Without the existence of stable protons composite particles like nuclei, atoms and molecules, but also the full complexity of nature could not have been developed. Therefore, there is a deep interest to understand the nucleon structure, in particular the origin of its stability. During the last decades large experimental and theoretical efforts have been made to understand this problem. Experimentally, large sets of data have been taken on hadron and lepton scattering and reactions. In comparison to hadrons, more detailed and reliable information has been expected from electromagnetic probes [1] . Therefore, data on electromagnetic form factors have been collected up to large momentum transfers in electron scattering from the proton [2] - [7] and neutron [8] - [11] .

Concerning the properties of nucleons, the proton has an exponentially falling charge density with a root mean square radius of about 1 fm, whereas the neutron has a vanishing charge density (integrated over full space) with regions of radius of positive and negative charge. Further, nucleons have a magnetic dipole moment, M1 = 2.793 for the protons and 1.913 for the neutron, which has given early evidence that nucleons are complex objects.

The root mean square radius of the charge density of the proton has been determined from electron scattering and lamb shifts in muonic hydrogen to be about 0.88 fm, but later revised to 0.84 fm [12] . But the electron form factors measured up to large momentum transfers could suggest an even smaller proton radius. A rather small radius has been deduced also from 4 GeV α-p scattering [MSD], but with larger uncertainties due to strong absorption effects. Another interesting fact, in an early empirical analysis of its electromagnetic structure by Kelly [13] , it has been found that the charge and magnetization densities of the proton should be different. This has been nicely confirmed by polarization experiments of electromagnetic form factors [14] [15] , but up to date this observation has never been satisfactorily explained.

Theoretically, the nucleon problem has been studied during decades in many different models and was subject of numerous conferences and specialized workshops. Its structure has been discussed in several versions of the quark model [16] [17] , in various Bag models [18] [19] , in descriptions inspired from quantum chromodynamics, see ref. [20] , by using the Dyson-Schwinger equation [21] , and in many empirical approaches. In particular, the confinement problem has not been understood [22] - [24] as well as the question of the enormous stability of the proton. In the flux-tube model [25] [26] it was assumed that interacting coloured bosons (gluons) form a plasma, by which the fermions are firmly confined, whereas in string models [27] the fermions should be bound by boson strings.

Significant progress in the understanding of particle properties has been made within the Standard Model of particle physics, see e.g. ref. [20] , which is composed of gauge invariant quantum theories of electromagnetic, strong and weak interactions. In particular, quantum chromodynamics, its strong interaction part, has been expected to be well suited for the description of the nucleon. Calculations on the proton have been performed on the lattice by adjusting several parameters, but without considering its important electromagnetic structure. A complete calculation of all nucleon properties, including magnetic moments or form factors is not possible, because in this model the radial degree of freedom is not developed explicitly. Therefore, more recent work on nucleon properties has been based on empirical descriptions, see e.g. [28] .

During the last decade, an alternative bound state formalism has been developed, based on a quantum field theory, in which in addition to fermions also boson fields have been introduced [Mo1]. This formalism [Mo2] satisfies relativity explicitly, because it contains the space (time) degree of freedom. In addition, it has very few parameters—determined all by basic conservation laws—and has been applied successfully to hadrons and atoms, but also to leptons and gravitational systems [Mo1-Mo6]. If also a quantitative description of the properties of nucleons can be achieved, this should give clear answers, why the proton is stable.

The underlying Lagrangian may be written in the form

$ℒ=\frac{1}{{\tilde{m}}^2}\left(\bar{\Psi }{D}_{\nu }\right)i{\gamma }^{\mu }{D}_{\mu }\left(D^{\nu }\Psi \right)-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$(1)

where $\tilde{m}$ is the mass parameter and Ψ are fermion fields, $\Psi ={\Psi }^{+},{\Psi }^{-}$. Further, vector boson fields $A_{\mu }$ with coupling g between the fermions are contained in the covariant derivatives $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$. The second term of the Lagrangian represents the Maxwell term with Abelian field strength tensors $F^{\mu \nu }$ given by $F^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$, which gives rise to both electric and magnetic coupling.

