The Natural Philosophy of Fundamental Particles

8/14/2019 The Natural Philosophy of Fundamental Particles

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The atural Philosophy of Fundamental Particles

Riccardo C. Storti1

Key Words: Balmer Series, Bohr Radius, Buckingham Theory, Casimir Force, ElectroMagnetics, Equivalence Principle, Eulers Constant, Fourier Series, Fundamental Particles, General Relativity, Gravity, Harmonics, Hydrogen Spectrum, ewtonian Mechanics, Particle Physics,Physical Modelling, Planck Scale, Polarizable Vacuum, Quantum Mechanics, Zero-Point-Field.

Abstract

Theoretical estimates and correlations, based upon the Electro-Gravi-Magnetics (EGM)

method, are presented for the Root-Mean-Square (RMS) charge radius and mass-energy of many

well established subatomic particles. The EGM method is a set of engineering equations and

techniques derived from the purely mathematical construct known as Buckinghams (Pi)Theory. The estimates and correlations coincide to astonishing precision with experimental data

presented by the Particle Data Group (PDG), CDF, D0, L3, SELEX and ZEUS Collaborations. Our

tabulated results clearly demonstrate a possible natural harmonic pattern representing all

fundamental subatomic particles. In addition, our method predicts the possible existence of several

other subatomic particles not contained within the Standard Model (SM). The accuracy and

simplicity of our computational estimates demonstrate that EGM is a useful tool to gain insight into

the domain of subatomic particles.

[email protected]

Delta Group Engineering P/L


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1 ITRODUCTIO

1.1 HARMONIC REPRESENTATION OF GRAVITATIONAL ACCELERATION

Electro-Gravi-Magnetics (EGM) is a term describing a hypothetical harmonic relationship

between Electricity, Gravity and Magnetism. The hypothesis may be mathematically articulated by

the application of Dimensional Analysis Techniques (DATs) and Buckingham Theory (BPT),

both being well established and thoroughly tested geometric engineering principles [1-3], viaFourier harmonics. [4] The hypothesis may be tested by the correct derivation of experimentally

verified fundamental properties not predicted within the Standard Model (SM) of particle physics.

Storti et. al. derived the EGM relationship in [5-23] where it was shown that a theoreticalrepresentation of constant acceleration at a mathematical point in a gravitational field may be

defined by a summation of trigonometric terms utilizing modified complex Fourier series in

exponential form, according to the harmonic distribution nPV = -N, 2 - N ... N, where N is an

odd number harmonic. Hence, the magnitude of the gravitational acceleration vector g (via the

equivalence principle applied in [5-23]) may be usefully represented by Eq. (1) as |nPV|,

g r M,( )G M.

r2

n

PV

2 i.

n PV.

e n PV. PV 1 r, M,( )

. t. i...

(1)

such that, the frequency spectrum of the harmonic gravitational field PV is given by Eq.(2), [8]

PV n PV r, M,n PV

r

32 c. G. M.

r.. KPV r M,( )

.

(2)

where,

Variable Description Units

PV(1,r,M) Fundamental spectral frequency. Hz

KPV Refractive index of a gravitational field in the Polarizable Vacuum

(PV) model of gravity, [5] only contributing significantly when a large

gravitational mass (i.e. a strong gravitational field) is considered. For

all applications herein, the effect is approximated to KPV(r,M) = 1.nPV Harmonic modes of the gravitational field.

None

r Magnitude of position vector from centre of mass. m

M Mass. kg

G Gravitational constant. m3kg

-1s

-2

Table 1,

Subsequently, the harmonic (Fourier) representation of the magnitude of the gravitational

acceleration vector at the surface of the Earth up to N = 21 is graphically shown to be,

Time

GravitationalAcceleration g

Figure 1: harmonic representation of gravitational acceleration,


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As N , the magnitude of the gravitational acceleration vector becomes measurably constant.

Hence, Eq. (1, 2) illustrate that the Newtonian representation of g is easily harmonized over the

Fourier domain, from geometrically based methods (i.e. DATs and BPT). Therefore, unifying (inprinciple) Newtonian, geometric (relativistic) and quantum (harmonized) models of gravity.

1.2 BOUNDARY CONDITIONS

1.2.1 FREQUENCY

Storti et. al. showed in [8] that the spectrum defined by Eq. (2) is discrete and finite. The

lower boundary value is given by PV(1,r,M), whilst the upper boundary value (also termed

the harmonic cut-off frequency) is given by Eq. (3),

r M,( ) n r M,( ) PV 1 r, M,( ).

(3)

supported by Eq. (4-7),

n r M,( ) r M,( )

12

4

r M,( )1 (4)

r M,( )

3

108U

mr M,( )

U r M,( ). 12 768 81

Um

r M,( )

U r M,( )

2

.. (5)

U m r M,( )3 M. c

2.

4 . r3.

(6)

U r M,( )h

2 c3. PV 1 r, M,( )

4.

(7)

where,

Variable Description Units

n Harmonic cut-off mode [mode number at ].

Harmonic cut-off function.

None

Um Mass-energy density of a solid spherical gravitational object.

U Energy density of mass induced gravitational field scaled to

the fundamental spectral frequency.

Pa

h Plancks Constant [6.6260693 x10-34

]. Js

c Velocity of light in a vacuum. m/s

Table 2,

Since the relationship between trigonometric terms at each amplitude and corresponding

frequency is mathematically defined by the nature of Fourier series, the derivation of Eq. (4, 5) is

based on the compression of energy density to one change in odd harmonic mode whilst preserving

dynamic, kinematic and geometric similarity in accordance with BPT.

The preservation of similarity across one change in odd mode is due to the mathematical properties of constant functions utilizing Fourier series as discussed in [8]. The subsequent

application of these results to Eq. (1) acts to decompress the energy density over the Fourier domain

yielding a highly precise reciprocal harmonic representation of g whilst preserving dynamic,

kinematic and geometric similarity to Newtonian gravity, identified by the compression technique

stated above.


