# Gravitational model of strong interaction

P1. The possible configuration of the lithium nucleus.

The gravitational model of strong interaction is the model, in which strong interaction is described by strong gravitation, the action of the gravitational torsion field and electromagnetic forces.

## The standard theory

In elementary particle physics the generally accepted model is the Standard model according to which strong interaction occurs on the scale, not much greater than the size of atomic nuclei. In this case, there are considered two different possible situations – strong interaction between nucleons (or between other hadrons), and strong interaction in the matter of hadrons. In the first case the pion-nucleon Yukawa interaction is often used, according to which the role of carriers of strong interaction between nucleons is played by virtual pions and other mesons. In the second case quantum chromodynamics (QCD) is involved, in which hadrons are composed of quarks, there are two quarks in each meson, and three quarks in baryons. Quarks interact by gluons and can not exist outside hadrons in the free form. Except two or three valence quarks, a hadron must contain gluon clouds surrounding quarks and seas of virtual particles such as quark-antiquark and electron-positron pairs, W and Z bosons. In QCD gluons are considered the carriers of strong interaction, and the interaction between nucleons is treated as a residual effect of the quark gluon fields that goes beyond hadrons. As a result, the forces between two nucleons must be much less than the forces between the quarks inside these nucleons.

At present the description of strong interaction between nucleons through virtual pions to some extent looks archaic and does not seem satisfactory. For example, it is unclear at what point between nucleons these virtual pions should appear, what is the mechanism of their origin and subsequent action. How can the momentum transfer from virtual pions either attract nucleons, or repel them depending on the distance between the nucleons? Due to what does the tensor component of nuclear forces, which is not purely radial, arise? From a philosophical point of view, reduction of the interaction between elementary particles to other elementary particles seems rather an artificial device, than description of the essence of phenomena.

QCD also has its own problems analyzed in the model of quark quasiparticles. Among the main problems are: introducing into the Standard theory many new unexplained entities and adjustable parameters; considering interactions as point events with quarks and bosons of point sizes and subsequent discrepancy in solutions; unobservability of free quarks and gluons, indicating that they are quasiparticles; color confinement as retention of color charge in hadrons, and asymptotic freedom of quarks at short distances between them; different masses of quarks with equal spins and two fixed values of the fractional elementary charge; the reason for collapse of massive quarks; specification of defragmentation method and hadronization of jets with obligatory conversion of various color quarks into colorless hadrons; the origin of quantum numbers of quarks, etc.

## The gravitational model

### The origin of mass

In the Standard model it is assumed that quarks, leptons, W and Z bosons acquire mass through the mechanism of spontaneous symmetry breaking and Higgs bosons. After that hadrons consisting of quarks also become massive. If we proceed from the theory of Infinite Hierarchical Nesting of Matter, the mass appears as an intrinsic property of material particles that occurs as a result of Le Sage's theory of gravitation. [1] [2] At the level of elementary particles strong gravitation influences the scattered matter, forming objects, containing different amounts of substance. Further, these objects evolve similarly to the main sequence stars, turning into low-mass particles like nuons and in nucleons. According to the substantial neutron model, neutrons appear first, and then protons and electrons emerge as a result of beta decay.

At the level of stars neutron corresponds to a neutron star, proton corresponds to a magnetar, pions correspond to neutron stars with minimum mass decaying in time, [3] and nuons correspond to white dwarfs. [4] Discreteness of masses of these objects is defined by a narrow range of masses, in which formation of these objects is possible in the field of strong (or, respectively, ordinary) gravitation according to the equation of their matter state.

The reason of mass and its inertia in the general case is interaction of gravitons with matter. This interaction leads to attraction of bodies and to the concept of gravitational mass. At the same time, in case of acceleration of bodies relative to the fluxes of gravitons inertial mass and the corresponding inertial force appear. Taking into account the similarity of matter levels and SPФ symmetry, such description of gravitation is universal for all levels of matter. Meanwhile, in general relativity, mass is considered as the consequence of curvature of spacetime around the matter and in elementary particle physics the Higgs mechanism is introduced to describe the mass. As we can see, in the latter case, representation of mass is ambiguous and depends strongly on the size and mass of the objects.

The estimate of the mass and sizes of nucleons can be obtained in the same way which was used in the study of properties of neutron stars based on the principles of quantum mechanics. [5] From comparison of the gravitational binding energy of the star and the quantum mechanical energy of matter (expressed by Planck constant) we obtain formulas for the mass and radius of the star. Similar formulas are used for nucleons as the analogues of neutron stars. It turns out that the mass and radius of nucleons are determined by quantum properties of their matter and depend on the value of the strong gravitational constant. For connection of the radius and mass of the proton the following formula is obtained: [6]

${\displaystyle R={\frac {d}{M^{1/3}}},\qquad \qquad (1)}$

where ${\displaystyle d}$ is a constant depending on the properties of the proton matter.

### Charge

The electromagnetic field, along with gravitational, is the fundamental force field. Natural objects, containing matter at low density and low energy of interaction, are generally neutral due to the compensation of positive and negative charges of the matter. Charged objects occur when the carriers of charge of a single sign are removed from (or added to) them. Characteristic and widespread process is acquisition of a positive charge by the proton and formation of a negatively charged electron in beta decay due to reactions of weak interaction occurring in the neutron matter. The analysis of energies in the proton shows that the proton charge has the maximum possible value when the density of zero electromagnetic energy is comparable to the energy density of strong gravitation. [6] Equality of the charge of proton and electron follows from the nature of neutron beta decay and takes place in every point of the Universe.

