_{1}

^{*}

Using the “Aureum Geometric Model” (AGM) of quarks, we formulate the structure equations describing mesons and, by a mathematic procedure, we calculate the theoretical spectrum of mass values of light mesons without strangeness.

In previous papers [

The aims of this paper are to calculate the mass values of light mesons using a way different to one of QCD potentials. Instead of incorporating the binding energy of the quarks in suitable potentials [^{2}]. This aspect would be similar to the interaction force in AGM, where the binding gluons in the geometric structure of two quarks (u, d) are equivalent to simple oscillators [_{i}, we can so admit that in the elastic parameters (k_{i}) are contained the mass defects; in this way the parameter k replaces the potential V(r)).

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 a table, see

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 ) ⊗ , ( B j ) ⊕ ] are the “base components” of the structure. The (F_{m}) is an application on the structure components (A, B), which gives us the mass values

Ä-operation ( a i , a j ) ∈ { a } | Å-operation ( a i , a j ) ∈ { a } |
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1) ( a i ⊗ a i ) = ( a ) | 1) ( a i ⊕ a i ) = 0 |

2) ( a i , a j ) ∈ { a } ( a i ⊗ a j ) = ( a j ⊗ a i ) | 2) ( a i ⊕ a j ) = ( a j ⊕ a i ) If ( a i ⊕ a j ) ⊕ = [ ( a i ⊕ a j ) ⊕ a k ] → ( a i ⊕ a j ) ⊕ ≠ ( a j ⊕ a i ) ⊕ |

3) ( n , m ) ∈ N ( n a i ⊗ a j ) = n ( a i ⊗ a j ) ( n a i ⊗ m a j ) = ( n × m ) ( a i ⊗ a j ) | 3) [ 2 ( a i ) ⊕ ( a j ) ] ≡ [ ( a i ) + ( a i ) ] ⊕ a j = [ ( a i ⊕ a j ) + ( a i ⊕ a j ) ] = 2 ( a i ⊕ a j ) |

4) transitive property if [ ( a ⊗ b ) , ( b ⊗ c ) ] → [ ( a ⊗ c ) ] | 4) transitive property if [ ( a ⊗ b ) , ( b ⊗ c ) , ( a ⊗ c ) ] if [ ( a ⊕ b ) , ( b ⊕ c ) ] → [ ( a ⊕ c ) ] |

5) [ ( a i ⊗ a j ) ⊗ a k ] = [ ( a i ⊗ a k ) ⊗ a j ] = [ ( a k ⊗ a j ) ⊗ a i ] | 5) [ ( a i ⊕ a j ) ⊕ a k ] = [ ( a i ⊕ a k ) ⊕ ( a j ⊕ a k ) ] |

⊗ _ ≡ [ ⊗ , ⊕ ] ↔ composed operation | |

6) If { [ A = ( a ⊕ b ) ] , [ B = ( c ⊕ d ) ] } → ( A ⊗ _ B ) = ( a ⊕ b ) ⊗ ( c ⊕ d ) | |

7) ( a ⊕ b ) ⊗ ( c ⊕ d ) = ( a ⊗ c ) ⊕ ( a ⊗ d ) ⊕ ( b ⊗ c ) ⊕ ( b ⊗ d ) | |

8) [ ( a i ⊗ a j ) ⊕ a k ] ≠ [ ( a j ⊗ a i ) ⊕ a k ] ↔ ( a i ⊗ a j ) ⊕ ≠ ( a j ⊗ a i ) ⊕ |

(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 (1)

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 ) } (2)

Note that the mass of two interpenetrating particles [ a ⊗ b ] will be obtained by the average value of individual masses [ 〈 m ( a , b ) 〉 ] , while the mass of two interacting particles [ a ⊕ b ] will be 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). 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 will be:

Δ m = Δ m g + Δ m e m

where Δm_{g} is the mass defect due to binding by gluons. Nevertheless, the Δm_{g} has been englobed 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 will 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 . ) (3)

where “i-degree” points out the degree of coupling of two and more couplings. Here the terms of degree superior to II˚ are omitted. 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 (4)

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 (AB), see _{m}, F_{Δm}) to obtain the properties of the operations (Ä, Å) for calculating the masses and mass defects (see

In article [

[ m ( u ) bare = ( 3 .51 ) MeV , m ( d ) bare = ( 5 .67 ) MeV ] (5)

These values are inside the range anticipated by literature [

[ m ( u ) dressed = ( 53 .31 ) MeV , m ( d ) dressed = ( 86 .26 ) MeV ] (6)

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. If the values in Equation (6) are that of bound quarks in pion by gluons, then we can think of incorporate the potential V(r) of gluons in QCD into mass values of the

(Ä-operation) in (F_{m}, F_{Δm}) representations | Å-operation in (F_{m}, F_{Δm}) representations |
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4) F m ( a i ⊗ a j ) = 〈 m ( a i , a j ) 〉 = 〈 m ( a i ) , m ( a j ) 〉 F m ( a i ⊗ a i ) = F m ( a i ) = m ( a i ) | 1) F m ( a i ⊕ a j ) = F m ( a i ) + F m ( a j ) = m ( a i ) + m ( a j ) F m ( a i ⊕ a i ) = F m ( a i ) + F m ( a i ) = m ( a i ) + m ( a i ) = 2 m ( a i ) |

2) F Δ m ( a i ⊗ a j ) = 0 F Δ m ( a i ⊗ a j ) ⊕ ≠ 0 with ( a i ⊗ a j ) ⊕ = ( a i ⊗ a j ) ⊕ a k | 2) F Δ m ( a ⊕ b ⊕ c ) = F Δ m ( a ) + F Δ m ( b ) + F Δ m ( c ) + F Δ m ( a ⊕ _ b ) a b + F Δ m ( a ⊕ _ c ) a c + F Δ m ( b ⊕ _ c ) b c |

3) F Δ m ( a i ⊗ a j ) ⊕ = F Δ m ( a i ) F Δ m ( ( a i ⊗ a j ) ⊕ a k ) = F Δ m ( ( a i ) ⊕ a k ) | 3) F Δ m [ ( a i ⊕ a j ) ⊕ a k ] = F Δ m [ ( a i ⊕ a k ) ⊕ ( a j ⊕ a k ) ] |

4) F m [ ( a i 1 ⊗ a j 2 ) ⊕ ( a i 2 ⊗ a j 1 ) ] = F m [ 2 ( a i ⊗ a j ) 12 ] | 4) F Δ m [ ( a i ⊗ a j ) ⊕ ( a k ⊗ a r ) ] = F Δ m [ ( a i ⊗ a j ) ⊕ ( a k ) ] + F Δ m [ ( a i ⊗ a j ) ⊕ ( a r ) ] |

5) F Δ m ( 2 a i ⊗ a j ) = F Δ m ( 2 ( a i ⊗ a j ) ) F Δ m ( n ( a i ⊗ a j ) ⊕ m ( a k ⊗ a r ) ) = ( n × m ) F Δ m ( ( a i ⊗ a j ) ⊕ ( a k ⊗ a r ) ) | 6) F Δ m [ a i ⊕ a j ] + F Δ m [ a i ⊕ a k ] = F Δ m [ a i ⊕ ( a j + a k ) ] |

Equation (6). So, the bound masses of quarks playing a fundamental rule inside pions. By pion (π º (u, d)) we can build the mesons’ structures. 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

However, this lattice, in turn, could have a background lattice of d-quark pairs {d, d}, see, in fact, the structure of neutral pion. In this way, both the { π r 0 } -lattice and {d, d} participating in building the mass spectrum of light mesons.

Here we report _{m}, F_{Δm}), (see

In equivalent two ways, see the ref. [_{Δ}_{m} of mass defects expresses these two possibilities. The structure equation contains the considerations on the various dynamics couplings and interpenetrations. The first way is considering without

Properties of pions in (Ä) operations | Properties of pions in (Å) operations |
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1) [ π ± = ( u ⊗ _ d ) ] 2) ( π ) 0 = [ ( π + ) ⊗ _ ( π − ) ] = [ ( u ⊕ d _ ) ⊗ _ ( u _ ⊕ d _ ) ] = [ ( π + ) ⊗ ( π − ) ] − [ ( u ⊕ u _ ) ⊗ ( d ⊕ d _ ) ] 3) ( π ± ) 0 = [ ( π + ) ⊗ ( π − ) ] virtual neutral pion 4) m [ ( π + ) ⊗ ( π − ) ] = ( π ± ) 0 = m ( π ± ) Equation (11) 5) m ( π + ⊗ π + ) = m ( π + ) 6) m ( π + ⊗ π − ) = 〈 m ( π + ) , m ( π − ) 〉 7) { Δ m ( π ) = [ m ( π ± ) − m ( π 0 ) ] } = ( 4.59 ) MeV 8) [ ( π 1 ± ⊗ π 2 ± ) ⊕ π 3 ± ] → Δ m ( π i ± ⊗ π j ± ) ⊕ = Δ m ( π j ± ) 9) 2 π ± ≡ 2 ( π 1 ± ⊗ _ π 2 ± ) 10) ( 2 π 1 ± ⊗ _ 2 π 2 ± ) = ( 2 × 2 ) ( π 1 ± ⊗ _ π 2 ± ) | 4) m ( π + ⊕ π − ) = m ( π + ) + m ( π − ) 5) { ( π r 0 ) = ( π + ⊕ π − ) } ; m ( π r 0 ) = 2 m ( π ± ) 6) F m [ ( π + ⊗ π − ) ⊕ ( π + ⊗ π − ) ] = F m [ 2 ( π ± ) 0 ] 8) Δ m [ ( π 1 ± ⊕ π 2 ∓ ) ⊗ π 0 ] = Δ m [ ( π 1 ± ⊕ π 2 ∓ ) π 0 ] [ Δ m ( π ) / 4 = ( 1.15 ) MeV ] |

11) Δ m ( π + ⊗ π − ) = 0 12) Δ m ( π i ± ⊗ π j ± ) ⊕ = Δ m ( π j ± ) | 12) [ ( π a ± ⊗ π b ± ) ⊗ π c ± ] ⊕ [ ( π a ± ⊗ π b ∓ ) ⊗ π c ∓ ] = [ ( π a ± ⊗ π b ± ) ⊕ ( π a ∓ ⊗ π b ∓ ) ] ⊕ [ π c ± ⊕ π c ∓ ] |

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 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.

