Proton Properties from Nested Surface Vortices

Abstract

A nested surface vortex structure may be used to explain several properties of free or chemically bound protons. The circular Unruh and zitterbewegung effects are combined to show that it is plausible for the mass of an unobserved ground-state proton to exist on a spherical surface. Such a model is consistent with general relativity. The charge of an unobserved ground-state proton is assumed to exist on two massless oppositely charged shells well outside that of its mass sphere. These two charge shells are assumed to exist on the two surfaces of a spindle torus. This spindle torus structure offers geometric explanations for proton isospin, g-factor, and charge radius. This geometric model involves mathematics typically encountered by undergraduate physics and chemistry students. Upon interaction with other particles, this ground-state proton model transforms into the valence quarks, gluon flux tubes, and initial sea quarks of the standard quantum chromodynamics model.

Full text

Chapter
Proton Properties from Nested
Surface Vortices
Steven Verrall, Kelly S. Verrall, Andrew Kaminsky,
Isaac Ozolins, Emily Friederick, Andrew Otto, Ivan Ngian,
Reagen McCormick and Pearl Scallon
Abstract
A nested surface vortex structure may be used to explain several properties of free
or chemically bound protons. The circular Unruh and zitterbewegung effects are
combined to show that it is plausible for the mass of an unobserved ground-state
proton to exist on a spherical surface. Such a model is consistent with general relativ-
ity. The charge of an unobserved ground-state proton is assumed to exist on two
massless oppositely charged shells well outside that of its mass sphere. These two
charge shells are assumed to exist on the two surfaces of a spindle torus. This spindle
torus structure offers geometric explanations for proton isospin, g-factor, and charge
radius. This geometric model involves mathematics typically encountered by under-
graduate physics and chemistry students. Upon interaction with other particles, this
ground-state proton model transforms into the valence quarks, gluon flux tubes, and
initial sea quarks of the standard quantum chromodynamics model.
Keywords: quantum vortex, zitterbewegung fermion, circular Unruh effect, general
relativity, intrinsic charm quarks, proton g-factor, proton charge radius
1. Introduction
Primordial nucleosynthesis formed the first atomic nuclei. This process ended
about 20 minutes after the Big Bang. The first stars and galaxies formed hundreds of
millions of years later. During the time between these processes, about 75% of the
mass of elemental matter was in the form of neutronless hydrogen-1 in its ionized,
atomic, and molecular forms. Ionized hydrogen-1 is a bare proton. The nucleus of
atomic hydrogen-1 is also a bare proton. Molecular hydrogen-1 consists of two protons
chemically bound by an electron cloud.
A completely accurate proton model has remained elusive since at least 1917 when
Ernest Rutherford first provided experimental evidence that all atoms contain pro-
tons. If the proton’s internal structure is not fully understood, humanity may lack
fundamental insights into the nature of matter. On the energy scale where matter
forms the building blocks of earthly life, protons appear to precisely maintain several
1

important parameters. These include net charge, rms charge radius, mass, magnetic
moment, spin, isospin, and parity.
In biological systems, almost all hydrogen is in the form of either chemically bound
protons or ionized protons. Magnetic resonance imaging (MRI) is a key diagnostic tool
in modern medicine because proton magnetic moment is unaffected by chemical
binding. Medical MRI relies on each proton’s magnetic moment precisely resonating
with radio-frequency waves to emit coherent radiation with compact direction, fre-
quency, and phase.
Quantum field theory (QFT) is the foundation of the Standard Model of particle
physics [1]. The Standard Model is not completely explained because several parame-
ters must be experimentally determined. QFT applies operators to create and annihi-
late particles [2, 3]. This circumvents potential physical mechanisms that create and
annihilate mass and charge.
This chapter summarizes a proposed mechanism where quantum networks of
interfering virtual vacuum momenta continually regenerate the mass and charge of
each free or chemically bound ground-state proton. This is called the ground-state
quantum vortex (GSQV) proton model [4, 5]. In this model, it is assumed that one real
spin-1 photon splits into two virtual circularly polarized spin-half photon vortices
during proton-antiproton pair production. An antiproton contains antimatter in the
form of antiquarks. Antimatter is of identical mass and opposite charge to matter.
Matter and antimatter readily annihilate each other.
All elemental matter consists of spin-half particles. In the GSQV proton model, the
mass-energy of a free or chemically bound spin-half proton is proposed to be gener-
ated by the toroidal revolution of a virtual photon [4]. The initial confinement of
mass-energy may be obtained by combining the zitterbewegung and circular Unruh
effects [4]. The toroidally revolving virtual photon is proposed to be circularly polar-
ized. This results in a virtual poloidal vortex component that Reference [4] associates
with isospin and assumes generating a charge. Reference [5] proposes a mechanism
where twin virtual poloidal circulations generate and maintain a GSQV proton’s
charge.
The GSQV proton model adds to QFT without replacing any of its long-established
aspects. It supports the fundamental validity of quantum chromodynamics (QCD)
[6, 7]. The GSQV proton model seamlessly merges with chiral effective field theory
(EFT) [8] and lattice QCD [9, 10] at higher energies. A free or chemically bound
proton, in its lowest energy (ground) state, is modeled as a completely coherent self-
synchronizing vortex structure. The quantum vortex structure is formed from toroi-
dally and poloidally circulating virtual fields in the form of standing waves. These
virtual fields arise from the real electromagnetic fields of real photons, which pre-
sumably formed the first protons in the early universe, and standing waves of virtual
quantum vacuum fields. The precise nature of the quantum vacuum remains an
unsolved problem. However, vacuum energy must manifest in a charge-neutral way,
and standing vacuum waves explain the experimentally supported Casimir effect [11].
The GSQV proton model is depicted in Figure 1. Proton mass-energy is concen-
trated in the relatively small blue central sphere, which Reference [4] calls a
zitterbewegung fermion. This finding is summarized in Sections 2 and 4. The sur-
rounding charge structures are massless and are proposed by Reference [5] to be
formed from standing vacuum waves. This finding is summarized in Section 5. At an
unspecified energy, minimally above the ground state, Reference [4] proposes that
the GSQV proton transforms into the quarks and gluons of established QCD theory.
This is depicted in Figure 2 and summarized in Section 3. Therefore, at higher
2
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