Secondary character of mass and charge of the electron in the hydrogen atom with respect to the proton follows from substantial electron model. In particular, the electron matter must have the charge equal to the charge of the proton, to ensure electrical neutrality of the atom. At the same time the mass of the electron must be of such value that its attraction to the nucleus in the strong gravitational field must be equal to the electric force between the charges of the nucleus and the electron. In addition, there are two almost identical forces: repulsion of the charged matter of the electron cloud from itself, and the centrifugal force of rotation. The sum of these four forces is zero in stationary rotation of the electron cloud, which allows determining the strong gravitational constant. In the process of nucleosynthesis of more massive atoms from hydrogen atom each time interaction of neutrons, protons and electrons takes place. It helps to understand the origin of the elementary charge and the necessity to use it in elementary particle physics and in atomic physics as the standard unit of charge. Thus, the origin of mass and charges of elementary particles does not require introduction of hypothetical fields like the Higgs field.

### Energy

Since at the level of elementary particles the main force is assumed strong gravitation, it allows us to calculate the total energy of hadron of the nucleon type, which is up to sign equal to the binding energy of the hadron matter: [3] [6]

${\displaystyle W={\frac {\delta \Gamma M^{2}}{2R}},\qquad \qquad (2)}$

Here ${\displaystyle \delta =0.62}$ for objects like nucleons and neutron stars, ${\displaystyle \Gamma }$ is the strong gravitational constant, ${\displaystyle M}$ and ${\displaystyle R}$ are the mass and radius of the hadron.

Relation (1) can be used to estimate approximately the particle radius by its known mass, rewriting (1) in the form ${\displaystyle RM^{1/3}=R_{p}M_{p}^{1/3}}$, where ${\displaystyle M_{p}}$ and ${\displaystyle R_{p}}$ are the mass and radius of the proton. If we substitute the obtained radii of the particles into (2), we shall obtain the gravitational binding energies of the particles shown in the Table. [6]

Characteristics of proton, pion and muon
Particle Mass-energy, MeV Mass, 10–27 kg Radius, 10–16 m Binding energy ${\displaystyle W}$, MeV
Proton p+ 938.272029 1.6726 8.7 938.272
Meson ${\displaystyle f_{0}^{0}}$ 600 1.1 10 354
Pion π+ 139.567 0.249 16.4 11
Muon μ+ 105.658 0.188 10900 0.095

The particle masses in Table are obtained by dividing the mass-energy converted from MeV to J, by the squared speed of light. The mass-energy corresponds to the rest energy in the special relativity, and is directly proportional to the mass (see the mass–energy equivalence). In contrast to this, the total energy of the particle is calculated by summing the potential energy of strong gravitation and the internal kinetic energy of particle matter, the absolute value of the total energy is equal to the binding energy or the energy required to scatter the particle matter to infinity with zero velocity. According to the Table the binding energy of pion matter in the strong gravitational field is one order of magnitude less than the rest energy, which is the consequence of low density of the pion matter.

The laws of conservation of energy and momentum in reactions with elementary particles in the special theory of relativity are as follows:

${\displaystyle \sum _{k=1}^{n}{\sqrt {p_{k}^{2}c^{2}+M_{k}^{2}c^{4}}}=\sum _{s=1}^{m}{\sqrt {p_{s}^{2}c^{2}+M_{s}^{2}c^{4}}},}$
${\displaystyle \sum _{k=1}^{n}{\vec {p}}_{k}=\sum _{s=1}^{m}{\vec {p}}_{s},}$

where before the interaction there are n particles, and after the interaction the number of particles is m, and there is a possibility that ${\displaystyle m\not =n}$, ${\displaystyle M_{k}}$ and ${\displaystyle p_{k}}$ are corresponding mass and momentum of particles, ${\displaystyle c}$ is the speed of light.

On the other hand, the energy balance can be written in such a way as to explicitly include the change in the total energy of particles: [6]

${\displaystyle \sum _{k=1}^{n}E_{k}+\sum _{k=1}^{n}T_{k}=\sum _{s=1}^{m}E_{s}+\sum _{s=1}^{m}T_{s}+E_{g}+E_{f},}$

where ${\displaystyle E_{k}=-{\frac {\delta \Gamma M_{k}^{2}}{2R_{k}}}}$ is the total energy of the k-th particle in the field of strong gravitation, ${\displaystyle R_{k}}$ is the radius of the k-th particle, ${\displaystyle T_{k}={\sqrt {p_{k}^{2}c^{2}+M_{k}^{2}c^{4}}}-M_{k}c^{2}}$ is the kinetic energy of the k-th particle in the special theory of relativity, and during interaction the energy of strong gravitation ${\displaystyle E_{g}}$ can be released (or, conversely, be added) which is associated with change in the radii and masses of particles, as well as the energy of the emerging electromagnetic and neutrino emission ${\displaystyle E_{f}}$.

