A Preonic Model of Quarks and Particles, Based on a Cold Genesis Theory

Abstract

The theoretic model explains the generating of heavy quarks and of heavy particles using the known quark s(strange) and two semi-light quarks: λ±(435 MeV) and v±(∼574 MeV) resulting from a cold genesis theory1 of astro-particles’ forming, (CGT), the masses of the quarks c• (charm) and b• (bottom) of the Standard Model but also the values used by de Souza: c = 1.7 GeV and b = 5 GeV, being re-obtained as tri-quark clusters in the form: [(qq¯)q]], by a simple de-excitation reaction, with the emission of a preonic boson obtained in CGT: z0 = 34me, respective: z2 = 4z0 and zμ = 6z0- for the quarks b• and b. A similar de-excitation reaction, with emission of a preonic boson zk = n⋅z0, (n = 1÷7), representing less than 2.7% from the particle’s mass, explains the experimentally obtained masses of heavy baryons and mesons and of some multi-quark particles, with discrepancy <1% generally. The heavy current quarks result by lighter current quarks with quasi-crystalline internal structure, with trigonal/hexagonal symmetry of preonic kernels of two preonic bosons: z2 = 4z0; zπ = 7z0, the top-quark resulting as: t = 17(bb¯) + b = (7x5)⋅m(b), with a kernel of regular hexagonal polyhedron form, given by kernels of b-quarks. The model is compatible with the generation model and explains the quarks pairs forming from relativist e−- e+ jets.

1. Introduction

In the Standard Model (S.M.), it is known the constituent quark model, with a valence current quark (u-up, d-down, s-strange) or (c-charm, b-bottom, t-top) with a current mass [1]: (1.8÷2.8; 4.3÷5.2; 92÷104) MeV/c², respective: (1.3; 4.2÷4.7; 156÷175) GeV/c² and a gluonic shell formed by gluons and sea-quarks [1], the resulted effective quark mass being the constituent quark mass: (336, 340, 486) MeV/c², respective: (1.55, 4.73, 177) GeV/c².

The electric charge of u-, c-, t- quarks is +(²/₃)e and the electric charge of d-, s-, b- quarks is –(¹/₃)e, the strong interaction of quarks being explained by so-named “color charge”, the gluons having two opposed color charges, the gluon field between a pair of color charges forming a narrow flux tube (as a ‘string’) between them, (the Lund string model [2]).

Conform to S.M., at the high-energy gluons the “breaking” of these strings into new quark–antiquark pairs can occurs, as part of the hadronization process, the upper limit for the gluon’s mass experimentally determined being 1÷1.3 MeV/c² [3].

Also, the S.M. considers approximately the same size order for the maximum radius of the electron- resulted as scattering center determined inside the electron with X-rays: ∼10⁻¹⁸ m [4, 5] with that of the scattering centers experimentally determined inside the nucleon: 0.43x10⁻¹⁸ m [6], considered as quarks in the S.M. and the current quarks are considered un-structured, even if they can transform through weak interactions. As consequence, the quarks of S.M. cannot explain the mass hierarchy of the elementary particles by the sum rule, without the Higgs mechanism of mass acquiring by coupling to the Higgs field- which explains also the gluons’ masses.

However, the Grand Unified theories (GUT) predict relations among the fermion masses, such as between the electron and the u-, d-quark, the muon and the strange quark and between the tau lepton and the bottom quark, but these relations were only partially obtained, (ex.: Georgi-Jarlskog mass relation, which for m_d, m_s, m_b gives discrepancy up to 21%, [7]).

An older model, proposed by A. O. Barut, proposed the explaining of the mesonic and baryonic particles by leptonic fermions: electrons, muons and neutrins, [8].

Another theoretic model which sustain the existence of a relation between fermions is the Generation Model, (G.M., [9]) which- using the Harari-Shupe rishons model and the rishons conservation rule, provides a unified classification scheme for leptons and quarks, simplified in comparison with the S.M. by using the conservation of only three additive quantum numbers: Q -charge, p- particle number and g -generation number.

The experimentally evidenced possibility of quark-antiquark –pairs obtaining by the interaction of relativist fluxes of negatrons and positrons is explained in the S.M. by the conclusion of e⁺ − e⁻ annihilation, with paired quarks forming from a high energy photon, resulting by the energy of the collided electrons.

It is sustained in the Standard Model, by the Einsteinian relativity, that “an alternative way to achieve pair production is to use an electron and a positron in the initial state thus producing an intermediate virtual photon. This virtual photon then decays into the final state pair. By raising the photon (or initial e⁺e⁻) energy just above 200 MeV, it becomes possible to produce pairs of muons and if the energy is raised further, to almost 4 GeV, then the threshold for tau pair production is passed. Eventually, as the energy increases further, the process can also be mediated by the intermediate Z^o weak boson. Quark pairs can be produced in exactly the same way, the main difference being that the final state particles materialize in the real world as streams or jets of particles, mainly hadrons, but with some photons and leptons. At lower energies, the light quarks are produced”, [10].

In our opinion, this model of quark pairs production has a serious naturalness problem, because the Eisteinian relativistic mass depends on the observer’s frame of reference, i.e. an imaginary observer placed in the inertial system of the moving electron (positron or negatron) cannot observe a mass increasing of that electron, so- their common conclusion cannot confirm the possibility of heavy virtual photon generating (with mass m_γ > 2m_e). Also, a mechanism of a real speed-dependent increasing of the effective mass is missing.

2. A quasi-crystal quark model as solution to the S.M.’s naturalness problem

In report with the previous case of naturalness problem of the S.M., it results as more natural alternative the possibility to explain the constituent quarks and the resulted elementary particles as clusters of negatron-positron pairs, named ‘gammons’ (γ(e⁻e⁺)) in a Cold Genesis pre-quantum theory of particles and fields, (C.G.T., [11, 12, 13, 14]), based on the Galilean relativity, which argued that preonic bosons and quarks can be formed also ‘at cold’, as Bose-Einstein condensate of ‘gammons’ which form quasi-stable basic preons z⁰ of mass ∼ 34 m_e, forming constituent quarks, (M. Arghirescu, 2006, [11, 12], p. 58).

This z⁰ -preon was deduced by calibrating the value: m_k = m_e/2α = 68.5m_e obtained by Olavi Hellman [15], by using the masses of the proton and of the Σ-baryon, [11, 12], the proton’ mass being given as ∼27x68 = 1836 m_e and the Σ-baryon’ mass being given as ∼69x34 = 2346 m_e.

The Olavi Hellman’s relation was deduced by a system of non-linear classical field equations having particle-like solutions, i.e. solutions which differ significantly from zero only in a region whose dimensions are of the same order of magnitude as the elementary length implied by the theory, this relation being obtained in the form:

$$\mathrm{M}=\left(\mathrm{k}‐1\right)\left(\raisebox{1ex}{$\mathrm{\hslash }$}\!\left/ \!\raisebox{-1ex}{$2l_{0}c$}\right.\right)=\left(k-1\right)\cdot m_e/2\alpha ;\left(\mathrm{k}\ge 2;{\mathrm{l}}_0=\raisebox{1ex}{${\mathrm{e}}^2$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}_{\mathrm{e}}{c}^2$}\right.={\mathrm{a}}_{\mathrm{c}}=2.82\mathrm{fm};\alpha =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$137$}\right.;\mathrm{\hslash }=\raisebox{1ex}{$\mathrm{h}$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.\right)$$E1

In a relative recent paper, (2015, [16]), a research team of Science’ Institute for Nuclear Research in Debrecen, Hungary, after some experiments for the detection of dark photons, announced that significant deviation from the internal pair creation during the (e⁺ − e⁻)- transition to the ground state of an excited Be8* nucleus was observed at large angles, which indicates that in an intermediate step, was formed a neutral super-light particle with a mass of ∼17 MeV/c², named X17, (∼34 m_e), the excited Be8^* state being obtained by proton interaction with a target of Li7, i.e. by a reaction:

$$\mathrm{Li}7+{\mathrm{p}}^{+}\to {\mathrm{Be}}^{\ast }\to \mathrm{Be}8+{\mathrm{b}}^0;{\mathrm{b}}^0\to {\mathrm{e}}^{+}+{\mathrm{e}}^{-},\mathrm{m}\left({\mathrm{b}}^0\right)\approx 34{\mathrm{m}}_{\mathrm{e}}$$E2

In another paper, [17], a team of American physicists from California concluded that the evidenced new boson- which was reconfirmed experimentally in 2019 [18], could be the evidence for the predicted X-boson of a fifth fundamental force, of lepton- quarks coupling.

The experimentally evidenced particle with 34m_e suggests that- in accordance with CGT, in the decay of Be8* to its ground state, the b⁰ –boson was emitted by an excited constituent nucleonic quark in the form of a neutral preon. Its stability can be explained by the conclusion that it is formed as cluster of an even number n = 7x6 = 42 quasielectrons, (integer number of degenerate “gammons”, γ ^*(e^*− e^*+)), with mass m_e^* ≈ 34/42 = 0.8095 m_e, i.e. reduced to a value corresponding to the charge e^* = ±(²/₃)e by a degeneration of the magnetic moment’s quantum vortex Γ_μ = Γ_A + Γ_B, given by ‘heavy’ etherons of mass m_s ≈ 10⁻⁶⁰ kg and ‘quantons’ of mass m_h = h⋅1/c² = 7.37x10⁻⁵¹ kg. The considered “gammons” were experimentally observed in the form of quanta of “un-matter” plasma, [19].

The m_e^* -value results in CGT by the conclusion that the difference between the masses of neutron and proton: (m_n -m_p ≈ 2.62 m_e) is given by an incorporate electron with degenerate magnetic moment and a linking ‘gammon’ σ_e (γ^*) = 2m_e^* ≈ 1.62 m_e, forming together a ‘weson’, w⁻ = (σ_e (γ^*) + e⁻), which explains the neutron in a dynamide model of Lenard- Radulescu type [11, 12], as being composed by a protonic center and a negatron revolving around it by the Γ_μ -vortex with the speed v_e^* ≪ c, at a distance r_e^* ≈ 1.36 fm [13], (close to the value of the nucleon’s scalar radius: ∼ 1.25 fm- used by the formula of nuclear radius: R_n ≈ r₀⋅A^1/3), at which it has a degenerate μ_e^S -magnetic moment and S_eⁿ -spin.

The conversion: p_r + e^’ → n_e + ν_e, (e⁻-capture) is explained by the conversion: σ_e (γ^*) → ν_e + ∈.

The degenerate value μ_e^p of the magnetic moment of a protonic positron (giving its charge, as in the Anderson’s model) or of a quasielectron results in CGT by the decreasing of its Compton radius r_μ^e proportional with the quantum density ρ_n of the protonic N^p-cluster in which is placed the electron’s super-dense kernel, (its centroid), from the value:

r_μ^e = 3.86x10⁻¹³m, to the value: r_i = r_μ^p = 0,59 fm, (virtual radius of the proton’s magnetic moment μ_p) given by the quantum mean density increasing, from the value: ${\overline{\mathrm{\rho }}}_e$ to the value: ${\overline{\mathrm{\rho }}}_n\cong \mathit{fd}\cdot \mathit{Np}\cdot {\overline{\mathrm{\rho }}}_e$, conform to equations:

$${\mu }_p=k_p\frac{m_e}{m_p}{\mu }_e=k_p\frac{{\rho }_e}{{\overline{\rho }}_n}{\mu }_e=k_p\frac{1}{f_d\cdot N^P}{\mu }_{\mathit{Bp}}=\frac{e.c\cdot r_{\mu }^p}{\mathit{2}};k_p=\frac{g_p}{g_e}=2.79=\frac{{\rho }_n\left(r^{+}\right)}{{\rho }_n^0}=e^{-\frac{r^{+}}{{\eta }_d}}$$E3

in which: k_P-the gyromagnetic ratio; ${\overline{\mathrm{\rho }}}_e;{\overline{\mathrm{\rho }}}_n$ – the mean density of electron and nucleon;

r_e⁺ = 0.96 fm -the position of the protonic positron’ centroid in report with the proton center; f_d - coefficient of the quasielectron’s mass degeneration, the positron’s vortex Γ_μ being diminished by the distribution of its vortical energy to all degenerate electrons of the particle’s N^p –cluster of ‘gammons’, giving the particle’s magnetic moment by an un-paired quantonic vortex of radius r_μ^p and of circulation: Γ_μ^p = 2πc⋅r_μ^p = g_p(m_e/m_p)⋅Γ_μ^e.

The used electron model supposes an exponential variation of its density, (conform to eq. (3)), given by photons of inertial mass m_f, vortically attracted around a dense kernel m₀ and confined in a volume of classic radius a = 1.41 fm, (the e-charge in electron’s surface).

The superposition of the (N^p + 1) quantonic vortices, Γ_μ^*, of the protonic quasielectrons, generates inside a volume with radius: d^a = 2.1 fm, a total dynamic pressure: P_n = (1/2)ρ_n(r)⋅c² which gives an exponential nuclear potential: V_n(r) = -υ_iP_n of eulerian form conform to (3), with: η* = 0.8 fm and υ_i(0.6 fm)-the ‘impenetrable’ volume [11, 12, 20]:

$$V_n\left(r\right)={\upsilon }_i{P}_n=V_{n0}\cdot e^{-r/\eta \ast };V_{n0}=-{\upsilon }_i{P}_{n0},$$E4

the nucleon resulting as formed by N^p ≈ 54x42 = 2268 quasi-electrons which give a proton density in its center of value: ρ_n^o ≈ N^p⋅ρ_e^o = 5.04x10¹⁷kg/m³, (ρ_e⁰ = 22.24 x10¹³ kg/m³), giving- with υ_i(a_i) = 0.9 fm³: V_s⁰ = 127.5 MeV and: V_s(d = 2 fm) ≈ 9 MeV – value specific to the mean binding energy per nucleon in the nuclei with the most strongly bound nucleons, (9.14 ÷9.15 MeV/nucleon for ⁵⁶Fe, ⁵⁸Fe, ⁶⁰Ni, ⁶²Ni).

