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European Journal of Applied Sciences – Vol. 12, No. 4
Publication Date: August 25, 2024
DOI:10.14738/aivp.124.17404.
Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components.
European Journal of Applied Sciences, Vol - 12(4). 293-303.
Services for Science and Education – United Kingdom
Fractal Representation of Composite Elementary Fermions with
Positive and Negative Mass Components
Judith Giannini
Independent Researcher/USA
ABSTRACT
The Fractal Rings and Composite Elementary Particles (FRACEP) Model provides
an alternate view to the elementary fermion picture. FRACEP represents a dual
universe where there are equal amounts of positive and negative matter (mass).
Its dual universe construction allows for composite particles of mixed (positive
and negative) mass with fractal-based components. The fractal dimension of these
composite particles is D = ~1.55, and is consistent with other estimates of D for the
elementary fermions and other fractal-based composite models. Initial efforts
developed an empirically-based two-parameter fit for the mass hierarchy formula,
relating the fractal dimension to the composite particle masses. The formula
predicts the cross-family (first-generation particles) masses, and the intra-family
masses (the three generations within each family). The mass predictions for the
structure-based composite particles, as well as, the mass hierarchy are consistent
with the Particle Data Group’s 2016/2020-update estimates for the elementary
fermions.
Keywords: Composite Elementary Particles, Fractals, Negative Mass, Negative Matter,
Preons, Mass Heirarchy, Dual Universe.
INTRODUCTION
Currently in particle physics, the elementary fermions are generally considered to have no
internal components. However, the spontaneous decay of most of them suggests the
possibility that they are composite in nature [1]. The Fractal Rings and Composite Elementary
Particles (FRACEP) Model, is an empirically-based effort to describe a possible construction
for the elementary fremions.
The idea of compositeness in the elementary fermions began taking shape, in the 1970s, with
a quantum-based preon theory describing possible particle substructure. Most of the early
models contained a basic set of two to four preons with an equal number of anti-preons in the
first-generation particles, lacked intrinsic mass, and required Higgs or some other mechanism
to provide the mass. The models often considered preons as having no internal structure [2-
3].
Harari [2] hypothesizes the higher generation fermions are excited states of the first- generation configurations. Shupe [3] explicitly notes that the constituent preons and anti- preons have masses that represent self-energy of the photon and gluon fields coupling to the
leptons and quarks, and that solitons have some of the desired properties for representing
fermionic preon-anti-preon pairs. Numerous models address the possibility of compositeness
in the particles [4-10]. Some predict particle masses considering simple preons as the
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substructure [11-12], while others like Salam [13] address composite preons or preons with a
fractal nature [14-16].
The general assumption is that all levels of the preons are positive mass, with simple, non- complex structures. The FRACEP model takes a broader view, considering them as coherent
composite quantities of matter with positive or negative mass that is associated with a Unified
Potential Field [17] defining the force they exert at any point in space.
Jammer indicates that the concept of matter and mass is an elusive one that’s meaning has
evolved over time [18]. From a classical point-of-view, mass is the amount of matter in a body,
and it is attractive with a force that is inversely proportional to the square of the separation
distance (expressed as the negative gradient of the potential). The general assumption is that
all matter and anti-matter needs to be positive [19], [20] (only one kind of polarity), but
Jammer notes (in Chapter 10) that Bondi pointed out that this assumption is purely empirical
indicating the idea of negative mass is not inconsistent with theory.
Quantum mechanics treats particles as having a dual nature with wave-particle duality.
Jammer notes (in Chapter 14) that it can be viewed as an analogous extension of the classical
picture in comparing the dynamical behavior of wave packets with that of Newtonian
particles. If one assigns “density” to the wave function (as a distribution over an infinite wave
front), integration over three dimensions would yield the mass of the particle’s wave packet.
The (assumed) inherent properties of the particles are assigned to be consistent with
empirical data, but Osmera [21] notes they are not generally related to any physical structure
or internal motions of the particles. This also generally assumes positive mass, though
negative energy states are not excluded, but Chen and Chou [22] studied the potential
usefulness of the negative mass concept in nonrelativistic quantum mechanics.