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1.2.2 POYNTING VECTOR

It was demonstrated by Haisch, Puthoff and Rueda in [24-26] that inertia may haveElectroMagnetic (EM) origins due to the Zero-Point-Field (ZPF) of Quantum-Electro-Dynamics

(QED), manifested by the Poynting vector, via the equivalence principle. Hence, it follows that

gravitational acceleration may also be EM in nature and the Polarizable Vacuum (PV) model of

gravity [27] is an EM polarized state of the ZPF with a Fourier distribution, assigning physical

meaning to Eq. (1).Subsequently, it follows that the energy density of a mass induced gravitational field may be

scaled to changes in odd harmonic mode numbers satisfying the mathematical properties of any

constant function described in terms of Fourier series utilizing Eq. (7) - such that,

U n PV r, M, U r M,( ) n PV 24

n PV4. (8)

Therefore, the Poynting vector2

of the polarized Zero-Point (ZP) gravitational field S

surrounding a solid spherical object with homogeneous mass-energy distribution is given by,

S n PV r, M, c U n PV r, M,. (9)

and may be graphically represented as follows,

Harmonic

ZPFPoyntingVector

S n PV RE, M E,

n PV

Figure 2,

where, RE and ME in Fig. (2) denote the radius and mass of the Earth respectively.

Fig. (2) illustrates that the Poynting vector of the ZP gravitational field increases with nPV.

Further work by Storti et. al. in [9] showed that >>99.99(%) of the effect in a gravitational field

exists well above the THz range. Hence, it becomes apparent that n and are important

characteristics of gravitational fields. We shall utilize these characteristics to quasi-unify particlephysics in harmonic form in the proceeding sections.

2 THE SIZE OF THE PROTO, EUTRO AD ELECTRO

In 2005, Storti et. al. derived the mass-energy threshold of the Photon utilizing n and theclassical Electron radius as shown in [12], to within 4.3(%) of the Particle Data Group (PDG)

value3 stated in [28], then proceeded to derive the mass-energies and radii of the Photon and

Graviton in [14] by the consistent utilization of n.

The method developed in [12] was re-applied in [13] to derive the sizes4 of the Electron,

Proton and Neutron. The motivation for this was to test the hypothesis presented in Section 1 by

direct comparison of the computed size values to experimentally measured fact. They believe that

2Per change in odd harmonic mode number.

3Consistent with experimental evidence and interpretation of data.

4 From first principles and from a single paradigm.


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highly precise computational predictions, agreeing with experimental evidence beyond the abilities

of the SM to do so, is conclusive evidence of the validity of the harmonic method developed.

To date, highly precise measurements have been made of the Root-Mean-Square (RMS)

charge radius of the Proton by [29] and the Mean-Square (MS) charge radius of the Neutron as

demonstrated in [30]. However, the calculations presented in [13] are considerably more accurate

than the physical measurements articulated in [29,30], lending support for the harmonic

representation of the magnitude of the gravitational acceleration vector stated in Eq. (1).

The basic approach utilized in [13] was to determine the equilibrium position between the polarized state of the ZPF and the mass-energy of the fundamental particle inducing space-time

curvature as would appear in General Relativity (GR). In other words, one may consider thecurvature of the space-time manifold surrounding an object to be a virtual fluid in equilibriumwith the object itself

5.

This concept is graphically represented in Fig. (3). A free fundamental particle with classical

form factor is depicted in equilibrium with the surrounding space-time manifold. The ZPF is

polarized by the presence of the particle in accordance with the PV model of gravity, which is (atleast) isomorphic to GR in the weak field. [27]

Figure 3: free fundamental particle with classical form factor,

In the case of the Proton, the ZPF equilibrium radius coincides with the RMS charge radius

r [Eq. (10)] producing the experimentally verified result rp by the SELEX Collaboration asstated in [29]6,

r

h m e.

16 c. 2. mp

3.

5

27 h. c.

4 . G.

m e4

mp

.. (10)

where, me and mp denote Electron and Proton rest-mass respectively.

In the case of the Neutron, the ZPF equilibrium radius coincides with the radial position of

zero charge density r [Eq. (11)] with respect to the Neutron charge distribution as illustrated in

Fig. (4). It is shown in [18] that r relates to the MS charge radius KS by a simple formula [Eq.

(12)] producing the experimentally verified result KX as presented in [30]7,

rh m e.

16 c. 2. m n

3.

5

27 h. c.

4 . G.

m e4

m n

..

(11)

where, mn denotes Neutron rest-mass.

5The intention is not to suggest that the space-time manifold is actually a fluid, it is merely to

present a method by which to solve a problem.6

r = 0.8306(fm), rp = 0.8307 0.012(fm).7 r = 0.8269(fm), KS = -0.1133(fm

2), KX = -0.113 0.005(fm2).


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r

dr

5

3

r

.

Charge Density

Maximum Charge Density

Minimum Charge Density

Neutron Charge Distribution

Radius

ChargeD

ensity

ch r( )

ch r0

ch rdr

r rdr

r

Figure 4: Neutron charge distribution,

KS

3 . r

2.

8

1 x( ) x3.

1 x x2

.

(12)

where, x is solved numerically8

within the MathCad environment by the following algorithm,

[18]

Given

ln x( )x2

x2

1

.1

3(13)

x Find x( ) (14)

Utilizing KS, KX may be converted to determine an experimental zero charge density radial

position value rX according to Eq. (15),

rX

r

KS

KS KX.. (15)

In the case of the Electron (as with the Proton), the ZPF equilibrium radius coincides with

the RMS charge radius r [Eq. (16)] producing an experimentally implied result9

as stated in [31],

r re1

2

ln 2 n re m e,. .

9

5

.

(16)

where, re and denote the classical Electron radius and Euler-Mascheroni constant [32]

respectively.

8x = 0.6829, rX = 0.8256 0.018(fm).

9 r 0.0118(fm), = 0.577215664901533.


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3 HARMOIC REPRESETATIO OF FUDAMETAL PARTICLES

3.1 ESTABLISHING THE FOUNDATIONS

Motivated by the physical validation of Eq. (10, 11), Storti et. al. conducted thoughtexperiments in [13] to investigate harmonic and trigonometric relationships by analyzing various

forms of radii combinations for the Electron, Proton and Neutron consistent with the DATs and

BPT derivations in [5-12] yielding the following useful approximations,

r m e,

r mp,

r m e,

r m n,2

(17)

r

r r

(18)

r

r

e

2

3.

(19)

where,

(i) and e denote the fine structure constant and exponential function respectively.(ii) Eq. (17) error:

(a) associated with (r,me)/(r,mp) = 2 is 8.876 x10-3

(%)

(b) associated with (r,me)/(r,mn) = 2 is 0.266(%).(iii) Eq. (18) error is 2.823(%).