The weak force of beta-emission is explained by the disintegration of a linking “gammon”: γ^*(e^*−e^*+) whose superdense centroids give the electronic (anti)neutrino, (the photonic shell being transformed into disintegration energy ∈_d), the couple: w^±(e^± γ^*) forming a “weson” which added to the u⁺-quark (or $\overline{u}$⁻) gives the d⁻-quark, (or⁻d⁺-quark).

The neutron’s disintegration is explained in CGT as a reaction of d-quark’s transforming:

$${\mathrm{d}}^{-}\to {\mathrm{u}}^{+}+{\mathrm{e}}^{-}+{\overline{\mathrm{\nu }}}_{\mathrm{e}}+{\in }_{\mathrm{d}};\left({\mathrm{w}}^{-}\left({\mathrm{e}}^{-}{\mathrm{\gamma }}^{\ast }\right)\to {\mathrm{e}}^{-}+{\overline{\mathrm{\nu }}}_{\mathrm{e}}+{\in }_{\mathrm{d}}\left(0.827\mathrm{MeV}\right)\right)$$E5

(by: σ_e (γ^*) → ${\overline{\mathrm{\nu }}}_e$ + ∈), the relativist speed v → c of the beta radiation electron being obtained with the vortex Γ_μ = Γ_A + Γ_B of the proton’s magnetic moment, whose etheronic part Γ_A explains the magnetic potential A and the atomic electrons’ perpetual rotation, the quantonic part Γ_B explaining the magnetic induction B of the proton’s magnetic field.

In this model, the quasi-crystalline structure of the preonic kernels and of quarks is maintained with the electronic centroids m₀ distanced at d_i ≈ 2x10⁻¹⁷ m, by the equilibrium between the electric and magnetic attraction between paired quasielectrons and a small repulsive field generated by internal photons’ destruction (by zero-point vibrations of the electronic centroids) and by heavy ‘naked’ photons (including also photons corresponding to X-rays and γ-rays- which can be emitted at nucleon’s vibration). The current quark’s radius results of value: r_q ≈ 0.21÷0.3 fm, conform to some older experiments [21].

The model explains the fact that some nuclear isomers such as Hf-178 (t½-31 y.) decay to a low-energy state by emitting gamma rays in concordance with the fact that the electron’s interaction with green laser pulses evidenced γ-rays emission [22, 23].

The number: n = 42 m_e* giving the z⁰-preon may be explained by considering a “quarcin” c₀^*+ = 7x3 = 21 m_e*, (with hexagonal symmetry), resulting that: z⁰ = (c₀^* + ${\overline{c}}_o^{\ast }$) = 34 m_e, (${\overline{c}}_o^{\ast }$-anti-quarcin), the reaction (2) being explained by the z⁰-preon’s decay, in the form:

$${\mathrm{z}}^0\left({{\mathrm{c}}_0}^{\ast }+{{\overline{\mathrm{c}}}_0}^{\ast }\right)\to {{\mathrm{c}}_0}^{+}+{{\overline{\mathrm{c}}}_0}^{+}\to {\mathrm{e}}^{+}+{\mathrm{e}}^{-}+\mathrm{\gamma };\left({{\mathrm{c}}_0}^{+}\to {{\mathrm{c}}_0}^0+{\mathrm{e}}^{\pm }\right).$$E6

(c₀⁺(²/₃e) -‘quarcins’, identifiable as high energy electrons [13]).

In CGT was deduced a quark model of cold forming quark, with effective (constituent) mass giving the particle’s mass by the sum rule, by considering as fundamental stable sub-constituent the basic preon z⁰ = 42 m_e^* ≅ 34 m_e which can form derived “zerons”, (preonic neutral bosons: 2z⁰; z₁(3z⁰); z₂(4z⁰); z_μ(6z⁰), z_π(7z⁰),), the light and semi-light quarks (m_qc² < 1 GeV) resulting by only two preonic bosons: z₂(4z⁰) = 136 m_e and: z_π(7z⁰) = 238 m_e.

The main CGT”s link with the S. M. consists in the fact that- considering the electron’s mass confined in its kernel, the z⁰-preon can be considered as formed by 3 pairs (u_c⁻u_c) of current u_c -quarks formed by a quasielectron (e^*±) surrounded by 3 gluonic ‘gammons’, i.e. with mass: m(u_c) = 7m_e^* ≈ 2.9 MeV/c², the current d-quark resulting with a mass:

m(d_c) = m(u_c) + w⁻(e⁻ γ ^*) = (2.9 + 2.62) MeV = 5.52 MeV, (in the limits accepted by S.M.).

Also, the degenerate gammons γ^*(0.827 MeV) can be considered a ‘gluol’ which is the CGT’s equivalent of the ‘gluon’ considered in the S.M. with a mass’ upper limit: 1.3 MeV.

The u_c –quarks forming by the gluol’s splitting is explainable by the attraction between the resulted ‘free’ quasielectrons and adjacent ‘gluols’, with the forming of a ($u_c{\overline{u}}_c$)-pair.

The cold quarks results as superpositions of preonic bosons z₂ = 4z⁰ and z_π = 7z⁰ with almost the same symmetry (Figures 1 and 2), resulting a constituent quark’ mass equation [13, 14]:

Figure 1.

The m₁^*_, z_π^* and r^*- quark pre-clusters forming from z⁰-preons.

Figure 2.

The cold forming of semi-light quarks by pre-clusters of m_1,2; z₂ and z_π.

$$q=\mathrm{m}+\mathrm{k}\cdot {\mathrm{z}}_{\pi }+\mathrm{n}\cdot \left(\mathrm{k}-2\right)\cdot {\mathrm{z}}_2;\mathrm{m}=\left({\mathrm{m}}_1^{+};{\mathrm{m}}_2^{-}\right);\mathrm{k}=0÷\mathrm{i}=3;\mathrm{n}<\mathrm{k},\left(\mathrm{n}=0÷2\right)$$E7

i.e.: -(k, n = 0) ⇒ q = m_1,2; (k = 1, n = 0) ⇒ q = r^±, (“rark”- un-stable quark); (k = 2) ⇒ q = p⁺, n⁻; − (k = 3, n = 0) ⇒ q = λ^+, λ⁻; (k = 3, n = 1) ⇒ q = s⁻, s⁺; (k = 3, n = 2) ⇒ q = v⁺; v⁻.

From eq. (7), the baryons mass results as combinations (q-q-q) according to equation:

$$M_b={\mathrm{M}}_{\mathrm{q}}+\mathrm{k}\cdot {\mathrm{z}}_{\pi }+\mathrm{n}\cdot \left(\mathrm{k}-6\right)\cdot {\mathrm{z}}_2;{\mathrm{M}}_{\mathrm{q}}=\sum _{\mathrm{i}=1}^3{\mathrm{m}}_{\mathrm{i}};{\mathrm{m}}_{\mathrm{i}}=\left({\mathrm{m}}_1^{+};{\mathrm{m}}_2^{‐}\right);\mathrm{k}=6÷9;\mathrm{n}\le 2$$E8

The particle’s mass results by eqs. (7) and (8) in the approximation of the sum rule applied to the particle’s cold forming, as consequence of the quantum fields’ superposition principle applied to the particle’s cold forming as sum of degenerate electrons, whose total vortical field Γ_v can explain also the nuclear force F_n = −∇V_n(r) conform to eq. (4), [11, 12, 20].

The particles cold forming by clusterizing of preons and quarks forming (‘mark’, ‘park’, ‘nark’, etc.) may result- according to CGT, in a “step-by-step” process [14] consisting in:

• quark pre-cluster forming, as un-collapsed quasi-crystal Bose-Einstein condensate of gammons or of preons, (at E_k = ½m_γv² → μ_c;μ_c- attractive potential, of magnetic or also electric interaction between gammonic electrons or between z⁰ -preons);

• quark (cluster) forming, as non-destructive (partial) collapsed quark pre-cluster; (at E_k = ½m_γv² < μ_c), and:

• elementary particle/dark boson forming as confined cluster of quarks with the current mass in the same baryonic impenetrable quantum volume and completed with a shell of ‘naked’ photons; (a small impenetrable volume of quasielectrons/preons and a repulsive field of zeroth vibrations impede the destructive collapse).

The known possibility to produce bosons and quarks ((q⁻-q⁺) pairs) by high energy (e⁻-e⁺) interactions is explained by the fact that –at distances r → a = 1.41 fm, it results that:

μ_c → e²/8πε₀a = m_ec², so (e⁻-e⁺)- pairs can be formed even at E_k → ½m_ec².

The selection rule for the quarks’ masses, which explains their relative stability, is given by the fact that the cold preonic pre-clusters of z₂₋ and z_π- bosons have almost the same symmetry (in form of star with three arms an of hexagon-Figure 1) and their superposition gives a quasi-stable pre-cluster (of charged pre-quarks or of pseudo-quarks with null charge), which- by cold non-destructive collapsing generated by attraction between electronic/preonic magnetic moments, gives a cold formed quark, with fractional charge.

Even if in CGT is considered that the integer e-charge of a formed particle is given by a single degenerate electron with degenerate magnetic moment, incorporated in the surface of the particle’s quantum volume, formally is useful to consider a fractional charge: (+²/₃)⋅e for the m₁⁺- quark (resulted formally by a z₂-zeron which loosed a degenerate quasielectron- Figure 1) and for p⁺-quarks, the cold quarks with negative charge (−¹/₃)⋅e: m₂⁻, n⁻, λ⁻, s⁻, v⁻, resulting as formed by a cold quark with positive charge (+²/₃)⋅e: m₁⁺, p⁺ and an attached electron with degenerate magnetic moment (→ n⁻) or also a z_π- zeron and one or two z₂– zerons (→λ_s⁻, s_s⁻, v⁻), they having also a positive charged correspondent: λ⁺, s⁺, v⁺, (without attached electron). In CGT the current quarks have a radius of ∼0.21 fm and a repulsive shell of ∼0.1 given by kinetized light photons.

The resulted structure of the fundamental elementary particles, considered as formed “at cold”, in the approximation of the sum rule, by cold quarks with effective mass m_k and fractional electric charge q^± = (+²/₃e or − ¹/₃e), formed as preonic clusters, is given by the sub-structures presented in the Table 1, [11, 12]. The disintegration of the negative quark into a positive quark occurs by the emission of the incorporated (degenerate) electron and of an electronic antineutrino resulted as coupled kernels of a destroyed pair of internal degenerate electrons, forming a linking “gammon”: σ_e (e^+* + e^*) → ν (${\overline{\mathrm{\nu }}}_e$), (as in eq. (5)).

Basic quarks: m1 = (z2- me*) = 135. 2 me,	m2^− = m1^+ +e^− + σe (e^+* + e^−*) = 137,8 me; m2^− → m1 + e^− + ${\overline{\mathrm{\nu }}}_e$; π^+(m1^−m2) → μ^+ + ν(2z^0)
Derived quarks: p^+ (n^−) = m1 (m2) + 2zπ; n = p^+ + w^−; n → p^+ + w^’; w^− = σe + e^−; σe → ${\overline{\mathrm{\nu }}}_e$ + ∈	(p; n) ≈ 611.2me; 613.8me; λ^± = p^+(n^−) + zπ; (λ^−) = 851.8 me; s^± = λ^± + z2; (s^−) = 987.8me; v^± = λ^± + 2z2; (v^−) = 1123.8 me; [p^+(n^−) + z2].
Mesons: ($q\overline{q})$	Baryons: (q-q-q); (q^+ ≡ q(^2/3e); q^− ≡ q(−^1/3e)
μ^± = 2z1 + e^± = 205 me; /μ^+ (206.7 me); (z1 = 3z^0; z2 = 4z^0; zμ = 6z^0; zπ = 7z^0)	-pr = 2p + n = 1836.2me; ne = 2n + p = 1838.8me; / exp.: pr^+(1836.1 me); ne (1838.7me);
π^o = $m_1+{\overline{m}}_1$= 270.4me;/π^0(264.2 me)	-Λ^o = s^− + n + p = 2212.8 me; / Λ^0 (2182.7 me)
π^+ = $m_1+{\overline{m}}_2$ = 273 me; / π^+(273.2 me); π^± → μ^± + νμ (2z0); νμ = zν = 2z0	-Δ^(++;+;0;−) = s^± + λ^± + p^+(n^−) = 2445.6; 2453.4 me; /exp.: Δ^±;0 = 2411 ± 4 me
K^+ = m1 + $\overline{\mathrm{\lambda }}$ = 987 me; / K^+ (966.3 me)	-Σ^+ = v^− + 2p = 2346.2me; Σ^− = v^− + 2n =2351.4me; /exp.: Σ^+(2327 me); Σ^− (2342.6 me)
K^o = m2 + $\overline{\mathrm{\lambda }}$ = 989.6 me; / K^o(974.5 me)	-Σ^o = v^− + n + p = 2348.8 me /exp. Σ^0 (2333 me);
η^o = m2 + $\overline{s}$ = 1125.6 me; / η^0 (1073 me); ϕ^0 = λ + $\overline{v}$ = 1975.6 me; / ϕ^0 (1995 me); N^0* = v + s + λ^+ = 2960.8 me / N^0*(2972 me)	-Ξ^o = 2 s^− + p = 2586.8 me; Ξ^− = 2 s^− + n = 2589.4 me; /exp.: Ξ^0 (2572); Ξ^− (2587.7 me); −Ω^− = 3v^− = 3371.4 me; / exp.: Ω^− (3273 me); N0^3* ≈ 3366 me; predicted: Ω^++ = 3v^+ = 3363.6 me

Table 1.

Elementary particles (theoretic mass (CGT)/experimentally determined mass).

Also, the charge of the muon is given- in the model, by an electron with degenerate magnetic moment attached to a neutral (collapsed) cluster z_μ = 6z⁰. According to CGT, the muon’s disintegration: μ^± → e^± + ν_μ is carried out by the mass conservation, in the form: μ^±(z_μ + e^±) → e^± + ν_μ(2z⁰) + z₂(4z⁰), which explains the fact that the electron and the muon have the same electric charge even if they have very different values of their masses.

Unlike the S.M., which considers the transforming: s⁻ → u⁺ + W⁻; W⁻ → e⁻ + ${\overline{\mathrm{\nu }}}_e$, in CGT we have a weak interaction reactions: s⁻ → p⁺ + z_π + z₂ + w⁻; w⁻ → e⁻ + σ_e → e⁻ + ${\overline{\mathrm{\nu }}}_e$.