Quantum field theory is a generalization of quantum mechanics, viewing particles as a
disturbance in the field, and mediating particle exchanged between any two particles
produces a force that is recognized as associated with a particle’s mass in the classical picture
[23]. This also generally assumes positive mass because chiral symmetry considerations allow
the negative sign of the mass to be mathematically rotated away.
Dismissing the idea of the existence of negative mass is tempting because it is unsettling to
readily accept something you cannot see. However, negative mass has been the subject of
considerable theoretical study since the 1800s as discussed in Jammer. Among the more
recent works, Bondi [24] developed a non-singular solution to Einstein’s equations showing a
repulsive force between bodies with mass densities of opposite sign. Bonner [25] considered
mechanics in a universe with negative mass and the influence of the negative mass on the
Schwarzschild black hole solution. Hoyle, Burbidge and Narlikai [26] noted that allowing only
positive mass results in the standard hot Big Bang model, and they developed a scale- invariant form of the gravity equations that reduce to general relativity under the right
conditions, leading to the creation of equal numbers of positive-mass and negative-mass
particles (pairs) during creation events.
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Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components. European Journal of
Applied Sciences, Vol - 12(4). 293-303.
URL: http://dx.doi.org/10.14738/aivp.124.17404
Most recently, Chang [27] explored the force of negative matter and its potential relation to
Dark Matter, using single-metric field equations describing repulsive interactions, and he
studied the relevance of fractal geometry in composite preon characterization of the fermions
[28]. In another approach, Petite and his colleagues [29] developed a bimetric model for a
dual universe describing the interaction of positive and negative masses where the two types
of matter have different light speed (that is, both positive-energy photons and negative- energy photons). Further, they studied the concept of negative masses in standard relativistic
quantum mechanics (the Dirac equation), showing that negative energies are acceptable
provided the masses are simultaneously negative [30].
The FRACEP model has similarities and differences from the traditional preon-based models.
Like the traditional models, it has preon-type components, referred to here as “complex- constituents” (CCs), that make up the fermions. (Because it is not formalized as a quantum- based model, there are not, at this time, quantum numbers associated with the CCs). Unlike
traditional models, the CCs can have complex structures containing both positive and negative
masses because they are built from two fundamental particles with inherent mass resulting
from total symmetry at their pair-creation (Gp with positive mass, and Gn with negative mass)
[31]. (This differs from the standard picture that assumes there is no inherent mass in the
fundamental particles.)
The fractal structures of the FRACEP particles have two levels of complexity in their CCs. The
carrier CCs are the lowest level of complexity, and are the basis of all the other structures. The
radical CCs have a higher level of complexity, and contain carrier CCs and other radical CCs.
This paper describes the fractal structure of the composite fermions, and considers a possible
empirically-based two-parameter fit for its mass hierarchy formula.
THE FRACEP MODEL AND THE BASIC CARRIER CCS
The FRACEP model [31], with its Unified Potential [17], represents a dual universe containing
equal amounts of both positive and negative mass (or perhaps more properly positive and
negative matter associated with a force that is attractive or repulsive). This dual universe
concept allows for particles of purely positive mass, purely negative mass, and mixed-mass
having both positive and negative mass elements. The Bright Universe (BU) is the universe we
see. The Dark Universe (DU) is the universe we cannot see. We hypothesize it contributes to
the cosmological dark matter and energy that compose most of our universe. The basic
elements in this construction are the carrier CCs (Table 1).
Table 1: The basic set of carrier CCs. The p-suffix indicates positive mass; the n-suffix
indicates negative mass. The B designation indicates Bright Universe particles; the D
designation indicates Dark Universe particles.
Universe Type Momentum Carriers Charge Carriers Spin Carriers Mass Type
Bright Universe
MGXp QBp SBp All positive
MRXp
QBn SBn Mixed
Dark Universe
QDp SDp
MGXn QDn SDn All negative
MRXn
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Whereas the Standard Model picture of the fermions assumes that spin and charge are
inherent properties of the particles, FRACEP treats them as components in the composite
structure. The three types of components are the carriers (momentum, spin and charge).