η’-meson ((957.78) MeV)

In ref. [^{0})_{r}, η^{*}) where the pion molecule (π^{0})_{r} overlapping to η^{*}-combination,

with [ η * = ( π 0 ) r ⊗ ( π 0 ) r ] ; we well have: [ η ′ = ( π 0 ) r ⊗ ( η * ) ] . We need to reduce the structure equation in sum of elementary components (A_{i}, B_{i}), see Equation (1).

It follows, by 7-property in

η ′ = { [ ( π 1 0 ) r ] ⊗ [ η * ] } = [ ( π 1 0 ) r ] ⊗ [ ( π 2 0 ) r ⊗ ( π 3 0 ) r ] = ( π 1 0 ) r ⊗ [ ( π + ⊕ π − ) 2 ⊗ ( π + ⊕ π − ) 3 ] η * = { [ ( π + ⊕ π − ) 1 ] ⊗ [ ( π + ⊕ π − ) 2 ⊗ ( π + ⊕ π − ) 3 ] η * } ( 1 ) = { [ ( π + ⊕ π − ) 1 ] ⊗ [ ( π + ⊗ π + ) 23 ⊕ ( π + ⊗ π − ) 23 ⊕ ( π − ⊗ π + ) 23 ⊕ ( π − ⊗ π − ) 23 ] η * } ( 1 ) (7)

We apply the F_{m} function:

m ( η ′ ) = F m ( { [ ( π + ⊕ π − ) 1 ] B ⊗ [ ( π + ⊗ π + ) 23 ⊕ ( π + ⊗ π − ) 23 ⊕ ( π − ⊗ π + ) 23 ⊕ ( π − ⊗ π − ) 23 ] η * } ) = F m ( B ⊗ η * )

Then in the F_{m}-representation, we can have

m ( η * ) = F m ( [ ( π + ⊗ π + ) 23 ⊕ ( π + ⊗ π − ) 23 ⊕ ( π − ⊗ π + ) 23 ⊕ ( π − ⊗ π − ) 23 ] η * ) = F m ( [ ( π + ) + 2 ( π + ⊗ π − ) + ( π − ) ] )

where we have used the 1_{(}_{Ä}_{)}-properties of F_{m} in

m ( η ′ ) = F m ( { [ ( π + ⊕ π − ) ] B ⊗ [ ( π + ) ⊕ 2 ( π + ⊗ π − ) ⊕ ( π − ) ] η * } ) = F m ( [ ( π + ⊗ π + ) ⊕ ( π + ⊗ 2 ( π ± ) 0 ) ⊕ ( π + ⊗ π − ) ] B 1 ⊕ [ ( π − ⊗ π + ) ⊕ ( π − ⊗ 2 ( π ± ) 0 ) ⊕ ( π − ⊗ π − ) ] B 2 ) = F m ( ( π + ) ⊕ 2 ( π ± ⊗ 2 ( π ± ) 0 ) ⊕ 2 ( π ± ) 0 ⊕ ( π − ) )

Moreover, we have [ ( π ± ) 0 = ( π + ⊗ π − ) ] and [ 2 ( π ± ) 0 = ( π + ⊗ π − ) ⊕ ( π + ⊗ π − ) ] F m and F m ( π + ⊗ π + ) = F m ( π + ) , see ref. [_{m}-function, see the 1_{(}_{Å}_{)}-property in

m ( η ′ ) * = F m { 2 ( π ± ⊗ 2 ( π ± ) 0 ) ⊕ ( 2 ( π ± ) 0 ) ⊕ ( π + ) ⊕ ( π − ) } ( 5 ) = ( 2 〈 m ( π ± , 2 ( π ± ) 0 ) 〉 ) + ( 2 〈 m ( π + , π − ) 〉 ) + m ( π + ) + m ( π − ) (8)

where it is:

〈 m ( π ± , 2 ( π ± ) 0 ) 〉 = [ m ( π ± ) + m ( 2 ( π ± ) 0 ) 2 ] = ( 209.36 ) MeV (9)

Thus the partial mass value is:

m ( η ′ ) * = 2 ( 209.36 ) MeV + 4 ( 139.57 ) MeV = ( 976 .99 ) MeV (10)

Mass defect of (η’) can be (see Equation (3)):

Δ m ( η ′ ) = Δ m ( ( π 0 ) r ⊗ η * ) = Δ m ( ( π 0 ) r ⊗ ( π 0 ) r ⊗ ( π 0 ) r )

Note that Δ m ( η * ) = Δ m ( η ) , see in ref. [

{ ( π 0 ) r 1 ⊗ ( π 0 ) r 2 ⊗ ( π 0 ) r 3 } = { ( π 0 ) r 1 ⊗ [ ( π 0 ) r 2 ⊗ ( π 0 ) r 3 ] η 23 } = { [ ( π 0 ) r 1 ⊗ ( π 0 ) r 2 ] η 12 ⊗ ( π 0 ) r 3 }

Thus, in Δm(η’) we need to insert 2Δm(η); it follows, with Δm(η^{*}) = Δm(η):

Δ m ( η ′ ) = Δ m ( ( π 0 ) r ⊗ η * ) = Δ m ( ( π + ⊕ π − ) ⊗ η ) = Δ m ( ( π + ⊗ η ) A 1 ⊕ ( π − ⊗ η ) A 2 ) = Δ m ( ( π + ⊗ η ) A 1 ) + Δ m ( ( π − ⊗ η ) A 2 ) + Δ m ( [ ( π + ⊗ η ) ⊕ _ ( π − ⊗ η ) ] B 12 ) = 2 Δ m ( η ± ) ( A 1 + A 2 ) + Δ m ( [ ( π + ⊗ η ) ⊕ _ ( π − ⊗ η ) ] B 12 ) (11)

Here, we apply the 2-property of

Besides, [ η ± = ( π ± ) ⊗ ( η ) ] and Δ m ( [ π ± ⊗ η ] ) = Δ m ( ( η ) π ± ) ≈ Δ m ( η ) = ( 10.35 ) MeV

This because η is a neutral particle and, thus the interaction between (π^{±}) and (η) during them interpenetration is not comparable with Δm(η). The value of B_{12}-component in Equation (11) we will be:

Δ m ( B 12 ) = Δ m ( ( [ π + ⊗ η ] ⊕ _ [ π − ⊗ η ] ) B 12 ) = Δ m ( ( η ⊗ _ [ π + ⊕ π − ] ) B 12 ) = Δ m ( [ ( π 0 ) r ⊗ ( π 0 ) r ] η ⊗ _ [ π + ⊕ π − ] ) = Δ m ( [ ( π + ⊕ π − ) a ⊗ ( π + ⊕ π − ) b ] η ⊗ _ ( π + ⊕ π − ) c ) (12)

Solving individual products:

Δ m ( [ ( π + ⊕ π − ) a ⊗ ( π + ⊕ π − ) b ] η ⊗ _ ( π + ⊕ π − ) c ) = Δ m ( [ ( π a + ⊗ π b + ) ⊕ ( π a + ⊗ π b − ) ⊕ ( π a − ⊗ π b + ) ⊕ ( π a − ⊗ π b − ) ] η ⊗ _ ( π c + ⊕ π c − ) ) = Δ m ( [ ( π a + ⊗ π b + ) ⊗ _ π c + ⊕ ( π a + ⊗ π b − ) ⊗ _ π c + ⊕ ( π a − ⊗ π b + ) ⊗ _ π c + ⊕ ( π a − ⊗ π b − ) ⊗ _ π c + ] η , c ) + Δ m ( [ ( π a + ⊗ π b + ) ⊗ _ π c − ⊕ ( π a + ⊗ π b − ) ⊗ _ π c − ⊕ ( π a − ⊗ π b + ) ⊗ _ π c − ⊕ ( π a − ⊗ π b − ) ⊗ _ π c − ] η , c ) (13)

Where we can note two terms at repulsive interaction

( π + ⊗ π + ) ⊗ _ π + ⇔ component at repulsive action ( π + ⊗ π − ) ⊗ _ π + = ( π ± ) 0 ⊗ _ π + ; ( π − ⊗ π + ) ⊗ _ π + = ( π ± ) 0 ⊗ _ π + ( π − ⊗ π − ) ⊗ _ π + = π − ⊗ _ ( π ± ) 0 ; ( π + ⊗ π + ) ⊗ _ π − = π + ⊗ _ ( π ± ) 0 ( π + ⊗ π − ) ⊗ _ π − = ( π ± ) 0 ⊗ _ π − ; ( π − ⊗ π + ) ⊗ _ π − = ( π ± ) 0 ⊗ _ π − ( π − ⊗ π − ) ⊗ _ π − ⇔ component at repulsive action

The components expressing an interpenetration between neutral pion (π^{±})^{0} and charged pion (π^{±}) have a zero value of mass defect. Only the two charged components have a negative mass defect. Therefore, the B_{12}-component has a final negative value of mass defect: Δm(B_{12}) < 0. We need calculating the numerical value of Δm(B_{12}), which, now, is:

Δ m ( B 12 ) = Δ m ( [ ( π a + ⊗ π b + ) ⊗ _ π c + ] r e p ⊕ [ ( π a − ⊗ π b − ) ⊗ _ π c − ] r e p ) (14)

In B_{12}-component there is an interaction (Å) between the three interpenetrated pions (π^{+}) and the three interpenetrated pions π^{−}; therefore, there is also a partial interaction between ( π c + ) and ( π c − ) : ( π c + ) ⊗ ( π c − ) .

The same happens between the two pairs [ ( π a + ⊗ π b + ) ↔ ( π a − ⊗ π b − ) ] of Equation (14); it follows: [ ( π a + ⊗ π b + ) ⊕ ( π a − ⊗ π b − ) ] .