### The structure of nucleons

According to the substantial neutron model and the substantial proton model, the difference between neutron and proton besides mass and charge is mainly in the difference in their internal electromagnetic structure. Thus, in neutron the space charge separation is assumed, the center of the neutron is positively charged, and the shell is negative. The rotation of the space charge creates a negative magnetic moment of neutron, directed opposite to the spin.

To connect the medium pressure ${\displaystyle p}$ and the mass density ${\displaystyle \rho }$ of nucleon in the first approximation, we can write down:

${\displaystyle p=K\rho ^{5/3},}$

where ${\displaystyle K=8.4\cdot 10^{4}}$ in SI units is the coefficient, found by the radius of nucleon, its mass and strong gravitational constant. [6] In the self-consistent model that takes into account the density distribution, the rest energy, magnetic moment and the conditions of limiting rotation, the ratio of the central proton mass density to its average density is 1.57. [7]

From the point of view of the matter state, instead of three quarks and indefinite number of gluons inside nucleons we expect up to ${\displaystyle 1.62\cdot 10^{57}}$ of smallest particles called praons. According to the theory of Infinite Hierarchical Nesting of Matter, praons have the same status in nucleons as nucleons themselves in neutron stars. This explains why in collisions even with highest energies we observe not gas of quarks and gluons, but jets of almost perfect liquid hadronic matter. [8] [9]

In the above picture de Broglie wavelength of moving nucleons and other elementary particles can be explained as a consequence of the relativistic transformation of wavelength of the internal oscillations of the fundamental fields potentials of the particles in the laboratory frame of reference.

### Deuteron

P2. Two examples of modern internucleon potential in 1S0 channel (the orbital angular momentum of nucleons is zero, nucleon spins are opposite, the total angular momentum of both nucleons is zero). Potentials AV18 and Reid 93. [10]

Deuteron is the simplest nucleus consisting of two nucleons – proton and neutron. In Figure P2 there are two examples of modern potential of nuclear force depending on distance ${\displaystyle r}$, which can be considered equal to the distance from the center of mass of the system of two nucleons to the center of one of the nucleons. From internucleon potential, understood as interaction energy of two nucleons, we can proceed to the force acting on the nucleon, according to the formula: ${\displaystyle F=-\nabla V_{c}(r)}$. The graph of this force is shown in Figure P3 for the potential AV18.

In the gravitational model of strong interaction the force ${\displaystyle F}$ between nucleons in the first approximation is presented as the difference between the force of spin repulsion ${\displaystyle F_{LL}}$ and the magnitude of gravitational force of attraction ${\displaystyle F_{\Gamma }}$: [4]

${\displaystyle ~F=F_{LL}-|F_{\Gamma }|={\frac {1}{2}}{\bigl |}\nabla (\mathbf {L} \cdot \mathbf {\Omega } ){\bigr |}-{\frac {\beta \Gamma M_{n}M_{p}}{4r^{2}}},}$

where ${\displaystyle Omega}$ is the gravitational torsion field from one nucleon, which acts on the effective spin ${\displaystyle L}$ of the other nucleon, ${\displaystyle M_{n}}$ and ${\displaystyle M_{p}}$ are the masses of neutron and proton, respectively, ${\displaystyle 2r}$ is the distance between the centers of nucleons, ${\displaystyle \beta }$ is the coefficient, the estimate of which for the case of two nucleons is ${\displaystyle \beta =0.26}$ as a result of exponential absorption of gravitons in the matter of nucleons, and for particles of lower density it is ${\displaystyle \beta =1}$.

P3. Dependence of the force ${\displaystyle F}$, obtained from the internucleon potential AV18, on the distance, in 1S0 channel. For comparison, the dependences are given of the magnitude of the gravitational force ${\displaystyle F_{\Gamma }}$ (with β = 1) and the estimated force of the spin interaction ${\displaystyle F_{LL}.}$

Spin ${\displaystyle L}$ includes the initial nucleon spin, and the spin induced by another nucleon by means of gravitational induction. The formula for the repulsive force of nucleon spins is: [4]

${\displaystyle F_{LL}={\frac {3\eta \Gamma L^{2}}{4c_{g}^{2}(2r)^{4}}}}$,

where ${\displaystyle c_{g}}$ is the speed of propagation of gravitational effect (the speed of gravity), which is close to the speed of light, ${\displaystyle 1.3<\eta <2.8}$ is the coefficient depending on the distance of nucleons interaction in the deuteron.

Since in Figure P3 at ${\displaystyle r=0.88}$ fm the force ${\displaystyle F}$ vanishes, we can estimate the distance between the surfaces of nucleons: ${\displaystyle s=2r-2R_{p}=0.02{\text{ fm}}}$. The described state with opposite nucleon spins has small binding energy of about 69 keV and therefore is unstable. If we locate nucleons in this state on the plane, the spins of nucleons will be perpendicular to the plane and opposite to each other.

From Figure P3 we see that when ${\displaystyle r=R_{p}}$, that is in nucleons’ contact, the spin repulsive force from the torsion field is equal to the gravitational force of attraction. At smaller distances a rapidly growing force of repulsion emerges. We can assume that the main contribution is made by the magnetic force of repulsion and the force of internal pressure in the matter of nucleons.