(iv) Eq. (19) error is 0.042(%).

3.2 IMPROVING ACCURACY

Since the experimental value of the RMS charge radius of the Proton is considered by the

scientific community to be precisely known10

, [29] the accuracy of Eq. (18, 19) may be improved

by re-computing the value of r and r. This action further strengthens the validity of Eq. (17)

by verifying trivial deviation utilizing the re-computed values.

Hence, it follows that numerical solutions for r and r, constrained by exact

mathematical statements [Eq. (16 19)], suggests that the gravitational relationship between the

Electron and Proton, as inferred by the result (r,me)/(r,mp) = 2, is harmonic. The

computational algorithm supporting this contention may be stated as follows,

Given

r

re

r m e,

r mp,

r

r r

r

r

e

2

3. 1

2ln 2 n re m e,

. .

9

5

2

(20)

r

rFind r r,

(21)

yields,

r

r

0.826838

0.011802fm( ).

(22)

10 To a degree of accuracy significantly greater than the Electron or Neutron.


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where,

(i) Eq. (17) error:

(a) associated with (r,me)/(r,mp) = 2 is 4.493 x10-7(%).

(b) associated with (r,me)/(r,mn) = 2 is 0.282(%).(ii) Eq. (18) error is 1.11 x10

-13(%).

(iii) Eq. (19) error is 0.026(%).

3.3 FORMULATING AN HYPOTHESIS

In the preceding calculations utilizing known particle mass and radii as a reference, it was

found that the harmonic cut-off frequency ratio of an Electron to a Proton was precisely 2. This

provokes the hypothesis that a simple harmonic pattern may exist describing the relationship of all

fundamental particles relative to an arbitrarily chosen base particle according to,

r1 M 1,

r2 M 2,St

(23)

Performing the appropriate substitutions utilizing Eq.(3 - 7), Eq. (23) may be simplified to,

M1

M 2

2r

2r1

5

. St 9

(24)

where, St represents the ratio of two particle spectra. Subsequently, r may be simply

calculated according to,

r r

5

1

29

m e

mp

2

.. (25)

3.4 IDENTIFYING A MATHEMATICAL PATTERN

Utilizing Eq. (24), Storti et. al. identify mathematical patterns in [15-17] showing that St

may be represented in terms of the Proton, Electron and Quarkharmonic cut-off frequencies derivedfrom the respective particle. Potentially, three new Leptons (L2, L3, L5 and associated Neutrinos:

2, 3, 5) and two new Quark / Bosons (QB5 and QB6) are predicted, beyond the SM as shown in

table (3).

The EGM Harmonic Representation of Fundamental Particles (i.e. table (3)) is applicable to

the size relationship between the Proton and Neutron (i.e. to calculate r from r and vice-versa

utilising St = 1) as an approximation only. For precise calculations based upon similar forms, thereader should refer to [13].

ote: although the newly predicted Leptons are within the kinetic range11

and therefore shouldhave been experimentally detected, there are substantial explanations discussed inSection 5.2.

Existing and Theoretical Particles ProtonHarmonics ElectronHarmonics QuarkHarmonics

Proton (p), Neutron (n) St = 1 St = 1/2 St = 1/14

Electron (e), Electron Neutrino (e) 2 1 1/7

L2, 2 (Theoretical Lepton, Neutrino) 4 2 2/7


Muon (), Muon Neutrino () 8 4 4/7


11 A region extensively explored in particle physics experiments.


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Tau (), Tau Neutrino () 12 6 6/7

Up Quark (uq), Down Quark (dq) 14 7 1

Strange Quark (sq) 28 14 2

Charm Quark (cq) 42 21 3

Bottom Quark (bq) 56 28 4

QB5 (Theoretical Quark or Boson) 70 35 5

QB6 (Theoretical Quark or Boson) 84 42 6

W Boson 98 49 7Z Boson 112 56 8

Higgs Boson (H) (Theoretical) 126 63 9

Top Quark (tq) 140 70 10

Table 3: harmonic representation of fundamental particles,

4 RESULTS

4.1 HARMONIC EVIDENCE OF UNIFICATION

Exploiting the mathematical pattern articulated in table (3), EGM predicts the RMS charge

radius and mass-energy of less accurately known particles, comparing them to expert opinion. Thevalues of St shown in table (3), predict possible particle mass and radii for all Leptons,

Neutrinos, Quarks and Intermediate Vector Bosons (IVBs), in complete agreement with the SM,

PDG estimates and studies by Hirsch et. al in [33] as shown in table (4),

Particle EGM Radii

x10-16

(cm)

EGM Mass-Energy

(computed or utilized)

PDG Mass-Energy Range

(2005 Values)

Proton (p) r = 830.5957

Neutron (n) r = 826.8379

Electron (e) r = 11.8055

Muon () r = 8.2165

Tau () r = 12.2415

Mass-Energy precisely known,

See: National Institute of Standards and Technology

(NIST) [34]

Note: m = 10-100

Electron Neutrino (e) ren 0.0954 men(eV) 3 - m men(eV) < 3Muon Neutrino () rn 0.6556 mn(MeV) 0.19 - m mn(MeV) < 0.19

Tau Neutrino () rn 1.9588 mn(MeV) 18.2 - m mn(MeV) < 18.2

Up Quark (uq) ruq 0.7682 muq(MeV) 3.5060 1.5 < muq(MeV) < 4

Down Quark (dq) rdq 1.0136 mdq(MeV) 7.0121 3 < mdq(MeV) < 8

Strange Quark (sq) rsq 0.8879 msq(MeV) 113.9460 80 < msq(MeV) < 130

Charm Quark (cq) rcq 1.0913 mcq(GeV) 1.1833 1.15 < mcq(GeV) < 1.35

Bottom Quark (bq) rbq 1.071 mbq(GeV) 4.1196 4.1 < mbq(GeV) < 4.4

Top Quark (tq) rtq 0.9294 mtq(GeV) 178.4979 169.2 < mtq(GeV) < 179.4

W Boson rW 1.2839 mW(GeV) 80.425 80.387 < mW(GeV) < 80.463

Z Boson rZ 1.0616 mZ(GeV) 91.1876 91.1855 < mZ(GeV) < 91.1897

Higgs Boson (H) rH 0.9403 mH(GeV) 114.4 + m mH(GeV) > 114.4

Photon () r = Kh m 3.2 x10-45(eV) m < 6 x10

-17(eV)

Graviton (g) rgg = 2(2/5)

r mgg = 2m No definitive commitment

L2 (Lepton) mL(2) 9(MeV)

L3 (Lepton) mL(3) 57(MeV)

L5 (Lepton)

rL 10.7518

mL(5) 566(MeV)

2 (L2 Neutrino) m2 men

3 (L3 Neutrino) m3 mn

5 (L5 Neutrino)

r2,3,5

ren,n,n m5 mn

Not predicted or considered


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QB5(Quark or Boson) mQB(5) 10(GeV)

QB6(Quark or Boson)

rQB 1.0052

mQB(6) 22(GeV)

Not predicted or considered

Table 4: RMS charge radii and mass-energies of fundamental particles,

where,

(i) K denotes a Planck scaling factor, determined to be (/2)1/3

in [17].

