As is well known in literature [6] [9], the protons are composed of three quarks: three centres of diffusion positioned in a triangular form in diffusion experiments with “bullet” electrons. These centres indicate the proton could have an internal geometric structure. Since between the Compton’s wave lengths of the Planck (Δ_pl) particle and the one of proton (Δ_p) there is a golden relation, at less than a scale factor, then we have, see the pentagon in Figure 1.

The vertices (A, B, D) are three quantum oscillators with which photon couples in the electromagnetic interactions. In pentagonal structure, it is evident that the diagonals [(AD), (BD)] are in a golden ratio with the base AB: ( ϕ ≈ 1.618 ) → [(AD/AB) = ϕ ]. Where ( ϕ ) is the “aureus” (golden) number. This rapport also implies quarks (u, d) are aureus triangles with [( ƛ u = ƛ y ), ( ƛ d = ƛ β )] and [ ( ƛ u / ƛ d ) = ( m d / m u ) = ϕ ≈ 1.618 ] with (m_d > m_u). Besides it is [ ƛ p = k p ƛ u ], where ( ƛ u ) is the Compton wavelength of “free” quark, while (k_p) is a coefficient of “elastic adaptation”, when (u, d) quarks reciprocally bind for origin the proton. Just k_p can be in relation with binding gluons of the (u, d) quarks into proton; we will point out [ V ( r ) QCD ↔ k p ] , where V(r) is colour potential in QCD theory [3] [10] [11]; so in this theory, or AGM, an elastic tension k replaces the potential V(r). We notice that this structure can be realizable only through “particular” quantum oscillators: the IQuO (Intrinsic Quantum Oscillator), which have a “sub-structure” made two sub-oscillators, where there are half-quanta see [7] [12] [13]. Furthermore, we recall that a geometric property of the golden triangles (u or d) is that one each triangle is composed in its turn of two golden triangles (u₁, d₁), see Figure 2 and ref. [7] [8].

This configuration occurs, see a-configuration, when u-quark (u) in a meson could attach themselves to IQuO-chain of the diagonal (AC) belonging to d-quark (d). These two features may origin a new strange state in quarks, which becomes so the strange quark (s).

We suggested that mesons [8] are geometric structures realized by coupling between the golden quarks (see Figure 3).

The first attempt of a formal “structure equation” is: [ ( π + ) = ( u ⊕ d _ ) , ( π − ) = ( u _ ⊕ d ) ] .

Where the components (π; u, d) are matrices with elements expressed by wave functions in the representation of quantum oscillators of the field [10] [11], see ahead. The sign (Å) point out formally dynamics coupling between quarks. Then, we have the configuration illustrated in Figure 3.

Where u is the u antiquark. The bonding (see gluons) between two free quarks ( u ↔ d ) increases the elastic tension between IQuO components of quarks [ ( k u , k d ) → ( k _ u , k _ d ) ] , which, in turn, increases the “free” frequencies [ ( ω u , ω d ) → ( ω _ u , ω _ d ) ] or mass [ ( m u , m d ) f r e e → ( m _ u , m _ d ) b o u n d s ] , see ref. [8] Equations (1) and (2); thus, the Compton wavelength decreases ( ƛ u < ƛ u ), ( ƛ d < ƛ d )], shrinking u and d quarks (see second drawing in Figure 3). The π-structure oscillates with only one frequency (ω_π) and period (τ_π), to which we associate (Δ_π, m_π). Speaking of elastic tensions in bound quarks, we can admit that in (k_i) are contained the mass defects (recall that k replaces the potential V(r)), so that we have [ m π = m _ u + m _ d ] . V(r), in turn, is related to coupling (Å) between IQuO, where [ ⊕ = ( ⊕ g + ⊕ e m ) ] with Å_g indicating the coupling by gluons and Å_em indicating the electromagnetic coupling. Therefore, even along sides, there are gluons (see the Joining-IQuO of Figure 3) connecting the vertex oscillators. This aspect is consistent with QCD [4]. In [8], we calculated the mass values of quarks inside the pion (u_π, d_π) by an equations’ system with solutions:

[ m ( u π ) = ( 53.31 ) MeV , m ( d π ) = ( 86.26 ) MeV ] (1)

We assign these mass values to the physical system composed by quarks (u, d) with gluons that form the pion, m (π ^±) = (139, 57) MeV. Note the u_π and d_π quarks with their gluons are called “dressed” quarks.

In the pion, we note different configurations related to the X-junction axis, Figure 4: two quarks have such a relative motion around X-axis because the vertices (A, D) are free of rotating [8]. These configurations, or orientations, can induce us to think about the spin of the pion. To individual quark, seen like Fermionic particle, one associates an intrinsic spin of quarks (s_q), while now we associate also an orbital spin (s_l) to rotations of u-quark (or d-quark) around the X-axis (see the experimental observations about proton spin [14] ). The hypothesis of structure does not conflict with experiments [15] of CERN (COMPASS), SLAC, and DESY (HERMES) and one can demonstrate the total spin: [ s π = s q + s l + s g ] .