The momentum-carriers carry the bulk of the matter in the particles, while the spin-carriers
and charge-carriers contain only a very small amount of matter compared to the momentum
carriers. The four types of momentum-carriers are purely self-similar fractal structures with
either positive or negative mass (but not mixed mass) that act as single particles. The general
particle, MGXp = Gp 9
X, is a 9th level spherical particle composed of fundamental particles
with only positive mass. The ring particle, MRXp = Gp 6 • 9
X, is a 9th level six-element ring, in
a plane, composed of spherical particles. It also is purely positive mass. These two structures
are the bulk of the BU mass content. The DU equivalents have only negative mass (MGXn = Gn
9
X, and MRXn = Gn 6 • 9
X). These two structures are the bulk of the DU mass content, and
likely contributor to the cosmological dark matter.
The four types of charge-carriers are either purely positive, purely negative, or mixed-mass
with a composite structure acting as a single particle containing three components: a central
momentum carrying particle group (either positive or negative mass), and two identical
charge carrying chains (either positive or negative mass). Positive mass chains (2-Gp 4
19)
lead to QBp or QDp (charge -1/3 e). Negative mass chains (2-Gn 4
19) lead to QBn or QDn
(charge +1/3 e).
The four types of spin-carriers are either purely positive, purely negative, or mixed-mass with
a composite structure acting as a single particle. Each spin-carrier is a five-element structure
composed of four identical momentum carrying particles (either positive or negative mass)
surrounding a single spin carrying component (either positive or negative mass). Positive
mass spin carrying components (2-Gp 5
16) lead to SBp or SDp (spin +1/2). Negative mass spin
carrying components (2-Gn 5
16) lead to SBn or SDn (spin -1/2).
These three carrier types are the basis of all of the radical CCs. The only inherent property of
the carriers is their mass which is obtained from the two fundamental particles that are used
to build them. The spin and charge effects are the result of the dynamic behavior of the
carriers.
THE FRACEP COMPOSITE PARTICLES AND THE RADICAL CCS
Because of their composite structure, the FRACEP fermions have a physical size. It is based on
the internal carrier and radical CC components, and is consistent with theory and scattering
experiment estimates of the elementary fermion size upper limits [32].
The radical CCs are groups of three CCs that are repeatedly used to build up the composite
particles. They act as single particles, and are functionally analogous to the traditional preons.
They can contain carrier CCs, fermions, or other radical CCs. The nested construction,
generally speaking, allows the larger three-element CCs to be composed of one level down
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Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components. European Journal of
Applied Sciences, Vol - 12(4). 293-303.
URL: http://dx.doi.org/10.14738/aivp.124.17404
(smaller in size and mass) three-element sub-CCs, and so forth down to the most basic fractal
structure of the carrier CCs.
Table 2 shows the CC structure of all of the BU composite fermions. Similar structures exist
for the anti-fermions (interchanging, in the CCs, the negative and positive charge-carriers
(QBp ↔ QBn, and QDp ↔ QDn), and at the same time, exchanging the positive and negative
spin-carriers (SBp ↔ SBn, and SDp ↔ SDn). In all cases, the momentum-carriers remain the
same for the particles and the anti-particles: MGXp and MRXp in the BU and MGXn and MRXn
in the DU. The CQ(i) in the u+ particle is one of 3 possible color charges – pairs of e-, e+, and
their dark counter parts giving an approximately zero mass for the color charges and the anti- color charges.
FRACTAL CHARACTERISTICS OF THE COMPOSITE PARTICLES
In one definition, Falconer [33] defines fractals as based on a set of characteristics that
includes fine structure detailed on arbitrarily small scales, usually with fractional dimension.
There is often some self-similarity, but for a natural structure, the features may only appear
fractal-like (in the mathematical set sense) over certain scales. He noted that “There are no
true fractals in nature”, and the distinction between “natural fractals” and mathematical
fractal sets is often blurred. It is in this sense that the FRACEP composite particles (natural
phenomena) are fractal-like and a measure of the fractal dimension is used in characterizing
the particles and components.