It may be also observed that the mass of the m_1,2 –quark is approximately equal to: m_e/α = 137m_e, corresponding to Olavi Hellman’s relation (1) by l₀ = a_c/2 = a = 1.41 fm –classical electron’s radius corresponding to its e-charge distributed in its surface S_e = 4πa² = k₁e.

A compound quark model which used a basic boson with the mass m(π_½) = m_e/α = 137m_e (α- the fine structure constant) close to the values: m(z₂) = 136 m_e and m(m₂⁻) = 137.8 m_e used in CGT, was proposed by D.-Yu Chung [24].

The strong force between quarks is explained in CGT by a ‘bag’ model resulted from the (multi)vortical model of nucleon, of cold genesis, [20].

3. The explaining of the heavy quarks as compound structures

From eq. (7) of the quark’s mass, it results in CGT that every quark has and a positive (q = +²/₃e) and a negative (q = −1/₃e) variant. Also, in report with the S.M. which identified only three quarks with mass < 1GeV/c², CGT deduced by eq. (7) other three quasi-stable quarks with mass < 1GeV/c²: m_1,2(69.5 MeV)- mesonics and: λ(435 MeV); v(574 MeV)- baryonics.

The masses of some “resonance” particles (*) result also in the variant of “cold” forming:

Δ^o* = 2v⁻ + p = 2858.8 m_e; Δ^−* = 2v⁻ + n = 2861.4 m_e; (known mass: 2850 m_e), and: Ξ^−* = 3 s⁻ = 2963.4 m_e; (known mass: 3004 m_e).

But for explain particles with mass heavier than that of the Ω⁻(1672) is necessary to explain the heavy quarks, with mass m_q > 1GeV/c², (of second and third generation).

It is believed generally that the baryon “resonances” are identified as excitations of the quarks, but other researches argue the possibility to obtain the energy spectrum of some baryon resonances as interaction between a baryon and vector mesons [25].

Also, M. de Souza [26] deduced the energy levels of baryon resonances with maximal discrepancy of ∼5% by a simple formula for the energy of the system of three quarks:

$${\mathrm{E}}_{\mathrm{n};\mathrm{m};\mathrm{k}}=\mathrm{h}{\mathrm{\nu }}_1\left(\mathrm{n}+1\right)+\mathrm{h}{\mathrm{\nu }}_2\left(\mathrm{m}+1\right)+\mathrm{h}{\mathrm{\nu }}_3\left(\mathrm{k}+1\right)$$E9

where n; m; k = 0; 1; 2; 3; 4; and hν₁, hν₂, hν₃ are the ground states of the corresponding energy levels of baryonic quarks, being equal to the rest mass’ energy of constituent quarks, (E_q = m_qc²) taken as being: E_q(m_u ≈ m_d) = 0.31 GeV, E_q(m_s) = 0.5 GeV, E_q(m_c) = 1.7 GeV, E_q(m_b) = 5 GeV, E_q(m_t) = 174 GeV, (u, d- up, down; s, c- strange, charm; b, t –bottom, top).

A selection rule for the heavy quarks masses, which- with high discrepancy, may be applied also for the values used by de Souza (which are close to but higher than those used by the Standard Model, m_q^•) was found by R. A. Carrigan Jr. [27] which found an exponential formula for the heavy quark masses, of the form:

$${\mathrm{m}}_{\mathrm{n}}^{•}\approx {\mathrm{m}}_1^{•}\times 3^{\mathrm{n}-1};{\mathrm{m}}_1^{•}={\mathrm{m}}_{\mathrm{s}}^{•}\approx 0.486\mathrm{GeV}/{\mathrm{c}}^2$$E10

as an approximation for the quark masses: m_c^• = 1.55 GeV/c² and m₃^• = m^•_b = 4.73 GeV/c²_, (values used by the Standard Model of the Quantum mechanics). Eq. (10) was explained by the forming of compound quarks with upper mass as tri-quark combinations:

$${\mathrm{q}}_{\mathrm{n}}={\left(\mathrm{q}\overline{\mathrm{q}}\mathrm{q}\right)}_{\mathrm{n}-1}$$E11

i.e- as triplets formed by quarks with adjacent lower mass, (m₂^• ≈ 3 m₁^•; m₃^• ≈ 3 m₂^•): two paired quarks and an un-paired quark giving the charge and the magnetic moment.

In concordance with G.M. it results that eq. (11) maintain the value p = 1/3 for all quarks.

From Figures 1 and 2, by a z⁰-preon with a ratio: length/diameter → 1, it results in CGT that the cold quark’s stability is given by the (quasi)crystalline arrangement of its cluster of preonic kernels (named ‘kerneloids’) of regular hexagonal polyhedron form, with the ratio: length/diameter = 1÷2, i.e.- with 3÷5 levels of preonic bosons z_π (7z⁰) and/or z₂(4z⁰) and a light m_1;2 -quark resulted from a z₂-boson by the loosing of a degenerate electron e^*.

By the principle of similitude, it results that a similar (quasi)crystalline arrangement may result by kernels of (semi)light quarks (m_q < 1GeV/c²) or of heavy quarks (c- or b-quarks), formed as composite quarks, by some (q-$\overline{q}$)-pairs and an un-paired q^± − quark, (eq. (11)).

We observe also that relation (10) between the values of (m_s^•, m_c^• and m_b^•) used by the Standard Model and the values (m_s, m_c and m_b) used by Souza (which are slightly higher) gives relative high discrepancy, i.e.: (m_c; m_c^•) ≠ 3m_s; (m_b; m_b^•) ≠ 3(m_c; m_c^•).

It was shown by author [28] that the values m_s, m_c and m_b used by de Souza [26] and those used by the S.M. [1] can be deduced by the CGT’s model, with eq. (10).

Using the notations: q^*; m_q^* for the cold quark of CGT and its mass, we observe that:

1. m_u ≈ m_d = 0.31 GeV/c² ≈ 612 m_e ≈ m(p; n) -the mass of the nucleonic quarks of CGT;

2. m_s = 0.5 GeV/c² = 978.5 m_e (≈ m_s^* = 987.8 m_e, ∼0.504GeV/c²) -the mass of s-quark resulting almost as in CGT;

3. m_c = 1.7 GeV/c² = 3326.8 m_e used by Souza can be obtained by eq. (10), by taking: m₃ = m_v⁺ ≈ 1121.2 m_e ≈ 0.574 GeV (instead of m_s), i.e- the mass of the superior quark named “vark”, used in CGT (in its positive (+²/₃e) or negative (−¹/₃e) charged variant), which is the adjacent inferior quark mass in report to m₄ = m_c and considering the quark c(m_c⁺) as de-excited state of the triplet with mass: m₄^* = m(c^*) = 3m_v^*(v⁺) = 3363.6 m_e, (1.718 GeV/c²), according to the next de-excitation reaction:

   $${\mathrm{c}}^{\ast \pm }\left[\left({\mathrm{v}}^{\pm }{\overline{\mathrm{v}}}^{\pm }\right){\mathrm{v}}^{\pm }\right]\to {\mathrm{c}}^{\pm }\left(\sim 1.702\mathrm{GeV}/{\mathrm{c}}^2\right)+{\mathrm{z}}^0\left(34{\mathrm{m}}_{\mathrm{e}}\right)$$E12

   -in concordance with the reaction (2). It results also a new Ω⁻ -variant: Ω⁻(3v⁻): Ω⁻(1702).

4. For m_b, we can obtain similarly the value used by de Souza [26], by eq. (10) in which: m₄^* = m_c and m₅^* = m(b^*±) = 3m_c ≈ 5.1 GeV/c², by the next de-excitation reaction:

   $${\mathrm{b}}^{\ast \pm }\left[\left({\mathrm{c}}^{\pm }{\overline{\mathrm{c}}}^{\pm }\right){\mathrm{c}}^{\pm }\right]\to {\mathrm{b}}^{\pm }\left(\sim 5\mathrm{GeV}/{\mathrm{c}}^2\right)+{\mathrm{z}}_3\left(204{\mathrm{m}}_{\mathrm{e}}\right);{\mathrm{z}}_3={\mathrm{z}}_{\mathrm{\mu }}=\left(2\times 3\right){\mathrm{z}}^0=2{\mathrm{z}}_1$$E13

   The considered reaction (13) is concordant with the reaction: π⁰ → z_μ(6z⁰) + ν_μ, and with the known reactions: π⁺ → μ ⁺ +ν_μ; μ^± → e^± + ν_μ(${\overline{\mathrm{\nu }}}_{\mathrm{\mu }}$)_, but considered as in CGT, by the mass conservation’ rule, i.e. - in the form: μ^± → z_μ + e^± → e^± + ν_μ(2z⁰) + z₂(4z⁰).

   The explanation of the value: m(b^* -b) = z_μ = 6z⁰, released at the b^*-cluster’s de-excitation, may be obtained by writing the mass of the resulted b-quark by eqs. (12) and (13), in the form:

   $$\mathrm{m}\left(\mathrm{b}\right)\approx 3\left(3\left({\mathrm{m}}_{\mathrm{v}}–{\mathrm{z}}^0/3\right)-2{\mathrm{z}}^0\right)=9\left({\mathrm{m}}_{\mathrm{v}}–{\mathrm{z}}^0\right);\left(\mathrm{v}\left(1123{\mathrm{m}}_{\mathrm{e}},)\right.\to {\mathrm{v}}^{·}\left(1089{\mathrm{m}}_{\mathrm{e}}\right)+{\mathrm{z}}^0\right)$$E14

   which shows that –according to the CGT’s model and in concordance with eq. (10), at the b-quark’ “hot” forming from v-quarks (resulted from nucleonic quarks), each constituent v-quark loose a z⁰-preon, as consequence of its meta-stability, the kernel of this z⁰-preon being emitted from the v-quark’s kernel.

   Applying this conclusion to Ω⁻(3v⁻) it results that: m(Ω⁻) = (3371.4 – 3x34) m_e = 3269.4 m_e, (compared with the experimental mass: 3273 m_e, resulting a discrepancy of 0.14%).

   The conclusion of eq. (14) is applicable and to the current b-quarks contained by the surface of the t-quark’s kernel and is specific to high energy interactions.

5. The t-quark results as collapsed cluster: t ^± = (7x5)m(b^±) ≈ 175GeV, (17($b\overline{b}$) + b^±), the structure of its kernel resulting by similitude with the preonic structure of the s-quark (CGT- Figure 2) as formed from a pre-cluster of regular hexagonal polyhedron form with length/diameter→ 1, in acceptable concordance with the experimentally obtained value of its constituent mass (176.8 ≈ 177 GeV/c², [29], p. 135), with upper stability given by the crystalline form of its current mass, formed by kernels of b-quarks, conform to the model.

The form: (7x3)m(b^±) = 105 GeV with un-paired b^± − quark and the forms:

h⁰ = (7x3)m(c) = 35.7 Mev/c² and (7x5)m(c) = 59.5 Mev/c² results also as possible.

The values of m_c and m_b used by de Souza and of the t-quark result by the eqs. (12) and (13) with discrepancies of 0.31%, 0.08%, and respective- of ∼1%.

The resulted compound quark model of heavy quarks suggests also the possibility of a cold genesis variant of these quarks, in the form: c^*($v\overline{v}v$) and b^*($c^{\ast }{\overline{c}}^{\ast }{c}^{\ast }$) and of some heavy baryons of low vibration of their quarks, formed by three heavy ‘cold’ quarks.

Because the quarks v^± and s^± of CGT have close masses, the quarks c^• ≈ 1.55GeV and b^• ≈ 4.73GeV of the S.M. [29] result from direct combinations between s^± and v^± or as derived from c-, b- quarks, approximately in the form:

$${\mathrm{c}}^{·}=\left(\left(\mathrm{s}\cdot \overline{\mathrm{s}}\cdot \mathrm{v}\right)-{\mathrm{z}}^0\right):{\mathrm{c}}^{\prime }\left({\mathrm{s}}^{-}{\overline{\mathrm{s}}}^{-}{\mathrm{v}}^{\pm }\right)\to {\mathrm{c}}^{·\pm }+{\mathrm{z}}^0\left(34{\mathrm{m}}_{\mathrm{e}}\right);\Rightarrow \mathrm{m}\left({\mathrm{c}}^{·}\right)\approx 1.557\mathrm{GeV}/{\mathrm{c}}^2$$E15

$${\mathrm{b}}^{·\pm }=\left(\left({\mathrm{c}}^{·\pm }{\overline{\mathrm{c}}}^{·\pm }{\mathrm{c}}^{\pm }\right)-{\mathrm{z}}_2\right):{\mathrm{b}}^{\prime }\left({\mathrm{c}}^{·\pm }{\overline{\mathrm{c}}}^{·\pm }{\mathrm{c}}^{\pm }\right)\to {\mathrm{b}}^{·\pm }+{\mathrm{z}}_2\left(136{\mathrm{m}}_{\mathrm{e}}\right);\Rightarrow \mathrm{m}\left({\mathrm{b}}^{·}\right)\approx 4.75\mathrm{GeV}/{\mathrm{c}}^2$$E16

the forms: c’, b’, being “isomeric” (excited, meta-stable) states. The explanation for the value m(b’ –b^•) = z₂(4z⁰) is the fact that the resulted b^• -quark’s structure is approximately:

$$\begin{array}{l}{\mathrm{b}}^{·}\approx \left[\left({\mathrm{c}}^{\prime }\left(\mathrm{s}\cdot \overline{\mathrm{s}}\cdot \mathrm{v}\right)-{\mathrm{z}}^0\right)\left({\overline{\mathrm{c}}}^{\prime }\left(\mathrm{s}\cdot \overline{\mathrm{s}}\cdot \overline{\mathrm{v}}\right)-{\mathrm{z}}^0\right)\left({\mathrm{c}}^{\wedge }\left(\mathrm{s}\cdot \overline{\mathrm{v}}\cdot \mathrm{v}\right)-{\mathrm{z}}^0\right)\right]=\left[{\mathrm{c}}^{·}{\mathrm{c}}^{·}\mathrm{c}\right]-{\mathrm{z}}_2=\mathrm{b}-2{\mathrm{z}}_{\mathrm{\pi }};\\ \left(\mathrm{v}\to \mathrm{s}+{\mathrm{z}}_2\Rightarrow \mathrm{c}\to {\mathrm{c}}^{·}+{\mathrm{\pi }}^0\left(2{\mathrm{z}}_2\right)\right).\end{array}$$E17

i.e.-formed by three compound quarks: c’,$\overline{c}$’, c^{^} - de-excited by emission of a z⁰-preon whose kernel is released from their quarcic kernel. So, the quarks s^•(486) c^•(1550); b^•(4730) used by the S.M. result from the CGT’s quarks: s⁻, c⁺, b⁻ by the reactions:

$$\mathrm{c}\left(1700\right)\to {\mathrm{c}}^{·}\left(1561\right)+{\mathrm{\pi }}^0\left(2{\mathrm{z}}_2\right);\mathrm{b}\left(5000\right)\to {\mathrm{b}}^{·}\left(4756\right)+{\mathrm{z}}_6\left(2{\mathrm{z}}_{\mathrm{\pi }}\right);{\mathrm{s}}^{\pm }\left(500\right)\to {\mathrm{s}}^{·\pm }\left(483\right)+{\mathrm{z}}^0.$$E18

Eq. (18) indicates that the S.M.’s quarks are quasi-stable ‘hot’ formed quarks, resulted from excited metastable quark: s⁻, c⁺, b⁻, and that excited quarks v, s loose a z⁰-preon.