From this Lagrangian a bound state formalism can be deduced by calculating matrix elements $ℳ\left(q\right)~{\widetilde{ℒ}}_1\left(q\right){ℒ}_1\left(q\right)$ in momentum space, in which the product of the overlapping fermion and boson fields Ψ and $A_{\mu }$ are replaced by dimensionless wave functions $\psi \left(q\right)$ and $w\left(q\right)$—which can be considered as the square root of their normalized probabilities in momentum space. For nucleons the fermion wave functions are given by $\psi \left(q\right)~\frac{1}{{\tilde{m}}^{9/2}}\left(\bar{\Psi }\Psi \right)\Psi$. The boson wave functions are given by $w\left(q\right)~\frac{1}{{\tilde{m}}^2}{A}_{\mu }{A}^{\nu }$, but one pair of boson fields has to be considered as interaction of vector structure $v_v\left(q\right)=\pm \alpha \frac{1}{\tilde{m}}{A}_{\mu }{A}^{\nu }$, acting between fermions and between bosons. The complete formalism is discussed in refs. [Mo1, Mo2] and can be applied to hadrons, leptons, atoms, but also to gravitational bound states.

Assuming spherical symmetric systems, the wave functions can be transformed to normal ( $r,\theta ,\Phi$) space by Fourier transformation $\varphi \left(r\right)=2\pi {\displaystyle \int q^2\text{d}q{j}_o\left(qr\right)\varphi \left(q\right)}$, where $j_o\left(qr\right)=\sin \left(qr\right)/\left(qr\right)$ and $\varphi \left(q,r\right)$ are fermion or boson wave functions.

Important to point out that for any basic bound state system there are always two fermion and boson wave functions ${\psi }_{s,v}\left(r\right)$ and $w_{s,v}\left(r\right)$ of scalar and vector structure (with scalar and vector coupling between their constituents) and of similar radial structure

${\psi }_{s,v}\left(r\right)~w_{s,v}\left(r\right).$(2)

Since the integration of fermions and bosons goes over 3-dim. and 2-dim. space, respectively, this yields normalization conditions $4\pi {\displaystyle \int r^2\text{d}r}{\psi }_{s,v}^2\left(r\right)=2\pi {\displaystyle \int r\text{d}r}{w}_{s,v}^2\left(r\right)=1$.

The form of the boson wave function of the scalar state is given by

$w_s\left(r\right)=w_{s_o}\exp \left\{-{\left(r/b\right)}^{\kappa }\right\},$(3)

whereas that of the vector state is given by

$w_v\left(r\right)=w_{v_o}\left[w_s\left(r\right)+\beta R\frac{\text{d}{w}_s\left(r\right)}{\text{d}r}\right].$(4)

The factors $w_{s_o}$ and $w_{v_o}$ are obtained from the normalization (see above), whereas $\beta R$ is given by $\beta R=-{\displaystyle \int r^2\text{d}r}{w}_s\left(r\right)/{\displaystyle \int r^2\text{d}r}\left[\text{d}{w}_s\left(r\right)/\text{d}r\right]$, so that both states are orthogonal, with the condition ${\displaystyle \int r\text{d}r}{w}_s\left(r\right)w_v\left(r\right)=0$.

The reasons, why the same form of wave functions is needed for very different systems, is due to two facts: first a geometric constraint

${\displaystyle \int \text{d}{r}^{\prime }}{w}_v\left(r^{\prime }\right)v_v\left(r-r^{\prime }\right)w_v\left(r^{\prime }\right)~w_s^2\left(r\right),$(5)

second, due to the two and three-dimensional structure of bosons and fermions in Equation (2).

Fermion binding energies are given for $N_f$ fermion bound states by matrix elements of the form

$E_{ng}^{s,v}=4\pi N_f{\displaystyle \int r^2\text{d}r}{\psi }_{s,v}\left(r\right)V_{ng}^{s,v}\left(r\right){\psi }_{s,v}\left(r\right),$(6)

with two boson potentials for n = 2, 3

$V_{2g}^{s,v}\left(r\right)=\frac{\alpha \left(2s+1\right){\left(\hslash c\right)}^2}{8\tilde{m}}\left(\frac{{\text{d}}^2{w}_s\left(r\right)}{\text{d}{r}^2}+\frac{2}{r}\frac{\text{d}{w}_s\left(r\right)}{\text{d}r}\right)\frac{1}{w_s\left(r\right)}+V_o$(7)

with s = 0 for scalar and s = 1 for vector states and

$V_{3g}^{s,v}\left(r\right)=\frac{{\alpha }^2\left(\hslash c\right)}{\tilde{m}}{\displaystyle \int \text{d}{r}^{\prime }}{w}_{s,v}\left(r^{\prime }\right)v_v\left(r-r^{\prime }\right)w_{s,v}\left(r^{\prime }\right),$ (8)

with an interaction $v_v\left(r\right)=-\alpha \left(\hslash c\right)w_v\left(r\right)$ for $w_v\left(r\right)\ge 0$ and $v_v\left(r\right)=0$ for $w_v\left(r\right)<0$.

