An unbound ground-state neutron model is developed following a similar rationale to the development of the refined GSQV proton model in Reference [9]. An unbound GSQV neutron is constructed from four rotating charge arcs, as shown in Figure 3 and Figure 4. The charge on each arc is quantized as the elementary charge, $\pm e$.

It is clear from Panel (a) of Figure 7 that this proposed neutron model contains three times as much positive charge, and three times as much negative charge, as that needed to form the neutron’s valence quarks. However the minimally excited GSQV proton model, shown in Panel (b) of Figure 7 and initially developed in Reference [9], includes a virtual up sea quark ($+2e/3$ charge) and a virtual antiup sea quark ($-2e/3$ charge). These virtual sea quarks are assumed to initiate a proton’s virtual pion cloud. Note that up and antiup quarks have approximately half the mass energy of down and antidown quarks. Therefore, an up-antiup quark pair represents a lower energy state than would a down-antidown quark pair. The GSQV neutron model, developed here, includes two pairs of virtual up-antiup sea quarks assumed to initiate a neutron’s virtual pion cloud.

The Gerasimov-Drell-Hearn (GDH) sum rule [28,100,101,102,103,104,105,106], initially developed in the 1960s, relates the proton and neutron anomalous magnetic moments. The GDH sum rule involves two adjustable parameters that may be calculated–to high precision–from nucleon masses, nucleon magnetic moments, and the fine-structure constant. Experimental verification of the GDH sum rule parameters has only recently been attained [28,105].

Recent analysis shows that single-pion photoproduction off the nucleon is the dominant contribution to the GDH sum rule [106]. Reference [106] found that non-single-pion photoproduction off the proton amounts to about 10%, whereas non-single-pion photoproduction off the neutron amounts to about 44%. This large difference may be partly explained by the minimally excited GSQV neutron containing twice as many virtual sea quarks as the minimally excited GSQV proton. The quarks above and below the dashed lines in Panel (a) of Figure 7, and above the dashed line in Panel (b) of Figure 7, may initiate pion production upon sufficient excitation. Therefore, Reference [106] provides experimental support for the GSQV neutron model plausibly containing two $+e$ and two $-e$ charge arcs.

In this paper, c is the speed of light, h is Planck’s constant, and $\hslash =h/2\pi$ is the reduced Planck constant. As in Reference [9], proton zitterbewegung radius is defined as where $m_p=1.67262192369\left(51\right)\times {10}^{-27}$ kg is proton mass [35,36] and ${\lambda }_p$ is proton Compton wavelength. Neutron Compton wavelength is defined as where $m_n=1.67492749804\left(95\right)\times {10}^{-27}$ kg is neutron mass [35,36]. Based on the refined GSQV proton model, developed in Reference [9], and Figure 3 and Figure 4 in this paper, neutron magnetic moment will be approximated by where $V_{ni}$ is the lemon volume formed by rotating the neutron’s two inner charge arcs about the neutron center, $V_{no}$ is the apple volume formed by rotating the neutron’s two outer charge arcs–plus uncharged extensions–about the neutron center, $Q_{n\mathrm{ex}}$ is the uncharged proportion of the neutron’s outer (apple) surface,

Following Reference [9], define and Following the Appendix of Reference [9], and Also following the Appendix of Reference [9], the charged proportion of the neutron’s outer (apple) surface, $(1-Q_{n\mathrm{ex}})$, will be the area of the surface of revolution formed by the outer charge arcs divided by the apple surface area:

Substituting Equations (4), (7), (8), and (9) into Equation (3) yields Substituting Equations (1) and (2) into Equation (10) and simplifying, which is about 0.16% less negative than the 2018 CODATA value [35,36] displayed later in this section as Equation (23). Note that the $m_p/m_n$ factor, in Equation (11), is due to the convention of defining the nuclear magneton, ${\mu }_N$, in terms of the proton mass. If the nuclear magneton was instead defined in terms of the neutron mass, the form of Equation (11) would imply a neutron g-factor independent of the nucleon masses.

Since the ratio of proton to neutron mass is close to unity, which is more than 7 times closer, than Equation (11), to the 2018 CODATA value [35,36] displayed later in this section as Equation (23). The surprising precision of Equation (12) may provide insights into the electron capture process of a proton-rich nucleus [107,108,109]. In an unstable proton-rich nucleus, suppose that the most loosely-bound proton experiences electron capture via the weak interaction [110]. Suppose that this process has the effect of the loosely-bound proton absorbing the former electron’s charge, via a W boson, but not its mass, $m_e$. The weak interaction is addressed in detail in Section 3.2. During electron capture, suppose the former electron’s mass energy cannot primarily be incorporated into the mass of the former proton. Energy dissipation pathways are therefore needed to facilitate the electron capture process. This may help explain the very low probabilities for neutral hydrogen and molecular hydrogen to undergo electron capture in extremely low-energy environments.

Suppose, that following the weak interaction [110], this loosely-bound former proton has the charge structure implied by Equations (3) and (10)–(12). Modifying Equation (11), if the mass of this loosely-bound former proton is close to $0.39717m_e$ less than that of an unbound proton, its magnetic moment will closely resemble that of an unbound neutron: The experimentally determined value of ${\mu }_n$ is displayed later in this section as Equation (23). This mass defect can be attributed to a relatively minor nuclear binding energy [111], which is consistent with a loosely-bound nucleon. Being electrically neutral, and possessing a substantial magnetic moment, this new nuclear neutron should be magnetically attracted to nearby nucleons and bind relatively strongly to the nucleus. Therefore, neutron binding may be initiated by semiclassical magnetic forces.