Table 2: The Radical CC and carrier CC structural components of the FRACEP fermions
that make up the core mass. The −, c+, −, and t+ particles also contain extra momentum
groups, { }, that are considered excitation masses that are not part of the core particles.
(The measurement of the elementary fermions recognizes the full excited states as the
particle mass.)
STRUCTURE RADICAL CC (RCn)
e− SBp
− SBp + (QBp + MG22p + QDn)
− − + MG22p + RC1 RC1 = [− + 3 MR22p + +]
e− 3 RC2 RC2 = [QBp + MG22p + SBp]
u+ (anti-RC2 + MR22p + anti-RC2)
+ − + MR22p + CQ(i)
d− u+ + MR22p + RC3 RC3 = [e+ + MR22p + e−]
− (RC4 + 2 MG22n + −) + {93 MR22p} RC4 = [RC3 + 6 MR22p + RC1]
c+ (− + 6 MR22p + RC5) + 48 MR22p + RC6
+ {14 MR24p + 14 MR22p}
RC5 = [+ + 6 MR22p + u+]
RC6 = [RC5 + 6 MR22p + anti-RC5]
s− c+(core) + MR22p + RC3
− (− + 6 MR22p + RC8) + MR24p + RC6
+ 93 MR22p + {18 MR24p + 60 MR22p}
RC7 = [− + 3 MR22p + +]
RC8 = [RC3 + 6 MR22p + RC7]
t+ (− + 6 MR22p + RC9) + 4 MR24p + RC10
+ {25 MR26p + 15 MR24p}
RC9 = [+ + 6 MR22p + c+]
RC10 = [RC9 + 6 MR22p + anti-RC9]
b− t+(core) + MR22p + RC3
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Gouyet [34] indicates there are several definitions of fractal dimension that usually giving the
same value, but not necessarily so in all cases. The method used here for determining the
fractal dimension of the particles and their components takes the familiar form D = log(m) /
log(r). For the purposes of this exercise, all masses are normalized by the fundamental
particle mass, m(Gp) = 1.72x10-22 MeV/c2; and all radii are normalized by the fundamental
particle radius, r(Gp) = 3.3x10-35 m.
The carrier CCs are the fundamental basis of all the composite particles and their components,
and they are fractal in the strict set sense of scale invariance. They vary in size from the
smallest CC in the structure, MG13p (4.37x10-10 MeV/c2 and 2.22x10-27 m), to the largest,
MG26p (1.11x10+3 MeV/c2 and 1.49x10-19 m), with its corresponding ring particle MR26p
(6.67x10+3 MeV/c2 and 5.96x10-19 m). The fractal dimension for the entire set is D = 1.58.
The ten radical CCs are fractal in their basis, but only fractal-like in their 3-component
construction which is nested through the group. Seen as a single coherent structure, the
radical CCs have masses that range from the smallest, RC2 (4.65x10-4 MeV/c2 and 1.96x10-
21 m), to the largest and most complex, RC10 (203.65 MeV/c2 and 2.83x10-19 m). The fractal
dimension for the entire set is D = 1.51.
The greatest level of complexity is in the composite particles which are fractal-like in their
components’ construction. The particles have masses [32] ranging from the smallest, ne-
(1.28x10-6 MeV/c2 and 5.68x10-25 m), to the largest, t+ (172572.11 MeV/c2 and 7.31x10-18
m). The computation of D for −, −, c+ and t+ using just the core was not significantly
different from the computation using the total mass (core plus {excitation mass}) and its
corresponding radius.
The fractal dimension for the particle set was generally D ~1.55, with some exceptions. The
down-quark family had a slightly smaller D ~ 1.53, and the largest D was for − at D = 2.0.
These values for the composite particles are consistent with other estimates for the electron.
In [16], the fractal dimension is between 1.431 and 2.033 depending on the phase of the
particle, while in [21], the value was 1.48, and in [35], it was 1.58.
