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European Journal of Applied Sciences – Vol. 11, No. 5
Publication Date: October 25, 2023
DOI:10.14738/aivp.115.15597
Giannini, J. (2023). Mass and Size Characterization of FRACEP Composite Elementary Fermions. European Journal of Applied
Sciences, Vol - 11(5). 212-221.
Services for Science and Education – United Kingdom
Mass and Size Characterization of FRACEP Composite Elementary
Fermions
Judith Giannini
Independent Researcher/USA
ABSTRACT
The FRACEP model characterizes the entire set of Standard Model elementary
fermions as composite particles having a preon-like fractal-based structure. The
internal components have several levels of complexity, beginning with the fractal
structures of the momentum carriers, and the dynamic charge carriers and spin
carriers. These base structures combine in a series of nested 3-element preon-like
components that build up the composite fermions (the subject of this paper) and
composite bosons. The entire structure is based on only two fundamental particles
forming the basis of a Dual Universe containing positive mass particles, negative
mass particles, and mixed mass particles. This system offers an intuitive
explanation for the observed decay of the particles that are otherwise assumed to
be fundamental because only positive mass particles or negative mass particles are
long-term stable. Repulsion within a mixed mass particle is only quasi-stable for
very short times. The estimate masses for the composite fermions are consistent
with the Particle Data Group’s 2016/2020-update estimates, and the estimated
radii are consistent with published expected upper size limits for the Standard
Model’s fundamental elementary fermions.
Keywords: Composite Elementary Particles, Standard Model Elementary Particles,
Fractals, Particle Radii Estimates, Negative Mass, Preons.
INTRODUCTION
The spontaneous decay of most of the Standard Model Elementary Particles (SMEP) suggests
the possibility of a composite nature in particles generally assumed to be fundamental. D'Souza
and Kalman [1] provide a good summary of the concerns of the completeness of the Standard
Model, and the issues leading to the interest in compositeness in its particles. FRACEP [2] was
developed as an arithmetically-based model that presents a composite structure with fractal- based components for the SMEP, providing an intuitive explanation for why the SMEP (assumed
all positive mass) decay.
The Standard Model is highly successful at predicting the behavior of the world of the small. It
has a quantum mechanical-based theoretical part treating the particles as fundamental and
describing particle interaction; and, an empirical part characterizing the SMEP set containing
24 fermion-class (spin-1/2) particles, and 26 boson-class (integer spin) particles. The
designations fermion and boson indicate both the particles and their corresponding anti- particles.
The fermions include six leptons divided equally among two families, plus an equal number of
anti-particles, and, six quarks divided equally among two families, plus their anti-particles.
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212-221.
URL: http://dx.doi.org/10.14738/aivp.115.15597
They have mass (acquired through the Higgs mechanism) and inherent characteristics of spin,
most with electric charge, and quarks with color charges.
The bosons are the field-exchange particles: 1) the photon for the electromagnetic field, the W+,
W-, and Z0 for the weak field, the eight gluons for the strong field, and the higgs that couples to
the Higgs field providing all the particles with mass. The bosons have Higgs provided mass and
the inherent characteristics of spin, some with electric charge, and some with color charge.
The FRACEP model characterizes the entire set of SMEP as composite particles built up from
top-level internal components that are preon-like repeated groupings of three fractal carrier
types: momentum-carriers, charge-carriers, and spin-carriers. The fractal structures are built
up from the two fundamental particles. Color charge is a component composed of composite
neutrinos and anti-neutrinos. Because the components are fractal structures, we consider the
composite versions of the SEMPs to be fractal-like structures as well. This paper focuses on the
preon-like fractal-based structure of the composite fermions. The discussion of the bosons is
reserved for another work.
THE FUNDAMENTAL PARTICLES AND SCALES
There are two fundamental particles that are used to build up all of the internal components of
the FRACEP composite particles [2], The Gp, and Gn have zero electric charge, zero spin, and
mass (m(Gp) = +1.724934x10-22 MeV/c2, and m(Gn) = -m(Gp)). The mass is inherent, and the
Higgs mechanism is not required for them because of their total symmetry at creation (in pairs).
This results in a Dual Universe concept – a Bright Universe (mostly positive mass), and a Dark
Universe (mostly negative mass), allowing a possible answer to the Dark Matter/Energy
problem in cosmology. FRACEP hypothesizes that the classical radius of Gp is the scale- invariant Planck length derived by Hoyle, r(Gp) = 3.307519x10 [3]. This reflects a symmetry in
nature. where the smallest mass particle (FRACEP’s Gp) is assumed to be the reciprocal of the
largest mass particle (the Planck particle, mass mp*= 5.97x1021 MeV/c2). FRACEP’s potential
computations [4] indicate the closest stable approach of any two Gp‘s is ~r(Gp). Also, the Gp
and Gn cannot approach closer than this because their repulsion only allows the configuration
to be quasi-stable in the oscillating region of the potential.
