_{1}

^{*}

In previous articles (Guido) we demonstrated that Quarks ( u, d) are represented by golden geometric structures of coupled quantum oscillators. In this article we show the geometric structure of the pion triplet and, in particular, via the structure equation of neutral pion, we identify its decays and we solve the spin question in hadrons thanks also to introduction of algebraic operations [?, ♁] on geometric structure. Moreover by means of the golden ratio between ( u, d), we determine the values of bare masses of quarks (3.51 MeV for u-quark and 5.67 MeV for d-quark) and those ones bounded in a pion (53.31 MeV for u-quark and 85.26 MeV for d-quark). Finally, using algebraic operations [?, ♁] we point out a new way to represent the processes of pions’ decay.

A problematic aspect in Hadrons (see literature [

m u = range [ 1 . 7 - 3 . 3 ] MeV

m d = range [ 4 . 1 - 5 . 8 ] MeV

Unlike leptons, these values are always referred to quarks seen confined inside hadrons and not observed directly as physical particles. So, we would still admit that quarks’ masses can be determined indirectly by their influence on hadronic properties.

Nowadays, one of the answers to the mystery of mass origin in hadrons is the fact that the mass remaining fraction is due to the force (gluons) binding the quarks within hadrons, i.e. Quantum Chromo Dynamics (QCD). This way one thinks that the values of the quark masses, obtained [

Nevertheless, all of this will be possible only if we affirm [

Nevertheless, for obtain the masses (and, in a following study, the nucleons ones as well) we need to revise the mass conception in physics and introduce a new idea of mass calculation (Ä-operation), which takes into account both interactions between quarks and a possible interpenetration of the quarks (this is an aspect purely quantum-mechanical like to superposition of waves). The latter may exist only if we admit different structures of coupled quantum oscillators (or quarks) in overlapping without exchanging energy. The global mass of hadron must therefore take into account all the possible configurations of quark components, both the ones with interpenetration and the ones with interactions.

The hypothesis of quark geometrical structure introduces a new paradigm (see GM) in the phenomenology of hadronic interactions, suggesting the quarks have an internal geometrical structure. Thanks to a few basic hypotheses, the new paradigm allows instead to describe the hadronic phenomenology in a more structured and simple way. A first evidence will be given by calculating the spectrum of light mesons with mass values very close (if not even equal) to the experimental ones. In this paper we formulate the structure equation of pion and its decays and we solve the spin question in hadrons. The interpenetration of quarks is connected to an intrinsic internal movement which well explains the total spin of pion as sum of more components of the spin connect to different parts of quark in movement.

Moreover thanks to golden form of quarks inside pion, we calculate the mass values of quarks (u, d) inside to pion, and, through structure equation of neutral pion, also those ones of bare masses.

Finally, using algebraic operations [Ä, Å] we point out a new way where to represent the processes of pions’ decay.

In ref. [_{0}). Moreover, talking about the Compton wavelength (Δ_{c}) or spin in massive particle, then we can speak of space inside them. All this could lead us to affirm the Space-Time (ST) being intrinsic to massive particles. Moreover, talking about Space-Time (ST) inside a massive particle, it’s exactly the same as talking about an internal structure of particle. If we want to talk about internal oscillations to particle, seen as structures, but at the same time to be coherent with the relativity and Quantum Mechanics, then only one assertion is possible: particles could be geometric structures of coupled quantum oscillators. Recall the quantum theory of fields where particles are sets of coupled quantum oscillators, then it’s intuitive to give a structure of coupled oscillators to massive elementary particles: this is the idea of hypothesis of structure [

To assign a geometric structure to coupled oscillators is equivalent to assign a proper frequency (ω_{0}) to them, or an oscillation period (τ) coinciding, in relativity, with the proper time.

The proper characteristic of a massive particle, associated with the proper time (τ) of an object, coincides with the proper mass (m_{0}). To speak of the mass or mass energy of a particle, it is the same as speaking the time of the clock that is inside them [ω_{0} ó τ ó m_{0}]. Then, based on QM [

{ E 0 = m c 2 E 0 = ℏ ω 0 } ⇒ { m = ℏ ω 0 c 2 } (1)

If the frequency (ω_{0}) origins the proper time (τ) of a massive particle [τ = h/mc^{2}], then for symmetry, exists a wavelength (Δ) that origins the “proper space” of the particle. Following De Broglie, it’s:

{ p 0 = m c p 0 = ℏ 2 π λ 0 ⇒ ƛ 0 = ℏ m c ≡ ƛ c (2)

The equation [(ω_{0}Δ) = (Δ/τ) = c] is the dispersion relationship in the Space-Time 4-dim of a reference frame “at rest” (S˚) where the wave is “a rest” or where an observer notes only a stationary oscillation and do not observes the “progression” of the wave.

Combining the equation of the relativistic energy of massive particle in movement (velocity (v)) in a reference frame (S) with the equations of Einstein (1) and De Broglie (2) we have:

{ E 2 = m 2 c 4 + p 2 c 2 ⇔ ω 2 = ω 0 2 + k 2 c 2 } (3)

This is the dispersion relationship of waves propagating in (S), as described by the Klein-Gordon equation:

{ ∂ 2 Ψ ( x , t ) ∂ t 2 = c 2 ∂ 2 Ψ ( x , t ) ∂ x 2 − ω 0 2 Ψ ( x , t ) } (4)

As is well known, this equation describes both the oscillations in a set of coupled pendulums through springs [

{ ∂ 2 Ψ ( x , t ) ∂ x 2 − ∂ 2 Ψ ( x , t ) c 2 ∂ t 2 = ( m c ℏ ) 2 Ψ ( x , t ) } ⇔ { ∇ 2 Ψ ( x , t ) = ( m c ℏ ) 2 Ψ ( x , t ) } (5)

We conjectured [_{0}) related to a particular elastic coupling, which is in addition to the one already existing between the oscillators of the massless scalar field (X). This “additional coupling” which produces the mass in a scalar field (X), has been referred to as a “massive coupling”. Then, we conjectured that the massive particle-field (X) is created by a “transversal coupling” (T_{0}) between the chains of oscillators of the scalar base field (X). This can be depicted figuratively as shown in

When we observe only the oscillation with frequency (ω_{0}) in all points (x) then we are at rest with the massive particle (m ó ω_{0} ó T_{0}) and this aspect is coincident to the that in which the springs do not are involved. Instead when the springs are involved the wave becomes progressive with frequency (ω) wave length (λ) and represent a massive particle with velocity (v) (see Equation (3)).

