The Quark Model for Leptons a Theoretical Basis for a Hadronic Structure of Leptons According to Quantum Chromodynamics Criteria

DOI: 10.36178/inv.phys.1103020

“INVESTIGATIONS” in Physics

The Quark Model for Leptons A Theoretical Basis for a Hadronic Structure of Leptons According to Quantum Chromodynamics Criteria

Mauro Santosuosso∗

Study Center for the Physical Investigation of Reality, V.le F. Cecconi, 17 - 00015 Monterotondo (RM), Italy April 3, 2021

Abstract Unlike preon models, in which it is hypothesized that the existence of constituents is more elementary than those we already know, a substantially different model is proposed here: The proposed model does not foresee new particles, but instead, focuses on a rather unconventional combination of the well-known quarks inside leptons. The idea arises from the observation of some “vacancies” in the baryon octet: All the triplets of quarks of the same flavor with total angular momentum J P = 1/2+ are denied by the Pauli exclusion principle. However, bypassing this prohibition because of the real possibility of introducing a form of “conditioned” symmetry, we arrive at a result – for down-type quark triplets only – in which two of the three quarks are strongly joined together in a di-quark, a two-color boson with an integer spin, which binds very closely to the third quark, together forming a charged colorless lepton. The neutral partners are given by the combination of the di-quark with the up-type quark, and with the related charged lepton, they form an isospin doublet. Thus, the di-quark boson becomes the carrier of the lepton number, which discriminates the three families. Studying the color forces according to the principles of quantum chromodynamics (QCD) shows how the quark – di-quark coupling is considerably higher than the quark – quark or quark – antiquark coupling, explaining why leptons today seem to be devoid of structure and unexpectedly justifying the smallness of their masses compared with those of the corresponding hadrons. Ultimately, lepton universality is considered. This manifests in weak in-

∗e-mail: [email protected]

teraction and decay processes. In this vein, two possible configurations are investigated, both of which are coherent with the quark mixing according to the Cabibbo – Kobayashi – Maskawa (CKM) matrix. As a conclusion, this work bridges the two classes of particles that structure the entire material universe – hadrons and leptons – taking steps toward realizing the much-desired theory of unification. Keywords : Baryon vacancy, conditioned symmetry, di-quark, hadronic leptons, two- color boson.

1 Introduction

The main motivation of a Grand Unified Theory (GUT) in physics is to reduce the known forces, which are apparently different from one another, to a single fundamental interaction. This can only be achieved by tracing the two main families of particles considered elementary – quarks and leptons – to a common denominator. One approach followed to achieve this goal has been to hypothesize the existence of more elementary constituents than quarks and leptons that make up their structure. From this perspective, we can understand all the so-called pre-quark or preon models, from the first one proposed [1, 2] up to the most recent ones [3–9], which have become current again after a decline in interest in superstring theory. We refer to this line of research if only to give a basis for comparison with what will be explained later; however, it is necessary to point out how different the spirit underlying this paper is from that which motivated allthe others. In fact, the departure could not be more different: Indeed, the model was not discovered following a study planned in view of the GUT, but instead, it was born from the observation of a curious “vacancy” at the baryon level – a particle that could have been there but that was not there – and from the author’s ignorance of the explanation that canonical physics attributes to this lack from the very beginning of quantum chromodynamics (QCD) [10–13]. This ignorance became a virtue when it allowed us to overcome an intrinsic limit to all preon models, that is, the view that the hypothetical subparticles must structure both quarks and leptons. It will be shown how the latter can instead be placed on the same plane as hadrons – as it is to be because they form stable atoms with them – while the well-known quarks structure both; the only difference is in the way in which they combine with each other in the two cases. Then, the canonical explanation, which was based on reasons of symmetry linked to the Pauli exclusion principle [14], falls because of a change of paradigm. The specific way in which quarks combine inside leptons, apparently not allowed by the aforementioned symmetry, proves to exist thanks to a different interpretation of this phenomenon. The remainder of the paper is structured as follows. Section 2 exposes the key idea of the model proposed here: It consists in the possibility of joining a pair of down-type quarks of the same generation into a two-color boson doublet stand for itself, by imposing a simple constraint condition on the exchange interaction between identical particles constituting the baryon vacancy. Section 3 is dedicated to the study of the color forces of this new member of the quark family in the specific case of the (d d) couple. This configuration resembles that of the Cooper electron pair in a superconducting fluid [15], and suggests the possibility that the interaction of the two-color boson with the third quark of the triplet is quite particular. This is what is described in Sections 3.1 and 3.2, in which the peculiarities of the gluon exchanges between the two-color boson and the down or up valence quarks

31 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 in the electron or electron neutrino configurations are analyzed in detail. Following this analysis, derived from considerations related to the confinement of color, the hypothesis arises that the great force of attraction between the two entities making up leptons can give rise to the extreme smallness of their masses compared with hadrons, as well as making “invisible” the internal structure of leptons such that they appear elementary when they are not. All this will require exactly identifying the symmetry group underlying the lepton model and verifying whether the running of the strong coupling constant confirms the known asymptotic trend observed in the hadronic case [16, 17]. Section 4 offers a brief overview of the main weak interactions between nucleons and leptons of the first generation, as well as the purely lepton collisions between the latter. Moreover, it highlights the particularity of the new model through the use of Feynman diagrams by identifying the two-color boson as responsible for the conservation of the lepton number. In Sections 5 and 6 the model is extended to the remaining two lepton families, those of mu and tau, and a double possibility of maintaining the universality of the weak interaction in decay processes is discussed according to the rules of the Cabibbo-Kobayashi- Maskawa (CKM) mixing matrix [18–20]. If the model is valid, only further studies and possible experimental results will be able to define which of the two paths reality has chosen. Finally, in Section 7, the article concludes with the proposal of some theoretical and experimental studies that could support the thesis discussed here.