It is explained why the particles’ masses result by the sum rule mainly by the variants of heavy quarks’ masses used by de Souza and by CGT, although experimentally were evidenced the mass variants of quarks characteristic to the S.M.

By (18), by: s^± → s^•± + z⁰, it results:

Δ^±;0 = [2445.6; (2453.4) -34] = [2411.6; (2418.4)] m_e;

Λ^o = (2212.8–34) m_e = 2178.8 m_e, (discr. < 1%); η^o = m₂ + $\overline{s}$^• = 1091.6 m_e, (discr. < 1.8%).

It results also that in the formula (9) it may be used also the rest energy of the quarks “v” (‘vark’) or λ(“lark”) in the first term, with hν₁ = m_qc² and that is more natural to interpret the numbers (n, m, k) = 0, 1, 2, 3, .., which multiply the baryonic resonance’ energy in eq. (9), as being numbers of neutral pseudo-quarks q_i⁰ added to the charged q_i^± − quark which gives the ground state according to a compound quark model, these neutral pseudo-quarks having the same rest mass, i.e.: q^±_i = ($q\overline{q}q$^±)_i-1; q⁰_i = (q⁻q⁻q⁺)_i-1.

This conclusion may explain also some mass differences of baryon resonances, resulted approximately equal to the mass of a boson z⁰, z₂, z_π, Σz₂; Σz_π.

For example, the neutral correspondent of the quark s^± is the scalar meson f₀(500)^o and the neutral correspondent of the quark c^± is the scalar meson ρ(1700)⁰ .

A next ‘cold’ heavy quark formed by clusterizing may result, according to the sum rule, by:

m₄^* = (m_f; m_f^*) ≈ (3m_b’; 3m_b^*), (f_q^± − “fark”, “frozen” quark, with mass: m(f_q) ≈ m(Ω_bbb)).

An ‘exotic’ heavy but un-stable quark may be formed by a q^± − quark surrounded by six q⁰-pseudoquarks or three pairs $q-\overline{q}$.

The t-quark’ forming from protonic jets: j(P) + j($\overline{P}$) is explained by the fact that the strong force F_N = −∇V_n determines a (quasi)crystalline (re)arrangement of the current quarks, by re-organizing of the internal structure of nucleonic quarks: m_u,d ≈ m_p,n ≈ 612 m_e.

If we interpret the energy: ΔE_q = m(δ_k)c², loosen at the de-exciting of the quarcic cluster q^c_k + 1, as binding energy between quarks, (by similitude with a nucleus), it may be obtained a semi-empiric relation for the mass of the heavy quarks c ≈ 1.7 GeV and b ≈ 5 GeV:

$$\mathrm{m}\left({{\mathrm{q}}_{\mathrm{n}}}^{\mathrm{c}}\right)=3^{\mathrm{n}-1}\left[{\mathrm{m}}_1–\left({\mathrm{z}}^0/3\right)ln\left(3^{\mathrm{n}-1}{3}^{\mathrm{n}-2}\right)\right];\mathrm{n}>1$$E19

with: m₁ = m_v = 0.574 GeV/c²; z⁰ = 34 m_e and ln(3ⁿ⁻¹3ⁿ⁻²) = ln3²ⁿ⁻³, which gives:

n = 2 → m(q₂^c)c² = 1.703 GeV; n = 3 → m(q₃^c)c² = 4.994 GeV; n = 4 → m(q₄^c)c² = 14.64 GeV.

The logarithmic part of the second term of the expression (19) indicates that the considered binding energy per v-quark increases with the number of v-quarks.

A relative similar semi-empiric relation may be found also for the quarks c^• ≈ 1.55 GeV and b^• ≈ 4.73 GeV, of the S.M., but in the form:

$$\mathrm{m}\left({{\mathrm{q}}_{\mathrm{n}}}^{·}\right)=3^{\mathrm{n}-1}\left[\left({{\mathrm{m}}_1}^{·}+{\mathrm{\delta }}^{·}\right)+\left({\mathrm{z}}^0/3\right)ln{3}^{\mathrm{n}-2}\right],\mathrm{n}>1;\left({{\mathrm{m}}_1}^{·}+{\mathrm{\delta }}^{·}\right)=\left(2{\mathrm{m}}_{\mathrm{s}}+{\mathrm{m}}_{\mathrm{v}}–{\mathrm{z}}^0\right)/3$$E20

with: m₁^• = m_s ≈ 500 MeV/c², ⇒ δ^• = 19 MeV/c², (or: m₁^• = m_s^• ≈ 486 MeV/c², ⇒ δ^• = 33 MeV/c²), giving: n = 2 → m(q₂^•)c² = 1.557 GeV ≈ m(c^•); n = 3 → m(q₃^•)c² = 4.728 GeV.

The expression (20) is characteristic to mass addition to the tri-quark cluster, (as consequence of the fact that was taken: m₁^• = m_s instead of: (2m_s + m_v)/3), as in the case of the Sakharov’s equation [30], (which adds a term of spin-spin interaction at the total mass of quarks) and shows a link between the masses of s-, c^•-, b^•- quarks and z⁰(34 m_e).

4. The explaining of the strong interactions between particles

It is possible to explain similarly – in CGT, also some strong interactions in which preonic z_k- bosons and bosonic pairs ($q-\overline{q}$) of the quantum vacuum, with total mass M ≤ Q/c², separated by the interaction energy Q, may generate heavier quarks in combination with the initial quarks, as in the next possible interaction:

$$\begin{array}{l}\text{(Experimental reaction)}:{\mathrm{\pi }}^{-}\left({\overline{\mathrm{m}}}_1+{\mathrm{m}}_2\right)+{\mathrm{p}}_{\mathrm{r}}\left(2{\mathrm{p}}^{+}+{\mathrm{n}}^{-}\right)+\mathrm{Q}\\ \to {\mathrm{\Lambda }}^{\mathrm{o}}\left(\mathrm{s}+\mathrm{n}+\mathrm{p}\right)+{\mathrm{K}}^{\mathrm{o}}\left({\mathrm{m}}_2+\overline{\mathrm{\lambda }}\right);\\ {\overline{\mathrm{m}}}_1+{\mathrm{p}}^{+}+\mathrm{Q}\to {\overline{\mathrm{m}}}_1+{\mathrm{m}}_1+{\mathrm{z}}_6+\mathrm{Q}={\mathrm{\pi }}^{\mathrm{o}}+{\mathrm{z}}_6+\mathrm{Q}\to \left({\mathrm{s}}^{-}+\overline{\mathrm{s}}\right);{\overline{\mathrm{s}}}^{+}\to \overline{\mathrm{\lambda }}+{\mathrm{z}}_2;\\ {\mathrm{s}}^{-}+{\mathrm{n}}^{-}+{\mathrm{p}}^{+}\to {\mathrm{\Lambda }}^{\mathrm{o}};\overline{\mathrm{\lambda }}+{\mathrm{m}}_2\to {\mathrm{K}}^{\mathrm{o}};\\ -\text{permitted reaction;}\left({\mathrm{z}}_2=136{\mathrm{m}}_{\mathrm{e}};{\mathrm{z}}_6=2{\mathrm{z}}_{\mathrm{\pi }}=476{\mathrm{m}}_{\mathrm{e}}\right).\end{array}$$E21

with (p; n) ≈ (u; d). The conservation of the ‘generation’ number g- in G.M., (or of the ‘strangeness’ S –in S.M.) corresponds to g(λ; v) = g(s) = 1; g(m₁; m₂) = g(u; d) = g(z_k) = 0 and to the possibility of quarks transforming: q₁^g ↔ q₂^g’ only for quarks with the same generation number: g = g’, but in CGT this indicates only the low probability of the quark’s transforming into another with different generation number: P(g’ ≠ g) ≪ 1, because in a weak interaction of type: q₁^g → q₂^g’ + Σz_k the sum ΣS is not mandatory conserved. For example, the reaction: π⁻(${\overline{m}}_1$ + m₂) + p_r(2p + n⁻) + Q_i → Λ^o(s + n + p) + π^o(m₁ + ${\overline{m}}_1$)- forbidden by the law of S- or g-number’s conservation, is theoretic possible in CGT by: p → m₁ + 2z_π;

Q_i → z_π + z₂; m₂ + 3z_π + z₂ → s⁻, (which not conserve ΣS or Σg), but is low probable.

A (semi)empiric relation for the particles lifetime, sustaining also the particles’ cold forming, is obtained in CGT by considering the μ^± − lepton (lifetime: τ₀ = 2.2x10⁻⁶ sec.) as single-particle and taking into account a vibration energy ε_v of the constituent quarks, which- according to CGT, generate a partial destruction of the particle’s intrinsic vorticity, with the loosing of a part: Δm_p of internal ‘naked’ photons, giving: τ_k ∼ 1/Δm_P(T).

At an ordinary temperature: T = 300 K of the particles’ environment, the lifetime τ_k(ε_v) of the baryons (n = 3 quarks and τ_B ≈ 10⁻¹⁰ sec.) and of the mesons (n = 2 quarks and τ_m ≈ 10⁻⁸ sec.) is dependent to its intrinsic temperature T_q conform to equation [11, 12]:

$${\tau }_k=\frac{{\tau }^0}{k_v\cdot {\mathit{10}}^{\mathit{2}n}}\approx \frac{{\tau }_0\cdot m_p}{\mathrm{\Delta }{m}_P\left(T\right)};{\tau }^0\cong 3x{\mathit{10}}^{-14}sec.;k_v=\frac{{\epsilon }_v}{{\epsilon }_v^o}=\frac{n\cdot {\nu }_i}{{\nu }_c^o}=\frac{n\cdot T}{T_N};{\mathrm{T}}_{\mathrm{N}}\cong 2x{10}^{12}\mathrm{K}$$E22

in which: ν_c⁰ represents the critical frequency of the phononic energy ε_v⁰ of quark vibration at which the proton’s disintegration take place: ν_c⁰ = ν_c(T_N ≈ 2x10¹²K) ≈ 4x10²² Hz.

Eq. (22) may explain the fact that the heavy baryons with composite heavy quarks can have a longer lifetime at T → 0 K but cannot have a long life at an ordinary temperature, [11, 12].

5. The nucleon’s stability and the quasi-crystalline structure of quarks

In CGT, similarly to the S.M.’s constituent quark model, it is considered that the electron’s mass is formed by a ‘kerneloid’ containing the (super)dense kernel m₀ of radius r₀ ≈ 10⁻¹⁸ m and by a shell of bosons which in the electron’s case are ‘naked’ photons, in concordance with the evidenced possibility to obtain a B-E condensate of photons, [31].

This electronic kerneloid is equivalent to an ‘impenetrable’ quantum volume (similar to that of the nucleon), having a radius r_ie ≈ 10⁻² fm- in accordance to some high-energy scattering experiments reported by Milonni et al. (1994, p.403 [32]).

The possibility to explain reactions of strong interactions between particles by heavier quarks transforming into lighter quark(s) and bosonic preon(s) specific to CGT but also by heavier quarks forming from these subcomponents, as in eq. (21), indicates that these sub-components maintain their higher stability also in strong interactions, by a quasi-crystalline arrangement of the electronic kerneloids k^e of their z⁰-preons, the resulted preonic kerneloids forming the quark’s kerneloid- which can be considered as being its current mass. The radius of the z⁰-preon’s kerneloid k^z results of value: r_z = 3.5x10⁻² fm, [13].

The preonic quasielectrons retain their photonic shell (also at the preon’s releasing) by the vortical field of the Γ^e_μ -vortices of the degenerate magnetic moments, maintained by their kernels, in accordance with a classic equation of electron’s intrinsic rest energy [11, 12]:

$${\mathrm{m}}_{\mathrm{e}}{\mathrm{c}}^2\approx \mathrm{½}\int {\mathrm{\epsilon }}_0{\mathrm{E}}^2\mathrm{dV}\left(\mathrm{r}\right)\approx \mathrm{½}\int {\mathrm{\mu }}_0{\mathrm{H}}^2\mathrm{dV}\left(\mathrm{r}\right);\left(\mathrm{E}=\mathrm{c}\cdot \mathrm{B};\mathrm{r}=0÷{\mathrm{r}}_{\mathrm{\mu }}=\mathrm{\hslash }/{\mathrm{m}}_{\mathrm{e}}\mathrm{c}\right)$$E23

which explains the electron’ mass m_e as saturation value: n⋅m_f of magnetically (vortically) confined ‘naked’ photons. These Γ_μ^e -vortices are maintained by the negentropy of the quantum vacuum given by etherono-quantonic winds (fluxes) which explain also the constancy of the magnetic moment of the free charged particles, in CGT [11, 12].

Eq. (23) explains the maintaining of the constituent mass also to quarks changed in strong interaction between interacting particles conform to the sum rule, as that of eq. (21).