MASS HIERARCHY FORMULA FOR THE COMPOSITE PARTICLES
Initial efforts determined a mass hierarchy formula for the composite particle set, relating D
and the cross-family masses (first-generation of each family). It also developed formulas for
the intra-family masses (the three generations within each family) starting with the predicted
cross-family masses. The formulas are empirically determined based on the lepton number,
the assigned family number, and a characteristic value of D = 1.55. The lepton number for
families f = 1 and f = 2 (neutrinos and electrons respectively) is l = 1, and for families f = 3 and
f = 4 (up-quarks and down-quarks respectively) is l = 0.
The Cross-Family Mass Formula
The cross-family mass formula is:
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Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components. European Journal of
Applied Sciences, Vol - 12(4). 293-303.
URL: http://dx.doi.org/10.14738/aivp.124.17404
m (f, 1) = A ‧ [ lf
2 + (l – 1) 2 (2 – f)] ‧ 10 – l [5 – f 2 + 2 (2 – f)] (1)
where A = 1.978 / D. This formula is a one-parameter fit that determines the mass of the first- generation particles only (Table 3). These particles have only a core mass with no extra
excitation mass components.
The Intra-Family Core Mass Formula
The intra-family mass formula is:
m (f, g) = m (f, 1) ‧ exp [T (f, g)]. (2)
Table 3: The mass hierarchy formula prediction for the first-generation FRACEP
composite fermions. All values are in MeV/c2. Column 2 is the assigned family number,
and column 3 is the lepton number (fermions are l = , and quarks are l = ).
Family f l m (f, l) FRACEP
Construction
Elementary
Fermions
neutrino 1 1 1.28x10−6
1.28x10−6 <0.17
electron 2 1 0.5105 0.5105 0.5110
up-quark 3 0 2.552 2.550 1.9 – 2.65
down- quark
4 0 5.1050 5.0970 4.5 – 5.15
This formula is a two-parameter fit that determines the mass of particles within each family
(Table 4). The formula does not address the number of possible generations. (Further study of
generation number limitations is required.) It required the cross-family value (1) as the
leading first-generation particle coefficient. The function T(f, g) is a linear fit, within each
family, followed by a fit across the families:
T (f, g) = f [Ca ‧ g + Cb] + [Da ‧ g + Db] (3)
Initial results present fits for the leptons and the quarks separately.
Table 4: The mass hierarchy fit parameters for generations 2 and 3 of the FRACEP
composite fermions, including: the construction mass; the family, generation, lepton
number; the T value; and the fit parameters for both l = 1, and l = 0. The fit parameters
are based on the FRACEP total mass (core mass + excitation mass components). It is
this total mass that compares with the elementary fermions.
FRACEP Mass f, g, l T (f, g) Ca, Cb Da, Db
for l = 1
0.16987
3.74
1, 2, 1
1, 3, 1
11.79906
14.89087 Ca = −0.26946,
Cb = −5.92755
Da = 3.36127,
105.66 Db = 11.54299
1776.81
2, 2, 1
2, 3, 1
5.33259
8.15494
for l = 0
1261.41 3, 2, 0 6.203104
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172,572.11 3, 3, 0 11.121686 Ca = −1.127436,
Cb = −1.03580
Da = 8.30089,
93.94 Db = −0.52666
4162.23
4, 2, 0
4, 3, 0
2.9124
6.703578
Table 5 shows the predictions and comparisons with FRACEP and the elementary fermions.
For families 1 and 4, core mass = total mass, but for generations 2 and 3 in families 2 and 3 the
additional excitation mass makes total mass > core mass. The core masses are: m (2, 2, 1) c =
10.87, m (2, 3, 1) c = 229.67, m (3, 2, 0) c = 96.38, and m (3, 3, 0) c = 4159.69. Future efforts
for a simpler, more theoretically-based relation are being considered.
Table 5: The formula prediction (m (f, g)) for the FRACEP composite particles, and the
measured elementary fermion values in MeV/c2. The FRACEP values and predictions
are for total mass (core plus an excitation mass). First-generation values are in Table 3.
