The Natural Philosophy Of Fundamental Particles

The atural Philosophy of Fundamental Particles Riccardo C. Storti1 Key Words: Balmer Series, Bohr Radius, Buckingham Π Theory, Casimir Force, ElectroMagnetics, Equivalence Principle, Euler’s 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 Buckingham’s “Π” (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.

1

[email protected] Delta Group Engineering P/L 1

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 (DAT’s) and Buckingham “Π” Theory (BPT), both being well established and thoroughly tested geometric engineering principles [1-3], via Fourier 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 theoretical representation 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 . 2

r

n PV

2 . i . π .n PV .ω e π . n PV

.. PV( 1 , r , M ) t i

(1)

such that, the frequency spectrum of the harmonic gravitational field “ωPV” is given by Eq.(2), [8] ω PV n PV, r , M

n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

(2)

where, Variable ωPV(1,r,M) KPV

nPV r M G

Description Units Fundamental spectral frequency. Hz 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 None all applications herein, the effect is approximated to KPV(r,M) = 1. Harmonic modes of the gravitational field. Magnitude of position vector from centre of mass. m Mass. kg 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,

Gravitational Acceleration

g

Time

Figure 1: harmonic representation of gravitational acceleration, 2

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. DAT’s and BPT). Therefore, unifying (in principle) 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 “ωPV(1,r,M)”, whilst the upper boundary value “ωΩ” (also termed the harmonic cut-off frequency) is given by Eq. (3), ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3)

supported by Eq. (4-7), n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

(4)

1

3

Ω ( r, M )

108.

U m( r , M )

12. 768 81.

U ω( r , M )

U m( r , M )

U ω( r , M )

3 .M .c

U m( r , M )

2

(5)

U ω( r , M )

2

(6)

3

4 .π .r

h . 4 ω PV( 1 , r , M ) 3 2 .c

(7)

where, Variable nΩ Ω Um Uω h c

Description Harmonic cut-off mode [mode number at ωΩ]. Harmonic cut-off function. Mass-energy density of a solid spherical gravitational object. Energy density of mass induced gravitational field scaled to the fundamental spectral frequency. Planck’s Constant [6.6260693 x10-34]. Velocity of light in a vacuum. Table 2,

Units None Pa

Js m/s

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.

3

1.2.2

POYNTING VECTOR

It was demonstrated by “Haisch, Puthoff and Rueda” in [24-26] that “inertia” may have ElectroMagnetic (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 ω n PV, r , M

U ω( r , M ) .

n PV

2

4

4

n PV

(8)

Therefore, the Poynting vector2 of the polarized Zero-Point (ZP) gravitational field “Sω” surrounding a solid spherical object with homogeneous mass-energy distribution is given by, S ω n PV, r , M

c .U ω n PV, r , M

(9)

ZPF Poynting Vector

and may be graphically represented as follows,

S ω n PV , R E , M E

n PV Harmonic

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 “nΩ” and “ωΩ” are important characteristics of gravitational fields. We shall utilize these characteristics to “quasi-unify” particle physics 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 “nΩ” and the classical 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 “nΩ”. 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 2

Per change in odd harmonic mode number. Consistent with experimental evidence and interpretation of data. 4 From first principles and from a single paradigm. 3

4

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 the curvature of the space-time manifold surrounding an object to be a “virtual fluid” in equilibrium with the object itself5. 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 (at least) 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 “rπ” [Eq. (10)] producing the experimentally verified result “rp” by the SELEX Collaboration as stated in [29]6, rπ

5

h .m e

4

. . m . 27 h c . e 2 3 4 .π .G m p 16.c .π .m p

(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 “rν” [Eq. (11)] with respect to the Neutron charge distribution as illustrated in Fig. (4). It is shown in [18] that “rν” relates to the MS charge radius “KS” by a simple formula [Eq. (12)] producing the experimentally verified result “KX” as presented in [30]7, rν

5

h .m e

4

. . m . 27 h c . e 2 3 4 .π .G m n 16.c .π .m n

(11)

where, “mn” denotes Neutron rest-mass.

