_{1}

^{*}

In this paper, we show a new theoretical procedure for calculating the nucleonic mass values. We develop this procedure on the geometric representation of (u, d) quarks, these seen as golden structures of coupled quantum oscillators (Aureum Geometric Model or AGM). Using AGM, we also build the geometric structures of nucleons (p, n), determining their structure equations and spins. Thank AGM, coherent to QCD, new aspects of the Quantum Mechanics emerge, opening to anew descriptive paradigm in Particle Physics.

One of the more problematic aspects in Hadronic literature [^{+}, π^{−}} and {d,d}-lattice of d-quarks. This hypothesis induces us to admit an appropriate structure equation [^{−}, Δ^{++}). In finally, in the conclusions section of this paper, we will show that the two paths, AGM and QCD, are physical equivalents.

In ref. [

By the “aurea” hypothesis, we have calculated the mass values of quarks inside the pion (u_{π}, d_{π}) by equations’ system:

{ m _ ( u π ) + m _ ( d π ) = m π ± m _ ( d π ) / m _ ( u π ) = ϕ (1)

where is f the golden number. This equation has solutions [m(u_{π}) = (53.31) MeV, m(d_{π}) = (86.26) MeV] with m(π^{±}) = (139.57) MeV. We assign these mass values to the physical system (pion) composed by quarks (u_{π}, d_{π}) and binding gluons: we speak of “dressed” quarks. Any observer detects the couple (u_{π}, d_{π}) as a unique quantum system (pion π) with non-separated components: we cannot detect an only oscillator of the structure. Besides, we obtained the mass values of bare quarks considering the mass defect between the charged pion and the neutral one, Δm(π^{±}, π^{0}), by equations’ system, see ref. [

{ ( 1 / 2 ) [ m ( u f ) + m ( d f ) ] = Δ m π 0 m ( d f ) / m ( u f ) = ϕ ⇒ { m ( u f ) = ( 3.51 ) MeV m ( d f ) = ( 5.67 ) MeV (2)

Also, the proton can be described by a structure equation relative to an “aurea” geometric structure: three golden triangles couple building a geometric figure which can change taking the form of a pentagon or triangle, see _{pl}) of the Planck-particle and the one of the proton (Δ_{p}) there is an aurea (golden) relation [(Δ_{p}/Δ_{pl}) µ f], see

Where (f) is the “aureus” (golden) number.

The configurations (a) and (b) belong in different times to the same particle because the u-quarks are in rotation around the sides (AD), (DB). Note the three vertices-IQuO can constitute the three scattering centers seen in the experiments of deep inelastic diffusion (e, p). In the first time, the geometric representation of quarks (AGM) can be seen differently to those of partons and QCD. In AGM, the quark is not a point-particle surrounded by clouds of gluons but a set of coupled oscillators (IQuO) by of junction oscillators (“junction gluons”). In its turn, each quark can be bound to other quarks by “binding gluons”. Nevertheless, studying more in-depth these new aspects, we can see the two models (QCD and AGM) having some points of connection. In fact, in QCD [^{+}), in the IQuO form. Tanks to the structure equation, we have calculated the mass of light mesons [

The structures are not rigid because the binding gluons allow the movement of quarks: the quarks reciprocally rotate one respect to other around to a common axis. As noted [_{i} Ä q_{j}), see the pion in ref. [^{±} = (u Ä d)]. The Ä operation is a combination of two operations (Å,Ä), where the (Å) is the representative operation of dynamics interaction, while the (Ä) is the operation of interpenetration. During this mutual crossing, quarks no energy exchange between them in those parts that interpenetrate (in the diagonal BC of

The different configurations, or orientations, can induce us to think about the orbital spin (s_{l}) of each quark. To individual quark, seen like Fermion particle, one associates an intrinsic spin of quarks (s_{q}), while now we associate also the gluons’ spin (s_{g}) to rotations of u-quark (or d-quark) around the X-axis (see the experimental observations about proton spin [_{q}) of quarks with their orbital motions (s_{l}) and their gluons (s_{g}). In this way, in AGM the total spin of the proton is the sum of different components of spin: [s_{π} = s_{q} + s_{l}+s_{g}]. Just in ref. [

s p = { s t o t ( q ) = [ ( ↑ ↓ ) q 1 , ( ↑ ↓ ) q 2 , ( ↑ ↓ ) q 3 ] s t o t ( l ) = [ ( ↑ ↓ ) q 1 , ( ↑ ↓ ) q 2 , ( ↑ ↓ ) q 3 ] s t o t ( g ) = [ ( ↑ ↓ ) q 1 , ( ↑ ↓ ) q 2 , ( ↑ ↓ ) q 3 ] } = { s t o t ( q ) + s t o t ( l ) + s t o t ( g ) } (3)

where [(s_{tot}(q) = ±1/2, ±3/2), s_{tot}(l) = ±1, s_{tot}(g) = m1]. In AGM, the structure of three coupled IQuO needs that [s_{l} = −s_{g}]: in this way, the proton spin is [s_{tot}(p) =s_{tot}(q) = ±1/2].