Therefore, we could exchange the terms of equation (see

Δ m ( B 12 ) = Δ m ( [ ( π a + ⊗ π b + ) ⊗ _ π c + ] r e p ⊕ [ ( π a − ⊗ π b − ) ⊗ _ π c − ] r e p ) ( 1 ) = Δ m r e p ( [ ( π a + ⊗ π b + ) ⊕ ( π a − ⊗ π b − ) ] B 1 ⊗ _ [ ( π c + ⊕ π c − ) ] B 2 ) ( 2 ) = Δ m r e p ( [ ( π a + ⊗ π b + ) ⊕ ( π a − ⊗ π b − ) ] B 1 ⊗ _ ( π r c 0 ) B 2 ) ( 3 )

Besides, note that the B_{2}-component [ ( π c + ⊕ π c − ) = π r 0 ] is neutral and partially could shield the attractive reciprocal interaction between the two couples ( π a b + ) and ( π a b − ), as also the reciprocal repulsive interaction between the internal pions of each couple: the 3-component expresses just this aspect. To express the shielding action of π r 0 on the B_{1}-component, we will write the 3-component in the following way:

Δ m ( B 12 ) = Δ m ( [ ( π a + ⊗ π b + ) ⊕ ( π a − ⊗ π b − ) ] B 1 ⊗ _ ( π r c 0 ) B 2 ) ( 3 ) = Δ m ( [ ( π a + ⊗ π b + ) ⊕ ( π a − ⊗ π b − ) ] ( π r 0 ) ) ( 4 ) (15)

By Equation (22-6) in ref. [

Δ m ( [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 − ⊗ π 2 − ) ] ( π r 0 ) ) = Δ m ( [ ( π 12 + ) ⊕ ( π 12 − ) ] ( π r 0 ) ) (16)

The shielding action of ( π r 0 ) lowers the value of mass defect, see in ref. [

Δ m ( [ ( π 12 + ) ⊕ ( π 12 − ) ] ( π r 0 ) ) = − ( Δ m ( π ) n m ) = − ( ( 4.60 ) 2 2 ) MeV (17)

This, (n = 4), because the neutral molecule ( π r 0 ) is a double particle which acting on two pions: the submultiple of Δm(π) is 2^{2}.

Thus, the total (η’)-mass by Equations [(11), (17)] and in ref. [_{Ci}) is:

m ( η ′ ) ≈ m ( η ′ ) * − Δ m = { ( 976.99 ) − [ 2 ( 10.35 ) − ( 1.15 ) ] } MeV = ( 957 .44 ) MeV (18)

Here too we obtain un value very close to the experimental one.

Also here we can build (see

The mass defect is, see the ref. [

Δ m ( η ′ ) = [ Δ m ( ( π 1 − ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 2 − ⊗ π 2 + ) ⊕ ) + Δ m ( ( π 3 − ⊗ π 3 + ) ⊕ ) ] G + [ Δ m ( ( π 2 − ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 1 − ⊗ π 2 + ) ⊕ ) ] Y + [ Δ m ( ( π 3 − ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 1 − ⊗ π 3 + ) ⊕ ) ] Y + [ Δ m ( ( π 3 − ⊗ π 2 + ) ⊕ ) + Δ m ( ( π 2 − ⊗ π 3 + ) ⊕ ) ] Y

− [ Δ m ( ( π 2 + ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 2 − ⊗ π 1 − ) ⊕ ) ] B − [ Δ m ( ( π 3 + ⊗ π 1 + ) ⊕ ) + Δ m ( ( π 3 − ⊗ π 1 − ) ⊕ ) ] B − [ Δ m ( ( π 3 + ⊗ π 2 + ) ⊕ ) + Δ m ( ( π 3 − ⊗ π 2 − ) ⊕ ) ] B − [ Δ m ( ( π i ± ⊗ π i ± ) ⊕ ) ] R

Δ m ( η ′ ) = { [ 3 ( 4 .6 ) ( i , i ) G + 6 ( 2.3 ) ( i , j ) Y ] ( ± , ∓ ) − [ 6 ( 1.15 ) ( i , j ) B + ( 1.15 ) ( i , i ) R ( ± , ± ) ] } MeV = ( 19 .55 ) MeV (19)

Obtaining the same value of mass defect of that Equation (18). As already find, also here we will have: m ( η ′ ) ≈ m ( η ′ ) * − Δ m = { ( 976.99 ) − ( 19.55 ) } MeV = ( 957 .44 ) MeV

The most probable decays [

[ η ′ → η + ( π + + π − ) ] ( 43 % ) , [ η ′ → η + ( π 0 + π 0 ) ] ( 23 % ) ,

[ η ′ → ρ + γ ] ( 30 % ) , [ η ′ → ω + γ ] ( 4 % )

Moreover, other decays are possible (see more forward) for available energy.

Note the spins of (η, η’) are at zero values: [ s ( η ) = 0 , s ( η ′ ) = 0 ] . This result

π 1 + | π 2 + | π 3 + | π 1 − | π 2 − | π 3 − | |
---|---|---|---|---|---|---|

π 1 + | ( π 1 + , π 1 + ) R | ( π 1 + , π 2 + ) | ( π 1 + , π 3 + ) | ( π 1 + , π 1 − ) | ( π 1 + , π 2 − ) | ( π 1 + , π 3 − ) |

π 2 + | ( π 2 + , π 1 + ) B | ( π 2 + , π 2 + ) R | ( π 2 + , π 3 + ) | ( π 2 + , π 1 − ) | ( π 2 + , π 2 − ) | ( π 2 + , π 3 − ) |

π 3 + | ( π 3 + , π 1 + ) B | ( π 3 + , π 2 + ) B | ( π 3 + , π 3 + ) R | ( π 3 + , π 1 − ) | ( π 3 + , π 2 − ) | ( π 3 + , π 3 − ) |

π 1 − | ( π 1 − , π 1 + ) G | ( π 1 − , π 2 + ) Y | ( π 1 − , π 3 + ) Y | ( π 1 − , π 1 − ) R | ( π 1 − , π 2 − ) | ( π 1 − , π 3 − ) |

π 2 − | ( π 2 − , π 1 + ) Y | ( π 2 − , π 2 + ) G | ( π 2 − , π 3 + ) Y | ( π 2 − , π 1 − ) B | ( π 2 − , π 2 − ) R | ( π 2 − , π 3 − ) |

π 3 − | ( π 3 − , π 1 + ) Y | ( π 3 − , π 2 + ) Y | ( π 3 − , π 3 + ) G | ( π 3 − , π 1 − ) B | ( π 3 − , π 2 − ) B | ( π 3 − , π 3 − ) R |

derives from the zero value of the spin [

Recall the interpenetration of two triangular structure (see the pion) has spin zero because the relative rotations in interpenetration are always opposites [

If we consider the superposition of two η-meson, as η’-meson, we have:

η ″ = { [ η * ] ⊗ [ η * ] } = [ ( π 0 ) r ⊗ ( π 0 ) r ] ⊗ [ ( π 0 ) r ⊗ ( π 0 ) r ]

Moreover, if we calculate the partial mass, then one finds the following value m = (1954.00) MeV, which is next to f_{2}-meson.

ρ-meson (775.26 MeV)

Note, generally, the existence of light mesons implies the presence of a background lattice of pion molecules {(π^{0})_{r}}. Nevertheless, this π-lattice, in turn, is built on another “primary” lattice {d, d}, as now we will show. This lattice can be important for binding more molecules of pions. Not only that, but we could think the presence of these lattices {d, d} and of molecules {(π^{0})_{r}} is connected to the “hadronization process” in strong interactions. The external energy flows into a hadron and “immediately” transformed, through these lattices, into further hadrons. In the same way, these lattices can also participate in building the more massive hadrons. Then we think that resonance states involve a “primary” background lattice {d, d}. In fact, into structure the ρ-meson we add the pair (d, d)_{π} and so having:

ρ = [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) B 1 ⊗ ( 2 π − ⊕ π r 0 ) B 2 ] A } (20)

where (π^{0})_{r} is the pion molecule and [ ( d , d _ ) π = ( d ⊗ d _ ) π ] is the pair of d_{π}-quark. Note that, see Equation (2) and

m ( d ⊗ d _ ) π = 〈 m ( d , d _ ) π 〉 = [ m ( d π ) + m ( d _ π ) 2 ] = m ( d π )

Besides the multiplicative number (2) in B_{i}-component becomes a multiplicative factor in F_{m}-function, see η’-meson and the 9_{(}_{Å}_{)}-property in

[ 2 π ± ≡ 2 ( π 1 ± ⊗ π 2 ± ) ]

We develop the structure equation:

ρ = [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) B 1 ⊗ ( 2 π − ⊕ π r 0 ) B 2 ] } = [ ( d , d _ ) π ] ⊗ { [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( 2 π 1 + ⊗ π r 2 0 ) A 2 ⊕ ( π r 1 0 ⊗ 2 π 2 − ) A 3 ⊕ ( π r 1 0 ⊗ π r 2 0 ) A 4 ] B } (21)

where [ 2 π + ⊗ 2 π − = 4 ( π + ⊗ π − ) ] , see the 10_{Ä}-property in _{m} function:

m ( ρ ) = F m ( [ ( d , d _ ) π ] ⊗ { [ 4 ( π 1 + ⊗ π 2 − ) ⊕ ( 2 π 1 + ⊗ π r 2 0 ) ⊕ ( π r 1 0 ⊗ 2 π 2 − ) ⊕ ( π r 1 0 ⊗ π r 2 0 ) ] B } ) = F m ( [ ( d , d _ ) π ] ⊗ { [ 4 ( π ± ) 12 0 ⊕ 2 ( 2 π ± ⊗ π r 0 ) 12 ⊕ ( π r 1 0 ⊗ π r 2 0 ) ] B } ) = F m ( { [ ( d , d _ ) π ⊗ 4 ( π ± ) 12 0 ] A 1 ⊕ [ ( d , d _ ) π ⊗ 2 ( 2 π ± ⊗ π r 0 ) 12 ] A 2 ⊕ [ ( d , d _ ) π ⊗ ( π r 1 0 ⊗ π r 2 0 ) ] A 3 } ) (22)

where [ ( π + ⊗ π − ) ≡ ( π ± ) 0 ] and (see 4-property in

m ( ρ ) = m { [ ( d , d _ ) π ⊗ 4 ( π ± ) 12 0 ] A 1 ⊕ [ ( d , d _ ) π ⊗ 2 ( 2 π ± ⊗ π r 0 ) 12 ] A 2 ⊕ [ ( d , d _ ) π ⊗ ( π r 0 ⊗ π r 0 ) ] A 3 } = { 〈 m ( ( d , d _ ) , 4 ( π ± ) 0 ) 〉 + 〈 m ( ( d , d _ ) , 2 ( 2 π ± ⊗ π r 0 ) ) 〉 + 〈 m ( ( d , d _ ) , ( π r 0 ⊗ π r 0 ) ) 〉 }