At distances from ${\displaystyle r_{1}=0.87\cdot 10^{-15}{\text{ m}}}$ to ${\displaystyle r_{2}=1.1\cdot 10^{-15}{\text{ m}}}$ the force from the torsion field decreases according to the dependence ${\displaystyle {\frac {1}{r^{n}}}}$, where ${\displaystyle ~4. If the nucleons (neutron and proton), considered as points were interacting only by torsion fields from their constant spins, the dependence on the distance in the formula for the force of spin interaction would have the form ${\displaystyle {\frac {1}{r^{4}}}}$. However, when two nucleons approach each other in deuteron formation, additional effects can occur. Firstly, in case of the same direction of spins, due to the effect of gravitational induction both nucleons will rotate each other as they approach, with increasing of their spins. Secondly, besides gravitational forces there are also electromagnetic ones, which are the forces of repulsion of magnetic moments in deuteron (and the magnitude of magnetic moments due to various effects can also change as the nucleons approach each other). All this leads to the fact that in the effective force ${\displaystyle F_{LL}}$ of nucleons repulsion the exponent increases up to the values greater than ${\displaystyle n=4}$. Thus, from a qualitative point of view, the gravitational forces of attraction and the forces of spins interaction and the electromagnetic forces can explain the nuclear forces at short distances.

At distances greater than ${\displaystyle r_{2}=1.1\cdot 10^{-15}{\text{ m}}}$ the force ${\displaystyle F_{LL}}$ in Figure P3, built as the sum of the force ${\displaystyle F}$ from internucleon potential and the magnitude of the gravitational force ${\displaystyle F_{\Gamma }}$, undergoes a strange kink, with a significant change in the speed of its decrease with distance. This is due to the inaccuracy of internucleon potential of the Standard model, according to which the interaction between nucleons occurs by means of special carriers – virtual mesons (in Figure P2 the areas are indicated where interactions with two pions 2π, with mesons ρ, ω, σ, and one meson π are taken into account) . Until now, the basis for calculation of the potential at these distances is the Yukawa potential of the following form:

${\displaystyle V_{y}(r)=-{\frac {g}{r}}\exp {\left(-{\frac {Mcr}{\hbar }}\right)},}$

where ${\displaystyle g}$ is some effective charge of strong interaction, ${\displaystyle M}$ is the mass of the particle – the carrier of interaction, ${\displaystyle \hbar }$ is the Dirac constant.

For the pion the quantity ${\displaystyle {\frac {Mcr}{\hbar }}}$ in the exponent is equal to one with ${\displaystyle r=1.4\cdot 10^{-15}{\text{ m}}}$, with masses of heavier mesons the distance decreases. The force from the Yukawa potential, which has the character of attraction, equals:

${\displaystyle F_{y}=-{\frac {g\left({\frac {Mcr}{\hbar }}+1\right)}{r^{2}}}\exp {\left(-{\frac {Mcr}{\hbar }}\right)}.}$

At such distances ${\displaystyle r}$, where ${\displaystyle {\frac {Mcr}{\hbar }}\approx 1}$, the force ${\displaystyle F_{y}}$ decreases slowly enough, in proportion ${\displaystyle {\frac {1}{r}}}$. This creates a kink in Figure P3 for the force of spins interaction ${\displaystyle F_{LL}}$. At the same time, the gravitational force ${\displaystyle F_{\Gamma }}$ changes in proportion ${\displaystyle {\frac {1}{r^{2}}}}$, that is, faster than the force ${\displaystyle F_{y}}$ from the Yukawa potential at the segment ${\displaystyle r>1.1\cdot 10^{-15}{\text{ m}}}$.

If we proceed from the gravitational and electromagnetic interactions, the formula for the total energy of interaction between nucleons in the deuteron must include gravitational energy, the energy of spins and spin-orbital gravitational interactions, the energy of spin-spin and spin-orbital interaction of the magnetic moments taking into account the possible effect of electromagnetic induction, increment of kinetic and rotational energy of nucleons, which may depend on the distance between the nucleons, due to the interaction of the gravitational and electromagnetic forces. Almost all of these energies can contribute to the creation of the effective force between nucleons.

In the deuteron we can assume that the nucleons are on the common axis of rotation, and the nucleon spins are directed in one side of the axis. Then, the magnetic moments of the proton and neutron are opposite, corresponding to the magnetic moment of the deuteron. In the absence of orbital rotation, the entire spin of the deuteron will consist of the nucleons spin. Considering only the main components of energies and forces, in equilibrium, at a small distance ${\displaystyle R}$ between the centers of the nucleons, the following relations hold:

${\displaystyle {\frac {0.26\Gamma M_{n}M_{p}}{R^{2}}}\approx {\frac {3\eta \Gamma L^{2}}{2c_{g}^{2}R^{4}}},}$
${\displaystyle -{\frac {0.26\Gamma M_{n}M_{p}}{R}}+2(U-U_{p})+\eta U_{0}+{\frac {L^{2}-L_{p}^{2}}{I}}+\epsilon _{d}\approx 0,}$

where ${\displaystyle R_{p}}$ is the proton radius as the measure of the nucleon radius, ${\displaystyle U=-{\frac {83\Gamma L^{2}}{252c_{g}^{2}R_{p}^{3}}}}$ is the internal energy of the torsion field in the deuteron per one nucleon, ${\displaystyle U_{p}=-{\frac {83\Gamma L_{p}^{2}}{252c_{g}^{2}R_{p}^{3}}}}$ is the internal energy of the torsion field of free nucleon, ${\displaystyle U_{0}={\frac {\Gamma L^{2}}{c_{g}^{2}R^{3}}}}$ is the energy of two spins in the field of each other, or doubled energy of one spin in the external torsion field from the second spin, ${\displaystyle {\frac {L^{2}-L_{p}^{2}}{I}}}$ is the change in rotational kinetic energy of nucleons, ${\displaystyle I}$ is the moment of inertia of the nucleon, ${\displaystyle \epsilon _{d}=2.24}$ MeV is the binding energy of the deuteron.