As noted, different relative orientations between u quarks and d quarks imply a relative rotation (spin) of one quark around the other quark, suggesting a mutual crossing of the quarks (see the overlapping of [I_DC]_u and [I_AC]_d in Figure 4. This crossing can have the meaning of a reciprocal interpenetration (Ä) between quarks (q_i Ä q_j). During this mutual crossing, quarks not exchange energy in those parts that interpenetrate, while, instead, there is an exchange of binding quanta in the diagonal BC of Figure 4. Note that the interpenetration between quarks implies a reciprocal intrinsic movement. This internal movement of components has a kinetic energy form, called by us interpenetration kinetic energy [8], which must be counted in the mass defect.

The interpenetration could explain the zero value of pion’s spin. In ref. [8] we showed that quarks have the orbital spins (s_l(u), s_l(d)) in a relative way opposite rotations: [ s l ( u ) = − s l ( d ) ] π → [ s l ( u ) + s l ( d ) ] π = 0 .

This opposite orbital rotation drags the bounding gluons of two quarks (g_u, g_d), which so have opposite orbital spins: [ s l ( g u ) = − s l ( g d ) ] π → [ s l ( g u ) + s l ( g d ) ] π = 0 .

The same occurs in gluons propagating along the IQuO of side BC; recall gluons as bosons with spin (s_g = 1), then would may be that the gluons in IQuO_BC (Figure 4) have opposite spins: [ s g ( u ) = − s g ( d ) ] π → [ s g ( u ) + s g ( d ) ] π = 0 .

The same could happen at the wave function (spinor) of quarks along the side BC:

[ s q ( u ) = − s q ( d ) ] π → [ s q ( u ) + s q ( d ) ] π = 0

Then, the global spin of a pion is zero.

In the interpenetration of quarks and their dynamics interactions, we used [8] a new Ä-operation of combination (or coupling) of quarks and particles: (a_i Ä b_j). The new operation (Ä) indicates a composition of two operations (Ä, Å) or [ĺ (Ä U Å)], where Ä-operation describes the proper interpenetration of the quarks and follows, in algebraic calculations, the properties of multiplication. Instead, Å-operation describes dynamics interactions and follows, in algebraic calculations, the properties of the sum (see Table 1).

The operation of combination (Å) could be used for express dynamics couplings ( u _ ↔ d ) (or ( u ↔ d _ )) in charged pion π (Figure 3). Nevertheless, the different relative orientations of quarks (u, d) induce us to admit interpenetrations between two quarks. Thus, we should write [ π ± = ( u ⊗ _ d ) ] .

In AGM, (u) and (d) represent the wave function associated with quark [8] [11] [13] in the representation of operators of annihilation and creation (a, a⁺), relative to IQuO or quantum oscillators at semi-quanta:

Ψ q ( x → , t ) = ( 1 V o l ) ∑ k → q k → ( t ) exp ( i k → ⋅ x → ) [ q k → ( t ) ] ≡ ( a k → + ( t ) a k → ( t ) ) IQuO = { ( ‖ u k → + ‖ ( t ) ‖ u k → ‖ ( t ) ) IQuO , ( ‖ d k → + ‖ ( t ) ‖ d k → ‖ ( t ) ) IQuO }

where ‖ u ‖ , ‖ d ‖ are matrices which express the structure of quantum oscillators (or IQuO) of quarks (u,d) in the IQuO-representation, see [7] [8] [12] [13].

We conjectured, see [8], that π⁰ neutral pions is originated by all the possible combinations of quarks’ couplings using both the Ä-operation and Å-operation or (Ä). Then, we defined (1-property Table 1)

( π 0 ) = [ ( π + ) ⊗ _ ( π − ) ] = [ ( u ⊕ d _ ) ⊗ _ ( u _ ⊕ d ) ] (2)

And we demonstrated [8] that it is:

Ψ ( π 0 ) ( ⊗ _ ) = [ ( π − ) ⊗ _ ( π + ) ] = { [ ( u ⊕ d _ ) ⊗ ( u _ ⊕ d ) ] ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] } Ψ _ = { [ ( π − ) ⊗ ( π + ) ] ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] } Ψ _ (3)

This equation represents the structural equation of neutral pion (π⁰) in no-local state (Y). Note { ( π 0 ) = [ ( π + ) ⊗ _ ( π − ) ] } ≠ [ ( π + ) ⊗ ( π − ) ] and thus { m [ ( π + ) ⊗ ( π − ) ] ≠ m ( π 0 ) } .

The interpenetration between two charged pions allows of assuming:

m [ ( π − ) ⊗ ( π + ) ] = m ( π ± ) 0 = m ( π ± ) (4)

The ( π ± ) 0 = [ ( π + ) ⊗ ( π − ) ] is called a virtual neutral pion.

In graphic mode, by ref. [8] we conjectured the following configurations (Figure 5).