Evaluating the Equation (12) terms in the square parentheses, Substituting $-\pi /3$ for the square-parentheses terms in Equation (12) yields where $q_u$ is the charge of an up, charm, or top quark. These compact equations offer only 2-digit precision. However, their precision improves to 5 digits if the number 4, in the denominator, is replaced by $\sqrt{33/2}$: While no rationale is offered to support the quantity $\sqrt{33/2}$, it is surprising that an equation involving just one 2-digit numeral can calculate an experimentally deterimined quantity to 5-digit precision.

Apart from the polar charge-exclusion zones on the GSQV neutron’s outer apple-shaped charge surface, Equations (3) and (10)–(13) assume a direction-independent steradian charge distribution. This was a key assumption applied when developing the GSQV proton model in Reference [9]. This assumption implies that the linear charge density, along each charge arc, depends only on distance from nucleon center. This results in a non-uniform linear charge density along each charge arc.

It will now be assumed that each charge arc’s linear charge density is slightly less non-uniform due to internal self-interaction that tends toward evening out charge distribution. This will not lead to a measurable electric quadrupole moment. This is because Larmor precession would still result in a uniform steradian charge distribution in the form of a Bloch sphere [96,97,98,99]. This slight charge redistribution will slightly change the magnetic moment contributed by each charge surface formed by the toroidally rotating charge arcs.

The neutron’s inner positively-charged lemon-shaped surface will be assumed to redistribute via a slight poleward migration. Since the magnetic moment of a current loop depends on loop cross-sectional area, the neutron’s inner positively-charged surface will now contribute a slightly smaller positive magnetic moment. Equation (3) will be modified by dividing the first term by $(1+{\delta }_n)$, where ${\delta }_n$ is dimensionless, positive, and much smaller than 1. Similarly, the neutron’s outer negatively-charged surface will be assumed to redistribute via a slight equatorial migration. The neutron’s outer negatively-charged surface will now contribute a slightly larger negative magnetic moment. Equation (3) will be modified by dividing the second term by $(1-{\delta }_n)$. An alternative implementation of adjustable parameter, ${\delta }_n$, would be to multiply the first term of Equation (3) by $(1-{\delta }_n)$ and to divide the second term of Equation (3) by $(1+{\delta }_n)$. This alternative was not selected because it results in a larger ${\delta }_n$ value.

Therefore, a precise calculation of GSQV neutron magnetic moment may be obtained from Substituting Equation (4) into Equation (17) yields This equation may be solved for ${\delta }_n$ by collecting terms to form a quadratic equation in terms of ${\delta }_n$: where and Using the 2018 CODATA value [35,36], and applying the quadratic formula yields The second solution is unphysical because ${\delta }_n$ is defined to be positive and much smaller than 1. Therefore ${\delta }_n=3.70115\left(55\right)\times {10}^{-4}$. Note that ${\delta }_n$ is more than 180 times smaller than $Q_{n\mathrm{ex}}$. This tiny proportion indicates that the charge distribution shift provided by ${\delta }_n$ is very slight.

In the refined GSQV proton model developed in Reference [9], the fine-tuned adjustable parameter, $R=0.76273\mathrm{fm}$, agrees with ${\lambda }_p/\sqrt{3}\approx 0.76292\mathrm{fm}$ to 3-digit precision, where is proton Compton wavelength. This paper models unbound GSQV nucleons as self-sustaining networks of one-dimensional harmonic waves in 3+1 spacetime dimensions. This involves assuming

Section 3.1.3 proposes that radial ensembles of virtual photon standing waves generate the charge arcs. It will be assumed that the radial systems of virtual photon standing waves interfere with the virtual photon standing waves–assumed to exist between the equatorial points of the proton inner arcs and the polar points of the neutron inner arcs.

Suppose the GSQV proton has both fundamental and second harmonic equatorial standing waves between the equatorial points of its inner charge arcs–shown in Figure 2. Since these equatorial points are ${\lambda }_p$ apart, the wavelength of the fundamental harmonic will be $2{\lambda }_p$ with momentum $h/2{\lambda }_p$. If the virtual standing waves generating the proton charge arcs are fundamental harmonics, they will be of wavelength $2{\lambda }_p/\sqrt{3}$ with momentum $\sqrt{3}h/2{\lambda }_p$. The rms value of the sum of interfering momenta $h/2{\lambda }_p$ and $\sqrt{3}h/2{\lambda }_p$ is given by which is the momentum of the second harmonic presumed to exist between the equatorial points of the GSQV proton’s inner charge arcs. This second-harmonic standing wave, with wavelength ${\lambda }_p$, was a key assumption used to develop the original GSQV proton model in Reference [9]. This paper therefore proposes that the ensemble of an unbound GSQV proton’s charge arc fundamental harmonics interfere with its equatorial fundamental harmonic to generate the second equatorial harmonic–quantized as the rms value of this interference.

Similarly, suppose that the neutron has both fundamental and second harmonic standing waves between the polar points of its inner charge arcs–shown in Figure 4. Since these polar points are $\sqrt{3}{\lambda }_n/2$ apart, the wavelength of the fundamental harmonic will be $\sqrt{3}{\lambda }_n$. Such a wave has momentum $h/\sqrt{3}{\lambda }_n$. If the virtual standing waves generating the neutron charge arcs are fundamental harmonics, they will be of wavelength ${\lambda }_n$ with momentum $h/{\lambda }_n$. The rms value of the sum of interfering momenta $h/\sqrt{3}{\lambda }_n$ and $h/{\lambda }_n$ is given by which is the momentum of the second harmonic presumed to exist between the polar points of the GSQV neutron’s inner charge arcs. This second-harmonic standing wave has wavelength $\sqrt{3}{\lambda }_n/2$, which is the polar-point separation shown in Figure 4. This paper therefore proposes that the ensemble of an unbound GSQV neutron’s charge arc fundamental harmonics interfere with its polar fundamental harmonic to generate the second polar harmonic–quantized as the rms value of this interference.