In the presented balance of forces in the first approximation only the force of gravitational attraction of nucleons and the force of spins repulsion ${\displaystyle F_{LL}}$ are taken into account. The solution for the balance of forces and energies is the value ${\displaystyle s<0.78\cdot 10^{-15}{\text{ m}}}$ between the adjacent surfaces of nucleons. [4] In this case, the equatorial speed of rotation of the nucleon surface is about 1.8 times higher than the speed of light, and is the analogue of the first cosmic velocity, found for planets and stars. The proposition that nucleons and more massive objects in general can not move faster than the speed of light, does not refer to the matter of nucleons. The particles of hadron matter inside nucleons on the average have the speed almost equal to the speed of light.

### Atomic nuclei

The analysis of the equilibrium state of nucleons in the deuteron allows us to formulate the following conditions for stability of atomic nuclei:

1. Adjacent nucleons must have zero relative rotation of the surfaces closest to each other. For example, if nucleons rotate as though located on the same axis, they must rotate synchronously. Otherwise, magnetic forces and torsion forces appear leveling the angular rotational velocity.
2. Another necessary condition is that the adjacent nucleons must have such direction of spins that there was a repulsive force between them.
3. The absence of diproton and dineutron shows that the combination with two adjacent protons or two neutrons on a common axis of rotation in the nucleus is hardly probable, at least for small nuclei.
P4. The structure of nuclei of hydrogen and helium isotopes. ${\displaystyle P_{n}}$ and ${\displaystyle L_{n}}$, ${\displaystyle P_{p}}$ and ${\displaystyle L_{p}}$ are the magnetic moments and spins of neutrons and protons, respectively.

Based on this, the models of triton (tritium nuclei, a heavy isotope of hydrogen) and of other basic nuclei – helium and lithium are built. Spin-spin forces between nucleons in atomic nuclei explain the Pauli exclusion principle, according to which identical fermions that are close, cannot have the same quantum numbers. In particular, spins of identical nucleons in the presented models of atomic nuclei are in opposite directions.

It is known that triton turns into the light helium nucleus with half-decay period 12.32 years due to beta decay. This can be presented in the way that the left neutron of the triton in Figure P4, undergoing beta decay, turns into a proton and moves to the position taken by the left proton in the nucleus of the light helium in Figure P4. In this case the direction of the nucleon spin is not changed. For the motion of the nucleon it needs momentum which arises from the emission of electron antineutrino in beta decay of the neutron. As it is shown in the substantial neutron model, antineutrino flies toward the spin of the decaying neutron and pushes the neutron in the opposite direction.

### Nuclear binding energy

P5. The dependence of the specific binding energy of atomic nuclei on the mass number ${\displaystyle A}$. [11]

With the help of the expression for the gravitational energy we can qualitatively show the need for the growth of specific nuclear binding energy (binding energy per nucleon) with growth of the mass of the nucleus. Since in the atomic nucleus the nucleon density only slightly depends on the number of nucleons, it means the approximate equality of volumes per one nucleon in different nuclei. It is usually assumed that the average distance between nucleons in nuclei is of the order of ${\displaystyle r=1.8\cdot 10^{-15}{\text{ m}}}$. If we denote the total binding energy of the nucleus by ${\displaystyle W}$, with a small number of nucleons this energy must be proportional to the gravitational energy ${\displaystyle U}$ of all the nucleons of the nucleus in the field of strong gravitation.

If the nucleus consists of ${\displaystyle A}$ nucleons, the nuclear mass is ${\displaystyle M_{N}=AM_{p}}$, the volume per one nucleon is equal to ${\displaystyle r^{3}=const}$, the volume of the nucleus is ${\displaystyle {\frac {4\pi R_{N}^{3}}{3}}=Ar^{3}}$, then the nucleus radius ${\displaystyle R_{N}}$ will be proportional to ${\displaystyle A^{1/3}}$. Hence the specific binding energy and the specific gravitational energy will be proportional to the quantity: ${\displaystyle {\frac {W}{A}}\approx {\frac {\mid U\mid }{A}}={\frac {\delta \Gamma M_{N}^{2}}{AR_{N}}}\approx A^{2/3}}$

