Rencontres du Vietnam-2018 Windows on the Universe
A fine-tuned interpretation of the charged lepton mass hierarchy in a microscopic cosmological model Vo Van Thuan Duy Tan University (DTU) 3 Quang Trung street, Hai Chau district, Danang, Vietnam
Vietnam Atomic Energy Institute (VINATOM-Hanoi) Email: [email protected]
ICISE-Quy Nhon, Vietnam, August 05-11, 2018 Contents
1. Time-Space Symmetry: Motivation–Cylindrical Geometry
2. Dual solutions of the gravitational equation for a microscopic cosmology
3. Charged lepton mass hierarchy in the first order approximation
4. The second approximation of tauon mass by minor curvatures
5. The third approximation by perturbative fine-tuning minor curvatures
6. Discussion 1- Time-Space Symmetry (1): Problem
Problems: The origin of the Mass Hierarchy of elementary fermions? Fermion masses are induced in interaction of genetic particles with Higgs field. All Yukawa couplings are determined independently. Interpretation of mass hierarchy within the Standard Model (SM) or even in moderate extension of SM are mostly phenomenological with qualitative predictions (e.g. models with flavor democracy, quark-lepton correspondences etc.). 1/ Quark mass hierarchy is parametrized by CKM mixing matrices being in consistency with QCD within SM; 2/ Neutrino mass hierarchy and their masses are parametrized by MNSP mixing matrices, being solved by extension of SM or by Beyond SM approaches; 3/ Mass hierarchy of charged leptons is as large as of both up-type and down-type quarks. Probably, this is a Beyond SM problem ! Exception: an excellent prediction by the Koide empirical formula, the origin of which is interpreted by Sumino in correlation with vacuum expectations of heavier family gauge bosons at 10E2-10E3 TeV.
Windows on the Universe, August 5-11, 2018 1- Time-Space Symmetry (2): Motivation
Objectives: To solve the mass hierarchy problem of charged leptons and to predict tauon mass with high accuracy. Motivations of the present research: Looking for an alternative Beyond SM ≡ Higher-dimensional space-time approach. 1/ Searching for a link between: GR and QM, To formulate a “microscopic” cosmological model (MicroCoM) to describe masses and mass hierarchy of leptons.
2/ Searching for a link between: The number of dimensions = The number of lepton generations. To add time-like EDs up to 3D-time: Time-Space Symmetry={3T-3X}: 3 orders of time-like curvatures induce masses of 03 charged leptons (electron, muon, tauon). To predict the mass ratios and to calculate the absolute masses of charged leptons.
Windows on the Universe, August 5-11, 2018 1- Time-Space Symmetry (3): References Phenomenological interpretations within SM or by moderate extension of SM. See, e.g. reviews: [1] H. Fritzsch and Z.Z.Xing, Phys.Rev.D61(2000)073016. [2] W.G.Hollik, U.J.S.Salazar, Nucl.Phys.B892(2015)364. Koide empirical formula (an excellent prediction of tauon mass): [3] Y.Koide, Phys.Lett.B 120(1983)161.
[4] Y. Sumino, Phys.Lett.B671(2009)477 (Effective field theory interpretation of Koide formula by family gauge symmetry U(3) broken at ퟏퟎퟑ − ퟏퟎퟑTeV). Beyond SM- an alternative: ED geometrical dynamics - Modern Kaluza- Klein Theories: 5D Space-Time-Matter (STM) theory is applied for QM: [5] P.S. Wesson, Phys.Lett.B 701(2011)379; Phys.Lett.B 706(2011)1; [6] Phys.Lett.B 722(2013)1; IJMPD 24(2015)1530001. Following induced matter approach we proposed a semi- phenomenological model with {3T-3X} Time-Space Symmetry (TSS): Bi-cylindrical GR Equation Formulation of the Microscopic Cosmology in 3T sub-space Charged lepton mass hierarchy Tauon Mass: [7] Thuan Vo Van, Foundations of Physics 47(2017)1559. [8] Vo Van Thuan, arXiv:1711.08346 [physics.gen-ph] to be published. Windows on the Universe, August 5-11, 2018 1- Time-Space Symmetry (4)- Cylindrical Geometry
In an ideal 6D flat time-space 푡1, 푡2, 푡3|푥1, 푥2, 푥3 considering orthonormal sub-spaces 3D-time and 3D-space:
ퟐ ퟐ ퟐ 풅푺 = 풅풕풌 − 풅풙풍 ; summation: 푘, 푙 = 1 ÷ 3. (1.1) Our physics works on its symmetrical “light-cone”: ퟐ Ԧퟐ 2 2 풅풌 = 풅풍 or 푑푡푘 = 푑푥푙 ; summation: 푘, 푙 = 1 ÷ 3 (1.2) Natural units (ħ = 푐 = 1) used unless it needs an explicit quantum dimension. It is equivalent to a 6D-vacuum with: 풅푺=0. Introducing a 6D isotropic plane wave equation: ퟐ ퟐ 흏 흍0 흏 흍0 ퟐ = ퟐ ; (1.3) 흏풕풌 흏풙풍
Where 흍0(풕풌, 풙풍 ) is a harmonic correlation of 푑푡 and 푑푥, containing only linear variables {풕풌, 풙풍}, serving a primitive source of quantum fluctuations in space-time. All chaos of displacements 푑푡 and 푑푥 can form averaged time- like and space-like global potentials 푽푻 and 푽푿.
Vo Van Thuan Windows 27th Rencontreon the Universe, de Blois August June 53,-11, 2015 2018 6 1- Time-Space Symmetry (5)- Bi-Cylindrical Geometry
Suggesting that the global potentials, originally, accelerating linear space- time into curved time-space which describes 3D spinning 휏Ԧ and 푠Ԧ in symmetrical orthonormal subspaces of 3D-time and 3D-space.
For a kinetic state (curved rotation + linear translation): {푡3,푥3} are accepted as longitudinal central axes of a symmetrical bi-cylindrical geometry (흍,흋,푡푘|흍,흋,푥푙); ퟐ ퟐ ퟐ The curved coordinates for 3D-space: 푥푗 ≡ 푥1, 푥2, 푧 with 풅풛 = 풅풙풏 + 풅풙ퟑ ; ퟐ ퟐ ퟐ Similarly, for 3D-time there are 푡푖 ≡ 푡1, 푡2, 푡 with 풅풕 = 풅풕ퟎ + 풅풕ퟑ . 푡0 and 푥푛 are local affine parameters in 3D-time and 3D-space, respectively.