This paper proposes that the charge structure of an unbound GSQV nucleon consists of a pair of positive charge arcs and a pair of negative charge arcs. These are shown in Figure 1 and Figure 2 for the GSQV proton and Figure 3 and Figure 4 for the GSQV neutron. The outermost point of each charge arc resides at the zitterbewegung equator. Each charge arc is proposed to toroidally revolve about the particle center with its outermost point moving at light speed. This charge motion is proposed to generate the nucleon magnetic moments.

A proton’s $+e$ net charge, and positive magnetic moment, is proposed to be generated from a pair of negative charge arcs, each with charge $-e/2$, revolving inside a pair of positive charge arcs, each with charge $+e$. These charge arcs are depicted in Figure 1 and Figure 2. A neutron’s zero net charge, and negative magnetic moment, is proposed to consist of a pair of positive charge arcs, each with charge $+e$, revolving inside a pair of negative charge arcs, each with charge $-e$. These charge arcs are depicted in Figure 3 and Figure 4.

An ensemble of virtual photon standing waves is proposed to connect each point on a charge arc to its radial center. Each radial center resides on the zitterbewegung equator–substantially closer to the nucleon center than the arc’s equatorial point. Each radial center coherently revolves about the particle center with the charge arc.

For unbound GSQV nucleons, the zitterbewegung effect [8,9,37,38,39,73,74,75,76,77,78,79,80,81] will be assumed to behave as a circulation of nucleon mass energy. Since unbound low-energy nucleons are spin-half particles, two loops of circulation represent a full zitterbewegung cycle. This mass-energy flow will be modeled as a type of power that will be called zitterbewegung inertial power (ZIP). This ZIP power may be interpreted as the rate at which the mass-energy of an unbound GSQV nucleon progresses forward in time. This may align with energetic causal set theory–where spacetime emerges from momentum space and the conservation of energy-momentum [14,15,16,17,18,19,20,21]. Unbound GSQV nucleon ZIP will be defined as where $E_N=h{f}_N$ is nucleon mass energy, and Compton frequency, $f_N$, is the number of full zitterbewegung cycles per second. Therefore

It will be assumed that an ensemble of virtual photons forms an ensemble of one-dimensional standing waves between every point on a charge arc and its radial center. Each charge arc is therefore continually reflecting and accelerating an ensemble of virtual photons. Each charge arc is a poloidal structure with poloidal radius $R_N$, which takes the value $R_N=R_p={\lambda }_p/\sqrt{3}$ for the proton or $R_N=R_n={\lambda }_n/2$ for the neutron.

For each unbound GSQV nucleon, vacuum energy poloidal acceleration density per poloidal radian will be assumed to be divided evenly among four charge arcs: where $a_{\mathrm{pol}}$ denotes the magnitude of poloidal centripetal acceleration, $\varphi$ denotes latitude, and Compton angular frequency, ${\omega }_N=2\pi f_N$, implies the key assumption that unbound GSQV nucleon charge and mass regenerate at the same rate. This assumption is explored in Section 3.1.5.

Since $R_N$ and ${\omega }_N$ are independent of $\varphi$, it is trivial to integrate Equation (31) to obtain the total poloidal acceleration for each charge arc: where $\Delta {\varphi }_{\mathrm{arc}}$ is the poloidal angular extent of the charge arc, and s is poloidal charge arc length. Compton angular frequency, ${\omega }_N$, may be interpreted as the temporal rate at which charge, $q_{\mathrm{arc}}$, is regenerated on arclength, s. This rate is tied to nucleon Compton or zitterbewegung frequency, $f_N={\omega }_N/2\pi$, and is independent of arclength, s.

Note that magnetic moment is generated by toroidal charge circulation. Poloidal acceleration, $a_{\mathrm{pol}}$, is due to the circular polarization of the revolving virtual photon assumed to initially generate the GSQV nucleon during pair production [9]. Therefore poloidal acceleration, $a_{\mathrm{pol}}$, is a circulation of virtual fields. This virtual field circulation is assumed to generate the four charge arcs of an unbound GSQV nucleon. The classical Larmor formula [112] will be repurposed to describe the amount of ZIP, $P_{\mathrm{pol}}$, that couples with the four poloidal virtual field accelerations to generate the four charge arcs: where ${\epsilon }_0$ is the electric permittivity of free space. Note that the classical Larmor formula is closely associated with the quantum Unruh effect [62,113,114,115,116,117,118].

Following Reference [9], $Q_{p\mathrm{ex}}$ is defined as the uncharged (polar) proportion of the GSQV proton’s outer surface. This outer surface is a surface of revolution formed by the GSQV proton’s outer charge arcs. It can be seen from Figure 6 that the uncharged proportion of the GSQV proton’s outer surface should resemble a flat cap at each pole. The quantity $(1-Q_{p\mathrm{ex}})$ represents the charged proportion of the GSQV proton’s outer surface. Since vacuum energy reflects only from the charged portion of each arc, it is reasonable to assume that for an unbound GSQV proton,

For each proton outer charge arc, $R_p={\lambda }_p/\sqrt{3}$ and $\Delta {\varphi }_{\mathrm{arc}}=\pi$. These quantities are depicted in Figure 1. Substituting into Equation (32) yields The relation $E_N=hc/{\lambda }_N=\hslash {\omega }_N$ implies ${\lambda }_N{\omega }_N=2\pi c$. Therefore since ${\omega }_N=2\pi f_N$.

Substituting Equation (36) into Equation (35) yields Squaring, Substituting Equation (38) into Equation (33), with $q_{\mathrm{arc}}=e$, yields Substituting Equation (34) into Equation (39) yields Rearranging, At low energies, the fine-structure constant may now be written as Presuming Equation (42) to be exact, the 2018 CODATA value [35,36], can be used to calculate This value is entirely reasonable, since it is just $2.1\%$ larger than that estimated in Reference [9]. Also note that $Q_{p\mathrm{ex}}$ is only $2.7\%$ larger than $\alpha$. Surprisingly, the low-energy fine-structure constant is found to depend only on the proportionate area of the GSQV proton’s charge-exclusion zone in 3+1 spacetime dimensions. At low energies, additional hidden dimensions may not be required to explain the coupling constant, $\alpha$, that enables the unrivaled precision of quantum electrodynamics [90,91,92].