THE CARRIER COMPONENTS
The Momentum Carriers
The momentum carriers are composed only of the fundamental particles. They have zero spin
and zero charge, and are the basis of the structure of all of the composite particles. The basic
general particle is spherical, composed of a 6-element ring plus three additional elements at
any fractal level such that the mass is m(MGXp) = m(Gp) • 9
X within a radius r(MGXp) = 4X
r(Gp). A similar structure based on Gn also exist.
The ring structure is a regular hexagonal composed of six particles at any fractal level such that
the mass is m(MRXp) = 6 • [m(Gp) • 9
X]. A similar structure based on Gn also exist. The ring is
confined to a two-dimensional plane. At any fractal level, its radius is r(MRXp) = 4X+1 r(Gp),
and its thickness equals the diameter of its particles, h(MRXp) = 2X+1 r(Gp).
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Fig. 1: The general particle and ring structure at the zero and first fractal levels.
All of the separation distances between pairs of elements along the ring are exactly the same
length. To maintain equal separation between particle pairs in the general particle, the three
additional elements must be out of the plane of the ring. Two of the particles are above the ring
plane and one is below [5].
The Charge-Carriers
The charge-carriers are fractal configurations composed only of the fundamental particles.
They have mass and charge, but zero spin, and are components in all of the composite particles
with the exception of the neutrino family particles, their anti-particles, and their dark
counterparts because those particles have zero charge.
The charge-carrier structure is composed of two charge-carrier-specific parts: 1) a charge- momentum part (MQp with only positive mass, or MQn with only negative mass), and 2) a
charge-effect part (QEp with only positive mass and a negative charge, or QEn with only
negative mass and a positive charge). MQp and MQn carry the bulk of the mass but no charge
or spin. The mass is m(MQp) = m(MG19p) + 2 m(MG13p) + 48 m(MR13p) + 121 m(MR16p). A
similar structure exists for MQn using Gn with m(MQn) = –m(MQp). QEp and QEn are dynamic
structures producing the observed charge, but have only a small amount of mass and no spin.
The charge effect is captured in a pair of linear chains where the mass of the pair is m(QEp) = 2
m (Gp) • 4
19, or m(QEn) = 2 m (Gn) • 4
19
.
The charge-carrier is assumed to be roughly spherical, with the components within MQp
organizing into a spherical configuration, while the QEp remains a chain. The radius of the
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Giannini, J. (2023). Mass and Size Characterization of FRACEP Composite Elementary Fermions. European Journal of Applied Sciences, Vol - 11(5).
212-221.
URL: http://dx.doi.org/10.14738/aivp.115.15597
charge carrier is r(QBp) ≈ 6.96x10-23 m [5]. There are four possible charge-carrier
combinations (B is for Bright Universe and D is for Dark Universe):
1. QBp = MQp + QEp with only positive mass and negative charge,
2. QBn = MQp + QEn with mixed mass and positive charge,
3. QDp = MQn + QEp with mixed mass and negative charge,
4. QDn = MQn + QEn with negative mass and positive charge.
The Spin Carriers
The spin-carriers are fractal configurations composed only of the fundamental particles. They
have mass and spin, but zero charge. All of the fermions have spin-carriers.
The spin-carrier structure is composed of two spin-carrier-specific parts: 1) a spin-momentum
part (MSp with only positive mass, or MSn with only negative mass), and 2) a spin-effect part
(SEp with only positive mass and a positive spin, or SEn with only negative mass and a negative
spin). MSp and MSn carry the bulk of the mass but no spin or charge. The mass is m(MSp) = 4
m(MG16p). A similar structure exists for MSn using Gn with m(MSn) = –m(MSp).
SEp and SEn are dynamic structures producing the observed spin, but have only a small amount
of mass and no charge. The spin effect is captured in a 5-element group where each element, in
the lowest fractal order, is a pair of Gp or Gn particles. The spin effect has 16 levels of the
recursive structures where the mass is m(SEp) = 2 m(Gp) • 5
16, and m(SEn) = 2 m (Gn) • 5
16
.