The quasi-crystalline arrangement of preonic kerneloids of quarks formed by clusterizing is ‘inherited’ from the quarcic non-collapsed quasi-crystalline pre-cluster formed by pre-clusters of z_{2 -} and z_{π -} preonic bosons, (Figures 1 and 2), the quarks confining force resulting in CGT by magneto-electric interaction between quasielectrons and by a pressure of kinetized photons giving a repulsive shell of radius 0.6 fm in accordance with a “bag” model of strong interaction with a bag’ radius r_i^* = a_i ≈ 0.6 fm [20], (as in the model of Toki & Hosaka).

The proton’s high stability- compared with the stability of heavier baryons and with that of non-leptonic mesons, is explained conform to eq. (22) by a stronger coupling of its quarks by magneto-electric interactions between pairs of quasielectrons also to the radial direction, (Figure 3), as consequence of the same size of the constituent quarks, which reduces the quark’s vibrations. Also, to the axial direction, the magnetic moments of the quasielectrons of a z⁰-preon are axially coupled, the central, un-paired Γ_μ^* -vortex-tube giving the preon’s magnetic moment; similarly, the axially coupled preons give the magnetic moment of the quarcic cluster of preonic bosons (z₂, z_π).

Figure 3.

Baryonic and preonic kerneloid.

From Figure 4 representing a preonic z_π -layer of a quarcic kerneloid it results that the calculated radius value: r_z = 3.5x10⁻² fm [13] of the preon’s kerneloid, ensures a mean distance: d_i ≈ (²/₃)⋅r_z ≈ 2.3x10⁻² fm between the electronic centroids m₀ on the radial direction, which gave a value: r_ie = 1.15 x10⁻² fm for the electron’s mechanical radius, (as in Ref. [32]), the minimal value of the preon’s length resulting of value: l_z = 6xd_i ≈ 0.14 fm.

Figure 4.

Preonic z_π -layer of quarcic kerneloid.

Because the quasi-crystalline structure of (u, d)- quark’s kerneloid have three layers- in CGT, (m_1;2; z_π; z_π -Figure 2), with (4; 7; 7) z⁰-preons, it results a minimal length of the (u; d)- quark’ kerneloid: l_q = 3l_z ≈ 0.42 fm, (at T → 0 K).

The minimal radius of the quark’s kerneloid (specific to its cold state) results of value:

r_q⁰ ≈ 3xr_z = 0.105 fm, value which- compared with the experimental radius value of the quark’s current mass: r_q ≈ 0.21 ÷0.3 fm [21, 33], indicates a small vibration liberty (l_v ≈ 0.1÷0.2 fm) of the z⁰-preos inside the quark’s kerneloid, equivalent to a repulsive shell δ_q.

The mechanical radius of the nucleonic impenetrable volume υ_i is given by three coupled quarcic kerneloids (Figure 3) and it results of value: r_iⁿ ≈ 2r_q + δ_q ≈ 0.45 ÷0.5 fm –close to the value of l_q, (r_iⁿ/l_q → 1) –in good accordance with the experiments of electrons and protons scattering to nucleons, [21], (∼0.45÷0.5 fm) and with the “bag” model of strong interaction resulted in CGT [20].

The last experimentally determined value for the quark’s radius: ∼4x10⁻¹⁹ m [6] corresponds in this case to the radius of the super-dense electronic centroid, [11, 12, 20], being close to the upper limit determined by X- rays scattering on electron [4, 5].

The quarcic cluster’s rotation, giving the quantum number l of orbital angular momentum, is generated by the magnetic moment’s vortex Γ_μ, according to CGT [11, 12].

6. The existence of new heavy quarks’ variants

By re-obtaining the constituent quark c^• ≈ 1.55 GeV of the S.M. by CGT, in the form:

c^• = [($s\overline{s})v$) – z⁰], it results that other composed heavy quarks of type: q_c = ($q_1{\overline{q}}_1{q}_2$) may be formed with the semi-heavy quarks: s^±, v^±, and by a de-excitation reaction.

From the de-excitation reactions (12)–(16) we observe also that the emitted preonic boson z_k at the ‘hot’ forming of compound quarks q_c is smaller for lighter composed quarks and bigger for heavier ones. With the notations:

δ₁ = (z_0); δ₂ = [z₁(3z⁰); z₂(4z⁰)]; δ₂^’ = (z₃ = z_μ = 6z⁰); δ₃ = (z₄ = z_π = 7z⁰); δ₃^’ = (z₅ = 2z₂ = π⁰),

q₁ = (s; v); q₂ = [c^•(1.55GeV); c(1.7GeV);c^{^}]; (q₃; q₃^’;q₃^*; q₃^*’) = (b^•; b; b^*; b^c); q₄ = f_q(b; b^•); q_t = t(b; b^•), it results as possible the variants (“flavors”) of composed quarks presented in Table 2.

q^c(compound)	q2(CGT)	q3 (CGT)	q3^* (CGT)	q3^’* (CGT)	q4 (CGT)	qt = (t;h)
q1 = (s; v)	c^•($s\overline{s}v$ -z0)	b^•($c^{•}{\overline{c}}^{•}c$ -z2)	b^•($c^{•}{\overline{c}}^{•}c$)	b^c($c^{\ast }{\overline{c}}^{\ast }{c}^{\ast }$)	fq ($b\overline{b}b$)	t(7x5)b
q2^’ = q^s	c($v\overline{v}v$ -z0)^s	b^^($c^{•}\overline{c}c$- z2)	b^^($c^{•}\overline{c}c)$	b^c’($c{\overline{c}}^{\ast }{c}^{\ast }$)	fq ($b^{•}{\overline{b}}^{•}{b}^{•}$)	t(7x5)b^•
^ - new	c^^($v\overline{v}s$ -z0)	b^^($c^{•}{\overline{c}}^{•}{c}^{•}$-z2)	b^^($c^{•}{\overline{c}}^{•}{c}^{•}$)	b^c(${\overline{c}}^{\ast }{c}^{\ast }$c^^)	fq ($b^{•}\overline{b}b)$	t(7x5)b^*
	c^^($s\overline{s}s$ -z0)	b($c\overline{c}c$ - z3)	b($c\overline{c}c$)	b^c’(${\overline{c}}^{^}{c}^{^}$c^*)	fq ($b^{•}{\overline{b}}^{•}b)$	h’(7x5)c
m(q^ck); GeV(/c^2)	1.557 (SM); (1.70)^s+; (1.704)^s−1.631; 1.483	4.744 (SM); 4.887; 4.601; (5.008)^− = b^s−(4.996)^+ = b^s+; (c ≈ 1.7); (c^• = 1.557)	4.814; 4.957; 4.671; (5.1)^+; (5.112)^−	5.154;5.16; 5.144;5.13; 5.084;; 5.014;	15.000; 14.232; 14.744; 14.488; 15-z5= =14.861	175; (166)^•; (180.4)^*−; (178)^*+; h’(59.5); t’($h^{\prime }{\overline{h}}^{\prime }{h}^{\prime }$) =178.5
δ(zk); (emitted)	δ1 = z0	δ2 (z2; z3); (z3 = zμ)			δ3(z4; z5)	?

Table 2.

Possible variants of compound heavy quarks (theoretical masses).

The used notations allow a general formula for the compound heavy quark forming, in the form of a weak de-excitation reaction (leptonic or non-leptonic):

$${{\mathrm{q}}^{\mathrm{c}}}_{\mathrm{k}+1}={{\mathrm{q}}^0}_{\mathrm{k}+1}{\left({\mathrm{q}}_{\mathrm{k}}{\overline{\mathrm{q}}}_{\mathrm{k}}{\mathrm{q}}_{\mathrm{k}}\right)}^{\prime }-{\mathrm{\delta }}_{\mathrm{k}};\mathrm{k}=1;2;3;\mathrm{m}\left({\mathrm{\delta }}_{\mathrm{k}}\right)\le \mathrm{m}\left({\mathrm{\pi }}^0\right);\mathrm{m}\left({\mathrm{\delta }}_{\mathrm{k}}\right)/\mathrm{m}\left({{\mathrm{q}}^0}_{\mathrm{k}+1}\right)\le 2\%$$E24

which do not depends on the quarks’ spins, in concordance with the experimentally resulted conclusion that the quarks’ spins have a low contribution to the particle’s spin, (4 ÷ 24%, [34]). Eq. (24) differs from the Sakharov’s equation [30] which- at the total mass of quarks, adds a term of spin–spin interaction of the quarks.

The quantity δ_k can be experimentally observed as gamma-quantum with energy E_γ ≥ 17 MeV corresponding to preonic bosons: z⁰, 2z⁰, z₁, z₂, or –respective, as μ⁰-, π⁰- mesons, the value of δ_k being proportional to the interaction energy, (δ_k ∼ E_i), the reaction (24) corresponding to a weak interaction of leptonic or non-leptonic type.

Also, the similitude with the semi-light quarks forming by bosonic preons (Figure 2), which explains the t-quark’s mass, indicates as possible also meta-stable states:

q_xc = (7xn)m(q^±) with n = 3÷6 kerneloids of q^± − quarks with m(q^±) - effective mass, the forms of type: q_hc = (7x3)m(q^±) or of type: q_hc = (7x5)m(q^±) = [17($q\overline{q}$) + q^±] being compound quarks and the forms with n = 4, 6, being pseudo-quarks, (neutral clusters).

Because- in the case of a t-quark, the de-excitation of at least the centrally contained b_t –quarks’ kernels is impeded by the others, we can take a value m(b_t) ≈ ($c\overline{c}$ c –z₂) ≈ 5.03 GeV which gives: m_t ≈ (7x5)m(b_t) = 176 GeV/c², (closer to the known value: 177 GeV/c²).

Heavier quarks (q₄, q_t) can also result according to eq. (24) or with the kernel formed as cluster of (7x5) kerneloids of s-, v-, c- or f_q-quarks, (ex.: h’(‘hark’) =17($c\overline{c})$+ c^± , m(h’) = 59.5 GeV), or in the form: [3( $q_1{\overline{q}}_1$ )+ q₂^±] with m(q₂ ) < m(q₁ ), (ex.: 3($b\overline{b})$+ c^± = 31.7 GeV).

A variant of t-quark results also in the form: t’ = ($h^{\prime }{\overline{h}}^{\prime }{h}^{\prime }$) = 178.5 GeV.

In Table 2 are presented the mass variants of compound heavy quarks resulting by eq. (24). The notations q^*, q (q^s) and q^• indicate cold quark, ‘hot’ meta-stable quark and respective-de-excited stable quark, (of S.M. type, resulted from a meta-stable quark, of Souza type).

7. The explaining of the heavy particles’ forming

Eq. (24) results as specific also to heavy particles’ ‘hot’ forming from de- excited heavy quarks, such as the in the case of the heavy pseudo-scalar or vector mesons (Tables 3 and 4 - Annex A and B) and of the baryons with J^{p 3}/₂, (Table 6), by a generalization of the form:

$${{\mathrm{M}}_{\mathrm{b}}}^{\mathrm{d}}={{\mathrm{M}}_{\mathrm{b}}}^0{\left({\mathrm{q}}_{\mathrm{i}}{\mathrm{q}}_{\mathrm{j}}{\mathrm{q}}_{\mathrm{k}}\right)}^{\prime }–\mathrm{m}\left({\mathrm{\delta }}_{\mathrm{Q}}\right);\mathrm{m}\left({\mathrm{\delta }}_{\mathrm{Q}}\right)\le \mathrm{m}\left({\mathrm{\pi }}^0\right),\mathrm{m}\left({\mathrm{\delta }}_{\mathrm{Q}}\right)/{{\mathrm{M}}_{\mathrm{b}}}^0\le 2.7\%.$$E25

m(δ_Q) = Σm(z_k) being given as sum of z_k -bosons emitted by the most excited q - quarks.

7.1 The heavy baryons

The most part of the heavy baryons with J^p½ (Table 5, Annex C) results with discrepancy of less than 1% as ground states ()^∕ of (q_i-q_j-q_k)’ combinations of de-excited quarks, without de-excitation reaction of the quarcic cluster, excepting some excited baryons with J^p½ as those having u- or d- quark(s) in their structure (considered in the p-; n- variant of CGT), for which the de-excited state ()^d is obtained by emission of at most (1÷2) z⁰-preons, with the exception of the double charmed Xi, whose de-excited state is obtained by emission of a z₂ bosonic preon, (the case of Ξ_cc⁺⁺(3621), resulted as [(pcc) - z₂]).

This fact indicates that by their magnetic attraction, the quarks with anti-parallel spins give an upper stability to the formed particle. The possibility to obtain the mass of the majority of baryons with J^p = ½ as simple sum of masses of component de-excited quarks indicates that the lost quantity δ_k is explained rather by eq. (24), as released preonic boson(s), than by eq. (19)- as matter corresponding to the binding energy.

The heavy baryons with m_qc² > 1 GeV and J^p3/2 (Table 6, Annex D), result in the model as de-excited states (q-q’-q”)^d obtained by the emission of z⁰- or z_k- “zerons” (k = 1÷4), Σm(z_k) representing less than 2.7% of its mass, (excepting Ξ^∗0(1532), resulted as: [(λ⁺vv) –z₁], for which m(z₁) ≈ 3.4%), being explained similarly also the recently discovered particles:

Σ_b⁻(6.097); X_b^−’(6.100) [37], as: (6.096)^d[(vvb)^’-z₁]; (6.113)^d[(vvb)^’-2z⁰] and: Z_c(3900); X_c(3872) as: [(v⁻c⁺c⁺)^’–z₂]; [(v⁻c⁺c⁺)^’ -z₃](3870)^d.

The higher instability of the non-de-excited baryons with J^p3/2 may be explained by the conclusion that the rotation of the quarcic cluster –giving the orbital angular momentum l, determines a lower magnetic attraction and a higher vibration of the current quarks.