= [ m ( d ) + 4 m ( 〈 ( π ± ) 0 〉 ) 2 + m ( d ) + 2 m ( 〈 2 π ± , π r 0 〉 ) 2 + m ( d ) + m ( 〈 π r 0 , π r 0 〉 ) 2 ] = 3 2 m ( d ) + 2 m ( π ± ) + [ m ( π ± ) + 1 2 m ( π r 0 ) ] + 1 2 m ( π r 0 ) = 3 2 m ( d ) + 3 m ( π ± ) + m ( π r 0 ) (23)

Its partial mass value is:

m ( ρ ) * = 3 2 m ( d ) + 3 m ( π ± ) + m ( π r 0 ) = [ 3 2 ( 86.26 ) + 3 ( 139.57 ) + 2 ( 139.57 ) ] MeV = ( 827 .24 ) MeV (24)

About mass defect by Equation (20) we have:

Δ m ( ρ ) ( d , d _ ) = F Δ m { ( d , d _ ) ⊗ [ ( 2 π + ⊕ π r 0 ) 1 ⊗ ( 2 π − ⊕ π r 0 ) 2 ] A } = F Δ m { [ ( 2 π + ⊕ π r 0 ) 1 ⊗ ( 2 π − ⊕ π r 0 ) 2 ] ( d , d _ ) } = F Δ m { [ ( 2 π 1 + ⊗ 2 π 2 − ) A 1 ⊕ ( 2 π 1 + ⊗ π r 2 0 ) A 2 ⊕ ( π r 1 0 ⊗ 2 π 2 − ) A 3 ⊕ ( π r 1 0 ⊗ π r 2 0 ) A 4 ] B } ( d , d _ ) (25)

where [ ( d , d _ ) ⊗ A = A ( d , d _ ) ] , see Equation (16). Processing the mass defect one obtains:

Δ m ( ρ ) ( d , d _ ) = F Δ m { [ ( 2 π 1 + ⊗ 2 π 2 − ) A 1 ⊕ ( 2 π 1 + ⊗ π r 2 0 ) A 2 ⊕ ( π r 1 0 ⊗ 2 π 2 − ) A 3 ⊕ ( π r 1 0 ⊗ π r 2 0 ) A 4 ] B } ( d , d _ ) = { 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 ) } ( d , d _ ) (26)

In a specific way, it is: [ Δ m ( A 1 ) = 0 , Δ m ( A 4 ) = 0 ] .

Here we cannot pose [ ( π r 0 ⊗ π r 0 ) = η ] and to have (Δm(η) ¹ 0), because η-meson do not belong to structure equation. Besides, by 5-property in

Δ m ( A 2 ) + Δ m ( A 3 ) = Δ m ( ( 2 π 1 + ⊗ π r 2 0 ) A 2 ) + Δ m ( ( π r 1 0 ⊗ 2 π 2 − ) A 3 ) = Δ m ( ( 2 π 1 + ⊗ ( π 2 + ⊕ π 2 − ) ) A 2 ) + Δ m ( ( 2 π 2 − ⊗ ( π 1 + ⊕ π 1 − ) ) A 3 ) = Δ m ( [ ( 2 π 1 + ⊗ π 2 + ) ⊕ ( 2 π 1 + ⊗ π 2 − ) ] A 2 ) + Δ m ( [ ( 2 π 2 − ⊗ π 1 + ) ⊕ ( 2 π 2 − ⊗ π 1 − ) ] A 3 ) A 3

= 4 Δ m ( [ ( π 1 + ⊗ π 2 + ) ⊕ ( π 1 + ⊗ π 2 − ) ] A 2 ) + 4 Δ m ( [ ( π 2 − ⊗ π 1 + ) ⊕ ( π 2 − ⊗ π 1 − ) ] A 3 ) = [ 4 ( 1.15 ) + 4 ( 1.15 ) ] MeV

By 5-property in

Δ m ( A 12 ) + Δ m ( A 13 ) = Δ m ( [ ( 2 π 1 + ⊗ 2 π 2 − ) A 1 ⊕ ( 2 π 1 + ⊗ π r 2 0 ) A 2 ] ) A 12 + Δ m ( [ ( 2 π 1 + ⊗ 2 π 2 − ) A 1 ⊕ ( π r 1 0 ⊗ 2 π 2 − ) A 3 ] ) A 13 = Δ m ( [ ( 2 π 1 + ⊗ 2 π 2 − ) A 1 ⊕ [ ( 2 π 1 + ⊗ π 2 + ) A ′ 1 ⊕ ( 2 π 1 + ⊗ π 2 − ) A ′ 2 ] A 2 ] ) A 12 + Δ m ( [ ( 2 π 1 + ⊗ 2 π 2 − ) A 1 ⊕ [ [ ( 2 π 2 − ⊗ π 1 + ) A ″ 1 ⊕ ( 2 π 2 − ⊗ π 1 − ) A ″ 2 ] ] A 3 ] ) A 13

= Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 1 + ⊗ π 2 + ) A ′ 1 ] A 2 ] ) A 12 + Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 1 + ⊗ π 2 − ) A ′ 2 ] A 2 ] ) A 12 + Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 2 − ⊗ π 1 + ) A ″ 1 ] A 3 ] ) A 13 + Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 2 − ⊗ π 1 − ) A ″ 2 ] A 3 ] ) A 13

= 8 Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 + ⊗ π 2 + ) A 2 ] ) A 12 + 8 Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 1 + ⊗ π 2 − ) A 2 ] ) A 12 + 8 Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 2 − ⊗ π 1 + ) A 3 ] ) A 13 + 8 Δ m ( [ ( π 1 + ⊗ π 2 − ) A 1 ⊕ ( π 2 − ⊗ π 1 − ) A 3 ] ) A 13 = 32 ( 1.15 ) MeV

Note [ Δ m ( A 12 ) = Δ m ( A 13 ) ] ; once more by 6-property in

Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 1 + ⊗ π 2 + ) A ′ 1 ] A 2 ] ) A 12 + Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 1 + ⊗ π 2 − ) A ′ 2 ] A 2 ] ) A 12 + Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 2 − ⊗ π 1 + ) A ″ 1 ] A 3 ] ) A 13 + Δ m ( [ 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ [ 2 ( π 2 − ⊗ π 1 − ) A ″ 2 ] A 3 ] ) A 13

= Δ m ( 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ { 2 ( π 1 + ⊗ π 2 + ) + 2 ( π 1 + ⊗ π 2 − ) + 2 ( π 2 − ⊗ π 1 + ) + 2 ( π 2 − ⊗ π 1 − ) } A 23 ) = Δ m ( 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ { [ 2 ( π 1 + ⊗ π 2 + ) + 2 ( π 2 − ⊗ π 1 − ) ] + [ 2 ( π 1 + ⊗ π 2 − ) + 2 ( π 2 − ⊗ π 1 + ) ] } A 23 ) = Δ m ( 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ { [ 4 ( π 12 ± ⊗ π 12 ± ) ] + [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] } A ′ 2 )

Thus the only component A ′ 2 represents two components (A_{2}, A_{3}):

A ′ 2 = ( A 2 + A 3 ) = ( [ 4 ( π 12 ± ⊗ π 12 ± ) ] + [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] A ′ 2 )

Then, by 3-property in

Δ m ( A 24 ) + Δ m ( A 34 ) = Δ m ( A ′ 2 ⊕ A 4 ) = Δ m ( A ′ 24 ) = F Δ m { [ { [ 4 ( π 12 ± ⊗ π 12 ± ) ] + [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] } A ′ 2 ⊕ ( π r 1 0 ⊗ π r 2 0 ) A 4 ] B } = F Δ m { [ 4 ( π 12 ± ⊗ π 12 ± ) ⊕ ( π r 1 0 ⊗ π r 2 0 ) ] B } + F Δ m { [ [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] A ′ 2 ⊕ ( π r 1 0 ⊗ π r 2 0 ) A 4 ] B } = F Δ m { [ 4 ( π 12 ± ⊗ π 12 ± ) ⊕ ( π r 1 0 ) ] B } + F Δ m { [ 4 ( π 12 ± ⊗ π 12 ± ) ⊕ ( π r 2 0 ) ] B }

+ F Δ m { [ [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] A ′ 2 ⊕ ( π r 1 0 ) ] B } + F Δ m { [ [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] A ′ 2 ⊕ ( π r 2 0 ) A 4 ] B } = F Δ m { [ 4 ( π 12 ± ⊗ π 12 ± ) ⊕ ( π 1 + ⊕ π 1 − ) ] B } + F Δ m { [ 4 ( π 12 ± ⊗ π 12 ± ) ⊕ ( π 2 + ⊕ π 2 − ) ] B } + F Δ m { [ [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] A ′ 2 ⊕ ( π 1 + ⊕ π 1 − ) ] B } + F Δ m { [ [ 4 ( π 1 ± ⊗ π 2 ∓ ) ] A ′ 2 ⊕ ( π 2 + ⊕ π 2 − ) ] B } = 16 ( 2.30 ) MeV q

The A_{14}-component is:

Δ m ( A 14 ) = Δ m ( { 4 ( π 1 + ⊗ π 2 − ) A 1 ⊕ _ ( π r 1 0 ⊗ π r 2 0 ) A 4 } A 14 ) = Δ m ( { 4 ( π 1 + ⊗ π 2 − ) ⊕ ( π r 1 0 ) } A 14 ) + Δ m ( { 4 ( π 1 + ⊗ π 2 − ) ⊕ ( π r 2 0 ) } A 13 ) = Δ m ( { 4 ( π 1 + ⊗ π 2 − ) ⊕ ( π 1 + ⊕ π 1 − ) } A 13 ) + Δ m ( { 4 ( π 1 + ⊗ π 2 − ) ⊕ ( π 2 + ⊕ π 2 − ) } A 13 ) = [ 4 ( 2.30 ) + 4 ( 2.30 ) ] MeV

where we have used once more by 4-property in

Δ m ( A 23 ) = Δ m ( [ ( 2 π 1 + ⊗ ( π r 0 ) 2 ) ⊕ _ ( ( π r 0 ) 1 ⊗ 2 π 2 − ) ] A 23 ) = Δ m ( [ ( ( π r 0 ) 2 ⊗ 2 π 1 + ) ⊕ _ ( ( π r 0 ) 1 ⊗ 2 π 2 − ) ] A 23 ) = Δ m ( [ ( ( π r 0 ) 1 ⊗ 2 π 1 + ) ⊕ _ ( ( π r 0 ) 2 ⊗ 2 π 2 − ) ] A 23 ) = Δ m ( ( π r 0 ) 12 ⊗ [ ( 2 π 1 + ) ⊕ _ ( 2 π 2 − ) ] A 23 ) = 4 Δ m ( [ ( π 1 + ⊕ π 2 − ) π r 0 ] A 23 ) = 4 ( 4.6 / 4 ) MeV = ( 4.6 ) MeV

where we have exchanged [ ( π r 0 ) 1 ↔ ( π r 0 ) 2 ] because of the dynamic interactions in couplings not change; besides, we need admitting that the neutral π r 0 -component shields the dynamics coupling between two charged pions. The shielding lowers the value of mass defect of the A_{23}-component of (1/4). It so follows:

Δ m ( ρ ) = Δ m ( A 1 ) + [ Δ m ( A 2 ) + Δ m ( A 3 ) + Δ m ( A 23 ) ] + Δ m ( A 4 ) + Δ m ( A 12 ) + Δ m ( A 13 ) + Δ m ( A 14 ) + Δ m ( A 234 ) = [ ( 0 ) A 1 + [ 8 ( 1.15 ) ] ( A 2 + A 3 ) + ( 0 ) A 4 + 32 ( 1.15 ) ( A 12 + A 13 ) + ( 4.6 ) A 23 + 16 ( 2.30 ) ( A 24 + A 34 ) + 8 ( 2.30 ) A 14 ] MeV ≈ ( 105 .8 ) MeV

In A_{i}-component the (d Ä d) could shield the electromagnetic interaction between two charged pions; see the Equations (15) and (17), then we can have

Δ m ( ρ ) ( d , d _ ) ≈ ( 105.8 / 2 ) MeV = ( 52.9 ) MeV

The presence of shielding neutral pair (d Ä d)_{π} instead of charged pions (π^{±}) leads us to conjecture, see Equations (32) and (46):

{ Δ m ( ( d , d _ ) ⊗ [ ( π + ⊗ π − ) ⊕ ( π − ⊗ π + ) ] ) = Δ m ( [ ( π + ⊗ π − ) ( d , d _ ) ⊕ ( π − ⊗ π + ) ( d , d _ ) ] ) = [ ( 1.15 ) 2 ] MeV Δ m ( ( d , d _ ) ⊗ [ ( π + ⊗ π − ) ⊕ ( π − ⊕ π + ) ] ) = Δ m ( [ ( π + ⊗ π − ) ( d , d _ ) ⊕ ( π − ⊕ π + ) ( d , d _ ) ] ) = [ ( 2.30 ) 2 ] MeV Δ m ( ( d , d _ ) ⊗ ( π + ⊕ π − ) ) = Δ m ( ( π + ⊕ π − ) ( d , d _ ) ) = [ ( 4.59 ) 2 ] MeV } (27)

Note that the interpenetration between the pair (d Ä d)_{π} and pions (π^{±}) lows the values (see in ref. [

Then, we will have in ρ-meson: Δ m ( ρ ) ( d , d _ ) ≈ ( 105.8 / 2 ) MeV = ( 52.9 ) MeV

Finally:

m ( ρ ) ≈ [ ( 827.24 ) MeV − ( 52.9 ) MeV ] = ( 774.34 ) MeV (28)

Besides, see the ref. [

m ( ρ 0 ) = m ( ρ 0 ) * − Δ m ( ρ 0 ) − O Δ m = ( 775 .49 ) MeV (29)

Very next to the experimental value [

{ [ ( 2 π + ⊕ π r 0 ) B 1 ⊗ ( 2 π − ⊕ π r 0 ) B 2 ] } = { [ ( 2 π 1 + ⊗ 2 π 1 − ) A 1 ⊕ ( 2 π 1 + ⊗ π r 2 0 ) A 2 ⊕ ( π r 3 0 ⊗ 2 π 1 − ) A 3 ⊕ ( π r 2 0 ⊗ π r 3 0 ) A 4 ] } = { [ ( 2 π 1 + ⊗ ( π 2 + ⊕ π 2 − ) ) A 2 ⊕ ( 2 π 1 − ⊗ ( π 3 + ⊕ π 3 − ) ) A 3 ] } ⊕ { [ ( 2 π 1 + ⊗ 2 π 1 − ) A 1 ⊕ ( ( π 2 + ⊕ π 2 − ) ⊗ ( π 3 + ⊕ π 3 − ) ) A 4 ] } = { [ ( 2 π 1 + ⊗ π 2 + ) ⊕ ( 2 π 1 + ⊗ π 2 − ) ] A 2 ⊕ [ ( 2 π 1 − ⊗ π 3 + ) ⊕ ( 2 π 1 − ⊗ π 3 − ) ] A 3 } ⊕ { [ 4 ( π 1 + ⊗ π 1 − ) A 1 ⊕ ( ( π 2 + ⊕ π 2 − ) ⊗ ( π 3 + ⊕ π 3 − ) ) A 4 ] }

By this structure, we build, see

However, we must not forget the presence of the pair (d, d) in structural equation. The presence of this pair has a double effect:

1) as we have already seen, the pair act as a shield between the interactions of pions

2) it makes the interactions no-commutative between two pions

[ ( ( π i ⊗ π j ) ⊕ ) ≠ ( ( π j ⊗ π i ) ⊕ ) ] .

This last aspect pushes us to consider all elements of matrix ‖ A i j ‖ : in the calculations, it is necessary to double the numerical values. Then we can use the following matrix, see

The mass defect will be:

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

Then we will have:

Δ m ( ρ ) = [ 18 ( 4.6 ) ( i , i ) G + 16 ( 2.3 ) ( 1 , 2 ) Y + 2 ( 2.3 ) ( 2 , 3 ) Y − 18 ( 1.15 ) ( ± , ± ) B − ( 1.15 ) R ] MeV = ( 102 .35 ) MeV Δ m ( ρ ) ( d , d _ ) = ( 51 .18 ) MeV m ( ρ ) = ( 827.24 ) MeV − ( 51 .18 ) MeV = ( 776 .06 ) MeV

2 π 1 + | π 2 + | π 3 + | 2 π 1 − | π 2 − | π 3 − | |
---|---|---|---|---|---|---|

2 π 1 + | 4 ( π 1 + , π 1 + ) | 2 ( π 1 + , π 2 + ) | 2 ( π 1 + , π 3 + ) | 4 ( π 1 + , π 1 − ) | 2 ( π 1 + , π 2 − ) | 2 ( π 1 + , π 3 − ) |

π 2 + | 2 ( π 2 + , π 1 + ) | ( π 2 + , π 2 + ) | ( π 2 + , π 3 + ) | 2 ( π 2 + , π 1 − ) | ( π 2 + , π 2 − ) | ( π 2 + , π 3 − ) |

π 3 + | 2 ( π 3 + , π 1 + ) | ( π 3 + , π 2 + ) | ( π 3 + , π 3 + ) | 2 ( π 3 + , π 1 − ) | ( π 3 + , π 2 − ) | ( π 3 + , π 3 − ) |

2 π 1 − | 4 ( π 1 − , π 1 + ) | 2 ( π 1 − , π 2 + ) | 2 ( π 1 − , π 3 + ) | 4 ( π 1 − , π 1 − ) | 4 ( π 1 − , π 2 − ) | 4 ( π 1 − , π 3 − ) |

π 2 − | 2 ( π 2 − , π 1 + ) | ( π 2 − , π 2 + ) | ( π 2 − , π 3 + ) | 2 ( π 2 − , π 1 − ) | ( π 2 − , π 2 − ) | ( π 2 − , π 3 − ) |

π 3 − | 2 ( π 3 − , π 1 + ) | ( π 3 − , π 2 + ) | ( π 3 − , π 3 + ) | 4 ( π 3 − , π 1 − ) | ( π 3 − , π 2 − ) | ( π 3 − , π 3 − ) |

4 π 1 + | π 2 + | π 3 + | 4 π 1 − | π 2 − | π 3 − | |
---|---|---|---|---|---|---|

4 π 1 + | 16 ( π 1 + , π 1 + ) | || | || | || | || | || |

π 2 + | 4 ( π 2 + , π 1 + ) | ( π 2 + , π 2 + ) | || | || | || | || |

π 3 + | 4 ( π 3 + , π 1 + ) | ( π 3 + , π 2 + ) | ( π 3 + , π 3 + ) | || | || | || |

4 π 1 − | 16 ( π 1 − , π 1 + ) | 4 ( π 1 − , π 2 + ) | 4 ( π 1 − , π 3 + ) | 16 ( π 1 − , π 1 − ) | || | || |

π 2 − | 4 ( π 2 − , π 1 + ) | ( π 2 − , π 2 + ) | ( π 2 − , π 3 + ) | 4 ( π 2 − , π 1 − ) | ( π 2 − , π 2 − ) | || |

π 3 − | 4 ( π 3 − , π 1 + ) | ( π 3 − , π 2 + ) | ( π 3 − , π 3 + ) | 4 ( π 3 − , π 1 − ) | ( π 3 − , π 2 − ) | ( π 3 − , π 3 − ) |

Here too it is necessary consider the [ ( O Δ m ) ( d , d _ ) = ( 1.15 ) MeV ] , we obtain so:

m ( ρ 0 ) = m ( ρ 0 ) * − Δ m ( ρ 0 ) − O Δ m = ( 775.76 ) MeV (30)

There is another possibility in combinations between basic pions of lattice { π r 0 } : a charged pion ( π ± ) can interpenetrate a molecule π r 0 : [ π ± → π r 0 ].