(ii) h denotes Planck length [4.05131993288926 x10-35

(m)].

(iii) rL and rQB denote the average radii of SM Leptons and Quark / Bosons

(respectively) utilized to calculate the mass-energy of the proposed new particles.ote:

(a) a formalism for the approximation of2, 3 and5 mass-energy is shown in [19].

(b) it is shown in [12,14,17] that the RMS charge diameters of a Photon and Graviton are h and

1.5h respectively, in agreement with Quantum Mechanical (QM) models.

4.2 RECENT DEVELOPMENTS

4.2.1 PDG MASS-ENERGY RANGES

The EGM construct was finalized by Storti et. al. in 2004 and tested against published PDG

data of the day [i.e. the 2005 values shown in table (4)]. Annually, as part of their continuousimprovement cycle, the PDG reconciles its published values of particle properties against the latestexperimental and theoretical evidence. The 2006 changes in PDG mass-energy range values notimpacting EGM are as follows:

1. Strange Quark = 70 < msq(MeV) < 120.2. Charm Quark = 1.16 < mcq(GeV) < 1.34.3. W Boson = 80.374 < mW(GeV) < 80.432.4. Z Boson = 91.1855 < mZ(GeV) < 91.1897.

Therefore, we may conclude that the EGM construct continues to predict experimentally verifiedresults within the SM to high computational precision.

4.2.2 ELECTRON NEUTRINO AND UP / DOWN / BOTTOM QUARK MASS

Particle physics research is a highly dynamic field supporting a landscape of constantly

changing hues. The EGM construct relates mass to size in harmonic terms. If one applies Eq. (24)

and utilizes the Proton as the reference particle in accordance with table (3), one obtains a single

expression with two unknowns, as implied by Eq. (25).

Since contemporary Physics is currently incapable of specifying the mass and size of most

fundamental particles precisely and concurrently, EGM is required to approximate values of either

mass or radius to predict one or the other (i.e. mass or size). Subsequently, the EGM predictionsarticulated in table (4) denote values based upon estimates of either mass or radius.

Hence, some of the results in table (4) are approximations and subject to revision as new

experimental evidence regarding particle properties (particularly mass), come to light. The 2006

changes in PDG mass-energy values affecting table (4) are shown below. In this data set, the EGMradii is displayed as a range relating to its mass-energy influence.

ote: the average value of EGM Up + Down Quark mass from table (4) [i.e. 5.2574(MeV)]remains within the 2006 average mass range specified by the PDG [i.e. 2.5 to 5.5(MeV)].


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Particle EGM Radii x10-16

(cm) EGM Mass-Energy

(utilized)

PDG Mass-Energy

Range (2006 Values)

Electron Neutrino (e) ren< 0.0811 men(eV) < 2

Up Quark (uq) 0.5469 < ruq< 0.7217 1.5 < muq(MeV) < 3

Down Quark (dq) 0.7217 < rdq< 1.0128 3 < mdq(MeV) < 7

Bottom Quark (bq) 1.0719 > rbq > 1.0863

PDG Mass-Energy

Range (2006 Values)

4.13 < mbq(GeV) < 4.27

Table 5: RMS charge radii and mass-energies of fundamental particles,

The predicted radii ranges above demonstrate that no significant deviation from table (4)

values exists. This emphasizes that the EGM harmonic representation of fundamental particles is a

robust formulation and is insensitive to minor fluctuations in particle mass, particularly in the

absence of experimentally determined RMS charge radii.

Therefore, we may conclude that the EGM construct continues to predict experimentally verifiedresults within the SM to high computational precision.

4.2.3 TOP QUARK MASS

Dilemma

The Collider Detector at Fermilab (CDF) and D-ZERO (D0) Collaborations have recently

revised their world average value of Top Quark mass from 178.0(GeV/c2) in 2004 [35] to,

172.0 in 2005 [36], 172.5 in early 2006, then 171.4 in July 2006. [37]

ote: since the precise value of mtq is subject to frequent revision, we shall utilize the 2005 valuein the resolution of the dilemma as it sits between the 2006 values.

Resolution

The EGM method utilizes fundamental particle RMS charge radius to determine mass.

Currently, Quark radii are not precisely known and approximations were applied in the formulation

of mtq displayed in table (4). However, if one utilizes the revised experimental value of m tq =172.0(GeV/c

2) to calculate the RMS charge radius of the Top Quark rtq, based on Proton

harmonics, it is immediately evident that a decrease in rtq of < 1.508(%) produces the new

world average value precisely. The relevant calculations may be performed simply as follows,

The revised Top Quark radius based upon the new world average Top Quark mass,

r

5

1

1409

172GeV

c2

.

mp

2

.. 0.9156 1016

cm.=

(26)

The decrease in Top Quark RMS charge radius [relative to the table (4) value] based upon thenew world average Top Quark mass becomes,

rtq

r

5

1

1409

172GeV

c2

.

mp

2

..

1 1.5076 %( )=

(27)

where, rtq denotes the RMS charge radius of the Top Quark from table (4).


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Therefore, since the change in rtq is so small and its experimental value is not precisely known,we may conclude the EGM construct continues to predict experimentally verified results within theSM to high computational precision.

ote: the 2006 value for revised m tq modifies the error defined by Eq. (27) to < 1.65(%).