In the interpenetration of quarks and their dynamics interactions, we used, see ref. [_{i} Ä b_{j}). The new operation (Ä) indicates a composition of two operations (Ä, Å) or [Ä º (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. In ref. [_{m}), (F_{Δm})]. In general, for every X-system (composed by more particles), we used [_{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})_{Ä}, (B_{j})_{Å}] are the “base components” of the structure which are combinations of coupling between base particles by of the operations (Ä,Å). The (F_{m}) is such 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 (4)

where (a, b, …, w, z) are the base particles which build the structure (hadron). 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 ) } (5)

Note that the mass of two interpenetrating particles [a Ä b] is obtained by the average value of individual masses [<m(a, b)>], while the mass of two interacting particles [a Å b] is obtained by the sum of the masses of each particle. To obtain the total mass of a structure, it needs to add eventual (m_{kin}) relativistic kinetic mass and mass defects (Δm). Nevertheless, we can consider the relativistic kinetic mass as a positive defect of mass: (Δm_{kin}) > 0.

We will have: [m_{tot} = m_{part} + Δm_{i})]. Where (Δm) > 0, (Δm) < 0.

The mass defect will be: Δm = Δm_{g} + Δm_{em}

Where Δm_{g} is the mass defect due to binding by gluons. Nevertheless, the Δm_{g} has been included in masses of the charged pion (see ref. [_{π}, d_{π})); therefore, we consider only electromagnetic mass defect: Δm = Δm_{em}.

To obtain the mass defects (Δm > 0, Δm < 0) we have used [_{Δm}) applied to structure equation for calculating the mass defects. Besides, 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 (6)

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 Å b), see the neutral pion in diagonal in ref. [_{ij} of mass defects (Δm_{i}) of particle X. The matrix A_{ij} is so defined,

Δm (X) | q_{1} | q_{2} | … | q_{n} |
---|---|---|---|---|

q_{1} | Δ m ( q 1 , q 1 ) ⊕ R | Δ m ( q 1 , q 2 ) ⊕ G | … | Δ m ( q 1 , q n ) ⊕ B |

q_{2} | Δ m ( q 2 , q 1 ) ⊕ G | Δ m ( q 2 , q 2 ) ⊕ R | … | Δ m ( q 2 , q n ) ⊕ G |

… | … | … | … | … |

q_{n} | Δ m ( q n , q 1 ) ⊕ B | Δ m ( q n , q 2 ) ⊕ G | … | Δ m ( q n , q n ) ⊕ R |

Where q_{i} is the base particles (here we have quarks) which compose the particle X. Besides, it is Δ m ( q i , q j ) ⊕ R , B , G = Δ m ( q i ⊕ q j ) R , B , G . The superscripts (R, G, B) are colors which point out positive mass defects (G) and negative (B), while (R) the elements of the diagonal to values positive or negative in according to specific cases.

In light mesons, we highlighted a mass spectrum built employing of lattices of base pions {π} and quarks {d, d} [

Recall [Ä = Ä U Å], see ref. [_{π}), m(d_{π})] in the pion. The structure equation needs to describe, thus, the interpenetration di each quark (q_{i}) with another quark (q_{j}) and with the pair of two “bonded” quarks (q_{j} Å q_{k}). In this way, the proton will be a “superposition” of all possible combinations made means (Ä º Ä U Å) with three quarks (u, d, u). These coupling combinations can be the followings:

[ ( u 1 d ) , ( u 1 u 2 ) , ( d u 2 ) ] A 1 , [ u 1 ( d u 2 ) , d ( u 1 u 2 ) , u 2 ( u 1 d ) ] A 2 _{ }

The property of operation (Ä) which describes the combinations of three quarks is:

[ A ⊗ _ B ⊗ _ C ] = { [ ( A ) ⊗ ( B ⊕ C ) ] ⊕ [ ( B ) ⊗ ( C ⊕ A ) ] ⊕ [ ( C ) ⊗ ( A ⊕ B ) ] } (7)

The geometric representation of proton admits two configurations,

Note the configuration 1(b) shows an electric charge equal to one, along the propagation side BC.

Also if we give a structure of coupling oscillators to the proton, the same one needs to associate a wave function to the proton and, thus, a field of quantum oscillators, expressed by annihilation operator (a) and creation (a^{+}). The representative quanta of the proton propagate along the axis passing through B and C (see

Let us go to determining the structure equation of the proton. Through geometric structure, see _{i}) interpenetrates the others two quark (q_{j}, q_{k}) in reciprocal interaction; this aspect needs to occur for each quark. Using Equation (7), we will have:

[ u 1 ⊗ _ d ⊗ _ u 2 ] = { [ ( u 1 ) ⊗ ( d ⊕ u 2 ) ] A 1 ⊕ [ ( d ) ⊗ ( u 2 ⊕ u 1 ) ] A 2 ⊕ [ ( u 2 ) ⊗ ( u 1 ⊕ d ) ] A 3 } A 3 (8)

This relation tells us that the quarks reciprocally interpenetrate, but at the same time, they interact between them. Therefore, the structure equation of the proton is:

( p ) = ( { 3 κ p [ u 1 ⊗ _ d ⊗ _ u 2 ] } ) (9)

where the number 3 point out the 3-dimensionality of a baryon (three free degrees) while to the mesons we assign a 2-dimensionality or two free degrees. The k_{p} coefficient is in connection to the global elasticity of the system of quantum oscillators (IQuO) of the geometric structure of the quarks into nucleons: we highlight that the increase of the number of vertices in a geometric structure makes increase the elastic tension between the quantum oscillators of structure.