We have two possibilities: ρ + = [ ( π + ⊗ π − ) ⊕ π + ] , ρ − = [ ( π − ⊗ π + ) ⊕ π − ] .

Then one can have: π r ± = [ π + ⊕ ( π − ⊗ π ± ) ] . So, it is possible having:

ρ ± = [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) B 1 ⊗ ( 2 π − ⊕ π r ± ) B 2 ] A } = [ ( d , d _ ) π ] ⊗ { [ 4 ( π + ⊗ π − ) ⊕ ( 2 π + ⊗ π r ± ) ⊕ ( π r 0 ⊗ 2 π − ) ⊕ ( π r 0 ⊗ π r ± ) ] B } = [ ( d , d _ ) π ] ⊗ { [ 4 ( π ± ) 0 ⊕ ( 2 π + ⊗ π r ± ) ⊕ ( π r 0 ⊗ 2 π − ) ⊕ ( π r 0 ⊗ π r ± ) ] B } = [ ( d , d _ ) π ] ⊗ { [ 4 ( π ± ) 0 ⊕ ( 2 π + ⊗ π r ± ) ⊕ ( 2 π − ⊗ π r 0 ) ⊕ ( π r 0 ⊗ π r ± ) ] B } (31)

Note that:

m ( π r ± ) = m ( [ π + ⊕ ( π − ⊗ π ± ) ] ) = m ( π + ) + m ( 〈 π + , π ± 〉 ) = m ( π + ) + ( m ( π + ) + m ( π ± ) 2 ) = 2 m ( π + ) = m ( π r 0 )

Appling the F_{m} with this relation, it is:

F m ( ρ ± ) = F m ( [ ( d , d _ ) π ] ⊗ [ 4 ( π ± ) 0 ⊕ ( 2 π + ⊗ π r ± ) ⊕ ( 2 π − ⊗ π r 0 ) ⊕ ( π r 0 ⊗ π r ± ) ] B ) = F m ( [ ( d , d _ ) π ] ⊗ [ 4 ( π ± ) 0 ⊕ 2 ( 2 π ± ⊗ π _ r ± ) ⊕ ( π r 0 ⊗ π r ± ) ] B ) ≡ F m ( { [ ( d , d _ ) π ⊗ 4 ( π ± ) 12 0 ] A 1 ⊕ [ ( d , d _ ) π ⊗ 2 ( 2 π ± ⊗ π _ r ± ) 12 ] A 2 ⊕ [ ( d , d _ ) π ⊗ ( π r 1 0 ⊗ π r ± ) ] A 3 } ) (32)

where F m ( π _ r ± ) ≡ F m ( π r ± ) ≡ F m ( π r 0 ) . This equation is equal to Equation (22), because m ( π r ± ) = m ( π r 0 ) . It follows partial masses are equal: m * ( ρ ± ) = m * ( ρ 0 ) . Thus we have: m ( ρ ± ) * = ( 827 .24 ) MeV .

Note that the structure equations are different only in the interpenetration of π^{±} with a charged pion π^{− }of the structure; therefore, an eventual mass defect could be only in this interpenetration:

{ ρ + ↔ [ ( π + ⊗ π − ) ⊕ π + ] or ρ − ↔ [ ( π − ⊗ π + ) ⊕ π − ] } ↔ { ρ ± ↔ [ ( π ± ⊗ π ∓ ) ⊕ π ± ] }

In Equation (26) we admitted Δm(A_{4}) = 0 with [ A 4 = ( π r 0 ) ⊗ ( π r 0 ) ] , now we have [ ( A 4 ) ± = ( π r 0 ⊗ π r ± ) ] ; therefore, we will have:

Δ m ( A 4 ) ( d , d _ ) = F Δ m ( [ ( π r 0 ⊗ π r ± ) ] A 4 ) ( d , d _ ) = F Δ m ( { ( π + ⊕ π − ) ⊗ [ ( π ± ⊗ π ∓ ) ⊕ π ± ] } A 4 ) ( d , d _ ) = F Δ m ( ( π + ⊕ π − ) ) ( d , d _ ) + F Δ m ( [ ( π ± ⊗ π ∓ ) ⊕ π ± ] ) ( d , d _ ) + O ′ Δ m = ( 1 2 ) F Δ m ( ( π + ⊕ π − ) ) + ( 1 2 ) F Δ m ( [ ( π ± ⊕ π ∓ ) ⊗ π ± ] ) + O ′ Δ m

= ( 1 2 ) F Δ m ( ( π + ⊕ π − ) ) + ( 1 2 ) F Δ m ( [ ( π ± ⊕ π ∓ ) π ± ] ) + O ′ Δ m = { ( 1 2 ) ( 4.60 ) MeV + ( 1 2 ) [ − ( 4.60 / 2 ) ] MeV } + O ′ Δ m = ( 1.15 ) MeV + O ′ Δ m (33)

The value (−4, 6/2) is due to anti-shield of subscript π^{±}. We calculate O ′ Δ m _{ }

O ′ Δ m = F Δ m ( { ( π + ⊕ π − ) ⊗ _ [ ( π ± ⊗ π ∓ ) ⊕ π ± ] } A 4 ) ( d , d _ ) = F Δ m ( { ( π + ⊕ π − ) ⊗ _ [ ( π ± ⊕ π ∓ ) π ± ] } ) ( d , d _ ) = F Δ m ( [ ( π + ⊕ π − ) ⊗ _ ( π ± ⊕ π ∓ ) π ± ] ) ( d , d _ ) = F Δ m ( [ ( π ± ⊕ π ∓ ) π ± ] ) ( d , d _ ) = ( 1 2 ) ( 4.60 ) π ± MeV = + ( 1 2 ) [ − ( 4.60 / 2 ) ] MeV = − ( 1.15 ) MeV

It follows: m ( ρ ± ) = m ( ρ ± ) * − Δ m ( ρ ± ) − O Δ m = ( 775.76 ) MeV .

So we obtain (for ρ^{±}) the same result (the mass defect) of ρ^{0}-meson. In this way, it could be here a perfect symmetry between ρ^{± }and ρ^{0}.

In this quarks’ model, one could so obtain the mass symmetry in the triplet (ρ^{±}, ρ^{0}), inside the experimental approximation. Besides, these procedures show correctly the existence of d-quark lattice or {d, d} where mesons are immersed, see the neutral molecule of pions. By structure equation, we have the following decay:

ρ = [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) B 1 ⊗ ( 2 π − ⊕ π r 0 ) B 2 ] A } with ( ρ 0 ) → ( π + + π − )

ρ ± = [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) B 1 ⊗ ( 2 π − ⊕ π r ± ) B 2 ] A } with ( ρ ± ) → ( π ± + π 0 )

This because [ ( d , d _ ) π ⊗ π r 0 ] ↔ ( π 0 ) and [ π r 0 = ( π + ⊕ π − ) ]

A second channel, with η-meson ( π r 0 ⊗ π r 0 ) , could be [ ρ → η + γ ] , where the (γ) is originated by (d, d), and another [ ρ → η + π ] , but they have low probability » (10)^{−4}. With high probability, there is only the first channel: in fact, the ρ-meson decays in two pions, see ref. [^{±}, ρ^{0}), see structure equation, is given by couple (d, d) because the system [(η) Å (π^{±})] has zero spin while the pair (d, d) being potentially a (matter-antimatter) pair shows a spin (s = 1) in correspondence to the virtual γ-photon:

S ( ρ ± ) = S ( d , d _ ) γ + [ S ( η ) + S ( π ± ) ] = S ( γ v ) = 1 .

where (γ_{v}) is virtual photon because of binding energy. Therefore ρ-mesons show a spin: [s = +1]. Note the ρ-mesons are vectorial mesons. Also in ρ^{±} meson, we treat an individual interaction between each pion (π_{i}) with other pions (π_{j}), [ ( π i ± ⊗ π j ± ) ⊕ ] , where we can consider that

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

We elaborate the structure equation of ρ-meson, see Equation (21):

ρ ± = { [ ( 2 π 1 + ⊗ π 2 + ) ⊕ ( 2 π 1 + ⊗ π 2 − ) ] ⊕ [ ( 2 π 1 − ⊗ π 3 + ) ⊕ ( 2 π 1 − ⊗ π 3 − ) ] } ⊕ { [ 4 ( π 1 + ⊗ π 1 − ) ⊕ ( ( π 2 + ⊕ π 2 − ) ⊗ ( π 3 ± ⊕ ( π 3 ∓ ⊗ π 4 ± ) ) ) ] } (34)

Note { ρ + ↔ [ ( π + ⊗ π − ) ⊕ π + ] or ρ − ↔ [ ( π − ⊗ π + ) ⊕ π − ] } ↔ { ρ ± ↔ [ ( π ± ⊗ π ∓ ) ⊕ π ± ] }

with

( π 3 ± ⊕ ( π 3 ∓ ⊗ π 4 ± ) ) = ( ( π 3 ± ⊕ π 3 ∓ ) π 4 ± )

The subscript π 4 ± lows the mass defect between the two pion ( π ± ⊕ π ∓ ) :

Δ m [ ( π ± ⊕ π ∓ ) π 4 ± = ( 2.3 ) MeV ]

Then we need to have a symmetric matrix; it follows, processing

The mass defect is:

Δ m ( ρ ± ) = [ 17 ( 4.6 ) ( i , i ) G + ( 4.6 / 4 ) ( π 4 0 ) G + 8 ( 2.3 ) ( 1 , 2 , 3 ) Y + 10 ( 2.3 / 2 ) ( 1 , 2 , 3 ) Y − 8 ( 1.15 ) ( ± , ± ) B + 10 ( 1.15 / 2 ) ( ± , ± ) B − 2 ( 1.15 ) 33 R ] MeV = ( 105.25 ) MeV Δ m ( ρ ± ) ( d , d _ ) = ( 52.63 ) MeV m ( ρ ± ) = ( 827.24 ) MeV − ( 53.20 ) MeV = ( 774.61 ) MeV

Besides, if we consider, as it is in the neutral ρ-meson, the term of the mass defects O ′ Δ m , with [ ( O Δ m ) d , d _ = − ( 1.15 ) MeV ] , this because ρ^{±}-meson has an electric charge (see its structure equation). So we will have:

m ( ρ ± ) − ( 1.15 ) MeV = [ ( 774 .61 ) + ( 1.15 ) ] MeV = ( 775.76 ) MeV

Here too, we obtain the symmetry between (ρ^{±}, ρ^{0}).