Planck charge, $q_P$, may now be written in terms of quantized electronic charge, e, and the low-energy fine-structure constant, $\alpha$, as where $e^2$ was divided by the right-hand expression in Equation (42). Taking the square root, where $q_u=2e/3$ is the charge of an up, charm, or top quark. Rearranging, Surprisingly up, charm, and top quark charge appears to depend only on Planck charge, the proportionate area of an unbound GSQV proton’s charge-exclusion zone, and $\pi$. It follows that where $q_d=e/3$ is the charge of a down, strange, or bottom quark.

Since $Q_{p\mathrm{ex}}\approx \alpha$, To the same level of precision, the actual relationships are Therefore Equation (49), which eliminates the $2/3$ and $1/3$ factors, overestimates quark charges by only 1 part in 10,000. However, note that Eqns. (47) and (48) also eliminate the $2/3$ and $1/3$ factors and are presumed exact.

Substituting $Q_{p\mathrm{ex}}\approx \alpha$ into Equation (42) yields which is accurate to 4 digits. Note that Equation (51) is much simpler, although less accurate, than Equation (59) of Reference [119]. The next section shows how the low-energy fine-structure constant can be calculated to 5-digit precision from the unbound nucleon charge arc quantum networks developed in this paper.

Extending the assumption applied in Equation (34), mass energy coupling with each type of charge arc will be defined as where ${\beta }_{\mathrm{arc}}\le 1$ depends on charge arc type. Note that each charge arc type exists as a pair, and that the same nucleon mass energy is assumed to simultaneously couple with all four charge arcs of an unbound GSQV nucleon.

Comparing Equation (34) to Equation (52), ${\beta }_{po}=1$ for the proton outer charge arcs. For each proton inner charge arc, $R_p={\lambda }_p/\sqrt{3}$ and $\Delta {\varphi }_{\mathrm{arc}}=2{\varphi }_{pl}$. These quantities are depicted in Figure 2. Substituting into Equation (32) and applying Equation (36) yields Squaring, Substituting Equation (54) into Equation (33), with $q_{\mathrm{arc}}=e/2$, yields Substituting Equation (52) into Equation (55) yields Rearranging, Comparing with Equation (41),

For each neutron outer charge arc, $R_n={\lambda }_n/2$ and $\Delta {\varphi }_{\mathrm{arc}}=\pi$. These quantities are depicted in Figure 3. The reason why the outer charge arcs, in both unbound GSQV nucleon models, should extend for $\pi$ radians is as follows: Charge is assumed to be regenerated by twin poloidal revolutions phase-locked with each other. Another assumption, is that at each moment in time, at most one inner and one outer charge arc can be regenerated. Therefore, whenever both phase-locked poloidal circulations are on outer arcs, only one can regenerate charge.

Substituting into Equation (32) and applying Equation (36) yields Squaring, Substituting Equation (60) into Equation (33), with $q_{\mathrm{arc}}=e$, yields Substituting Equation (52) into Equation (61) yields Rearranging, Comparing with Equation (41),

For each neutron inner charge arc, $R_n={\lambda }_n/2$ and $\Delta {\varphi }_{\mathrm{arc}}=2\pi /3$. These quantities are depicted in Figure 4. Substituting into Equation (32) and applying Equation (36) yields Squaring, Substituting Equation (66) into Equation (33), with $q_{\mathrm{arc}}=e$, yields Substituting Equation (52) into Equation (67) yields Rearranging, Comparing with Equation (41),

Average neutron mass energy coupling with its charge arcs, where the value of $Q_{p\mathrm{ex}}$ is obtained from Equation (44). Average proton mass energy coupling with its charge arcs, where ${\varphi }_{pl}={cos}^{-1}(1-\sqrt{3}/2)$. When rounded to 5 digits, $\overline{{\beta }_p}$ is the same as $3/5$. The quantity $Q_{p\mathrm{ex}}$ will now be recalculated from ${\varphi }_{pl}$ and the assumption $\overline{{\beta }_p}=3/5$: Rearranging, Substituting Equation (74) into Equation (42) yields which implies This agrees to five-digit precision with the established experimental value $7.2973526\times {10}^{-3}$ [35,36] rounded to the same level of precision. This is more than four times more accurate than calculated from Equation (59) of Reference [119]. Note that the fine-structure constant is known to increase with energy, and that the value obtained from Equation (75) and shown in Equation (76) is slightly larger than the experimentally established ground-state value. This is not the case with Equation (77). Could Eqns. (73) to (76) correspond to an energy above which an unbound GSQV nucleon transforms into a minimally excited state, as depicted in Figure 7?

The up and down quark charges may be estimated by substituting Equation (74) into Equations (47) and (48): and which are accurate to at least 5 digits compared to the exact SI base unit values, defined in 2019 [35,36]. Note that Equations (47) and (48) are simpler than Equations (78) and (79) and are presumed exact.

For most of the known unstable particles, decay times are well established [5]. However, very little is known about particle formation times [120]. It is plausible that a nucleon’s charge and mass creation, annihilation, and regeneration all occur on the timescale equal to the length of the nucleon’s zitterbewegung cycle. This is the time, $t_z$, for light to travel a nucleon Compton wavelength, ${\lambda }_N$: which is the same order of magnitude as the strong interaction timescale. This may conceptually align with energetic causal set theory–where stable particles may continually regenerate themselves at each moment in time [14,15,16,17,18,19,20,21].