Note (see Figure 5) there are two possible configurations: [(π⁰)_a, (π⁰)_b].

There are so different possibilities of configuration of quarks around the axes of propagation, with consequent rotations (spin). If the spins of (u Å d) and (u Å d) are respectively zero, then one can think the spin of neutral pion is zero.

The coupling between two quarks involves the combination of all possible configurations of these two quarks. In this way, the total mass ( m t o t ≡ m ⊗ _ ) is the sum of all masses associated with each combination of interpenetration (m_Ä) and interaction (m_Å) [8]. It follows: [ m ⊗ _ = m ⊗ + m ⊕ ] .

In general, for every X-system (composed by more particles), we use a partial mass Function (F_m) applied to structure equation of X-system, with X [ ( A 1 , A 2 , ⋯ , A n ) ⊗ ; ( B 1 , B 2 , ⋯ , B n ) ⊕ ] , where [ ( A i ) ⊗ , ( B j ) ⊕ ] are the “base components” of the structure [8]. The (F_m) is an application on the structure components (A, B), which gives us the mass values (m_i) of these components of base. The (F_m) operates on X, in the following way:

F m ( X ) = { ∑ ( i , j ) = 1 n F m [ ( A i ) ( ⊗ ) , ( B j ) ( ⊕ ) ] } = [ ∑ i = 1 n m ( A i ) ( ⊗ ) ] A + [ ∑ j = 1 m m ( B j ) ( ⊕ ) ] B = [ m ( a ⊗ b ) A 1 + ⋯ + m ( w ⊗ z ) A n ] A + [ m ( a ⊕ b ) B 1 + ⋯ + m ( w ⊕ z ) B m ] B (5)

We will have the following applications:

{ F m ( A ( ⊗ ) ) = F m [ ( a ) ⊗ ( b ) ] A = 〈 m ( a , b ) 〉 = 〈 m ( a ) , m ( b ) 〉 F m ( B ( ⊕ ) ) = F m [ ( a ) ⊕ ( b ) ] B = m ( ( a ) ⊕ ( b ) ) = m ( a ) + m ( b ) } (6)

To obtain the total mass of a structure, it needs to add eventual (m_kin) relativistic kinetic mass and mass defects (Δm). To the exception of some cases (which we will specify) ( m k i n ) ≪ m 0 , therefore we will have: [ m t o t = m p a r t ± Δ m i ] . The mass defect is: Δ m = Δ m g + Δ m e m . Here Δm_g is the mass defect due to binding by gluons. Nevertheless, the Δm_g has been englobed in masses of the charged pionor in quark (u_π, d_π)); therefore, we consider only electromagnetic mass defect: Δm = Δm_em.

To obtain the mass defects (Δm > 0, Δm < 0) we use a Function (F_Δm) of mass defect applied to structure equation so defined:

F Δ m ( A 1 , A 2 , ⋯ , A n ) = { ∑ ( i , j ) = 1 n F Δ m [ ( A i ) ( ⊗ ) , ( B j ) ( ⊕ ) ] } = [ ∑ i = 1 Δ m ( A i ) ( ⊗ ) + ∑ j = 1 Δ m ( B i ) ( ⊕ ) ] + [ ∑ ( i = 1 , j = 1 ) n Δ m ( A i ⊕ B j ) ] ( I ° -degree )       + [ ∑ ( i = 1 , j = 1 , k = 1 i ≠ j ≠ k ) n Δ m ( A i ⊕ ( A j ⊕ A k ) ) ] ( II ° -deg . )       + [ ∑ ( i = 1 , j = 1 , k = 1 i ≠ j ≠ k ) n Δ m ( B i ⊕ ( B j ⊕ B k ) ) ] ( II ° -deg . ) (7)

It needs to consider that:

Δ m [ ( a ) ⊗ ( b ) ] A i = { 0 Δ m ( a , b ) ( a ∩ b ) ≠ 0 } Δ m [ ( a ) ⊕ ( b ) ] B i = Δ m ( a , b ) interaction (8)

where the (a, b) point out “base particles” of the (A_i, B_j)-component, as i.e. the pions π or the quarks q. Note mass defect is zero if there is the only interpenetration between the two particles (a, b) without interacting parts. Instead, the mass defect cannot be zero if there are some parts of (a, b) dynamically interacting (a Ç b), see the neutral pion in diagonal (AC or FH in Figure 5), where the d-quark and u-quark, exchange energy quanta along the diagonal. Now, we can consider the properties of Table 1 in the space of the functions (F_m, F_Δm) to obtain the properties of the operations (Ä, Å) for calculating the masses and mass defects of mesons (see Table 2).

Equivalent two ways can treat the binding energy between quarks inside pions. The function F_Δm of mass defects expresses these two possibilities.