2 Baryon vacancy

If we carefully observe the flavor combinations of quarks forming baryons, either those of the stable nucleons or those of the multiple unstable hadrons, we cannot escape that some of those with total angular momentum J = 1/2 (in ℏ units) are missing. In fact, no combinations with three quarks of the same flavor form baryons – neither stable nor unstable – while there are, as resonances, those with total angular momentum J = 3/2. For the moment, if we limit our consideration to the flavors u and d, what has just been said is equivalent to admitting that there is no trace of the baryons

(u u u)1/2 (d d d)1/2, whereas resonances

∆++ = (u u u)3/2 ∆− = (d d d)3/2 have been known since the early 1950s. After the theorization of the quark model by M. Gell-Mann and G. Zweig [10, 11], it was precisely the need to explain these resonances in light of the Pauli exclusion principle [14] that required the introduction of a new quantum number – that of color [12, 13]. The lack of combinations with J = 1/2 has since been attributed to the impossibility of realizing an overall antisymmetric wave function for the exchange of any pair of quarks, as required by the spin and statistic theorem in the precise

32 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 case of three identical fermions. Let us analyze the details of this assertion. The baryon wave function can be broken down into four components:

ΨB = φorb χsp ϕfl ψcol which represent the orbital, spin, flavor, and color parts, respectively. It can be reasonably assumed that the baryon is stable in the ground state, with orbital angular momentum L = 0; therefore, the φorb component will be symmetrical for the exchange of any two quarks. Evidently, the ϕfl component can only be symmetrical because the flavor of the three quarks is the same. Instead, the ψcol must necessarily be antisymmetric under all possible exchanges because only color singlets are observed in nature. It follows that 1 ψcol = √ (RGB + GBR + BRG − RBG − GRB − BGR), (1) 6 where R stands for red, B for blue and G for green. This forces the spin wave function to be symmetrical under the exchange of any two fermions. However, since it is possible to obtain a χsp that is only partially symmetrical – that is, symmetrical due to the exchange of only two quarks, but not of the third – with a total angular momentum J = 1/2, this leads to the impossibility of forming the baryons (u u u)1/2 and (d d d)1/2. The conclusion we have reached seems to leave no room for objection: It has repeatedly been confirmed by facts because none of the supposed particles has ever appeared in any scattering event among those that have occurred over the years in experiments with accelerators, nor has it materialized in the hadronic showers because of cosmic rays. Despite the prior evidence, nature may be in for a surprise: It may be using symmetry in a subtler way than we grant it, with permission or total prohibition. Indeed, by reversing the above argument, we can read the whole matter again from the following perspective: Nature allows the existence of systems composed of three quarks of the same flavor with J = 1/2 and a spin wave function χsp with mixed symmetry – that is, a configuration that is symmetric only for the exchange of two quarks of thetriplet – provided that the third quark has no exchange interaction with the other two. Instead of throwing the whole thing away, the necessary constraints are placed. Then, thanks to the symmetry that isolates the two partners of the doublet from the remaining quark, the couple will form a union so close as to become an entity in itself (a boson, as we shall see); however, this entity still be strongly linked to the third quark thanks to the color force. Thus, we have transformed what seemed to be an impossibility of principle into a conditional possibility (symmetry under a condition). It is now a question of discovering whether at least one of the aforementioned triplets can be identified with an already existing and known particle. If we consider the combination (d d d) with total charge Q = −1 and total angular momentum J = 1/2 under the guise of that missing baryon, we recognize a perfectly stable particle that we know very well because it has always existed in nature – the electron. Of course, our knowledge of physics does not seem to agree at all with this hypothesis because we know that the electron is a lepton, that is, a type of elementary particle that is very different from hadrons, devoid of internal structure, and unaware of the color force. However, it is a sign of perplexity that there are two distinct families of particles in nature – leptons and quarks – which structure the same material world without having a common origin. Therefore, our suggestion is to put aside the sense of revulsion for this only apparently risky hypothesis

33 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 for a moment and explore the consequences of this statement, leaving the conclusions to the end. We may find that they are not at all contradictory to the physics weknow, proving to be promising to obtain that small advance toward the much-desired theory of unification that has not yet been achieved. From what has been said above, the baryon triplet that presumably forms the electron cannot be of the same type as those that make up the internal structure of the two nucleons because symmetry does not allow such a configuration. Instead, it must bea configuration in which two of the three quarks d are in an orbital angular momentum with l = 0, a total spin integer, and wave functions ϕfl and ψcol which are symmetric and antisymmetric, respectively. The couple constitutes a new elementary entity that will be named a di-quark1 and to which, from now on, the symbol “d” is assigned: This will have a double color deriving from the fusion of the two primary colors of the single quarks – which is then the complementary color to that of the solitary d quark – and because it has an integer spin, it will be a boson. The result is a color – bicolor doublet (d d) which, unlike in the case of mesons, contains a two-color di-quark instead of the anticolor antiquark. Ultimately, the final particle is colorless and perfectly stable, with spin J = 1/2 and electric charge Q = −1 as the quantum numbers of the electron. The sketch in Fig. 1 gives an idea of the transformation.

Figure 1: The sketch illustrates the formation of the two-color di-quark d starting from a pair of down quarks of different colors. Finally, an electron is obtained.