EDs turn into dynamical functions of other space-time dimensions:
흍 = 흍(풕ퟎ, 풕ퟑ, 풙풏, 풙풍) and 흋 = 휴풕 − 풌풋풙풋 = 휴ퟎ풕ퟎ +휴ퟑ 풕ퟑ − 풌풏풙풏 − 풌풍풙풍; (1.4)
Here 흍 = 흍(T).흍 (X) is variable-separable and 흋 is linear dependent. Possible to express them in a bi-cylindrical geometry.
Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 7 1- Time-Space Symmetry (6): Bi-Cylindrical Metrics
In observation of an individual lepton (흉풏 , 풔풏 = ±1/2), due to interaction of a Higgs-like potential 푉푇 the time-space symmetry being spontaneously broken for forming energy-momentum (풅풔ퟎ ≫ 풅흈풔풑풊풏 ≫ 풅흈푷푵푪 ≫ 풅풔푪푷푽) leads to an Asymmetrical Bi-cylindrical Geometry: ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ 풅휮푨 ≡ 풅푺 + 풅흈 = (풅풔ퟎ +풅풔푪푷푽 ) − (풅흈풔풑풊풏 +풅흈푷푵푪 ) = 풅풕 − 풅풛 = ퟐ ퟐ ퟐ ퟐ = [풅흍 풕ퟎ, 풕ퟑ + 흍 풕ퟎ , 풕ퟑ 풅흋 풕ퟎ , 풕ퟑ + 풅풕ퟑ ] − ퟐ ퟐ ퟐ ퟐ −[풅흍 풙풏, 풙ퟑ +흍 풙풏 , 풙ퟑ 풅흋 풙풏, 풙ퟑ + 풅풙ퟑ ] . (1.5) The Asymmetrical Geometry (5) for charged leptons : ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ 풅푺풆 ≈ 풅풔ퟎ − 풅흈푺풑풊풏 = 풅풕 − 풅풛 ≡ 풅풕 − 풅풙풋 ; (1.6) It resembles the special relativity (SR), however, here coordinates {t,z} are curved.
Fig.1.1. Asymmetrical bi-cylindrical geometry:
-Time-like curvature 풅풔ퟎ: odd term, being strong and almost absolute;
-Space-like curvature 풅흈풔풑풊풏: even term being not absolute (quasi-curvature)
Vo Van Thuan Windows 27th Rencontreon the Universe, de Blois August June 53,-11, 2015 2018 8 1- Time-Space Symmetry (7): Breaking Symmetry Phenomenological assumptions 푑푆 and 푑휎 are time-like and space-like intervals introduced for compensating the curvatures to maintain a conservation of linear translation (CLT) in a relation to (1) and (2). From a semi-phenomenological view:
ퟐ ퟐ ퟐ ퟐ ퟐ 풅푺 = 풅풔풐풅풅 − 풅풔풆풗풆풏 = 풅풔ퟎ − 풅풔푪푷푽 ퟐ ퟐ ퟐ ퟐ ퟐ 풅흈 = 풅흈풐풅풅 − 풅흈풆풗풆풏 = 풅흈푷푵푪 − 풅흈푺풑풊풏 . Table 1. Semi-phenomenological based Data for curved geometries of leptons:
Dynamics Higgs-like Weak CPV potential Spatial source potential interaction spinning
Corresponding 풅풔풐풅풅 풅흈풐풅풅 풅풔풆풗풆풏 풅흈풆풗풆풏 Interval For heavy Major Weak Super weak Minor leptons (e, μ, τ)
Corresponding 풉흉 풉풔 흉 (time-like) s (space-like) Rotational character Iso-Helicity Helicity Pseudo-spin Spin The Asymmetrical Geometry (5) for charged leptons : ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ 풅푺풆 ≈ 풅풔ퟎ − 풅흈푺풑풊풏 = 풅풕 − 풅풛 ≡ 풅풕 − 풅풙풋 ; (1.6) It resembles the special relativity, however, here coordinates {t,z} are curved.
Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 20182015 9 2- A duality of higher-dimensional gravitational equation (1)
Applying Geometry (1.6) with signatures {- - - +++} of bi-cylindrical 풎 curvatures, the gravitational equation in vacuum (푻풊 =0) reads: ퟏ 푹풎 − 휹풎푹 = ퟎ; (2.1) 풊 ퟐ 풊 In an apparent vacuum 훬 = 0, Eq. (2.1) leads to {3T,3X}-Ricci vacuum equation: 훽 휎 푅훼 푇 + 푅훾 푋 = 0. (2.2) Where Ricci tensors with 훼, 훽 ∈ 3퐷-time = 3T and ones with 훾, 휎 ∈ 3퐷-space = 3X.
