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.

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 protons. 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 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, frequency, 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 parameters must be experimentally determined. QFT applies operators to create and annihilate 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 generated 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 polarized. 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 toroidally and poloidally circulating virtual fields in the form of standing waves. These virtual fields arise from the real electromagnetic fields of real photons, which presumably 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 concentrated in the relatively small blue central sphere, which Reference [4] calls a zitterbewegung fermion. This finding is summarized in Sections 2 and 4. The surrounding 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 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, 13, 14, 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 $\pi$. 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, 17, 18].

Figure 1.

The GSQV proton model consists of a central zitterbewegung fermion (blue sphere), of radius $r_{\mathit{pz}}=\frac{{\lambda }_p}{4\pi }$, orbited by four massless charge arcs. The outer two charge arcs are quantized to the value of the elementary charge, $+e$. 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 $2r_{\mathit{pa}}$. The distance between the two equatorial points of the inner charge arcs is always proton Compton wavelength, ${\lambda }_p$. Therefore, $r_{\mathit{pa}}={\lambda }_p\left(\frac{1}{\sqrt{3}}-\frac{1}{2}\right)$, which is slightly less than $r_{\mathit{pz}}$. The red dots represent uncharged polar regions where virtual poloidal flow is split into two equal magnitudes.

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 $+e$ 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 $-e/2$ charge 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.

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 coupling may be represented in the form of a virtual optimal Möbius band [5, 19, 20, 21, 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, ${\lambda }_p$ is proton Compton wavelength. Each proton charge arc is assumed to be regenerated by half a poloidal turn, at radius $R_p={\lambda }_p/\sqrt{3}$, each zitterbewegung cycle. Reference [4] associates the half-poloidal turn with quantum mechanical isospin. The $\sqrt{3}$ factor 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 revolutions of the central zitterbewegung fermion [4, 5]. A proton zitterbewegung cycle may be characterized by Compton wavelength, ${\lambda }_p$, and Compton frequency, $f_p=c/{\lambda }_p$, where $c$ is 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, 24, 25, 26, 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

$$r_{\mathit{pz}}=\frac{\mathrm{\hslash }}{2m_{p}c}=\frac{{\lambda }_p}{4\pi },$$E1

where $\mathrm{\hslash }$ is Planck’s reduced constant and $m_p$ is proton mass. Note that

$${\lambda }_p=\frac{h}{m_{p}c}\text{and}\mathrm{\hslash }=\frac{h}{2\pi },$$E2

where $h$ is Planck’s constant. In Figure 1, the proton zitterbewegung fermion is depicted as a blue sphere. Its radius, $r_{\mathit{pz}}$, 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, $T_{\text{circ}}$ [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],

$$a_{\mathit{pz}}=\frac{c^2}{r_{\mathit{pz}}}=\frac{2m_p{c}^3}{\mathrm{\hslash }}.$$E3

Reference [4] shows that for a zitterbewegung fermion,

$$T_{\text{circ}}=\frac{\mathrm{\hslash }{a}_{\mathit{pz}}}{4\sqrt{3}{\mathit{ck}}_B}=\frac{m_p{c}^2}{2\sqrt{3}{k}_B},$$E4

where $k_B$ is the Boltzmann constant. This implies that proton mass-energy,

$$E_p=m_p{c}^2=2\sqrt{3}{k}_B{T}_{\text{circ}}.$$E5

The median energy of a particle in a thermal bath at temperature, $T$, is given by $3.50302k_{B}T$ [29]. This appears to imply that proton mass-energy, $E_p$, is approximately $2\sqrt{3}/3.50302\approx 98.9\%$ 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 $E_p$ and 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, ${\lambda }_p$. 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, ${\lambda }_p$, 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 $+2e$ charge of the outer GSQV proton charge arcs evenly divides into three up quarks of charge $+2e/3$. As proposed in Reference [5], it is likely that the two $+e$ charge arcs merge into a $+2e$ charge surface [4] before decomposing into three up quarks of charge $+2e/3$.