energies, there should be no conflict with the established chiral EFT [8] and lattice
QCD [9, 10] models. The GSQV proton model may help resolve recently discovered
discrepancies occurring at the lowest energies [12–15]. Section 6 summarizes the
Reference [5] finding that up, charm, and top quark charge appears to depend only on
Planck charge, the proportionate area of a GSQV proton’s polar charge-exclusion
zone, and π. The charge-exclusion zones are indicated by the red dots in Figure 1.
Section 7 summarizes how Reference [5] applies the GSQV proton model to calculate
properties of intrinsic charm quarks. Section 8 summarizes how Reference [5] models
the proton magnetic moment. Section 9 summarizes how Reference [5] calculates
proton charge radii statistically consistent with the most accurate experimental
estimates [16–18].
When in its ground state, Reference [5] proposes that proton charge and mass are
coupled and continually regenerate each other. For each proton charge arc, this cou-
pling may be represented in the form of a virtual optimal Möbius band [5, 19–22].
Reference [5] proposes that this implies the geometry of a GSQV proton is optimal,
which may explain why free or chemically bound protons do not decay. In Figure 1,λp
is proton Compton wavelength. Each proton charge arc is assumed to be regenerated
by half a poloidal turn, at radius Rp¼λp=ffiffiffi
3
p, each zitterbewegung cycle. Reference [4]
Figure 1.
The GSQV proton model consists of a central zitterbewegung fermion (blue sphere), of radius rpz ¼λp
4π, orbited by
four massless charge arcs. The outer two charge arcs are quantized to the value of the elementary charge, þe. The
inner two charge arcs are quantized to the value of negative half the elementary charge, e=2. Each charge arc
orbits with its equator always moving at light speed, c. The inner and outer charge arcs occasionally align with the
separation 2rpa. The distance between the two equatorial points of the inner charge arcs is always proton Compton
wavelength, λp. Therefore, rpa ¼λp1
ffiffi3
p1
2

, which is slightly less than rpz. The red dots represent uncharged polar
regions where virtual poloidal flow is split into two equal magnitudes.
3
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

associates the half-poloidal turn with quantum mechanical isospin. The ffiffiffi
3
pfactor is the
aspect ratio of the optimal Möbius band and is applied extensively in Sections 5–9.
Since protons are spin-half particles, each zitterbewegung cycle involves two rev-
olutions of the central zitterbewegung fermion [4, 5]. A proton zitterbewegung cycle
may be characterized by Compton wavelength, λp, and Compton frequency,
fp¼c=λp, where cis the speed of light.
2. Proton mass as quantized circular Unruh energy
References [4, 5] propose that the mass-energy of a free or chemically bound low-
energy proton is equivalent to confined quantized circular Unruh energy [23–27]. The
Unruh effect is known to be fundamentally local [28]. This circular Unruh energy is
generated by the light-speed internal circulation of what Reference [4] calls a
zitterbewegung fermion. To external observers, a zitterbewegung fermion consists of
a uniformly distributed ensemble of light-speed circulations of point-like objects on
the surface of a sphere of radius
rpz ¼ℏ
2mpc¼λp
4π, (1)
where ℏis Planck’s reduced constant and mpis proton mass. Note that
Figure 2.
Minimally excited free or chemically bound proton. Each circle represents a quark. Dashed outline represents an
anticolor. The three quarks in the right-hand column transition from the two outer þe charge arcs of the GSQV
proton. These momentarily form a color-neutral quark triplet. The two quarks in the left-hand column transition
from the two inner e=2charge arcs of the GSQV proton. These momentarily form a color-neutral quark doublet.
The three quarks below the dashed line connect via gluon flux tubes to form the proton’s color-neutral valence
quark triplet. The two quarks above the dashed line are unbound intrinsic sea quarks.
4
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

λp¼h
mpcand ℏ¼h
2π, (2)
where his Planck’s constant. In Figure 1, the proton zitterbewegung fermion is
depicted as a blue sphere. Its radius, rpz, is clearly much smaller than the proton
charge radius. Reference [4] proposes that each ensemble member of a
zitterbewegung fermion is entangled with the rest of the ensemble.
In QFT, it is well established that any acceleration causes an increase in vacuum
energy. This increase in vacuum energy can be described as Unruh temperature.
Circular motion is caused by centripetal acceleration, which is associated with circular
Unruh energy, Tcirc [23]. To inertial observers, a free or chemically bound proton’s
zitterbewegung fermion consists of an ensemble of point-like objects with
zitterbewegung acceleration [4],
apz ¼c2
rpz ¼2mpc3
ℏ:(3)
Reference [4] shows that for a zitterbewegung fermion,
Tcirc ¼ℏapz
4ffiffiffi
3
pckB¼mpc2
2ffiffiffi
3
pkB
, (4)
where kBis the Boltzmann constant. This implies that proton mass-energy,
Ep¼mpc2¼2ffiffiffi
3
pkBTcirc:(5)
The median energy of a particle in a thermal bath at temperature, T, is given by
3:50302kBT[29]. This appears to imply that proton mass-energy, Ep, is approximately
2ffiffiffi
3
p=3:50302 ≈98:9% of the median thermal excess vacuum energy due to internal
zitterbewegung acceleration. References [5, 29] propose that the median energy is the
most physically meaningful thermal spectrum peak. Reference [5] proposes that the
approximately 1.1% discrepancy between Epand the median thermal excess vacuum
energy may explain the cosmic origin of quark masses.
The distance between the equatorial points of the two inner e=2 charge arcs,
depicted in Figure 1, is set to the proton Compton wavelength, λp. References [4, 5]
propose that vacuum standing waves exist between the equatorial points of the
inner e=2 charge arcs. Section 5 summarizes this proposed phenomenon. Since these
standing waves precisely match the proton’s Compton wavelength, λp, Reference [4]
proposes that they trigger the quantization of the Unruh energy of the central
zitterbewegung fermion.
3. Proton quark formation
As proposed in Reference [4], a minimally excited GSQV proton may couple with
the Higgs field and transfer mass-energy from the GSQV nucleon’s central
zitterbewegung fermion to its system of revolving formerly massless charge arcs.
References [4, 5] propose that the þ2echarge of the outer GSQV proton charge arcs
evenly divides into three up quarks of charge þ2e=3. As proposed in Reference [5], it
5
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

