_{1}

^{*}

Highlighting a golden triangular form in
*u* and
*d *quarks (Quark Geometric Model), we build the geometric structures of light meson
*η* and individualize its decays and spin. By the structure equations describing mesons, we determine a mathematic procedure to calculate the theoretical value of the mass of light mesons
*η*.

One of the more problematic aspects in Hadronic literature [

In a precedent set of studies [

Nevertheless, for obtaining the masses (and, in the following study, the nucleons ones as well), we need to introduce a new idea of mass calculation (Ä-operation). This idea of calculation takes into account both interactions between quarks and a possible interpenetration [

As is well known in literature [_{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

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 [_{1}, d_{1}), see

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 [

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 [

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. [_{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 _{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, _{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 [

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 _{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

The interpenetration could explain the zero value of pion’s spin. In ref. [_{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} (

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 [_{i} Ä b_{j}). The new operation (Ä) indicates a composition of two operations (Ä, Å) or [Äº (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

The operation of combination (Å) could be used for express dynamics couplings ( u _ ↔ d ) (or ( u ↔ d _ )) in charged pion π (

In AGM, (u) and (d) represent the wave function associated with quark [^{+}), 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 [

We conjectured, see [^{0} 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

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

And we demonstrated [

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

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

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

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

This equation represents the structural equation of neutral pion (π^{0}) 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. [

Note (see ^{0})_{a}, (π^{0})_{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}) [

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

Equivalent two ways can treat the binding energy between quarks inside pions. The function F_{Δ}_{m} 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

(Ä-operation) in (F_{m}, F_{Δm}) representations | Å-operation in (F_{m}, F_{Δm}) representations |
---|---|

1) 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 ) ] |

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 [^{*}) partial mass of π^{0} 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

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 (A_{1}, A_{2}) 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 [

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 [

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

It is physically possible that the pions [π^{0}, 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 (π^{0}), 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 [π^{0}; (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

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 (u_{2}, u_{2}) fix their respective vertices (G, H) on the diagonal of quarks (d_{2}, d_{2}), then the structure can have more stability. The fixing (see section 2.1 and _{3}, d_{3}), (u_{3}, d_{3})]_{ss}. We point out these quarks with a denomination of sub-quarks (u, d). The representative figure of η-meson is (

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 (u_{1}, u_{1}) for the rotation of u-quarks around two Y-axis, but this energy could be minimal (K_{kin}) ≪ m_{0}c^{2}). The spin of pair (u_{1}, u_{1}) is zero like that of neutral pion because this is to base of η-meson (see its structure). Processing the “structure equation” one has:

Properties of pions in (Ä) operations | Properties of pions in (Å) operations |
---|---|

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 ± ) | 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 ] 7) Δ m [ ( π 1 ± ⊕ π 2 ∓ ) ⊗ π 0 ] = Δ m [ ( π 1 ± ⊕ π 2 ∓ ) π 0 ] |

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 (π^{0})_{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

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

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

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

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 (π^{0}):

[ 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)

Combinations | Coupling |
---|---|

[ ( π 1 ± ⊗ π 2 ± ) i ⊕ ( π 1 ± ⊗ π 2 ± ) j ] | coupling of I˚ degree |

{ ( π 1 ± ⊗ π 2 ± ) i ⊕ [ ( π 1 ± ⊗ π 2 ± ) j ⊕ ( π 1 ± ⊗ π 2 ± ) k ] } | coupling of II˚ degree |

[ ( π 1 ± ⊗ π 2 ± ) i ⊕ ( π 1 ± ⊗ π 2 ± ) j ] ⊕ [ ( π 1 ± ⊗ π 2 ± ) k ⊕ ( π 1 ± ⊗ π 2 ± ) r ] | coupling of III˚ degree |

{ ( π 1 ± ⊗ π 2 ± ) i ⊕ [ ( π 1 ± ⊗ π 2 ± ) j ⊕ ( π 1 ± ⊗ π 2 ± ) k ⊕ ( π 1 ± ⊗ π 2 ± ) r ] } | coupling of IV˚ degree |

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

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

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 (A_{34}) admits four possible combinations of exchange ( π i ↔ π j ); then we determine four equivalent combinations (S_{1}, S_{2}, S_{3}, S_{4}) 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 (S_{1}, S_{2}, S_{3}, S_{4}) 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 (A_{13}) and Δm (A_{23}). 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. [

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 _{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

η | π 1 + | π 2 + | π 1 − | π 2 − |
---|---|---|---|---|

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

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

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

π 2 − | ( π 2 − , π 1 + ) ⊕ Y | ( π 2 − , π 2 + ) ⊕ G | ( π 2 − , π 1 − ) ⊕ B | ( 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 ^{+}, π^{−}, π˚, γ) with the following combination or channels:

1) (π^{+}, π^{−}, π˚)_{(23%)}, 2) (π^{0}, π^{0}, π^{0})_{(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

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

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).

In this paper, we have shown that quarks, mesons and proton have a geometric structure. The phenomenology of interactions points out that particles transform in other particles: we ask us how it is possible that coupling photons create quarks’ pairs, see the reaction ( e − + e + → q + q _ ). A comprehensive answer could then be to assign a geometric structure (of coupled quantum oscillators) to all elementary particles (that is not composed of sub-particles) and that it is possible to transform a structure into another. Thus there would be a mechanism of topological transformations on geometrical structures which would transform the one into any other. In this way, we introduce a new paradigm in the phenomenology of particles and the interactions, which opens up new perspectives for resolving the various problems of the Standard Model.

The first perspective is the one of the “internal structure” of quarks that is realizable only through “particular” quantum oscillators. In a particle-structure, the vertex-oscillators and junction oscillators must have the “hooks” to the extremity [

A quantum oscillator with a sub-structure constituted by sub-unit of oscillation, or “sub-oscillators”, and “semi-quanta” is an oscillator of type “IQuO” [

The second perspective is the one that a particle more massive can be structured by particles of base, see the η-meson. In the next study we will show that all mesons more massive than pion are composite structure of pion molecules. Not only, but we think that also the nucleons and other hadrons are geometric structures of coupled quantum oscillators. The mathematics procedure which has allowed of calculating the mass of η-meson, the same will allows us of calculating the mass of other mesons ( ρ , ω , ϕ ) which make, with the pions, the light component but strangeness of the octet of fundamental mesons.

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

Guido, G. (2020) The Theoretical Value of Mass of the Light η-Meson via the Quarks’ Geometric Model. Journal of High Energy Physics, Gravitation and Cosmology, 6, 368-387. https://doi.org/10.4236/jhepgc.2020.63030