5

The 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 rπ = 0.8306(fm), rp = 0.8307 ± 0.012(fm). 7 rν = 0.8269(fm), KS = -0.1133(fm2), KX = -0.113 ± 0.005(fm2). 5

Neutron Charge Distribution

Charge Density

rν

r dr

ρ ch( r )

r dr

ρ ch r 0

5. 3

rν

ρ ch r dr

r Radius

Charge Density Maximum Charge Density Minimum Charge Density

Figure 4: Neutron charge distribution, KS

2 3 . π . r ν ( 1 x) . x3 . 2 8 1 x x

(12)

where, “x” is solved numerically8 within the “MathCad” environment by the following algorithm, [18] Given 2

x

ln( x) . 2

x x

1

(13)

1 3

(14)

Find( x)

Utilizing “KS”, “KX” may be converted to determine an experimental zero charge density radial position value “rX” according to Eq. (15), rX

rν KS

. K .K S X

(15)

In the case of the Electron (as with the Proton), the ZPF equilibrium radius coincides with the RMS charge radius “rε” [Eq. (16)] producing an experimentally implied result9 as stated in [31], 9

r ε r e.

1. 2

5

ln 2 .n Ω r e , m e

γ

(16)

where, “re” and “γ” denote the classical Electron radius and Euler-Mascheroni constant [32] respectively.

8 9

x = 0.6829, rX = 0.8256 ± 0.018(fm). rε ≥ 0.0118(fm), γ = 0.577215664901533. 6

3

HARMOIC REPRESETATIO OF FUDAMETAL PARTICLES

3.1

ESTABLISHING THE FOUNDATIONS

Motivated by the physical validation of Eq. (10, 11), Storti et. al. conducted thought experiments in [13] to investigate harmonic and trigonometric relationships by analyzing various forms of radii combinations for the Electron, Proton and Neutron consistent with the DAT’s and BPT derivations in [5-12] – yielding the following useful approximations, ω Ω r ε,m e

ω Ω r ε,m e

ω Ω r π, m p

ω Ω r ν,mn

rε rπ

α

(17)

2

(18)

π rν

rε

2

.e

3

(19)

rπ

where, (i) (ii)

(iii) (iv) 3.2

“α” and “e” denote the fine structure constant and exponential function respectively. Eq. (17) error: (a) associated with “ωΩ(rε,me)/ωΩ(rπ,mp) = 2” is “8.876 x10-3(%)” (b) associated with “ωΩ(rε,me)/ωΩ(rν,mn) = 2” is “0.266(%)”. Eq. (18) error is “2.823(%)”. Eq. (19) error is “0.042(%)”.

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 “rν” and “rε”. 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 “rν” and “rε”, constrained by exact mathematical statements [Eq. (16 – 19)], suggests that the gravitational relationship between the Electron and Proton, as inferred by the result “ωΩ(rε,me)/ωΩ(rπ,mp) = 2”, is harmonic. The computational algorithm supporting this contention may be stated as follows, Given α

r ε ω Ω r ε, m e r e ω Ω r π, m p

rν rε

rε rπ

rε rν

rπ

9

2

.e

3

1. 2

5

ln 2 .n Ω r e , m e

γ

2 π

(20)

(21)

Find r ν , r ε

yields, rν

0.826838

rε

0.011802

10

.( fm)

(22)

To a degree of accuracy significantly greater than the Electron or Neutron. 7

where, (i)

(ii) (iii) 3.3

Eq. (17) error: (a) associated with “ωΩ(rε,me)/ωΩ(rπ,mp) = 2” is “4.493 x10-7(%)”. (b) associated with “ωΩ(rε,me)/ωΩ(rν,mn) = 2” is “0.282(%)”. Eq. (18) error is “1.11 x10-13(%)”. Eq. (19) error is “0.026(%)”.