Let us note the relationship between the Compton’s wavelength of the proton and that of Planck’s particle [_{(}_{pl}_{,p}_{)} the experimental numerical ratio between the Compton’s wave length (Δ_{pl}) and (Δ_{p}), experimentally it’s:

( n ( p l , p ) ) = ( ƛ p ƛ p l ) = ( m p l m P ) = [ 2 .176450 × 10 − 8 1 .672623 × 10 − 27 ] = 1 .301 × 10 19 (6)

the power (10)^{(p)} can be a representative scale factor (s).

Recall: ϕ 2 = ( 1.618 ) 2 = ( 2.618 ) 2

Where ( ϕ ) is the “aureus” (golden) number. Note [ n ( p l , p ) / 10 19 ] ~ [ ϕ 2 / 2 ] and [ 2 n ( p l , p ) = ϕ 2 s ] . Between Compton’s wave length (Δ_{pl}) and (Δ_{p}) there is so a golden relation, at less than a s-scale factor (10)^{p}. Recall the golden segments (see

By property of the golden segments it’s:

{ ƛ β = ( ƛ γ ϕ ) ƛ γ = ϕ 2 ƛ α (7)

These relations there are in a pentagon between the side and apothem. As is well known in literature, the protons are composed of three quarks: three centres of diffusion positioned in a triangular form in diffusion experiments with “bullet” electrons. These centres indicate (

By the Hypothesis of Structure, we could conjecture a proton having a “pentagonal” geometric structure where the three component quarks are coincident with three constituent triangles, see

The scale factor (10)^{p} of proton relative to Plank’s particle can be due to expansion of universe which maintains invariant the relation between physical quantities.

Then we could state proton like golden particle; by

processes (e + p), investigating the internal structure of the proton, three diffuser center are highlighted in triangular form. Still considering diffusion processes, we have conjectured (see

( ϕ = 1.618 ) → [ ( AD / AB ) = ϕ ] .

This implies also quarks (u, d) are aureus triangles. We have (see Equation (7)):

{ ƛ u = ƛ γ ƛ d = ƛ β }

A geometric property of the golden triangles (u or d) is that one each triangle is composed in its turn of two golden triangles, see

Because the supporting diagonals in the proton are essential for its structure, it’s conjectured that the side (AD)_{p} (in _{p} = h/m_{p}c), thus: [Δ_{p} = k_{p}Δ_{u}].

Where (Δ_{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’ll point out [V(r)_{QCD} ó k_{p}], where V(r) is gluonic potential in QCD theory [

[ ( ƛ u / ƛ d ) = ( m d / m u ) = ϕ ≈ 1.618 ] with ( m d > m u ).

These internal oscillators have been indicated [

Straight away we notice that this structure is realizable only through “particular” quantum oscillators. This “particularity” is underlined into necessity that the vertex—oscillators and junction oscillators must have a structure of “hooks”: this induces us to talk about a “sub-structure” into quantum oscillator which is highlighted only into quantum oscillator coupled to other oscillators. It’s evident that the sides of this structure are made with “additional coupling” between quantum oscillators: there is so one only frequency of oscillation for whole structure (ω_{0 }ó m_{0}). As we have already note in previous articles [

Not only but more components in an oscillator encourages us to believe that the energy of the “quanta” should be distributed between these oscillating components. The presence of more components in an oscillator causes the splitting of its quanta of energy into two, and more, sub-oscillators: this introduces the idea of half-quanta (“semi-quanta”) or individually half-quantum (“semi-quantum”).

Recall in quantum oscillator the energy levels: [ ε n = ( n + 1 2 ) h ν ] . For (n = 1) is [ ε 1 = ( 1 + 1 2 ) h ν ] ; with two sub-oscillators the probability function P_{1}(x) of energy distribution,

Note [ε_{0} = (1/2)hω)] indicates a semi-quantum, while [ε = (1)hω) = 2(1/2)hω] represents one quantum composed by two “entangled” semi-quanta. We so can think that the “IQuO” inside quarks are quantum oscillators at “semi-quanta”. 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” [^{+}):

[ Ψ ( t ) ] ≡ ( a ^ r ′ + ( t ) a ^ r ′ ( t ) ) = { a ^ r ′ + ( t ) = ( • ^ + ) el [ exp ( i r ′ ω t ) ] + ( o ^ + ) in [ exp ( i ( r ′ ω t − π / 2 ) ) ] a ^ r ′ ( t ) = ( o ^ ) el [ exp ( − i r ′ ω t ) ] + ( • ^ ) in [ exp ( − i ( r ′ ω t − π / 2 ) ) ] with Ψ → ( t ) ≡ [ a → + ( t ) + a → ( t ) ] = [ ( a el + ( t ) + a in + ( t ) ) + ( a el ( t ) + a in ( t ) ) ]

where [(·), (o)] are the components of [a, a^{+}] and operators of full semi-quantum (·) and empty semi-quantum (o). To treat the particles as IQuO structures can explain the origin of some fundamental physics greatness as the electric charge, spin, isospin and color charge [_{1}) with another of IQuO (I_{2}) makes a coupling between two IQuO (I_{1}) and (I_{2}): [(I_{1}) Ç (I_{2})]. In ref. [

The representative matrix [_{(}_{n=2*)} (with n = 2* because it is composed by 3 sub-oscillators with 3 full semi-quanta (·)) will be represented by a column matrix with three elements:

( ( Ψ ) ) c l = ( ( o ^ el e − i ρ + • ^ in e − i ρ ° ) ( • ^ el e − i α + o ^ in e − i α ° ) c ( o ^ el + e − i σ + • ^ in + e − i σ ° ) ) 3

where [(α° = α + π/2), (ρ° = ρ − π/2), (σ°= σ ± π/2)].

The index (cl) point out the clockwise associated to d-quark, which is represented (omitting the junctions’ IQuO) by an overlapping of three IQO-V [(I_{A}(+) I_{B}(+) I_{C})]:

{ ( ( o el ) A ( • in ) A ( • el ) A ( o in ) A ( o el + ) A ( • in + ) A ) c l + ( ( • el ) B ( o in ) B ( o el + ) B ( • in + ) B ( • el ) B ( o in ) B ) c l + ( ( • el + ) C ( o in + ) C ( • el ) C ( o in ) C ( o el ) C ( • in ) C ) c l }

(omitting the phases)

By matrices we can make the various coupling between quarks and so to build the structure representatives of hadrons.

Finally, we do not must thinking to a rigid structure of quarks: different orientations in space are possible (see

Note: talking about a geometric structure of coupling oscillators in quarks, implicitly it’s giving mass values not zero to them. Therefore, this quark model describes only “massive quarks”. This will be more evident in next section.

Mesons are hadrons composed of quark-antiquark pairs and the most elementary is the pion.

A first attempt of structure equation is: [(π^{+}) = (u Å d), (π^{−}) = (u Å d)].

The sign (Å) point out the dynamics coupling between quarks; it could involve both gluonic coupling and electromagnetic: [Å = Å_{g} + Å_{em}].

