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For Charged Leptons 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 .
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