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, ω Ω r 1, M 1 ω Ω r 2, M 2

(23)

St ω

Performing the appropriate substitutions utilizing Eq.(3 - 7), Eq. (23) may be simplified to, M1

2

.

M2

r2 r1

5

St ω

9

(24)

where, “Stω” represents the ratio of two particle spectra. Subsequently, “rε” may be simply calculated according to, 5

r ε r π.

3.4

1 . me 9 2 mp

2

(25)

IDENTIFYING A MATHEMATICAL PATTERN

Utilizing Eq. (24), Storti et. al. identify mathematical patterns in [15-17] showing that “Stω” may be represented in terms of the Proton, Electron and Quark harmonic cut-off frequencies derived from the respective particle. Potentially, three new Leptons (L2, L3, L5 and associated Neutrino’s: ν2, ν3, ν5) and two new Quark / Boson’s (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 “rπ” from “rν” and vice-versa utilising “Stω = 1”) as an approximation only. For precise calculations based upon similar forms, the reader should refer to [13]. &ote: although the newly predicted Leptons are within the kinetic range11 and therefore “should have been experimentally detected”, there are substantial explanations discussed in Section 5.2. Existing and Theoretical Particles Proton (p), Neutron (n) Electron (e), Electron Neutrino (ν νe) L2, ν2 (Theoretical Lepton, Neutrino) L3, ν3 (Theoretical Lepton, Neutrino) Muon (µ µ), Muon Neutrino (ν νµ ) L5, ν5 (Theoretical Lepton, Neutrino) 11

Proton Harmonics Stω = 1 2 4 6 8 10

Electron Harmonics Stω = 1/2 1 2 3 4 5

A region extensively explored in particle physics experiments. 8

Quark Harmonics Stω = 1/14 1/7 2/7 3/7 4/7 5/7

12 6 Tau (ττ), Tau Neutrino (ν ντ) Up Quark (uq), Down Quark (dq) 14 7 Strange Quark (sq) 28 14 Charm Quark (cq) 42 21 Bottom Quark (bq) 56 28 QB5 (Theoretical Quark or Boson) 70 35 QB6 (Theoretical Quark or Boson) 84 42 W Boson 98 49 Z Boson 112 56 Higgs Boson (H) (Theoretical) 126 63 Top Quark (tq) 140 70 Table 3: harmonic representation of fundamental particles, 4

RESULTS

4.1

HARMONIC EVIDENCE OF UNIFICATION

6/7 1 2 3 4 5 6 7 8 9 10

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. The values of “Stω” shown in table (3), predict possible particle mass and radii for all Leptons, Neutrinos, Quarks and Intermediate Vector Bosons (IVB’s), in complete agreement with the SM, PDG estimates and studies by “Hirsch et. al” in [33] as shown in table (4), Particle Proton (p) Neutron (n) Electron (e) Muon (µ µ) Tau (ττ) Electron Neutrino (ν νe) Muon Neutrino (ν νµ ) Tau Neutrino (ν ντ) Up Quark (uq) Down Quark (dq) Strange Quark (sq) Charm Quark (cq) Bottom Quark (bq) Top Quark (tq) W Boson Z Boson Higgs Boson (H) Photon (γγ) Graviton (γγg) L2 (Lepton) L3 (Lepton) L5 (Lepton) ν2 (L2 Neutrino) ν3 (L3 Neutrino) ν5 (L5 Neutrino)

EGM Radii x10-16(cm) rπ = 830.5957 rν = 826.8379 rε = 11.8055 rµ = 8.2165 rτ = 12.2415 ren ≈ 0.0954 rµn ≈ 0.6556 rτn ≈ 1.9588 ruq ≈ 0.7682 rdq ≈ 1.0136 rsq ≈ 0.8879 rcq ≈ 1.0913 rbq ≈ 1.071 rtq ≈ 0.9294 rW ≈ 1.2839 rZ ≈ 1.0616 rH ≈ 0.9403 rγγ = ½Kλλh rgg = 2(2/5)rγγ rL ≈ 10.7518 rν2,ν3,ν5 ≈ ren,µn,τn