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 $-e$ charge shell [4] and divide into a down $\left(-e/3\right)$ and antiup $\left(-2e/3\right)$ quark pair. To maintain color neutrality, these two quarks must be a color-anticolor pair. The left-hand column 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, $p=U/3V$, where $U$ is trapped energy and $V$ is volume. For the GSQV proton’s spherical zitterbewegung fermion, $U=m_p{c}^2$ and $V=\frac{4}{3}\pi r_{\mathit{pz}}^3$. Therefore

$$p=\frac{U}{3V}=\frac{m_p{c}^2}{4\pi r_{\mathit{pz}}^3}.$$E6

The total outward force exerted on the zitterbewegung fermion’s surface, $F_{\text{out}}=\mathit{pA}$, where $A=4\pi r_{\mathit{pz}}^2$. Therefore

$$F_{\text{out}}=\frac{m_p{c}^2}{r_{\mathit{pz}}}=m_p{a}_{\mathit{pz}},$$E7

where $a_{\mathit{pz}}$ 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 $F_{\text{in}}=F_{\text{out}}$. In the GSQV proton model, $F_{\text{in}}$ 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, $F_{\text{in}}$, needed to stabilize the zitterbewegung spherical shell. This is equivalent to inward gravitational pressure,

$$p_g=\frac{m_p{c}^2}{4\pi r_{\mathit{pz}}^3}.$$E8

Reference [4] supposes that the point-like objects described in Section 2 are actually spheres of radius $\sqrt{2}{l}_P$, where $l_P\approx 1.616\times {10}^{-35}\mathrm{m}$ is the Planck length. Each circling sphere will have a cross-sectional area $2\pi l_P^2$ and always be located on the zitterbewegung spherical shell. The inward gravitational force, $F_g$, due to inward gravitational pressure, $p_g$, on each circling sphere, will be assumed to be a cross-sectional area, $2\pi l_P^2$, multiplied by inward gravitational pressure:

$$F_g=2\pi l_P^2{p}_g=\frac{m_p{c}^2{l}_P^2}{2r_{\mathit{pz}}^3}=\frac{{\mathit{Gm}}_p{m}_P{l}_P}{2r_{\mathit{pz}}^3}=\frac{{\mathit{Gm}}_p^2}{r_{\mathit{pz}}^2},$$E9

since $c^2={\mathit{Gm}}_P/l_P$ and $m_P{l}_P/r_{\mathit{pz}}=\mathrm{\hslash }/{\mathit{cr}}_{\mathit{pz}}=2m_p$, where $G$ is the universal gravitational constant and $m_P$ is the Planck mass. Each circling sphere therefore experiences a force, due to curved spacetime, as if it were a point-like proton mass, $m_p$, 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 $\le r_{\mathit{pz}}$. 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 $r_{\mathit{pz}}$, is equivalent 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 particles. 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.

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 experiment 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, ${\lambda }_p$, exists across the equatorial diameter of the inner charge surface. This implies that a single standing wave, of wavelength ${\lambda }_p$, 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 wavelength. Since the inner-arc equatorial points are ${\lambda }_p$ apart, the fundamental harmonic will be of wavelength $2{\lambda }_p$ and momentum $h/2{\lambda }_p$. Reference [5] assumes that the poloidal ensembles of virtual standing waves consist of fundamental harmonics of wavelength $2{\lambda }_p/\sqrt{3}$ and momentum $\sqrt{3}h/2{\lambda }_p$. The $\sqrt{3}$ factor is due to assumed charge and mass coupling via a virtual optimal Möbius band geometry. The rms value of the sum of interfering momenta $h/2{\lambda }_p$ and $\sqrt{3}h/2{\lambda }_p$ is given by

$$p_{\mathrm{rms}}=\sqrt{{\left(\frac{h}{2{\lambda }_p}\right)}^2+{\left(\frac{\sqrt{3}h}{2{\lambda }_p}\right)}^2}=\frac{h}{2{\lambda }_p}\sqrt{1^2+{\left(\sqrt{3}\right)}^2}=\frac{h}{{\lambda }_p}.$$E10