This all seems to make sense, but another pressing problem needs to be solved imme- diately. In fact, we know that the fundamental family of leptons to which the electron belongs has a second partner, the electron neutrino νe. In light of what has been done for the electron, it is not difficult to guess the shape of its internal composition: Being neutral, it will have as a precursor a triplet of which the quark combination is the same as the neutron, that is, (u d d); then, as for the electron, the fusion of the two d quarks into a two-color di-quark will yield the compound (u d); the resultant scheme is shown in Fig. 2. Of course, the triplet (u d d) does not represent a real neutron: Transformation is not a decay; actually, such a decay would be impossible because neither the mass nor the momentum of the initial neutron could be preserved. (Later, we shall see the Feynman diagram relating to the beta decay of the neutron, and we shall understand that there are no contradictions).

1The term, of course, is not new (see ref. [21]), but the lepton context in which it is being used is original.

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Before we proceed, there are a couple of points worth making: 1) We know that electron and electron neutrino form a weak isospin doublet, like proton and neutron, with I = 1/2. The value I3 = +1/2 is assigned as the third component of the isospin to the neutrino instead of the electron I3 = −1/2. However, while the quark model is satisfactory in explaining this in the case of nucleons – the exchange of the u quark with that d transforms one particle into the other – in the case of leptons, it is assumed as a not-so-evident functional principle. The di-quark model elegantly intervenes to justify all this; in fact, by exchanging the up quark with the down one, we pass from the neutrino to the electron, and the choice of signs for the projection of the isospin is consistent in the two cases, as shown in the box below:   u ( )   u 1 p+ = u ν = I = + e 3 2 d d

  d ( )   d 1 n = u e− = I = − . 3 2 d d

2) According to the proposed model, for all intents and purposes, electrons and neutrinos can be considered hadrons; therefore, they can also be assigned a baryon number. The quark boson will logically have a baryon number equal to B = +2/3, and will lead to the unit value B = +1 for both particles. It is important to note that this generalization confirms the full conservation of B in leptonic and semi-leptonic processes, making the baryon number a universal property.

Figure 2: Formation of the electron neutrino because of the merging of a pair of down quarks in a d.

3 The properties of the di-quark

Let us now analyze the protagonist of this story, the two-color di-quark d. From the point of view of chromodynamics, it accomplishes an anti-triplet state, as its color wave function is antisymmetric. In fact, within the SU(3) symmetry of color, the two identical

35 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 quarks belong to the representation 3 ⊗ 3, which is factored as

3 ⊗ 3 = 3 ⊕ 6, where the first term to the right of the equals sign is precisely the antisymmetric triplet and the second is the symmetric sextet. Only the first combination produces a negative cF color factor, leading to an attraction of the two d quarks of different colors; the sextet is instead repulsive, and therefore, it does not give rise to stable states. Let us see this in detail. Given the following wave function: 1 ψcol = √ (RB − BR), 2 the color factors for the two diagrams 1 and 2 of Fig. 3 – describing the possibility of exchange between quarks because of their indistinguishability – are the same for both. Writing this point in reference to the first, we have 1 [ ] [ ] c = c(3)†λαc(1) · c(4)†λαc(2) , F 4 which implies the sum over the eight Gell-Mann matrices λα. Its value is equal to 2 c = − F 3 and leads to an attractive short-distance potential with the following form: 2 α V = − S , d 3 r where αS is the strong coupling constant given by g2 α = S S 4π in terms of the color charge g S. According to the Pauli principle, the wave function Ψd = φorb χsp ϕfl ψcol of the di-quark d must be antisymmetric overall. Since we found that ψcol is antisymmetric, while ϕfl and φorb are both symmetric – the former because the two quarks have the same flavor, the second because they will most likely beina zero orbital angular momentum – it follows that the spin wave function χsp must also be symmetric, and this leads to the triplet S = 1. Therefore, the di-quark d is a two-color boson with spin equal to 1, with electric charge q = −2/3 and parity P = +1. We can speak of a boson, that is, an independent entity, precisely because of the isolation discussed in the previous paragraph because of the stringent demands of symmetry. The elementary particle containing the di-quark realizes the color singlet by means of the third quark – d for the electron and u for the neutrino – the color of which must necessarily complement that of the d boson (In this regard we shall introduce three new symbols that make the writing synthetic, specifically, G¨ = red-blue, R¨ = blue-green and B¨ = red-green; the umlaut on the letters indicates that the color is double). In the case of antimatter, all the arguments made so far can be repeated by replacing the d quarks with their respective antiquarks: The progenitor baryon vacancy is (d¯ d¯ d¯)1/2, and the di-quark

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Figure 3: Gluon interaction between two different colored d quarks contemplating diagrams 1 and 2. d¯ that is formed is an antiboson having spin S = 1, electric charge q = +2/3 and parity P = +1, which is the same as for the boson. Up to now, we have talked only about d, and we have not wondered if there is also the analogous combination that derives from the hypothetical baryon vacancy (u u u)1/2. To consider this, it is necessary to make the following observation: The possibility of the existence of the di-quark is strongly conditioned by the existence of a lepton that has the right electric charge of the vacancy in question, but unlike the case of the triplet (d d d)1/2 – which has exactly the same charge as the electron – there is no known stable lepton with a charge of Q = +2; thus, we must accept that the di-quark “uu” does not form in nature. A reason for this could lie in the following argument: Since the quark u has a double electric charge of d, the combination uu is disadvantaged compared to that d because the electrostatic repulsion between the two would have a value four times superior to the other, ending by nullifying the attraction because of the color charge at short distances. However, this argument is only assertive and has no claim to truth; there could be much deeper reasons than this that forbid the existence of uu that we cannot grasp at this stage of the theory’s embryonic development. It is remarkable that the non-existence of the doublet uu is supported by two other causes, the first of which is linked to isospin and the second to the color force, described as follows: 1) Logically, the di-quark d should be the component I3 = −1 of an isospin triplet I = 1, which also provides for the combination uu with projection along the third axis equal to 1/2 I3 = +1. This is analogous to admitting that a baryon with the combination (d uu) could exist, that is, a light proton that is very similar to the electron. However, this baryon does not exist (nor can the hypothesis be followed that it could be the positron because the constituent quarks are all particles and not the corresponding antiparticles). 2) We must accept the impossibility of obtaining a neutral lepton with the di-quark uu;

