot explain why there are three generations of quarks and lepton, and makes no predictions relating quark and lepton masses [

● must be modified to accommodate neutrino mass to account for oscillations between neutrino states as they propagate through space;

● treats fundamental particles as point particles with angular momentum ℏ / 2 , or ℏ . Angular momentum is usually defined for rotating objects extended in space and, regarding point particles with angular momentum, we might ask what is rotating.

● is formulated in quantum field theory, based on continuum mathematics with infinite degrees of freedom. However, the holographic principle [^{122}] of bits of information will ever be available to describe the observable universe, suggesting quantum field theory approximates an underlying discrete theory.

● involves point fermions with infinite energy density and physical theories involving infinities are logically complicated and debatable [

This analysis, treating Standard Model particles with mass as spheres with finite radii, addresses these important issues. In particular, it explains why three, and only three fermions are in each charge state and relates first generation quark masses to electron mass.

Masses m i , in (MeV/c^{2}), of charged Standard Model fermions [

Neutrino masses are estimated using known [^{±} with mass 80.4 GeV/c^{2}, Z with mass 91.2 GeV/c^{2}, and Higgs with mass 125 GeV/c^{2}.

This analysis describes fundamental fermions as spherical shells with radius l 4 rotating around an axial core centered on the axis of rotation, with half of any fermion charge on the shell surface at each end of the axis of rotation. Fundamental fermions are then represented by Godel solutions of Einstein’s equations, with average matter density ρ equal average fermion mass density, pressure

Fermion | Mass m_{i} (MeV/c^{2}) | Wavelength l_{i} (F) | Charge |
---|---|---|---|

Electron | 0.511 | 386 | e |

Up quark | 2.16 | 91.4 | 2e/3 |

Down quark | 4.67 | 42.3 | −e/3 |

Strange quark | 93 | 2.12 | −e/3 |

Muon | 105.7 | 1.87 | e |

Charm quark | 1270 | 0.155 | 2e/3 |

Tau lepton | 1777 | 0.111 | e |

Bottom quark | 4180 | 0.0472 | −e/3 |

Top quark | 173,000 | 0.00114 | 2e/3 |

( 1 2 ) ρ c 2 from negative vacuum energy density − ( 1 2 ) ρ c 2 , and effective internal gravitational constant G f determined by ω = 2 π G f ρ . Charge is on the shell at the axis of rotation, so rotation does not cause radiative loss of energy by accelerated charge. Rotation axis orientation is unknown until z component of fermion angular momentum is measured, so fermion mass appears sinusoidally distributed on a disk of radius (l/4) perpendicular to the line of sight.

Fundamental fermions, considered as spheres with size characterized by Compton wavelengths l, have three associated geometric quantities, volume ~l^{3}, surface area ~l^{2}, and diameter ~l. Mass and pressure distribution in fundamental fermions identifies three wavelengths in each charge state as solutions of a cubic equation A l 3 + B l 2 + C l = 0 . Describing mass and pressure distribution in terms of surface and linear elements requires minimum shell thickness and core radius near the Planck length l P = ℏ G c 3 . In each charge state n e 3 , with n = 0 , 1 , 2 or 3, total fermion mass is the sum of mass equivalent of pressure, m 2 , in the volume, mass equivalent of surface pressure π 4 S l 2 , and core mass L l . So 4 3 π ρ ( l 4 ) 3 = 4 3 π ρ 2 ( l 4 ) 3 + 4 π S ( l 4 ) 2 + 2 L ( l 2 ) . Written as A l 3 − B l 2 − C l = 0 , with A = π 96 ρ , B = π S 4 , and C = 2 L , the discriminant B 2 C 2 − 4 A C 3 is positive regardless of the sign of B and the equation has three real roots corresponding to three fermion Compton wavelengths in a charge state. Nickalls [

Fermion spheres with radius l 4 and core radius r have moment of inertia I = 2 5 m 2 ( l 4 ) 2 + 2 3 π 4 S l 2 ( l 4 ) 2 + 1 2 L l r 2 , with negligible last term because r ≪ l . Angular velocity ω = ℏ 2 I = 8 c ( 0.2 l − l N ) and tangential speed of points on the spherical shell equator as multiple of the speed of light v T c = ω l 4 c = 2 l ( 0.2 l − l N ) . v T c > 1 for lowest mass fermions in each charge state, with closed time-like curves within those Godel solutions acceptable in fundamental fermions unchanging from creation to annihilation. From ω = 2 π G f ρ , G f G = 3 l 4 l P 2 ( 0.2 l − l N ) 2 .

Ground state fundamental fermions constituents of atoms and molecules, differ from other generation fundamental fermions by having core mass less than total mass, tangential speeds v T c > 1 , and larger internal gravitational constants. With fine structure constant e 2 ℏ c = 1 137 , electrostatic potential energy of fundamental fermions from repulsion between equal surface charges at the rotation axis is ( n e 6 ) 2 / ( l 2 ) = n 2 m e 2 9 ℏ c = n 2 m 1233 . If electrostatic potential energy is the same for all charged ground state fundamental fermions and electron mass m e = 0.511 MeV , up quark mass m u = 4 m e = 2.04 MeV and down quark mass m d = 9 m e = 4.60 MeV , well within quark mass error bars [

Treating massive Standard Model bosons as spheres with finite radii and an internal gravitational constant is simpler than for fundamental fermions. W^{±} and Z bosons as uniform spheres with moment of inertia I = 2 5 m r 2 and radius r = l 4 have angular momentum ℏ = I ω , so ω = 40 ℏ m l 2 . For Godel systems with internal gravitational constant G i , ω = 2 π G i ρ , resulting in G i G = 25 3 ( l l P ) 2 , where Planck length l P = 1.62 × 10 − 20 F . Higgs bosons are treated as static Einstein solutions of general relativity with matter energy density ρ c 2 and positive vacuum energy density 1 2 ρ c 2 , opposite the negative vacuum energy density of Godel solutions. The Friedmann equation for the radius of those closed, homogeneous, isotropic systems with internal gravitational constant G i is ( d R d t ) 2 − 8 π G i 3 [ ρ c 2 ( R 0 R ) 3 + 1 2 ρ c 2 ] ( R c ) 2 = − c 2 , with ( d R d t ) = 0 , R = R 0 = l H 4 , and Higgs Compton wavelength l H , resulting in G i G = 1 12 ( l H l P ) 2 .

Considering fundamental fermions as spherical shells rotating around an axial core, identified with Godel solutions to Einstein’s equations, in each charge state,

● three fermion wavelengths specify volume, surface, and core mass as multiples of total fermion mass, and

● each fermion mass and wavelength specify angular velocity, tangential speed of points on the shell equator, and effective internal fermion gravitational constant.

Electrostatic potential energy in ground state fundamental fermions, from repulsion between equal surface charges at the rotation axis, is identical if up quark mass = 4 m e = 2.04 ( MeV / c 2 ) and down quark mass = 9 m e = 4.60 ( MeV / c 2 ) .

Inquiry on the nature of mass by the late UCSF Professor Leon Kaufman, and on the nature of spin by my wife, Lou Mongan, led to this analysis.

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

Mongan, T.R. (2020) Standard Model Particles with Mass Treated as Spheres with Finite Radii. Journal of Modern Physics, 11, 1993-1998. https://doi.org/10.4236/jmp.2020.1112126