This is the momentum of the second harmonic, of wavelength ${\lambda }_p$, 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 $+e$ charge arcs and a pair of $-e/2$ charge arcs. These are shown in Figure 1. 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

$$P_p=E_p{f}_p,$$E11

where $E_p={\mathit{hf}}_p$ is proton mass-energy and $f_p$ is proton Compton frequency. Therefore

$$P_p={\mathit{hf}}_p^2.$$E12

Reference [5] proposes that each charge arc is continually reflecting and accelerating virtual photons. Acceleration density per poloidal radian is assumed divided evenly among four charge arcs:

$$\frac{{\mathit{da}}_{\mathrm{pol}}}{\mathit{d\varphi }}=\frac{1}{4}{R}_p{\omega }_p^2,$$E13

where $a_{\mathrm{pol}}$ denotes the magnitude of poloidal centripetal acceleration, $\varphi$ denotes latitude, and proton Compton angular frequency, ${\omega }_p=2\pi f_p$, implies the key assumption that GSQV proton mass and charge regenerate at the same rate.

Reference [5] shows that Eq. (13) may be integrated to obtain

$$a_{\mathrm{pol}}=\frac{1}{4}{R}_p{\omega }_p^2\mathrm{\Delta }{\varphi }_{\mathrm{arc}},$$E14

where $\mathrm{\Delta }{\varphi }_{\mathrm{arc}}$ 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 $q_{\mathrm{arc}}$ is defined by Reference [5] as the charge of each arc. Reference [5] applies the classical Larmor formula to describe the amount of ZIP, $P_{\mathrm{pol}}$, which generates the four charge arcs:

$$P_{\mathrm{pol}}=4\frac{q_{\mathrm{arc}}^2{a}_{\mathrm{pol}}^2}{6{\mathit{\pi \epsilon }}_0{c}^3},$$E15

where ${\epsilon }_0$ is the electric permittivity of free space.

References [4, 5] define $Q_{p\mathrm{ex}}$ as the uncharged proportion of the GSQV proton’s outer surface, which resembles a flat cap at each pole. It follows that $\left(1-Q_{p\mathrm{ex}}\right)$ represents the charged proportion of the GSQV proton’s outer surface. For the outer charge arcs, it is reasonable to assume [5] that

$$P_{\mathrm{pol}}=\left(1-Q_{p\mathrm{ex}}\right)P_p=\left(1-Q_{p\mathrm{ex}}\right){\mathit{hf}}_p^2.$$E16

Substituting $R_p={\lambda }_p/\sqrt{3}$ and $\mathrm{\Delta }{\varphi }_{\mathrm{arc}}=\pi$ into Eq. (14) yields

$$a_{\mathrm{pol}}=\frac{{\mathit{\pi \lambda }}_p}{4\sqrt{3}}{\omega }_p^2.$$E17

Reference [5] shows that

$${\lambda }_p{\omega }_p^2=4{\pi }^2{\mathit{cf}}_p.$$E18

Therefore

$$a_{\mathrm{pol}}=\frac{{\pi }^3}{\sqrt{3}}{\mathit{cf}}_p.$$E19

Squaring yields

$$a_{\mathrm{pol}}^2=\frac{{\pi }^6}{3}{c}^2{f}_p^2.$$E20

For a proton outer charge arc, $q_{\mathrm{arc}}=e$. Substituting Eq. (20) into Eq. (15) yields

$$P_{\mathrm{pol}}=\frac{2{\pi }^5{e}^2{f}_p^2}{9{\epsilon }_{0}c}.$$E21

Substituting Eq. (16) into Eq. (21) yields

$$\left(1-Q_{p\mathrm{ex}}\right)h=\frac{2{\pi }^5{e}^2}{9{\epsilon }_{0}c}.$$E22

Rearranging,

$$e^2=\frac{9{\epsilon }_0\mathit{hc}}{2{\pi }^5}\left(1-Q_{p\mathrm{ex}}\right).$$E23