37 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 the two only solutions would in fact be given by combining it with two d or one d and two d quarks. However, both must be discarded because they conflict with QCD rules, favoring repulsion and not cohesion, and introducing a chimera in the model.

3.1 The shape of the electron We now want to try to characterize the color force that holds quark and di-quark together inside the electron, and if possible, give an estimate of it. Therefore, we return to the color wave function (1) of a generic baryon and rewrite it by highlighting the anti-triplets representing the two-color di-quarks d. Thus, we obtain ( ) 1 1 1 1 ψcol = √ G · √ [RB − BR] + B · √ [RG − GR] + R · √ [BG − GB] , 3 2 2 2 which can be simplified – using the three new symbols introduced above –as

1 ¨ ¨ ¨ ψcol = √ (GG + BB + RR). (2) 3

This looks a lot like the ψcol of a meson,

1 ¯ ¯ ¯ ψcol = √ (GG + BB + RR) 3 if we match G¯ → G,¨ B¯ → B¨, and R¯ → R¨; however, despite the similarity, G,¨ B¨, and R¨ bosons behave quite differently from the antiquarks because they are made up ofmatter and have a double color instead of an anticolor. In all probability, the force with which they attract the fermionic partner is also double compared to that released by an antiquark constituting the meson. Then, we can hypothesize an interaction between them like the one shown in Fig. 4.

Figure 4: Scattering event quark – di-quark for the couple BB¨.

The gluon in the above picture is only a symbol that does not currently have a coun- terpart in QCD; to characterize it, one would have to introduce an additional component to the SU(3) color symmetry group: As it stands, it cannot explain the force that holds together d or u quarks with d. Indeed, it does not contemplate the direct exchange of

38 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 gluons that couple a color particle with a two-color one. However, we can exploit the basic insight schematized in the figure in trying to reconstruct the process in the context of conventional QCD and then calculate the color factor that belongs to it. If nothing else, this will give a rough idea of the difference that there is between hadrons and leptons and which makes the latter apparently unrelated to the former, to the point that physics has built two distinct worlds for them that communicate only through electroweak interaction. Let us see how the model proposed here performs in solving this hiatus. First, we hypothesize that the coupling is due to the simultaneous exchange, between boson and fermion, of two closely related gluons. In addition to the direct process, there is also a second one with the vertices of the gluons inverted. The respective Feynman diagrams 1 and 2 are shown in Fig. 5 for a color configuration chosen at random among those possible. The Feynman amplitude M1 of process 1, which contemplates different color configurations, is written as ( ) ( ) [ ] † g i g 1 M = i[¯u(3)c −i S λbγν −i S λaγµ c u(1)] · − 1 i 2 k/ − m 2 l q4 ( ) ( ) † g † g × [¯u(4)c −i S λaγµ c u(2)] · [¯u(6)c −i S λbγν c u(5)], j 2 m k 2 n

b ν a µ and analogously, in M2, the factor λ γ is swapped with λ γ ; the sum over the repeated indices is implied. The values from 1 to 6 name the four-momentum p1 ÷ p6 of the respective outer fermionic lines, while we expressly require that the four momentum of the two gluons be equal to each other and equal to q : We assume that, to remain intact during the interaction with the isolated quark, the boson d must simultaneously exchange gluon packets of the same intensity for both colors of which it is composed. To satisfy the conservation laws, we must impose the following constraints on the energy-momentum four vectors:

k = p1 − q

p3 = p1 − 2q

p4 + p6 = p2 + p5 + 2q.

′ ′ The last relation can be written as p = p + 2q, where p = p4 + p6 and p = p2 + p5. This highlights that the boson di-quark moves as an entity in itself. Finally, the indices i, j, k, l, m, n = 1, 2, 3, subject to the constraints i ≠ j ≠ k and l ≠ m ≠ n, distinguish the color components c, which take, as usual, the following values:       1 0 0       c1 = 0 c2 = 1 c3 = 0 . 0 0 1

After a few simple algebraic steps with gamma matrices, the total Feynman amplitude M = M1 + M2 is written, as g4 1 1 M = S [¯u(3)γµ γνu(1)] · [¯u(4)γ u(2)] 16 q4 k/ − m µ × · †{ a b} · † a · † b [¯u(6)γνu(5)] [ci λ , λ cl] [cjλ cm] [ckλ cn].

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Figure 5: Two-gluon scattering event. The contribution of the first Feynman diagram must be added together with that of the second. The four momenta of the two gluon propagators are identical in both configurations.