EGM Mass-Energy (computed or utilized)

PDG Mass-Energy Range (2005 Values)

Mass-Energy precisely known, See: National Institute of Standards and Technology (NIST) [34] Note: δm = 10-100 men(eV) ≈ 3 - δm mµn(MeV) ≈ 0.19 - δm mτn(MeV) ≈ 18.2 - δm muq(MeV) ≈ 3.5060 mdq(MeV) ≈ 7.0121 msq(MeV) ≈ 113.9460 mcq(GeV) ≈ 1.1833 mbq(GeV) ≈ 4.1196 mtq(GeV) ≈ 178.4979 mW(GeV) ≈ 80.425 mZ(GeV) ≈ 91.1876 mH(GeV) ≈ 114.4 + δm mγγ ≈ 3.2 x10-45(eV) mgg = 2mγγ mL(2) ≈ 9(MeV) mL(3) ≈ 57(MeV) mL(5) ≈ 566(MeV) mν2 ≈ men mν3 ≈ mµn mν5 ≈ mτn 9

men(eV) < 3 mµn(MeV) < 0.19 mτn(MeV) < 18.2 1.5 < muq(MeV) < 4 3 < mdq(MeV) < 8 80 < msq(MeV) < 130 1.15 < mcq(GeV) < 1.35 4.1 < mbq(GeV) < 4.4 169.2 < mtq(GeV) < 179.4 80.387 < mW(GeV) < 80.463 91.1855 < mZ(GeV) < 91.1897 mH(GeV) > 114.4 mγ < 6 x10-17(eV) No definitive commitment

Not predicted or considered

QB5 (Quark or Boson) rQB ≈ 1.0052 mQB(5) ≈ 10(GeV) Not predicted or considered QB6 (Quark or Boson) mQB(6) ≈ 22(GeV) Table 4: RMS charge radii and mass-energies of fundamental particles, where, (i) “Kλ” denotes a Planck scaling factor, determined to be “(π/2)1/3” in [17]. (ii) “λh” 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 of ν2, ν3 and ν5 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 “λh” and “1.5λh” 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 “continuous improvement cycle”, the PDG reconciles its published values of particle properties against the latest experimental and theoretical evidence. The 2006 changes in PDG mass-energy range values not impacting 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 verified results 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 predictions articulated 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 EGM radii 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)].

10

Particle

EGM Radii x10-16(cm)

EGM Mass-Energy (utilized)

PDG Mass-Energy Range (2006 Values) men(eV) < 2 Electron Neutrino (ν νe) ren < 0.0811 PDG Mass-Energy Up Quark (uq) 1.5 < muq(MeV) < 3 0.5469 < ruq < 0.7217 Range (2006 Values) Down Quark (dq) 3 < mdq(MeV) < 7 0.7217 < rdq < 1.0128 Bottom Quark (bq) 1.0719 > rbq > 1.0863 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 verified results 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 value in 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 “mtq = 172.0(GeV/c2)” 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, 5

GeV 172. 2 1 . c r π. 9 mp 140

2

= 0.9156 10

16 .

cm

(26)

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

1 = 1.5076 ( % )

5

GeV 172. 2 1 . c r π. 9 mp 140

2

where, “rtq” denotes the RMS charge radius of the “Top Quark” from table (4). 11

(27)