The asymptotic low-energy value of the fine-structure constant [1, 2],

$$\alpha =\frac{e^2}{2{\epsilon }_0\mathit{hc}}.$$E24

Substituting Eq. (23) into Eq. (24) yields

$$\alpha =\frac{9}{4{\pi }^5}\left(1-Q_{p\mathrm{ex}}\right).$$E25

Presuming Eq. (25) to be exact, the 2022 Committee on Data of the International Science Council (CODATA) value [31],

$$\alpha =7.2973525643\left(11\right)\times {10}^{-3},$$E26

can be used to calculate

$$Q_{p\mathrm{ex}}=1-\frac{4}{9}{\pi }^5\alpha =1-{\mathit{\alpha \pi }}^5{\left(\frac{q_u}{e}\right)}^2=7.49620822\left(15\right)\times {10}^{-3},$$E27

where $q_u=2e/3$ is the charge of an up, charm, or top quark. Note that this $Q_{p\mathrm{ex}}$ value is only $2.1\%$ larger than that estimated in Reference [4]. This numerically supports the assumption $R_p={\lambda }_p/\sqrt{3}$, which is tied to the assumption that charge and mass are coupled via a virtual optimal Möbius band geometry [5].

Planck charge, $q_P$, may now be written in terms of $e$ and $\alpha$ as

$$q_P^2=\frac{e^2}{\alpha }=\frac{4{\pi }^5{e}^2}{9\left(1-Q_{p\mathrm{ex}}\right)},$$E28

where $e^2$ was divided by Eq. (25). Taking the square root,

$$q_P=\frac{2e}{3}\sqrt{\frac{{\pi }^5}{\left(1-Q_{p\mathrm{ex}}\right)}}=q_u\sqrt{{\pi }^5/\left(1-Q_{p\mathrm{ex}}\right)}.$$E29

Rearranging,

$$q_u=q_P\sqrt{\left(1-Q_{p\mathrm{ex}}\right)/{\pi }^5},$$E30

Surprisingly up, charm, and top quark charge appears to depend only on Planck charge, the geometric constant $\pi$, and the charged proportion of the GSQV proton’s outer surface, $\left(1-Q_{p\mathrm{ex}}\right)$. It follows that

$$q_d=\frac{q_P}{2}\sqrt{\left(1-Q_{p\mathrm{ex}}\right)/{\pi }^5},$$E31

where $q_d=e/3$ is the charge of a down, strange, or bottom quark.

7. Intrinsic charm quark formation

Reference [5] proposes that the intrinsic up-antiup ($\mathrm{u}\overline{\mathrm{u}}$) virtual sea quark pair, depicted in Figure 2, may occasionally transform into a charm quark, c, and an anticharm quark, $\overline{\mathrm{c}}$. This $\mathrm{c}\overline{\mathrm{c}}$ quark pair may momentarily form bound states, such as $\mid \text{uudc}\overline{\mathrm{c}}>$, 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

$$2\left(\frac{{\lambda }_p}{\sqrt{3}}-2r_{\mathit{pa}}\right)=2{\lambda }_p\left(1-\frac{1}{\sqrt{3}}\right).$$E32

This is shorter than the wavelength of the fundamental harmonic ensemble regenerating the charge arcs, $2R_p=2{\lambda }_p/\sqrt{3}$, explained in Section 5 and Reference [5]. The proportion of shortening is clearly

$$\sqrt{3}\left(1-\frac{1}{\sqrt{3}}\right)=\sqrt{3}-1.$$E33

Reference [5] proposes that this shortening is additive and equivalent to proton mass-energy momentarily increasing by the factor ${\left(\sqrt{3}-1\right)}^{-1}$. This is evaluated as

$$\frac{m_p}{\sqrt{3}-1}\approx 1281.70\mathrm{MeV}/{\mathrm{c}}^2,$$E34

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 $1279\mathrm{MeV}/{\mathrm{c}}^2$. This value is well inside the 2022 recommended Particle Data Group range: $m_c=1270\pm 20\mathrm{MeV}/{\mathrm{c}}^2$ [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], $r_{\mathit{cz}}$, of the charm quark “running” mass, $m_c$ [17].