The presence should be noted of the anti-commutator of Gell-Mann lambda matrices, which is factored into one term proportional to the identity matrix and in another proportional to the completely symmetric structure tensor dabc. Therefore, the color factor is equal to ( ) 1 ∑8 4 − † c · † a · † b cF = [ci δabI + 2 dabcλ cl] [cjλ cm] [ckλ cn]. 16 a,b,c=1 3 To calculate this formula, it is necessary to define the initial and final color states of the quarks entering the scattering process. We distinguish two possible situations: The first admits the existence of only the diagonal terms of the development – thatis,those that keep the same color between the initial and final states of the process. In this case, we speak of a strong constraint, because there is no exchange among the terms GG,¨ BB¨ and RR¨ of wave function (2). The second possibility provides that the three color configurations can be mixed together; this makes the process more similar tothe meson one. In the latter case, we speak of a weak constraint because the absence of exchange interaction between quark and di-quark, that is, the absence of exchange of

40 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 physical coordinates between them, does not necessarily prohibit the mixing of color between analogous components of the different doublets. As in this second⟨ case,⟩ itis necessary to consider nine possible configurations instead of just three, namely GG¨|GG¨ , ⟨ ⟩ ⟨ ⟩ GG¨|BB¨ , ··· RR¨|RR¨ , the color factor will be different from that of the first case.To obtain the desired result, one can simply calculate⟨ one of⟩ each of⟨ the three⟩ diagonal color factors and the six mixed ones, for example, GG¨|GG¨ and GG¨|BB¨ , because each member of same family will have equal values. Then,    [ ] (⟨ ⟩) 0 1  4 2   c GG¨|GG¨ = − (001) I − √ λ8 0 F 16 3 3 1     0 1     1 ×(010)λ8 1 · (100)λ8 0 = − 18 0 0 is obtained for the first case, and (⟨ ⟩) 1 † † † c GG¨|BB¨ = − {[c λ6c ] · [c λ4c ] · [c λ1c ] F 16 2 3 3 1 1 2 † 7 · † 5 · † 1 + [c2λ c3] [c3λ c1] [c1λ c2] − † 7 · † 4 · † 2 [c2λ c3] [c3λ c1] [c1λ c2] † † † 1 + [c λ6c ] · [c λ5c ] · [c λ2c ]} = − 2 3 3 1 1 2 4 S − is obtained for the second. In conclusion, we shall have cF = 1/18 in the case of strong W − constraint (S = strong) and cF = 5/9, in the case of weak constraint (W = weak), where the value of the latter is 10 times higher than the that of the former. We also need to consider another process of the second order along with the one just seen, that is, the process constituted by the interaction of three gluons, at the ends of which, there are the fermionic vertices of the three quarks of different colors. The corresponding Feynman diagram is shown in Fig. 6, and the relative amplitude M is written as ( ) [ ] [ ] † g i 1 M = i[¯u(3)c −i S λaγµ c u(1)] · (−g f abcV ) · − · − i 2 l S µνρ k2 q4 ( ) ( ) † g † g × [¯u(4)c −i S λbγν c u(2)] · [¯u(6)c −i S λcγρ c u(5)]. j 2 m k 2 n However, if we carry out the same calculation performed on the diagonal components, we find a result that is identically null. This derives from the tensor of the structure constants abc of the SU(3)C group, f , since the latter is completely antisymmetric; indeed, when two of its indices are equal to each other, the resulting value is zero. However, we need at least two of the indices to be both equal to 3 or 8 to satisfy the assumed color configuration, and this always leads to null values. For the mixed components, the situation changes, but the final result is identical – that is, zero. It is shown that, next to the six components to be considered, there are as many in which the colors of the two quarks of the d boson are exchanged, producing values equal and opposite to those of the first six. The sum still gives zero.

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Figure 6: Three gluon quark – di-quark scattering event inside the electron.

Let us now evaluate the consequences of the calculation just made on the short-distance potential Vd-d that hypothetically holds the electron together. Both in the case of strong and weak constraints, the resulting force is attractive, and a result is obtained that is proportional to the square of the coupling constant, in that α2 V = −C S , d-d r C − S C − W with = 4πcF for the strong bond and = 4πcF for the weak one, which produces a result ten times greater than the first. Unfortunately, without any experimental data regarding the internal structure of the electron in terms of energy, it is not possible to distinguish between these two alternatives. However, we know that the strong coupling 2 constant αS(Q ) grows rapidly as the transferred four-momentum decreases and is already close to unity around 200 MeV. Therefore, at the typical four-momentum equivalent to 2 the electron mass, which is well below that value, it can be assumed that αS(Q ) is of the order of 10. Entering the coupling constant with twice the hadronic power in the potential formula will produce a result that could reasonably be a few orders of magnitude greater than usual, leading to a bond of the constituents of the internal structure of the lepton, which is strong enough not to have managed to break it so far. The same conclusion can also be reached by means of the purely deductive argument given below. The confinement of quarks in hadrons – even if still unproven analytically – can be traced back to the color force if only one is convinced of the fact that the inevitable simultaneous presence of its three representatives is itself the key. (This is also true in the case of mesons in which the anticolor – that is, the lack of the positive color – is equivalent to the simultaneous presence of the other two that make the particle colorless). If we now assume that two quarks of different colors are closely united (as happens inthe di-quark), the third color will be attracted by the resultant of them with much greater force than it is by the single separate components. Then, in the electron model proposed