5 DISCUSSIO

5.1 EXPERIMENTAL EVIDENCE OF UNIFICATION

Table (3, 4, 5) display mathematical facts demonstrating that all fundamental particles may

be represented as harmonics of an arbitrarily selected reference particle, in complete agreement with

the SM. Considering that the EGM method is so radically different and quantifies the physical

world beyond contemporary solutions, one becomes tempted to disregard table (3, 4, 5) in favor of

concluding these to be coincidental.

However, it is inconceivable that such precision from a single paradigm spanning the entire

family of fundamental particles could be coincidental. The derivation of the Top Quark mass-

energy is in itself, an astonishing result which the SM is currently incapable of producing.

Moreover, the derivation of (a), EM radii characteristics of the Proton and Neutron (rE, rM

and rM) (b), the classical RMS charge radius of the Proton (c), the 1st term of the Hydrogen atomspectrum A and (d), the Bohr radius rx: all from the same paradigm, [18-20] strengthens theharmonic case.

Additionally, Storti et. al. demonstrate in Quinta Essentia, A Practical Guide to Space-Time Engineering, Part 3: pg. 54 (see: Ref.) that the probability of coincidence is


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5.2 THE ANSWERS TO SOME IMPORTANT QUESTIONS

5.2.1 WHAT CAUSES HARMONIC PATTERNS TO FORM?

(a) ZPF Equilibrium

A free fundamental particle is regarded by EGM as a bubble of energy equivalent mass.

Nature always seeks the lowest energy state: so surely, the lowest state for a free fundamental particle should be to diffuse itself to non-existence in the absence of something acting to

keep it contained?

This provokes the suggestion that a free fundamental particle is kept contained by the

surrounding space-time manifold. In other words, free fundamental particles are analogous to

neutrally buoyant bubbles floating in a locally static fluid (the space-time manifold). EGM is an

approximation method, developed by the application of standard engineering tools, which finds the

ZPF equilibrium point between the mass-energy equivalence of the particle and the space-time

manifold (the ZPF) surrounding it - as depicted by Fig. (3).

(b) Inherent Quantum Characteristics

If one assumes that the basic nature of the Universe is built upon quantum states ofexistence, it follows that ZPF equilibrium is a common and convenient feature amongst free

fundamental particles by which to test this assumption. Relativity tells us that no absolute frames of

reference exist, so a logical course of action is to define a datum as EGM is derived from a

gravitational base. In our case, it is an arbitrary choice of fundamental particle.

To be representative of the quantum realm, it follows that ZPF equilibrium between free

fundamental particles should also be analogous to quantum and fractional quantum numbers as

one finds with the Quantum Hall Effect. Subsequently, the harmonic patterns of table (3) form

because the determination of ZPF equilibrium is applied to inherently quantum characteristic

objects i.e. fundamental particles.Hence, it should be no surprise to the reader that comparing a set of inherently quantum

characterized objects to each other, each of which may be described by a single wavefunction at its

harmonic cut-off frequency, results in a globally harmonic description. That is, the EGM harmonic

representation of fundamental particles is a quantum statement of ZPF equilibrium as one would

expect. In-fact, it would be alarming if table (3), or a suitable variation thereof, could not be

formulated.

Therefore, harmonic patterns form due to inherent quantum characteristics and ZPF equilibrium.

5.2.2 WHY HAVENT THE NEW PARTICLES BEEN EXPERIMENTALLY DETECTED?

EGM approaches the question of particle existence, not just by mass as in the SM, but by

harmonic cut-off frequency (i.e. by mass andZPF equilibrium). Storti et. al. showed in [9]

that the bulk of the PV spectral energy

13

at the surface of the Earth exists well above the THzrange. Hence, generalizing this result to any mass implies that the harmonic cut-off period14

T

defines the minimum detection interval to confirm (or refute) the existence of the proposed L2, L3,

L5 Leptons and associated 2, 3, 5 Neutrinos. In other words, a particle exists forat least the

period specified by T i.e. its minimum lifetime.Quantum Field Theory (QFT) approaches this question from a highly useful, but extremely

limited perspective compared to the EGM construct. QFT utilizes particle mass to determine the

13>> 99.99(%).

14 The inverse of .


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14

minimum detection period (in terms of eV) to be designed into experiments. To date, this approach

has been highly successful, but results in the conclusion that no new Leptons exist beyond the SM

in the mass-energy range specified by the proposed Leptons. Whilst QFT is a highly useful

yardstick, it is by no means a definitive benchmark to warrant termination of exploratory

investigations for additional particles.

Typically in the SM, short lived particles are seen as resonances in cross sections of data

sets and many Hadrons in the data tables are revealed in this manner. Hence, the SM asserts that the

more unstable particles are, the stronger the interaction and the greater the likelihood of detection.The EGM construct regards the existing Leptons of the SM as long-lived particles. It also

asserts that the SM does not adequately address the existence or stability of the extremely short-livedLeptons proposed. This assertion is supported by the fact that detection of these particles issubstantially beyond current capabilities due to:

1. The minimum detection interval (with negligible experimental error) being < 10-29(s).2. The possibility that the proposed Leptons are transient (intermediate) states of particle

production processes which decay before detection. For example, perhaps an Electron

passes through an L2 phase prior to stabilization to Electronic form (for an appropriate

production process). Subsequently, this would be not be detected if the transition process is

very rapid and the accelerator energies are too low.

3. The possibility of statistically low production events.Hence:

1. The proposed Leptons are too short-lived to appear as resonances in cross-sections.2. The SM assertion that the more unstable particles are, the stronger the interaction and the

greater the likelihood of detection is invalid for the proposed Leptons.

Therefore, contemporary particle experiments are incapable of detecting the proposed Leptons atthe minimum accelerator energy levels required to refute the EGM construct.

5.2.3 WHY SHOULD ONE BELIEVE THAT ALL FUNDAMENTAL PARTICLES MAY BE

DESCRIBED AS HARMONIC MULTIPLES OF EACH OTHER?

Because of the precise experimental and mathematical evidence presented in table (3, 4, 6).

These results were achieved by construction of a model based upon a single gravitational paradigm.Moreover, Storti et. al. also derive the Casmir force in [11] from [5-10] utilizing Eq. (1 - 3).

5.2.4 WHY IS EGM A METHOD AND NOT A THEORY?

EGM is a method and not a theory because: (i) it is an engineering approximation and (ii),the mass and size of most subatomic particles are not precisely known. It harmonizes all

fundamental particles relative to an arbitrarily chosen reference particle by parameterizing ZPF

equilibrium in terms of harmonic cut-off frequency .