Each of the four momentum components (MG16p) are spherical, while the spin effect
component (SEp) is approximately a thin plate. This gives the 5-element structure a roughly
doughnut shape where the central hole has a radius r(SEp) ≈ 1.42x10-25 m, and each of the
encircling momentum components has a radius r(MSp) ≈ 1.42x10-25 m. For the sake of
composite particle size determination, the structure can be viewed as roughly spherical with
a radius of r(SBp) ≈ 5.68x10-25 m [5]. There are four possible spin-carrier combinations (B
is for Bright Universe and D is for Dark Universe):
1. SBp = MSp + SEp with only positive mass and positive spin,
2. SBn = MSp + SEn with mixed mass and negative spin,
3. SDp = MSn + SEp with mixed mass and positive spin,
4. SDn = MSn + SEn with only negative mass and negative spin.
Fig. 2: The spin-carrier for SBp. The other three cases have similar configurations.
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THE FERMION COMPOSITE STRUCTURE
The composite fermions have fractal-based, preon-like components. With the exception of the
electron and the electron-neutrino and their dark counterparts, all of the composite fermions
have mixed mass causing them to be unstable, as is observed in the SMEP. In FRACEP, the Bright
Universe (BU) contains the SMEP we recognize. It has four families with three generations each
– the particles in each generation being more massive than those in the previous ones. The
composite fermions retain this hierarchical structure.
The BU 1st generation particles include: ne-, e-, u+, d-, and their anti-particles. A similar set of
Dark Universe (DU) particles exist (equal and opposite to the BU in every property and
characteristic). The BU 2nd generation particles include: nm-, m-, c+, s-, and their anti-particles.
A similar set of Dark Universe (DU) particles exist (equal and opposite to the BU in every
property and characteristic). The BU 3rd generation particles include: nt-, t-, t+, b-, and their
anti-particles. A similar set of Dark Universe (DU) particles exist (equal and opposite to the BU
in every property and characteristic).
The Neutrino Family (q = 0e, s = +1/2)
The ne-is the spin carrier without any additional momentum-carriers. It is the simplest particle,
containing no charge- or extra momentum-carriers – only spin. It has only positive mass. Along
with its DU counter-part (which contains only negative mass), these are the only stable
neutrino-family particles because they are not mixed mass. Each of the higher generations is
the previous generation neutrino plus extra components (radicals) that act as single particle- like structures:
• Gen 1. ne- = SBp
• Gen 2. nm- = ve- + Rq0
• Gen 3. nt- = nm- + MG22p + R2v(m).
The radicals Rq0 = MG22p + (QBp + QDn), and R2n(m) = nm- + 3MR22p + nm+ are mixed-mass
components (negative mass in red). The result is that the generations 2 and 3 are not long-term
stable. This is true of the BU and the DU higher generation neutrinos.
The Electron Family (q = -1e, s = +1/2)
The e- is simplest member of this family. It contains three groups with a spin-carrier, a charge- carrier, and a momentum-carrier in each group (Rsqp = ne- + QBp + MG22p). The e- has only
positive mass. Along with its DU counter-part (which contains only negative mass), these are
the only stable electron-family particles because they are not mixed mass.
The higher generation particles have a more complicated growth pattern than the neutrino
family, and contain more complicated radicals (that act as single particle-like structures).
Unlike the neutrino family and the electron, itself, the m- and t- (and their anti-particles and
dark counterparts) have a defined core with additional {bulk momentum groups} that are not
part of the particle core. It is suggested that this {bulk momentum} may act something like an
excitation mass for the particles. The SM recognizes the core plus the {bulk momentum} as the
particle in a fundamental state.
• Gen 1. e- = 3 Rsqp
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Giannini, J. (2023). Mass and Size Characterization of FRACEP Composite Elementary Fermions. European Journal of Applied Sciences, Vol - 11(5).
212-221.
URL: http://dx.doi.org/10.14738/aivp.115.15597
• Gen 2. m- = [ core = ⸨Rd- + 6 MR22p + R2n(m) ⸩ + nm- + 2 MG22n]
+ {bulk momentum = 93 MR22p}
• Gen 3. t- = [ core = ⸨Rd- + 6 MR22p + R2n(t) ⸩
+ ⸨ R+(nm, u) + 6 MR22p + R-(nm, u) ⸩ + MR24p
+ ⸨3 MG22p + nt- + 3 MR22p + 93 MR22p ⸩]
+ {18 MR24p + 60 MR22p}
The five radicals used include: Rsqp, Rd- = ne+ + MR22p + e-, R2n(m), R2n(t) = nt- + 6 MR22p
+ nt-, and R-(nm, u) = nm- + 6 MR22p + u-.