Recalling 휓 = 휓(푦) and 휑 = 휑(푦) are functional, where: 푦 ≡ 푡, 푧 ∈ 푡, 푥푗 ≡ {푡0, 푡3, 푥푛, 푥푙} ∈ {푡푖, 푥푗}; 푦3 ≡ {푡3, 푥3} being implicitly embedded in 3D-time: 푡3 ∈ {푡푘} and in 3D-space: 푥3 ∈ {푥푙} , respectively. Prior assuming the Hubble law of the cosmological expansion is applied to the Bi-Cylindrical Geometry (1.6) of microscopic space-time: 휕ψ 휕푦 1 = 푣푦 = 퐻푦ψ and therefore = ; (2.3) 휕푦 휕ψ 퐻푦ψ
Where 퐻푦 is the “micro-Hubble constant”, then the expansion rate 푣푦 increases proportional to the “micro-scale factor” ψ;
Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 20152018 10 2- A duality of higher-dimensional gravitational equation (2)
GR equation (2.2) with only diagonal terms leads to two independent sub- ퟑ ퟑ equations: 푹ퟑ 푻 + 푹ퟑ 푿 = ퟎ ; (2.4) 흍 흍 푹흍 푻 + 푹흍(푿) = ퟎ ; (2.5)
Sub-equation (2.4) defines conservation of linear translation (CLT) of Eq. (1.3): 흏ퟐ흍 흏ퟐ흍 − ퟐ + ퟐ = ퟎ. (2.6) 흏풕ퟑ 흏풙ퟑ which leads to a Lorentz-like condition for compensating longitudinal fluctuations: 2 2 휔3 − 푘3 ψ = 0. (2.7)
The sub-equation (2.5): in a time-space symmetrical representation, accounting Lorentz-like condition (2.7) reads: ퟐ ퟐ 흏ퟐ흍 흏흋 흏ퟐ흍 흏흋 ퟐ − 흍 = ퟐ − 훙, (2.8) 흏풕 흏풕ퟎ 흏풙풋 흏풙풏
Where 푦= {푡푖, 푥푗} as for summation of time-like and space-like variables; due to the 휕2ψ 휕2ψ 휕2ψ 휕2ψ 휕2ψ 휕2ψ 3D-local orthogonality: 2 = 2 + 2 and 2 = 2 + 2 . (2.9) 휕푡 휕푡0 휕푡3 휕푥푗 휕푥푛 휕푥푙
Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 11 2- A duality of higher-dimensional gravitational equation (3) For a homogeneity condition Eq.(2.8) is getting a symmetrical equation of bi- geodesic acceleration of deviation ψ : 2 2 휕2ψ 휕휑 휕2ψ 휕휑 2 − ψ = 2 − ψ . (2.10) 휕푡0 휕푡0 휕푥푛 휕푥푛 Due to 3D local geodesic conditions in 3D-time and in 3D-space, both sides in (2.10) are getting independent: ퟐ 흏ퟐ훙 흏흋 In 3D-time: ퟐ − 흍 = ퟎ. (2.11-T) 흏풕ퟎ 흏풕ퟎ ퟐ 흏ퟐ훙 흏흋 In 3D-space: ퟐ − 훙 = ퟎ . (2.11-X) 흏풙풏 흏풙풏
Then it leads to de Sitter-like exponential sub-solutions being able to describe Hubble-like expansion in microscopic 3D-time or 3D-space, correspondingly. Those conditions will be applied in a so-called “microscopic” cosmological model (MicroCoM) for lepton mass hierarchy. The separated geodesic conditions in 3D sub-spaces ensure that due to a symmetry-breaking, their scales are able to renormalized independently without violation of an invariant formalism. In a result Non-zero mass terms appeared.
Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 20182015 12 2- A duality of higher-dimensional gravitational equation (4) Recall that Geometry (1.6) is formulated due to a Higgs-like potential, + producing time-like polarization 푽푻푷 ⇒ 퐏 and a week space-like PNC: ퟐ ퟐ ퟐ 흏흋 흏흋 ퟐ + ퟐ 흏흋 ퟐ 2 (푉푇푷) = 푉푇 − + + ψ ≡ 풇풆 흌 + 흓ퟎ ψ ⇒ (푃 ) = + ψ ≡ (풇풆흓ퟎ) ψ = 푚0ψ ; (2.12) 흏풕ퟎ 흏풕ퟎ 흏풕ퟎ where 흌 is Higgs field and ϕퟎ is Higgs vacuum; 풇풆 is Higgs-electron interaction coupling . Transformation from 6D time-space to 4D space-time is performed. Then Geodesics (2.8) turns to a formal 4D Asymmetrical equation:
ퟐ 흏ퟐ흍 흏ퟐ흍 휕휑 ퟐ − ퟐ = [횲푻− −횲푳 흍; (2.13) 흏풕 흏풙풋 휕푥푛 풆풗풆풏
where : Effective strong potentials V푻 of a time-like “cosmological constant” 횲푻 with a residue of ퟐ ퟐ 흏흋 흏흋 P-odd component 휦푳 fulfilled breaking symmetry: [횲푻−횲푳]흍 ≡ + − 푳 흍. 흏풕ퟎ 흏풙풏 A transformation to imaginary variables 풚 → 풊. 풚 leads (2.8) to a wave-like solution. Accordingly, with Lorentz-like condition (2.7), the wave-like solution reads:
ퟐ ퟐ ퟐ ퟐ 흏 흍 흏 흍 흏흋 ퟐ 흏흋 − ퟐ + ퟐ = + − 퐵푒 풌풏. μ풆 풆풗풆풏 − 푳 흍; (2.14) 흏풕 흏풙풋 흏풕ퟎ 흏풙풏 where 퐵푒 is a calibration factor and μ풆 is magnetic dipole moment of electron; its orientation is in correlation with spin vector 풔 and being P-even.
Vo Van Thuan Windows 27th Rencontreon the Universe, de Blois August June 53,-11, 2015 2018 13 2- A duality of higher-dimensional gravitational equation (5)
Re-scaling (2.14) with Planck constant and Compton length, then adopting the quantum operators, as a rule for transformation from the superluminal phase frame 흏 흏 흏 흏 to the subluminal realistic frame: → 풊. ħ = 푬 and → −풊. ħ = 풑ෝ풋 흏풕 흏풕 흏풙풋 흏풙풋 Equation (2.14) leads to a generalized KGF equation with a wave-like 흍 ≡ 흍풘: ퟐ ퟐ ퟐ 흏 흍 ퟐ 흏 흍 ퟐ −ħ ퟐ + ħ ퟐ − 풎 흍 = ퟎ ; (2.15) 흏풕 흏풙풋 2 2 ퟐ Or in momentum representation (흍 → 흍 ≡ 흍풑): (퐸 −풑풋 ) 흍풑= 풎 흍풑; (2.16) ퟐ ퟐ ퟐ ퟐ ퟐ ퟐ where : 풎 = 풎ퟎ −휹풎 = 풎ퟎ − 풎푺 − 풎푳 풎ퟎ = ħΩ0 is the conventional rest mass, defined by 횲 푻; 풎푺 as a P-even contribution links with an external rotational curvature in 3D-space; 풎푳 ≪ 풎푺 is a tiny mass factor generated by 횲 푳, related to a P-odd effect of parity non-conservation (PNC).