Then, we have the configuration illustrated in

Where u is the u antiquark. The double circles indicate the coupling between vertices-oscillators (B ó B’, C ó C’).

The coupling between the u quark and d quark forms so a quadrangular structure (ABDC). The bond (see gluons) between two free quarks (u ó 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})_{free} → (m_{u}, m_{d})], see Equation (1); thus the Compton wavelength decreases ( ƛ _ u < ƛ u ), ( ƛ _ d < ƛ d )], shrinking the u and d quarks (see second drawing in _{π}) and period (τ_{π}), to which we associate (Δ_{π}, m_{π}). Each quark (u, d) contributes to total mass (m_{π}) with its mass value (m_{u}, m_{d}), maintaining the golden ratio [ ( m _ d / m _ u ) = ϕ ] . Speaking of elastic tensions in bound quarks we can admit that in the (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}].

The representative quantum of (u-quark)_{π} oscillates (in the Space-Time (ST)) with proper time [ τ _ u = ( ƛ _ u / c ) ] π and that one of (d-quark)_{π} with [ τ _ d = ( ƛ _ d / c ) ] π , while in bound system (π) the representative quantum oscillates (in ST) with proper frequency (ω_{π}), with elastic constant (k_{π}), wave length ƛ π and proper time [ τ _ π = ( ƛ _ π / c ) ] π .

Note, instead, the representative quantum of (u-quark)_{π} covers the route (BDC ó l_{u}) in time [Δτ_{u} = (l_{u}/v_{u})]_{π} and that one (ABC ó l_{d}) of (d-quark)_{π} in [Δτ_{d} = (l_{d}/v_{u})]_{π}, in bound system (π) the representative quantum covers the route (ABDC ó l_{π}) in time [Δτ_{π} = (l_{π}/v_{π})]_{π}.

In this case we’ll have that the periods of route in u-quark and in d-quark must be coincident with that of π-structure: [Δτ_{u} = Δτ_{d} = Δτ_{π}]. This indicates that the velocities of propagation of quanta in the structures (u, d, π) are different because are different the elastic tensions of respective sides; recall (see ^{2}= (T/μ)] with (T ó k) and (μ) is mass density of the chain of coupled oscillators. It follows: (k_{u} < k_{d} < k_{π}) ó (v_{u} < v_{d} < v_{π}).

Where [k_{u} ó (BDC) óω_{u}], [k_{d} ó (ABC) óω_{d}], [k_{π} ó (ABDC) óω_{π}].

As it occurs in Hidrogen atom, where the oscillation period (τ_{H}) along an orbital is joined to rotation period Δτ_{H}, so it needs happen in pion: [(τ_{π}) ó Δτ_{π} ó (τ_{π})].

The experimental mass of pion (m_{π}) has frequency (ω_{π}) and period (τ_{π}); we must point out that: Δτ_{π} = (τ_{π}).

By π-structure we point out Δ_{π} = Δ_{u} = k_{u}Δ_{u}, where Δ_{u} is the Compton wavelength of the free u quark (m_{u} is bare mass, k_{u} internal elasticity of free quark); the Δ_{π} is the Compton wavelength (side BC) of quadrangular structure (ABDC)_{π} and is coincident to Δ_{u}, the Compton wavelength of the u quark bounded inside to pion. Therefore Δ_{π} > Δ_{π}, where Δ_{π} = h/m_{π}c, indicating that the junction IQuO of side CD can be composed of more quantum oscillators [

To the frequency ω_{π} we associate k_{π} elastic coefficient, which is related to the gluonic potential V(r) of the QCD and thus in turn is related to coupling (Å), where [Å = (Å_{g} + Å_{em})]. Therefore, even along sides, there are gluons (see the Joining-IQuO of

In the pion, different configurations are related to the X-junction axis,

Where I_{A}, I_{B}, I_{C}, I_{D}, are vertex IQuO, I_{AB}, I_{BC}, I_{CA}, I_{BD}, and I_{DC} are junction IQuO. In configuration 1, I_{D} is an IQuO vertex non-coupled with IQuO of the side AC. The I_{D} IQuO is free. To tie the two quarks (u, d), it is necessary to add a junction IQuO between the respective bases’ oscillators. We suspect that the IQuO oscillators of the junction between the sides (BC)_{d}, and (B’C’)_{u} could be gluons (see _{q}), while we associate an orbital spin (s_{l}) to rotations of u-quark (or d-quark) around the X-axis (see the experimental observations about proton spin). The hypothesis of structure does not conflict with experiments of CERN (COMPASS), SLAC, and DESY (HERMES), where the spin of the proton is the sum of intrinsic spins (s_{q}) of quarks with their orbital motions (s_{l}); note in rotations around X-axis are involved also binding gluons therefore we add theirs

gluonic orbital motion (s_{g}), as in spin of proton. This model is so consistent with experimental observations [

[ s π = s q + s l + s g ]

The possibility of reciprocal rotations of two quarks (u and d) around the X-axis (as orbital motions) induces us to see the quarks as free, and this last word leads us to the behaviour of Asymptotic Freedom of quarks. There are, in fact, no gluons between the vertices (I_{A} and I_{D}).

In addition, we may think that the X-axis in _{x}) along the X-axis. In this way, the X-axis becomes a chain of quantum oscillator pairs (I_{B}-I_{C})_{i}, of the fermion type, with junction IQuO (I_{BC})_{i} of global length (Δ_{π}): [(I_{B}-I_{BC}-I_{C})_{i}…(I_{B}-I_{BC}-I_{C})_{(i+n)}…(I_{B}-I_{BC}-I_{C})_{k}…].

As _{B}-I_{BC}-I_{C})_{i} is part of a quadrangular structure (ABDC) of coupled IQuO. Note only quanta of the pion field propagate along this chain with scalar wave function Ψ(x, t) (see the Heisenberg relation and Equation (5)).