The structure equation contains the considerations on the various dynamics couplings and interpenetrations: by this equation, we obtain the mass value if we consider the binding energies. The Equation (7) expresses all possible couplings

between pions (quarks): component Δm (A) takes into account the eventual dynamics coupling in the interpenetration between two pions, the Δm (B) the specific dynamics coupling and Δm (A Å B) takes in consideration the coupling between pairs of pions or quarks. Into evolving the calculations on the mass defects we can use two procedures. The first way is considering without dynamical couplings all the interpenetrations between two pions, but not the ones between two pairs of pions. Vice versa, the second way is considering without dynamical couplings all the interpenetrations between two pairs of pions, but not the ones between two pions. This last aspect implies that all interactions of the different pairs are attributable to the individual couplings between two pions. In this second case, we can represent all possible dynamics couplings by a matrix: ‖ A i j ‖ = [ ( π i ± ⊗ π j ± ) ⊕ ] .

We will show the two possible ways in the calculations of meson masses.

By structure equation [8] the (m^*) partial mass of π⁰ is:

m ( π 0 ) * = F m ( { [ ( π − ) ⊗ ( π + ) ] A 1 ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] A 2 } ) = 〈 m ( π − , π + ) 〉 A 1 + 〈 m ( ( u u _ ) ⊕ , ( d _ d ) ⊕ ) 〉 A 2 (9)

We have

〈 m ( π + , π − ) 〉 A 1 = [ m ( π + ) + m ( π − ) 2 ] A 1 〈 m ( ( u u _ ) ⊕ , ( d _ d ) ⊕ ) 〉 A 2 = [ 2 m ( u ) + 2 m ( d ) 2 ] A 2 (10)

The values of Δm mass defects (see equation (7) and 2_Å-property in Table 2) would be:

Δ m ( π 0 ) * = F Δ m ( { [ ( π − ) ⊗ ( π + ) ] A 1 ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] A 2 } ) = F Δ m ( [ ( π − ) ⊗ ( π + ) ] A 1 ) + F Δ m ( [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] A 2 )     + F Δ m ( [ ( π − ) ⊗ ( π + ) ] A 1 ⊕ _ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] A 2 ) ( B = A 1 ⊕ A 2 ) (11)

where the Ä-operation point out the coupling between two no-separated components (A₁, A₂) and with

a ) F Δ m [ ( π + ) ⊗ ( π − ) ] A 1 = 0 b ) F Δ m [ ( u u _ ) ⊕ ⊗ ( d _ d ) ⊕ ] A 2 = Δ m ( ( u u _ ) ⊕ , ( d _ d ) ⊕ ) ≠ 0 c ) F Δ m { [ ( π + ) ⊗ ( π − ) ] A 1 ⊕ _ [ ( u u _ ) ⊕ ⊗ ( d _ d ) ⊕ ] A 2 } ( B = A 1 ⊕ A 2 ) ≠ 0 (12)

In [8] we demonstrated that the global mass value of pion is:

m ( π 0 ) = m ( π 0 ) * − Δ m = m ( π ± ) − ( 1 / 2 ) [ ε u ( γ ) + ε d ( γ ) ] ( u , d ) 0 = m ( π ± ) − ε γ ( π ) ( u , d ) 0

where ( ε γ ) is annihilation energy of pairs [ ( u ⊕ u _ ) , ( d ⊕ d _ ) ] , with [ Δ m γ = ε γ = ( 4.59 ) MeV ] . This energy, as already it has been said, is coincident with mass at rest of free quarks (see the bare mass):

[ m b ( u ) = ( 3.51 ) MeV , m b ( d ) = ( 5.67 ) MeV ] (13)

These values [8] are inside the range anticipated by literature [16] [17].

It is so evident that the values of masses of quarks, both bare and dressed inside pion, could be used to obtaining the mass spectrum of light mesons. To make this, it needs so to admit the presence of a lattice of bound “virtual pions” {π⁺, π⁻}, with an elementary cell having the form of the { ( π r 0 ) = ( π + ⊕ π − ) } , which represents a “molecule” of pions, you see Figure 4. In this way, both the { π r 0 } -lattice and {d, d} participating in building the mass spectrum of light mesons. By Equations (5) and (7) we will show the calculations to find the mass value of light mesons with notable precision as also the mass value of fundamental nucleons (a work in preparation). We will use the equations of Table 3, which represent the application of the properties of Table 1 and Table 3.

It is physically possible that the pions [π⁰, π⁺, π⁻] may overlap simultaneously, then they can give origin to massive mesons if the internal quanta are sufficient in number. The pion triplet will be called “meson of basic level” or first-level of structure.

As is well known, the first massive meson, after (π⁰), is the meson η. The η-meson places to the 2-level” of structure. We can conjecture a possible state of not-separated overlapping of mesons [π⁰; (π⁺, π⁻)]. In the first moment, we can consider the following combination or “structure equation”:

η = [ ( π + ⊕ π − ) 1 ⊗ ( π + ⊕ π − ) 2 ] = [ ( π 0 ) r 1 ⊗ ( π 0 ) r 2 ]

where ( π 0 ) r = ( π + ⊕ π − ) is a “molecule” of pions, composed by 4 quarks. This is built by a dynamics coupling [ ⊕ = ⊕ g + ⊕ e m ] of two charged pions; the electric charge is zero and mass { m ( π 0 ) r = m [ ( π + ) ⊕ ( π − ) ] = m [ ( π + ) + ( π − ) ] = 2 m ( π ± ) } , see Table 3.