The Å-operation in eq. 8 not cancels the no-locality of superposition of three states (Y_{A}) associated with A_{i}-components (only an outer observation or interaction could reduce the (Y)). The electric charge is, instead, already connected to an eigenstate of the wave function, relatives to its phase [

p = 3 κ p { [ u 1 ⊗ ( d ⊕ u 2 ) ] A 1 ⊕ [ d ⊗ ( u 2 ⊕ u 1 ) ] A 2 ⊕ [ u 2 ⊗ ( u 1 ⊕ d ) ] A 3 } = { 3 κ p [ p a ⊕ p b ⊕ p c ] } (10)

where (a, b, c) are the indices of the following relations:

p a = [ u 1 ⊗ ( d ⊕ u 2 ) ] a = [ ( u 1 ⊗ d ) ⊕ ( u 1 ⊗ u 2 ) ] a p b = [ d ⊗ ( u 2 ⊕ u 1 ) ] b = [ ( d ⊗ u 2 ) ⊕ ( d ⊗ u 1 ) ] b p c = [ u 2 ⊗ ( u 1 ⊕ d ) ] c = [ ( u 2 ⊗ u 1 ) ⊕ ( u 2 ⊗ d ) ] c (11)

Now, we calculate partial mass of the proton, using the properties of operations (Ä, Å), see the tables in ref. [_{m} in Equations (4) and (5). In first, it is:

F m ( p a ) = m ( p a ) = m [ ( u 1 ⊗ d ) ⊕ ( u 1 ⊗ u 2 ) ] a = m ( 〈 u 1 , d 〉 ) + m ( 〈 u 1 , u 2 〉 ) F m ( p b ) = m ( p b ) = m [ ( d ⊗ u 2 ) ⊕ ( d ⊗ u 1 ) ] b = m ( 〈 d , u 2 〉 ) + m ( 〈 d , u 1 〉 ) F m ( p c ) = m ( p c ) = m [ ( u 2 ⊗ u 1 ) ⊕ ( u 2 ⊗ d ) ] c = m ( 〈 u 2 , u 1 〉 ) + m ( 〈 u 2 , d 〉 ) (12)

With sum:

m ( p a , b , c ) = m ( p a ) + m ( p b ) + m ( p c ) = 2 m ( 〈 u 1 , d 〉 ) + 2 m ( 〈 u 2 , d 〉 ) + 2 m ( 〈 u 1 , u 2 〉 ) (13)

This because is [ A ⊗ B ] = [ B ⊗ A ] . Nevertheless, note the quarks masses (u,d) of proton could be different to one of pion: [ m ( u p ) , m ( d p ) ] ≠ [ m ( u π ) , m ( d π ) ]

The couplings [[u_{1}(du_{2}), d(u_{1}u_{2}), u_{2}(u_{1}d)]_{A}_{2}] in proton induces us to think that the pair (u_{p}, d_{p}) needs considering with elastic tension more strong than pair (u_{π}, d_{π}):

( u p , d p ) = k p ( u π , d π ) , ( k p > 0 )

We have already said that the elastic tension (k) between the IQuO of sides and the ones of vertices is function of them number; then, it is intuitive considering the parameterk given by the rapport between n_{v} number vertices

[(A(d), B(d), C(d), D(u_{1}), E(u_{2})] of geometric structure (in proton is n_{v} = 5) and number of total (n_{el}) of the “coupling” vertices (elastic vertices) between two quarks:

· In b-config.: [(B(d,u_{1}), C(d,u_{1}); (B(d,u_{2}), C(d,u_{2}); (B(u_{1},u_{2}), C(u_{1},u_{2})] ó (n_{elv}) = 6

· In a-config.: [(B(d,u_{1}), C(d,u_{2}); (B(d,u_{2}); B(u_{1},u_{2},d)] ó (n_{elv}) = 6

Where B(u_{1}, u_{2}, d) is a triple vertex. It follows: [k_{p} = n_{v}/(n_{elv}) = 5/6]. Then, by mass values of solutions of Equation (1), the proton mass is:

m ( p ) = { 3 κ p [ m ( p a ) + m ( p b ) + m ( p c ) ] } = 3 κ p [ 2 m ( 〈 u 1 , d 〉 ) + 2 m ( 〈 u 2 , d 〉 ) + 2 m ( 〈 u 1 , u 2 〉 ) ] = 3 κ p [ 4 m ( 〈 u , d 〉 ) + 2 m ( 〈 u 1 , u 2 〉 ) ] = 3 κ p [ 4 ( 69.79 ) + 2 ( 53.31 ) ] MeV (14)

Having that [k_{p} = (5/6)], it follows:

m ( p ) = 3 ( 5 6 ) [ 4 m ( 〈 u , d 〉 ) + 2 m ( 〈 u , u 〉 ) ] = [ 10 ( 69.79 ) + 5 ( 53.31 ) ] MeV = ( 964 .45 ) MeV (15)

In the structure equation, one can identify the various defects of mass. The electromagnetic mass defects will be Δm_{g}(u_{p}, d_{p}) and Δm_{g}(u_{p}, u_{p}).