For depolarized fields, applying (2.11) and ignoring 횲 푳, i.e. 풎 → 풎ퟎ, and 푥푗 → 푥푙, Equation (2.15) is identical to the traditional KGF equation (for spin-zero particles):
ퟐ ퟐ ퟐ 흏 흍 ퟐ 흏 흍 ퟐ −ħ ퟐ + ħ ퟐ − 풎ퟎ 흍 = ퟎ ; (2.17) 흏풕 흏풙풍 Generalized KGF Equation (2.15) serves as a QM motion equation of spinning particle, in particular, as the squared Dirac equation of electron. The curved geometry (1.5) is more general embedding the flat 4D Minkowski space-time for accommodation of the SM of Quantum Field theories.
Vo Van Thuan Windows 27th Rencontreon the Universe, de Blois August June 53,-11, 2015 2018 14 2- A duality of higher-dimensional gravitational equation (6)
Interpretation of QM in 4D space-time proves that the extended space-time can serve for accommodation of QM as well as SM of elementary particles.
In a duality to solution of Gravitational General Relativity Equation the 3D- time-like local geodesic solution (2.11-T): ퟐ 흏ퟐ훙 흏흋 흏ퟐ훙 ퟐ − 흍 = ퟐ − 횲푻흍 = ퟎ. (2.11-T) 흏풕ퟎ 흏풕ퟎ 흏풕ퟎ in a homogeneity and isotropic condition leads to Hubble-like expansion in the microscopic time-space and formulates a Microscopic Cosmological Model (MicroCoM), in analogue to the standard model (SM) of macroscopic cosmology.
In MicroCoM the curvatures from 1D to 3D are formulated in a time- like micro-volume which are evolving toward the future along a linearly imitated longitudinal time axis dt. The proposed MicroCoM is able to solve the mass hierarchy problem of elementary particles, in particular, of charged leptons by linking the time-like curvatures with proper masses.
Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 2015 2018 15 3- Microscopic cosmological model (MicroCoM) for lepton hierarchy (1) In 4D space-time assuming that all leptons, as a material points, are to involve in a common basic time-like cylindrical geodesic evolution with a + + internal 1D circular curvature of the time-like circle 푺ퟏ(휑 ), where 휑 is azimuth rotation in the plane {푡1, 푡2} about 푡3 and its sign “+” means a fixed time-like polarization from the past to the future; Developing more generalized 3D spherical system, described by nautical + angles {휑 , 휃푇, 훾푇}, where 휃푇 is a zenith in the plane {푡1, 푡3} and 훾푇 is another zenith in the orthogonal plane {푡2, 푡3} .
Fig 3.1. Nautical angles to a time-like cylinder.
Vo Van Thuan Windows 27th onRencontre the Universe, de Blois August June 5-3,11, 2015 2018 16 3- MicroCoM for lepton hierarchy (2)
For a n-hyper spherical surface: its intrinsic curvature 푪풏 is a product of all its principle sectional curvatures (n extrinsic 푺ퟏ): −ퟏ −풏 푪풏 = 푪ퟏ. 푪풏−ퟏ = 흍 . 푪풏−ퟏ = 흍 ; 푛 = 1,3 ; (3.1)
In according to general relativity, the energy density 흆풏 correlates with its curvature and the density 흆ퟏ of lightest lepton as: 흐 흐ퟎ ퟏ ퟏ 흆 = ퟎ = = 흆 ; (3.2) 풏 흍풏 흍 흍풏−ퟏ ퟏ 흍풏−ퟏ
Where the factor 흐ퟎ is assumed a universal lepton energy factor (universal, because all 3 generations are involved in cylindrical condition and having the same lepton energy factor 흐ퟎ).
Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 2015 2018 17 3- MicroCoM for lepton hierarchy (3)
Electron oscillating on a fixed line-segment of the time-like amplitude 휱, + formulating 1D proper (or co-moving) “volume”: 푽ퟏ 휑 = 휱 = 흍푻; where 푻 is the 1D time-like Lagrange radius. For instance, 휱 plays a role of the time-like microscopic Hubble radius and the wave function 흍 plays a role of the time-like scale factor. The mass of electron defined by 1D Lagrange “volume” will be: 흐 풎 = 흆 푽 = 흆 휱 = ퟎ 흍푻 = 흐 푻 = 흐 푾 ; (3.3) ퟏ ퟏ ퟏ ퟏ 흍 ퟎ ퟎ ퟏ
where 푾ퟏ is the dimensionless Lagrange volume of electron.
For muon and tauon except the basic time-like cylindrical curved evolution 휑+, it needs to add ED curvatures made by evolution in simplest configurations of hyper-spherical “surfaces”:
i/ 푺ퟏ (휃푇) and 푺ퟏ (훾푇) or ii/ 푺ퟐ(휃푇, 훾푇). Those curvatures are seen with fixed 휱 from the cylindrical basis. Fig 3.2. Linearization of time axis of electron
Vo Van Thuan Windows 27th onRencontre the Universe, de Blois August June 5-3,11, 2015 2018 18 3- MicroCoM for lepton hierarchy (4)
The “co-moving volumes” 푽풏 휱 with fixed 푺풏−ퟏ(휱) are calculated as: 휱 휱 (3.4) 푽풏 휱 = ퟎ 푺풏−ퟏ 푣 풅풗 = 푺풏−ퟏ 휱 ퟎ 풅풗 = 푺풏−ퟏ휱 = 푽ퟏ푺풏−ퟏ (휱 is” fixed” from the 4D-spacetime observation due to not being able to see the additional curvatures, instead of this, observing only their flat footprints at the same maximal level 휱). For homogeneity condition the simplest “2D-rotational co-moving volume” is: + + + 푽ퟐ 휑 , "휃푇 + 훾푇" ≡ 푽ퟐ 휑 , 휃푇 + 푽ퟐ 휑 , 훾푇 = + ퟐ = ퟐ. 푽ퟐ = 푽ퟏ 휑 푺ퟏ 휃푇 + 푺ퟏ 훾푇 = 휱. ퟐ푺ퟏ = ퟒ흅휱 Accordingly, the lepton mass of 2D time-like curved particle (muon) is: ퟏ 흐 풎 = 흆 푽 = 흆 휱. ퟐ푺 = ퟎ ퟒ흅휱ퟐ = 흐 ퟒ흅푻ퟐ = 흐 푾 ; (3.5) ퟐ ퟐ ퟐ ퟏ 흍 ퟏ 흍ퟐ ퟎ ퟎ ퟐ The next simplest 3D-rotational co-moving volume is: + + ퟑ 푽ퟑ 휑 , "휃푇 ⊗ 훾푇" = 푽ퟏ 휑 푺ퟐ 휃푇, 훾푇 = 휱. 푺ퟐ = ퟒ흅휱 Accordingly, the lepton mass of 3D time-like curved particle (tauon) is: ퟏ 흐 풎 = 흆 푽 = 흆 휱. 푺 = ퟎ ퟒ흅휱ퟑ = 흐 ퟒ흅푻ퟑ = 흐 푾 ; (3.6) ퟑ ퟑ ퟑ ퟏ 흍ퟐ ퟐ 흍ퟑ ퟎ ퟎ ퟑ We could use the precise experimental data of electron and muon masses to determine 흐ퟎ and 푻 in according to (3.3) and (3.5) as two free parameters, and then to calculate the tauon mass by (3.6), as a prediction.
Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 3,5- 11,2015 2018 19 3- MicroCoM for lepton hierarchy (5)
Using 푻 = ퟏퟔ. ퟒퟓퟒ, and the lepton energy factor 흐ퟎ = ퟑퟏ. ퟎퟓퟔ 푘푒푉 calibrated to experimental values of 풎풆 and 풎흁 we can predict the mass of tauon 풎흉 then to come to mass ratios of all three charged lepton generations: 풎풆: 풎흁: 풎흉 = 풎ퟏ: 풎ퟐ: 풎ퟑ=1:206.8:3402.2=0.511:105.7:1738.5 (MeV); (3.7) Let’s compare with experimental: C.Patrignani et al., Particle Data Group,Chin.Phys. C40 (2016)
The result (as for the 1rst order of approximation) is resumed in the Table 2: n-Lepton 1-electron 2-muon 3-tau lepton Density,휌푛 휖0 휖0 휖0 휓 휓2 휓3 2 3 Comoving volume, 푽푛 휱 4휋휱 4휋휱 2 3 Formulas of mass, 풎풏 휖0푇 휖04휋푇 휖04휋푇 Calculated mass ratio 1 206.77 3402.18 푻 ≈ ퟏퟔ. ퟒퟓퟒ;
흐ퟎ = ퟑퟏ. ퟎퟓퟔ 풌풆푽 Experimental leton 0.5109989461(31) 105.6583745(24) 1776.86(12) mass, 풎풏 (푀푒푉) Calculated lepton 0.5110* 105.66* 1738.51 mass, 풎풏 (푀푒푉)
*) Same experimental values 풎풆 and 풎흁 for calibration. The deviation of prediction from the experimental tau-lepton mass is - 2,16%.
Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 20 4- 2nd approximation of tauon mass by minor curvatures (1)
Fine-tuning by contribution from minor curvatures 퐶푘 to the major curvature 퐶푛 where k Electron mass is rewritten in 2nd order approximation as: 풎ퟏ ퟐ = 풎ퟏ 푻ퟐ = 휺ퟐ. 푻ퟐ. (4.1) Formula of muon mass is upgraded as: 퐶1 푚2 2 = 푚2 푇2 1 + 훿 ; (4.2) 퐶2 푎 푎 where 푚 푇 = ε . 4휋푇2 ; 훿 is a symbolized scale of the order of ratio . 2 2 2 2 푏 푏 As 퐶1 and 퐶2 are of different dimensions, they are re-normalized by their 푪 푽 .푪 푾 corresponding co-moving volumes: 휹 ퟏ ≡ ퟏ ퟏ = ퟏ, which leads to a 푪ퟐ ퟐ푽ퟐ.푪ퟐ 푾ퟐ ratio of dimensionless Lagrange volumes for comparison. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 21 4- 2nd approximation of tauon mass by minor curvatures (2) In general, 퐶푘 and 퐶푛 of different dimensions are re-normalized by corresponding dimensionless Lagrange volumes as follows: 푾ퟏ 풎ퟏ 푻ퟐ 풎ퟐ ퟐ = 풎ퟐ 푻ퟐ ퟏ + = 풎ퟐ 푻ퟐ ퟏ + = 푾ퟐ 풎ퟐ 푻ퟐ = 풎ퟐ 푻ퟐ + 풎ퟏ 푻ퟐ . (4.3) Tauon mass is corrected up to 퐶2 as: 퐶1 퐶2 푚3 2 = 푚3 푇2 1 + 훿 + 훿 (4.4) 퐶3 퐶3 퐶 푽 .푪 ퟏ 푾 where in particular: 훿 2 ≡ ퟐ ퟐ = ퟐ, which leads to: 퐶3 푽ퟑ.푪ퟑ ퟐ 푾ퟑ 풎ퟏ 푻ퟐ ퟏ 풎ퟐ 푻ퟐ 풎ퟑ ퟐ = 풎ퟑ 푻ퟐ ퟏ + + = 풎ퟑ 푻ퟐ ퟐ 풎ퟑ 푻ퟐ ퟏ = 풎 푻 + 퐦 푻 + 풎 푻 ; (4.5) ퟑ ퟐ ퟏ ퟐ ퟐ ퟐ ퟐ 3 where: 푚3 푇2 = ε2. 4휋푇2 . Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 22 4- 2nd approximation of tauon mass by minor curvatures (3) 1 The factor of 푚 푇 in Equation (4.5) of 푚 2 implies that because 2 2 2 3 the principal muon mass consists of double 푉2 co-moving volume as: + + 푚2 푇2 = 푊1휌1[푆1 휃푇 + 푆1(훾푇)]~ 퐶2.[ 푉2(휑 , 휃푇) + 푉2(휑 , 훾푇)] different factors of 퐶2 contribution for muon and tauon mean that in Equation (4.3) 푪ퟐ refers to muon mass (~ double 푽ퟐ), while in Equation (4.5) 푪ퟐ relates to a correction to tauon mass, taking a single 푽ퟐ only. In the result, both corrected configurations of muon in (4.3) and of tauon in (4.5) contain equally a structural term 풎ퟏ 푻ퟐ to meet the requirement that they are involved in the same basic time-like cylindrical geodesic evolution like electron. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 3,5- 11,2015 2018 23 4- 2nd approximation of tauon mass by minor curvatures (4) Two new free parameters 푇2 and ε2 are determined based on experimental electron and muon masses as: 1 푇 = (푅 − 1) = 16.37451965. 2 4휋 21 ε2 = 31.20695794 (keV) where 푅21 is the experimental mass ratio of muon to electron. Now Equation (4.5) for calculation of tauon mass in the second approximation leads to: 풎ퟑ ퟐ = ퟏퟕퟕퟒ. ퟖퟐ (MeV). The uncertainty of this theoretical prediction is ignorable, because it depends only on experimental errors of electron and muon masses. The calculation in the second order approximation deviates from the experimental tauon mass by 0.11% which is by 18.8 times better than the prediction in the first approximation (2.16%). Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 24 5- 3rd approximation by perturbative fine-tuning minor curvatures (1) The next infinite perturbative orders of minor curvatures 푪풌 to the major curvature 푪풏. Electron mass is modified as: 풎ퟏ ∞ = 풎ퟏ 푻∞ = 휺∞. 