Now, we calculate the mass of quarks inside the pion: (u_{π}, d_{π}) are the bound quarks. We consider ( m d / m u ) = ϕ ≈ 1.618 (see Equation (7)). Then, we conjecture that the golden ratio is also present in bound quarks in a pion:

{ m _ ( u π ) + m _ ( d π ) = m π ± m _ ( d π ) / m _ ( u π ) = ϕ (8)

This equation has solutions

{ m _ ( u π ) = m π ± ( 1 + ϕ ) = m π ± 2.618 = ( 53 .31 ) MeV m _ ( d π ) = m ( u π ) ϕ = ( 86 .26 ) MeV (9)

where m(π^{±}) ≈ (139.57) MeV; see ref. [

{ m _ ( u π ) + m _ ( d π ) = m π ± } ⇒ { ( ℏ ƛ ( u π ) c ) + ( ℏ ƛ ( d π ) c ) = ( ℏ ƛ π c ) } ⇒ { ( ℏ ƛ _ u c ) + ( ℏ ƛ _ d c ) = ( ℏ ƛ π c ) } ⇒ { ƛ _ d + ƛ _ u ƛ _ u ƛ _ d = 1 ƛ π } ⇒ { ƛ π = ( ƛ _ u ƛ _ d ƛ _ d + ƛ _ u ) = ƛ _ u ( 1 1 + ϕ ) } ⇒ { ƛ π = ƛ _ u / ϕ 2 } ⇒ { ƛ π = ƛ _ π / ϕ 2 }

where Δ_{π} = Δ_{u} and Δ_{π} < Δ_{u} = Δ_{π}. We indicate the side BC (_{π}) to a quadrangular structure (ABDC): BC = Δ_{π}.

k_{π} is the parameter of calibration that adapts the pion to the bound u quark, which be part of it: k π = ( ƛ _ π / ƛ π ) = ( ϕ 2 ) . We assign the mass values of Equation (9) to the physical system composed by quarks and gluons that form the pion; the u_{π} and d_{π} quarks with their gluons are called dressed quarks. The u_{π} and d_{π} quarks are always detected by any observer like a unique quantum system with non-separated components.

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

In the interpenetration of quarks and their dynamics interactions we’ll use a new Ä-operation of combination (or coupling) of quarks and particles: (a_{i} Ä b_{j}). The new operation (Ä) indicates a composition of two operations (Ä, Å) or [Ä º (ÄUÅ)], where Ä-operation describes the proper interpenetration of the quarks and it follows the properties of multiplication, while Å-operation describes dynamics interactions and follows the properties of sum. We use the Ä-operation in that systems in which there are presents interpenetrations between components and dynamics interactions: we speak of dynamics interpenetrations.

The coupling (u ó d) (or (u ó d)) in a pion π (^{±} = (u Å d)]. The (Å) sign point up a dynamics coupling between. Nevertheless the different relative orientations of quarks (u, d) induce us to admit interpenetrations between two quarks. Thus, we should write [π^{±} = (u Ä d)]. The reciprocal interpenetrations determine the aspects of the interpenetrating kinetic energy and spin. The intrinsic energy of movement makes increasing the mass of two quarks so as the dynamics coupling, through gluons of side AC (see _{bare} → k_{π}].

Finally, the interpenetration could explain the zero value of pion’s spin. In fact, quarks’ could have a relative spin (s(u), s(d)) consequent to the reciprocal interpenetration between a u quark and d quark; the reciprocal interpenetration of quarks implies relative, opposite rotations, meaning

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

Then, the spin of a pion is zero.

Now, we ask us about the structure of neutral pion (π^{0}). To have a neutral configuration with two quarks (u, d) it needs to combine two pairs [(u, d), (u, d)].

The wave function of pion is [π^{0} = a(uu + dd)] and it point out a no-separated state (or entanglement state) of quarks (u, u, d, d). Therefore, the neutral pion is a unique elementary particle: it follows that its components [(u, d) and (u, d)] must be reciprocally “interpenetrated” for originating a unique physical object (see Section 3.3). We conjecture that neutral pions are originated by all the possible combinations of quarks’ couplings using both the Ä-operation and Å-operation. We’ll define:

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

Thus, neutral pion is so a dynamics interpenetration of a pair of charged pions. The combination of quarks by operation (Ä) involves both reciprocal interpenetration (Ä) and the dynamical coupling (Å).

The interpenetration of two charged pions makes individual the neutral pion: this could implicate [m(π^{0}) ≈ m(π^{±})]. Now, we can depict the configurations the neutral pion (π^{0}) = [(π^{+}) Ä (π^{−})]; we conjecture the following structure (

Note (see ^{0})_{a}, (π^{0})_{b}].

In a-configuration quarks (u,u) are attacked to (d,d) respectively in (AD) side and (BC), while in b-configuration both quarks are attacked in (FH) diagonal. In two configurations the vertexes (L, M) are frees in movement.

Even if speaking about “interpenetration” of quarks, in diagonal (AC) the quarks (d) and (d) are dynamically coupled so as in diagonal (FH); the same occurs between the quarks (u, u) in b-configuration. All this determines an exchange of quanta along [(AC), (FH)]. Thus, in interpenetration between quarks there can be some parts (or sides) in interaction (that of junction). Besides, there are 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. As a matter of fact, if we take in considerations the rotations of two pions then different possibilities there can be: the total spin can have values (0, ±1). As we’ll see the value (±1) point out the r-meson while the value zero the π^{0}-meson. In this last meson, (d, d) quarks are both in opposite relative rotation (see the charged pions in par. 3.1) along diagonal (AC); the same must occurs to (u) and (u) for obtain always spin zero [s(π^{0}) = 0]. Also in b-configuration: here both (d, d) and (u, u)

are in opposite relative rotation. Note in b-configuration that u-quark is interpenetrated to d-quark in (π^{−}), to same way the quarks (u, d) in (π^{+}) so like the (d, d) quark are reciprocally interpenetrated. When the quanta of quark pairs, casually cross, then, in short time, it is produced an annihilation: this is more probable in b-configuration ((π^{0})_{b}). The a-configuration is instead more fitting for make a base in mesons having major mass.

Another possibility, see a-configuration, can be that the quarks (u, u) could attach themselves to IQuO-chain of the diagonal (AC) belonging to quarks (d, d). This can occur when the quanta of u-quark (u-quark), crossing the C-vertex, propagate along in (AC) and overlap to ones of IQuO-chain (AC) of d-quark (d-quark). The same can occur in b-configuration: the vertices L and M attach in diagonals FG and EH. These two features may origin a new strange s-state in quarks which becomes so the strange quark (s). Note two possibilities of being of strange quarks: (s_{l}-quark, s_{s}-quark). This last feature could determine different k-mesons or kaons.

Because there are two configurations then one can have two axes of propagation of the neutral pion: the diagonals [(AC)_{X}, (FH)_{Y}]. Nevertheless, the propagation axes become also coupling axes where the quark-antiquark pairs annihilate (along X-axis) after a phase shift reciprocal (recall the electromagnetic field is a gauge field); this phase shift could also make rotate the X-axis which becomes S-axis (see _{1}, d_{1}) (u_{2}, d_{2}). In literature [^{0}) → (γ + γ)], electromagnetic decay, see

It’s possible even, with low probability (1.2%), that one of two photon (see Dalitz) decays in a pair (e^{+} + e^{−}) along the S-axis.