Particle f, g, l m (f, g) FRACEP Construction Measured Values
for l = 1
−
−
1, 2, 1
1,3, 1
0.16987
3.74
0.16987
3.74
<0.17
<18.2
−
−
2, 2, 1
2,3, 1
105.66
1776.81
105.66
1776.81
105.66
1776.74 – 1776.98
for l = 0
c+
t+
3, 2, 0
3, 3, 0
1261.41
172572.11
1261.41
172572.11
1205 – 1290
172460 – 173060
s- b- 4, 2, 0
4, 3, 0
93.94
4162.23
93.94
4162.23
88 – 104
4160 – 4210
DISCUSSION
In quantum mechanics, the elementary particles are treated as quantum objects with wave- particle duality. Their inherent properties are assigned to be consistent with empirical data,
but Osmera [21] noted they are not generally related to any physical structure or internal
motions of the particles. Kempkes [35] showed that fractal behavior was demonstrated in
electrons residing on a Cu (111), but it was not presented as an alternative to the traditional
non-composite nature of electrons.
As a dual universe construction, the FRACEP model explores the nature of the elementary
fermions assuming internal preon-like, fractal-based, positive and negative mass components.
Its construction is empirically-derived with the possibility for future development of a formal
theoretical framework. Golmankhaneh [36] has investigated fractal calculus as it relates to the
Schrödinger equation offering one possible path for further FRACEP development.
The proposed composite construction of FRACEP offers a possible way to relate structure
(and internal motion) to the physical properties of the elementary fermions because of the
carrier CC’s dynamic behavior. The question of the mechanism by which the spin-carrier
provides the spin effect in under consideration. It is well known that there is no sensible way
to explain the intrinsic spin effect in the presumed point-like fundamental particle in quantum
mechanics, but the issue of the origin of spin is the subject of research. Pons and his
colleagues [37] propose a physical explanation for spin using non-local hidden-variable
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Applied Sciences, Vol - 12(4). 293-303.
URL: http://dx.doi.org/10.14738/aivp.124.17404
(NLHV) theory by relating a g-factor for the spin phenomena to physical sub-structures in the
particle (due to the hidden variables) that is otherwise not allowed in point-like particles.
Butto [38] proposes describing the electron as a superfluid frictionless vortex that has all of
the observed properties of the particle; and, Sebens [39-40] describes the electron as an
excitation in the Dirac field that experiences a real rotation of energy and charge, postulating
that to be the origin of spin.
Another possibility for the origin of spin is found in Einstein-Cartan Theory that describes
intrinsic angular momentum with torsion. Petti [41-42] indicates that the stress-energy
tensor includes an asymmetric contribution that is related to a spin tensor. The related
torsion field vanishes outside the rotating matter where the spin-spin self-interaction inside
the matter appears in the Dirac fields as simple rotating masses. This theory could provide an
explanation of the spin effect in the FRACEP spin-carriers where the collection of small
rotating objects within the spin-carrier could appear as a continuous spinning unit outside the
matter group.
In addition, FRACEP provides: 1) an empirically-derived framework for the mass hierarchy
within the families and across the families; 2) an intuitive mechanism for the observed
particle instability and decay (only the non-mixed-mass particles are long-term stable); and
3) a possible explanation for the puzzling dark matter and energy that dominate our galaxy.
The evidence for this picture is intriguing, but not conclusive, and further study is needed. A
number of unanswered questions have been swept under the rug by the simplicity of this
heuristic model (structurally detailed as it may be). We have not focused on a theoretical
development, or addressed the issues of the inherent mass in FRACEP’s fundamental
particles, or the bonding mechanism that maintains the substructures beyond the obvious
attraction provided by the Unified Potential [17]. Despite this, there is the suggestion of a
composite nature for the elementary fermions – though no proof of the internal structure is
available at this time, and unambiguous evidence in this area are needed.
References
[1]. L.A. D'Souza, C.S. Kalman, Preons: Models of Leptons, Quarks and Gauge Bosons as Composite Objects,
World Scientific Pub. Co., Pte. Ldt., Singapore, 1992.