The most de-excited states of heavy baryons with J^p3/2 are obtained by emission of a “zeron” z₁ = 3z⁰, i.e.- of a z^o- preon per each q-quark. A good argument for the proposed model is also the fact that- by eq. (24) applied for heavy J^3/2- baryons, it can be explained also the particles: Z_c(3900); X_c(3872), (considered as tetraquarks) and: B^*_j(5732) (considered as meson [35, 36]), as forms: Z_c^’(v⁻ c⁺ c⁺)^’–z₂ = (ccs)^d = 3904.5 MeV/c²; X_c’(v⁻ c⁺ c⁺)^’ –z₃ = 3869.7 MeV/c², and: B^*_j = 5748(b λ n) –z⁰ = 5731 MeV/c² (discrepancy < 0.2%). Small differences, under 17 MeV, between baryon resonances of the same family, such as between Δ(1900), Δ(1905), Δ(1910), Δ(1920), Δ(1930), may be explained by partial destroying of z⁰- preons at relative high interaction energies, with the loosing of some “gammonic” pairs of electronic kerneloids from its structure. An argument can be the fact that- at the c-quark’s ‘hot’ transforming, are emitted more electrons than muons, [38].

Also, heavy neutral bosons, quasi-stable at T → 0 K, can result in form of clusters with mass: M_Z = ∑_K(n_1⋅z_π + n_2⋅z₂); (n_1;2 ≤ 5), conform to CGT.

7.2 The heavy mesons

The theoretically resulted masses of heavy mesons are presented in Tables 3 and 4, which show that these mesons may be explained as pairs (q_k⁻q_l) by considering the both variants of the charm and of the bottom quarks: c^s = c and c^•; b^s = b and b^•, (the Souza’s and the S.M.’s mass variants), the pair of heavier quarks being de-excited by the emission of a preonic ‘zeron’ z_k, in accordance with eq. (25) but considered for M^d($q_k{\overline{q}}_l$).

For example, differences between vector mesons, such as: ω⁰(0.7826 GeV) and ϕ⁰(1.0195GeV), (difference: ∼ 464 m_e ≈ 2 z_π = 476 m_e) can be explained by the structure: ω⁰(($d\overline{s}$) – z⁰) = 0.795 GeV and ϕ⁰(($s\overline{v}$) –z₁(3z⁰)) = 1.022 GeV.

The discovered particle X(5568), with un-clarified structure (supposed to be a tetra-quark [36]) results by the model as charged tetra-quark X^±[b${\overline{\mathrm{\lambda }}}^{\pm }{m}^{\pm }{\overline{m}}^{\pm }$)] or as heavy meson: X_b($v\overline{b}$) = 5574 MeV, which decays by a reaction of type: X⁺ [b$\overline{\mathrm{\lambda }}(m_1{\overline{m}}_2$) ] → B_s^0’($b\overline{\mathrm{\lambda }}$) + π⁺ or: X⁻ [b$\overline{\mathrm{\lambda }}(m_2{\overline{m}}_1$)] → B_s^0’$(b\overline{\mathrm{\lambda }}$) + π⁻, respective – by a reaction: X_b^± ($v\overline{b}$) → B_s^0’($\mathrm{\lambda }\overline{b}$) + π⁰(2z₂), resulted in CGT by the v-quark’s transforming: v^± → λ^± + 2z₂, (Table 1).

It results also that neutral W-, Z-bosons can be explained as neutral clusters, in the form: W⁰ = (2x2³)b ≈ 80 GeV; Z⁰ = 2x3²b ≈ 90 GeV (doublets of cubic or squared clusters of b-quarks), the cubic form: 3³b = 27b = 135 GeV resulting as possible heavy quark.

Also, a neutral variant of the recently obtained new value for the t-quark’s mass: 199 GeV/c² [39] may be explained as a pseudo-quark with regular hexagonal polyhedron form of its kerneloidic cluster, in the form: m(t^•) = (7x6)m(b^•), (b^• ≈ 4.75 GeV).

The resulted meson ($h^{\prime }{\overline{h}}^{\prime }$) = 119GeV has a mass close to those of the boson H⁰ (125.7GeV).

The heavy τ- particle (∼1.77GeV), considered as heavy lepton in S.M., with mass resulted from the empirical Koide’s formula, is obtained as exotic meson: τ($m_{1,2}{\overline{c}}^{\pm }$) = (1.7695 GeV).

7.3 The multi-quark particles

Some particles evidenced with a multi-quark type structure may be also explained similarly, by de-excitation reactions, specific to a reaction of weak type, (Annex E, Table 7).

Θ⁺(1540) (considered pentaquark ($\mathit{\text{uudd}}\overline{s}$)) results as: Θ⁺[($p\overline{n}{s}^{•}\overline{\mathrm{\lambda }}$] by: s → s^• + z⁰.

The determined masses of relative recently discovered ‘charmed’ particles (july 2022, [40]) of penta-quark and tetra-quark type: -P_Ψ^Λ(4338 MeV/c²) /($\mathit{\text{sucd}}\overline{c}$) and -T^a_cs(2900)⁺⁺/($\overline{s}c\overline{u}d$) with a neutral variant: T^a_cs(2900)⁰/($\overline{s}c\overline{d}u$), whose constituent quarks considered by S.M. give by the sum’ rule: 4262 MeV/c², respective: 2712 MeV/c², (discrepancy higher than 1.7%, respective 6%), can be explained by the sum’ rule with the mass/structure: P_Ψ^Λ($\mathit{\text{ducd}}\overline{c}$) = (2x313 + 312 + 2x1700) = 4338 MeV/c² and:

T^a_cs($\overline{v}c\overline{d}u$) = (574 + 1700 + 312 + 313) = 2899 MeV/c², (with discrepancy ≈ 0, in CGT).

Also, it result as possible octo-quark particles in cubic form, as: ½W⁰ = (2³)b^± ≈ 40 GeV.

8. Concordance with experiments

The main experiments that sustain indirectly the used explanatory model of CGT, are:

1. the transforming of gamma quantum of 1 MeV into a pair (e⁺e⁻) by an electrostatic energy E_e ≥ 2m_ec² and the experimentally obtaining of “gammons” in the form of quanta of “un-matter” plasma, [19], (which argue the model of gamma quantum of 1 MeV as pair of degenerate electrons and the possibility of particles’ forming from “gammons”);

2. the experimentally obtaining of a Bose-Einstein condensate of photons, (a “super-photon”), by a German team (2010, [31]), indirectly proving the existence of the rest mass of photons (considered in CGT), as in the case of the ‘dark photon’ theorized in the Q. M.);

3. the “stopped light” experiment [41], that evidenced a ultra-slow passing of a photon through a Bose-Einstein condensate, and the field-like nature of the “dark” energy evidenced in astrophysics [42] (which sustains the vortical photon/electron model of CGT);

4. the decay: t → W+ j(b⁻b), [43], (evidencing that b-quarks are components of t-quark);

5. the experimentally obtaining of W, Z-bosons and $q-\overline{q}$ pairs by (e⁻ -e⁺) or (e⁻-p⁺)-interactions and the charm-quark decaying into muons and electrons [36], (which sustain the possibility of particles’ forming as clusters of pairs of degenerate electrons).

6. the same size order for the experimentally determined radius’ values of the scattering center(s) inside the electrons and inside the nucleons, (electronic dense centroids, in CGT).

9. Conclusions

• The possibility to explain the mass spectrum of a large number of particles by preonic compound quarks, by the approximation of the sum rule, with the heavy quarks: c, b, formed as triplets: q_n = ($q_k{\overline{q}}_k{q}_k$)_n-1 or q_n = ($q_k{\overline{q}}_k{q}_l$)_n-1 of quarks q_k, q_l with adjacent lower mass: v^±(573 MeV) or/and s^±(∼500 MeV) resulted in CGT (as clusters of z⁰–preons) and with participation of the (semi)light quark λ^± (435 MeV) of CGT, sustains indirectly the particles’ model of CGT and the possibility of a cold genesis scenario in conditions of B-E condensate’ pre-clusters forming, (T → 0 K; E_k = p²/2m_q → μ _q), and argues the conclusion that the entire mass of the particles comes from constituent quarks. Also, the resulting model indicates the cold forming of a quasi-equal number of negatrons and positrons in the Proto-universe’s period (before Big-Bang), by chiral quantum fluctuations, (vortices).

  In this case, in CGT, the genesis of z⁰-preon (X17-boson) from degenerate electrons results as being fundamental in the particles’ natural explaining, (as the Higgs boson in the S.M).

  Also, by the fact that the cold quarks result as superpositions of preonic bosons z₂ = 4z⁰ and z_π = 7z⁰ with almost the same symmetry, the resulting model explains naturally not only the fact that quarks can be transferred in strong interaction from a particle to its interaction partner with the same constituent mass (carrying only its bosonic shell), but also the possibility of paired quarks forming from relativist jets of negatrons and positrons, in the sense that the explanation of the S.M. for this phenomenon, supposing the possibility of a quarks pair forming from a pair of relativist electrons with increased mass via an intermediary virtual photon is identified in CGT as a formal explanation, based on a relation that depends on the reference frame of the observer and not to a microphysical phenomenon, the Einsteinian speed-depending mass increasing resulting as apparent in CGT, (being explained as apparent effect generated by a real decreasing of the values of the electric field, E_L ∼ γ_c⁻³; E_T ∼ γ_c⁻¹, (γ_c = 1/√(1 –v²/c²), and of the magnetic field: B ∼ γ_c⁻¹, [44]).

• The fact that the quarks: ‘sark’, ‘chark’,‘bark’, (‘strange’, ‘charm’, ‘bottom’- in S.M.) were experimentally evidenced in the mass variant of S.M. can be explained by the conclusion that they result from the Souza’s/CGT’s variant by loosing of z⁰-, π⁰-, z₆(2z_π)- boson.

• The CGT’s model do not exclude the possibility of small additions (< 2%) of gammons and z⁰-preons to the mass of quarks also at some excited heavy particles as in the case of Z_b(10650) [38], (Z^*[(($u\overline{u}{\overline{b}}^{-}{b}^{+}$)⁺+2z⁰] = (10624)^d + 34 = 10,658 MeV) -relative similar to the m₁-quark or to the proton’s case- which acquiring a negatron and a linking ‘gammon’ σ_e (γ^*) becomes neutron, in CGT, (the limit 2% being given by: (m₂⁻- m₁⁺)/m₂ ≈ 2.6/137, explained by eq. (5), by the mass of the “weson” w⁻ = σ_e (γ^*) + e⁻ = 2.6 m_e).

• It results also that - if the z⁰-, z_k- preonic neutral bosons or/and pseudo-quarks q⁰ (theoretically considered in CGT) are in a cold state (T → 0 K), being neutral particles they interact weakly with photons of electromagnetic radiation and they can explain in this case a part of the Universe’s dark matter.

• The main CGT’s link with the Standard Model consists in the fact that- considering the electron’s mass confined in its kernel, the z⁰-preon (identified as being the experimentally evidenced X17-boson) can be considered as formed by 3 pairs ($u_c{\overline{u}}_c$) of current u_c -quarks formed by a quasielectron (e^*±) surrounded by 3 gluonic ‘gammons’, (‘gluols’- in CGT), i.e. with mass: m(u_c) = 7m_e^* ≈ 2.9 MeV/c², the current d-quark resulting with a mass:

  m(d_c) = m(u_c) + w⁻(e⁻ γ ^*) = (2.9 + 2.62) MeV = 5.52 MeV, (in the limits accepted by S.M.), with the difference that in CGT the radius r₀ ≤ 10⁻¹⁸ m is the radius of the electron’s kernel.

  Also, the degenerate ‘gammon’ γ^*(0.827 MeV) can be considered a ‘gluol’ which is the CGT’s equivalent of the ‘gluon’ considered in the S.M. with a mass’ upper limit: 1.3 MeV, but in the CGT’s model of quark the cluster of ‘gluols’ generates a saturated state, as a “glasma” in the S.M.

  The u_c –quarks forming by the gluol’s splitting is explainable by the attraction between the resulted ‘free’ quasielectrons and adjacent ‘gluols’, with the forming of a ($u_c{\overline{u}}_c$)-pair.

  Another similitude consists in the fact that the ‘weson’ w^± of CGT and the W^±-boson of SM, even if they have different nature, they transform similarly, (w^± → e^± + ν_{e (}${\overline{\mathrm{\nu }}}_e$) + ∈), the second generation variant, (→μ^± + ν_μ(${\overline{\mathrm{\nu }}}_{\mathrm{\mu }}$)), being specific to the transforming of the couple: (m_1,2z₂) or (⁻m_1,2z₂) incorporated in the preonic cluster of a semi-light quark, conform CGT.

• Regarding the generation model (G.M.), the previous observation explains the difference of generation number (g) between (e, ν_e)- and (μ + ν_μ)- leptons and between (u, d)- and (s, c)-quarks, as consequence of the fact that the first leptonic doublet results by the transforming of a current quark (u_c, d_c), (i.e.- leptonic) and the second leptonic doublet results by the transforming of a constituent quark, (i.e. -non-leptonic). But in CGT the conserving of Σg indicates only the low probability: P(g’ ≠ g) ≪ 1 of quark’s transforming of type: q₁^g ↔ q₂^g’ + Σz_k with g’ ≠ g, because- in these weak interaction reactions, the strangeness’ conserving is not mandatory, they being allowed by the conservation of Σp.

  Also, the sum rule used in CGT corresponds to the conservation of the particle number, p, (without the use of the rishons model), which explains also the form (21) of the strong interaction process by the conclusion that the neutral bosons γ^*, z_k, (gammons, ‘zerons’) have p = 0.

  So, it results that the G.M. is compatible with CGT’s model as specific formalism, the natural restrictions of the CGT’s model being based on the total charge and mass-energy conservation rule, (with mass→energy conversion but without the relativist speed-depending mass increasing), which explains also the selection rule for the quarks’ masses.

  It also results the possibility to extend naturally the list of quarks, with the advantage of the explaining of the particles’ mass spectrum by the sum rule, by de-excitation reactions.