is likely that the two þecharge arcs merge into a þ2echarge surface [4] before
decomposing into three up quarks of charge þ2e=3.
The structure shown in Figure 1 rotates toroidally about the GSQV proton axis on
surfaces of revolution. Apart from the flat polar caps, the rest of the toroidally rotating
structure exists on the two surfaces of a spindle torus. The inner surface is called a
lemon and the outer surface is called an apple. Further details can be found in
Reference [4].
Quarks have long been known to be spin-half particles. Therefore, according to the
Pauli exclusion principle, these three up quarks cannot be in the same quantum state.
QCD invokes three fundamental color charge types to adhere to the Pauli exclusion
principle. The right-hand column in Figure 2 represents the outer GSQV proton shell
transforming into a blue, green, and red color-neutral up-quark triplet.
Since a QCD proton is color neutral, and the GSQV proton’s outer shell transforms
into a color-neutral quark group, the GSQV proton’s inner shell must also transform
into a color-neutral quark group. At energies slightly above the proton’s ground state,
Reference [5] proposes that the two e=2 inner charge arcs merge into a echarge
shell [4] and divide into a down e=3ðÞand antiup 2e=3ðÞquark pair. To maintain
color neutrality, these two quarks must be a color-anticolor pair. The left-hand col-
umn in Figure 2 represents the inner GSQV proton shell transforming into a blue
down quark and an antiblue (shown as dashed blue) antiup quark. However, any
color-anticolor combination is possible.
It is important to remember that color charge cycles extremely rapidly—to the
degree that the color charge of each valence quark is effectively the superposition of
all three colors and the color charge of each sea quark is the superposition of all three
colors and all three anticolors.
Reference [5] proposes that the curvature of the charge shells prevents the
initial formation of gluon flux tubes between quarks on the same shell. Initially,
two gluon flux tubes are proposed to form between inner and outer valence
quarks. Each of the two outer valence quarks is assumed to connect to the same
inner valence quark via a gluon flux tube. This initial double gluon flux tube
structure is then proposed to immediately transform into the three-way
symmetric gluon flux tube structure described by the Standard Model of particle
physics [30].
4. Ground-state strong force and gravitation equivalence
Reference [4] models the interior of the GSQV proton’s zitterbewegung fermion as
flat Minkowski spacetime. Excess vacuum energy, equaling proton mass-energy, is
assumed contained inside the zitterbewegung fermion’s spherical shell.
In flat Minkowski spacetime, trapped energy will exert pressure, p¼U=3V, where
Uis trapped energy and Vis volume. For the GSQV proton’s spherical zitterbewegung
fermion, U¼mpc2and V¼4
3πr3
pz. Therefore
p¼U
3V¼mpc2
4πr3
pz
:(6)
The total outward force exerted on the zitterbewegung fermion’s surface,
Fout ¼pA, where A¼4πr2
pz. Therefore
6
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

Fout ¼mpc2
rpz ¼mpapz, (7)
where apz is zitterbewegung centripetal acceleration defined by Eq. (3). For the
zitterbewegung fermion to be stable, there must be a balancing total inward force of
value Fin ¼Fout. In the GSQV proton model, Fin plays the role of the strong force
needed to confine proton mass-energy.
One of the foundational assumptions, used to derive the curved spacetime feature
of general relativity, is the equivalence of gravitational and inertial mass. Inertial mass
is itself equivalent to confined energy. It will therefore be assumed that the spacetime
curvature of general relativity causes the total inward force, Fin, needed to stabilize
the zitterbewegung spherical shell. This is equivalent to inward gravitational pressure,
pg¼mpc2
4πr3
pz
:(8)
Reference [4] supposes that the point-like objects described in Section 2 are
actually spheres of radius ffiffiffi
2
plP, where lP≈1:616 1035m is the Planck length. Each
circling sphere will have a cross-sectional area 2πl2
Pand always be located on the
zitterbewegung spherical shell. The inward gravitational force, Fg, due to inward
gravitational pressure, pg, on each circling sphere, will be assumed to be a cross-
sectional area, 2πl2
P, multiplied by inward gravitational pressure:
Fg¼2πl2
Ppg¼mpc2l2
P
2r3
pz ¼GmpmPlP
2r3
pz ¼Gm2
p
r2
pz
, (9)
since c2¼GmP=lPand mPlP=rpz ¼ℏ=crpz ¼2mp, where Gis the universal gravita-
tional constant and mPis the Planck mass. Each circling sphere therefore experiences a
force, due to curved spacetime, as if it were a point-like proton mass, mp, in the
Newtonian gravitational field of another point-like proton mass at the center of the
zitterbewegung fermion.
The same effect would occur if the point-like proton mass, at the center of the
zitterbewegung fermion, was uniformly distributed on a thin spherical shell of radius
≤rpz. Such a structure would exert no interior gravitational field, and therefore no
interior spacetime curvature, but would exert the same external gravitational field as that
of a point-like proton mass. It is therefore possible to model the zitterbewegung fermion’s
interior as flat Minkowski spacetime, which was assumed when deriving Eq. (6).
Proton mass uniformly distributed on a thin spherical shell, of radius rpz, is equiv-
alent to the mass distribution of the superposition of all zitterbewegung fermion
ensemble members. Therefore, each ensemble member is effectively gravitationally
entangled with the rest of the ensemble.
In particle physics, the strong and weak forces are often referred to as interactions.
This is because they can be mathematically described as an exchange of virtual parti-
cles. The same is true of the mathematical description of electromagnetism in particle
physics. The terms “force”and “interaction”can generally be interchanged. The term
CP-symmetry refers to the combination of charge conjugation symmetry and parity
symmetry. For more than 50 years, it has been established both experimentally and
theoretically that the weak force can violate CP-symmetry.
7
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