During this entanglement, the maximum angular separation, ${\theta }_{\mathrm{ent}}$, between the zitterbewegung equator and each $R_p$ vector will be

$${\theta }_{\mathrm{ent}}={sin}^{-1}\left(\frac{r_{\mathit{cz}}}{R_p}\right)={sin}^{-1}\left(\frac{\sqrt{3}{m}_p}{4\pi m_c}\right).$$E35

Reference [5] shows that all possible entanglements will occur on an angular area that is

$$\frac{\left(1-cos{\theta }_{\mathrm{ent}}\right)}{2}=\frac{1}{2}\left(1-\sqrt{1-\frac{3m_p^2}{16{\pi }^2{m}_c^2}}\right)\approx 0.26\%$$E36

of the total spherical angular area. This implies the $\mid \text{uudc}\overline{\mathrm{c}}>$ bound state is present about 0.26% of the time. Reference [5] calculates the fraction of proton momentum carried by intrinsic charm quarks as

$$\frac{m_c}{m_p}\left(1-cos{\theta }_{\mathrm{ent}}\right)=\frac{m_c}{m_p}-\sqrt{{\left(\frac{m_c}{m_p}\right)}^2-\frac{3}{16{\pi }^2}}=0.704\left(11\right)\%.$$E37

The uncertainty has been propagated from the 2022 recommended Particle Data Group range: $m_c=1270\pm 20\mathrm{MeV}/{\mathrm{c}}^2$ [17]. Reference [32] reports an experimental estimate $\left(0.62\pm 0.28\right)\%$. 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:

$${\mu }_p=\left(\frac{2V_{\mathit{po}}}{\left(1+{\delta }_p\right)\left(1-Q_{p\mathrm{ex}}\right){\left(R_p+r_{\mathit{pa}}\right)}^2}-\frac{V_{\mathit{pi}}}{\left(1-{\delta }_p\right){\left(R_p-r_{\mathit{pa}}\right)}^2}\right)\frac{{\mu }_N}{4\pi r_{\mathit{pz}}},$$E38

where ${\mu }_N$ is the nuclear magneton unit typically used to express magnetic moments of atomic nuclei. The quantity $V_{\mathit{pi}}$ is the lemon volume formed by toroidally rotating the GSQV proton’s two inner charge arcs about the GSQV proton axis. The quantity $V_{\mathit{po}}$ 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],

$$V_{\mathit{pi}}=\frac{4}{3}\pi R_p^3\left[{sin}^3{\varphi }_{\mathit{pl}}-\frac{3}{4}cos{\varphi }_{\mathit{pl}}\left(2{\varphi }_{\mathit{pl}}-sin2{\varphi }_{\mathit{pl}}\right)\right]$$E39

and

$$V_{\mathit{po}}={\pi }^2{R}_p^2\left(r_{\mathit{pa}}+\frac{4R_p}{3\pi }\right)+2\pi r_{\mathit{pa}}^2{R}_p,$$E40

where

$${\varphi }_{\mathit{pl}}={cos}^{-1}\left(\frac{r_{\mathit{pa}}}{R_p}\right)={cos}^{-1}\left(1-\frac{\sqrt{3}}{2}\right)$$E41

and

$$r_{\mathit{pa}}=R_p-\frac{{\lambda }_p}{2}={\lambda }_p\left(\frac{1}{\sqrt{3}}-\frac{1}{2}\right).$$E42

Substituting Eqs. (1) and (42) into Eq. (38) yields

$${\mu }_p=\left(\frac{6V_{\mathit{po}}}{\left(1+{\delta }_p\right)\left(1-Q_{p\mathrm{ex}}\right){\left(4-\sqrt{3}\right)}^2}-\frac{V_{\mathit{pi}}}{1-{\delta }_p}\right)\frac{4{\mu }_N}{{\lambda }_p^3}.$$E43

Noting $R_p={\lambda }_p/\sqrt{3}$, it can be seen from Eqs. (39)–(42) that both $V_{\mathit{pi}}$ and $V_{\mathit{po}}$ are proportional to ${\lambda }_p^3$. A nucleus g-factor is twice the value of its magnetic moment when expressed with the nuclear magneton unit. Since both $Q_{p\mathrm{ex}}$ and ${\delta }_p$ are unitless, Eq. (43) implies proton g-factor is independent of proton mass.