R+ (nm, u) is the anti-radical to R- (nm, u). The Rd- radical is used throughout all of the
remaining fermions. The mixed-mass radicals and particles (nm-, nt-, and MG22n) in
generations 2 and 3 means that they are not long-term stable. This is true of their anti-particles
and their DU counterparts.
The Up-Quark Family (q = +2/3e, s = +1/2)
The quarks are the next level of complication. The u+ is the simplest member of the up-quark
family. Its Rsqn is the anti-radical of the one used in the electron, giving the up-quarks their
positive electric charge, +2/3e, rather than the -1e of the electron family.
The u+ includes a color-charge component, giveing the other quarks their color charge. Like the
electron family, the higher generations of the family have a defined core with additional {bulk
momentum groups} that are not part of the particle core.
• Gen 1. u+ = 2 Rsqn + 2 MR22p + nm- + CQ(i)
• Gen 2. c+ = [ core = ⸨ R+(nm, u) + 6 MR22p + R-(nm, u) ⸩ + 48 MR22p
+ ⸨nt- + 6 MR22p + R+(nm, u) ) ⸩ ] + {14 MR24p + 14 MR22p }
• Gen 3. t+ = [ core = ⸨ R+(nt, c) + 6 MR22p + R-(nt, c) ⸩ + 4 MR24p
+ ⸨ nt- + 6 MR22p + R+(nt, c) ⸩ ] + {25 MR26p + 15 MR24p }
The five radicals used include: Rsqn = ne+ + QBn + MG22n, R-(nm, u) and its anti-radical, and
R+(nt, c) = nt+ + 6 MR22p + c+, and its anti-radical. The mixed-mass radicals and particles
(nm- and nt-) in all generations of the up-quark family means that they are not long-term stable.
This is true of their anti-particles and their DU counterparts.
The Down-Quark Family (q = -1/3e, s = +1/2)
The down-quark family maintains a symmetry throughout the growth in the three
generations, adding the Rd- radical and an MR22p connecting group (without any other
additional components or {bulk momentum groups}) to the up-quark member at each
generation:
• Gen 1. d- = u+ + MR22p + Rd-
• Gen 2. s- = c+(core) + MR22p + Rd-
• Gen 3. b- = t+(core) + MR22p + Rd- The down-quark family members are mixed-mass, and are all not long-term stable. This is
true of their anti-particles and their DU counterparts.
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COMPOSITE FERMION SIZE-ESTIMATES
SMEP Size Studies and Estimates
Within the Standard Model, the SMEP are treated as fundamental point sources with no
internal components. It is generally accepted that their radii are less than 10-
18 m. Theoretical
studies and scattering experiments have explored ways of defining the properties of the
particles to determine the upper limits of their size.
For example, Shulman [6] models the characteristics of the charged leptons, and considers
whether they are consistent with point source particles. He notes that insignificant size
should correspond with a great mass, but the classical radius of the electron (R0 = e2
/ m c2
=
2.8179403267x10-
15 m) is consistent with its measured mass (0.5110 MeV/c2). Further, he
notes that the presence of the spin and magnetic moment in the electron are evidence of a
rotating mass – again inconsistent with a point source.
Ghosh [7] considers the dynamic and static properties of the electron, discussing various
approaches to defining its structure. He concluded that charge radius (quantized as RE < a
2
R0) had yet to be calculated precisely, but was expected to be on the order of 10-
19 m or less.
While Hirsch [8] indicated that experimental evidence for neutrino oscillations implies that
they are the first elementary particles whose properties cannot be fully described within the
Standard Model, and concluded the limits on the neutrinos’ radii were consistent with r(nm)
< 8.25x10-
19 m and r(nt) < 3.15x10-
18 m when combining astrophysical observations with
ground-based experiments.
Storti [9] provides the most comprehensive compilation of radii results for all of the leptons
and quarks, deriving the mass-energy properties using Electro-Gravi-Magnetics, and
determining the associated RMS charge radii (on the order of 10-
18 m for all of the particles).
Their radii results indicate agreement with scattering experiments, and mass predictions
were consistent with the Particle Data Group expected values for 2004 [10].
Zarnecki [11] presented more recent results for the quark radius based on high–precision
HERA data of electron-quark scattering, indicating an upper limit on the effective quark
radius of 0.43x10-18 m at the 95% C.L. This new value (2016 estimates) is approximately half
of the previous values presented by Sorti (2011 estimates).