For having two photons with spin [s(γ) = ± 1] it needs happen:

[s(u_{1}) + s(d_{1}) = (±1/2) + (m1/2); s(u_{2[}) + s(d_{2}) = (m1/2) + (±1/2)] in t initial

[s(u_{1}) + s(d_{1}) = (±1/2) + (±1/2); s(u_{2[}) + s(d_{2}) = (m1/2) + (m1/2)] in t initial

The first possibility admits not annihilation:

[s(u_{1}) + s(u_{2})] + [s(d_{1}) + s(d_{2})] → [s(γ_{u}) + s(γ_{d}) = (0) + (0)] impossible

The second possibility admits the following annihilation:

[s(u_{1}) + s(u_{2})] + [s(d_{1}) + s(d_{2})] → [s(γ_{u}) + s(γ_{d}) = (±1) + (m1)]

But here the two photons would have different energies because m(u_{π}) ¹ m(d_{π}), see Equation (9). By experiments with neutral pion decay one observes two photon of equal energy. For explain this experimental aspect it need well to analyze the structure equation of π^{0}.

Now, it’s necessary to explain mathematically the (Ä)-operation in neutral pion.

The interpenetration implies a no-separated state of all possible combinations of the quarks: [(uu), (ud), (du), (dd)].

These combinations recall the Ä-tensorial product between two vectors with components (u, u; d, d); if we represent charged pions as components of (π^{0}) “vector”, then we’ll have:

π 0 ≡ ( π + , π − ) = [ ( u d _ ) , ( d u _ ) ] (10)

where it’s:

π − = ( C ^ π + ) = C ^ ( u d _ ) = σ 1 [ ( u d _ ) ] * = ( 0 1 1 0 ) ( u _ d ) = ( d u _ ) (11)

where Ĉ is the operator of charge conjugation and s_{1} the Pauli’s matrix.

Here, (u) and (d) represent the wave function associated to quark [^{+}), see reference [_{t}), it follows:

( π + ⊗ t π − ) = ( u d _ ) ⊗ t ( d u _ ) = ( u d u u _ d _ d d _ u _ ) (12)

This combination represents not a pion expressed by equation: (π^{0}) = [(π^{+}) Ä (π^{−})]. Then, one could use the matrix-vector [π º (u, d)]; neutral pion could be:

π 0 ≡ ( π q , π q + ) ≡ ( π q , C ^ π q ) = [ ( u d ) , ( d _ u _ ) ] (13)

If we use (Ä_{t}), it will be [

π 0 = ( π q ⊗ t π q + ) = ( π q ⊗ t C ^ π q ) = ( u d ) ⊗ t ( d _ u _ ) = ( u d _ u u _ d d _ d u _ ) (14)

We point out this matrix as that representative of neutral pion.

The wave function of neutral pion could be obtained from determinant if this matrix:

Ψ ( π 0 ) = | π 0 | = | u d _ u u _ d d _ d u _ | = { [ ( u d _ ) ⋅ ( d u _ ) ] − [ ( d d _ ) ⋅ ( u u _ ) ] } (15)

But this function is different from the one of literature, because we have introduced in theory of quarks their interpenetration. Nevertheless, in Equation (15) the interpenetration is not highlighted. Then, it needs to associate the (Ä)-operation to the tensorial product (Ä_{t}): we must consider the dynamics combinations [(ud) → (u Å d)] and those of interpenetration [(ud) → (u Ä d)]. Therefore, it should be:

π 0 = ( π q ⊗ _ π q + ) = ( π q ⊗ _ C ^ π q ) = ( u d ) ⊗ _ ( d _ u _ ) = ( u ⊕ d _ u ⊕ u _ d ⊕ d _ d ⊕ u _ ) ⊗ (16)

where the Ä-index in the determinant point out that the multiplication operation of the elements of matrix is substituted for Ä-operation.

The interpenetration is highlighted in wave function Ψ(π) obtained from determinant:

Ψ ( π 0 ) = | π 0 | ⊗ _ = | u ⊕ d _ u ⊕ u _ d ⊕ d _ d ⊕ u _ | ⊗ = { [ ( u ⊕ d _ ) ⊗ ( d ⊕ u _ ) ] − [ ( d ⊕ d _ ) ⊗ ( u ⊕ u _ ) ] } (17)

From definition of pion it follows generally:

( π 0 ) Ψ _ = [ ( π − ) ⊗ _ ( π + ) ] Ψ _ = [ ( u _ ⊕ d ) ⊗ ( u ⊕ d _ ) ] Ψ _ = { [ ( u _ ⊗ u ) ⊕ ( u _ ⊗ d _ ) ⊕ ( d ⊗ u ) ⊕ ( d ⊕ d _ ) ] } Ψ _ (18)

where Ψ is a superposition state and thus no-separated state or not local.

Here, we treated the Ä-operation like an arithmetical product (x) while the Å-operation with that of sum. In a superposition state, the quark interpenetrations of couples [(u Ä u), (u Ä d), (d Ä u), (d Ä d)] involve the following couplings of interpenetration:

1) ( u ⊗ u _ ) ⊕ ( d _ ⊗ u _ ) ⊕ ⋯ → ( u ⊕ d _ ) ⊗ ( ⋯ ⊕ u _ )

2) ( d _ ⊗ d ) ⊕ ( u ⊗ d ) ⊕ ⋯ → ( d _ ⊕ u ) ⊗ ( ⋯ ⊕ d )

This because if there are the couplings (u Ä u) and (u Ä d) then it follows, see transitive property, (u Ä d); the same for others pairs. Combining the points (1) and (2) we’ll have

[ ( u ⊗ u _ ) ⊕ ( u ⊗ d ) ⊕ ( d _ ⊗ u _ ) ⊕ ( d _ ⊗ d ) ] = [ ( u ⊕ d _ ) ⊗ ( d ⊕ u _ ) ] ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊗ d ) ]

This equation point out one of algebraic properties of Ä-operation and Å. Below here we list some properties:

1 ) ( a ⊗ a ) = a ; ( a ⊕ a ) = 0 2 ) ( a ⊗ b ) = ( b ⊗ a ) ; ( a ⊕ b ) = ( b ⊕ a ) 3 ) if ( a 1 ≡ a 2 ≡ a ) ⇒ ( a 1 ⊕ a 2 ) = 2 a 4 ) if ( a ⊗ b ) and ( b ⊗ c ) ⇒ ( a ⊗ c ) ( transitiveproperty ) a ) ⊗ _ ≡ ( ⊗ , ⊕ ) ⊗ _ ≡ ( operationofcombination )

5 ) if { [ A = ( a ⊕ b ) ] , [ B = ( c ⊕ d ) ] } ⇒ ( A ⊗ _ B ) = ( a ⊕ b ) ⊗ ( c ⊕ d ) 6 ) ( a ⊕ b ) ⊗ ( c ⊕ d ) = ( a ⊗ c ) ⊕ ( a ⊗ d ) ⊕ ( b ⊗ c ) ⊕ ( b ⊗ d ) 7 ) if { ( a ⊗ b ) , ( a ⊗ c ) , ( a ⊗ d ) } ⇒ ( a ⊗ c ) ⊕ ( a ⊗ d ) ⊕ ( b ⊗ c ) ⊕ ( b ⊗ d ) = [ ( a ⊕ b ) ⊗ ( d ⊕ c ) ] ⊕ [ ( a ⊕ c ) ⊗ ( b ⊗ d ) ] (19)

Generally, we’ll write:

( π 0 ) ( ⊗ _ ) = [ ( π − ) ⊗ _ ( π + ) ] = { [ ( u ⊕ d _ ) ⊗ ( u _ ⊕ d ) ] ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] } Ψ _ = { [ ( π − ) ⊗ ( π + ) ] ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] } Ψ _ (20)

or This the structure equation of neutral pion (π^{0}) in no local state (Ψ).