42 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 here, we can hypothesize that its parts are bound together by a static color force that is far more intense than that which binds a quark to another quark or an antiquark. In fact, it happens that, in a process of scattering between high-energy baryons, a single quark is moved so far from its center that the other two are seen as a single bicolor entity; the static force we have talked about, similar to an elastic force acting at a long distance, manifests itself by opposing this. However, if we are in a low-energy regime, that force, unlike what happens for a hadron, is always there in the electron. To understand whether this force decreases as the momentum increases, it would be necessary to identify exactly the symmetry group of the gluon gauge field in question and then calculate all the corrective contributions resulting from the gluon and fermion loops in the various orders, verifying the so-called asymptotic freedom [16, 17]. The running of the strong coupling constant, which is linked precisely to this point, could have a different or even inverse trend to the known one, at least as long as the di-quark exists as such. Whatever the case may be, it is precisely the high quark – di-quark bond value inherent in the model that would justify a behavior so different from that of the hadron that itis believed to have no internal structure. Accelerators like LEP or SLAC have probably not exceeded the threshold that would have allowed the identification of the single constituents of the compound. This could also explain the peculiarity of the small value of the electron mass compared with that of the proton. The strong binding energy of the pair (d d) would produce a potential hole (an energy gap) with a marked minimum, creating a very stable state of equilibrium. This would make the mass of the two elementary components – quark and di-quark – almost zero, explaining why the equivalent mass of the electron seems to derive only from the energy of its electrostatic field. From the point of view of dynamics, the energy gap could indicate the establishment of a particular state of “superfluid” motion of the d boson, which does not admit any interference with the fermion partner as long as this motion persists. In contrast, the bond between the two remains very strong because of the confinement of color. It is the analog of what happens in a superconducting metal below the critical temperature: Cooper pairs [15] can move like bosons in macroscopic orbits without the slightest friction with the lattice (without any interaction), but this does not mean that these constituents are somehow separable from the rest of the metal; they are a part of the integral. Thanks to the superfluid state, the two partners would always be in a state of perfect s-orbital with zero angular momentum, confirming the measurement of the electron’s sphericity: It has recently been tested [22] up to a length of the order of 10−29 centimeters. Eventually, since we assume a hadronic origin for the electron, it is necessary to evaluate the following statements. Electrons, unlike baryons, do not produce hadronization in inelastic collisions, nor can they produce nucleation as protons and neutrons do. Why? Let us first answer the second question: Neutrinos, which should perform an action similar to that of neutrons in favoring the nucleation process, always escape at the speed of light; they are unstoppable, and therefore, the equivalent cannot be formed of deuterium or tritium. If anything, the mutual approach of two electrons is disadvantaged, either because the attraction of the pair (d d) is much more solid than that of the single quarks in the nucleons or because the pion, appointed to bind the two with the strong force, would be so much more massive than the electrons as to frustrate the enterprise. However, in terms of hadronization, it is assumed that it can occur in Møller scattering at very high energies, but this has not yet been investigated.

43 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021

3.2 The shape of the electron neutrino After understanding how the hadronic model is applied to the electron, let us turn to understanding the differences with respect to its partner, the electron neutrino; in fact,all the arguments made for one are transferable to the other. In particular, the color factors calculated in the strong and weak constraint regimes are the same from the moment that no reference to the flavor of the single quark is made in the formulas, and therefore, it can be either d or u. Regarding the attractive potential Vu-d, while similar to the other in terms of color force, it is assumed to be different because of the electromagnetic component of quarks that compose neutrinos; we shall clarify this point in a moment. First, however, there is a question that has so far been overlooked. The combination (u d) seems to cause perplexity about the symmetry constraints that allowed us to derive the one (d d) corresponding to it. Since the flavor of the u quark is different from that ofthetwod that form the di-quark, these constraints would no longer have reason to exist because the Pauli exclusion principle does not apply to it. Indeed, the combination (u d d)1/2 is not a baryon vacancy at all but exists as a neutron. However, quantum mechanics has accustomed us to the fact that if there is a probability that a physical state exists, then that state will manifest. The state of di-quark is a possible state that occurs in the electron as long as the free down quark remains extraneous to the physical exchange with the other two, but then it will also manifest – and even more so – in the neutrino, in which u already naturally satisfies the constraint because it has no exchange component with them. If the di-quark is indeed in a superfluid state, it is insensitive to excitations. Clearly, there is a big difference between this configuration and the (u d d) configuration of a neutron; in the latter, all three pairs of the antitriplet have the same probability of forming. This leads to a particle with a mass significantly greater than that of a neutrino because of the high energy of interaction between its constituent quarks.

Figure 7: Total angular momentum J of a left-handed electron neutrino resulting from the combination of spins S and s with orbital angular momentum l.

Let us now deal with the electromagnetic problem left open a moment ago: the peculiarity of the neutrino with respect to the electron lies in the fact that, in it, the electric charges +2/3 of the u quark and −2/3 of the di-quark d cancel each other out exactly, and so do the respective magnetic moments, if, as it is plausible to assume, the fermion and boson masses both tend to zero; therefore, the neutrino completely loses the ability to interact with matter in electromagnetic terms. The weak force remains, while the strong force cannot go beyond the limits of its size. Since the potential hole with which the color force attracts the boson and fermion inside νe is even deeper than in an electron – there is

44 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 now electromagnetic attraction between the components – the mass of the entire particle virtually disappears and the electron neutrino is forced to travel at the speed of light. (Of course, this is not exactly how things are because a mass, albeit a small one, is attributed to it after the experiments of flavor oscillation). In Fig. 7, the composition scheme of the orbital angular momentum l with the spins S of the boson and s of the fermion is shown to give J = 1/2 in the direction opposite to that of the motion of the left-handed neutrino. Since its motion is ultra-relativistic, the orbit of the pair of particles in the ground state is really that of Bohr, with an angular momentum equal to one. The direction of velocity might be opposite to the one chosen, because there is no known reason to deny it; however, in nature, this case does not occur: As we know, there are only left-handed neutrinos. However, we can define a neutrino parity by remembering that the di-quark has Pd = +1; this is given by l P = Pd · Pu · (−1) = −1 and that of the antineutrino is opposite and equal to +1.