The potential $V_{2g}^{s,v}\left(r\right)$ can be identified with the “confinement” potential (with a quite linear shape towards large radii), needed in meson spectroscopy [22] - [24] . $V_o$ is a small constant, see ref. [Mo2], which may be related to a binding of bosons in the vacuum. The potential $V_{3g}^{s,v}\left(r\right)$ represents a boson-exchange potential.

Kinetic energies are given by

$T_{ng}^{s,v}=\frac{4\pi }{2}{\displaystyle \int r^3\text{d}r}{\psi }_{s,v}\left(r\right)V_{ng}^{Ts,v}\left(r\right)\frac{\text{d}{\psi }_{s,v}\left(r\right)}{\text{d}r}$(9)

with

$V_{1g}^{Ts,v}\left(r\right)=\frac{\alpha \left(2s+1\right){\left(\hslash c\right)}^3}{4{\tilde{m}}^2}\left(\frac{{\text{d}}^2{w}_s\left(r\right)}{\text{d}{r}^2}+\frac{2}{r}\frac{\text{d}{w}_s\left(r\right)}{\text{d}r}\right),$(10)

where $x=1,3$ for scalar and vector state, respectively, and

$V_{2g}^{Ts,v}\left(r\right)=\frac{{\alpha }^2{\left(\hslash c\right)}^2}{\tilde{m}}{w}_{s,v}\left(r\right)w_{s,v}\left(r\right).$(11)

In addition, there is an energy of acceleration of the form

$\text{Δ}{E}^{s,v}=\frac{4\pi }{2}\alpha \left(\hslash c\right){\displaystyle \int r^4\text{d}r}{\psi }_{s,v}\left(r\right)w_{s,v}\left(r\right)\frac{{\text{d}}^2{\psi }_{s,v}\left(r\right)}{\text{d}{r}^2}.$(12)

In simple hadronic systems, as discussed in ref. [Mo2], this term corresponds to spurious motion and does not contribute to the mass. But in complex systems, as the present one, the total binding energy is given by

$M_{s,v}=-\left[E_{ng}^{s,v}+T_{ng}^{s,v}+\text{Δ}{E}^{s,v}\right].$(13)

In addition to the fermion contributions there are boson matrix elements, leading to energies

$E_g^{s,v}=2\pi {\alpha }^2{N}_g{\displaystyle \int r\text{d}r}{w}_{s,v}\left(r\right)v_v\left(r\right)w_{s,v}\left(r\right),$(14)

kinetic energies

$T_g^{s,v}=2\pi \frac{{\alpha }^2\left(\hslash c\right)}{4}{\displaystyle \int r^2\text{d}r}{w}_{s,v}\left(r\right)\frac{1}{r}\frac{\text{d}{w}_{s,v}\left(r\right)}{\text{d}r}$(15)

and a contribution from acceleration

$\text{Δ}{E}_g^{s,v}=2\pi \frac{{\alpha }^2\left(\hslash c\right)}{8}{\displaystyle \int r^2\text{d}r}{w}_{s,v}\left(r\right)\frac{{\text{d}}^2{w}_{s,v}\left(r\right)}{\text{d}{r}^2}.$(16)

Then, the total boson energy is

$E_{g,tot}^{s,v}=E_g^{s,v}+T_g^{s,v}+\text{Δ}{E}_g^{s,v}.$(17)

Important to note, the total fermion and boson energy of particles has to cancel each other, $E_{f,tot}+E_{g,tot}=0$, see ref. [Mo2], which indicates a coupling to the vacuum.