Immediately, note there are two double pairs [ ( u 1 d _ 1 ) π + , ( u _ 1 d 1 ) π − ] , [ ( u 2 d _ 2 ) π + , ( u _ 2 d 2 ) π − ] .

This structure of couplings can be somewhat unstable; if the quarks (u₂, u₂) fix their respective vertices (G, H) on the diagonal of quarks (d₂, d₂), then the structure can have more stability. The fixing (see section 2.1 and Figure 2) gives origin to configuration with strange quarks (s, s): if quanta number is sufficient, then two new pairs of quarks are created [(u₃, d₃), (u₃, d₃)]_ss. We point out these quarks with a denomination of sub-quarks (u, d). The representative figure of η-meson is (Figure 6).

Recall, by literature, the wave function of η-meson: [ η = c 1 ( u u _ + d d _ ) + c 2 ( s s _ ) ] .

Note that in the structure equation, the quarks (s, s) are “hidden”. Moreover, the fixing in (G, H) determines a zero value of the interpenetration kinetic energy (K_int) of the d-quarks and sub-quarks. There is a (K_int) in pair (u₁, u₁) for the rotation of u-quarks around two Y-axis, but this energy could be minimal (K_kin) ≪ m₀c²). The spin of pair (u₁, u₁) is zero like that of neutral pion because this is to base of η-meson (see its structure). Processing the “structure equation” one has:

η = [ ( π 1 0 ) r ⊗ _ ( π 2 0 ) r ] = [ ( π + ⊕ π − ) 1 ⊗ ( π + ⊕ π − ) 2 ] = [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ⊕ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] (14)

The indices (1, 2) point out the two molecules of pions (π⁰)_r.

To obtain the partial mass, we will apply the mass Function (F_m) to the structural equation of the η-system. In processing the structure equation, we use the properties of Ä-operation and that of mass function F_m (see Table 2). Then the partial mass m* (see Equations (5) and (6) is:

m ( η ) * = F m { [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ⊕ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] } = F m { [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ] B 1 } ⊕ F m { [ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] B 1 } = F m { [ ( π 1 + ⊗ π 2 − ) ⊕ ( π 2 + ⊗ π 1 − ) ] B 1 } + F m { [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] B 2 } = m { [ ( π 1 + ⊗ π 2 − ) + ( π 2 + ⊗ π 1 − ) ] B 1 } + m { [ ( π 1 + ⊗ π 2 + ) + ( π 1 − ⊗ π 2 − ) ] B 2 }

= m ( 2 ( π + ⊗ π − ) 12 ) A 1 + m ( 2 ( π ± ⊗ π ± ) 12 ) A 2 = 2 ( 〈 m ( π + , π − ) 〉 ) + 2 ( 〈 m ( π ± , π ± ) 〉 ) = 2 m ( π ± ) + 2 m ( π ± ) = ( 558.28 ) MeV (15)

where [ F m ( B 1 ⊕ B 2 ) = m ( B 1 ⊕ B 2 ) ] , see Equation (6), and ( B i ) ⊕ → ( A i ) ⊗ .

Note, see Equation (10) and Table 2, that F m ( A ⊗ B ) = 〈 m ( A ) , m ( B ) 〉 ; if (A = B), then F m ( A ⊗ A ) = 〈 m ( A ) , m ( A ) 〉 = m ( A ) , thus it follows:

F m ( A ⊗ A ) = F m ( A ) ↔ ( A ⊗ A ) F m = m ( A ) .

To obtain the mass defects, we should use the Function (F_Δm) (see Equation (7)) applied to the following structure equation, with Å-operation of dynamics coupling:

η = [ ( π 1 0 ) r ⊗ _ ( π 2 0 ) r ] = [ ( π + ⊕ π − ) 1 ⊗ ( π + ⊕ π − ) 2 ] = [ ( π 1 + ⊗ π 2 + ) a ⊕ ( π 1 + ⊗ π 2 − ) b ⊕ ( π 1 − ⊗ π 2 + ) c ⊕ ( π 1 − ⊗ π 2 − ) d ] (16)

where we have used the 2-property of Table 1. Note that this equation explicates all dynamic couplings. Then, the equation of mass defect should be:

Δ m ( η ) = F Δ m ( [ ( π 1 + ⊗ π 2 + ) a ⊕ ( π 1 + ⊗ π 2 − ) b ⊕ ( π 1 − ⊗ π 2 + ) c ⊕ ( π 1 − ⊗ π 2 − ) d ] )

To obtain the global mass defect, we should consider all possible dynamic couplings (Å) between the four components (a, b, c, d), see Table 4.