We can admit that: Δ m γ ( q i , q j ) p = 3 κ p Δ m γ ( q i , q j ) π . Recall that [

Δ m γ ( u π , d π ) = m ( π ± ) − m ( π 0 ) = Δ m ( ε π ± ( γ ) ) = ( 4.59 ) MeV / c 2 (16)

With

Δ m ( ε π ± ( γ ) ) = 1 2 [ m γ ( u ) + m γ ( d ) ] = m γ ( 〈 u , d 〉 ) (17)

where m_{g}(q) is the electromagnetic mass of quark, that is the quark mass generating photons in annihilation processes. Then, m_{g}(q) represents the free mass of quarks or, also, the bare mass, see Equation (2). We will have here:

{ m γ ( u ) = ( 3.51 ) MeV / c 2 = Δ m γ ( u , u ) m γ ( d ) = ( 5 .68 ) MeV / c 2 = Δ m γ ( d , d ) } (18)

where we have so calculated Δm_{g}(u, u) and Δm_{g}(d, d). Note that these values are coincident with the values of the superior extremes of mass interval reported in experimental literature [_{π}(u), m_{π}(d)]). So, using the mass values [m_{p}(u), m_{p}(d)], we incorporate the mass defects relative to gluon interactions between quarks consequently.

We could calculate the global mass defect Δm_{g}(p) in two equivalent way [

1) Using the F_{Δm}-function

2) Using the matrix A_{ij} of the mass defects Δm_{q}

Here, we choose the matrix A_{ij} of mass defects. Then, we build the matrix A_{ij} of Δm_{g}(q) of all couplings with interpenetration between quarks [(q_{i} Ä q_{j})], which compose the structure equation in p-nucleon and interacting with other quark, [(q_{i} Ä q_{j})_{Å}].

If the interpenetrating quark pair interacts to other quark, [ ( q i ⊗ q j ) ⊕ q j ] , then it is: [ ( q i ⊗ q j ) ⊕ ] ≠ [ ( q j ⊗ q i ) ⊕ ] . This aspect gives the no-commutation of field operators (q_{i}, q_{j}). The matrix is, see

Δm^{*}(p) | u_{1} | d_{1} | u_{2} |
---|---|---|---|

u_{1} | Δ m ( u 1 , u 1 ) ⊕ R | Δ m ( u 1 , d 1 ) ⊕ G | Δ m ( u 1 , u 2 ) ⊕ B |

d_{1} | Δ m ( d 1 , u 1 ) ⊕ G | Δ m ( d 1 , d 1 ) ⊕ R | Δ m ( d 1 , u 2 ) ⊕ G |

u_{2} | Δ m ( u 2 , u 1 ) ⊕ B | Δ m ( u 2 , d 1 ) ⊕ G | Δ m ( u 2 , u 2 ) ⊕ R |

For calculating the mass defect, we sum all mass defects relative to elements of the matrix. We are turning to Equations (16)-(18):

Δ m * ( p ) = [ 4 Δ m ( ( u ⊗ d ) ⊕ ) ] G + [ 2 Δ m ( ( u 1 ⊗ u 2 ) ⊕ ) ] B + [ Δ m ( ( u 1 ⊗ u 1 ) ⊕ ) + Δ m ( ( u 2 ⊗ u 2 ) ⊕ ) + Δ m ( ( d ⊗ d ) ⊕ ) ] R = { [ ( 18.36 ) G − ( 7 , 02 ) B ] MeV + [ ( R ( u 1 ) + R ( u 2 ) + R ( d ) ) ⊕ ] R } = ( 11.34 ) MeV + R ( q i ) (19)

Recall the Δ m γ ( q i , q j ) p > 0 , while Δ m γ ( q i , q i ) p < 0 .

For finding the values of mass defects R(u_{i}) and R(d), we highlight the spin values of three quarks (u, u, d) and we give the coefficients (r_{i}) to function R of the mass defects:

R ⊕ = [ ( R ( u 1 ) + R ( u 2 ) + R ( d ) ) ] = [ ( r 1 Δ m ( u ( ↑ ) ⊗ u ( ↑ ) ) u 1 + r 2 Δ m ( u ( ↓ ) ⊗ u ( ↓ ) ) u 2 + r 3 Δ m ( d ( ↑ ) ⊗ d ( ↑ ) ) d ) ⊕ ] R (20)

The arrows of the subscripts in terms of (u) and (d) point out the spin, see Equation (3).

In ref. [_{ij} matrix in mesons, we gave the values of (1, 15) MeV. In the proton, we assume the same value (see the couplings (q_{i}, q_{j}) as in meson), also if we can admit some corrections to this value. To give a numerical value of mass defect to the terms (q_{i} Ä q_{i})_{R} would imply the interpenetration of a quark with itself. Now, we need to give a plausible explanation for this paradoxical aspect. A quark in rotation around to an axis implies different configurations of itself to the time flow. Recall the QM admit a not space locality, see the superposition of states in the experiments of the undulatory mechanics with two fissures. If the time is a coordinate equivalent to one of space (see the relativity), then we need also talk about not time locality (note that we are also talking about a time operator in QM). In this way we can admit some kind of time superposition of a particle with itself, which, however, must not violate the energy conservation: so, a particle cannot interfere negatively with itself (surely, if the particle is free). The positive superposition in time with itself might imply the interpenetration with itself of a particle. This interpenetration might have a dynamic component with a negative mass defect. All this is possible only in proper space rotations (spin) and not in space translations. This quantum aspect of proper rotation and of not time locality of quarks leads us to conjecture that the elements in red color (in diagonal) of the mass defect matrix can represent the kinetic energy of rotation of quarks around to the side AB, see _{Kin} < 0. Recall, in this way, the influence of the spin of a particle in them interactions. About the proton spin, see ref. [

