Hierarchical Fine-Structures in Quarks and beyond

Hierarchical Fine-Structures in Quarks and beyond 1.1 For decades, the biennial updates on quarks from PDG has gradually and faithfully converged to a hierarchical structure, scaled by the Fine-Structure Constant (α) for Top, Charm & Down, while its circular variant (α·π) for Bottom, Strange & Up. Extending downwards reveals a seed mass mgluon requiring two column table with five lines - labelled by the respective hierarchical pair in question:

Pair

Major

Minor

Q2 ; Q3

mcharm    mtop

Q1 ; Q2

mdown  12    mcharm

G ; Q1

mgluon  32    mdown

E ;G

melectron 

Q1 ; E

mdown 

2 27 27 3

mstrange      mbottom

  1  mgluon

mup      mstrange mgluon      mup melectron 

 melectron

mup 

2 27 27 2

  1  mgluon  melectron

Observe chained recursive construction we section with dotted line at Electron crossing melectron , and solid line at Electron reset, and hyphened line brings up the 1st generation Quarks. On examination, we notice a familiar structure from atomic spectra that relates the Rydberg energy to the Electron mass:

mdown  12   2  mtop

mrydberg  12   2  melectron

;

mcharm  2   1  mdown ;

m pion  2   1  melectron 

Here the charged Pion is to an Electron as Charm Quark to a Down Quark. The present status of convergence is from PDG 2016, and gives Top Quark to 0.5% precision, Bottom Quark to 1%, Charm- to 2%, Strange- to 6%, Down- to 10%, and Up Quark to 20% being the lightest quark but having the largest uncertainty rendering error-scaling as m ~ 1/m as expected for a noisy background:

Quark

Valuesix  digits

PDG2016

 mg

172.728 GeV

173.2  0.8

Bottom

mb   31 3  mg

4.17805 GeV

4.18  0.04

Charm

mc 

 mg

1.26045 GeV

1.27  0.03

Strange ms   21 2  mg

95.7831 MeV

96  3

 mg

4.59899 MeV

4.7  0.4

mu  1  mg

2.19586 MeV

2.3  0.5

Top

Down Up

Equation mt 

md 

4 3 3

4 3 2

2 3

To terminate the page, we evaluate inter-generation ratios of each quark pair - yielding powers of :

md 2    mu 3

;

mc 4   2  ms 3

;

mt 4   3  mb 3

With the next biennial PDG update on quarks due in 2018, we rest our case and turn to the Leptons. Academy of Industry and Arts © 3. April 2017 ~ Gudlaugur Kristinn Ottarsson ~ [email protected] ~ Page 1

Hierarchical Fine-Structures in Quarks and beyond 2.1 Hierarchy and the Fine Structure Constant in Particle Physics: The Fine Structure Constant (FSC) is presently celebrating its centennial anniversary since emerging in the electrostatic Coulomb energy; VCoulomb , and the energy of the 1s1 electron in Hydrogen; ERydberg :

VCoulomb 

  c r

;

ERydberg  12   2  me  c 2 ;  

e2 4   0   c

Similarly – while perhaps without general awareness, FSC further gave a geometrical hierarchy to the Hydrogen Rydberg radius, the electron Compton radius and the Classical electron radius as disclosed:

rCompton  / me  c

;

rCompton / rRydberg  rClassical / rCompton  

Presently the classical radius of the electron is just a theoretical curiosity, as re ~ 3·rCompton ~ 0.15am is closer to its true value. On the other hand is charge radius of the proton that recently was shown by this author to yield to rp = 4·ħ/c·mp = 0.841235642··· fm – while the charge radius of the pion (±) is about half its Compton radius, or r± = ½·ħ/c·m± = 0.706905··· fm. The next noteworthy entry of FSC into particle physics was via failure of Dirac’s equation to render the magnetic moment of the electron, until Schwinger gave the value; 1 + ½·(/) = 1.00116140··· , that was later refined via Feynman’s QED to; 1 + ½·(/) - ⅓·(/)² = 1.00115961··· , where the 2nd order term is present author’s rationalization of the QED calculation; A1(4) = 0.328478··· , and thus identified as a Taylor expansion of 2 - ln(1+x)/x. This miraculously rationalizes the 3rd order QED coefficient to an integer, leaving only small hadronic contributions of 4th order we denote by he, as follows:

e 

 / 3 4 e          2  ln 1    3     he     2  me       

e   2  m

 

 / 3 4          2  ln 1    3     h           

 / 3 4 e          2  ln 1    3     h     2  me       

2        0  1  1      5      

      1  1  1      10    

2

  