The dependence of the specific binding energy on the number of nucleons in the nucleus as ${\displaystyle A^{2/3}}$ in general is confirmed: if for the deuteron it is equal to 1.1 MeV/nucleon, then for the nuclei with ${\displaystyle A=20}$ the specific binding energy is equal to 8 MeV/nucleon. With further increase of ${\displaystyle A}$ the specific binding energy reaches the value of 8.7 MeV/nucleon, and then slowly decreases (approximately up to about 7.6 MeV/nucleon for the nucleus of uranium). Such a dependence is explained by the growing influence of the electrical energy of protons repulsion with the growth of ${\displaystyle A}$, which reduces the binding energy. In addition, from the Le Sage’s theory of gravitation the saturation effect of strong gravitational energy follows with ${\displaystyle A>17}$. [4] As a consequence of this effect in case of further increase in the mass of the nucleus the nucleons added to the nucleus will have the same energy, and the potential of the gravitational field remains almost unchanged. The gravitational pressure in the nucleus is fixed and stops growing, correspondingly, the average distances between the nucleons stop changing too. This result is consistent with the fact that with ${\displaystyle A\approx 50}$ the maximum of the specific binding energy of atomic nuclei is reached.

All hadrons, including mesons and baryons, can be divided into three groups. The first group includes the simplest quasi-stable hadrons like pions and nucleons, having long lifetime. These hadrons can be considered as independent particles experiencing decay due to weak interaction (except for stable proton). The second group consists of long-lived strange, charmed and beautiful particles, and the third group includes resonances, whose lifetime is almost equal to the time of particles’ transit near each other at their close interaction. As shown in the model of quark quasiparticles, strange particles can be represented as composite hadrons of simple hadrons of the first group. [6] For example, hyperon Λ is assumed to consist of proton and pion rapidly rotating next to each other along one axis, held by strong gravitation and spin torsion fields. In calculating the equilibrium conditions, the equations for the forces and energies are used, similar to those presented above for the deuteron.

Hypothetical composition of other strange hadrons is as follows: hyperon Σ is a compound of neutron with pion; Ξ includes proton and two pions; three or four pions with proton make up Ω-baryon. K-mesons are compounds of three pions and have the following compositions:

${\displaystyle K_{L}^{0}=\pi ^{-}\pi ^{0}\pi ^{+},}$
${\displaystyle K_{S}^{0}=\pi ^{-}\pi ^{+}\pi ^{0},}$
${\displaystyle K^{-}=\pi ^{-}\pi ^{+}\pi ^{-},}$
${\displaystyle K^{+}=\pi ^{+}\pi ^{-}\pi ^{+}.}$

Different combinations of pions in neutral kaon can explain different lifetimes of ${\displaystyle K_{L}^{0}}$ and ${\displaystyle K_{S}^{0}}$ states. In contrast to atomic nuclei, compounds of nucleons and pions (or pions with each other) can not be stable, and over time they disintegrate. The same applies to charmed and beautiful hadrons.

There are many works in which resonances are presented not as interactions of quarks but as dynamically bound short-lived states of simpler hadrons. For example, hyperon Λ(1405) is considered as dynamic bound state of nucleon and kaon, [12] and scalar mesons f(980) and a(980) are considered as a molecule of kaon and antikaon. [13] Hadronic molecules of kaon, antikaon and nucleon are discussed in [14] by solving the Schrödinger equation for the wave function of three particles and using two interaction potentials assumed in the model. There are strong proofs that many resonant states N, Δ, Λ, Σ, Ξ, Ω are dynamically bound states of vector mesons (of ρ and ω type) with baryons belonging to baryon octet with nucleons and to decuplet with Δ. [15]

## Weak interaction

In gravitational model of strong interaction masses as well as charges of elementary particles are explained by the properties and the structure of particles’ matter, and by the action of strong gravitation and electromagnetic forces in this matter. Under the action of these forces ordering of matter particle takes place, and this matter has the ability to transform slowly in reactions of weak interaction. Thus weak interaction at the level of elementary particles is reduced again to weak interaction, but at the level of minute particles that make up the matter of elementary particles.

The examples of description of reactions of weak interaction in the matter of elementary particles are shown in the model of quark quasiparticles, in the substantial neutron model and in the substantial proton model. In particular it is shown that neutrinos of one basic level of matter are two-component and consist of fluxes of electron neutrinos and antineutrinos of the lower basic level of matter.

This means that weak interaction can be explained not with the help of special field quanta of the type of W and Z bosons, but represent as the property of matter to change naturally in conditions of maximum possible density of matter and energy. Thus, it is expected that during the time of the order of 2·1015 years neutron stars must undergo β--decay, with formation of magnetars and ejection of negatively charged shells (similarly to neutrons decomposing with formation of protons, electrons and electron antineutrinos).

## Coupling constants

To compare gravitational, weak, electromagnetic and strong interactions the energies of corresponding forces are usually considered, acting on proton matter taking into account its mass and charge, in the field of other proton. For energies we can write down:

${\displaystyle U_{G}=-{\frac {GM_{p}^{2}}{r}},\qquad \qquad U_{W}={\frac {G_{F}M_{p}^{2}c^{2}}{\hbar ^{2}r}}\exp {\left(-{\frac {M_{W}cr}{\hbar }}\right)},}$
${\displaystyle U_{e}={\frac {e^{2}}{4\pi \varepsilon _{0}r}},\qquad \qquad U_{s}=-{\frac {g_{N\pi }^{2}}{4\pi r}}\exp {\left(-{\frac {M_{\pi }cr}{\hbar }}\right)},}$

where ${\displaystyle G}$ is the gravitational constant, ${\displaystyle M_{p}}$ is the proton mass, ${\displaystyle r}$ is the distance between the centers of protons, ${\displaystyle G_{F}}$ is Fermi constant of weak interaction, ${\displaystyle c}$ is the speed of light, ${\displaystyle M_{W}}$ is the mass of virtual W or Z boson, which is considered the carrier of weak interaction, ${\displaystyle e}$ is the proton charge, equal to the elementary charge, ${\displaystyle \varepsilon _{0}}$ is the electric constant, ${\displaystyle g_{N\pi }}$ is the charge of strong interaction, ${\displaystyle M_{\pi }}$ is the mass of virtual particle (mostly pion), which is supposed carrier of strong interaction.