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 the SM to high computational precision. &ote: the 2006 value for revised “mtq” 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” massenergy 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 (rπE, rπM and rνM) (b), the classical RMS charge radius of the Proton (c), the 1st term of the Hydrogen atom spectrum “λA” and (d), the Bohr radius “rx”: all from the same paradigm, [18-20] strengthens the harmonic case. Additionally, Storti et. al. demonstrate in “Quinta Essentia, A Practical Guide to SpaceTime Engineering, Part 3: pg. 54 (see: Ref.)” that the probability of coincidence is “<< 10-38” based upon the results shown in table (6), Particle / Atom EGM Prediction Experimental Measurement (%) Error -16 Proton (p) rπ = 830.5957 x10 (cm) [13] rπ = 830.6624 x10-16(cm) [29] < 0.008 < 0.062 rπE = 848.5274 x10-16(cm) [18] rπE = 848 x10-16(cm) [41,42] -16 -16 < 0.825 rπM = 849.9334 x10 (cm) [18] rπM = 857 x10 (cm) [41,42] -16 -16 rp = 874.5944 x10 (cm) [18] rp = 875.0 x10 (cm) [34] < 0.046 -16 -16 Neutron (n) rν = 826.8379 x10 (cm) [13] rX ≈ 825.6174 x10 (cm) [18] < 0.148 -26 2 KS = -0.1133 x10 (cm ) [18] KX = -0.113 x10-26(cm2) [30] < 0.296 rνM = 878.9719 x10-16(cm) [18] rνM = 879 x10-16(cm) [41,42] < 0.003 Top Quark (tq) mtq(GeV) ≈ 178.4979 [16,21] < 3.64 mtq(GeV) ≈ 172.0 [36] Hydrogen (H) < 0.131 λA = 657.3290(nm) [20] λB = 656.4696(nm) [43] rBohr = 0.0529(nm) [34] < 0.353 rx = 0.0527(nm) [20] Table 6: experimentally verified EGM12 predictions, where, (i) “rπE” and “rπM” denote the Electric and Magnetic radii of the Proton respectively. (ii) “rνM” denotes the Magnetic radius of the Neutron. (iii) “λA” and “λB” denote the first term of the Hydrogen atom spectrum (Balmer series). (iv) “rp = 875.0 x10-16(cm)” and “rBohr = 0.0529(nm)” are not experimental values, they denote the classical RMS charge radius of the Proton and the Bohr radius, i.e. the official values listed by NIST. [34] &ote: numerical simulations generating all values in table (3, 4, 6) can be found in [21-23].

12

Refer to “Appendix B” for mathematical definitions of EGM predictions. 12

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 of existence, 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 HAVEN’T 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 “ωΩ” (i.e. by mass and ZPF equilibrium). Storti et. al. showed in [9] that the bulk of the PV spectral energy13 at the surface of the Earth exists well above the “THz” range. Hence, generalizing this result to any mass implies that the harmonic cut-off period14 “TΩ” defines the minimum detection interval to confirm (or refute) the existence of the proposed “L2, L3, L5” Leptons and associated “ν2, ν3, ν5” Neutrinos. In other words, a particle exists for at least the period specified by “TΩ” – 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 14

“>> 99.99(%)”. The inverse of “ωΩ”. 13

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 shortlived Leptons proposed. This assertion is supported by the fact that detection of these particles is substantially 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 at the 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 “ωΩ”. The formulation of table (3) is a robust approximation based upon PDG data. Other interpretations 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 “Stω” in table (3) might differ. However, in the absence of exact experimentally measured mass and size information, there is little motivation 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. “ωΩ”, 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 14

2. 3. 4. 5.

charge radius as a positive quantity. Conventional techniques favor the non-intuitive form of a negative squared quantity. “ωΩ” is capable of producing “a Top Quark” mass value – the SM cannot. EGM produces the results defined in table (6). Extremely short-lived Leptons [i.e. with lifetimes of “< 10-29(s)”] cannot exist, or do not exist for a plausible harmonic interpretation. Any other harmonic interpretation, in the absence of exact mass and size values determined experimentally, 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 NOT CONSTITUENT 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 experimentally implicit 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) in equilibrium with the polarized ZPF (the objects own gravitational field). Therefore, since fundamental particles with classical form factor denote fundamental states (or systems: Quarks in the Proton and &eutron) of polarized ZPF equilibrium, it follows that only the three families will be predicted.

