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 con- stituents 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 mo- mentum 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 cou- pling 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] Copyright © the Author 2021. Published by SCEPHIR. M. Santosuosso The Quark Model for Leptons VoL. 1, No. 2, (30-52) 2021 teraction and decay processes. In this vein, two possible configurations are in- vestigated, 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 con- sidered 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 perspec- tive, 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 chromo- dynamics (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 lep- tons, 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 ~ 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.