Substituting Equation (42) into Equation (52) yields where represents the coupling between nucleon mass and each type of charge arc. Section 3.1.4 derived the ${\beta }_{\mathrm{arc}}$ values: ${\beta }_{po}=1$; ${\beta }_{pi}={\varphi }_{pl}^2/{\pi }^2$; ${\beta }_{no}=3/4$; and ${\beta }_{ni}=1/3$. Substituting these values into Equation (83), using the 2018 CODATA value, $\alpha =7.2973525693\left(11\right)\times {10}^{-3}$[35,36], and ${\varphi }_{pl}={cos}^{-1}(1-\sqrt{3}/2)$ yields the nucleon mass-to-charge coupling constants: and It is apparent from Equations (71) and (72) that an unbound GSQV neutron’s mass energy couples with its charge arcs only about 10% less than an unbound GSQV proton’s mass energy couples with its charge arcs. However, Equations (84) to (87) show that nucleon mass energy couples much more with outer charge arcs than inner charge arcs, and that this coupling ratio is more than twice as much for the unbound GSQV proton as it is for the unbound GSQV neutron: Also note that which implies that GSQV neutron outer charge arcs are about 34 times less coupled with mass energy compared to GSQV proton outer charge arcs.

When minimally excited above its ground state, about 1-2% of an unbound GSQV nucleon’s mass energy may couple with the Higgs field and transfer from the GSQV nucleon’s central zitterbewegung fermion to its system of revolving formerly-massless charge arcs. As proposed in Reference [9], for the proton, this process may initially form three valence quarks and a virtual $\mathrm{u}\overline{\mathrm{u}}$ quark pair that initializes the formation of a virtual pion cloud.

A neutral pion exists as the slightly imbalanced superposition of bound $\mathrm{u}\overline{\mathrm{u}}$ and $\mathrm{d}\overline{\mathrm{d}}$ quark pairs [110]. The imbalance is due to unequal bare quark masses. Since bare down quarks are more massive than bare up quarks [5], it will be assumed that the GSQV proton’s precursor, to a virtual neutral pion, is purely the lower energy $\mathrm{u}\overline{\mathrm{u}}$ quark pair. Note that this paper assumes that the virtual $\mathrm{u}\overline{\mathrm{u}}$ quark pairs shown in Figure 7 are initially unbound sea quarks.

In addition to a minimally excited unbound GSQV proton containing an initial unbound $\mathrm{u}\overline{\mathrm{u}}$ virtual quark pair, which may bind and initialize a virtual neutral pion, the same unbound $\mathrm{u}\overline{\mathrm{u}}$ virtual quark pair may occasionally transform into a charm quark, c, and an anticharm quark, $\overline{\mathrm{c}}$. This unbound $\mathrm{c}\overline{\mathrm{c}}$ quark pair may momentarily bind with the proton’s three valence quarks to form bound states such as $\left|\mathrm{uudc}\overline{\mathrm{c}}\right.〉$[1,40,41,42,43,44,45,46,47,48,49,50]. The instrinsic charge values do not change, since up and charm quarks are like-charged. However, a substantial amount of mass energy will need to be temporarily borrowed from, and returned to, the local vacuum. Therefore, a minimally excited unbound GSQV proton model may partly explain the experimental evidence for the proton’s probabilistically small intrinsic $\mathrm{c}\overline{\mathrm{c}}$ component [1,40,41,42,43,44,45,46,47,48,49,50].

The unbound GSQV proton model may explain charm quark mass as follows: With both unbound GSQV nucleon models, charge is assumed to be regenerated by twin poloidal revolutions phase-locked with each other. Another assumption, is that at each moment in time, at most one inner and one outer charge arc can be regenerated. Therefore, whenever both phase-locked poloidal circulations are on an outer arc, only one can regenerate charge. This explains how the polar charge-exclusion zones form. With the unbound GSQV proton model, the uncharged polar poloidal circulation is divided into two paths as shown in Figure 6.

For the unbound GSQV neutron model, this phase locking prevents the twin poloidal vectors from overlapping. However, for the unbound GSQV proton model, the two poloidal vectors will occasionally overlap. As illustrated in Figure 2, two overlaps will occur almost simultaneously–half a poloidal cycle apart. Suppose each of these twin overlapping lengths momentarily forms a fundamental harmonic standing wave of vacuum energy. Each of these standing waves will be of wavelength which is proportionately shorter than the wavelength of the fundamental harmonic ensemble regenerating the charge arcs, $2R_p=2{\lambda }_p/\sqrt{3}$, by the factor The unbound GSQV proton’s virtual optimal Möbius band relationship between charge and mass generation is assumed to hold. This momentary shortening of the $R_p$ vector will be assumed additive and equivalent to proton mass energy, $m_p\approx 938.272{\mathrm{MeV}/\mathrm{c}}^2$[5,35,36], increasing by the factor ${(\sqrt{3}-1)}^{-1}$. This will appear as a momentary increase in proton mass energy by the amount Since there are two momentary overlaps, the proton mass increase should appear to be twice this amount. This is more than enough mass energy to generate the charm and anticharm quark "running" masses [5].

Since the charge associated with this momentary increase in proton mass energy already exists in the form of a virtual unbound $\mathrm{u}\overline{\mathrm{u}}$ quark pair, the bare mass of an up quark [5] will be subtracted. Up quark bare mass at the charm quark "running" mass renormalization scale will be applied [5]. At the energy of the charm quark "running" mass, the up quark mass is close to 25% larger than that at $2.0\mathrm{GeV}$[5]. In Section 3.3, Equation (103) estimates up quark mass as $m_u=2.{30}_{-0.10}^{+0.23}{\mathrm{MeV}/\mathrm{c}}^2$ on the $2.0\mathrm{GeV}$ scale. This becomes $m_u=2.{88}_{-0.13}^{+0.29}{\mathrm{MeV}/\mathrm{c}}^2$ when increased by 25%. We therefore estimate the mass energy of each intrinsic charm or anticharm quark as Both Equation (92) and Equation (93) are well within the 2022 recommended Particle Data Group range: $m_c=1270\pm 20{\mathrm{MeV}/\mathrm{c}}^2$ [5].