Note {(π^{0}) = [(π^{+}) Ä (π^{−})]} ¹ [(π^{+}) Ä (π^{−})] and thus {m[(π^{+}) Ä (π^{−})] ¹ m(π^{0})}.

The interpenetration between two charged pions allows of assume

m [ ( π − ) ⊗ ( π + ) ] = m ( π ± ) (21)

The (π^{±})^{0} =[(π^{+}) Ä (π^{−})] is called bare neutral pion.

The “coupling” between two quarks is so made by combination of all possible configurations of these two quarks. In this way the total mass (m_{tot} º m_{Ä}) is the sum of all masses associated to each combination of interpenetration (m_{Ä}) and interaction (m_{Å}), with [(m_{Å}) = m_{0} + m_{kin} ± (Δm)] where (m_{0}) is proper mass, (m_{kin}) is interpenetrating kinetic energy and (Δm) is interaction energy between the component masses. It follows: [m_{Ä} = m_{Ä} + m_{Å}].

Then now we can calculate the pion mass (mass at rest) by structure Equation (20) of neutral pion:

m ( π 0 ) = m ( { [ ( π − ) ⊗ ( π + ) ] A 1 ⊕ [ ( u ⊕ u _ ) ⊗ ( d _ ⊕ d ) ] A 2 } ) (22)

Particular Where (A_{i}) are the components of physical system (π^{0}).

The A_{1}-component [π^{−} Ä π^{+}] implies the interpenetrating of two charged pions. Also the two pairs (u Å u) and (d Å d) are interpenetrated, see [(u Å u) Ä (d Å d)], while each pair ([u Å u], [d Å d]) indicates a “attractive” coupling electromagnetic (Å_{em}) between u quark and u-quark (the same for d-quark and d-quark) but “without” annihilation (that is before of the decay); in electromagnetic energy it’s:

{ [ ( u ⊕ u _ ) e m = ε γ ( u u _ ) ] , [ ( d ⊕ d _ ) e m = ε γ ( d d _ ) ] }

The masses of interpenetrating quarks cannot be summed between them: the interpenetrating merges the quarks in a unique object. Nevertheless, we cannot ignore the interactions [u Å u]_{em}, [d Å d]_{em} in calculating the total mass. To obtain the partial mass, or without mass defect, we’ll use the structure Equation (20) and its equivalent forms. For every X-system (composed by more particles), we’ll 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})_{Ä}, (B_{j})_{Å}] 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 (m_{i}) associated to 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 ) ( ⊗ ) ] + [ ∑ j = 1 m m ( B j ) ( ⊕ ) ] = [ m ( a ⊗ b ) A 1 + ⋯ + m ( w ⊗ z ) A n ] + [ m ( c ⊕ d ) B 1 ⊕ ⋯ ⊕ m ( r ⊗ s ) B m ] (23)

We’ll have the following applications:

{ F m ( A ( ⊗ ) ) = F m [ ( a ) ⊗ ( b ) ] A = 〈 m ( a , b ) 〉 F m ( B ( ⊕ ) ) = F m [ ( a ) ⊕ ( b ) ] B = m ( a ) + m ( b ) } (24)

Note that the mass of two interpenetrating particles [a Ä b] will be obtained by average value of individual masses [ 〈 m ( a , b ) 〉 ] , while mass of two interacting particles [a Å b] will be obtained by 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 exception of some cases (which we’ll specify) (m_{kin}) ≪ m_{0}, therefore we’ll have: [m_{tot} = m_{part} ± Δm_{i})].

The mass defect will be:

Δ m = Δ m g + Δ m e m

where Δm_{g} is gluonic mass defect. Nevertheless the Δm_{g} has been englobed in masses of charged pion (see Equation (8)): therefore we consider only electromagnetic mass defect Δm = Δm_{em}.

To obtain the mass defects (Δm > 0, Δm < 0) we’ll 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 = 1 , j = 1 , k = 1 i ≠ j ≠ k ) n Δ m ( A i ⊕ ( A j ⊕ A k ) ) ] + [ ∑ ( i = 1 , j = 1 , k = 1 i ≠ j ≠ k ) n Δ m ( B i ⊕ ( B j ⊕ B k ) ) ] (25)

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 (26)

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 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,

By structure equation and Equation (24) one finds the partial mass of π^{0}:

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 (27)

where we have components

F m ( π + ⊗ π − ) A 1 = 〈 m ( π + , π − ) 〉 A 1 = [ m ( π + ) + m ( π − ) 2 ] A 1 F m [ ( u u _ ) ⊕ ⊗ ( d _ d ) ⊕ ] A 2 = [ m ( u ⊕ u _ ) + m ( d _ ⊕ d ) 2 ] A 2 = [ 2 m ( u ) + 2 m ( d ) 2 ] A 2 (28)

The first (A_{1}) is the average value of the mass of charged pions. The second component (A_{2}) would be the mass associated to interpenetration between pairs [(uu), (dd)] but in each pair the quarks are interacting [(u Å u), (d Å d)], see b-configuration. Therefore, the A_{2}-component is a dynamics component (A_{2}-component), so as the A_{12}-component composed by the Å-application operating on the two components (A_{1}, A_{2}): A_{12} → A_{12} = B_{1} = (A_{1} Å A_{2})

The values of mass defects (see (25)) would be then:

I ) F Δ m [ ( π + ) ⊗ ( π − ) ] A 1 = 0 II ) F Δ m [ ( u u _ ) ⊕ ⊗ ( d _ d ) ⊕ ] A 2 = Δ m ( ( u u _ ) ⊕ , ( d _ d ) ⊕ ) ≠ 0 III ) F Δ m { [ ( π + ) ⊗ ( π − ) ] A 1 ⊕ [ ( u u _ ) ⊕ ⊗ ( d _ d ) ⊕ ] A 2 } B 1 ≠ 0 (29)