4 Nucleon - lepton weak interaction and lepton scattering processes

Let us now turn to verify that the proposed model is consistent with what physics already affirms regarding the weak interaction processes between nucleons and leptons, as well as with the decay of the neutron and the electron capture by the proton. We refer + − + + to Fig. 8 relating to the processes nνe ⇄ p e and ne ⇄ p ν¯e: As can be seen from the Feynman diagrams illustrated in it, the two processes are of a weak type and can go in both directions, from right to left or from left to right; for this reason, vector boson W− is shown without an explicit direction.

Figure 8: First-order scattering processes with the exchange of a vector boson W−.

− + The interesting thing, as mentioned above, is that the pair νee and e ν¯e are weak isospin doublets, the components of which are transformed into one another because of the medi- ation of the vector boson W. Indeed, d and u reverse each other in the first combination,

45 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 while −d¯ and u¯ do so in the second. (The negative sign of the antiquark down is needed because we must use the conjugate representation 2∗ of SU(2)). What can be deduced from the diagrams is just what has already been anticipated at the end of Section 2; that is, between the pairs of leptons, a transformation of flavor of the quarks takes place similar to that which occurs in the nucleons. Therefore, the hadronic model presented here provides a deeper reason for why the electron and neutrino form an isospin doublet compared with the standard model (SM), which does not provide any in reality.

Figure 9: Neutron decay.

− Let us now consider neutron decay, as shown in Fig. 9: The pair e ν¯e created together with the proton because of the decay is made up of the following elements as a whole: (d u¯ d d¯). We can then reflect on this. Suppose we hypothesize a decay process like the following: n → pπ− (This clearly exists only when neutron is energized because of a collision with a heavy particle, for example, a proton). The negative pion is formed by the quark – antiquark pair (du),¯ and the energy required to obtain it is ∼ 110 times higher than that released by the natural decay of the neutron. However, if the model is to be justified, it must include the energy of the pairdu,¯ as well as the energy of the duo boson – antiboson (d d¯). Despite this, it is far inferior to the other, confirming that the quark – di-quark lepton configuration achieves a much more strongly bound state than does π− meson, reducing the energy of the system until decay into electron and antineutrino becomes possible. To conclude this section, let us turn to the electron – antineutrino scattering processes in its two main variants (see Fig. 10). Analogous Feynman diagrams are valid for electron – neutrino scattering and for the corresponding positron – neutrino and positron– antineutrino. As can be seen from the pictorial representations, the neutral vector boson does not change the color of the subparticles between final states, while the charged one does. This reveals how each lepton can be in an overlap of the three possible color – bicolor combinations through the weak force, and therefore, through SU(2) symmetry.

46 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021

Figure 10: Electron – antineutrino scattering with neutral and charged current exchange.

5 Conservation of the lepton number: quark structure of mu, tau, and their respective neutrinos

What defines the lepton number and determines its conservation in the scattering processes or in the decay processes because of the weak force? The di-quark model answers this question well, recognizing precisely the author of this conservation law in the ¯ d boson: If we assign the value Le = 1 to it (Le = −1 in the case of antiparticle d), we find the law intact as the electron and electron neutrino that contain it both have a positive lepton number, while the positron and antineutrino, which contain the antiboson, have a negative lepton number. The simultaneous presence of boson and antiboson produces a null lepton number, as in the neutron decay seen above. All this is almost trivial. However, we cannot stop at the sole explanation of the pair of stable leptons in terms of subparticles; if we want to reach the goal we have set, to discover the substructure common to all leptons, we must try to solve the nature of mu, tau, and their respective neutrino partners according to the proposed model. However, since the conservation of the muon and tauon numbers proceed similarly to the electron numbers, this is enough to put us on the right track. If the di-quark d is the entity that gives specificity to the first of the three lepton families – that is to say, the least massive–it will be necessary that there are two bosons that similar to it, but different at the same time, as well as heavier – which give specificity to the remaining two generations. The choice seems almost obligatory since there are only two quarks with an electric charge q = −1/3, in addition to d which can form the aforementioned boson state: The combinations will be (ss) for the muon family and (bb) for the tauon one, and the symbols indicating the two- color di-quarks will henceforth be assumed to be “s” and “b” respectively. This is reflected in that there are two baryon vacancies, (s s s)1/2 and (b b b)1/2, that can be considered the precursors of µ− and τ −: From these vacancies, just as happened for the electron, the two lepton combinations derive (s s) and (b b); to construct the corresponding neutrinos, we replace the isolated s and b quarks with u. The summary diagram of the quark structure

47 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 of the three lepton families is given below: ( ) ( ) ( ) u u u ν = ν = ν = e d µ s τ b

( ) ( ) ( ) d s b e− = µ− = τ − = . d s b

The respective antiparticles of each of these six entities are formed with antiquarks and antibosons. The same rule used for the lepton number Le can now be transposed to Lµ and Lτ and apply, as we shall see shortly, to the muon and tauon decay processes that confirm their conservation. It should be remembered that the conservation principle of the lepton number does not find justice in SM, since it requires the existence of three different conserved chargesand the simultaneous presence of three distinct gauge fields, those of γe, γµ and γτ . There is no trace of any of these! The di-quark configuration overcomes this difficulty by identifying the conserved “charges” in the three different two-color bosons; in contrast, the three gauge fields will probably be sought among the gluon fields whose confinement doesnot allow direct vision today. However, as elucidated in the next section, we shall see that it is not possible to leave the combinations between quarks and di-quarks summarized in the above proposed box as they are; they will have to be modified appropriately if the model is to maintain the lepton universality, a universality that has been proven to date with a high degree of precision in the experimental field.