The expressions for the corresponding coupling constants follow from the relations for the energy:

${\displaystyle \alpha _{G}={\frac {GM_{p}^{2}}{\hbar c}}=5.907\cdot 10^{-39},\qquad \qquad \alpha _{W}={\frac {G_{F}M_{p}^{2}c}{\hbar ^{3}}}\approx 1.0\cdot 10^{-5},}$
${\displaystyle \alpha _{e}={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}=7.297\cdot 10^{-3},\qquad \qquad \alpha _{s}={\frac {g_{N\pi }^{2}}{4\pi \hbar c}}\approx 14.6.}$

The coupling constant of electromagnetic interaction ${\displaystyle \alpha _{e}}$ is called the fine structure constant. The charge of strong interaction ${\displaystyle g_{N\pi }}$ and the Fermi constant ${\displaystyle G_{F}}$ are axiomatically introduced into the Standard theory to describe the experimental results. If we proceed from the notion of strong gravitation, the interaction energy of two nucleons and the coupling constant will equal: [6]

${\displaystyle U_{pp}=-{\frac {\beta \Gamma M_{p}^{2}}{r}},\qquad \qquad \alpha _{pp}={\frac {\beta \Gamma M_{p}^{2}}{\hbar c}}=13.4\beta ,}$

where ${\displaystyle \Gamma }$ is the strong gravitational constant, ${\displaystyle \beta =0.26}$ for the case of two nucleons.

This shows that the coupling constant ${\displaystyle \alpha _{pp}}$ of strong gravitation is of the same order of magnitude as the coupling constant ${\displaystyle \alpha _{s}}$ of strong interaction. In atomic nuclei the equilibrium of nucleons is achieved due to attraction from the strong gravitational field and repulsion from gravitational torsion fields, and the coupling constants of both components of the gravitational field (gravitational field strength of strong gravitation and gravitational torsion field) are leveled in magnitude.

## Unification of interactions

In the Standard model the gauge approach of quantum field theory is used, when for each type of interaction (gravitational, electromagnetic, weak and strong) their own fields and gauge bosons-quanta, that carry the interaction, are introduced. Gravitons, photons, W and Z bosons, gluons, and particles such as Higgs boson appear this way. At present, weak and electromagnetic interactions of elementary particles, despite their significant differences, are discussed by electroweak theory based on the unified mathematical formalism. In future it is planned to add in the common scheme strong (Grand Unified Theory) and gravitational interaction (theory of everything).

The disadvantage of this approach is its focus only on description of the observed processes, without penetration into the essence of phenomena. So far, there are no specific mechanisms that explain how the forces of attraction or repulsion between particles emerge due to the action of any gauge bosons, such as photons. There is a gap between the facts that single accelerated charges generate real electromagnetic emission that transfers energy and theoretically is considered as a set of photons or partial waves, and there is no such emission near fixed charges although the adjacent charges somehow interact with each other. To create a unified picture it is necessary that photons could explain not only free electromagnetic emission but also the static electromagnetic field. However, in case of purely electrostatic and magnetostatic fields there are no electromagnetic energy fluxes, the Poynting vector at each point is zero, and thus the possible direction of the motion of photons can not be determined. Introducing into the theory the idea of virtual particles does not help, because interactions with virtual photons, gluons, W and Z bosons, quark-antiquark pairs, electron-positron fields, etc. can not be considered as the ultimate solution.

As shown above, strong interaction at the level of elementary particles can be reduced to the action of strong gravitation and spin gravitational forces from torsion fields, with addition from electromagnetic forces. The same forces and the corresponding interaction will be at the level of the stars, with substitution of strong gravitation by ordinary gravitation. It is considered that gravitation at the macro level in general must be described by relativistic theories such as general theory of relativity (GTR) or covariant theory of gravitation (CTG). But in GTR gravitation is reduced to curvature of spacetime and is not a physical force, whereas in CTG the cause of gravitational force is assumed the action of gravitons, considered in Le Sage’s theory of gravitation. Gravitons act on the matter regardless of the number and density of this matter and the effects of general relativity emerging in strong fields near massive bodies, such as the effect of time dilation, are reduced to the influence of gravitons on the properties of electromagnetic waves (photons) used for measurements. The latter means the dependence of the results of measurements of length and time on the measurement procedure by means of light signals, with the constant picture of interaction of gravitons with matter. In this case, the effects of GTR, including the hypothetical black holes, are external, and do not reflect the real essence of gravitational interaction.