The formulation of table (3) is a robust approximation based upon PDG data. Otherinterpretations are possible, depending on the values utilized. For example, if one re-applies the

method presented in [16] based upon other data, the values of St in table (3) might differ.

However, in the absence of exact experimentally measured mass and size information, there is littlemotivation to postulate alternative harmonic sequences, particularly since the current formulation

fits the available experimental evidence extremely well.

If all mass and size values were exactly known by experimental measurement, the main

sequence formulated in [16] (or a suitable variation thereof) will produce a precise harmonic

representation of fundamental particles, invariant to interpretation. Table (3) values cannot be

dismissed due to potential multiplicity before reconciling how:

1. , which is the basis of the table (3) construct, produces Eq. (10, 11) as derived in [13].These generate radii values substantially more accurate than any other contemporary

method. In-fact, it is a noteworthy result that EGM is capable of producing the Neutron MS


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charge radius as a positive quantity. Conventional techniques favor the non-intuitive form of

a negative squared quantity.

2. is capable of producing a Top Quark mass value the SM cannot.3. EGM produces the results defined in table (6).4. Extremely short-lived Leptons [i.e. with lifetimes of < 10-29(s)] cannot exist, or do not

exist for a plausible harmonic interpretation.

5. Any other harmonic interpretation, in the absence of exact mass and size values determinedexperimentally, denote a superior formulation.

Therefore, EGM is a method facilitating the harmonic representation of fundamental particles.

5.2.5 WHAT WOULD ONE NEED TO DO, IN ORDER TO DISPROVE THE EGM METHOD?

Explain how experimental measurements of charge radii and mass-energy by international

collaborations such as CDF, D0, L3, SELEX and ZEUS in [29,35-40,44], do not correlate to EGM

calculations.

5.2.6 WHY DOES THE EGM METHOD PRODUCE CURRENT QUARK MASSES AND NOTCONSTITUENT MASSES?

The EGM method is capable of producing current and constituent Quark masses, only

current Quark masses are presented herein. This manuscript is limited to current Quark masses

because it is the simplest example of ZPF equilibrium applicable whereby a particle is treated as a

system and the equilibrium radius is calculated.

Determination of the constituent Quark mass is a more complicated process, but the method

of solution remains basically the same. For example, Storti et. al. calculate an experimentallyimplicit value of the Bohr radius in [20] by treating the atom as a system in equilibrium with the

polarized ZPF.

5.2.7 WHY DOES THE EGM METHOD YIELD ONLY THE THREE OBSERVED

FAMILIES?

This occurs because it treats all objects with mass as a system (e.g. the Bohr atom) inequilibrium with the polarized ZPF (the objects own gravitational field). Therefore, sincefundamental particles with classical form factor denote fundamental states (or systems: Quarks inthe Proton and eutron) of polarized ZPF equilibrium, it follows that only the three families will be

predicted.


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5.3 PERIODIC TABLE OF ELEMENTARY PARTICLES

Assuming QB5,6 to be Intermediate Vector Bosons (IVB's), we shall conjecture that the

periodic table of elementary particles may be constructed as follows,

TTyyppeess ooffMMaatttteerr

GGrroouupp II GGrroouupp IIII GGrroouupp IIIIII

Up 14

+2/3,1/2,[R,G,B]

uuqq1.5 < muq(MeV) < 3

Charm 42

+2/3,1/2,[R,G,B]

ccqq

1.1833(GeV)

Top 140

+2/3,1/2,[R,G,B]

ttqq

172.0(GeV)

QQuuaarrkkss

Down 14

-1/3,1/2,[R,G,B]

ddqq3 < mdq(MeV) < 7

Strange 28

-1/3,1/2,[R,G,B]

ssqq

113.9460(MeV)

Bottom 56

-1/3,1/2,[R,G,B]

bbqq4.13 < mbq(GeV) < 4.27

Electron 2

-1,1/2

ee= 0.5110(MeV)

Muon 8

-1,1/2

= 105.7(MeV)

Tau 12

-1,1/2

= 1.777(GeV)

SSttaann

ddaarrdd

MMooddeell

LLeepp

ttoonnss

Electron Neutrino 20,1/2

ee

< 2(eV)

Muon Neutrino 80,1/2

< 0.19(MeV)

Tau Neutrino 120,1/2

< 18.2(MeV)

L2 4

-1,1/2

LL22

9(MeV)

L3 6

-1,1/2

LL33

57(MeV)

L5 10

-1,1/2

LL55

566(MeV)

EEGGMM

LLeeppttoonnss

L2 Neutrino 4

0,1/2

22

men

L3 Neutrino 6

0,1/2

33

mn

L5 Neutrino 10

0,1/2

55

mnSSttaannddaarrdd MMooddeell aanndd EEGGMM BBoossoonnss

Photon N/A

1,Charge,

3.2 x10-45

(eV)

Gluon ?

1,Colour,1

ggll< 10(MeV)

QB6 84

1, Weak Charge,10-6

QQBB66

22(GeV)

Z Boson 112

1,Weak Charge,10-6

ZZ

91.1875(GeV)

Graviton N/A

2,Energy,10-39

gg

= 2m

QB5 70

1, Weak Charge,10-6

QQBB55

10(GeV)

W Boson 98

1,Weak Charge,10-6

WW

80.27(GeV)

Higgs Boson 126

0,Higgs Field,?

HH

> 114.4(GeV)

Table 7: predicted periodic table of elementary particles,

LLeeggeennddQQuuaarrkkss LLeeppttoonnss BBoossoonnss

Name St

Charge(e),Spin,Colour

SSyymmbboollMass-Energy

Name St

Charge(e),Spin


Name

Spin,Source,*SC


(i) *Where, SC denotes coupling strength at 1(GeV). [45]

(ii) The values of St in table (7) utilize the Proton as the reference particle. This is due to itsRMS charge radius and mass-energy being precisely known by physical measurement.

Table 7: particle legend,


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6 COCLUSIO

A concise mathematical relationship, based upon homogeneity concepts inherent in

Buckingham Theory, augmented with Fourier series, has been used to combine gravitational

acceleration and ElectroMagnetism into a method producing fundamental particle properties to

extraordinary precision. This also results in the representation of fundamental particles as harmonic

forms of each other. Additionally, the solution herein predicts the existence of new fundamental

particles not found within the Standard Model suggesting the following:1. An exciting avenue for community exploration, beyond the Standard Model.2. The potential for new Physics at higher accelerator energies.3. The potential for unification of fundamental particles.4. Physical limitations on the value of two extremely important mathematical constants [i.e.

and ] at the quantum mechanical level subject to uncertainty principles.