According to long-established QCD theory, it may also be possible for the strong
force to violate CP-symmetry. However, no experiment involving only the strong
force has violated CP-symmetry. As QCD theory provides no fundamental reason for
CP-symmetry to be conserved, this is known as the strong CP problem. No experi-
ment has shown gravity to violate CP-symmetry. The connection made in Reference
[4], and this section, between the strong force and gravity may imply that the strong
force is CP-symmetric because gravity is CP-symmetric.
5. Proton charge arcs as quantum networks
In Figure 1, each inner charge arc shares its center with an outer charge arc.
Reference [5] proposes that each inner-outer charge arc pair is generated by an
ensemble of virtual photon standing waves with a uniform poloidal distribution. This
finding is summarized in Section 6.
The charge arcs, proposed in Reference [5], are assumed to toroidally rotate about
the proton axis and generate the charge surfaces proposed in Reference [4]. Reference
[4] also proposes that an ensemble of standing waves, precisely matching proton
Compton wavelength, λp, exists across the equatorial diameter of the inner charge
surface. This implies that a single standing wave, of wavelength λp, exists between the
equatorial points of the proton inner charge arcs as shown in Figure 1.
Reference [5] proposes that this equatorial standing wave is actually the
superposition of the fundamental and second harmonics. Inside an infinite square
well, the fundamental harmonic is a half wavelength and the second harmonic is a full
wavelength. Reference [5] proposes that each poloidal ensemble of virtual photon
standing waves interferes with the equatorial fundamental harmonic to generate the
equatorial second harmonic.
Note that photon momentum is equal to Planck’s constant, h, divided by wave-
length. Since the inner-arc equatorial points are λpapart, the fundamental harmonic
will be of wavelength 2λpand momentum h=2λp. Reference [5] assumes that the
poloidal ensembles of virtual standing waves consist of fundamental harmonics of
wavelength 2λp=ffiffiffi
3
pand momentum ffiffiffi
3
ph=2λp. The ffiffiffi
3
pfactor is due to assumed charge
and mass coupling via a virtual optimal Möbius band geometry. The rms value of the
sum of interfering momenta h=2λpand ffiffiffi
3
ph=2λpis given by
prms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
2λp

2
þffiffiffi
3
ph
2λp

2
s¼h
2λpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12þffiffiffi
3
p

2
r¼h
λp
:(10)
This is the momentum of the second harmonic, of wavelength λp, presumed to
exist between the equatorial points of the GSQV proton’s inner charge arcs. This key
assumption was used to develop the original GSQV proton model [4]. Reference [5]
therefore proposes that the second equatorial harmonic is quantized as the rms value
of the interference between the fundamental equatorial harmonic and the poloidal
distributions of fundamental harmonics.
6. Charge arc generation and proton charge-exclusion zone
Reference [5] proposes that the charge structure of a GSQV proton consists of a
pair of þecharge arcs and a pair of e=2 charge arcs. These are shown in Figure 1.
8
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

Each charge arc’s equatorial point is farthermost from the proton axis. The equatorial
point on each charge arc is proposed to toroidally revolve about the proton axis at light
speed. It is this charge motion that generates the proton magnetic moment.
Reference [5] proposes that a standing wave ensemble connects each charge arc to
its center and defines proton zitterbewegung inertial power (ZIP) as
Pp¼Epfp, (11)
where Ep¼hf pis proton mass-energy and fpis proton Compton frequency.
Therefore
Pp¼hf 2
p:(12)
Reference [5] proposes that each charge arc is continually reflecting and accelerat-
ing virtual photons. Acceleration density per poloidal radian is assumed divided
evenly among four charge arcs:
dapol
dϕ¼1
4Rpω2
p, (13)
where apol denotes the magnitude of poloidal centripetal acceleration, ϕdenotes
latitude, and proton Compton angular frequency, ωp¼2πfp, implies the key assump-
tion that GSQV proton mass and charge regenerate at the same rate.
Reference [5] shows that Eq. (13) may be integrated to obtain
apol ¼1
4Rpω2
pΔϕarc, (14)
where Δϕarc is the angular extent of the charge arc. This poloidal acceleration is
assumed to generate the four charge arcs of a GSQV proton. The quantity qarc is defined
by Reference [5] as the charge of each arc. Reference [5] applies the classical Larmor
formula to describe the amount of ZIP, Ppol, which generates the four charge arcs:
Ppol ¼4q2
arca2
pol
6πε0c3, (15)
where ε0is the electric permittivity of free space.
References [4, 5] define Qpex as the uncharged proportion of the GSQV proton’s
outer surface, which resembles a flat cap at each pole. It follows that 1 Qpex

represents the charged proportion of the GSQV proton’s outer surface. For the outer
charge arcs, it is reasonable to assume [5] that
Ppol ¼1Qpex

Pp¼1Qpex

hf 2
p:(16)
Substituting Rp¼λp=ffiffiffi
3
pand Δϕarc ¼πinto Eq. (14) yields
apol ¼πλp
4ffiffiffi
3
pω2
p:(17)
9
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

Reference [5] shows that
λpω2
p¼4π2cf p:(18)
Therefore
apol ¼π3
ffiffiffi
3
pcf p:(19)
Squaring yields
a2
pol ¼π6
3c2f2
p:(20)
For a proton outer charge arc, qarc ¼e. Substituting Eq. (20) into Eq. (15) yields
Ppol ¼2π5e2f2
p
9ε0c:(21)
Substituting Eq. (16) into Eq. (21) yields
1Qpex

h¼2π5e2
9ε0c:(22)
Rearranging,
e2¼9ε0hc
2π51Qpex

:(23)
The asymptotic low-energy value of the fine-structure constant [1, 2],
α¼e2
2ε0hc :(24)
Substituting Eq. (23) into Eq. (24) yields
α¼9
4π51Qpex

:(25)
Presuming Eq. (25) to be exact, the 2022 Committee on Data of the International
Science Council (CODATA) value [31],
α¼7:2973525643 11ðÞ103, (26)
can be used to calculate
Qpex ¼14
9π5α¼1απ5qu
e