The quantity ${\delta }_p$ is a positive dimensionless parameter with a value less than 1/1000. In Eqs. (38) and (43), ${\delta }_p$ has 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 subjectively applies classical electromagnetism to the interior of a quantum mechanical particle. This is beyond known physics. As such, ${\delta }_p$ is used as an adjustable parameter. While we do not calculate ${\delta }_p$ from 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 ${\delta }_p$, may be obtained by rearranging Eq. (43):

$$a{\delta }_p^2+b{\delta }_p+c=0,$$E44

where

$$a=-{\left(4-\sqrt{3}\right)}^2\left(1-Q_{p\mathrm{ex}}\right)\frac{{\lambda }_p^3{\mu }_p}{4{\mu }_N},$$E45

$$b={\left(4-\sqrt{3}\right)}^2\left(1-Q_{p\mathrm{ex}}\right)V_{\mathit{pi}}+6V_{\mathit{po}},$$E46

and

$$c=b-12V_{\mathit{po}}-a.$$E47

Applying the quadratic formula,

$${\delta }_p=\frac{-b\pm \sqrt{b^2-4\mathit{ac}}}{2a}=2.7353468\left(15\right)\times {10}^{-4}\text{and}\approx 2.64,$$E48

where the 2022 CODATA value of ${\mu }_p$, displayed as Eq. (49), is input to Eq. (45). Note that the 2022 CODATA uncertainties in $\alpha$ and ${\mu }_p$ contribute almost equally to the uncertainty in ${\delta }_p$. The second solution is unphysical, so ${\delta }_p=2.7353468\left(15\right)\times {10}^{-4}$ provides a calculated proton magnetic moment, ${\mu }_p$, as precise as the 2022 CODATA value [31]:

$${\mu }_p=2.79284734463\left(82\right){\mu }_N.$$E49

Reference [5] shows how ${\delta }_p=2.7353468\left(15\right)\times {10}^{-4}$ can 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, $r_A=0.73\pm 0.17\mathrm{fm}$. Based on Figure 1, the GSQV proton’s effective polar charge radius is

$$r_A\approx R_p=\frac{{\lambda }_p}{\sqrt{3}}\approx 0.763\mathrm{fm},$$E50

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 ${\delta }_p$, 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 $R_{\mathit{Eo}}$ would yield a spherically symmetric charge distribution with electric potential

$$U\left(R_{\mathit{Eo}}\right)=\frac{\mathit{ke}}{R_{\mathit{Eo}}}$$E51

at distance $R_{\mathit{Eo}}$ from the proton center, where $k$ is Coulomb’s constant. The GSQV proton model is approximately spherical. Therefore, electric potential on its outer surface, at average distance

$$r_s=\sqrt[3]{\frac{3}{4\pi }{V}_{\mathit{po}}}$$E52

from the proton center, may be approximated by

$$U\left(r_s\right)\approx \frac{\mathit{ke}}{r_s}.$$E53

This implies

$$\frac{U\left(r_s\right)}{\mathit{ke}}\approx \frac{1}{r_s}.$$E54

For a sphere,

$$\frac{\mathit{rA}}{V}=3,$$E55

where $r$ is radius, $A$ is surface area, and $V$ is volume. Since a perfect sphere has a minimum surface area to volume ratio for its size, $\mathit{rA}/V$ should be greater than 3 for a perturbed sphere. The Appendix of Reference [4] shows how to calculate the surface area of volume $V_{\mathit{po}}$:

$$A_{\mathit{po}}=2{\pi }^2{R}_p\left(r_{\mathit{pa}}+\frac{2R_p}{\pi }\right)+2\pi r_{\mathit{pa}}^2.$$E56

For the GSQV proton model of Section 8,

$$\frac{r_s{A}_{\mathit{po}}}{V_{\mathit{po}}}\approx 3.0059.$$E57

To a first approximation, combining Eqs. (54) and (57) yields

$$\frac{U\left(r_s\right)}{\mathit{ke}}\approx \frac{A_{\mathit{po}}}{3V_{\mathit{po}}}.$$E58

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,

$$A_{\mathrm{eff}}=A_{\mathit{po}}\left(1-Q_{p\mathrm{ex}}\right).$$E59

This is assumed equivalent to

$$V_{\mathrm{eff}}\approx V_{\mathit{po}}\left(1-\frac{Q_{p\mathrm{ex}}}{3}\right),$$E60

which implies an effective charge radius,

$$r_{\mathrm{eff}}=\sqrt[3]{\frac{3}{4\pi }{V}_{\mathrm{eff}}}\approx \sqrt[3]{\frac{3}{4\pi }{V}_{\mathit{po}}\left(1-\frac{Q_{p\mathrm{ex}}}{3}\right)}\approx 0.8409\mathrm{fm}.$$E61

This value is consistent with both the 2022 Particle Data Group recommended value, $r_p=0.8409\left(4\right)\mathrm{fm}$ [16, 17], and the 2022 CODATA recommended value, $r_p=0.84075\left(64\right)\mathrm{fm}$ [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.