FRACEP Composite Particle Size Estimates
The FRACEP version of the SEMP provide a different way of looking at the structure of the
elementary fermions. The composite nature of the particles provides a way of determining
size based on the closeness and arrangement of the components within the structure [5]. The
radii of the particles are classical, in the sense of defining the physical extent of the composite
structure. The masses, for the determined structures, agree with the 2020 estimates of the
Particle Data Group [12].
The particles in the table are arranged in order of increasing mass – with the core mass
determining the order for the m-, c+, t- and t+. The physical radii for the composite particles
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Giannini, J. (2023). Mass and Size Characterization of FRACEP Composite Elementary Fermions. European Journal of Applied Sciences, Vol - 11(5).
212-221.
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increase with mass (for all of the particles), while remaining less than the generally accepted
value of <10-18 m. This is true for the non-mass-excited particle-cores, as well as, for the
excited particles (with the exception of the t+ which is the largest mass configuration). The
values are less than the Sorti modeled charge radii which are approximately uniform for all
particles. Finally, the composite quark radii are consistent with Zarnecki’s upper limit for the
quarks [11].
CONCLUSIONS
The fractal structure of the composite elementary particles has updated masses that are
consistent with the Particle Data Group’s 2016/2020-update estimates. The sizes of the
composite structures, assuming the groupings of the repeated components in the larger
Table 2. The updated masses and radii of the carrier components and the composite
elementary fermions. The particles m-, c+, t-, and t+ have a compact core plus additional
momentum-carriers that are not part of the core. The additional mass possibly indicates a core
in something like an excited state. The FRACEP size shows both total and core estimates. The
Gp, the momentum-, the spin-, and the charge-carriers have no analog in the SMEP. The SMEP
values are the 2020 estimates of the Particle Data Group [12].
FRACEP Particles Mass
(MeV/c2)
SMEP
(MeV/c2)
FRACEP
Classical Radius (m)
Gp 1.724934x10-22 r0 = 3.30752x10-35
Momentum (MGXp) 9
X • m(Gp ) 0.866 N1/3 3 (4X) r0
N = carriers in clump
ne-
(Spin, SBp)
1.28x10-6 <15x10-6 5.68x10-25
Charge (QBp) 4.65x10-4 6.96x10-23
nm- 0.17 <0.17 8.74x10-22
e- 0.51 0.510999
+ 4x10-8
2.25x10-21
up (u+) 2.55 1.9 – 2.65 1.6x10-20
nt- 3.74 <18.2 1.6210-20
down (d-) 5.1 4.5 – 5.15 2.98x10-20
muon (m-)
core = 10.873
{total = 105.66}
105.65835
+ 4.9x10-8
core = 3.04x10-20
{ 5.35x10-20 }
charm (c+)
core = 91.38
{total = 1261.41}
1250 – 1290 core = 6.6x10-20
{ 3.97x10-19 }
strange (s-) 93.94 88 – 104 1.05x10-19
tau (t-)
core = 229.67
{total = 1776.81}
1776.74
– 1776.98
core = 9.64x10-20
{ 4.37x10-19 }
top (t+)
core = 4159.69 172460
– 173060
core = 8.15x10-19
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{total =
172572.11}
{ 7.31x10-18 }
bottom (b-) 4162.23 4160 – 4210 1.23x10-18
more complex particles, agree with the upper limits of size based on experimental and
modeling efforts.
One notable difference is the composite particle mass hierarchy is consistently generation 1
< generation 2 < generation 3 for all families. For the SMEP, this pattern is not followed by the
up-quark family, where generation 1 < generation 3 < generation 2. It appears that the
generations 2 and 3 up-quark particles are the FRACEP mass-excited versions (total mass =
core particle mass plus the {bulk momentum} mass). Despite this, the agreement offers
support for a composite nature for the elementary fermions – though not proof of the internal
structure is available at this time. Experiments providing unambiguous evidence of the
internal structure are needed.
The FRACEP composite structure offers a possible mechanism for the observed decay of the
(assumed fundamental) SMEP. There is no intuitive explanation for why all positive mass
particles should decay. Because the FRACEP particles are mixed mass, spontaneous decay
should be expected. Investigation of the expected scattering behavior is required. As the only
all positive particles, the electron and electron neutrino should be expected to scatter as
solitons, while the rest of the particles (all mixed-mass) are expected to scatter as coherent
groups that break up into smaller groups.
Finally, the Dual Universe resulting from the positive and negative mass FRACEP fundamental
particles could provide an answer to nature of the Dark Matter problem in cosmology. At
present, it is understood that 75% of the matter in the universe is missing, and candidates for
the nature of this unseen matter are still unverified.
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