It is possible to calculate all mass defects of these couplings; note that into mass values of pions are contained the binding energies of gluons; therefore, mass defects will be relative to electromagnetic coupling (Å_em). The mass defect most relevant is that relative to coupling [ ( π + ⊕ π − ) ] in neutral pion (π⁰):

[ Δ m ( π ) = [ m ( π ± ) − m ( π 0 ) ] = ( 4.59 ) MeV ] or [ Δ m ( π + ⊕ π − ) = ( 4.59 ) MeV ]

All the combinations of pairs [ ( π 1 ± ⊗ π 2 ± ) i ⊕ ( π 1 ± ⊗ π 2 ± ) j ] have values of mass defect that are sub-multiples of Δm(π):

Δ m ( A i ⊕ A j ) = ( Δ m ( π ) n ) (17)

where n is an integer, n ≥ 1; the number of quarks or pions involved in couplings can determine the number n, because the binding gluons are distributed on the n “sides” of coupling between quarks. Here it is necessary a selection rule of various couplings, because mesons are couplings two by two of quark-antiquark; then, we consider only couplings of the first degree (I˚) and, some cases, of the second (II˚).

If we omit superior degrees (III˚ or IV˚) of Å-coupling, the same, we can obtain values of the mass defect with excellent approximation because the mass defects of combinations of degrees (III˚, IV˚) are very smalls. We point out with (O_Δm) the values of mass defects omitted, which

O Δ m = ( Δ m ( π ) n _ r ) (18)

where (n, r) are integers (r > 1, n > 1); r is correlated to the degree of coupling between pion pairs, while n is correlated to pion pairs: for pion pairs (n = 2) and (r = 2), it is:

O Δ m = ( Δ m ( π ) 2 2 ) = ( ( 4.59 ) MeV 4 ) = ( 1.15 ) MeV (19)

Then the mass defect is, see Equation (7):

Δ m ( η ) = F Δ m { ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 1 + ⊗ π 2 + ) A 3 ⊕ ( π 1 − ⊗ π 2 − ) A 4 } = Δ m ( A 1 ) + Δ m ( A 2 ) + Δ m ( A 3 ) + Δ m ( A 4 ) + Δ m ( A 12 )     + Δ m ( A 13 ) + Δ m ( A 14 ) + Δ m ( A 23 ) + Δ m ( A 24 ) + Δ m ( A 34 ) (20)

If we take into account the Equation (12a), it derives that:

[ Δ m ( A 1 ) = Δ m ( A 2 ) = Δ m ( A 3 ) = Δ m ( A 4 ) = 0 ]

By this hypothesis, one admits that into interpenetration between two pions there is not a mass defect if between these there is not dynamics couplings (2-property Table 2):

Δ m ( π i ± ⊗ π j ± ) = Δ m ( π i ± ⊗ π j ∓ ) = 0

Therefore, mass defect will be obtained by:

Δ m ( η ) = Δ m ( A 12 ) + Δ m ( A 13 ) + Δ m ( A 14 ) + Δ m ( A 23 ) + Δ m ( A 24 ) + Δ m ( A 34 )

With

Δ m ( A 13 ) = Δ m ( A 23 ) ; Δ m ( A 14 ) = Δ m ( A 24 ) Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 + ⊗ π 2 + ) A 3 ] ) = Δ m ( [ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 1 + ⊗ π 2 + ) A 3 ] ) Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] ) = Δ m ( [ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] )

Nevertheless, note that two interpenetrated pions with equal charge [ ( π 1 ± , π 2 ± ) ] , but interacting with other pions [ ( π 1 ± ⊗ π 2 ± ) ⊕ π 3 ± ] , can be an only one particle (see boson properties): Δ m ( ( π i ± ⊗ π j ± ) ⊕ ) = Δ m ( π i ± )

See 3-property in Table 2 and 8-property in Table 3. It follows:

Δ m ( A 34 ) = Δ m ( [ ( π 1 + ⊗ π 2 + ) A 3 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] ) ≡ Δ m ( [ ( π 12 + ) ⊕ ⊕ ( π 12 − ) ⊕ ] ) = Δ m ( [ ( π + ) ⊕ ( π − ) ] ) = ( 4.59 ) MeV

In Equation (20), we observe that Δm (A₃₄) admits four possible combinations of exchange ( π i ↔ π j ); then we determine four equivalent combinations (S₁, S₂, S₃, S₄) using the operation of exchange S which acts on the pions (bosons):

S ^ ( π 1 − ↔ π 1 + ) ( A 34 ) = S ^ ( π 1 − ↔ π 1 + ) [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] A 34 = [ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ] S 1

S ^ ( π 1 + ↔ π 2 − ) ( A 34 ) = S ^ ( π 1 + ↔ π 2 − ) [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] A 34 = [ ( π 2 − ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 1 + ) ] S 2

S ^ ( π 2 + ↔ π 1 − ) ( A 34 ) = S ^ ( π 2 + ↔ π 1 − ) [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] A 34 = [ ( π 1 + ⊗ π 1 − ) ⊕ ( π 2 + ⊗ π 2 − ) ] S 3