The twin charge-regenerating vectors are poloidally phase locked. However, they are generally in different planes which rotate toroidally with the charge arcs. This is because the equators of the inner and outer charge arcs rotate toroidally at the same speed (light speed), which implies a different toroidal frequency for each charge arc type. As shown in Figure 2, the twin charge-regenerating vectors align perfectly only on the zitterbewegung equator.

Suppose the twin charge-regenerating vectors become entangled whenever both vector tips, which is where charge is regenerated, are no farther from the zitterbewegung equator than the zitterbewegung radius, $r_{cz}$[9], of the charm quark "running" mass, $m_c$[5]. Also suppose that charm quark mass-energy is generated during such entanglement. During this charm quark mass-energy generation, the maximum angular separation, ${\theta }_{\mathrm{ent}}$, between each $R_p$ vector and the zitterbewegung equator will be It will be assumed that this angular separation will uniformly manifest as any combination of poloidal and toroidal misalignments. It is well known that such a uniform angular distribution will cover angular area $2\pi (1-cos{\theta }_{\mathrm{ent}})$. Since there are $4\pi$ steradians in a sphere, this angular area is of the total spherical angular area, where the identity $cos\left({sin}^{-1}\left(x\right)\right)=\sqrt{1-x^2}$ has been applied. This suggests that the $\left|\mathrm{uudc}\overline{\mathrm{c}}\right.〉$ bound state occurs about 0.26% of the time. Since this momentary state includes two charm quark masses, and each charm quark "running" mass [5] is significantly larger than the proton mass, we estimate the fraction of the proton momentum carried by intrinsic charm quarks as where the uncertainty is propagated from the 2022 recommended Particle Data Group range: $m_c=1270\pm 20{\mathrm{MeV}/\mathrm{c}}^2$[5]. Our estimate is consistent with the value $(0.62\pm 0.28)\%$, reported in Reference [1], and at least an order of magnitude more precise. Note that all quantities calculated from proton mass, $m_p$, are time-averaged. The momentary intrinsic charm and anticharm quarks are assumed to exchange energy and momentum with the time-averaged proton energy and momentum. The Heisenberg uncertainty principle allows intrinsic fluctuations of proton energy and momentum on short timescales [121,122].

Except for the unbound GSQV neutron’s outer charge arc pair, all other unbound nucleon charge arc pairs must merge before decomposing into the quark charge values shown in Figure 7. Such merging presumably momentarily forms charge shells [9]. It can be seen from the right-hand column of Panel (a) of Figure 7 that the two neutron outer charge arcs do not necessarily need to merge before decomposing into the four quarks shown. 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.

The remaining 98-99% of a minimally excited unbound GSQV nucleon’s mass energy is proposed to transform into gluon flux tubes and quark relativistic kinetic energy [123]. This paper proposes that the curvature of the unbound GSQV charge arcs and charge shells prevents the initial formation of gluon flux tubes between quarks on the same arc or shell. It will therefore be assumed that initially two gluon flux tubes form between inner and outer valence quarks. In particular, each of two outer valence quarks connects to the same inner valence quark via a gluon flux tube. It will be assumed that this initial double gluon flux tube structure immediately transforms into the three-way symmetric gluon flux tube structure described by the Standard Model of particle physics [123].

The nucleon’s spin angular momentum may partially migrate from the gluon flux tubes to the valence quarks [9]. This partial migration may be interaction-dependent and provide a conceptual clue to the nucleon spin crisis [9,124,125,126].

For the minimally excited unbound GSQV proton, Panel (b) of Figure 7 illustrates the transition from massless charge arcs to massive quarks. The proton’s two up valence quarks originate from its outer charge arcs, while the proton’s down valence quark originates from its inner charge arcs. Comparing Figure 1 and Figure 2, clearly the proton’s inner arcs are less symmetrical than its outer arcs. Therefore, the proton’s up-quark probability distribution should be more symmetrical than its down-quark probability distribution. This expected effect has been confirmed to occur in recent lattice QCD computations [34].

For the minimally excited unbound GSQV neutron, Panel (a) of Figure 7 illustrates the transition from massless charge arcs to massive quarks. The neutron’s two down valence quarks originate from its outer charge arcs, while the neutron’s up valence quark originates from its inner charge arcs. The minimally excited unbound GSQV neutron contains two unbound $\mathrm{u}\overline{\mathrm{u}}$ quark pairs that are proposed to initialize the formation of a virtual pion cloud.

The long-established neutron decay mechanism is for a down valence quark to decay to an up valence quark via the emission of a ${\mathrm{W}}^{-}$ boson or absorption of a ${\mathrm{W}}^{+}$ boson [127]. Both processes are equivalent and emit an electron and an electron antineutrino. The unbound GSQV nucleon models offer the following similar, but more detailed, mechanism. Suppose that the weak interaction [110] involves one of the unbound GSQV neutron’s outer charge arcs either transforming into a ${\mathrm{W}}^{-}$ boson or absorbing a ${\mathrm{W}}^{+}$ boson. Note that in the minimally excited GSQV neutron model, illustrated in Panel (a) of Figure 7, each outer charge arc initially transforms into an unbound $\overline{\mathrm{u}}\mathrm{d}$ quark pair. The Standard Model allows a W boson to both form from, and decay into, a first-generation quark pair [123]. Examples include leptonic decay of a charged pion and charged kaon decay into three charged pions. It is therefore plausible that the weak interaction [110] applies to the absolute ground state of the GSQV neutron model as well as its minimally excited state illustrated in Panel (a) of Figure 7.