2¼7:49620822 15ðÞ103, (27)
where qu¼2e=3 is the charge of an up, charm, or top quark. Note that this Qpex
value is only 2:1% larger than that estimated in Reference [4]. This numerically
10
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

supports the assumption Rp¼λp=ffiffiffi
3
p, which is tied to the assumption that charge and
mass are coupled via a virtual optimal Möbius band geometry [5].
Planck charge, qP, may now be written in terms of eand αas
q2
P¼e2
α¼4π5e2
91Qpex

, (28)
where e2was divided by Eq. (25). Taking the square root,
qP¼2e
3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π5
1Qpex

v
u
u
t¼quffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π5=1Qpex

r:(29)
Rearranging,
qu¼qPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Qpex

=π5
r, (30)
Surprisingly up, charm, and top quark charge appears to depend only on Planck
charge, the geometric constant π, and the charged proportion of the GSQV proton’s
outer surface, 1 Qpex

. It follows that
qd¼qP
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Qpex

=π5
r, (31)
where qd¼e=3 is the charge of a down, strange, or bottom quark.
7. Intrinsic charm quark formation
Reference [5] proposes that the intrinsic up-antiup (uu) virtual sea quark pair,
depicted in Figure 2, may occasionally transform into a charm quark, c, and an
anticharm quark, c. This cc quark pair may momentarily form bound states, such as
∣uudcc>, with the proton’s three valence quarks [32]. Reference [5] proposes that
twin poloidal revolutions, phase-locked with each other, continually regenerate
charge. Reference [4] associates poloidal revolution with both isospin and charge
generation. The two vectors shown in Figure 1 always have the same poloidal angle,
relative to the equator, as they circulate. These two poloidal vectors occasionally
overlap. The overlaps are indicated by the heavy dashed lines in Figure 1. Two such
overlaps will occur in each poloidal cycle, which is the same time duration as a
zitterbewegung cycle because charge and mass are assumed to regenerate at the same
rate.
Reference [5] proposes that each of these overlapping lengths exists momentarily
as a standing wave of vacuum energy. If each wave is a fundamental harmonic, it will
be of wavelength
2λp
ffiffiffi
3
p2rpa

¼2λp11
ffiffiffi
3
p

:(32)
11
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

This is shorter than the wavelength of the fundamental harmonic ensemble
regenerating the charge arcs, 2Rp¼2λp=ffiffiffi
3
p, explained in Section 5 and Reference [5].
The proportion of shortening is clearly
ffiffiffi
3
p11
ffiffiffi
3
p

¼ffiffiffi
3
p1:(33)
Reference [5] proposes that this shortening is additive and equivalent to proton
mass-energy momentarily increasing by the factor ffiffiffi
3
p1

1. This is evaluated as
mp
ffiffiffi
3
p1≈1281:70MeV=c2, (34)
which is slightly more energy than that needed to generate either a charm or
anticharm quark “running”mass [17]. The momentary proton mass-energy
increase should appear to be twice this amount because there are two momentary
overlaps in each poloidal cycle. This process therefore provides more than enough
additional energy to generate both the charm and anticharm quark “running”masses
[17]. Since this charm-anticharm quark pair forms from an up-antiup quark pair,
Reference [5] goes on to calculate charm quark “running”mass as about 1279 MeV=c2.
This value is well inside the 2022 recommended Particle Data Group range:
mc¼1270 20MeV=c2[17].
Even though the twin charge-regenerating vectors are poloidally phase-locked,
they are not toroidally synchronized. Reference [5] supposes that the twin
charge-regenerating vectors become entangled, and generate additional mass-energy,
whenever both vector tips are sufficiently close to the zitterbewegung equator.
Sufficiently close is assumed to be closer than the zitterbewegung radius [4], rcz, of the
charm quark “running”mass, mc[17].
During this entanglement, the maximum angular separation, θent, between the
zitterbewegung equator and each Rpvector will be
θent ¼sin 1rcz
Rp

¼sin 1ffiffiffi
3
pmp
4πmc
!
:(35)
Reference [5] shows that all possible entanglements will occur on an angular area
that is
1cos θent
ðÞ
2¼1
21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
13m2
p
16π2m2
c
s
0
@1
A≈0:26% (36)
of the total spherical angular area. This implies the ∣uudcc>bound state is present
about 0.26% of the time. Reference [5] calculates the fraction of proton momentum
carried by intrinsic charm quarks as
mc
mp
1cos θent
ðÞ¼
mc
mpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mc
mp

2
3
16π2
s¼0:704 11ðÞ%:(37)
12
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

The uncertainty has been propagated from the 2022 recommended Particle Data
Group range: mc¼1270 20MeV=c2[17]. Reference [32] reports an experimental
estimate 0:62 0:28ðÞ%. Our calculation, displayed as Eq. (37), is consistent with this
experimental range and about 25 times more precise.
8. Proton magnetic moment from charge arcs
Reference [5] proposes the following equation for proton magnetic moment:
μp¼2Vpo
1þδp

1Qpex

Rpþrpa

2Vpi
1δp

Rprpa

2
0
@1
A
μN
4πrpz
, (38)
where μNis the nuclear magneton unit typically used to express magnetic
moments of atomic nuclei. The quantity Vpi is the lemon volume formed by toroidally
rotating the GSQV proton’s two inner charge arcs about the GSQV proton axis. The
quantity Vpo is the volume formed by toroidally rotating the GSQV proton’s two outer
charge arcs, with flat end caps, about the GSQV proton axis. Applying calculus results
derived in the Appendix of Reference [4],
Vpi ¼4
3πR3
psin 3ϕpl 3
4cos ϕpl 2ϕpl sin 2ϕpl