Heavy mesons (MeV/c^2 ); -experimental mass- [35]	Theoretic mass, (de-excited meson, CGT)^d , MeV/c^2 ; (p; n) ≈ 312; 313	Observations /; Predictions , (MeV/c^2)
η’ (957.6); f^0(980); ^1/3(($u\overline{u})$ +($d\overline{d})$+ ($s\overline{s})$)	(965)^d = η”(1000)’- 2z^0 ; ( )’ -excited; (983)^d = η”(1000)’- z^0 ;	η”(($s+\overline{s})$’ = 1000; ($s+\overline{\mathrm{\lambda }})$= (935)^’
ηc(2980.3) ($c\overline{c})$; D^*sJ(2860); ηb (9300) ($b\overline{b})$	2992.5: $\left(c^{•}{\overline{c}}^{•}\right)$^d = $\left(c^{•}{\overline{c}}^{•}\right)$’ -zπ(7z^0); 9338.4$:\left(b^{•}{\overline{b}}^{•}\right)$^d = $\left(b^{•}{\overline{b}}^{•}\right)$’- zπ	[($c\overline{c})$’ -zπ] ≈ 3278; (c→ c^•); $\left(b^{•}{\overline{b}}^{•}\right)$= 9460; (b→b^• + z6)
D^+;D^0(∼1865); ($c\overline{d})$; ($c\overline{u})$	(1863)^d = ( $c^{•}\overline{n}){}^d$ = ($c\overline{n})$’ - π^0 ; (c→c^•)	[c($v\overline{v}v$^±)$\overline{n}$]^*≈ 2031
X(5568) (? )^x [36]	5574 ($v\overline{b})$^d →Bs($\mathrm{\lambda }\overline{b})$’ + π^0; (v→λ+π^0)	(x-un-clear structure)
Ds^+(1968.4) ($c\overline{s})$	1968 ( $c^{•}\overline{\mathrm{\lambda }})$^d = ( $c^{•}\overline{\mathrm{\lambda }})$’ – z^0	[c($v\overline{v}v$^±)$\overline{s}$]^* ≈ 2218
B^+;B^0(∼5279) ($u\overline{b})$; ($d\overline{b})$	5278 ($p\overline{b}$)^d = ($p\overline{b}$)’ –z1(3z^0)	($p{\overline{b}}^{•}$); ($n{\overline{b}}^{•}$) ≈ 5042
Bs^0(5366.3), ($s\overline{b})$ Bs^0’(5372); (Λb(5425)^r	(5365.5)^d = ($\mathrm{\lambda }\overline{b})$’ –z2(4z^0); Bs($\mathrm{\lambda }\overline{b})$’ = 5435; (5382)^d = Bs($\mathrm{\lambda }\overline{b})$’- z1	($s\overline{b})$’– z2 = 5430.5 (Λb); ($s{\overline{b}}^{•}$)’ – z2 = 5244
Bc^+(6276±4) ($c\overline{b})$	(6297)^d = $\left(c^{•}{\overline{b}}^{•}\right)$= ($c{\overline{b}}^{•}$)’-π^0(2z2)	($c{\overline{b}}^{•}$)’= 6440; ($c\overline{b})$= 6700

Table 3.

Theoretic masses of heavy pseudo-scalar mesons, (CGT).

Heavy mesons (MeV/c^2); -experimental mass- [35]	Theoretic mass, (de-excited meson, CGT)^d, MeV/c^2;	Observations + Predictions (MeV/c^2)
ρ^+(775); ρ^0 (775.26); ρ^−(775); ($u\overline{d})$; ($d\overline{d})$ ; ($d\overline{u})$	(777.3)^d = ($p\overline{s})$’ – 2z^0; (778.3)^d = ($n\overline{s})$’ – 2z^0	z^0 = 17.374 MeV/c^2; z2 = 69.5; z3 = zμ = 6z^0
ω(782.65) ; ($u\overline{u}+d\overline{d})$/√2	(795.6)^d = ($n\overline{s})$’ – z^0	813 = ($n\overline{s})$
ϕ (1019.46) ; ($n\overline{s})$	(1022)^d = ($s\overline{v})$’ -z1(3z^0)	(1009.5) = ($s\overline{s})$
J/ψ (3096.9); Θ^0c(3.099); ($c\overline{c})$	(3096.7)^d = ($c^{•}{\overline{c}}^{•}$ )’ - z^0	m(c^•) = 1557 (CGT)
ϒ(1S) (9460.3) ; ($b\overline{b})$	(9462.6)^d = ($b^{•}{\overline{b}}^{•}$ )’ - z^0	m(b^•) = 4744 (CGT)
K^∗+(891.6)($u\overline{s});$K^∗0(895.8)($d\overline{s})$	(∼900)^d =$(s\overline{\mathrm{\lambda }}$^±)’- 2z^0	870; 871$,(\mathrm{\lambda }\overline{\mathrm{\lambda }}$^±)’
D^*+(2010.3)($c\overline{d})$ D^*0(2007)($c\overline{u})$;	2012($c\overline{n})$’ ; 2011($c\overline{p})$’	( )’-excited state
D^∗+s (2112.1) ($c\overline{s})$	(2117.6)^d = ($c\overline{\mathrm{\lambda }})$’ - z^0	m(c) ≈ 1700
B^∗+; B^∗0(5325.2) ($u\overline{b})$ ; ($d\overline{b})$	(∼5331)^d = ( ${\mathrm{\lambda }}^{\pm }\overline{b})$’- z3	5312($p\overline{b});$ 5313($n\overline{b})$
B^∗0s(5415.4) ($s\overline{b})$	( 5417.6)^d = ($\mathrm{\lambda }\overline{b})$’ - z^0	m(b) ≈ 5000
B^∗+c (unknown) ; ($c\overline{b})$	6700 ($c\overline{b})$’ ; 6557( $c^{•}\overline{b})$’	($c^{•}{\overline{b}}^{•}$ )’ = 6300

Table 4.

Theoretic masses of heavy vector mesons, (CGT).

Baryons, J^p ½ experimental mass (GeV/c^2), [29, 36, 37], CDF collab.2009^*	Theoretic mass, (cold baryon, CGT)* p^*; n^*(∼0.312;); λ^−*(0.435); s^−*(0.504); v^−*(0.574); c^*+(1.718); b^*−(5.166); b^*+(5.154)	Ground st./De-excited baryon; ()-ground st.; ()^d -de-exc.(theor.); s^−(0.504); c^±(∼1.7); b^±(∼5.0); -predicted baryons ()^p; (GeV)
N (0.938÷0.939); (udd)	∼0.939; (ppn); (pnn),	(^r) = recent dis.; ()’-excited st.
-Δ^(++;+;0;−) (1.232)	∼1.25; (s^± + λ^± + p^+(n^−))*	(1.25)’ – z^0 = (1.233)^d; (s → s^•)
-Λ^0 (1.116); Λ^0*(∼1.6) (uds)	∼1.13 (n + p + s)*; 1.583 (v + 2 s)^*	(1.13)’-z^0 = 1.113; 1.583(v + 2 s)^’
-Σ^+; Σ^−; Σ^0 (1.189; 1.192; 1.197); (uus; uds; dds)	∼1.199; ∼1.2; (v + 2p)^*; (v + p + n)^*; (v + 2n)^*	(1.199)’; (1.2) -discrepancy (CGT): 0.25%
- K^0*(1.522); Ξ^−* (1.535) (dss)	1.514 (v + s + λ)*; (s s s)^* = 1.51	- (1.51);
-Λc^+ (2.286) (udc); -Ds0 (2317)^r [36]	2.343 (pnc)^* = (pnc^*); (mc^* = 3mv^*)	-(2.325)’ (≈ m(Ds0); (pnc)’ - z^0 = (2.308)^d; d.0.4%
Λc (2627)	2.653 (λsc)^*	(λsc) = 2.635; −discr. 0.3%
-Λb^+(5.619)(udb); Λb^0(6.071)^r [37]	5.791(pnb)^*;(nnb)^*; 6.228(svb^+)^*	(5.625); −dis. 0.1%; (6.074)
-Σc^++(2.454)(uuc); Σc^+(2.453)(udc)	2.465 (pλ^+c)^*; 2.466 (pλ^−c)^*	∼(2.448); −discrep. 0.3%
-Σc^0(2.4537) (ddc)	2.467 (nλ^−c)^*;	(2.449); −discrep. 0.3%
-Σb^+(5.811) (uub)	5.79 (ppb)^*; 5.913 (pλ^+b)^*	(ps^+b) ≈ 5.812; −dis. 0.05%
-Σb^0(5813.5) (udb)	5.791(pnb)^*; 5.914 (pλ^−b)^*; (ps^−b)^*	(ps^−b) =5.813; (pnb)^d = 5.62
-Σb^−(5.815) (ddb)	5.79(nnb)^*;5.91(nλ^−b)^*; 5.96(nsb)^*	(ns^−b) =5.814; −dis. 0.12%
-Ξc^+(2.467) (usc); - Ξc^0 (∼2.469) (dsc)	2.526 (psc)^*; 2.527 (nsc)^*	(nsc)’ = (2.513)’; (nsc)’–2z^0 = (2.479)^d;(dis.0.5%)
-Ξc^’+ (2.575); (usc); (‘) = “prime”	2.604 (pvc)^*	(2.586); −discrep. 0.4%
Ξc^’0 (2.578); (dsc); Λc(2593)	2.605 (nvc)^*	(2.587); − discrep. 0.35%
-Ξcc^++ (3.621); (3.627); (ucc); Λ(3621); ψ’(3686)	3.748 (pcc)^*; (s^±c^•c^•)^* = 3.648; (pcc)^’ – z^0 = (3694,6)^d	(3.712)’; (3.618)(s^±c^•c^•); (vc^•c^•)’–z2 = (3.618)^d; (dis. ≈ 0)
Ξcc^+ (unknown) (dcc)	3.749 (ncc)^*	(3.713)^’;(ncc)’– z2 = (3.643)^d
Ξb^0 (5.788) (usb)	(psb)^* = 5.978;	(5.812)^’ –z^0; (pvb) = 5.886
Ξb^0 (5.792)^0; Ξb^− (5.796)^− (dsb)	(nsb^±)^* ≈ 5.979;	Ξb(5.796)^d ≈ Ξb^*(5.813)^’–z^0
Ξb^0’ (usb); Ξb^−‘(dsb); (unknown)	[(p;n)vb]^* ≈ 6.053; 5.914[(p;n)λb]^*	[(p;n)vb) ≈ (5.887)’; (5.748)’
Ξb^−(5.935)^r(ssb);Λb^’(5.92)^r(usb) [37]	(λsb)^* = 6.1	(5.935)^’ (λsb)^’; −discr. → 0
Ξbb^0; Ξbb^− (unknown); (ubb), (dbb)	10.644 (pbb)^*; 10.645 (nbb)^*	(10.312)^’
Ξcb^+; Ξcb^0 (unknown);(ucb), (dcb)	7.196 (pcb)^*; 7.197 (ncb)^*	(7.012)^’; (7.013)’
Ξcb^+’; Ξcb^0’ (unknown); (ucb),(dcb)	7.317 (λ^+cb)^*; 7.319 (λ^−cb)^*	(7.135)^’; (7.136)^’
-Ωc^0 (2.695); (2.698) (ssc^+)	2.718 (ssc)^*;	(2.7)(ssc); (λλc) =2.57
Ωb^*−(6.054),(ssb); Λb^0**(6.072)^r, (*)	(svb)^* = 6.240; (ssb)^* = 6.166;	(6.074)’(sv^−b^−)^’- z^0 = (6.056)^d
-Ωcc^+(scc); Zc(3900); ψ2(3823)^r	3.936 (scc)^*; (λcc)^* = 3.871	Zc(scc)^’ = 3.9; (λcc)^’ = 3.835
-Ωcb^0 (unknown) (scb)	7.384 (scb)^*; 7.458 (vcb)^*	(7.2)^’; (7.274)^’
-Ωbb^− (unknown) (sbb)	10.832 (sbb)^*; (vbb)^* = 10.906	(10.5)^’; (vbb)^’ = (10.574)’
-Ωccb^+; −Ωcbb^0,(ccb); (cbb), (unkn.)	8.602 (ccb)^*; 12.046 (cbb)^*	(8.4)^’; (11.7)^’
-Λb^’(6.146)^r; (6.152)^r (ssb)	6.302 (vvb)^*;	(vvb)^’ = (6.148)^’;
D3^*0(2.760)^r;Ω c(2.768)^r; B^*j(5.732)	2.796 (v^− s^− c^+)*; 5.902(b^* λ n)	(2.774); (5.748)’- z^0 = (5.730)^d
-Ac^0; Ac^− (unknown -predicted)^p	(λvc^+)^* = 2.727; (λvc^−)^* = 2.73;	(λvc)^’ ≈ (2.7)’; (2.71)^p
-Λc(2.86)^*); (Ωc^*)?; (Ωcc^*)?	(vvc)^* = 2.866; (vcc)^* = 4.01	(2.85)^p(vvc)^’; (3.974)^p(vcc)^’
-(Ωb^*)?^− (unknown- predicted)	6.036 (λλb)^*; 6.175 (λvb)*	(5.87)^p(λλb)’; (6.01)^p(λvb)’
? Rhcc; Rhcb (unknown-predicted)	(h’c c)^* = 62.936; (h’c b)^* = 66.384	(62.9)^p; (66.2)^p
? Rhhc; Rhhb (unknown-predicted)	(h’h’c)^* = 120.7; (h’h’b)^* = 124.16	(h’h’c)^’ = (120.7)^p; (124.)^p

Table 5.

Theoretic masses of de-excited (“hot” formed) baryons, J^p ½, (CGT).