푻∞. (5.1) Formula of muon mass is upgraded as: 푞 푞 ∞ 퐶1 ∞ 퐶1 푚2 ∞ = 푚2 푇∞ 1 + σ푞=1 훿 = 푚2 푇∞ σ푞=0 훿 ; (5.2) 퐶2 퐶2 After re-normalization it leads to: 푞 ∞ 푚1 푇∞ 흆ퟐퟏ 풎ퟐ ∞ = 푚2 푇∞ σ푞=0 = 풎ퟐ 푻∞ + 풎ퟏ 푻∞ ; (5.3) 푚2 푇∞ 흆ퟐퟏ−ퟏ 2 where 푚2 푇∞ = ε∞. 4휋푇∞. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 3,5- 11,2015 2018 25 5- 3rd approximation by perturbative fine-tuning minor curvatures (2) The summations converge in infinity to finite quantities as: ∞ 1 휌푖푗 σ푞=0 푞 = ; (5.4) 휌푖푗 휌푖푗−1 푚푖 푇∞ where for i > j: 휌푖푗 = > 1 푚푗 푇∞ Tauon mass is corrected in infinity perturbative orders as: ∞ 푝 ∞ 푞 퐶1 퐶1 푚3 ∞ = 푚3 푇∞ + 푚1 푇∞ 훿 . 훿 + 퐶2 퐶3 푝=0 푞=0 푞 1 ∞ 퐶2 + 푚2 푇∞ σ푞=0 훿 ; (5.5) 2 퐶3 which leads to: 흆ퟐퟏ 흆ퟑퟏ 풎ퟑ ∞ = 풎ퟑ 푇∞ + 풎ퟏ 푇∞ ∗ + 흆ퟐퟏ − ퟏ 흆ퟑퟏ − ퟏ ퟏ ퟐ.흆ퟑퟐ + 풎ퟐ 푇∞ ; (5.6) ퟐ ퟐ.흆ퟑퟐ−ퟏ 3 where: 푚3 푇∞ = ε∞. 4휋푇∞ . Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 3,5- 11,2015 2018 26 5- 3rd approximation by perturbative fine-tuning minor curvatures (3) Two new free parameters 푇2 and ε2 are determined based on experimental electron and muon masses as: 1 푇 = .휌 = 16.37413102, {where 휌 = f(푅 ) is determined from the ∞ 4휋 21 21 21 푚2 푒푥푝 푚2 ∞ 휌21 experimental ratio 푅21 = = = 휌21 + } 푚1 푒푥푝 푚1 ∞ 휌21−1 and: ε∞ = 31.20769862 (keV). By Equation (5.6), in the third approximation: 풎ퟑ ∞ =1776.40 (MeV). This theoretical prediction has also ignorable uncertainty due to high precision of the experimental electron and muon masses. The fine-tuning approximation in infinite perturbation is 83.4 times better than the prediction in the first approximation. In accordance with the notion of the curved 3D-time, it is noticed that all ∞ 1 ratios 휌푖푗 are enough large which make all summations σ푞=0 푞 휌푖푗 fast converged at powers of a perturbative order not higher than the major curvature order in each formula, i.e. 풒 ≤ 풏 ≤ ퟑ. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 3,5- 11,2015 2018 27 6- Discussion (1) Similar to STM theory (Wesson et al.), our time-space symmetrical (TSS) model shows that 4D-Quantum Mechanics originates from the Higher-dimensional General Relativity: TSS geometrical dynamical approach clarifies QM phenomena (meaning of quantum operators, derivation of KGF equation, Heisenberg inequalities, wave-particle duality, origin of Bohm quantum potential, Schrodinger Zitterbewegung)… The extended space-time can accommodate the 4D-SM of QFT. also serves a basis for a Microscopic Cosmological model: Hubble expansion mechanism is applied in microscopic 3D-time subspace that leads to different time-like configurations with hyper-spherical curvatures. Applying the model with a maximal time-like dimension (3D) : By extending cylindrical curvature to additional 2D and 3D time-like hyper-spherical configurations: • mass ratios of charged leptons are estimated satisfactory. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 3,5- 11,2015 2018 28 6- Discussion (2) The only quantitative prediction of tauon mass has been achieved by Koide empirical formula based on electron and muon masses: 2 푚 + 푚 + 푚 = ( 푚 + 푚 + 푚 )2 푒 휇 휏 3 푒 휇 휏 which leads to the quantity 풎흉 푲풐풊풅풆 = ퟏퟕퟕퟔ. ퟗퟕ MeV being in an excellent agreement within 1.σ with experimental tauon mass 풎흉 풆풙풑 = ퟏퟕퟕퟔ. ퟖퟔ ± ퟎ. ퟏퟐ MeV. Some geometrical interpretation of Koide formula was proposed by Kocik (arXiv: 1201.2067v1[physics.gen-ph]) where mass correlations are expressed through Descartes-like circles or with their corresponding squared curvatures. However, no more physics could be developed after this point. Sumino assumed the family gauge symmetry U(3) with new gauge bosons at 10E2-10E3 TeV scale to maintain the Koide formula, which due to breaking leads to the SM as an effective field (EFT). The problem: it seems to require higher symmetries at very high energies, which takes time for the next accelerator generation. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 29 6- Discussion (3) In opposite, our TSS based MicroCoM demonstrates an explicit physical interpretation, which serves a solution to the problem of charged lepton mass hierarchy : the 3D of time-like sub-space is a constraint of the number of lepton generations (exacting number 3); the basic common time-like cylindrical evolution ensures the causality by one-directional evolution (toward the future) and together with two universal free parameters (푻∞ and 훆∞) explains why three lepton generations have similar properties. A theoretical calculation by TSS-MicroCoM in perturbative approximation leads to prediction : 풎ퟑ ∞ = ퟏퟕퟕퟔ. ퟒퟎ MeV; which is a fairly passable consistency with experimental tauon mass : 풎흉 풆풙풑 = ퟏퟕퟕퟔ. ퟖퟔ ± ퟎ. ퟏퟐ MeV. From another perspective, as the deviation of calculation is still 3.83σ, It needs further research for any new hyper-fine adjustment of the present theoretical calculation. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 30 The Literature Pagoda in Hanoi Thank You for Attention! Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 20152018 31 Appendix 1: Why Bi-Cylindrical Geometry ? The Bi-cylindrical Formalism is implemented in the following steps: Formulation of bi-cylindrical geometry of {3T-3X} time-space symmetry where the two 3D sub-spaces are orthonormal to each other. Vacuum solutions of general relativity equation in such geometry Transformation of the bi-cylindrical variables in to the functions 흍 = 흍(풚) and 흋 = 흋(풚) of 푦 ≡{풕ퟎ, 풕ퟑ, 풙풏, 풙ퟑ} in {3T-3X} time-space symmetrical geometry. Most convenient functions are exponential for imitation of both Hubble- like expansion and quantum waves. Those functions are naturally separable for their variables. A Higgs-like interaction for violating the time-space symmetry: A Lorentz-like condition is introduced for cancelation of all longitudinal fluctuations, which conserves the linear translational equation (CLT) in transformation from a higher dimensional (6D-) geometry to a lower (4D-) realistic geometry. Accordingly, the separated geodesic conditions in 3D sub-spaces ensure that due to a symmetry-breaking, their scales are able to renormalized independently following an invariant formalism. Non-zero mass terms appeared. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 32 Appendix 2 - Why Dual Solution ? Duality of the solution of 6D-Bi-Cylindrical GR Equation : Bi-cylindrical geodesic equation (2.13); Wave-like solution (2.14): Dual sub-solutions describe the same physical substance. Serving for Quantum Mechanical Interpretations: From Wave-like Equation a generalized QM equation is derived: KGF; From separated 3D-local geodesic conditions (in 3T and 3X): Heisenberg inequalities are derived. Qualitative explanation of QM phenomena: i/ Physical meaning of the QM energy-momentum operators; ii/ Wave-particle duality; iii/ Bohm quantum potential; iv/ Schrodinger ZBW (Zitterbewegung). Then, for Formation of a microscopic cosmological model with a cylindrical basis from 3D-local geodesic equation in 3T the Hubble-like expansion is in homogeneous and isotropic conditions. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 33 Appendix-3: Calculation of curvature tensors Christoffel symbols: by applying (9) following are found valid: ψψ 휑휑 ψ 𝑔 휕𝑔휑휑 1 휕ψ φ φ 𝑔 휕𝑔휑휑 1 휕ψ 훤휑휑 = − = − ; 훤ψφ= 훤휑ψ = = 2 2 휕ψ 퐻푦 휕푦 2 휕ψ ψ 퐻푦 휕푦 33 2 휑휑 3 𝑔 휕𝑔휑휑 1 휕(ψ ) 휕ψ φ φ 𝑔 휕𝑔휑휑 훤휑휑 = − = − = −ψ ; 훤3φ= 훤휑3 = = 2 휕푦3 2 휕푦3 휕푦3 2 휕푦3 1 휕(ψ2) 1 휕ψ 2 = . 2ψ 휕푦3 ψ 휕푦3 Ricci tensors for the bi-3D cylindrical geometry: φ 2 2 휕훤φψ 휕푦 φ φ 1 휕 ψ 1 휕ψ 푅ψψ = − − 훤 훤 = − 3 2 2 + 4 2 ; 휕푦 휕ψ ψ휑 ψφ ψ 퐻푦 휕푦 ψ 퐻푦 휕푦 φ 2 휕훤φ3 φ φ 1 휕 ψ 푅33 = − − 훤3휑훤3φ = − 2 ; 휕푦3 ψ 휕푦3 ψ 3 2 2 2 휕훤φφ 휕푦 휕훤φφ φ ψ φ 3 1 휕 ψ 1 휕ψ 휕 ψ 푅φ휑 = + − 훤 훤휑휑 − 훤φ3훤휑휑 = − 2 2 + 2 2 − ψ 2 . 휕푦 휕ψ 휕푦3 φψ ψ퐻푦 휕푦 ψ 퐻푦 휕푦 휕푦3 ퟐ ퟐ ퟐ 풊풎 풎 풎 ퟐ 흏 흍 ퟐ 흏흍 ퟐ 흏 흍 Obviously, 푹 = 품 푹풊풎 = 휹풊 푹풊 = − ퟑ ퟐ ퟐ + ퟒ ퟐ − ퟐ . 흍 푯풚 흏풚 흍 푯풚 흏풚 흍 흏풚ퟑ 휎 휎 훽 훽 Its space-time representation reads: 푅 = 훿훾 푅훾 푋 − 훿훼 푅훼 푇 . Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 34 Appendix-4: Outputs from the dual solutions of GR (1) The meaning of quantum energy momentum operators: Mathematical transformation from the geodesic equation (2.13) with an exponential solution to a wave-like solution (2.14) is performed by transformation of variables: 풕 → −풊풕 and 풙풋 → 풊풙풋 in similar to that of quantum dynamic operators. This is not only a mathematical formalism, but also a significant physical operation, equivalent to transformation from external to internal investigation. Indeed, for the phase 휑 = Ω푡 − 푘푗푥푗 = 푐표푛푠푡 in the internal phase continuum: the phase velocity 푑푥푗 Ω is superluminal, i.e. 