{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:10:17Z","timestamp":1760058617586,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T00:00:00Z","timestamp":1744675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Bulgarian Ministry of Education and Science"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this work is to propose an explanation of the inverse mass hierarchy of the low-lying nonet of the scalar mesons in the framework of the massless Nambu\u2013Jona-Lasinio UR(3)\u00d7UL(3) quark model. The proposed explanation is based on symmetry principles. The collective meson states are described via quark\u2013antiquark pairs, whose condensates lead simultaneously to spontaneous breaking of chiral and flavour symmetry. It is shown that, due to flavour symmetry breaking, two iso-doublets of K0*(700) mesons play the role of Goldstone bosons. It is also proven that there exists a solution with degenerate masses of the a0(980) and f0(980) mesons and a zero mass of the f0(500) meson.<\/jats:p>","DOI":"10.3390\/sym17040600","type":"journal-article","created":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T10:49:54Z","timestamp":1744714194000},"page":"600","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Explanation of the Mass Pattern of the Low-Lying Scalar Nonet"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5841-3316","authenticated-orcid":false,"given":"Mihail","family":"Chizhov","sequence":"first","affiliation":[{"name":"Faculty of Physics, Sofia University \u201cSt. Kliment Ohridski\u201d, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria"}]},{"given":"Emanuil","family":"Chizhov","sequence":"additional","affiliation":[{"name":"Institute of Astronomy with NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria"}]},{"given":"Momchil","family":"Naydenov","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Sofia University \u201cSt. Kliment Ohridski\u201d, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4628-4195","authenticated-orcid":false,"given":"Daniela","family":"Kirilova","sequence":"additional","affiliation":[{"name":"Institute of Astronomy with NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,15]]},"reference":[{"key":"ref_1","first-page":"41","article-title":"A New method in the theory of superconductivity. I","volume":"34","author":"Bogoliubov","year":"1958","journal-title":"Sov. Phys. JETP"},{"key":"ref_2","first-page":"51","article-title":"A New method in the theory of superconductivity. III","volume":"34","author":"Bogoliubov","year":"1958","journal-title":"Sov. Phys. JETP"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1103\/PhysRev.117.648","article-title":"Quasiparticles and Gauge Invariance in the Theory of Superconductivity","volume":"117","author":"Nambu","year":"1960","journal-title":"Phys. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/BF02812722","article-title":"Field Theories with Superconductor Solutions","volume":"19","author":"Goldstone","year":"1961","journal-title":"Nuovo Cimento"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1103\/PhysRev.127.965","article-title":"Broken Symmetries","volume":"127","author":"Goldstone","year":"1962","journal-title":"Phys. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1103\/PhysRevD.15.267","article-title":"Multi-Quark Hadrons. I. The Phenomenology of Q2Q\u00af2 Mesons","volume":"15","author":"Jaffe","year":"1977","journal-title":"Phys. Rev. D"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"053D02","DOI":"10.1093\/ptep\/ptaa040","article-title":"Inverse Mass Hierarchy of Light Scalar Mesons Driven by Anomaly-Induced Flavor Breaking","volume":"5","author":"Kuroda","year":"2020","journal-title":"Prog. Theor. Exp. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1143\/PTP.109.969","article-title":"UA(1) breaking effects on the light scalar meson spectrum","volume":"109","author":"Naito","year":"2003","journal-title":"Prog. Theor. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1134\/S1547477120020144","article-title":"On the Problem of Spontaneous Isotopic Symmetry Breaking in Nambu\u2013Jona-Lasinio Models","volume":"17","author":"Naydenov","year":"2020","journal-title":"Phys. Part. Nucl. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1103\/PhysRev.122.345","article-title":"Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I","volume":"122","author":"Nambu","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1103\/PhysRev.124.246","article-title":"Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II","volume":"124","author":"Nambu","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0003-4916(77)90169-5","article-title":"Path Integral Formulation of Mean Field Perturbation Theory","volume":"109","author":"Bender","year":"1977","journal-title":"Ann. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4551","DOI":"10.1103\/PhysRevD.18.4551","article-title":"Renormalization of Four Fermion Theories in a Mean Field Expansion","volume":"18","author":"Tamvakis","year":"1978","journal-title":"Phys. Rev. D"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2755","DOI":"10.1103\/PhysRevD.14.2755","article-title":"A New Approach to Collective Phenomena in Superconductivity Models","volume":"14","author":"Eguchi","year":"1976","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1134\/S0021364020150011","article-title":"Mass prediction for the last discovered member of the axial-vector nonet with quantum numbers JPC = 1+\u2212","volume":"112","author":"Chizhov","year":"2020","journal-title":"JETP Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1134\/1.1804212","article-title":"Vector meson couplings to vector and tensor currents in extended NJL quark model","volume":"80","author":"Chizhov","year":"2004","journal-title":"JETP Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"072005","DOI":"10.1103\/PhysRevD.98.072005","article-title":"Observation of h1(1380) in the J\/\u03c8\u2192\u03b7\u2032KK\u00af\u03c0 Decay","volume":"98","author":"Ablikim","year":"2018","journal-title":"Phys. Rev. D"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/600\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:15:10Z","timestamp":1760030110000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/600"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,15]]},"references-count":17,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040600"],"URL":"https:\/\/doi.org\/10.3390\/sym17040600","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,4,15]]}}}