6 Lepton decays: Universality and the CKM matrix

Let us consider muon decay (see Fig. 11) and verify that the known conservation laws are satisfied. Now that we have reduced mu to a hadron and ascertained thatits decay – mediated by W− – transforms s quark into u, we wonder how it is possible to maintain the lepton universality [18, 19] because we know that this transformation is strongly suppressed in relation to that from d to u. Instead of the Fermi coupling constant GF typical of this decay, we should find a much smaller one, for example, the one that 0 belongs to the process Λ → peν¯e which is analogous to the first regarding the exchange of quarks (see Fig. 11). In light of the new model, it is no longer possible to hide behind the elementarity of leptons without structure, accepting that the value of the coupling constant is an intrinsic fact of their physical nature, and therefore, inexplicable: They too are now made of the same quarks of baryons! At this point, at least two paths can be followed, both of which seem to be coherent: In the event of the correctness of the model, only future experiments will be able to decide which of the two might be the way that nature follows. Let us consider them separately. The first path is to admit that the mixing matrix CKM is also active for leptons, and that the single quarks that enter the constitution of tau, mu, and electron are in a superposition of the three flavor states according to this matrix. At least for the moment, we hypothesize that the di-quark bosons do not participate directly in this mixing; this seems plausible because they are carriers of the lepton number, which is kept separately

48 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 for the three different families. Then, the final scheme of the lepton structure isasgiven below: ( ) ( ) ( ) u c t ν = ν = ν = e d µ s τ b

( ) ( ) ( ) d′ s′ b′ e− = µ− = τ − = . d s b Here, the weak eigenstates are conventionally given in terms of the mass eigenstates by     d′ d  ′    s  = VCKM s , b′ b and VCKM is precisely the CKM matrix. Accepting the challenge of neutrinos that seem to have become too massive – but mass is a state of affairs of which the meaning still eludes us – we make a choice that seems risky, but which is in fact the most consistent with the SM and agrees with the descent of the three charged leptons from their respective baryon vacancies. As far as the electron is concerned, even if it seems difficult to admit that its down quark is mixed with the s and b quarks of the higher generations – such a mixing seems to create instability – it should be emphasized that it is not possible to do otherwise. Only in this way is the GIM mechanism [20] safe, and in the processes of elastic scattering between electron and mu or electron and tau, the exchange component because of neutral currents with the flavor change is canceled.

Figure 11: Muon decay compared with the analogous decay of the hyperon Λ0.

The second path considers that, in nature, there are no different types of charged leptons of the same generation that can be physically distinguished; for example, there is only one mu, not “variants” of the same one – as happens for n and Λ0 baryons, where 0 decays n → peν¯e and Λ → peν¯e are experimentally distinguishable. In light of this, it can also be assumed that quark mixing for leptons is “frozen” and does not happen at all. Turning to the example of mu is like saying that of the three allowed combinations,

49 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021

(d s), (s s) and (b s) only the first really exists as a particle, while the last two donot. The scheme of the structure of the three generations would ultimately be resolved as ( ) ( ) ( ) u u u ν = ν = ν = e d µ s τ b

( ) ( ) ( ) d d d e− = µ− = τ − = . d s b

Lepton universality would be safe because there would be no alternative decay channels: The W± bosons would always and only exchange the flavors of the u and d quarks with each other. The decays of tau (see Fig. 12) show the coherence of this system and its intuitive simplicity; we can see how the conservation of the lepton numbers Le, Lµ and Lτ is due to that of the three di-quarks d, s and b respectively.

Figure 12: Tau decay according to electron and muon channels

7 Conclusions

The work shown in these pages is a harbinger of proposals for subsequent studies, both in the experimental and theoretical fields. From the theoretical perspective, it suggests the following points: 1) The search for a completion of color SU(3) with a gauge group or subgroup that contemplates the interactions color – bicolor (the adjoint representation 3 of SU(2) could fit); 2) Computation of the perturbative terms at the various orders linked to the strong cou- 2 pling constant αS(Q ) of the aforementioned interaction to verify the asymptotic freedom in the explicit case of leptons; and 3) The launch of a radical investigation into the ultimate nature of quarks and color charges, which finally sheds light on the relationship between them and the spacetime in which they emerge, that actually substantiates them. Beyond this threshold, in fact, the much misunderstood meaning of elementary particle is hidden.

50 M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021

In terms of the experimental perspective, it must be admitted that the search for pre-quarks – that is, for those elementary components that are supposed to constitute the internal structure of the quarks – directs particle physics, in our opinion, toward a path that is already far too worn. Indeed, to confirm whether quarks are composed of preons, it induces physicists to build particle accelerators that typically generate hadronic collisions (pp or pp). If we accept the thesis of this paper instead, we agree that the lepton structure is what is to be investigated, whereas quarks are fundamental entities. Thus, the experimental proposal that derives from this work is to address the construction of electron or muon colliding beam accelerators that should work at energies well above those reached by previous accelerators (LEP, SLAC, etc.) to investigate whether there exists an underlying di-quark structure.

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