ω-meson (782.7 MeV)

Further proof of existence of lattices {d, d} and {π^{0}} is in the neutral ω-meson which could be defined by the connection:

ρ ± | 4 π 1 + | π 2 + | π 3 + ⊗ π 4 + | 4 π 1 − | π 2 − | π 3 − ⊗ π 4 − |
---|---|---|---|---|---|---|

4 π 1 + | 16 ( π 1 + , π 1 + ) | \\ | \\ | \\ | \\ | \\ |

π 2 + | 4 ( π 2 + , π 1 + ) | ( π 2 + , π 2 + ) | \\ | \\ | \\ | \\ |

π 3 + ⊗ π 4 + | 4 ( π 3 + , π 1 + ) π 4 + | ( π 3 + , π 2 + ) π 4 + | ( π 3 + , π 3 + ) 2 π 4 + | \\ | \\ | \\ |

4 π 1 − | 16 ( π 1 − , π 1 + ) | 4 ( π 1 − , π 2 + ) | 4 ( π 1 − , π 3 + ) π 4 + | 16 ( π 1 − , π 1 − ) | \\ | \\ |

π 2 − | 4 ( π 2 − , π 1 + ) | ( π 2 − , π 2 + ) | ( π 2 − , π 3 + ) π 4 + | 4 ( π 2 − , π 1 − ) | ( π 2 − , π 2 − ) | \\ |

π 3 − ⊗ π 4 − | 4 ( π 3 − , π 1 + ) π 4 − | ( π 3 − , π 2 + ) π 4 − | ( π 3 − , π 3 + ) π 4 0 | 4 ( π 3 − , π 1 − ) π 4 − | ( π 3 − , π 2 − ) π 4 − | ( π 3 − , π 3 − ) 2 π 4 − |

ω = [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ 2 π 0 ) ⊗ ( 2 π − ⊕ 2 π 0 ) ] } .

We have:

ω = [ ( d , d _ ) π ] ⊗ { [ ( 2 π 1 + ⊕ 2 π 1 0 ) B 1 ⊗ ( 2 π 2 − ⊕ 2 π 2 0 ) B 2 ] A } = [ ( d , d _ ) π ] ⊗ { [ 4 ( π 1 + ⊗ π 2 − ) ⊕ ( 2 π 1 + ⊗ 2 π 2 0 ) ⊕ ( 2 π 1 0 ⊗ 2 π 2 − ) ⊕ ( ( 2 π 1 0 ) ⊗ ( 2 π 2 0 ) ) ] B } = [ ( d , d _ ) π ] ⊗ { [ 4 ( π ± ) 12 0 ⊕ 2 ( 2 π ± ⊗ 2 π 0 ) 12 ⊕ ( 2 π 12 0 ) ] B } = { [ ( d , d _ ) π ⊗ 4 ( π ± ) 0 ] A 1 ⊕ [ ( d , d _ ) π ⊗ 2 ( 2 π ± ⊗ 2 π 0 ) ] A 2 ⊕ [ ( d , d _ ) π ⊗ ( 2 π 12 0 ) ] A 3 } (35)

where we use the 3_{Ä}-property and 4, in _{m}. In fact, note that

m ( ( 2 π 1 0 ) ⊗ ( 2 π 2 0 ) ) = m ( 〈 ( 2 π 1 0 ) , ( 2 π 2 0 ) 〉 ) = m ( 2 π 1 0 ) + m ( 2 π 1 0 ) 2 = m ( 2 π 1 0 )

The partial mass is:

m ( ω ) * = m { [ ( d , d _ ) π ⊗ 4 ( π ± ) 0 ] A 1 ⊕ [ ( d , d _ ) π ⊗ 2 ( 2 π ± ⊗ 2 π 0 ) ] A 2 ⊕ [ ( d , d _ ) π ⊗ ( 2 π 0 ) ] A 3 } = { 〈 m ( ( d , d _ ) , 4 ( π ± ) 0 ) 〉 + 〈 m ( ( d , d _ ) , 2 ( 2 π ± ⊗ 2 π 0 ) ) 〉 + 〈 m ( ( d , d _ ) , ( 2 π 0 ) ) 〉 } = [ m ( d ) + m ( 4 π ± ) 0 2 + m ( d ) + 2 m ( 〈 2 π ± , 2 π 0 〉 ) 2 + m ( d ) + m ( 2 π 0 ) 2 ]

= 3 2 m ( d ) + 2 m ( π ± ) + m ( π ± ) + m ( π 0 ) + m ( π 0 ) = 3 2 m ( d ) + 3 m ( π ± ) + 2 m ( π 0 ) = 3 2 ( 86.26 ) + 3 ( 139.57 ) + 2 ( 134.97 ) MeV = ( 818 .04 ) MeV (36)

The mass defect can calculate with more facility by matrix ‖ A i j ‖ .

Then we can write the structure equation in the following way:

ω = [ ( d , d _ ) π ] ⊗ { [ ( 2 π 1 + ⊕ 2 π 2 0 ) ⊗ ( 2 π 3 − ⊕ 2 π 4 0 ) ] } = { [ ( 2 π 1 + ⊗ 2 π 3 − ) ⊕ ( 2 π 1 + ⊗ 2 π 4 0 ) ⊕ ( 2 π 2 0 ⊗ 2 π 3 − ) ⊕ ( ( 2 π 2 0 ) ⊗ ( 2 π 4 0 ) ) ] } ( d , d _ ) = ( ω B ) ( d , d _ )

where

ω B = { [ ( 2 π 1 + ⊗ 2 π 3 − ) ⊕ ( 2 π 1 + ⊗ 2 π 4 0 ) ⊕ ( 2 π 2 0 ⊗ 2 π 3 − ) ⊕ ( ( 2 π 2 0 ) ⊗ ( 2 π 4 0 ) ) ] } = { [ ( 2 π 1 + ⊗ 2 π 3 − ) ⊕ 4 ( π 4 0 ) π 1 + ⊕ 4 ( π 2 0 ) π 3 − ⊕ ( ( 2 π 2 0 ) ⊗ ( 2 π 4 0 ) ) ] } = { [ ( 2 π 1 + ⊗ 2 π 3 − ) ⊕ 4 ( π 4 0 + ) ⊕ 4 ( π 2 0 − ) ⊕ ( ( 2 π 2 0 ) ⊗ ( 2 π 4 0 ) ) ] }

Recall that in the calculations it is necessary to double the numerical values, see matrix ‖ A i j ‖ , see

We need to consider all elements of the matrix because in interaction with shield particles could be [ ( ( π i ⊗ π j ) ⊕ ) ≠ ( ( π j ⊗ π i ) ⊕ ) ] ; The mass defect is:

Δ m ( ω B ) = [ 8 ( 4.6 ) ( − , + ) + 16 ( 2.3 ) ( ± , 0 ± ) + 8 ( 1.15 ) ( 0 ± , 0 ± ) ] ( G ) MeV + [ 32 ( 1.15 ) ( 0 , ± ) + 24 ( 0.58 ) ( 0 , 0 ± ) ] Y MeV + [ − 16 ( 2.3 ) ( ± , 0 ± ) − 8 ( 1.15 ) ( 0 − , 0 − ) ] ( B ) MeV + [ ≈ 0 ] R MeV = ( 87 .52 ) MeV

2 π 1 + | 2 π 4 0 + | 2 π 3 − | 2 π 2 0 − | 2 π 2 0 | 2 π 4 0 | |
---|---|---|---|---|---|---|

2 π 1 + | 8 ( π 1 + , π 1 + ) | \\ | \\ | \\ | \\ | \\ |

2 π 4 0 + | 8 ( π 4 0 + , π 1 + ) | 8 ( π 4 0 + , π 4 0 + ) | \\ | \\ | \\ | \\ |

2 π 3 − | 8 ( π 3 − , π 1 + ) | 8 ( π 3 − , π 4 0 + ) | 8 ( π 3 − , π 3 − ) | \\ | \\ | \\ |

2 π 2 0 − | 8 ( π 2 0 − , π 1 + ) | 8 ( π 2 0 − , π 4 0 + ) | 8 ( π 2 0 − , π 3 − ) | 8 ( π 2 0 − , π 2 0 − ) | \\ | \\ |

2 π 2 0 | 8 ( π 2 0 , π 1 + ) | 8 ( π 2 0 , π 4 0 + ) | 8 ( π 2 0 , π 3 − ) | 8 ( π 2 0 , π 2 0 − ) | 8 ( π 2 0 , π 2 0 ) | \\ |

2 π 4 0 | 8 ( π 4 0 , π 1 + ) | 8 ( π 4 0 , π 4 0 + ) | 8 ( π 4 0 , π 3 − ) | 8 ( π 4 0 , π 2 0 − ) | 8 ( π 4 0 , π 2 0 ) | 8 ( π 4 0 , π 4 0 ) |

Δ m ( ω B ) ( d , d _ ) = ( 43 .76 ) MeV m ( ω ) = ( 827.24 ) MeV − ( 43 .76 ) MeV = ( 783 .48 ) MeV

The mass defect in the elements in red is almost zero for the presence of neutral pions which shield the repulsive actions. If we consider O_{Δm} will be:

m ( ω ) = m * ( ω ) − O Δ m = ( 783.48 ) MeV − ( 1.15 ) MeV = ( 782.33 ) MeV (37)

In perfect accord with experimental value [

ω → [ π + + π − + π 0 ] 90 % ; ω → [ π 0 + γ ] 8 % ; ω → [ π + + π − ] 2 % _{ }

Here, the photon (γ) is originated by lattice {d, d}. Spin of (ω), see structure equation, is given by couple (d, d) which shows a spin(s = 1), because potentially it is in correspondence to the virtual γ-photon: S ( ω ) = S ( d , d _ ) γ = S ( γ v ) = 1

Therefore ω-mesons is vectorial mesons (J^{p} = 1^{−}). To think to particles made of quarks in interpenetration with lattices {π^{0}}, the pair{d, d} can induce us to see the mesons with a structure more articulated or “molecules” of quarks.