REFERECES

[1] B.S. Massey, Mechanics of Fluids sixth edition, Van Nostrand Reinhold (International), 1989,Ch. 9.

[2] Rogers & Mayhew, Engineering Thermodynamics Work & Heat Transfer third edition,Longman Scientific & Technical, 1980, Part IV, Ch. 22.

[3] Douglas, Gasiorek, Swaffield, Fluid Mechanics second edition, Longman Scientific &Technical, 1987, Part VII, Ch. 25.

[4] K.A. Stroud, Further Engineering Mathematics, MacMillan Education LTD, Camelot PressLTD, 1986, Programme 17.

Note: [5 - 20] refer to: http://www.deltagroupengineering.com/Docs/Metric_Engineering.pdf

Riccardo C. Storti, Quinta Essentia: A Practical Guide to Space-Time Engineering, Part 3, MetricEngineering & The Quasi-Unification of Particle Physics, ISBN 978-1-84753-942-7, In Press.

[5] Ch. 3.1, Dimensional Analysis, Pg(85 - 95):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum I,

Physics Essays: Vol. 19, No. 1: March 2006.

[6] Ch. 3.2, General Modelling and the Critical Factor, Pg(97 - 105):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum II,

Physics Essays: Vol. 19, No. 2: June 2006.

[7] Ch. 3.3, The Engineered Metric, Pg(107 - 114):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum III,

Physics Essays: Vol. 19, No. 3: September 2006.

[8] Ch. 3.4, Amplitude and Frequency Spectra, Pg(115 - 123):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum IV,


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Physics Essays: Vol. 19, No. 4: December 2006.

[9] Ch. 3.5, General Similarity, Pg(125 - 143):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum V,

Physics Essays: Vol. 20, No. 1: March 2007.

[10] Ch. 3.6, Harmonic and Spectral Similarity, Pg(145 - 157):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum VI,

Physics Essays: Vol. 20, No. 2: June 2007.

[11] Ch. 3.7, The Casimir Effect, Pg(159 - 166):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum VII,

Physics Essays: Vol. 20, No. 3: September 2007.

[12] Ch. 3.8, Derivation of the Photon Mass-Energy Threshold, Pg(169 - 173):This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Derivation of thePhoton mass-energy threshold, The Nature of Light: What Is a Photon?, edited by C.

Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE,

Bellingham, WA, 2005) [pg. 207 - 213].

[13] Ch. 3.9, Derivation of Fundamental Particle Radii (Electron, Proton and Neutron),

Pg(175 - 182).

This chapter has been submitted to Physics Essays as: Riccardo C. Storti, Todd J.Desiato, Derivation of Fundamental Particle Radii (Electron, Proton and Neutron).

[14] Ch. 3.10, Derivation of the Photon and Graviton Mass-Energies and Radii,

Pg(183 - 187):

This chapter appears in: Riccardo C. Storti, Todd J. Desiato, Derivation of thePhoton & Graviton mass-energies & radii, The Nature of Light: What Is a Photon?,

edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol.

5866 (SPIE, Bellingham, WA, 2005) [pg. 214 - 217].

[15] Ch. 3.11, Derivation of Lepton Radii, Pg(189 - 193).

[16] Ch. 3.12, Derivation of Quark and Boson Mass-Energies and Radii, Pg(195 - 203).

[17] Ch. 3.13, The Planck Scale, Photons, Predicting New Particles and Designing anExperiment to Test the Negative Energy Conjecture, Pg(205 - 216).

[18] App. 3.G, Derivation of ElectroMagnetic Radii, Pg(255 - 262).

[19] App. 3.H, Calculation of L2, L3 and L5 Associated Neutrino Radii, Pg(263).

[20] App. 3.I, Derivation of the Hydrogen Atom Spectrum (Balmer Series) and an

Experimentally Implicit Definition of the Bohr Radius, Pg(265 - 268).


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[21] App. 3.K, Numerical Simulations, MathCad 8 Professional, Complete Simulation,

Pg(283 - 363).

[22] App. 3.L, Numerical Simulations, MathCad 8 Professional, Calculation Engine,

Pg(367 - 386).

[23] App. 3.M, Numerical Simulations, MathCad 12, High Precision Calculation Results,

Pg(389 - 393).

[24] Alfonso Rueda, Bernard Haisch, Contribution to inertial mass by reaction of the vacuum toaccelerated motion, Found.Phys. 28 (1998) 1057-1108: http://www.arxiv.org/abs/physics/9802030

[25] Alfonso Rueda, Bernard Haisch, Inertia as reaction of the vacuum to accelerated motion,Phys.Lett. A240 (1998) 115-126: http://www.arxiv.org/abs/physics/9802031

[26] Bernard Haisch, Alfonso Rueda, Hal Puthoff, Advances in the proposed electromagneticzero-point field theory of inertia, presentation at 34th AIAA/ASME/SAE/ASEE Joint Propulsion

Conference, July 13-15, 1998, Cleveland, OH, 10 pages: http://www.arxiv.org/abs/physics/9807023

[27] Puthoff et. Al., Polarizable-Vacuum (PV) approach to general relativity, Found. Phys. 32,927 - 943 (2002): http://xxx.lanl.gov/abs/gr-qc/9909037

[28] Particle Data Group, Photon Mass-Energy Threshold: S. Eidelman et Al. Phys. Lett. B 592, 1(2004): http://pdg.lbl.gov/2006/listings/s000.pdf

[29] The SELEX Collaboration, Measurement of the -

Charge Radius by -

- Electron Elastic

Scattering, Phys.Lett. B522 (2001) 233-239: http://arxiv.org/hep-ex/0106053

[30] Karmanov et. Al., On Calculation of the Neutron Charge Radius, Contribution to the ThirdInternational Conference on Perspectives in Hadronic Physics, Trieste, Italy, 7-11 May 2001, Nucl.

Phys. A699 (2002) 148-151: http://arxiv.org/abs/hep-ph/0106349

[31] P. W. Milonni, The Quantum Vacuum An Introduction to Quantum Electrodynamics,Academic Press, Inc. 1994. Page 403.