s p = { s t ( q ) = [ ( ↑ ) u 1 , ( ↓ ) u 2 , ( ↑ , ↓ ) d 1 ] ≡ s q ( u 1 ) + s q ( u 2 ) + s q ( d 1 ) = s q ( d 1 ) = ± ( 1 / 2 ) s t ( l ) = [ ( ↑ ) u 1 , ( ↓ ) u 2 , ( ↑ , ↓ ) d 1 ] ≡ s l ( u 1 ) + s l ( u 2 ) + s l ( d 1 ) = s l ( d 1 ) = ± ( 1 / 2 ) s t ( g ) = [ ( ↓ ) u 1 , ( ↑ ) u 1 , ( ↓ , ↑ ) d 1 ] ≡ s g ( u 1 ) + s g ( u 2 ) + s g ( d 1 ) = s g ( d 1 ) = ∓ ( 1 / 2 ) } = s q ( d 1 ) = ± ( 1 / 2 ) (21)

Note that [s(l) + s(g) = 0] and [(s_{tot}(q) = ±1/2, s_{tot}(l) = ±1), (s_{tot}(g) = m1)]: the orbital spin is always opposite to gluon spin. From Equaiton (21), the contribution of the quarks’ spin to total spin of proton is (1/3), see the literature. Besides, all quarks are free in their orbital movement, see the literature, even if there is a binding connection between (q_{i}, q_{j}). Then, we will have:

( R ( u 1 ) + R ( u 2 ) + R ( d ) ) ′ = [ ( r ′ 1 Δ m ′ ( u ( ↑ ) ⊗ u ( ↑ ) ) u 1 + r ′ 2 Δ m ′ ( u ( ↓ ) ⊗ u ( ↓ ) ) u 2 + r ′ 3 Δ m ′ ( d ( ↑ ) ⊗ d ( ↑ ) ) d ) ] ' = [ ( ( − 1 ) u 1 + ( + 1 ) u 2 + ( − 1 ) d ) ] R ( 1.15 ) MeV = [ ( 0 u , u − 1 d ) ] R ( 1.15 ) MeV (22)

where we have calculated the eigenvalues of [ R ( q ) = r Δ m ( q i ⊗ q i ) ⊕ ] :

R ′ ( q ) = r ′ [ Δ m ′ ( q i ⊗ q i ) ] with eigenvalues r’ = ±1 and [ Δ m ′ ( q i ⊗ q i ) ] = ( 1.15 ) MeV

Exactly, we have

1) r ′ Δ m ′ ( u 1 ⊗ u 1 ) = − r ′ Δ m ′ ( u 2 ⊗ u 2 ) ′ because the spins are opposite

2) r ′ Δ m ′ ( u 1 ⊗ u 1 ) ′ = − r ′ Δ m ′ ( d ⊗ d ) ′ because quarks are reciprocally in relative movement

Then we have:

Δ m * ( p ) = { [ ( 11.34 ) ] G B − [ ( 1.15 ) R ] } MeV = ( 10.19 ) MeV (23)

Nevertheless, the value (1, 15) MeV is only approximate, see ref [_{p}, see Equation (15):

Δ m ( p ) = 3 κ p [ Δ m * ( p ) ] = ( 5 2 ) [ ( 10.19 ) ] MeV = ( 25.48 ) MeV (24)

Then, the total mass of the proton is:

m t o t ( p ) = m ( p ) − Δ m γ ( p ) = [ ( 964.45 ) − ( 25 .48 ) ] MeV = ( 938 .97 ) MeV (25)

Value very next to the experimental.

The state with [(s_{tot})_{q} = ±1/2] is the one of proton, while the state with [s_{to}_{t}(q) = ±3/2] is the resonance N^{+}(3/2)_{(1520)MeV}: ( p + γ ) → N + → ( p + γ ) , see ref. [

s N + = { s t ( q ) = [ ( ↑ , ↓ ) u 1 , ( ↑ , ↓ ) u 2 , ( ↑ , ↓ ) d 1 ] ≡ s q ( u 1 d 1 u 2 ) = ± ( 3 / 2 ) s t ( l ) = [ ( ↑ , ↓ ) u 1 , ( ↑ , ↓ ) u 2 , ( ↑ , ↓ ) d 1 ] ≡ s l ( d 1 ) = ± 3 s t ( g ) = [ ( ↓ , ↑ ) u 1 , ( ↓ , ↑ ) u 1 , ( ↓ , ↑ ) d 1 ] ≡ s g ( d 1 ) = ∓ 3 } = s q = ± ( 3 / 2 ) (26)

In the next paper, we will show the structure equation of resonance N^{+}(3/2) and its calculation of mass.