In the above formalism there are only three parameters for electric binding, the shape and slope parameters $\kappa$ and b and the coupling constant $\alpha$. For magnetic bound states there is another parameter, a relative velocity factor (v/c). It turned out that for all systems studied $\kappa =1.375$ and $\alpha =2.158$ lead to optimal results. The remaining parameters b and (v/c) are well determined by energy and momentum conservation for scalar and vector states

$\left[{\langle q_f^2\rangle }^{1/2}-{\langle q_g^2\rangle }^{1/2}\right]\left(v/c\right)~\left(E_{f,tot}+E_{g,tot}\right),$(18)

where ${\langle q_f^2\rangle }^{1/2}={\left[{\displaystyle \int q^4\text{d}q}{ℳ}_{3g}\left(q\right)/{\displaystyle \int q^2\text{d}q}{ℳ}_{3g}\left(q\right)\right]}^{1/2}$ is the root square momentum of fermions and ${\langle q_g^2\rangle }^{1/2}={\left[{\displaystyle \int q^3\text{d}q}{ℳ}^g\left(q\right)/{\displaystyle \int q\text{d}q}{ℳ}^g\left(q\right)\right]}^{1/2}$ that of bosons. The matrix elements ${ℳ}_{3g}\left(q\right)$ and ${ℳ}^g\left(q\right)$ are the Fourier transformed values of $M_{3g}\left(r\right)=4\pi N{V}_{3g}\left(r\right){\psi }^2\left(r\right)$ and $M^g\left(r\right)=2\pi N{v}_v\left(r\right)w^2\left(r\right)$.

Since ${\langle q_f^2\rangle }^{1/2}$ and ${\langle q_g^2\rangle }^{1/2}$ are different, the energies have to be modified by recoil corrections $rec=\mp f_{rec}\left({\langle q_f^2\rangle }^{1/2}-{\langle q_g^2\rangle }^{1/2}\right)/\left({\langle q_f^2\rangle }^{1/2}+{\langle q_g^2\rangle }^{1/2}\right)$, with sign negative for fermions and positive for bosons. Because we are dealing here with two states, we simply use $E_f^{corr}=\left(1-{f^{\prime }}_{rec}\right)E_f$ and $E_g^{corr}=\left(1+{f^{\prime }}_{rec}\right)E_g$ with ${f^{\prime }}_{rec}$ ad-justed separately to scalar and vector states.

An additional constraint is provided by a mass-radius condition, which is given by

$Ra{t}_{2g}=\frac{{\left(\hslash c\right)}^2{\left(v/c\right)}^2}{\tilde{m}\left(M_s/2\right)\langle r_{g_s}^2\rangle }=1.$(19)

Interestingly, full relativity is automatically included in the present formalism: the mean momentum is equal to the mass, see Equation (18), and consequently the Fourier transformation of the momentum yields a correlation between space and time. The validity of space-time is also directly evident from the mass-radius condition (19).

For systems with a mass of about 0.94 GeV simple $\left(f\bar{f}\right)f$ bound state solutions¹ as those discussed in ref. [Mo2] do not exist, because energy-momentum conservation is not fulfilled. Further, the measured electromagnetic form factors of proton and neutron are different, which suggests a mixing of states. Since in the present formalism the mass of the vector state $M_v$ is slightly higher than three times the mass $M_s$ of the scalar state, a mixing of a scalar $3\cdot \left(f\bar{f}\right)f$ bound state with a vector $\left(f\bar{f}\right)f$ state (with quite comparable masses) appears to be realistic. Writing the densities in the form ${\rho }_s\left(r\right)={\psi }_s^2\left(r\right)$ and ${\rho }_v\left(r\right)={\psi }_v^2\left(r\right)$, the charge components of proton and neutron can be written by

${\rho }_{prot}^e\left(r\right)=a_e{\rho }_s^e\left(r\right)+\left(1-a_e\right){\rho }_v^e\left(r\right)$(20)

${\rho }_{neut}^e\left(r\right)=\left(1-a_e\right){\rho }_s^e\left(r\right)-a_e{\rho }_v^e\left(r\right),$(21)

whereas the magnetic components are given by

${\rho }_{prot}^m\left(r\right)=a_m{\rho }_s^m\left(r\right)+\left(1-a_m\right){\rho }_v^m\left(r\right)$(22)

${\rho }_{neut}^m\left(r\right)=\left(1-a_m\right){\rho }_s^m\left(r\right)+a_m{\rho }_v^m\left(r\right).$(23)

Mixing terms do not contribute, since ${\psi }_s\left(r\right){\psi }_v\left(r\right)=0$.