Baryons experimental mass (MeV), [29] J^p 3/2	Theor. mass, GeV; (de-excited quarks)^’: λ^−(0.435); u; d = p; n (0.312); s(0.5); v^−(0.574); c (1.7); b (∼5.0)	Theoretic mass, (GeV); of de-excited baryon ()^d; + predicted baryons- z1(3z^0); z2(4z^0); z3(6z^0); ()’-excited state
-N^0*; Λ^0* (1520); udd; sdu	(v + s + λ)’ ≈ (s s s)^* = 1.514	(1.514)
-Λb^0 (5912); −Λb^0 (5920); (?);	(λ s b)^’ = 5.935	(λ s b)^’- z^0 ≈ 5.917
Σs^∗−(1385) dds; Σc^∗++(2518) uuc	(v s p)^’ = 1.390; (ps^+c)^’ = 2.511	1.390; 2.511
Σ^∗+c(2517.5); Σ^∗0c(2518.8)	(ps^−c)^’ = 2.512; (ns^−c)^’ = 2.513	2.512; 2.513
Σ^∗+b (5832.1), uub; Σ^∗0b, udb	(pvb)’ = 5.886; (nvb^+)’ = 5.887	((p;n) v b^+)’ – z1 ≈ (5.835)^d
Σ^∗−b (5835.1) ddb	(n v b^−)’ = 5.899	(n v b^−)’ – z1 = (5.847)^d
Ξ^∗0 (1531.8) uus	(λ^+ v v)’ = 1.58; (s^+s v) = 1.57	(λ^+ v v)’ - z1 = (1.531)^d
Ξ^∗− (1535) uds	(λ^− v v)’ = 1.584	(λ^− v v)’ - z1 = (1.532)^d
Ξ^∗+c(2645.9); Ξ^∗0c(2645.9),(u/d)sc	(s^± + s^− + c)^’ = 2.7	(s^±s^− c^+)^’-z1 ≈ (2.647; 2.648)^d
Dj^*(2760); Dj(2740)	(s^± + v^− + c^*)^’ = 2.774	(s^± v^−c^*)^’-z^0;2z^0 ≈ 2.76; 2.74
Ξcc(3519); Xc1(3510) ucc;	(λ^± c^• c^•)’ = 3.549 ≈ Xc2(3556)	((3.549)’ - 2z^0 = (3.515)^d
Ξ^∗+cc; Ξ^∗++cc (unknown), (d/u)cc; ψ (3770) ucc; X(3842.7)	(λ^± + c + c)^’ = 3.836; (s^±c c)^’ = 3.900;	(λ^±c c)^’ - z1 = (3.783)^d; (s^± c c)^’ - z1 = (3.848)^d
Ξ^∗0b (5945.5); Ξ^∗−b(5955)^r (u/d)sb	(s^− + s^− + b^±)’ = 5.996; 6.01	(2 s^− + b^±)^’-z1 ≈ (5.94; 5.95)^d
Ξ^∗0bb; Ξ^∗−bb (unknown) (u/d)bb	(λ^± + b^− + b^−)’ = 10.459	(s^± + 2b^+)’ – z2 = (10.431)^d
Ξ^∗+cb (unknown) ucb	(λ^− + c^+ + b^+)’ = 7.135	(s^−c^+b^+)’ - z1 = (7.147)^d
Ξ^∗0cb (unknown) dcb	(λ^− + c^− + b^+)’ = 7.139	(s^−c^−b^+)’ - z1 = (7.151)^d
Ω^−(1672.45) sss	(v^− v^− v^−)’ = 1.722	(v^− v^− v^−)’ – z 1 = (1.670)^d
Ω^∗0c (2766) ssc	(v^− + s^− + c)^’ = 2.774	discr.0.3% (isomeric.)
Ω^∗−b (unknown) ssb	(v^− + s^− + b^−)^’ = 6.074	(v^− s^− b^−)^’-z3 = (5.9697)^d
Ω^∗+cc (unknown) scc; Zcs(3985); Xc2(3930); Zc(3900); Xc(3872) [36]	(v^− + c^+ + c^+)^’ = 3.974 (Zcs); (3.974)’ -2z^0 = Xc2(3.939)^d	(v^−c^+c^+)^’–z1 = (3.922)^d; (v^− c^+ c^+)^’ –z2 = (3.9045)^d; (v^− c^+ c^+)^’ –z3 = (3.8697)^d
Ω^∗0cb (unknown) scb	(v^− + c^+ + b^−)^’ = 7.274	(v^−c^+ b^−)^’ – z3 = (7.1697)^d
Ω^∗−bb (unknown) sbb	(v^− + b^− + b^−)^’ = 10.574	(v^−b^− b^−)^’ – z3 = (10.470)^d
Ω^++ccc (unknown) ccc	(c^+ + c^+ + c^+)^’ = 5.1; (c^+• + c^+• + c^+•)^’ = 4.67	(c^+ c^+ c^+)^’ – z3 = (5.0)^d; (c^+•c^+•c^+•)^’– z2 = (4.6)^d
Ω^∗+ccb (unknown) ccb	(c^+ + c^+ + b^−)^’ = 8.4; (c^+• + c^+• + b^−•)’ = 7.858	(c^+ c^+ b^−)^’ -z3 = (8.296)^d; (c^+• c^+•b^−•)^’-z3 = (7.753)^d
Ω^∗0cbb (unknown) cbb	(c^+ + b^− + b^−)^’ = 11.7	(c^+ b^− b^−)’ -z3 = (11.595)^d
Ω^∗0bbb (unknown) bbb	(b^+ + b^− + b^−)^’ ≈ 15.0	m(Ω^∗0bbb)’ ≈ m (f^±)
Σb^−(6.097)^r; Xb^−’(6.100)^r [37]; Ξb^−’(6.100)^r; ()^r-recently discover.	(s^− + v^− + b^−)^’ = 6.074; (v^− + v^− + b^−)^’ = 6.148	(6.096)^d = (vvb)^’-z1 = Ξb^∗π^0 (vvb)^’- 2z^0 = (6.113)^d

Table 6.

Theoretic masses of de-excited (“hot” formed) baryons, J^p3/₂, (CGT).

Multi-quark particle (CGT); (q-q-q…q) or (q-‾q-q -‾q); (c = c^+(^2/3e)) ; b = b^−(-^1/3e)); (t = t^+(^2/3)e; t ≈ (7x5)b	Theoretic mass, GeV/c^2,; (de-excited q. ()^d./excited bar.()^’; u;d; (p;n)^d (∼0.312)^d; λ^−(0.435)^d; s^− (∼0.504)^d; v^−(0.574)^d ; c^+(1.7)^d ; b^−(∼5.0)^d; m(t) = (7x5)m(b) = 175	Closest experimental; value ( )^e , GeV; [36, 37]; (particle/mass value); -observations-
$\left[p\overline{d}s\overline{\mathrm{\lambda }}\right]$; $\left[p\overline{d}{s}^{•}\overline{\mathrm{\lambda }}\right]$	(1.56)’; [(1.56)’- z^0] = (1.543)^d;	K^+(1.555); Θ^+(1.54) ; (s→s^•)
[$p\overline{p}c\overline{v}$^−]^+ ; [$p\overline{p}b\overline{v}$^+]^−; [$p\overline{n}c\overline{v}$^−]^++; [$n\overline{p}c\overline{v}$^−]^0	(2.898); [(6.198)’^−- z1] = (6.146)^d; (2.899) -2z^0 = (2.864)^d	Xc(2.9)^e; Λb^0(6.146)^d [37]; T^acs (2.9)^e [40] ; D^*sJ(2860)^−
[$p\overline{s}s\overline{v}$^−]^+; [$s\overline{p}c\overline{v}$^+]^− ;	(1.886)’ ; [(1.886) –z^0] = (1.868.6)^d	Φ^−−(1860)
[$\overline{p}p\overline{c}b$^−]^− ; [$\overline{p}p{b}^{+}\overline{b}$^−]^+	(7.325) ; [(10.624)^’bb ± (z^0;2z^0)]	Zb^±(10.610); (10.65) [36]
$[c\overline{c}{s}^{-}{v}^{-}{b}^{-}$];[$b\overline{b}{s}^{-}{v}^{-}{c}^{+}]$^0	(9.474)^’ ; (12.81)^’	Υ^−(1S)(9.46)^e
[$s\overline{s}c\overline{v}$^−]^+; [$s\overline{s}b\overline{v}$^+]^−; [$s\overline{s}c\overline{\mathrm{\lambda }}$^−]^+; [$\mathrm{\lambda }\overline{\mathrm{\lambda }}c\overline{s}$^−]^+	(3.274); (6.574); (3.135)’; (3.135)^’ – z^0 = (3.118)^d; (3.135)’–2z^0/3z^0 ≈ (3.100)/(3.083)^d; (3.070 )’ – z^0 = (3.053)^d	Ωc^0(3.119)^e; Θ^0c(3.099); J/ψ(1S)(3.097)^e; Ωc^0(3.090)^e ; Ωc^0(3.066)^e; Ξc(3.055)^e; Ωc(3.05)^e; Ξc(3.07)’→Ξc(2965)^e + μ;
[$s\overline{s}c\overline{b}$^−]^+ ; [$v\overline{v}c\overline{s}$^−]^+	(7.7)^’ ; (3.348)^’; (2.948)’ ; (2.948)’ – z^0 = (2.931)^d	Ξc^0(2939)^r ; Ξc^0(2923)^r
[$\overline{v}v\overline{b}s$^+]^+ ; [$v\overline{v}c\overline{b}$^−]^+	(6.648)^’ ; (7.848)^’
[$\overline{c}c\overline{v}s$^+]^+ ; [$\overline{c}c\overline{v}v$^−]^0	(4.474)’ – z^0 = (4.456)’; (4.548)^d; (4.474)’- zμ = (4.37)^d; (4.548)’-z1	Pc(4.457)^e -z^0 = Pc(4.44)^e Zc(4.380)^e; X(4.500)^e [37]
[$\overline{c}c\overline{c}c$^−]^0; [$\overline{s}b\overline{\mathrm{\lambda }}\mathrm{\lambda }$^−]^0	(6.8)^’; (6.370)’ – z^0 = (6.352)^d	X(6.900)^e; Ωb(6.35)^e; (6.34)^e
[$\overline{c}c\overline{d}p$^+]^+ ; [$\overline{b}b\overline{d}v$^−]^0	(4.025)^’ ; (10.887)^’; (6.248)^’; (6.230)^d ≈ (6.248)’ - z^0	Zc(4.020)^e ; Yb(10.890)^e [36]; Ξb(6.227)^r
[$\overline{c}c\overline{b}s$^−]^0 ; [$\overline{c}c\overline{b}v$^−]^0	(8.9)^’ ; (8.974)^’
[$\overline{c}\mathrm{\lambda }\overline{\mathrm{\lambda }}\mathrm{\lambda }$^−]^− ; [c$\overline{c}\mathrm{\lambda }\mathrm{\lambda }\mathrm{\lambda }]$^−; [c$\overline{c}v\mathrm{\lambda }n]$^−*; [c$\overline{c}npn]$^0	(3.005)^’ ; (4.705)^’ ; (4.148)^’(c‾c d ‾λ) ; (4.338)^0(d u c d‾c)^d	Dj(3.0)^e; Xc(4.700)^e, [37]; X(4.140)^e-2z^0 ≈ Zc(4.100)^e; PΨ^Λ(4.338)^0 [40]
[$b\overline{\mathrm{\lambda }}{\overline{m}}_2{m}_1]{}^{+}$; [$\overline{b}b\overline{v}s$^−]^0	(5.574)^’ ; (11.074)^’ ; (11.074)’– z1(3z^0) = (11.022)^d	X(5.568)^e → Bs^0π^±; Υ^−(11.020)^e
${\left[v\overline{v}c\overline{c}{\overline{m}}_2{m}_1\right]}^{+}$; [c$\overline{c}n\mathrm{\lambda }s]$^−	(4.687)^’; 4.687’–z^0 = (4.670)^d; 4.687’–z1(3z^0) =(4.635)^d ; (4.648)^’;^’ (4.648)^’ – z^0 = (4.631)^d	X(4.685)^e ; Y(4.660)^e; X(4.630)^e [37]
[$\overline{c}c\overline{n}p$^+]^+ ; [$\overline{c}c\overline{s}s$ ]^0	(4.026)^’– z^0 = (4.009)^d ; (4.400)^’	Zcs(4.003)^+; ψ(4.415)^e
[$\overline{c}c\overline{s}v$ ]^0 ; [$\overline{c}c\overline{v}v$ ]^0	(4.474)^’– z^0/2z^0 = (4.457)^d; (4.474)^’– z1 = (4.422)^d; (4.548)^’	Pc(4.450); Pc(4.440)^e; Zcc(4.430)^e ;
[$\overline{c}c\overline{\mathrm{\lambda }}p$]^+; [$\overline{c}c\overline{\mathrm{\lambda }}\mathrm{\lambda }$]^0; [$\overline{c}c\overline{\mathrm{\lambda }}s$]^0	(4.147)^’ ; (4.270)^’ ; (4.335)^’; (4.270)^’ – z1 = (4.218)^d	ψ(4.160); X(4.274); Zcs(4.220)^+
$\left[s\overline{s}{c}^{•}\overline{s}\right]$; $\left[s^{•}{\overline{s}}^{•}{c}^{•}{\overline{s}}^{•}\right]$; [$\overline{c}s\overline{s}n$ ]^−	(3.057)’; (3.057)’ – z1 = (3.005)^d; (3.012)’; (3.012)’ – z1 = (2.995)^d	Ωc(3.050)^e, Ωc(3.000)^e,; (3.012)’ – z2 = (.2977)^d ; Ξc(2.965)^e
$\left[c^{•}vvv\overline{v}\right]$^0 ; $\left[c^{+}vvv\overline{v}\right]$^0	(3.996)’;	Zcs(3.985)^+;
[c$\overline{v}v\overline{v}v\overline{n}]$^+;[c$\overline{v}s\overline{v}v\overline{p}]$^0	(4.309)’; (4.234)’- z^0 = (4.217)^d	Pc(4.312)^+ ; Zcs(4.220)^+
[c$\overline{\mathrm{\lambda }}\mathrm{\lambda }\overline{s}s\overline{s}]$^+; [c$\mathrm{\lambda }\mathrm{\lambda }sss]$^−	(4.070)^’– z^0/2z^0 = (4.053)/(4.035)^d	ψ(4.040);
[$\overline{b}b\overline{s}c$ ]^+ ; [$\overline{b}b\overline{v}c$ ]^+	(12.218)^’ ; (12.292)^’
[s $\overline{s}vcb]$^0; [v $\overline{v}scb]$^0	[b $\overline{b}pn\mathrm{\lambda }vvc]$^0	(8.292)^’ ; (8.366)^’ ; (13.833)^’
[p $\overline{p}vcb]$^0; [p $\overline{p}v\overline{s}c\overline{b}]$^+	(7.9) ; (8.4)

Table 7.

Theoretic masses of non-excited and de-excited multi-quark particles, (explained or predicted by CGT).