(39)
and
Vpo ¼π2R2
prpa þ4Rp
3π

þ2πr2
paRp, (40)
where
ϕpl ¼cos 1rpa
Rp

¼cos 11ffiffiffi
3
p
2
 (41)
and
rpa ¼Rpλp
2¼λp
1
ffiffiffi
3
p1
2

:(42)
Substituting Eqs. (1) and (42) into Eq. (38) yields
μp¼6Vpo
1þδp

1Qpex

4ffiffiffi
3
p
2Vpi
1δp
0
@1
A
4μN
λ3
p
:(43)
Noting Rp¼λp=ffiffiffi
3
p, it can be seen from Eqs. (39)–(42) that both Vpi and Vpo are
proportional to λ3
p. A nucleus g-factor is twice the value of its magnetic moment when
expressed with the nuclear magneton unit. Since both Qpex and δpare unitless,
Eq. (43) implies proton g-factor is independent of proton mass.
13
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

The quantity δpis a positive dimensionless parameter with a value less than
1/1000. In Eqs. (38) and (43), δphas the effect of perturbing the charge distribution
slightly away from the equator on the outer charge arcs and slightly toward the
equator on the inner charge arcs.
The reason for these slight perturbations may be explained subjectively as the self-
interaction of the toroidally rotating charge arcs with their electromagnetic fields.
However, it must be stressed that this proposed self-interaction mechanism subjec-
tively applies classical electromagnetism to the interior of a quantum mechanical
particle. This is beyond known physics. As such, δpis used as an adjustable parameter.
While we do not calculate δpfrom first principles, Reference [5] shows that it can be
calculated from a similar adjustable parameter defined for a GSQV neutron model,
neutron mass, and the sum of the up and down quark masses.
A quadratic equation, in terms of δp, may be obtained by rearranging Eq. (43):
aδ2
pþbδpþc¼0, (44)
where
a¼ 4ffiffiffi
3
p

21Qpex

λ3
pμp
4μN
, (45)
b¼4ffiffiffi
3
p

21Qpex

Vpi þ6Vpo, (46)
and
c¼b12Vpo a:(47)
Applying the quadratic formula,
δp¼bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b24ac
p
2a¼2:7353468 15ðÞ104and ≈2:64, (48)
where the 2022 CODATA value of μp, displayed as Eq. (49), is input to Eq. (45).
Note that the 2022 CODATA uncertainties in αand μpcontribute almost equally to the
uncertainty in δp. The second solution is unphysical, so δp¼2:7353468 15ðÞ104
provides a calculated proton magnetic moment, μp, as precise as the 2022 CODATA
value [31]:
μp¼2:79284734463 82ðÞμN:(49)
Reference [5] shows how δp¼2:7353468 15ðÞ104can be input into a GSQV
neutron model that calculates a neutron magnetic moment about two orders of
magnitude more precisely than the most accurate experiments.
9. Effective charge radius
Reference [18] experimentally estimated proton polar axial charge radius, rA¼0:73
0:17 fm. Based on Figure 1, the GSQV proton’seffectivepolarchargeradiusis
14
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

rA≈Rp¼λp
ffiffiffi
3
p≈0:763fm, (50)
which is well within the range reported by Reference [18].
Reference [4] originally developed the GSQV proton model by assuming a charge
distribution that is the radial projection of two uniformly charged concentric spheres.
Due to the tiny value of δp, this key assumption still accurately reflects the GSQV
proton charge distribution implied by Eqs. (38) and (43). The polar charge-exclusion
zones on the GSQV proton’s outer charge surface provide the most significant
deviation from this key assumption.
If a GSQV proton did not have charge-exclusion zones, radially projecting all
charge out to REo would yield a spherically symmetric charge distribution with electric
potential
UR
Eo
ðÞ¼
ke
REo
(51)
at distance REo from the proton center, where kis Coulomb’s constant. The GSQV
proton model is approximately spherical. Therefore, electric potential on its outer
surface, at average distance
rs¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
4πVpo
3
r(52)
from the proton center, may be approximated by
Ur
s
ðÞ≈ke
rs
:(53)
This implies
Ur
s
ðÞ
ke ≈1
rs
:(54)
For a sphere,
rA
V¼3, (55)
where ris radius, Ais surface area, and Vis volume. Since a perfect sphere has a
minimum surface area to volume ratio for its size, rA=Vshould be greater than 3 for a
perturbed sphere. The Appendix of Reference [4] shows how to calculate the surface
area of volume Vpo:
Apo ¼2π2Rprpa þ2Rp
π

þ2πr2
pa:(56)
For the GSQV proton model of Section 8,
rsApo
Vpo
≈3:0059:(57)
15
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

To a first approximation, combining Eqs. (54) and (57) yields
Ur
s
ðÞ
ke ≈Apo
3Vpo
:(58)
This implies that a change in charge volume would have about a third of the effect,
on a muon or electron orbital, as a change in outer surface charge area. The GSQV
proton outer surface charge area,
Aeff ¼Apo 1Qpex

:(59)
This is assumed equivalent to
Veff ≈Vpo 1Qpex
3

, (60)
which implies an effective charge radius,
reff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
4πVeff
3
r≈ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
4πVpo 1Qpex
3

3
s≈0:8409 fm:(61)
This value is consistent with both the 2022 Particle Data Group recommended
value, rp¼0:8409 4ðÞfm [16, 17], and the 2022 CODATA recommended value,
rp¼0:84075 64ðÞfm [31].
10. Conclusions
This chapter summarizes our previously reported findings [4, 5] that a nested
surface vortex structure can explain several properties of free or chemically bound
protons. We call this the GSQV proton model. This geometric model can be visualized
in the usual 3 spatial dimensions, with mathematics not beyond that typically
encountered by undergraduate physics and chemistry students. Additional details can
be found in References [4, 5]. Reference [5] includes a GSQV neutron model and
proposes novel mechanisms that link proton and neutron properties.
Sections 2 and 4 summarized the Reference [4] finding that GSQV proton mass-
energy may be concentrated in a relatively small central sphere called a
zitterbewegung fermion. This aspect of the model is consistent with general
relativity. Section 5 summarized the Reference [5] proposal that the surrounding
massless charge structures are formed from standing vacuum waves. These charge
structures take the form of arcs rotating about the proton axis. This rotation is set to
light speed at the equator, and causes the proton charge to be distributed on the inner
lemon and outer apple surfaces of a spindle torus. The polar dimples of the spindle
torus are uncharged.
At higher energies, this model transforms into the valence quarks, gluon flux
tubes, and initial sea quarks of the standard quantum chromodynamics model. This
was summarized in Section 3. With the established chiral EFT and lattice QCD
models, recently discovered discrepancies with experiment occur at the lowest
energies. The GSQV proton model may therefore help resolve these discrepancies.
16
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