For the basis states, slope parameters b and (v/c) (with common shape parameter $\kappa =1.375$ and coupling constant $\alpha =2.158$), radii and masses are given in Table 1 , as well as the number of bosons $N_g$ to get similar (absolute) boson and fermion energies. Further, the recoil corrections ${f^{\prime }}_{rec}$ were 0.30 and −0.02 for electric binding of scalar and vector states, and 0.26 and 0.14 for magnetic binding, respectively.

Resulting densities and potentials are shown in Figure 1 . In the upper part the densities are displayed, the boson-exchange and confinement potentials $V_{3g}^{s,v}\left(r\right)$ and $V_{2g}^s\left(r\right)$ are given in the middle and lower part, respectively. In the middle part the scalar density (rescaled) is shown again by dot-dashed line, which indicates that the geometric condition (5) is really satisfied.

The analysis of nucleons in a bound state formalism based on a fermion-boson symmetric Lagrangian has shown that a quantitative description of important nucleon properties could be achieved, assuming a mixing of a $3\cdot \left(f\bar{f}\right)f$ state with scalar coupling between its constituents with a corresponding $\left(f\bar{f}\right)f$ bound state of vector coupling. All basic parameters are constrained by boundary conditions on energy and momentum conservation; further, two mixing parameters are adjusted to get neutron charge neutrality and the correct magnetic moments of proton and neutron. The mixing of these basic states gives rise to quite different proton and neutron density distributions, first observed by Kelly but not really understood up to present.

Of particular interest is the high stability of the nucleon. Responsible for this property is the dynamically generated confinement potential (and the related second order boson potential), giving rise to negative/positive extra energies for electric binding of the proton/neutron and opposite for magnetic binding of these particles. This indicates that binding and repulsion (electric and magnetic) is present at the same time for both particles, which blocks the decay of these particles. Further, the detailed extra binding is different for both particles, which allows decay of the neutron to proton and electron. Small vacuum energies are present to get complete cancellation of all terms. This shows a rather complex mixing of the different components, very different from the binding of leptons, in which just a cancellation of the fermion and boson energies takes place.

Confinement has been indeed expected as the origin of the high stability of the proton in many empirical pictures, where it is expected to provide an attractive flux tube in extensions of the constituent quark model or leads to stable string tensions in higher dimensional string models.

Other systems as “strange” baryons ${\Sigma }^{\pm }$ and Λ may have a structure similar to nucleons. Thus, it would be interesting to see, whether these particles can be understood in a similar way and why these particles are not stable.

I am grateful to several colleagues, in particular to Benoit Loiseau for his help to establish details of the theory and to Thomas Sefzick for setting up the Web-presentation https://h2909473.stratoserver.com for on-line calculations of leptons and mesons.

[MSD] H.P. Morsch, W. Spang and P. Decowski, Folding study of α-p scattering: Systematics of elastic scattering, effective interaction and inelastic excitation of N* resonances, Phys. Rev. 67, 064001 (2003)

[Mo1] H.P. Morsch, Acceleration in a fundamental bound state theory and the fate of gravitational systems, J. Adv. Math. and Comp. Sc. 28(3): 1-13 (2018)

[Mo2] H.P. Morsch, Bound state description of particles from a quantum field theory of fermions and bosons, compatible with relativity, J. High Energy Physics, Gravity and Cosmology 10, 562 (2024)

[Mo3] H.P. Morsch, Unique structure of particle bound states, Brit. J. Math. and Comp. Sc. 17(6): 1-11 (2016)

[Mo4] H.P. Morsch, Fundamental bound state description of light atoms and the fine structure constant $\alpha ~1/137$, Boson J. Modern Phys. 3,1: 197 (2017)

[Mo5] H.P. Morsch, Lepton bound state theory based on first principles, J. Adv. Math. and Comp. Sc. 36(3): 118 (2021)

[Mo6] H.P. Morsch, Origin of gravitation and description of galaxy rotation in a fundamental bound state approach, Global J. Sci. Front. Research. A 18, 4 v.1, p. 25 (2018)

¹f indicating massless fermions (quantons). f is used here instead of q, not to be confused with quarks, massive fermions in quantum chromodynamics.