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 $\pi$. 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 experimental estimates.

Acknowledgments

The authors thank Zhijun Jia for his advice on how to write an appropriate introduction 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 University 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 relationship 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

$A$	sphere surface area
$A_{\mathrm{eff}}$	proton outer surface charge area
$A_{\mathit{po}}$	proton outer surface area
$a_{\mathrm{pol}}$	poloidal centripetal acceleration
$a_{\mathit{pz}}$	proton zitterbewegung acceleration
c	speed of light in vacuum $=299792458\mathrm{m}/\mathrm{s}$
c	charm quark
$\overline{\mathrm{c}}$	anticharm quark
CODATA	Committee on Data of the International Science Council
CP	charge parity
d	down quark
e	elementary charge $=1.602176634\times {10}^{-19}\mathrm{C}$
$E_p$	proton mass-energy
EFT	effective field theory
$F_g$	gravitational force
$F_{\text{in}}$	total inward force
$F_{\text{out}}$	total outward force
$f_p$	proton Compton frequency
G	universal gravitational constant
GSQV	ground-state quantum vortex
h	Planck constant $=6.62607015\times {10}^{-34}\mathrm{J}/\mathrm{Hz}$
$\mathrm{\hslash }$	reduced Planck constant $=h/2\pi$
k	Coulomb constant
$k_B$	Boltzmann constant
$l_P$	Planck length
$m_c$	charm quark mass
$m_P$	Planck mass
$m_p$	proton mass
MRI	magnetic resonance imaging
$p_g=p$	proton gravitational pressure
$P_p$	proton zitterbewegung inertial power
$P_{\mathrm{pol}}$	amount of $P_p$ that regenerates the four charge arcs
$p_{\mathrm{rms}}$	rms momentum
$q_{\mathrm{arc}}$	arc charge
$q_d$	charge of a down, strange, or bottom quark
$q_P$	Planck charge
$q_u$	charge of an up, charm, or top quark
QCD	quantum chromodynamics
QFT	quantum field theory
$Q_{p\mathrm{ex}}$	uncharged proportion of GSQV proton outer surface
r	sphere radius
$r_A$	proton polar axial charge radius
$r_{\mathit{cz}}$	charm quark zitterbewegung radius
$r_{\mathrm{eff}}$	proton effective charge radius
$R_{E\mathrm{o}}$	radius of spherically symmetric charge distribution
$R_p$	proton charge arc radius
$r_p$	proton rms charge radius
$r_{\mathit{pa}}$	half minimum equatorial separation between charge arcs
$r_{\mathit{pz}}$	proton zitterbewegung radius
$r_s$	radius of sphere with volume $V_{\mathit{po}}$
$T$	temperature
$T_{\text{circ}}$	temperature of circular Unruh energy
U	potential energy
u	up quark
$\overline{\mathrm{u}}$	antiup quark
V	sphere volume
$V_{\mathrm{eff}}$	proton effective charge volume
$V_{\mathit{pi}}$	proton lemon volume
$V_{\mathit{po}}$	proton outer volume
ZIP	zitterbewegung inertial power
$\alpha$	fine-structure constant
${\delta }_p$	dimensionless fine-tuning parameter
${\epsilon }_0$	electric permittivity of free space
${\theta }_{\mathrm{ent}}$	maximum angular separation
${\lambda }_p$	proton Compton wavelength
${\mu }_N$	nuclear magneton
${\mu }_p$	proton magnetic moment
$\pi$	ratio of circle circumference to diameter
$\varphi$	poloidal latitude
${\varphi }_{\mathit{pl}}$	maximum poloidal latitude of an inner charge arc
$\mathrm{\Delta }{\varphi }_{\mathrm{arc}}$	angular extent of an outer charge arc
${\omega }_p$	proton Compton angular frequency