Panel (b) of Figure 7 shows that the minimally excited unbound GSQV proton model does not contain an intrinsic $\overline{\mathrm{d}}$ quark (charge $+e/3$). The absence of an intrinsic $\overline{\mathrm{d}}$ sea quark may be associated with proton stability. An $\overline{\mathrm{d}}$ quark would presumably be needed to either form and emit a ${\mathrm{W}}^{+}$ boson from an $\mathrm{u}\overline{\mathrm{d}}$ quark pair, or annihilate with an $\overline{\mathrm{u}}\mathrm{d}$ quark pair that decomposes from an absorbed ${\mathrm{W}}^{-}$ boson. Note that the electron capture [107,108,109,110] and positron emission [128] processes may generally involve a W boson interacting with a loosely bound proton’s inner charge region, which is more likely to occur with the assistance of other bound nucleons [111]. Section 3.1.1 discusses some details of how this paper’s GSQV nucleon models explain aspects of electron capture.

Once an unbound GSQV neutron has emitted a ${\mathrm{W}}^{-}$ boson, or absorbed a ${\mathrm{W}}^{+}$ boson, it will resemble an inside-out unbound GSQV proton. Such an inside-out configuration should be electromagnetically unstable. All other parameters remaining unchanged, Equation (18) will now be modified by removing $-e$ from the neutron’s outer charge arcs. This halves the charge on the outer arcs, so that the former neutron’s magnetic moment becomes which is of opposite magnetic polarity to the neutron. This magnetic moment is also considerably weaker than that of either free nucleon.

Suppose that electrostatic effects dominate the dynamics of this magnetically weakened environment. The former neutron’s inner charge region will have a far greater charge density than its outer charge region. It will be assumed that this causes the inner positive charge region to expand outward, which should result in an increasingly positive magnetic moment. This outward expansion could conceivably stop once the perturbed nucleon’s magnetic moment becomes strong enough to contain the charge structures via the unbound GSQV nucleon’s magnetic self-interaction. Details of such self-interaction are subjectively described in Reference [9]. It will be assumed that the expansion of the inner positive charge region will not stop until after it has completely passed through the negative charge region. This new structure is assumed to stabilize once it becomes identical to that of an unbound GSQV proton–with its twin virtual optimal Möbius bands [10,11,12,13]. See Section 3.1 and Appendix A for details.

For the decay process of a minimally excited GSQV neutron, it will be assumed that a similar electromagnetic restructuring occurs. In addition, the trilateral gluon flux tube structure must disconnect from the down valence quark that combines with an intrinsic antiup sea quark. This unbound $\overline{\mathrm{u}}\mathrm{d}$ quark pair will presumably either form and emit a ${\mathrm{W}}^{-}$ boson or annihilate with a $\mathrm{u}\overline{\mathrm{d}}$ quark pair that decomposes from an absorbed ${\mathrm{W}}^{+}$ boson. The trilateral gluon flux tube structure then presumably connects with an intrinsic up sea quark, causing it to transform into a valence up quark.

The Standard Model assumes that a valence quark may change flavor by emitting or absorbing a ${\mathrm{W}}^{+}$ or ${\mathrm{W}}^{-}$ boson. Other quarks are not directly involved. The ${\mathrm{W}}^{+}$ and ${\mathrm{W}}^{-}$ bosons share the same mass, $m_{\mathrm{W}}$, which may be calculated from other Standard Model parameters. Reference [129] recently calculated the following Standard Model prediction of the W boson mass: which is statistically consistent with the ATLAS Collaboration’s recent experimental determination [130], Note that Reference [129] performs a global fit of electroweak data within the Standard Model without directly incorporating the first-generation quark masses.

This paper proposes the following unbound neutron decay process as an extension of the Standard Model:

Note that this paper does not attempt to explain the absolute or relative values of the nucleon masses. This paper proposes that the W boson of the Standard Model should be amended by adding an up quark mass and a down quark mass. During leptonic decay of a charged pion [131], it is plausible that the W boson mass actually includes the bare masses of the pion’s two quarks: one up or antiup; one down or antidown. For most W boson interactions, the principle applied by this paper is that quark flavor change necessarily involves up and down sea quarks. This implies that a second or third generation quark transforms into a first generation quark before emitting or absorbing a W boson. The converse also applies and each generation change will involve an exchange of mass and vacuum energies.

This paper assumes that the intrinsic properties of unbound nucleons are determined at the energy scale of nucleon-antinucleon pair production, which is about $2\mathrm{GeV}$. At this energy scale, the sum of the up and down bare quark masses is approximately $6.9{\mathrm{MeV}/\mathrm{c}}^2$[5], with a much smaller absolute uncertainty than of the W boson mass estimates in Equations (98) and (99). This paper therefore proposes amending Equation (98) by adding $6.9{\mathrm{MeV}/\mathrm{c}}^2$: Note that the means of Equations (99) and (100) differ by only $5.1{\mathrm{MeV}/\mathrm{c}}^2$, which is less than both the statistical uncertainty of Equation (99) and the uncertainty of Equation (100). Clearly the neutron decay model proposed by this paper agrees substantially better with the recent analysis of the ATLAS Collaboration [130] than does the Standard model. The ATLAS Collaboration is expected to produce future W boson mass estimates with substantially lower uncertainty. It is possible that these future estimates could agree substantially more with the Standard Model than the model proposed by this paper. Such an outcome could falsify key aspects of the modeling proposed by this paper.

It is often speculated that the down quark mass, at the precise energy threshold of nucleon-antinucleon pair production, may be precisely 9 electron masses [5]: which closely agrees with the central estimate in Equation (103).