15

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,

EGM Leptons

Standard M od e l Leptons

Quarks

Types of Matter Group I Group II Group III Up 14 Charm 42 Top 140 +2/3,1/2,[R,G,B] +2/3,1/2,[R,G,B] +2/3,1/2,[R,G,B] uq cq tq 1.5 < muq(MeV) < 3 ≈ 1.1833(GeV) ≈ 172.0(GeV) Down 14 Strange 28 Bottom 56 -1/3,1/2,[R,G,B] -1/3,1/2,[R,G,B] -1/3,1/2,[R,G,B] dq sq bq 3 < mdq(MeV) < 7 4.13 < mbq(GeV) < 4.27 ≈ 113.9460(MeV) 12 Electron 2 Muon 8 Tau -1,1/2 -1,1/2 -1,1/2 e µ τ = 0.5110(MeV) = 105.7(MeV) = 1.777(GeV) Electron Neutrino 2 Muon Neutrino 8 Tau Neutrino 12 0,1/2 0,1/2 0,1/2 νe νµ ντ < 2(eV) < 0.19(MeV) < 18.2(MeV) L2 4 L3 6 L5 10 -1,1/2 -1,1/2 -1,1/2 L2 L3 L5 ≈ 9(MeV) ≈ 57(MeV) ≈ 566(MeV) L2 Neutrino 4 L3 Neutrino 6 L5 Neutrino 10 0,1/2 0,1/2 0,1/2 ν2 ν3 ν5 ≈ men ≈ mµn ≈ mτn Standard Model and EGM Bosons Photon N/A Gluon 84 Z Boson 112 ? QB6 1, Weak Charge,10-6 1,Colour,1 1,Weak Charge,10-6 1,Charge,α gl QB6 Z γ -45 < 10(MeV) ≈ 22(GeV) ≈ 91.1875(GeV) ≈ 3.2 x10 (eV) Graviton N/A QB5 70 W Boson 98 Higgs Boson 126 -39 -6 -6 2,Energy,10 1, Weak Charge,10 1,Weak Charge,10 0,Higgs Field,? QB5 W H γg ≈ 10(GeV) ≈ 80.27(GeV) > 114.4(GeV) = 2mγγ Table 7: predicted periodic table of elementary particles, Quarks

Legend Leptons

Name

Bosons

Stω Name Stω Name Stω Charge(e),Spin,Colour Charge(e),Spin Spin,Source,*SC Symbol S ym b ol Symbol Mass-Energy Mass-Energy Mass-Energy (i) *Where, “SC” denotes coupling strength at “1(GeV)”. [45] (ii) The values of “Stω” in table (7) utilize the Proton as the reference particle. This is due to its RMS charge radius and mass-energy being precisely known by physical measurement. Table 7: particle legend, 16

6

COCLUSIO

A concise mathematical relationship, based upon homogeneity concepts inherent in Buckingham “Π” 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. “π” and “γ”] 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 Press LTD, 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, Metric Engineering & 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-GraviMagnetics (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-GraviMagnetics (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-GraviMagnetics (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-GraviMagnetics (EGM), Practical modelling methods of the polarizable vacuum – IV, 17

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-GraviMagnetics (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-GraviMagnetics (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-GraviMagnetics (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 the Photon 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 the Photon & 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 an Experiment 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). 18

[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 to accelerated 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 electromagnetic zero-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 Σ- Charge Radius by Σ- - 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 Third International 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, Nature 429 (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

19

[38] The CDF & D0 Collaborations, W Mass & Properties, FERMILAB-CONF-05-507-E. http://arxiv.org/abs/hep-ex/0511039 [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 Z Resonance, CERN-PH-EP/2005-041, SLAC-R-774: http://arxiv.org/abs/hep-ex/0509008 [41] “Hammer and Meißner et. Al”., Updated dispersion-theoretical analysis of the nucleon electromagnetic 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: http://arxiv.org/abs/hep-ex/0401009 [45] “James William Rohlf”, Modern Physics from α 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.