As we know in the interpenetration of two object there is not interaction if there are not parts in interaction; the components of structure equation which express an interaction between parts (along the diagonal (HF)) are the A_{2}-component and A_{12}-component, but the A_{1} is not. Thus, there is mass defect only in A_{2}-component and A_{12}. The A_{12}-component describes two charged pions coupled, along diagonal [(AC) or (HF)], to the two pairs [(u Å u), (d Å d)]: this could give a “binding energy”. The coupling (A_{1} Å A_{2}) could be both gluonic (Å_{g}) and electromagnetic (Å_{em}), but gluonic energy has been incorporated in elastic tensions (k_{i}). Another binding energy is in the interpenetration between pairs [(u Å u), (d Å d)], see A_{2} component: this is an electromagnetic energy originated by pair interaction [(uu), (dd)] before of annihilations. This electromagnetic energy will be equal to one of annihilation:

〈 m ( ( u u _ ) ⊕ , ( d _ d ) ⊕ ) 〉 A 2 = 〈 m ( ε γ ( u u _ ) ⊕ e m , ε γ ( d _ d ) ⊕ e m ) 〉 A 2

The A_{2}-component is the bare component (u, d)_{0} of pion’s quarks while A_{1}-component is the bound component of quarks of the charged pions due to gluons → (u, d)_{π}. Because the quarks of pairs [(uu), (dd)]_{A}_{2} are the same quarks of charged pions, it follows that the binding energy of the A_{12}-component is equal to that of A_{2}-component in Equation (27):

Δ m ( ( u u _ ) ⊕ e m , ( d _ d ) ⊕ e m ) A 2 = Δ m { ( π + ) ⊗ ( π − ) ⊕ [ ( u u _ ) ⊕ e m ⊗ ( d _ d ) ⊕ e m ] } B = 〈 m ( ε γ ( u u _ ) ⊕ e m , ε γ ( d _ d ) ⊕ e m ) 〉

Finally, the global mass value of pion is obtained as:

m ( π 0 ) = m ( π 0 ) * − Δ m = ( 1 / 2 ) [ m ( π + ) + m ( π − ) ] + ( 1 / 2 ) [ ε u ( γ ) + ε d ( γ ) ] ( u , d ) 0 − [ Δ m A 2 + Δ m A 12 ] e m = ( 1 / 2 ) [ m ( π + ) + m ( π − ) ] + ( 1 / 2 ) [ ε u ( γ ) + ε d ( γ ) ] ( u , d ) 0 − 2 { ( 1 / 2 ) [ ε u ( γ ) + ε d ( γ ) ] } = m ( π ± ) − ( 1 / 2 ) [ ε u ( γ ) + ε d ( γ ) ] ( u , d ) 0 = m ( π ± ) − ε γ ( π ) ( u , d ) 0 (30)

where (ε_{γ}) is annihilation energy of pairs [(u Å u), (d Å d)]. This energy, as already it has been said, is coincident with mass at rest of free quarks (see the bare mass). It’s clear that the binding energy flows in intermediary gluons until to decay. We experimentally know [Δm_{γ} = ε_{γ} = (4.59) MeV] from difference between of masses of pions: [Δm(π) =[m(π^{±}) − m(π^{0})] where m(π^{0}) ≈ (134.97) MeV [

Let’s go explain the decay at two photon of equal energy in neutral pion. For explain this experimental aspect it need looking well the structure equation.

For happen this decay we must conjecture that it exist an intermediary field (H_{W}) which transform the quarks and origins the decay; in this way the neutral pion comes projected in the representation of Field H_{W}:

( π 0 ) + { H W } → [ ( π 0 ) ] H W → ( γ + γ )

where {H_{W}} is a lattice of two interpenetrated field of type H_{W}: {H_{W}}= (H_{W} Ä H_{W}).

(H_{W}, H_{W}) operate as:

{ [ H w u = d ] , [ H w d = u ] [ H _ w u _ = d _ ] , [ H _ w d _ = u _ ] } (31)

That is H_{W} operates on quarks of neutral pion. Here (u, d) are matrices

also the Field H_{w} is represented by matrices which can change the quarks (u ó d). In next study we’ll give the matrix form of H_{w}. Using the structure equation we’ll have:

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

= { [ ( u ⊕ d _ ) ⊗ H W ] ⊕ [ ( u _ ⊕ d ) ⊗ H W ] } π 0 ⊗ { [ ( u ⊕ d _ ) ⊗ H _ W ] ⊕ [ ( u _ ⊕ d ) ⊗ H _ W ] } π 0 = { [ ( u ⊗ H W ) ⊕ d _ ] ⊕ [ ( d ⊕ H W ) ⊕ u _ ] } π 0 ⊗ { [ ( d _ ⊗ H _ W ) ⊕ u ] ⊕ [ ( u _ ⊗ H _ W ) ⊕ d ] } π 0 = { [ d ⊕ d _ ] ⊕ [ u ⊕ u _ ] } ⊗ { [ u _ ⊕ u ] ⊕ [ d _ ⊕ d ] } π 0 (33)

This H_{w}-Field is a sort of “Higgs Field” which operates like a vectorial boson W but the decay time is equal to the one electromagnetic.

Note that the H_{w} is a charged field because for transform uód it needs that it has an electric charge (±1): H_{w} → [(H_{w})^{−}, (H_{w})^{+}]. Moreover, because H_{w} transform [Ψ_{u}(spinorial) ó Ψ_{d}(spinorial)] could be a scalar field.

We calculate the “mass-energy” of final lattice of quarks, using the formula of mass function.

F m ( A 1 ⊗ A 2 ) = F m { [ d ⊕ d _ ] ⊕ [ u ⊕ u _ ] } A 1 ⊗ { [ u _ ⊕ u ] ⊕ [ d _ ⊕ d ] A 2 } π 0 = [ m ( A 1 ) + m ( A 2 ) 2 ] π 0 = [ ( ε ( γ d ) + ε ( γ u ) ) + ( ε ( γ d ) + ε ( γ u ) ) 2 ] π 0 = [ ( ε ( γ d ) + ε ( γ u ) ) π 0 2 + ( ε ( γ d ) + ε ( γ u ) ) π 0 2 ] = [ ε ( γ ) π 0 2 + ε ( γ ) π 0 2 ] ⇒ [ γ ( π 0 / 2 ) + γ ( π 0 / 2 ) ] (34)

We have so obtained two equal photon of decay.

Also the particle-field H_{w} has geometric structure. In next study we’ll give both its structure equation and geometric structure.

Finally, we show as the graphic representation di Feymann can be deepened by algebraic formalism of operations (operators) [Ä, Å].