We might think to follow geometric structures of the neutral state with three quarks (d, u, d) (

Note, in the b-configuration, along the diagonal (A’B’), the sum of the electric charge of quarks is zero. In previous works [

( n ) = 3 κ n ( { ( d ⊗ d _ ) A ⊗ _ [ d 1 ⊗ _ u ⊗ _ d 2 ] B } ) (27)

With configuration, see

The k_{n} is the elasticity coefficient modulating the masses of quarks (u, d) respect to one of pion (u_{π}, d_{π}): [ m ( u n , d n ) = κ n m ( u π , d π ) ]

The vertices number [A, B, C, D, F, G, E] is n_{v} = 7, while the number of total (n_{el}) of the “coupling” elastic vertices of quarks is:

[(A(d_{1}, u), C(d_{1}, u); A(d_{1}, d_{2}), C(d_{1}, d_{2}); A(d_{2}, u), C(d_{2}, u); A(d_{3}, d_{3}), C(d_{3}, d_{3})] or number 8

We find: [k_{n} = n_{v}/(n_{el}) = 7/8].

Recall, the increasing of number of vertices in a geometric structure makes increase the elastic tension between the quantum oscillators of structure. Processing the structure Equation (27), we will have:

n = 3 κ n { [ ( d ⊗ d _ ) ] A ⊗ _ { [ d 1 ⊗ ( u ⊕ d 2 ) ] B 1 ⊕ [ u ⊗ ( d 2 ⊕ d 1 ) ] B 2 ⊕ [ d 2 ⊗ ( d 1 ⊕ u ) ] B 3 } } = 3 κ n { [ ( d ⊗ d _ ) ] A ⊗ _ [ N a ⊕ N b ⊕ N c ] B } (28)

The mass calculation uses F_{m}-function. In this case, with the presence of the pair (d, d), we project the operation (A Ä B) in F_{m}-function space:

[ ⊗ _ F m = ( ⊗ U ⊕ ) F m = ⊕ ⊗ ]

That is the operation (Å_{Ä}) point out interaction with interpenetration of components. Then, it is:

m ( n ) = 3 κ n F m ( [ ( d ⊗ d _ ) ] A ⊗ _ [ n a ⊕ n b ⊕ n c ] B ) = 3 κ n m ( [ ( d ⊗ d _ ) ] A ⊕ ⊗ { [ d 1 ⊗ ( u ⊕ d 2 ) ] B 1 ⊕ [ u ⊗ ( d 2 ⊕ d 1 ) ] B 2 ⊕ [ d 2 ⊗ ( d 1 ⊕ u ) ] B 3 } ) ⊗

= 3 κ n { m ( d ⊗ d _ ) A + m ( { [ d 1 ⊗ ( u ⊕ d 2 ) ] B 1 ⊕ [ u ⊗ ( d 2 ⊕ d 1 ) ] B 2 ⊕ [ d 2 ⊗ ( d 1 ⊕ u ) ] B 3 } ) } ⊗ = [ 3 κ n m ( d ) ] A + { 3 κ n [ m ( n a ) + m ( n b ) + m ( n c ) ] } B = 3 κ n m ( n ) A + 3 κ n m ( n ) B (29)

With

m ( n a ) = m [ ( d 1 ⊗ u ) ⊕ ( d 1 ⊗ d 2 ) ] a = m ( 〈 d 1 , u 〉 ) + m ( 〈 d 1 , d 2 〉 ) m ( n b ) = m [ ( u ⊗ d 2 ) ⊕ ( u ⊗ d 1 ) ] b = m ( 〈 u , d 2 〉 ) + m ( 〈 u , d 1 〉 ) m ( n c ) = m [ ( d 2 ⊗ d 1 ) ⊕ ( d 2 ⊗ u ) ] c = m ( 〈 d 2 , d 1 〉 ) + m ( 〈 d 2 , u 〉 ) (30)

The state with spin s = 1/2 is the neutron, see

If the neutron spin is (1/2) and u-quark rotates around axis AC, then the two d-quarks have opposite spin to it. Recall the (d, d)_{g} pair is at spin s = 1; thus the two couples [(d, d),(d, d)_{g}] have an overall zero spin. It follows:

s n = { s t ( q ) = [ ( ↓ ↓ ) ( d 1 , d 2 ) , ( ↑ , ↓ ) u 1 ] , [ ( ↑ ↑ ) ( d , d _ ) γ ] ≡ [ s γ ( d , d _ ) + s q ( d 1 , d 2 ) ] + s q ( u 1 ) = ± ( 1 / 2 ) s t ( l ) = [ ( ↓ ↓ ) ( d 1 , d 2 ) , ( ↑ , ↓ ) u 1 ] , [ ( ↑ ↑ ) ( d , d _ ) γ ] ≡ s l ( u 1 ) = ± ( 1 / 2 ) s t ( g ) = [ ( ↑ ↑ ) ( d 1 , d 2 ) , ( ↓ , ↑ ) u 1 ] , [ ( ↓ ↓ ) ( d , d _ ) γ ] ≡ s g ( u 1 ) = ∓ ( 1 / 2 ) } = s q ( u 1 ) = ± ( 1 / 2 ) (31)

In all two cases, the two quarks (d_{1}, d_{2}) cannot interpenetrate each other, because

They have parallel spins: then, it is (d_{1} Ä d_{2}) º 0 and, thus, [m(d_{1} Ä d_{2}) = 0], see _{1} Ä d_{2}) = 0] it follows, see Equation (30):

m ( n a ) = m [ ( d 1 ⊗ u ) ] a = m ( 〈 d 1 , u 〉 ) m ( n b ) = m [ ( u ⊗ d 2 ) ⊕ ( u ⊗ d 1 ) ] b = m ( 〈 u , d 2 〉 ) + m ( 〈 u , d 1 〉 ) m ( n c ) = m [ ( d 2 ⊗ u ) ] c = m ( 〈 d 2 , u 〉 ) (32)