Section 6 summarized the Reference [5] finding that up, charm, and top quark
charge depends only on Planck charge, the proportionate area of a GSQV proton’s
polar charge-exclusion zone, and π. Section 7 reviewed how the GSQV proton model
can calculate properties of intrinsic charm quarks. Section 8 reviewed the proton
magnetic moment calculation. Section 9 summarized the Reference [5] finding that
effective proton charge radii are statistically consistent with the most accurate exper-
imental estimates.
Acknowledgements
The authors thank Zhijun Jia for his advice on how to write an appropriate intro-
duction and for his encouragement and endless enthusiasm. The authors also thank
Kori Verrall for her helpful discussions and instruction on the optimal Möbius band.
The authors also thank Dean Ju Kim and Associate Dean Gubbi Sudhakaran for their
helpful technical discussions and key administrative support. Most of the original
concepts were developed, while S.V. was partly supported by UWL Faculty Research
Grant 23-01-SV and A.O. was supported by UWL URC grant #35F22. The CPC was
funded by the authors.
Conflicts of interest
During the course of this research, Andrew Kaminsky graduated from the Univer-
sity of Wisconsin at La Crosse and was an employee at Benchmark and later PDA
Engineering. These employment relationships had no influence on this research.
During the course of this research, Isaac Ozolins graduated from the University of
Wisconsin at La Crosse and was an employee at ThermTech. This employment rela-
tionship had no influence on this research.
During the course of this research, Andrew Otto graduated from the University of
Wisconsin at La Crosse and was an employee at St. Croix Health. This employment
relationship had no influence on this research.
Nomenclature
Asphere surface area
Aeff proton outer surface charge area
Apo proton outer surface area
apol poloidal centripetal acceleration
apz proton zitterbewegung acceleration
cspeed of light in vacuum ¼299792458m=s
c charm quark
c anticharm quark
CODATA Committee on Data of the International Science Council
CP charge parity
d down quark
eelementary charge ¼1:602176634 1019C
Epproton mass-energy
17
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

EFT effective field theory
Fggravitational force
Fin total inward force
Fout total outward force
fpproton Compton frequency
Guniversal gravitational constant
GSQV ground-state quantum vortex
hPlanck constant ¼6:62607015 1034 J=Hz
ℏreduced Planck constant ¼h=2π
kCoulomb constant
kBBoltzmann constant
lPPlanck length
mccharm quark mass
mPPlanck mass
mpproton mass
MRI magnetic resonance imaging
pg¼pproton gravitational pressure
Ppproton zitterbewegung inertial power
Ppol amount of Ppthat regenerates the four charge arcs
prms rms momentum
qarc arc charge
qdcharge of a down, strange, or bottom quark
qPPlanck charge
qucharge of an up, charm, or top quark
QCD quantum chromodynamics
QFT quantum field theory
Qpex uncharged proportion of GSQV proton outer surface
rsphere radius
rAproton polar axial charge radius
rcz charm quark zitterbewegung radius
reff proton effective charge radius
REo radius of spherically symmetric charge distribution
Rpproton charge arc radius
rpproton rms charge radius
rpa half minimum equatorial separation between charge arcs
rpz proton zitterbewegung radius
rsradius of sphere with volume Vpo
Ttemperature
Tcirc temperature of circular Unruh energy
Upotential energy
u up quark
u antiup quark
Vsphere volume
Veff proton effective charge volume
Vpi proton lemon volume
Vpo proton outer volume
ZIP zitterbewegung inertial power
αfine-structure constant
18
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

δpdimensionless fine-tuning parameter
ε0electric permittivity of free space
θent maximum angular separation
λpproton Compton wavelength
μNnuclear magneton
μpproton magnetic moment
πratio of circle circumference to diameter
ϕpoloidal latitude
ϕpl maximum poloidal latitude of an inner charge arc
Δϕarc angular extent of an outer charge arc
ωpproton Compton angular frequency
Author details
Steven Verrall
1
*
†
, Kelly S. Verrall
2†
, Andrew Kaminsky
1,3,4†
, Isaac Ozolins
1,5†
,
Emily Friederick
1†
, Andrew Otto
1,6†
, Ivan Ngian
1†
, Reagen McCormick
1†
and Pearl Scallon
1†
1 University of Wisconsin at La Crosse, La Crosse, WI, USA
2 Independent Researcher, La Crosse, WI, USA
3 PDA Engineering, Burnsville, MN, USA
4 Benchmark, Winona, MN, USA
5 ThermTech Inc., Waukesha, WI, USA
6 St. Croix Health, St. Croix Falls, WI, USA
*Address all correspondence to: steven.verrall@gmail.com
†These authors contributed equally.
© 2024 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of
the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited.
19
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975