In the weak fields, where the dependence on the measurement procedure can be neglected, GTR turns to gravitoelectromagnetism and CTG – to Lorentz-invariant theory of gravitation (LITG). It turns out that in LITG and in gravitoelectromagnetism the equations of gravitational field are almost exactly the same and are similar in form to the equations of the electromagnetic field in electrodynamics, which is emphasized in Maxwell-like gravitational equations. Apparently, the similarity of equations for both fields is not accidental. Relationship between electromagnetism and gravitation may be that part of gravitons are photons emitted by charged praons – the carriers of matter, from which the matter can be formed, which is part of the matter of elementary particles. [6]

Representation of gravitons in the form of photons was used to represent the formula of the gravitational force, as a result of Compton scattering of photons in matter. [16] On the other hand, emission of energy of supernovae in the formation of neutron stars is almost entirely realized through neutrino emission, and not in the form of expected gravitational waves. That, and the similar penetrating ability of particles give the reason to assume that neutrinos of one level of matter are gravitational quanta or gravitons of higher levels of matter. Comparison of the energy density of neutrino emission in supernova and the similar neutrino emission from the matter of the neutron which is formed, taking into account the coefficients of similarity in the theory of similarity of matter levels, shows that the gravitons of ordinary gravitation can be neutrinos generated by matter, the carriers of which correspond to elementary particles in the same way as elementary particles correspond to stars, or can be smaller by one basic level of matter. [6] If gravitons are both photons and neutrinos, then there is may be the case when the neutrino is a type of electromagnetic radiation. The difference between photon and neutrino is that neutrino is two-component emission with opposite polarization of the components. In this case, photon and neutrino are composed of the corresponding fluxes of tiny quanta of emission.

In the above picture electromagnetic and gravitational fields are fundamental and similar to each other in the form of equations for the field and the acting forces, and in the origin. For example, gravitational quanta of one level of matter are able to compress the matter into massive objects of higher level of matter up to such density that as a result emission of new, more powerful gravitational and electromagnetic quanta is possible, as well as emission of charged particles such as cosmic rays. After that, the emerged relativistic particles, neutrinos and photons interact in a similar way with the matter of higher levels of matter, and so the process goes on. At the same time, there is a reverse process of energy dissipation and splitting of quanta. Adding to the fluxes of gravitons by Le Sage’s model the fluxes of charged particles of different signs allows us not only to derive the Newton’s formula of gravitation and to explain the origin of gravitational force, but also to understand the origin of the electrostatic force between two charges. [17] [2] [18] Strong and weak interactions by their nature are not fundamental field interactions, characteristic of each object separately. This is due to the fact that the first of them depends on the combination of fundamental fields in the interaction of objects with each other, and the second – on the interaction of internal fields in the matter of objects with the total gravitational and electromagnetic fields of these objects, or with external emission, leading to matter transformation.

It was shown that the electromagnetic and gravitational fields, acceleration field, pressure field, dissipation field, strong interaction field, weak interaction field, and other vector fields acting on the matter and its particles, are the components of general field. [19] [20]

## References

1. Fedosin S.G. Model of Gravitational Interaction in the Concept of Gravitons. Journal of Vectorial Relativity, Vol. 4, No. 1, pp.1-24 (2009).
2. Fedosin S.G. The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model. Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197.
3. Fedosin S.G. (1999), written at Perm, pages 544, Fizika i filosofiia podobiia ot preonov do metagalaktik, ISBN 5-8131-0012-1.
4. Sergey Fedosin, The physical theories and infinite hierarchical nesting of matter, Volume 1, LAP LAMBERT Academic Publishing, pages: 580, ISBN 978-3-659-57301-9.
5. Landau L.D. On the theory of stars. – Phys. Z. Sowjetunion, Vol. 1, p. 285 (1932).
6. Comments to the book: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
7. Fedosin S.G. The radius of the proton in the self-consistent model. Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012).
8. "'Perfect' Liquid Hot Enough to be Quark Soup". Brookhaven National Laboratory News. 2010. Retrieved 2010-02-26.
9. "LHC Experiments Bring New Insight into Primordial Universe". Brookhaven National Laboratory News. 2010. Retrieved 2010-12-09.
10. Ishii N., Aoki S., Hatsuda T. The Nuclear Force from Lattice QCD. arXiv: nucl-th / 0611096 v1, 28 Nov 2006.
11. Яворский Б.М., Детлаф А.А., Лебедев А.К. Справочник по физике. – М.: Оникс, Мир и образование, 2006, 1056 с.
12. R. H. Dalitz and S. F. Tuan, Phys. Rev. Lett. Vol. 2, p. 425 (1959); Annals Phys. 10, 307 (1960).
13. J.D. Weinstein and N. Isgur, Phys. Rev. Lett. 48, 659 (1982); Phys. Rev. D 41, 2236 (1990).
14. Daisuke Jido, Yoshiko Kanada-En'yo. A new N* resonance as a hadronic molecular state. 25 Jun 2009.
15. Oset, E. at al. Dynamically generated resonances. 20 Jun 2009.
16. Michelini Maurizio. The Physical Reality Underlying the Relativistic Mechanics and the Gravitational Interaction. – arXiv: physics / 0607136 v1, 14 Jul 2006.
17. Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
18. Fedosin S.G. The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model. Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357.
19. Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.
20. Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025.