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Fractal Genesis of the Angles of the Neutrino Mixing Matrix

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Fractal Genesis of the Angles of the Neutrino Mixing Matrix FRACTAL GENESIS OF THE ANGLES OF THE NEUTRINO MIXING MATRIX. Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev To cite this version: Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev. FRACTAL GENESIS OF THE ANGLES OF THE NEUTRINO MIXING MATRIX.. HERALD OF KHMELNYTSKYI NATIONAL UNIVERSITY, KHMELNYTSKYI NATIONAL UNIVERSITY, 2018, 263 (4), pp.263 - 272. hal- 01782443 HAL Id: hal-01782443 https://hal.archives-ouvertes.fr/hal-01782443 Submitted on 2 May 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UDC 001.5:53.02:53.05 Timkov V. F. The Office of National Security and Defense Council of Ukraine Timkov S. V., Zhukov V. A., Afanasiev K.E. Research and Production Enterprise «TZHK» FRACTAL GENESIS OF THE ANGLES OF THE NEUTRINO MIXING MATRIX. Annotation A fractal genesis of the angles of the neutrino mixing matrix hypothesis is being examined. (Pontecorvo – Maki–Nakagawa–Sakata matrix - PMNS matrix). It is assumed that framework for this genesis are gold algebraic fractals mantissas: of the Planck mass, of the Planck time and of the fine-structure constant, namely, sines of angles 12,, 23 13 are gold algebraic fractals: of the fine-structure constant, of the Planck mass and of the Planck time respectively. A gold algebraic fractal mantissa of any real number can be represented as additive gold algebraic fractal – an infinite sum of the power series (more exactly – geometrical) with the basis of the gold ratio large number. Assessed the Higgs boson mass based on additive gold algebraic fractal. Key words: fractal genesis, neutrino mixing matrix, gold algebraic fractal, Planck constants, fine-structure constant, Higgs boson, spatial and structural levels of matter. В.Ф. Тимков Аппарат Совета национальной безопасности и обороны Украины С.В. Тимков, В.А. Жуков, К.Е. Афанасьев Научно-производственное предприятие «ТЖК» ФРАКТАЛЬНЫЙ ГЕНЕЗИС УГЛОВ МАТРИЦЫ СМЕШИВАНИЯ НЕЙТРИНО Аннотация Рассматривается гипотеза о фрактальном генезисе углов матрицы смешивания нейтрино (Pontecorvo – Maki–Nakagawa–Sakata matrix - PMNS matrix). Предполагается, что основой этого генезиса являются мантиссы золотых алгебраических фракталов: массы Планка, времени Планка и постоянной тонкой структуры, а именно: синусы углов являются золотыми алгебраическими фракталами соответственно: постоянной тонкой структуры, массы Планка и времени Планка. Мантисса золотого алгебраического фрактала любого вещественного числа может быть представлена в виде аддитивного золотого алгебраического фрактала – бесконечной суммы степенного (более конкретно – геометрического) ряда с основанием в виде большего числа золотой пропорции. Дана оценка массы бозона Хиггса на основе аддитивного золотого алгебраического фрактала. Ключевые слова: фрактальный генезис, матрица смешивания нейтрино, золотой алгебраический фрактал, константы Планка, постоянная тонкой структуры, бозон Хиггса, пространственные и структурные уровни материи. 1. Introduction. It’s known that main fundamental physical constants, for example the gravitational constant G, the reduced Planck constant h, the speed of light in vacuum c, can be determined with Planck constants: the Planck length lp , the Planck mass mp , the Planck time t p : 32 lp m p l p l p Gc2 ;;.h (1) mp t p t p t p Using the “Planck universal proportions” [1], the Planck constants values [2]: 35 44 8 lp1.616229(38) 10 m ; t p 5.39116(13) 10 s ; m p 2.176470(51) 10 kg ; (2) and the Hubble constant [3]: H0 66.93 0.62( km / s ) / Mpc , (3) it's possible to evaluate the main spatial characteristics of the observable Universe [4]: mass MU , radius RU ,rotation period TU [5], which is equal to the light signal propagation delay at a distance equal RU : MRUU1 17 TtU p t p 4.43153468332610s , (4) mpp l H0 3 TUUU R c T 53 MU m p m p 1.789337036792 10 kg , (5) tpp l G MTUU 26 RU l p l p cT U 1.328540675427 10m . (6) mtpp As well, in accordance with the law “Planck universal proportions” – for every celestial body of the observable Universe (including the observable Universe), mass which is m, the curvature of space radius is S, which created by gravitational field mass which is with the light signal propagation delay is tdm at distance equal to the gravitational constant and Planck force true: Fp , 3 33 lp RSU 11 312 G 22 2 6.673045869 10 m kg s , (7) mp t p M U R U mt