{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T05:32:18Z","timestamp":1768973538763,"version":"3.49.0"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T00:00:00Z","timestamp":1715644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"INFN within the Iniziativa Specifica QG-sky","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"INFN within the Iniziativa Specifica QG-sky","award":["PRIN 2022BP52A MUR"],"award-info":[{"award-number":["PRIN 2022BP52A MUR"]}]},{"name":"the European Union, Next Generation EU, PNRR project \u201cNational Centre for HPC, Big Data and Quantum Computing\u201d","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"the European Union, Next Generation EU, PNRR project \u201cNational Centre for HPC, Big Data and Quantum Computing\u201d","award":["PRIN 2022BP52A MUR"],"award-info":[{"award-number":["PRIN 2022BP52A MUR"]}]},{"name":"\u201cThe Holographic Universe for all Lambdas\u201d Lecce-Naples","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"\u201cThe Holographic Universe for all Lambdas\u201d Lecce-Naples","award":["PRIN 2022BP52A MUR"],"award-info":[{"award-number":["PRIN 2022BP52A MUR"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>One of the most important and unanswered problems in particle physics is the origin of the three generations of quarks and leptons. The Standard Model does not provide any hint regarding its sequential charge assignments, which remain a fundamental mystery of Nature. One possible solution of the puzzle is to look for charge assignments, in a given gauge theory, that are inter-generational, by employing the cancellation of the gravitational and gauge anomalies horizontally. The 331 model, based on an SU(3)C\u00d7SU(3)L\u00d7U(1)X does this in an economical way and defines a possible extension of the Standard Model, where the number of families has necessarily to be three. We review the model in Pisano, Pleitez, and Frampton\u2019s formulation, which predicts the existence of bileptons. Another characteristics of the model is to unify the SU(3)C\u00d7SU(2)L\u00d7U(1)X into the 331 symmetry at a scale that is in the TeV range. Expressions of the scalar mass eigenstates and of the renormalization group equations of the model are also presented.<\/jats:p>","DOI":"10.3390\/e26050420","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T10:26:36Z","timestamp":1715682396000},"page":"420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The SU(3)C \u00d7 SU(3)L \u00d7 U(1)X (331) Model: Addressing the Fermion Families Problem within Horizontal Anomalies Cancellation"],"prefix":"10.3390","volume":"26","author":[{"given":"Claudio","family":"Corian\u00f2","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica e Fisica, Universit\u00e0 del Salento and INFN Sezione di Lecce, National Center for HPC, Big Data and Quantum Computing, Via Arnesano, 73100 Lecce, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1084-8649","authenticated-orcid":false,"given":"Dario","family":"Melle","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Fisica, Universit\u00e0 del Salento and INFN Sezione di Lecce, National Center for HPC, Big Data and Quantum Computing, Via Arnesano, 73100 Lecce, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2889","DOI":"10.1103\/PhysRevLett.69.2889","article-title":"Chiral dilepton model and the flavor question","volume":"69","author":"Frampton","year":"1992","journal-title":"Phys. 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