In first we consider the decay of charged pion:

( π − ) H W = [ ( u _ ) ⊕ ( d ) ] H W = { [ ( u _ ) ⊕ ( d ) ] ⊗ { H W } } H W = { [ ( u _ ⊕ d ) ] ⊗ [ H W ⊗ H _ W ] } π − = { [ ( u _ ⊕ d ) ] ⊗ H W } ⊗ { [ ( u _ ⊕ d ) ] ⊗ H _ W } π − = { [ ( d ⊕ H W ) ⊕ u _ ] ⊗ { [ ( u _ ⊗ H _ W ) ⊕ d ] } } π − = { [ u ⊕ u _ ] ⊗ [ d _ ⊕ d ] } π − = ( γ + γ ) impossible

Note the not conservation of electric charge and spin (if the H_{w} is scalar). Then in charged pion the H_{w}-field becomes the vectorial boson W^{±}. In this case the decay of charged pion will be a weak decay:

( π − ) W = [ ( u _ ) ⊕ ( d ) ] W = { [ ( u _ ) ⊕ ( d ) ] ⊕ { W } } W = { [ ( u _ ⊕ d ) ] ⊕ [ W + ⊕ W _ − ] } π − = { [ ( u _ ⊕ d ) ] ⊗ W + ⊕ W _ − } π − = { [ ( u _ ⊕ W + ) ⊕ d ] ⊕ W _ − } π − = [ d _ ⊕ d ] ⊕ W _ − = ( γ ) ⊕ W _ − = ( W _ − ) * (35)

Here the equation says us that we can have different decays:

(W^{−})^{*} → [(μ^{−}) + n_{e}] > (99%) decay probability; (W^{−})^{*} → [(e^{−} + n_{e}) + γ] ≪ 1%.

The annihilation energy of quark (u, d) belonging to neutral pion will be coincident with mass energy of quarks. In this way it is possible calculating the quarks’ masses.

A system of two equations can give the bare masses of quarks (u, d) if admit their masses in golden relation:

{ ( 1 / 2 ) [ m ( u f ) + m ( d f ) ] = Δ m π 0 m ( d f ) / m ( u f ) = ϕ ⇒ { ( 1 / 2 ) [ m ( u f ) + m ( u f ) ϕ ] = Δ m π 0 m ( d f ) / m ( u f ) = ϕ { ( 1 / 2 ) m ( u f ) [ 1 + ϕ ] = Δ m π 0 m ( d f ) / m ( u f ) = ϕ ⇒ { m ( u f ) = [ 2 Δ m π 0 ( 1 + ϕ ) ] m ( d f ) = ϕ m ( u f ) ⇒ { m ( u f ) = ( 3.51 ) MeV m ( d f ) = ( 5.67 ) MeV (36)

These values are inside the range anticipated by literature (see Section 3.2).

It’s so evident that the values of masses of quarks, both bare and dressed inside pion, could be used for obtaining the mass spectrum of light mesons. If the values in Equation (9) are that of bound quarks in pion by gluons, then we can think of having incorporated the potential V(r) of gluons in QCD into mass values. So the bound masses of quarks playing a fundamental rule inside pions. In this way the system of pion comes a basic system with mass determined by Yucawa’s Hypothesis. By pion (π º (u, d)) it’s possible to build the hadrons’ structures. For making this, it needs to admit the presence of a lattice of “virtual pions” {π^{0}}, with elementary cell having the form of the (π^{0}), you see ^{0}}-lattice and {d,d} participating to build the mass spectrum of light mesons. By Equations (23) and (25) we’ll show the calculations (in next paper) for finding the mass value of light mesons with notable precision as also the mass value of fundamental nucleons.

Another aspect is surfaced in this work: there are not static quarks inside hadron (i.e. pion) but they are in UN particular movie where the gluons play a not secondary role. In fact, we already have noted, in framework of hypothesis of structure, the possibility having quarks in rotation, see the “interpenetration”. The hypothesis of structure is not “conflict” with experiments where the spin of proton (also meson) is the sum of intrinsic spins of quarks with the orbital motions of quarks and gluons; this is because the rotation movements the triangles-quarks with the binding gluons can be seen as orbital motions:

[ s proton = s quark + s gluon + s quark orbit + s gluon orbit ] .

The GM model foresees that gluons, just because they are junction oscillators, are “located” inside and around quarks. This aspect is in agreement with experimental data on the observation of proton moment where the presence of gluons in the proton is given by the fraction of moment carried by the quarks: (P_{quark}/P_{p}) ≈ (1/2).

The fact that the missing moment is brought by gluons seems confirmed by the cross sections in hadron collisions. In collisions at high energies the existence of gluons seems to be in the presence of 3 coplanar hadronic jets [^{+}e^{−}):

[ e + + e − → q + q + g → X hadrons ]

Nevertheless, we could think to another possibility:

[ e + + e − → q + q + ( d d ) → X hadrons ]

where the pairs (dd) belongs like lattice {d,d}. Both Quark and gluons, are never visible in the detector because they can only exist confined within the hadrons.

Another aspect is surfaced in section () with lattice {H_{w}, H_{w}}: two photons of π^{0}-decay are with equal energy if there is present as background the lattice {H_{w}, H_{w}}. This lattice could be highlighted in the hadron production accompanied by the production of the W, Z bosons:

p p → q q − + X + Z → l + l − + X

Moreover, the appearance of vector bosons in hadronic jets or the transformations in beta decay induce us to introduce a hypothesis of structure in the particles that is not however a hypothesis of composition in even more elementary particles but, considering that any field can be composed of coupled quantum oscillators, a hypothesis of particles as oscillators coupled with structure well-defined geometric. 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 → hadrons

How is it possible for photons to create quarks (or also lepton)? If we postulate that photons are the intermediaries of the EM force, we ask as it is possible that they can origin leptons or also quarks, particles with an “additional” agent charge, as the color charge. 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. Thus there would be a mechanism of topological transformations on geometrical structures that would thus be transformed the one into each other. This aspect has been met in conjecture of field H_{w} which transforms u-quark ó d-quark and d-quark ó u-quark. The H_{w} is a sort of Higg’s Field but with aspect of vector Boson W because transform the quarks the one into other. The decay time is very short (≈10^{−17} sec) and it’s an electromagnetic decay but not weak also if vectorial bosons (W) are present. As it’s easy understanding the long times in β-decay are due to existence of neutrino: the weak processes with “neutrinoless” are very more shorts.

Finally, in this paper a new paradigm (the particles’ geometrics structure) is introduced in the phenomenology of any interaction which opens up new perspectives for resolving the various problems of this. On this basis, we can discuss about “internal structure” of quarks [

To treat the particles as IQuO structures can explain the origin of some fundamental physics greatness as the electric charge, spin, isospin and color charge [

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

Guido, G. (2019) The Bare and Dressed Masses of Quarks in Pions via the of Quarks’ Geometric Model. Journal of High Energy Physics, Gravitation and Cosmology, 5, 1123-1149. https://doi.org/10.4236/jhepgc.2019.54065