2        3  1  1       20     

The logarithm thus sweeps up all the transcendental coefficients of QED, leaving only the hadronic contributions. The terminating objects on the right wield holy number; 1 = ½·(1+√5) as the reader is invited to replicate - by expanding the differences into continued fractions to render a new window into the anatomy of leptons by isolating their decay mechanism that we identify as neutrino presence. For readers unfamiliar with the number-theoretical method of continued fractions, it renders a powerful tool to explore algebraic nature of real numbers such as rationality, transcendentality or the presence of holy numbers with their distinct fingerprints such as 1 = CF[1; 1, 1, 1, 1, 1, 1, 1, ….]. We close by calculating the moments, with he = 7.54 for the Electron to give 1.00115965218078 @ 0.13 ppb, while h = 0 for the Muon to give 1.00116592046 @ 0.44 ppm, and h = 0 for the Tauon to give us 1.00117715 @ 62 ppm - indeed with all three moments faithfully rendering the current state of the art! Academy of Industry and Arts © 3. April 2017 ~ Gudlaugur Kristinn Ottarsson ~ [email protected] ~ Page 2

Hierarchical Fine-Structures in Quarks and beyond 2.2 Natural Recursions, Hierarchy and Holy Numbers in Particle Physics: In section 2.1, we absorbed transcendental QED terms of magnetic moments of leptons into a logarithmic object, that in fact recently surfaced as the 'null solution' of the primary natural recursion; Hn+1 = x·Hn + ·Hn-1 that also satisfies the following differential equation:

 2  dxd H n  3  x  dxd H n  H n  n 2  H n 2

2

 2  x 2  4   ; H 0 ( x)   1  ln( x   ) The Holy function Hn(x) is in fact a special case of the Associated Legendre function, which solves the Maxwell Polar Operator for azimuthally symmetric charge distributions, as well as populating the core of the Fermi-Dirac distribution function and seeds the generalized Riemann Zeta functions as the reader is invited to investigate in this authors papers “On Natural Recursions”1) and “Electromagnetic & Magnetoelectric Structures”2) from 2016 and 2013 respectively.

2.3 Mass Hierarchy in Leptons: We will now employ our number theoretical and analytical methods to evaluate mass ratios of Leptons, where the mass of the Electron is presently known to 6 ppb, Muon mass to 23 ppb, while the Tauon trails at 68 ppm, indeed the reverse situation of Section 1.1 where precision improved with mass:

me 4  2      mo   3 2   

;

m me



4  3 4  2   2    5 3  2   

;

m 4  5 9  3    3   m   8 5  3   

The 2nd term in the Muon mass ratio was obtained by summing its emerging power series to infinity where the first few terms gave a geometric series in (2/3) yielding the result above, while the corresponding term in the Tauon mass ratio was made to inherit that structure, but with (3/5) which gave the closest fit. However, by rewinding this hierarchy downwards yielded our hypothetical analogue for an Electron mass ratio to seed mass mo - in suit with our seed mass mg for Quarks in Section 1.1. The seed mass mo is indeed a welcomed newcomer as it serves as a bridge down to the minuscule Neutrino masses with mo = 442.677708 ± 0.000003 eV, and to terminate this section, we would like to make contact with the now 36 year old mass formula of Yoshio Koide that can be simplified due to its scale invariance in terms of just two mass ratios denoted by a and b - and indeed extended to accommodate both Leptons and Quarks via rational parameter k and its binary conjugate:

a  b 1  4k2 a  b  a b

;

a  b 1 a  b  a b

 4k 2

;

k k  2

Inserting our Lepton masses into the left equation, with a = m/me and b = m/me gives k = 0.999999944 while our Minor Quarks yield to k = 6/5 – 0.00042, or to 0.035%, and our Major Quarks give k = 5/3 – 0.0033, or to 0.2%, while together they give k  k = 1.9953, or also to 0.2% - which is indeed quite amazing in the light of present Quark mass uncertainties we exposed in Section 1.1. Finally, with the Neutrino masses presently totally out of analytical reach, we hypothesize them to yield to k = 2/k = 2. Academy of Industry and Arts © 3. April 2017 ~ Gudlaugur Kristinn Ottarsson ~ [email protected] ~ Page 3

Hierarchical Fine-Structures in Quarks and beyond Hierarchical Fine-Structures in Quarks and beyond. Now updated with Magnetic Moments- and Mass Hierarchy of Leptons. Also extended the Yoshio Koide formula from 1981 by rescaling and parametrisation to accommodate both Leptons and Quarks and to prepare for Neutrino masses as the binary conjugate of the Electron family. A new name is suggested for the extended equation: “The YK/GK mass equation for Leptons and Quarks”.

Working Paper · January 2017 to April 2017 DOI: 10.13140/RG.2.2.18352.76806 · Affiliation: Academy of Industry and Arts, Version: #0.1, State: In Progress.

Academy of Industry and Arts © 3. April 2017 ~ Gudlaugur Kristinn Ottarsson ~ [email protected] ~ Page 4