20

APPEDIX A 1

T Ω ( r, M )

(A1)

ω Ω ( r, M )

Illustrational (only) wavefunction “ψ” [Eq. (A2)] based on Proton harmonics, sin St ω .2 .π .ω Ω r π , m p .t

ψ St ω , t

(A2)

1. T Ω r π ,m p 2

ψ( 1, t) ψ( 2, t) ψ( 4, t)

5 .10

0

29

1 .10

28

1.5 .10

28

2 .10

28

2.5 .10

28

3 .10

28

3.5 .10

28

ψ( 6, t)

t

Proton, Neutron Electron, Electron Neutrino L2, v2 L3, v3

Figure A1, 1 . T Ω r π ,m p

16

ψ( 8, t) ψ ( 10 , t ) ψ ( 12 , t )

0

5 .10

30

1 .10

29

1.5 .10

29

2 .10

29

2.5 .10

29

ψ ( 14 , t )

t

Muon, Muon Neutrino L5, v5 Tau, Tau Neutrino Up and Down Quark

Figure A2,

21

3 .10

29

3.5 .10

29

4 .10

29

4.5 .10

29

1 . T Ω r π ,m p

56

ψ ( 28 , t ) ψ ( 42 , t ) ψ ( 56 , t )

0

1 .10

30

2 .10

30

3 .10

30

4 .10

30

5 .10

30

6 .10

30

30

7 .10

8 .10

30

9 .10

30

3 .10

30

1 .10

29

1.1 .10

29

1.2 .10

29

1.3 .10

29

ψ ( 70 , t )

t

Strange Quark Charm Quark Bottom Quark QB5

Figure A3, 1 . T Ω r π ,m p

168

ψ ( 84 , t ) ψ ( 98 , t ) ψ ( 112 , t ) ψ ( 126 , t )

0

5 .10

31

1 .10

30

1.5 .10

30

2 .10

30

2.5 .10

ψ ( 140 , t )

t

QB6 W Boson Z Boson Higgs Boson Top Quark

Figure A4,

22

30

3.5 .10

30

4 .10

30

4.5 .10

30

APPEDIX B Graviton mass-energy, [14] mgg = 2mγγ

(B1)

Photon mass-energy, [14] 3

h .

m γγ

re

π .r e 2 .c .G.m e

3

2

512.G.m e

.

n Ω r e, m e

.

2

c .π

ln 2 .n Ω r e , m e

γ

2

(B2)

Photon RMS charge radius – 1st representation, [14] 5

2

m γγ

r γγ r e .

m e .c

(B3)

2

Photon RMS charge radius – 2nd representation (in terms of the Planck scale), [17] G.h . r µ

r γγ K ω .

c

3

(B4)

rτ

Graviton RMS charge radius, [14] 5

r gg

4 .r γγ

(B5)

Fine structure constant – 2nd representation, [15] rµ

α

rε

rτ

.e

(B6)

rν

Neutron charge distribution, [18] r

ρ ch ( r )

KS

2. 3

3

π .r ν

5.

. e 2

x

rν

2

1. 3

1

e

r x .r

2 ν

(B7)

x

Neutron Magnetic radius, [18] Given r dr rν r ν . ρ ch r νM

(B8)

ρ ch ( r ) d r rν

r νM

Find r νM

Proton Electric radius, [18] Given r dr r ν . ρ ch r πE

(B9)

ρ ch ( r ) d r rν

r πE

Find r πE

23

Proton Magnetic radius, [18] Given ∞ r ν . ρ ch r πM

(B10)

ρ ch ( r ) d r r dr rν

r πM

Find r πM

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

1. 2

r νM

(B11)

rν

1st term of the Balmer series by the EGM method, [18] λA

λ PV 1 , K ω .r Bohr , m p

(B12)

2 .n Ω K ω .r Bohr , m p

EGM wavelength, [8] c

λ PV n PV, r , M

(B13)

ω PV n PV, r , M

Bohr radius, [43] r Bohr

ε 0 .h

2

π .m e .Q e

2

24

(B14)

APPEDIX C Illustration examples of Dimensional Analysis Techniques and Buckingham “Π” Theory (for unfamiliar readers) have been taken from [46] and shown below,

25

26

27

28

29

30

31

32