[32] Mathworld, http://mathworld.wolfram.com/Euler-MascheroniConstant.html

[33] Hirsch et. Al., Bounds on the tau and muon neutrino vector and axial vector charge radius,Phys. Rev. D67: http://arxiv.org/abs/hep-ph/0210137

[34] National Institute of Standards and Technology (NIST): http://physics.nist.gov/cuu/

[35] The D-ZERO Collaboration, A Precision Measurement of the Mass of the Top Quark, Nature429 (2004) 638-642: http://arxiv.org/abs/hep-ex/0406031

[36] Progress in Top Quark Physics (Evelyn Thomson): Conference proceedings for PANIC05,

Particles & Nuclei International Conference, Santa Fe, New Mexico (USA), October 24 28, 2005.

http://arxiv.org/abs/hep-ex/0602024

[37] Combination of CDF and D0 Results on the Mass of the Top Quark, Fermilab-TM-2347-E,

TEVEWWG/top 2006/01, CDF-8162, D0-5064: http://arxiv.org/abs/hep-ex/0603039


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[38] The CDF & D0 Collaborations, W Mass & Properties, FERMILAB-CONF-05-507-E.


[39] The L3 Collaboration, Measurement of the Mass and the Width of the W Boson at LEP, Eur.

Phys.J. C45 (2006) 569-587: http://arxiv.org/abs/hep-ex/0511049

[40] The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group,

the SLD Electroweak & Heavy Flavor Groups, Precision Electroweak Measurements on the ZResonance, CERN-PH-EP/2005-041, SLAC-R-774: http://arxiv.org/abs/hep-ex/0509008

[41] Hammer and Meiner et. Al., Updated dispersion-theoretical analysis of the nucleonelectromagnetic form factors, Eur. Phys.J. A20 (2004) 469-473:

http://arxiv.org/abs/hep-ph/0312081

[42] Hammer et. Al, Nucleon Form Factors in Dispersion Theory, invited talk at the Symposium"20 Years of Physics at the Mainz Microtron MAMI", October 20-22, 2005, Mainz, Germany,

HISKP-TH-05/25: http://arxiv.org/abs/hep-ph/0602121

[43] Spectrum of the Hydrogen Atom, University of Tel Aviv.

http://www.tau.ac.il/~phchlab/experiments/hydrogen/balmer.htm

[44] The ZEUS Collaboration, Search for contact interactions, large extra dimensions and finite

quark radius in ep collisions at HERA, Phys. Lett. B591 (2004) 23-41:


[45] James William Rohlf, Modern Physics from to Z, John Wiley & Sons, Inc. 1994.

[46] J. F. Douglas, Solving Problems in Fluid Mechanics, Vol. 2, Third Edition, Longman

Scientific & Technical, ISBN 0-470-20776-0 (USA only), 1986.


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APPEDIX A

T r M,( )1

r M,( )(A1)

Illustrational (only) wavefunction [Eq. (A2)] based on Proton harmonics,

St t, sin St 2. . r mp,

. t. (A2)

0 5 .1029

1 .1028

1.5 .1028

2 .1028

2.5 .1028

3 .1028

3.5 .1028

Proton, Neutron

Electron, Electron Neutrino

L2, v2

L3, v3

1 t,( )

2 t,( )

4 t,( )

6 t,( )

1

2T r mp,.

t

Figure A1,

0 5 .1030

1 .1029

1.5 .1029

2 .1029

2.5 .1029

3 .1029

3.5 .1029

4 .1029

4.5 .1029

Muon, Muon Neutrino

L5, v5

Tau, Tau Neutrino

Up and Down Quark

8 t,( )

10 t,( )

12 t,( )

14 t,( )

1

16T r mp,.

t

Figure A2,


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0 1 .1030

2 .1030

3 .1030

4 .1030

5 .1030

6 .1030

7 .1030

8 .1030

9 .1030

1 .1029

1.1 .1029

1.2 .1029

1.3 .1029

Strange Quark

Charm Quark

Bottom Quark

QB5

28 t,( )

42 t,( )

56 t,( )

70 t,( )

1

56T r mp,.

t

Figure A3,

0 5 .1031

1 .1030

1.5 .1030

2 .1030

2.5 .1030

3 .1030

3.5 .1030

4 .1030

4.5 .1030

QB6

W Boson

Z Boson

Higgs Boson

Top Quark

84 t,( )

98 t,( )

112 t,( )

126 t,( )

140 t,( )

1

168T r mp,.

t

Figure A4,


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APPEDIX B

Graviton mass-energy, [14]

mgg = 2m (B1)

Photon mass-energy, [14]

mh

re3

3 re.

2 c. G. m e.

.512 G. m e

.

c 2.

2

.n re m e,

ln 2 n re m e,.

2

.

(B2)

Photon RMS charge radius 1st

representation, [14]

r re

5

m

m e c2.

2

. (B3)

Photon RMS charge radius 2nd

representation (in terms of the Planck scale), [17]

r KG h.

c3

.r

r

.

(B4)

Graviton RMS charge radius, [14]

rgg

54 r.

(B5)

Fine structure constant 2nd

representation, [15]

r

r

e

r

r. (B6)

Neutron charge distribution, [18]

ch r( )2

3

KS

3

r

5. x2

1.

. e

r

r

2

1

x3

e

r

x r

.

2

.. (B7)

Neutron Magnetic radius, [18]

Given

r ch rM.

r

rdr

r

r ch r( )d

(B8)

rM Find r M

Proton Electric radius, [18]

Given

r ch rE.

r

rdr

r ch r( )d (B9)

rE Find r E


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Proton Magnetic radius, [18]

Given

r ch rM.

rdr

r

r ch r( )d

(B10)

rM Find r M

Classical RMS charge radius of the Proton by the EGM method, [18]

rP rE1

2rM r

.

(B11)

1st

term of the Balmer series by the EGM method, [18]

A

PV 1 K rBohr., mp,

2 n K rBohr. mp,

.

(B12)

EGM wavelength, [8]

PV n PV r, M,c

PV n PV r, M,(B13)

Bohr radius, [43]

rBohr

0 h2.

m e. Q e

2.(B14)


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APPEDIX C

Illustration examples of Dimensional Analysis Techniques and Buckingham Theory

(for unfamiliar readers) have been taken from [46] and shown below,


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