Thus:

m ( n a , b , c ) B = [ m ( n a ) + m ( n b ) + m ( n c ) ] B = [ 2 m ( 〈 d 1 , u 〉 ) + 2 m ( 〈 u , d 2 〉 ) ] B (33)

and

m ( n ) B = { [ m ( n a ) + m ( n b ) + m ( n c ) ] } = [ 2 m ( 〈 d 1 , u 〉 ) + 2 m ( 〈 u , d 1 〉 ) ] = [ 4 ( 69.79 ) ] MeV = ( 279.16 ) MeV (34)

It follows

m ( n ) = 3 κ n [ m ( n ) A + m ( n ) B ] = 3 κ n [ m ( d ) + m ( n ) B ] = 3 κ n [ ( 86.26 ) + ( 279.16 ) ] MeV = 3 ( 7 8 ) ( 365.42 ) MeV = ( 959.23 ) MeV (35)

As in the proton, also here we insert the interpenetration of a quark with itself. This last aspect comes highlighted by the matrix A_{ij}.We build the matrix A_{ij} of all couplings with interpenetration between quarks [(q_{i} Ä q_{j})], which compose the structure equation in n-nucleon and interacting with other quark, [(q_{i} Ä q_{j})_{Å}]. As in proton, it is: [ ( q i ⊗ q j ) ⊕ ] ≠ [ ( q j ⊗ q i ) ⊕ ] .

The matrix also admits the elements A_{ii}¹ 0. Then, the matrix is, see

Δm^{*}(n) | d_{1} | u_{1} | d_{2} | d_{3 } | d_{4} |
---|---|---|---|---|---|

d_{1} | (d_{1}d_{1})^{R} | (d_{1}u_{1})^{G}^{1} | (d_{1}d_{2})^{B}^{1} | (d_{1}d_{3})^{B}^{1} | (d_{1}d_{4})^{G}^{2} |

u_{1} | (u_{1}d_{1})^{G1} | (u_{1}u_{1})^{R} | (u_{1}d_{2})^{G}^{1} | (u_{1}d_{3})^{G}^{1} | (u_{1}d_{4})^{B}^{2} |

d_{2} | (d_{2}d_{1})^{B}^{1} | (d_{2}u_{1})^{G}^{1} | (d_{2}d_{2})^{R} | (d_{2}d_{3})^{B}^{1} | (d_{2}d_{4})^{G}^{2} |

d_{3} | (d_{3}d_{1})^{B}^{1} | (d_{3}u_{1})^{G}^{1} | (d_{3}d_{2})^{B}^{1} | (d_{3}d_{3})^{R} | (d_{3}d_{4})^{G}^{2} |

d_{4} | (d_{4}d_{1})^{G}^{2} | (d_{4}u_{1})^{B}^{2} | (d_{4}d_{2})^{G}^{2} | (d_{4}d_{3})^{G}^{2} | (d_{4}d_{4})^{R} |

Here we have omitted the subscript (Å), see

Δ m * ( n ) = [ Δ m ( 6 ( u ⊗ d ) ) G 1 + Δ m ( 6 ( d _ ⊗ d ) ) G 2 ] + [ Δ m ( 6 ( d i ⊗ d j ) ) B 1 + 2 Δ m ( ( u ⊗ d _ ) ) B 2 ] + [ 2 Δ m ( ( d i ⊗ d i ) 1 , 2 ) + Δ m ( ( d i ⊗ d i ) γ ) + Δ m ( ( d _ i ⊗ d _ i ) γ ) + Δ m ( ( u i ⊗ u i ) ) ] R (36)

For the spin (s = 1/2), see the table of the neutron spin, we can have the term R’:

R ′ ( n ) = [ 2 Δ m ( ( d i ⊗ d i ) 1 , 2 ) + Δ m ( ( d i ⊗ d i ) γ ) + Δ m ( ( d _ i ⊗ d _ i ) γ ) + Δ m ( ( u i ⊗ u i ) ) ] R ′ = [ r ′ 1 Δ m ′ ( d ( ↓ ) ⊗ d ( ↓ ) ) d 1 + r ′ 2 Δ m ′ ( d ( ↓ ) ⊗ d ( ↓ ) ) d 2 + r ′ 3 Δ m ′ ( d ( ↑ ) ⊗ d ( ↑ ) ) γ

+ r ′ 4 Δ m ′ ( d _ ( ↑ ) ⊗ d _ ( ↑ ) ) γ + r ′ 5 Δ m ′ ( u ( ↓ ) ⊗ u ( ↓ ) ) ] = [ ( − 2 ( d 1 , d 2 ) + 1 γ − 1 γ − 1 u ) ( 1.15 ) R ] MeV = [ ( − 3 ) ( 1.15 ) R ] MeV (37)