S ^ ( π 2 + ↔ π 2 − ) ( A 34 ) = S ^ ( π 2 + ↔ π 2 − ) [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] A 34 = [ ( π 1 + ⊗ π 2 − ) ⊕ ( π 1 − ⊗ π 2 + ) ] S 4

Each of these combinations (S₁, S₂, S₃, S₄) must contribute to overall mass defect (quantum aspect). Note that

Δ m ( [ ( π 1 + ⊗ π 2 − ) ⊕ ( π 1 − ⊗ π 2 + ) ] ) ⇒ Δ m ( [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] ) S 4

Where have exchanged of place ( π 1 + ↔ π 1 − ); then we can think that:

Δ m ( [ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ] ) = ( 1 4 ) Δ m ( [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] ) = ( 1 4 ) ( 4.60 ) MeV = ( 1.15 ) MeV (21)

Thus, in η-meson, it is:

Δ m ( A 12 ) = Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ] ) 1 = ( 1.15 ) MeV Δ m ( A 13 ) = Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 + ⊗ π 2 + ) A 3 ] ) 2 = ( 1.15 ) MeV Δ m ( A 14 ) = Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] ) 3 = ( 1.15 ) MeV Δ m ( A 23 ) =   Δ m ( [ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 1 + ⊗ π 2 + ) A 3 ] ) 4 = ( 1.15 ) MeV Δ m ( A 24 ) = Δ m ( [ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] ) 5 = ( 1.15 ) MeV Δ m ( A 34 ) = Δ m ( [ ( π 1 + ⊗ π 2 + ) A 3 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] ) 6 = ( 4.60 ) MeV (22)

Then we can have:

Δ m ( η ) = F Δ m ( { ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 1 + ⊗ π 2 + ) A 3 ⊕ ( π 1 − ⊗ π 2 − ) A 4 } ) = F Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ] ⊕ [ ( π 1 + ⊗ π 2 + ) A 3 ⊕ ( π 1 − ⊗ π 2 − ) A 4 ] ) = F Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ] ⊕ [ ( π 12 + ) A 3 ⊕ ( π 12 − ) A 4 ] ) = F Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ] ⊕ [ ( π 12 + ⊕ π 12 − ) A 34 ] ) = F Δ m ( ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 − ⊗ π 2 + ) A 2 ⊕ ( π 12 + ⊕ π 12 − ) A 3 ) (23)

It needs determining the Δm (A₁₃) and Δm (A₂₃). Note that if

Δ m ( [ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ] ) = ( 1.15 ) MeV Δ m ( [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] ) = Δ m ( [ ( π 1 + ) ⊕ ( π 2 − ) ] ) = ( 4.6 ) MeV

Then we can think:

Δ m ( [ ( π 1 − ⊗ π 2 + ) ⊕ ( π 1 + ⊕ π 2 − ) ] ) = ( Δ m ( π ) n ) ( n = 2 ) = ( 2.3 ) MeV (24)

The overall mass defect will be:

Δ m ( η ) = Δ m ( A 1 ) + Δ m ( A 2 ) + Δ m ( A 3 ) + Δ m ( A 12 ) + Δ m ( A 13 ) + Δ m ( A 23 ) = [ ( 0 ) A 1 + ( 0 ) A 2 + ( 4.6 ) A 3 + ( 1.15 ) A 12 + ( 2.3 ) A 13 + ( 2.3 ) A 23 ] MeV = ( 10.35 ) MeV

The global mass is (omitting (O_Δmi)):

m ( η ) ≈ m ( η * ) − Δ m η = [ ( 558.28 ) − ( 10.35 ) ] MeV = ( 547 .95 ) MeV (25)

This value is very next to that experimental, see ref. [16] where m(η) = (547,86) MeV.

Since any coupling ( A i ⊕ A j ) between pion pairs is always attributable to the couplings between individual pions, ( π i ± ⊗ π j ± ) , we can treat the mass defect taking in account only reciprocal dynamics couplings between two pions and no between pairs. Therefore, we could consider

Δ m ( A 12 ) = Δ m ( A 13 ) = Δ m ( A 14 ) = Δ m ( A 23 ) = Δ m ( A 24 ) = Δ m ( A 34 ) = 0 [ Δ m ( A 1 ) , Δ m ( A 2 ) , Δ m ( A 3 ) , Δ m ( A 4 ) ] ≠ 0

We build the matrix A_ij of all couplings with interpenetration between pions [ ( π i ± ⊗ π j ± ) ⊕ ] , which compose the structure equation in η-meson. The matrix, see Table 5, is symmetric A_ij = A_ji, therefore we consider only half elements, the ones in colour:

We pose, see Equation (22), the binding energy [ ( π i ± ⊗ π i ∓ ) ⊕ ] G = ( 4.6 ) MeV in pions belonging to the same molecule (Green colour). Besides, we will have the binding energy [ ( π i ± ⊗ π j ∓ ) ⊕ ] Y = ( 2.3 ) MeV , because in pions belonging to two different molecules the binding is weaker (Yellow colour). Instead, there is repulsive energy between pions with same charge (Blue colour): [ ( π i ± ⊗ π j ± ) ⊕ ] B = − ( 1.15 ) MeV .