This paper will apply this hypothesis and also propose that the down quark mass, at the precise energy threshold of nucleon-antinucleon pair production, is precisely the difference between the charged and uncharged pion masses [5,132]: Note that Equations (104) and (105) differ by more than 11 standard deviations, and so are unlikely to be consistent with each other. At the precise energy threshold of nucleon-antinucleon pair production, suppose that virtual neutral pions are in equilibrium with virtual photons. Also suppose that the formation of a virtual charged pion pair requires the temporary borrowing of vacuum energy equal to the mass energy of an unbound bare $\mathrm{d}\overline{\mathrm{d}}$ quark pair. A pair of virtual charged pions will then be formed from the d quark (charge $-e/3$) binding with an intrinsic $\overline{\mathrm{u}}$ quark and the $\overline{\mathrm{d}}$ quark (charge $+e/3$) binding with an intrinsic u quark. If this occurs, it is plausible that from which Equation (105) follows. Note the complete absence of antidown quarks (charge $+e/3$) and absence of sea down quarks (charge $-e/3$) in Figure 7, where no pions are present. Pions are assumed to form at higher energies.

In the minimally excited GSQV nucleon models, depicted in Figure 7, it is possible–but not necessary–that each unbound up and antiup quark possesses relativistic energy equaling the mass energy of a neutral pion. This may explain how these quarks are able to initiate charged pion production. Reference [106] found that non-single-pion photoproduction off the proton amounts to about 10%, whereas non-single-pion photoproduction off the neutron amounts to about 44%. This large difference may be partly explained by the minimally excited GSQV neutron containing twice as many charged pion-production sites, in the form of virtual up and antiup sea quarks, as the minimally excited GSQV proton.

Section 3.1.1 defined and calculated a parameter, ${\delta }_n$, which represents a slight perturbation of the unbound GSQV neutron’s charge distribution along each charge arc. This perturbation will be assumed due to the interaction between the radial vacuum standing wave ensemble, maintaining each charge arc, and the charge arc itself.

During Steps 3 & 4 of the unbound neutron decay process described in Section 3.2.4, new charge arcs form with radius, $R_{n^{\prime }}$, equal to the Compton wavelength of the decaying neutron’s temporarily reduced mass divided by $\sqrt{3}$. It will be assumed, that during Steps 3 & 4, the charge perturbation parameter changes with the inverse square of the charge arc radius. The reason for this inverse square relationship is as follows. The wavelength of the radial standing wave ensembles will increase linearly with charge arc radius, which will cause their energies to decrease linearly with arc radius. In addition, the energy density of these radial standing wave ensembles, along each charge arc, will dilute linearly with charge arc radius. The charge perturbation parameter is assumed proportional to this energy density, which declines with the square of charge arc radius.

It will be assumed that the charge perturbation parameter, ${\delta }_{n^{\prime }}$, does not change further during Steps 5 & 6. Therefore define unbound proton charge perturbation parameter, since Compton wavelength is inversely proportional to mass and this paper assumes $m_u+m_d=\frac{3}{2}{m}_d$.

Reference [9] combines the zitterbewegung and circular Unruh effects to propose that proton mass energy is quantized as approximately $2\sqrt{3}/0.2855\approx 98.9\%$ of the median thermal excess vacuum energy due to the circular Unruh effect. The approximate value $0.2855$ is obtained by numerically integrating the Planck spectrum and the relationship $k_{B}T/0.2855\approx 3.503k_{B}T$, where $k_B$ is the Boltzmann constant and T is temperature. A more precise numerical integration yields median thermal energy to be approximately $3.50302k_{B}T$ [133]. Reference [133] argues that the median energy is the most physically meaningful numerical criterion for the thermal spectrum peak, and that this should be more widely taught.

In the early universe, unbound nucleons initially formed during the hadron epoch. Early in the hadron epoch, each unbound nucleon was completely surrounded by the quark-gluon plasma (QGP) from which it formed [134]. Prior to neutrino decoupling, protons were in equilibrium with neutrons and antiprotons were in equilibrium with antineutrons [134].

This paper proposes that each proton-neutron pair, and each antiproton-antineutron pair, of quantum vortices generated more than enough circular Unruh energy for the mass energy of each nucleon. It is proposed that these quantum vortices generated additional circular Unruh energy amounting to the bare mass energies of the nucleon pair’s six valence quarks. This excess energy is proposed to replace the mass energies of the QGP quarks from which the nucleon pair formed. It is therefore proposed that where $T_p$ and $T_n$ respectively represent the circular Unruh temperatures of the proton and neutron. Reference [9] shows that $T_p=\frac{m_p{c}^2}{2\sqrt{3}{k}_B}$, which implies $T_n=\frac{m_n{c}^2}{2\sqrt{3}{k}_B}$. Substuting into Equation (108) yields Rearranging and substituting proton and neutron mass values in units of ${\mathrm{MeV}/\mathrm{c}}^2$ [5,35,36] yields which is less than $2\%$ above the central estimate in Equation (101). Since this paper assumes $m_u=m_d/2$, which is only about $2\%$ above the $m_d$ values calculated from Eqns. (103), (104), and (105). The consistency between Eqns. (103), (104), (105), and (111) is encouraging. The additional $2\%$ calculated by Equations (110) and (111) may be due to an intrinsic heavy quark component [1,40,41,42,43,44,45,46,47,48,49,50].

Following Reference [9], Section 3.1.1, and Section 3.3.2, proton magnetic moment will be calculated from where $V_{pi}$ is the lemon volume formed by rotating the proton’s two inner charge arcs about the proton center, $V_{po}$ is the volume formed by rotating the proton’s two outer charge arcs, with flat end caps, about the proton center, $Q_{p\mathrm{ex}}$ is the uncharged proportion of the proton’s outer surface, $R_p={\lambda }_p/\sqrt{3}$, and Substituting Equations (1) and (113) into Equation (112) yields