[2]. H. Harari, A Schematic Model of Quarks and Leptons, Phys. Lett. 86B (1979) 83-86.
[3]. M.A. Shupe, A Composite Model of Leptons and Quarks, Phys. Lett. 86B (1979) 87-92.
[4]. G. ’t Hooft, Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking, Proc. 1979 Cargese
Institute on Recent Developments in Gauge Theories, Plenum Press, New York, 1980, pp. 135-137.
[5]. H. Terazawa, Subquark Model of Leptons and Quarks. Phys. Rev. 22D (1980) 184-199.
[6]. H. Terazawa, M. Yasue, K. Akama and M. Hatashi, Observable Effects of the Possible Sub-Structure of
Leptons and Quarks. Phys. Lett. 112B (1982) 387-392.
[7]. H. Terazawa, HIGGS Boson Mass in the Minimal Unified Subquark Model, Uzhhorod University Scientific
Herald. Series Physics 32 (2012) 56-58.
Page 10 of 11
Services for Science and Education – United Kingdom 302
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
[8]. H. Terazawa and M. Yasue, Composite Higgs Boson in the Unified Subquark Model of All Fundamental
Particles and Forces, J. Mod. Phys. 5 (2014) 205-208.
[9]. A.E. Nelson, M. Park, D. G. E. Walker, Composite Higgs Models with a Hidden Sector, 21 Oct. 2018,
arXiv:1809.09667v2.
[10]. A. Nelson, Composite Higgs, Quarks and Leptons, A Contemporary View, Symposium on Fundamental
Physics in Memory of Sidney Drell, 12 Jan 2018, arXiv:0312287,0504252.
[11]. H. Mansour, On the Preon Model, Open Journal of Microphysics, 9 (2019) 11-14.
https://doi.org/10.4236/ojm.2019.92002.
[12]. J.J. Bevelacqua, A First-Order Mass Formula for Quarks in Terms of Constituent Preons, J. of Nuclear and
Particle Physics, 9(1) (2019) 1-4, DOI: 10.5923/j.jnpp.20190901.01.
[13]. A. Salam, Gauge Unification of Fundamental Forces, Rev. Mod. Phys. 52 (1980) 525-538.
[14]. V. Burdyuzha, Fractal Universe, Preon Structure of Particles, and Familon Model of Dark Matter,
Astronomy Reports, June (2014), DOI: 10.1134/S106377291406002X.
[15]. B. Tatischeff, what do fractals learn us concerning the masses of fundamental particles, of hadrons, and of
nuclei? Concerning also disintegration life-times? XXI International Baldin Seminar on High Energy
Physics Problems, September 10-15, 2012, JINR, Dubna, Russia, arXiv:1303.5230v1 [nucl-ex] 21 Mar 2013.
[16]. Yi-Fang Chang, Higgs Mechanism, Mass Formulas of Hadrons, Dark Matter and Fractal Model of Particles,
Int. J. Mod. Theoretical Phys. 3 (2014) 1-18.
[17]. J. Giannini, Feasibility of Constructing a Unified Positive and Negative Mass Potential, Int. J. Mod.
Theoretical Phys. 8(1) (2019) 1-16.
[18]. M. Jammer, Concepts of Mass in Classical and Modern Physics, Harvard University Press, Cambridge MA
1961.
[19]. L.I. Schiff, Gravitational Properties of Anti-matter, Proc. National Academy of Science, Washington, DC 45
(1959) 69-80.
[20]. S. Weinberg, Space Inversion and Time-Reversal. In the Quantum Theory of Fields, vol. 1; Cambridge Univ.
Press: Cambridge, UK, 2005.
[21]. P. Osmera, Fractal Dimension of Electron, Proceedings of MENDEL2012, Brno, Czech Republic, 2012
(ResearchGate # 259921014).
[22]. Yu-Hsin Chen and Sheng D. Chao, Negative Mass Can Be Positively Useful in Quantum Mechanics, J. Chin.