ϕ -meson (1019.41 MeV)

Another meson with by lattice {π^{0}} is the ϕ -meson: ϕ 0 = { [ ( d , d _ ) ⊗ ( 2 π 0 ⊕ η ) ] ⊕ ( η ) }

Then

ϕ 0 = { [ ( d , d _ ) ⊗ ( 2 π 0 ⊕ η ) ] ⊕ η } = { [ ( ( d , d _ ) ⊗ 2 π 0 ) ⊕ ( ( d , d _ ) ⊗ ( π r 0 ⊗ π r 0 ) η ) ] ⊕ ( π r 0 ⊗ π r 0 ) η } (38)

The partial mass is

m ( ϕ ) * = m ( 〈 ( ( d , d _ ) , 2 π 0 ) 〉 ) + m ( 〈 ( ( d , d _ ) , ( η ) ) 〉 ) + m ( η ) = ( 1 / 2 ) [ 2 m ( d ) + m ( 2 π 0 ) + 3 m ( η ) ] = [ ( 86.26 ) + ( 134.98 ) + ( 821.91 ) ] MeV = ( 1043.15 ) MeV (39)

There could be a mass defect Δ m ( ϕ ) . We adapt the equation

ϕ 0 = { [ ( d , d _ ) ⊗ ( 2 π 0 ⊕ η ) ] ⊕ η } = { [ ( 2 π 0 ⊕ η ) ( d , d _ ) ] ⊕ η } = { [ ( 2 π a 0 ⊕ η c ) ( d , d _ ) ] ⊕ η c } = { [ ( 2 π a 0 ⊕ ( π r d 0 ⊗ π r e 0 ) b ) ( d , d _ ) ] ⊕ ( π r f 0 ⊗ π r g 0 ) c } = { [ ( 2 π a 0 ⊕ ( π r e 0 ) π r d 0 ) ( d , d _ ) ] ⊕ ( π r f 0 ) π r g 0 } = { [ ( 2 ( π 1 + ⊗ _ π 1 − ) b ⊕ ( π 2 + ⊕ π 2 − ) π 0 ) ( d , d _ ) ] ⊕ ( π 3 + ⊕ π 3 − ) π 0 } (40)

The matrix is, see

ϕ | 2 π 1 + | ( 1 / 4 ) π 2 + | ( 1 / 2 ) π 3 + | 2 π 1 − | ( 1 / 4 ) π 2 − ( d , d _ ) | ( 1 / 2 ) π 3 − |
---|---|---|---|---|---|---|

2 π 1 + | 4 ( π 1 + , π 1 + ) | \\ | \\ | \\ | \\ | |

( 1 / 4 ) π 2 + ( d , d _ ) | ( 1 / 2 ) ( π 2 + , π 1 + ) ( d , d _ ) | ( 1 / 16 ) ( π 2 + , π 2 + ) ( d , d _ ) _{ } | \\ | \\ | \\ | |

( 1 / 2 ) π 3 + | ( π 3 + , π 1 + ) | ( 1 / 8 ) ( π 3 + , π 2 + ) | ( 1 / 4 ) ( π 3 + , π 3 + ) | \\ | \\ | |

2 π 1 − | 4 ( π 1 − , π 1 + ) | ( 1 / 2 ) ( π 1 − , π 2 + ) | ( π 1 − , π 3 + ) | 4 ( π 1 − , π 1 − ) | ||

( 1 / 4 ) π 2 − | 2 ( 1 / 4 ) ( π 2 − , π 1 + ) ( d , d _ ) | ( 1 / 16 ) ( π 2 − , π 2 + ) ( d , d _ ) | ( 1 / 8 ) ( π 2 − , π 3 + ) ( d , d _ ) | ( 1 / 2 ) ( π 2 − , π 1 − ) ( d , d _ ) | ( 1 / 16 ) ( π 2 − , π 2 − ) ( d , d _ ) | \\ |

( 1 / 2 ) π 3 − | ( π 3 − , π 1 + ) | ( 1 / 8 ) ( π 3 − , π 2 + ) | ( 1 / 4 ) ( π 3 − , π 3 + ) | ( π 3 − , π 1 − ) | ( 1 / 8 ) ( π 3 − , π 2 − ) | ( 1 / 4 ) ( π 3 − , π 3 − ) |

The mass defect will be:

Δ m ( ϕ ) = [ 69 16 ] G ( 4.6 ) MeV + [ 26 8 ] Y ( 2.3 ) MeV − [ 26 8 ] B ( 1.15 ) MeV − [ ( 1.15 ) ] R MeV = [ 69 + 26 16 ] G ( 4.6 ) MeV − [ 26 8 ] B ( 1.15 ) MeV − [ ( 1.15 ) ] R MeV = [ ( 27.31 ) − ( 3.74 ) − ( 1.15 / 4 ) ] MeV = ( 23.28 ) MeV

Then, the mass is:

m ( ϕ ) = m ( ϕ ) * − Δ m ( ϕ ) = { ( 1043.15 ) − [ ( 23.28 ) ] } MeV / c 2 = ( 1019 .87 ) MeV / c 2 (41)

Spin of ( ϕ ), see structure equation, is given by couple (d, d) because the system { [ ( 2 π 0 ⊕ η ) ] ⊕ ( η ) } has zero spin while the pair (d, d) shows a spin (s = 1) in correspondence to the virtual γ-photon: S ( ρ ± ) = S ( d , d _ ) γ + [ S ( η ) + S ( π ± ) ] = S ( γ v ) = 1 .

Therefore ϕ -mesons show a spin: [s = +1]. Note the ϕ -mesons are vectorial mesons.

If we look with attention the structure equations of mesons, one can deduce a representative spectrum built by pion “quanta” which overlap each other, see the ref. [

Where (π^{0}) is the neutral pion, while (π^{0})_{r} is [π^{+} Å π^{−}]. We could think to 6-level others mesons as i.e. x = { 3 ( d , d _ ) ⊗ ⋯ [ ( π 0 ) r ⊗ ⋯ ( η ⊕ η ⊕ η ) ] ⋯ } .

By this model, it is possible to calculate all others mesons with higher mass. It needs to think the structure equation, calculating the global mass defect and thus

Meson | Level | Structure Equation | Theory Mass (MeV) | Exper. Mass (MeV) |
---|---|---|---|---|

π^{0} | 1 | [ π + ⊗ π − ] ⊕ [ ( u ⊕ u _ ) ⊗ ( d ⊕ d _ ) ] | (134.97) | (134.97) |

η^{0} | 2 | ( π 0 ) r ⊗ ( π 0 ) r = ( π + ⊕ π − ) ⊗ ( π + ⊕ π − ) | (547.95) | (547.86) |

η' | 3 | { ( π 0 ) r ⊗ ( η ) } = ( π 0 ) r ⊗ ( π 0 ) r ⊗ ( π 0 ) r | (957.44) | (957.78) |

ρ^{ } | 3 | [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) ⊗ ( 2 π − ⊕ π r 0 ) ] } | (775.76) | (775.26) |

ρ^{±} | 3 | [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ π r 0 ) ⊗ ( 2 π − ⊕ π r ± ) ] } | (775.76) | (775.26) |

ω | 4 | [ ( d , d _ ) π ] ⊗ { [ ( 2 π + ⊕ 2 π 0 ) ⊗ ( ( 2 π − ⊕ 2 π 0 ) ] } | (782.33) | (782.65) |

ϕ 0 | 4 | ϕ 0 = { [ ( d , d _ ) ⊗ ( 2 π 0 ⊕ η ) ] ⊕ ( η ) } | (1019.87) | (1019.41) |

finding the global mass value. Note the following

Furthermore, this quarks’ theory allows of calculating the masses of nucleons, as it will be shown in the next study. In essence, a meson is a lattice of quark pairs (see {d, d}, {(π^{0})_{r}}) and internal gluons {g} of coupling [

In the next study, we will show the equation structure of the nucleons and barions without strangeness. The structure equations will allow us to calculate their masses.

The hypothesis of structure allows us of describe the various transformations of particles, as the hadron production accompanied by the production of the W, Z bosons, in the beta decay or the production of hadronic jets:

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

How is it possible for photons to create quarks? A comprehensive answer could then be to assign a geometric structure (of quantum oscillators) to all elementary particles (that is not composed of sub-particles) and that it is possible to transform a structure into another. In this way, we introduce a new paradigm in the phenomenology of any interaction, which opens up new perspectives for resolving the various problems of this. On this basis, we can discuss the “internal structure” of quarks with the physical awareness that this structure is realizable only through “particular” quantum oscillators [

To treat the particles as IQuO structures can explain the origin of some fundamental physics greatness as the electric charge [

So, the IQuO idea constitutes a new paradigm in physics that allows us to describe with depth the physical phenomenon of particles and to open new descriptive scenarios of interactions between particles.

The author declares no conflicts of interest regarding the publication of this paper.

Guido, G. (2020) The Theoretical Spectrum of Mass of the Light Mesons without Strangeness via the Quarks’ Geometric Model. Journal of High Energy Physics, Gravitation and Cosmology, 6, 388-415. https://doi.org/10.4236/jhepgc.2020.63031