The repulsive action is weaker than that attraction because η-meson is a binding state until to decay. Finally, to all elements of the diagonal (Red colour), we also assign an overall mass defect value given by the coupling term that follows

∪ k ( Δ m ( π i ± ⊗ π i ± ) ) k = F Δ m { ( π 1 + ⊗ π 1 + ) ⊗ _ ( π 2 + ⊗ π 2 + ) ⊗ _ ( π 1 − ⊗ π 1 − ) ⊗ _ ( π 2 − ⊗ π 2 − ) } R

Being all repulsive terms, we will assign an overall mass defect with a negative value:

∪ ( [ ( π i ± ⊗ π i ± ) ⊕ ] R ) = − ( 1.15 ) MeV

In synthesis, we will have

[ Δ m ( ( π 1 − ⊗ π 1 + ) ⊕ ) = Δ m ( ( π 2 − ⊗ π 2 + ) ⊕ ) ] G = ( 4 .6 ) MeV [ Δ m ( ( π 2 − ⊗ π 1 + ) ⊕ ) = Δ m ( ( π 2 + ⊗ π 1 − ) ⊕ ) ] Y = ( 2.3 ) MeV [ Δ m ( ( π 2 + ⊗ π 1 + ) ⊕ ) = Δ m ( ( π 2 − ⊗ π 1 − ) ⊕ ) ] B = − ( 1.15 ) MeV [ ∪ k Δ m ( ( π i ± ⊗ π i ± ) ⊕ ) k ] R = − ( 1.15 ) MeV (26)

The indexes (G, Y, B, R) are the colours of elements of matrix ‖ A i j ‖ . The total mass defect is

Δ m ( η ) = ∑ ( ( i , j ) ∈ ‖ A i j ‖ ) Δ m ( π i ⊗ π j ) ⊕

Summing we will have:

Δ m ( η ) = [ Δ m ( ( π 1 − ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 2 − ⊗ π 2 + ) ⊕ ) ] G     + [ Δ m ( ( π 2 − ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 2 + ⊗ π 1 − ) ⊕ ) ] Y     − [ Δ m ( ( π 2 + ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 2 − ⊗ π 1 − ) ⊕ ) ] B − [ Δ m ( ( π i ± ⊗ π i ± ) ⊕ ) ] R Δ m ( η ) = { [ 2 ( 4 .6 ) ( i , i ) G + 2 ( 2.3 ) ( i , j ) Y ] ( ± , ∓ ) − [ 2 ( 1.15 ) ( i , j ) B + ( 1.15 ) ( i , i ) R ( ± , ± ) ] } MeV = ( 10.35 ) MeV (27)

It follows:

m ( η ) ≈ m ( η * ) − Δ m η = [ ( 558.28 ) − ( 10.35 ) ] MeV = ( 547 .95 ) MeV

We obtained the same result of the procedure with F_Δm. In this way, we can say that the two procedures are equivalent to calculate the mass defect.

Note in Figure 6 there is a principal axis (axis of propagation) and two secondary. There are four possible particles (π⁺, π⁻, π˚, γ) with the following combination or channels:

1) (π⁺, π⁻, π˚)_(23%), 2) (π⁰, π⁰, π⁰)_(33%), 3) (γ, γ)_(39%), 4) (π⁺, π⁻, γ)_(5%), 5) ((e⁻ + e⁺), γ), 6) (π⁺, π⁻)

So, see structure Equation (15), the possible decays (see even Figure 6) of η, will be (talk [16] about the probabilities):

1) only principal X-axis [ ( η ) → ( γ + γ ) ] ( B12 ) ; this channel is possible only if the quarks ((u₁, u₁)) are attached in diagonal along X-axis (see b-configuration in Figure 5)

2) with principal axis and secondary

· (X-axis, Z-axis) [ ( η ) → ( 3 π 1 0 ) ] ( B1 ) + ( B12 ) .

· (X-axis, Y-axis) [ ( η ) → ( π + + π − ) + ( π ° ) ] ( B2 ) + ( B1 ) .

· (X-axis, Y-axis) [ ( η ) → ( π + + π − ) + ( γ ) ] .

· (X-axis, Y-axis) [ η → ( e − + e + ) + γ ] .

(see the Dalitz decay) and other possible secondary decays.

Note that a decay [ η → ( π + + π − ) + 2 π ° ] is not admitted by the energy conservation (see the mass defects) in rest frame referent of (η) because

[ m ( π + ) + m ( π − ) + 2 m ( π ° ) ] > m ( η ) .

Note that a decay [ η → ( π + + π − ) ] is not possible for the presence of three axis (X, Y, Z).