Then, it is

Δ m * ( n ) = [ Δ m ( 6 ( u ⊗ d ) ) G 1 + Δ m ( 6 ( d _ ⊗ d ) ) G 2 ] + [ Δ m ( 6 ( d i ⊗ d j ) ) B 1 + 2 Δ m ( ( u ⊗ d _ ) ) B 2 ] + R ( n ) = [ 6 ( 4 .59 ) G 1 + 6 ( 5 .68 ) G 2 ] MeV − [ 6 ( 5.68 ) B 1 + 2 ( 4 .59 ) B 2 ] MeV + [ − ( 3.45 ) R ] MeV = ( 14.91 ) MeV (38)

The pair (d, d) in the structure of neutron shields the electromagnetic interactions between quarks, therefore it weakens the mass defect. The presence of the pair d-quark along diagonal, see

Δ m * * ( n ) = [ Δ m * ( n ) ] ( d , d _ ) = ( 1 2 ) [ Δ m * ( n ) ] = ( 7.46 ) MeV (39)

Like the proton, in the neutron being explicit the elasticity parameter k_{n}, it is, see Equation (24):

Δ m ( n ) = 3 κ n [ Δ m * * ( n ) ] ( d , d _ ) = 3 ( 7 8 ) ( 7.46 ) MeV = ( 19.58 ) MeV (40)

Thus, it is:

m t o t ( n ) = m ( n ) − Δ m γ ( n ) = { ( 959.23 ) − ( 19.58 ) } MeV = [ ( 939.65 ) ] MeV / c 2 (41)

Next to that experimental m(n) = (939.57) MeV.

In this paper, some aspects introduce a new paradigm in the phenomenology of the interactions. Describing the particles as a set of coupling quantum oscillators allows us of understanding more in-depth the Standard Model (SM) and its descriptions about the interactions, decay, spin and dynamics of particles. Besides, the AGM adds the possibilities of calculating [

· The interpenetration between quarks;

· The connection between interpenetration and spin;

· The interpenetration of a quark with itself;

· The not-locality of a quark in time or the time as a quantum operator.

AGM allows explaining some aspects of the phenomenology of hadronic interactions which in contemporary literature do not have a conclusive understanding yet. For instance:

1) The spin question of the proton (mesons) has a clarifying explanation in Equations (3) and (21).

2) The moment questions of protons (parton model) [_{quark}/P_{p}) » (1/2). This result is a consequence of the presence of gluons in the proton. The cross-sections confirm the fact that the gluons bring a missing moment in hadron collisions. AGM explains this result, considering that along the side BC in _{p}) is composed by the moment of gluons (P_{g}) of the junction and (P_{q}) moment of quarks (u, d, u).

3) Some researchers, see ref. [

4) The problematic phenomenology of the hadron jets [

( e + e − ) : e + + e − → q + q + g → X h a d r o n s (*)

Recall both Quark and gluons, are never visible in the detector because they can only exist confined within the hadrons. In the theory of AGM is possible to interpret the jets and their hadronization as consequence of presence of lattice {d,d} U {u,u}. In fact, we write the reaction (*) as:

( e + + e − ) → [ ( γ + γ ) ⊗ _ { d , d _ } ⊗ { u , u _ } ] → q + q + g → X h a d r o n s

while, in annihilation process with proton-antiproton:

( p + ⊗ _ p − ) ⊗ _ [ { d , d _ } ⊗ { u , u _ } ] → [ N + + N − ] { d , d _ } + [ X h a d r o n s ] { d , d _ } _{ }

Which we show in

By structure equations and geometric aspect of hadron, it is possible to find the decay probabilities.

The interpenetration of a quark with itself Δm(q_{i} Ä q_{i})_{R} implies to introduce the not-local time. This, because a quark q is in all times allowing to proper rotation also as it happens to spin, where the quark q is in all orientations allowing to the proper rotation. This aspect is confirmed by mass defect in Δm(q_{i} Ä q_{i})_{R} so as the spin, confirmed in all interactions between particles. In this way, one needs to admit the existence of time operator, see the Heinseberg relations in time version: ΔEΔt ³ 2πh The same definition of a virtual particle is so correlated to

not-local time and, thus, time operator or the time as an observable.

Further, AGM allows explaining some problematic aspects in the hadron production accompanied by the production of the W, Z bosons:

p p → q q − + X + Z → l + l − + X

The appearance of vector bosons in hadronic jets or the transformations in beta decay induce us to introduce a hypothesis of structure in the boson particles. For example, one of the issues not yet fully clarified is the transformation of leptonic particles into quarks as happens in the production of hadronic jets:

( e − + e + ) → γ + γ → q + q → h a d r o n s

How is it possible for photons to create quarks? Recall that photons are the intermediaries of the EM force. Then, we ask us because they can produce particles like quarks, provided with an “additional” agent charge, which is the color charge.

A comprehensive answer could then be to assign a structure also to a “real” elementary particle (that is not composed of sub-particles) and that it is possible to transform a structure into another only if the particles are geometric structures. Thus, there would be a mechanism of topological transformations on geometrical structures that would thus be transformed one into each other. In this way, we introduce a new paradigm in the phenomenology of strong interactions that open new perspectives for resolving the various problems of these interactions.

Note speaking of structures of quantum oscillators into particles implies (see

The authors declare no conflicts of interest regarding the publication of this paper.

Guido, G. (2021) Theoretical Spectrum of Mass of the Nucleons: New Aspects of the QM. Journal of High Energy Physics, Gravitation and Cosmology, 7, 123-143. https://doi.org/10.4236/jhepgc.2021.71006