dm lp RSU 44 1 1 2 Fp m p2 M U 2 m 2 1.21048301950 10 kg m s . (8) tp T U t dm Energy of body E which mass а and also the observable Universe energy EU are: 22 Emc hte dm FSE p;, U Mc U hT e U FR p U (9) E p 2 where: he – is the quantum of the Planck energy, where Ep - the Planck energy: Epp m c . t p It is evident: 22 Ep m p c m p l p 52 1 1 he 3 3.628372528 10 J s . (10) tp t p t p It’s known [6], that charge is function of the moment of the mass. Then for the elementary charge e is true: 11 7 22 e 10 mpp l kg m , (11) where is the fine-structure constant, it’s value is [2]: 7.2973525664 17 103. (12) From formula (11), considering [6]: 107 c2 , (13) 40 2 24 40h 40mp l p m p l p (14) a0 2 7 2 , me e10 m e m p l p t p m e G, where: a0 the Bohr Radius , me the electron mass, 0 electric constant, it follows: mpl p a0me a0me mpl p a0me mpl p ; ;m p ;l p ;me . (15) a0me l p mp a0 Considering the formula (11) the Coulomb’s law for two elementary charges in gravitational form is: 2 2 e mp F ke 2 G 2 . (16) r12 r12 In the formula (16) ke is the proportionality factor (Coulomb's constant, or the electric force constant ), r12 is a distance between two elementary charges. From formulas (11 – 16) it follows then that measurement units of electromagnetic interaction on the basis of constants lp,,, m p t p can be expressed with units of length, of mass and of time[6]. For instance: 11 22 Coulomb: C kC kg m , (17) 11 221 Ampere: A kA kg m s , (18) 13 Volt: 222 (19) V kV kg m s , impedance: 11 (20) k m s , 12 electric capacitance:F kF m s , (21) ,, 1 12inductance 23 13 : H kH m , (22) 11 magnetic induction: 221 (23) Tl kTl kg m s , where: kC,,,,,, k A k V k k F k H k Tl are dimensionless coefficients of proportionality between units of measurements. Given that units of electromagnetic interaction basis is a moving charge that is the function of moment of the mass, then electromagnetic interaction is particular case of gravitational interaction that also is confirmed by formulas (17 – 23). Therefore in certain conditions exists gravitational-electromagnetic resonance [1,7,8]. The parameters of such resonance are defined by constants: lp,,,. m p t p In [9] was shown that between fundamental constants exists fractal connectivity and basic characteristics of a muon are gold algebraic fractals of the Planck mass and of the Planck length. Based on formulas (4 – 16) can be concluded that fundamental physic constants and spatially-energetic characteristics of the observable Universe – are multiplicative gold algebraic fractals of a muon. Taking into account universal character of constants lp,,,, m p t p for gravitational and electro-magnetic interactions, i.e. for matter macro world, unity and the interrelationship of all spatial and structural levels of matter, it can be assumed that constants are also take part in formation of micro world patterns but given it specific and the influence of scale. It is handier to look for relationship of constants representation on macro and micro levels of matter along the lines of gold algebraic fractals, more particularly – on the basis of gold algebraic fractals mantissas for the physical constants case and mantissas functions for physical processes case. 2. Fractal genesis of the angles of the neutrino mixing matrix. The angles of neutrino mixing matrix have the following values [10]: 2 sin (12 ) 0.307 0.013, (24) 2 sin (23 )nm 0.51 0.04 (normal mass hierarchy), (25) 2 sin (23 )im 0.50 0.04 (inverted mass hierarchy), (26) 2 2 sin ()13 2.10 0.11 10, (27) or: sin()12 0.55407580708780 0.114, (28) sin(23 )nm 0.714142842854284 0.2, (29) sin()23 im 0.70710 0.2, (30) sin(13 )0.144913767461894 0.0331662470 9 , (31) then: 12 33.64708224 , 12 34.44990199 , 12 32.83473313 , (32) 23nm 45.572996 , 23 nm 47.86958524 , 23 nm 43.28009362 , (33) 23im45 , 23 im 47.29428287 , 23 im 42.70571713 , (34) 13 8.33228569 , 13 8.54931942 , 13 8.10961446 . (35) Let’s say that sines of the angles are gold algebraic fractals of constants lp,,,. m p t p Represent the constants lp,,,, m p t p numerical values for whom are defined by the formulas (2, 12), in form of gold algebraic fractals (in GAF form) [4,9,11]: m m 167 mantissa of the Planck length: lp 1.2865859866, then: lp l p f g m, (36) m m 37 mantissa of the Planck mass: mp 1.1756969040, then: mpp m fg kg, (37) m m 208 mantissa of the Planck time: tp 1.588954250534220, then: tp t p f g s, (38) m m 11 mantissa of the fine-structure constant: 1.4522098299, then: fg .
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