References
[1] Oerter R. The Theory of Almost
Everything: The Standard Model, the
Unsung Triumph of Modern Physics.
New York: Pi Press; 2006. 327 p
[2] Peskin M, Schroeder D. An
Introduction to Quantum Field Theory.
Boca Raton: CRC Press; 1995. 866 p.
DOI: 10.1201/9780429503559
[3] Sebens CT. The fundamentality of
fields. Synthese. 2022;200:1-28.
DOI: 10.1007/s11229-022-03844-2
[4] Verrall SC, Atkins M, Kaminsky A,
et al. Ground state quantum vortex
proton model. Foundations of Physics.
2023;53. DOI: 10.1007/s10701-023-
00669-y
[5] Verrall SC, Kaminsky A, Verrall KN,
et al. Unbound low-energy nucleons as
semiclassical quantum networks. 2024.
DOI: 10.20944/preprints202405.0932.v1
[Preprint]
[6] Greensite J. An Introduction to the
Confinement Problem. 2nd ed. Cham,
Switzerland: Springer Nature; 2020. 271
p. DOI: 10.1007/978-3-030-51563-8
[7] Achenbach P, Adhikari D,
Afanasev A, et al. The present and
future of QCD. Nuclear Physics A. 2024;
1047. DOI: 10.1016/j.nuclphysa.2024.
122874
[8] Bernard V, Kaiser N, Ulf-G M. Chiral
dynamics in nucleons and nuclei.
International Journal of Modern Physics
E: Nuclear Physics. 1995;4:193-344.
DOI: 10.1142/S0218301395000092
[9] NPLQCD Collaboration. Magnetic
moments of light nuclei from lattice
quantum chromodynamics. Physical
Review Letters. 2014;113. DOI: 10.1103/
PhysRevLett.113.252001
[10] Bhattacharya S, Cichy K,
Constantinou M, et al. Moments of
proton GPDs from the OPE of nonlocal
quark bilinears up to NNLO. Physical
Review D. 2023;108. DOI: 10.1103/
PhysRevD.108.014507
[11] Lamoreaux SK. The Casimir force:
Background, experiments, and
applications. Reports on Progress in
Physics. 2005;68. DOI: 10.1088/
0034-4885/68/1/R04
[12] Zheng X, Deur A, Kang H, et al.
Measurement of the proton spin
structure at long distances. Nature
Physics. 2021;17:736-741. DOI: 10.1038/
s41567-021-01198-z
[13] Deur A, Chen JP, Kuhn SE, et al.
Experimental study of the behavior of
the Bjorken sum at very low Q
2
. Physics
Letters B. 2022;825. DOI: 10.1016/j.
physletb.2022.136878
[14] Li R, Sparveris N, Atac H, et al.
Measured proton electromagnetic
structure deviates from theoretical
predictions. Nature. 2022;611:265-270.
DOI: 10.1038/s41586-022-05248-1
[15] Ruth D, Zielinski R, Gu C, et al.
Proton spin structure and generalized
polarizabilities in the strong quantum
chromodynamics regime. Nature
Physics. 2022;18:1441-1446.
DOI: 10.1038/s41567-022-01781-y
[16] Antognini A et al. Proton structure
from the measurement of 2S-2P
transition frequencies of muonic
hydrogen. Science. 2013;339:417-420.
DOI: 10.1126/science.1230016
[17] Workman RL et al. Review of
particle physics. PTEP. 2022;2022.
DOI: 10.1093/ptep/ptac097
20
Vortex Dynamics –Theoretical, Experimental and Numerical Approaches

[18] Cai T, Moore ML, Olivier A, et al.
Measurement of the axial vector form
factor from antineutrino-proton
scattering. Nature. 2023;614:48-53.
DOI: 10.1038/s41586-022-05478-3
[19] Wunderlich W. Über ein
abwickelbares Möbiusband. Monatshefte
für Mathematik. 1962;66:276-289.
DOI: 10.1007/BF01299052
[20] Halpern B, Weaver C. Inverting a
cylinder through isometric immersions
and embeddings. Transactions of the
American Mathematical Society. 1977;
230:41-70. DOI: 10.1090/S0002-9947-
1977-0474388-1
[21] Fuchs D, Tabachnikov S.
Mathematical Omnibus: Thirty Lectures
on Classic Mathematics. Vol. 46.
Providence, RI, USA: American
Mathematical Society; 2007. 465 p.
DOI: 10.1090/mbk/046
[22] Schwartz RE. The optimal paper
Moebius band. Annals of Mathematics.
DOI: 10.48550/arXiv.2308.12641
[Preprint]
[23] Biermann S, Erne S, Gooding C, et al.
Unruh and analogue Unruh
temperatures for circular motion in 3+1
and 2+1 dimensions. Physical Review D.
2020;102. DOI: 10.1103/PhysRevD.
102.085006
[24] Lochan K, Ulbricht H, Vinante A,
Goyal SK. Detecting acceleration-
enhanced vacuum fluctuations with
atoms inside a cavity. Physical Review
Letters. 2020;125. DOI: 10.1103/
PhysRevLett.125.241301
[25] Zhang J, Yu H. Entanglement
harvesting for Unruh-DeWitt detectors
in circular motion. Physical Review D.
2020;102. DOI: 10.1103/PhysRevD.
102.065013
[26] Zhou Y, Hu J, Yu H. Entanglement
dynamics for Unruh-DeWitt detectors
interacting with massive scalar fields:
The Unruh and anti-Unruh effects.
Journal of High Energy Physics. 2021;88.
DOI: 10.1007/JHEP09(2021)088
[27] Zhou Y, Hu J, Yu H. Steady-state
entanglement for rotating Unruh-
DeWitt detectors. Physical Review D.
2022;106. DOI: 10.1103/PhysRevD.
106.105028
[28] Anastopoulos C, Savvidou N.
Coherences of accelerated detectors and
the local character of the Unruh effect.
Journal of Mathematical Physics. 2012;
53. DOI: 10.1063/1.3679554
[29] Heald M. Where is the “Wien peak”?
American Journal of Physics. 2003;71:
1322-1323. DOI: 10.1119/1.1604387
[30] Pvoh B, Rith K, Scholz C, et al.
Particles and Nuclei: An Introduction to
the Physical Concepts. 7th ed.
Heidelberg, Germany: Springer; 2015.
458 p. DOI: 10.1007/978-3-662-46321-5
[31] Mohr P, Tiesinga E, Newell D,
Taylor B. Codata Internationally
Recommended 2022 Values of the
Fundamental Physical Constants
[Internet]. Gaithersburg, MD, USA:
National Institute of Standards and
Technology (NIST); 2024. Available
from: https://physics.nist.gov/constants
[Accessed: June 4, 2024]
[32] NNPDF Collaboration. Evidence for
intrinsic charm quarks in the proton.
Nature. 2022;608:483-487.
DOI: 10.1038/s41586-022-04998-2
21
Proton Properties from Nested Surface Vortices
DOI: http://dx.doi.org/10.5772/intechopen.1005975