Chem. Soc., 65 (2018) 664-666.
[23]. A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, Princeton NJ 2003.
[24]. H. Bondi, Negative Mass in General Relativity. Rev. Mod. Phys., 29 (1957) 423-428.
[25]. W.B. Bonner, Negative Mass in General Relativity. Gen. Rel. and Grav., 21 (1989) 1143-1157.
[26]. F. Hoyle, G. Burbidge, and J.V. Narlikai, A Different Approach to Cosmology; Cambridge U. Press:
Cambridge, UK, 2000.
Page 11 of 11
303
Giannini, J. (2024). Fractal Representation of Composite Elementary Fermions with Positive and Negative Mass Components. European Journal of
Applied Sciences, Vol - 12(4). 293-303.
URL: http://dx.doi.org/10.14738/aivp.124.17404
[27]. Yi-Fang Chang, Field Equations of Repulsion Force between Positive-Negative Matter, Inflation Cosmos
and Many Worlds. Int. J. Mod. Theoretical Phys., 2 (2013) 100-117.
[28]. Yi-Fang Chang. Final Simplest Model of Smallest Particles and Possibly Developed Directions of Particle
Physics. Phys Sci & Biophys J 2021, 5(2) 000196.
[29]. J.P. Petit and G.D. Agostini, Cosmological Bimetric Model with Interacting Positive and Negative Masses
and Two Different Speeds of Light, in Agreement with the Observed Acceleration of the Universe. Mod.
Phys. Lett. A, 29(2014) 1450182 (15 pages).
[30]. N. Debergh, J.-P. Petit and G. D'Agostini, On evidence for negative energies and masses in the Dirac
equation through a unitary time-reversal operator, Journal of Physics Communications, November 2018,
https://www.researchgate.net/publication/328705449, (14 pages).
[31]. J. Giannini, Fractal Composite Quarks and Leptons with Positive and Negative Mass Components, Int. J.
Mod. Theoretical Phys, 8(1) (2019) 41-63.
[32]. J. Giannini, Mass and Size Characterization Of FRACEP Composite Elementary Fermions, European Journal
of Applied Sciences 11(5) (2023) 212-221, DOI:10.14738/aivp.115.15597.
[33]. K. Falconer, Fractal Geometry, third ed., John Wiley & Sons Ltd, West Sussex, UK, 2014.
[34]. J.-F. Gouyet, Physics and Fractal Structures, first ed., Eng. Trans, Springer-Verlag, NY, 1996.
[35]. S.N. Kempkes et al., Design and Characterization of Electrons in a Fractal Geometry, Nature Physics 15
(2019) 127-131.
[36]. A.K. Golmankhaneh1, S. Pellis, and M. Zingales, Fractal Schrödinger equation: implications for fractal sets,
J. Phys. A: Math. Theor. 57 (2024) 185201 (19pp).
[37]. D.J. Pons, A.D. Pons, and A.J. Pons, A Physical Explanation for Particle Spin, Journal of Modern Physics. 10
(2019) 835-860. https://doi.org/10.4236/jmp.2019.107056.
[38]. N. Butto, A New Theory for the Essence and Origin of Electron Spin. Journal of High Energy Physics,
Gravitation and Cosmology, 7 (2021) 1459-1471. https://doi.org/10.4236/jhepgc.2021.74088.
[39]. C.T. Sebens, The Fundamentality of Fields. Synthese 200 (2022) 380. https://doi.org/10.1007/s11229-
022-03844-2.
[40]. C.T. Sebens, Particles, fields, and the measurement of electron spin. Synthese 198 (2021) 11943–11975.
https://doi.org/10.1007/s11229-020-02843-5.
[41]. R.J. Petti, On the Local Geometry of Rotating Matter, General Relativity and Gravitation, 18(5) (1986)
[42]. R.J. Petti, Do Some Virtual Bound States Carry Torsion Trace? International Journal of Geometric Methods in
Modern Physics, 19(5) (2022). DOI: 10.1142/S0219887822500761 http://arxiv.org/abs/2202.12734.