푣푝ℎ푎푠푒 = = > 푐. It is equivalent to converting the role of 푑푡 푘푗 space time in the internal superluminal frame comparing with the external subluminal space-time. Wave-particle duality: Subjecting the same microscopic substance: i/ the monotonic exponential solution describes motion/evolution of a material point, as a localized particle; while ii/ the wave-like solution transforming into QM equation (KGF) describes the motion/evolution of same particle, but as a wave-like substance. Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 2015 2018 35 Appendix-4: Outputs from the dual solutions of GR (2) Based on 3D-local geodesic deviation acceleration conditions (2.11-T) and (2.11-X), we can understand some important QM phenomena: Bohm quantum Potential: ퟐ ퟐ ퟐ 흏푺 ퟐ ħ 흏 흍 = 퐵푒 ħ. 풌풏. μ풆 풆풗풆풏 = ퟐ = −ퟐ풎푸푩; (2.11-B) 흏풙풏 흍 흏풙풏 which is proportional to Bohm’s quantum potential 푸푩. Schrödinger’s Zitterbewegung: The existence of the spin term in Generalized QM Klein-Gordon-Fock equation (2.15) is reminiscent of ZBW of free electron. When we describe a linear translation of the freely moving particle by Equation (2.17), the ZBW term is almost compensated by the geodesic condition (2.11-X) except a tiny P-odd term (However the latter is hard to observe). Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 2015 2018 36 Appendix-5: Heisenberg Indeterminism (1) A. Coordinate-momentum inequality: ퟏ 흏흍 ퟐ The local geodesic condition (2.10) leads to: 풅 . 풅풙풏 = 풅흋 ≥ ퟎ ; (H1) 흍 흏풙풏 −ퟏ 흏흍 ퟐ ∆풑 . ∆풙 ≥ ∆풑풏 . ∆풙풏 > 흍 풅 풊. ħ . 풅풙풏 = 풊. ħ . 풅흋 ≥ ퟎ; (H2) 흏풙풏 Accepting the conditions: i/ Quantization of azimuth: 흋= n.2π; ii/ For Poisson/Gaussian distribution of quantum statistics: < 흋 >풎풊풏= 2π and 풅흋 ≈ 흈흋 = 2π = standard deviation. Then, from (H2): ∆풑 . ∆풙 > 2π ħ. (H3) B. Time-energy inequality: ퟏ 흏흍 ퟐ Following 3D-time local geodesic condition (2.10) : 풅 . 풅풕ퟎ = 풅흋 ≥ ퟎ ; (H4) 흍 흏풕ퟎ −ퟏ 흏흍 ퟐ ∆푬 . ∆풕 ≥ ∆푬ퟎ . ∆풕ퟎ > 흍 풅 풊. ħ . 풅풕ퟎ = 풊. ħ . 풅흋 ≥ ퟎ ; (H5) 흏풕ퟎ With the same conditions (i) and (ii): ∆푬 . ∆풕 > 2π ħ. (H6) The inequalities (H3) and (H6) show that the QM indeterminism takes origin from the curvatures of space and time. Vo Van Thuan Windows 27th on Rencontre the Universe, de Blois August June 5-11,3, 2015 2018 37 Appendix-5: Heisenberg indeterminism (2) For a local geodesic in closed 3D-time: ퟏ 흏흍 ퟐ 풅 . 풅풕ퟎ = 풅흋 ≥ ퟎ ; 흍 흏풕ퟎ Multiplying both sides on the quantum scale unit 푖. ħ, and turning to finite differentials we get the time-energy indetermination: ∆푬 . ∆풕 ≥ ∆푬ퟎ . ∆풕ퟎ > 풅푬ퟎ . 풅풕ퟎ = −ퟏ −ퟏ 흏흍 = 흍 풅 푬ퟎ. 흍 풅풕ퟎ = 흍 풅 풊. ħ 풅풕ퟎ = 흏풕ퟎ = 풊. ħ . 풅흋ퟐ ≥ ퟎ ; 푨 where due to involving in the internal curvature 푬 (풏) = 풎 (풏) = 풏 ퟎ ퟎ 흍풏 then, in average: ퟏ 풅 푬 . 흍 = 흍풅 푬 + 푬 풅 흍 = 흍풅푬 ± (흍풅푬 ) = 흍풅푬 . ퟎ ퟎ ퟎ ퟎ 풏 ퟎ ퟎ Similarly we can get the space-momentum indetermination. Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 38 Appendix-6: A scenario similar to the Standard cosmological model (A hypothesis) During the Big-Bang inflation, we suggest the following scenario of MicroCoM, similar to the Standard Cosmological model of the Universe: The micro-scale factor 흍 increases exponentially ( time-like Hubble constant 20 −1 −20 푯푻 = Λ푻 = 7.764*10 푠푒푐 and the instant of inflation Δ푡1 = 1.926 ∗ 10 sec after 1 sec from the Big-Bang). For the next time-life of the Universe =13.7 *109 years, based on the idea of time-space symmetry, it is assumed for 3D-time (exactly as in 3D-space): 흍 ~ 푡1/2 for radiation dominant era and 흍 ~푡2/3 for matter dominant era. 휱 In a result, the time-like Lagrange radius 푻 decreases from 푻0 = = ퟏ for Δ푡1 흍ퟎ 휱 then steps up to the present value 푻 = ≈ 16.5. 흍 For leptons born after the inflation era, assuming following anthropic principle (very qualitatively) that the Hubble radius of any quantum fluctuations should adapt the contemporary value 휱, while the scale factor 흍 being governed by a contemporary chaotic Higgs-like potential in such a way, that is to meet the contemporary time-like Lagrange radius 푻 (for today, 푻 =16.5). Vo Van Thuan Windows 27th Rencontre on the Universe, de Blois August June 53,-11, 2015 2018 39