{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T12:43:23Z","timestamp":1775047403223,"version":"3.50.1"},"reference-count":63,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T00:00:00Z","timestamp":1668384000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a phenomenological model of two-zeros Majorana neutrino mass matrix based on the A4 symmetry, where the structure of mixing matrix is a unimodular second scheme of trimaximal TM2, and the charged lepton mass matrix is diagonal. We show that, among seven possible two-zero textures with A4 symmetry, only two textures, namely the texture with Mee=0 and Me\u03bc=0 and its permutation, are acceptable in the non-perturbation method, since the results associated with these two textures are consistent with the experimental data. We obtain a unique relation between our phases, namely \u03c1+\u03c3=\u03d5\u00b1\u03c0, and an effective equation sin2\u03b813=23R\u03bd where R\u03bd=\u03b4m2\u0394m2. Then, only by using the experimental ranges of R\u03bd, we obtain the allowable range of the unknown parameter \u03d5 as the phase of TM2 mixing matrix, which leads to obtaining not only the ranges of all neutrino oscillation parameters of the model (which agree well with experimental data) but also with the masses of neutrinos, the Dirac and Majorana phases and the Jarlskog parameter, and to predict the normal neutrino mass hierarchy. Finally, we show that all the predictions regarding our two specific textures agree with the corresponding data reported from neutrino oscillation, cosmic microwave background and neutrinoless double beta decay.<\/jats:p>","DOI":"10.3390\/sym14112410","type":"journal-article","created":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T02:31:15Z","timestamp":1668479475000},"page":"2410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Two-Zero Textures Based on A4 Symmetry and Unimodular Mixing Matrix"],"prefix":"10.3390","volume":"14","author":[{"given":"Neda","family":"Razzaghi","sequence":"first","affiliation":[{"name":"Department of Physics, Qazvin Branch, Islamic Azad University, Qazvin 341851416, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3455-1954","authenticated-orcid":false,"given":"Seyed Meraj Mousavi","family":"Rasouli","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"}]},{"given":"Paulo","family":"Parada","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7170-8952","authenticated-orcid":false,"given":"Paulo","family":"Moniz","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113012","DOI":"10.1103\/PhysRevD.64.113012","article-title":"Softly broken A4 symmetry for nearly degenerate neutrino masses","volume":"64","author":"Ma","year":"2001","journal-title":"G. Phys. Rev. D"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0370-2693(02)03153-2","article-title":"Underlying A4 Symmetry for the Neutrino Mass Matrix and the Quark Mixing Matrix","volume":"552","author":"Babu","year":"2003","journal-title":"Phys. Lett. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37301","DOI":"10.1103\/PhysRevD.72.037301","article-title":"Aspects of the Tetrahedral Neutrino Mass Matrix","volume":"72","author":"Ma","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2701","DOI":"10.1103\/RevModPhys.82.2701","article-title":"Discrete flavor symmetries and models of neutrino mixing","volume":"82","author":"Altarelli","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"151802","DOI":"10.1103\/PhysRevLett.99.151802","article-title":"Predictive Flavor Symmetries of the Neutrino Mass Matrix","volume":"99","author":"Hirsch","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.nuclphysb.2006.02.015","article-title":"Tri-Bimaximal Neutrino Mixing, A4 and the Modular Symmetry","volume":"741","author":"Altarelli","year":"2006","journal-title":"Nucl. Phys. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"85005","DOI":"10.1088\/0954-3899\/36\/8\/085005","article-title":"A simplest A4 model for tri-bimaximal neutrino mixing","volume":"36","author":"Altarelli","year":"2009","journal-title":"J. Phys. G"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"13011","DOI":"10.1103\/PhysRevD.84.013011","article-title":"Tribimaximal mixing from small groups","volume":"84","author":"Parattu","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/JHEP03(2012)036","article-title":"A4 models of tri-bimaximal-reactor mixing","volume":"3","author":"King","year":"2012","journal-title":"J. High Energy Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/JHEP08(2012)021","article-title":"Discrete Flavour Groups. \u03b813 and Lepton Flavour Violation","volume":"8","author":"Altarelli","year":"2012","journal-title":"J. High Energy Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1002\/prop.201200117","article-title":"Tri-bimaximal neutrino mixing and discrete flavour symmetries","volume":"61","author":"Altarelli","year":"2013","journal-title":"Fortschr. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1016\/j.physletb.2013.09.058","article-title":"A new A4 model for lepton mixing","volume":"726","author":"Ferreira","year":"2013","journal-title":"Phys. Lett. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0370-2693(02)01336-9","article-title":"Tri-Bimaximal Mixing and the Neutrino Oscillation Data","volume":"530","author":"Harrison","year":"2002","journal-title":"Phys. Lett. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"181801","DOI":"10.1103\/PhysRevLett.112.181801","article-title":"Observation of Electron Neutrino Appearance in a Muon Neutrino Beam","volume":"112","author":"Abe","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"191802","DOI":"10.1103\/PhysRevLett.108.191802","article-title":"Observation of Reactor Electron Antineutrinos Disappearance in the RENO Experiment","volume":"108","author":"Ahn","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"52008","DOI":"10.1103\/PhysRevD.86.052008","article-title":"Double Chooz Collaboration. Reactor \u03bd\u00afe disappearance in the Double Chooz experiment","volume":"86","author":"Abe","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"61801","DOI":"10.1103\/PhysRevLett.112.061801","article-title":"Spectral Measurement of Electron Antineutrino Oscillation Amplitude and Frequency at Daya Bay","volume":"112","author":"An","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","unstructured":"Hu, B.Z. (2014). Daya Bay Collaboration. New Results from the Daya Bay Reactor Neutrino Experiment. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"93018","DOI":"10.1103\/PhysRevD.89.093018","article-title":"Neutrino masses and mixings: Status of known and unknown 3\u03bd parameters","volume":"89","author":"Capozzi","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"90001","DOI":"10.1088\/1674-1137\/38\/9\/090001","article-title":"Review of particle physics","volume":"38","author":"Olive","year":"2014","journal-title":"Chin. Phys. C"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1103\/PhysRevD.22.2227","article-title":"Neutrino masses in SU(2)\u2297U(1) theories","volume":"22","author":"Schechter","year":"1980","journal-title":"Phys.Rev. D"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/S0370-2693(01)00952-2","article-title":"How to Describe Neutrino Mixing and CP Violation","volume":"517","author":"Fritzsch","year":"2001","journal-title":"Phys. Lett. B"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Particle Data Group, and Yao, W.M. (2006). Review of Particle Physics. J. Phys. G, 33, 1.","DOI":"10.1088\/0954-3899\/33\/1\/001"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/JHEP02(2021)071","article-title":"Global reassessment of the neutrino oscillation picture","volume":"2021","author":"Forero","year":"2021","journal-title":"J. High Energ. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1143\/PTPS.183.1","article-title":"Non-Abelian Discrete Symmetries in Particle Physics","volume":"183","author":"Ishimori","year":"2010","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/JHEP10(2013)043","article-title":"Masses, mixing angles and phases of general Majorana neutrino mass matrix","volume":"2013","author":"Adhikary","year":"2013","journal-title":"J. High Energy Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"93007","DOI":"10.1103\/PhysRevD.88.093007","article-title":"Relating neutrino masses and mixings by discrete symmetries","volume":"88","author":"Hernandez","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"53014","DOI":"10.1103\/PhysRevD.86.053014","article-title":"Lepton mixing and discrete symmetries","volume":"86","author":"Hernandez","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"31901","DOI":"10.1103\/PhysRevD.70.031901","article-title":"A4 symmetry and neutrinos with very different masses","volume":"70","author":"Ma","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"93002","DOI":"10.1103\/PhysRevD.81.093002","article-title":"Deviations from tribimaximal mixing due to the vacuum expectation value misalignment in A4 models","volume":"81","author":"Barry","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"93003","DOI":"10.1103\/PhysRevD.86.093003","article-title":"Nonzero \u03b813 and violation CP in a model with A4 flavor symmetry","volume":"86","author":"Ahn","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0370-2693(02)01817-8","article-title":"Zeroes of the neutrino mass matrix","volume":"536","author":"Frampton","year":"2002","journal-title":"Phys. Lett. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"45030","DOI":"10.1103\/PhysRevD.86.045030","article-title":"New simple A4 neutrino model for nonzero \u03b813 and large \u03b4CP","volume":"86","author":"Ishimori","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1250134","DOI":"10.1142\/S0217751X12501345","article-title":"Scotogenic A4 Neutrino Model for Nonzero \u03b813 and Large \u03b4CP","volume":"27","author":"Ma","year":"2012","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"12003","DOI":"10.1088\/1742-6596\/627\/1\/012003","article-title":"A prediction of \u03b4CP for a normal neutrino mass ordering in an extended standard model with an A4 flavour symmetry","volume":"627","author":"Dinh","year":"2015","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_36","unstructured":"Dinh, D.N., Ky, N.A., Van, P.Q., and Van, N.T.H. (2016). A see-saw scenario of an A4 flavour symmetric standard model. arXiv."},{"key":"ref_37","first-page":"001","article-title":"On neutrino and charged lepton masses and mixings: A view from the electroweak-scale right-handed neutrino model. J","volume":"9","author":"Hung","year":"2015","journal-title":"High Energy Phys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Razzaghi, N., Rasouli, S.M.M., Parada, P., and Moniz, P. (2022). Generating CP Violation from a Modified Fridberg-Lee Model. Universe, 8.","DOI":"10.3390\/universe8090448"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/0370-2693(81)90559-1","article-title":"Left-handed neutrino mass scale and spontaneously broken lepton number","volume":"99","author":"Gelmini","year":"1981","journal-title":"Phys. Lett. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"73015","DOI":"10.1103\/PhysRevD.78.073015","article-title":"Unique horizontal symmetry of leptons","volume":"78","author":"Lam","year":"2008","journal-title":"Phys. Rev. D"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.physletb.2006.08.007","article-title":"Magic neutrino mass matrix and the Bjorken\u2013Harrison\u2013Scott parameterization","volume":"640","author":"Lam","year":"2006","journal-title":"Phys. Lett. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"15020","DOI":"10.1103\/PhysRevD.96.015020","article-title":"Implications of texture zeros for a variant of tribimaximal mixing","volume":"96","author":"Kumar","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"36004","DOI":"10.1103\/PhysRevD.94.036004","article-title":"Zeros in the magic neutrino mass matrix","volume":"94","author":"Gautam","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/0550-3213(79)90362-6","article-title":"Quark masses and flavor mixing","volume":"155","author":"Fritzsch","year":"1979","journal-title":"Nucl. Phys. B"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0217751X04016969","article-title":"Flavor mixing and CP violation of massive neutrinos","volume":"19","author":"Xing","year":"2004","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/S0370-2693(02)01354-0","article-title":"Texture zeros and Majorana phases of the neutrino mass matrix","volume":"530","author":"Xing","year":"2002","journal-title":"Phys. Lett. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2863184","DOI":"10.1155\/2018\/2863184","article-title":"Texture one zero Dirac neutrino mass matrix with vanishing determinant or trace condition","volume":"2018","author":"Singh","year":"2018","journal-title":"Adv. High Energy Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1088\/1126-6708\/2005\/07\/034","article-title":"Minimal mass matrices for Dirac neutrinos","volume":"507","author":"Hagedorn","year":"2005","journal-title":"J. High Energy Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1350040","DOI":"10.1142\/S0217751X13500401","article-title":"Texture zeros for Dirac neutrinos and current experimental tests","volume":"28","author":"Liu","year":"2013","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"113011","DOI":"10.1103\/PhysRevD.85.113011","article-title":"One-zero textures of Majorana neutrino mass matrix and current experimental tests","volume":"85","author":"Lashin","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/JHEP09(2011)083","article-title":"Two-zero textures of the Majorana neutrino mass matrix and current experimental tests","volume":"9","author":"Fritzsch","year":"2011","journal-title":"J. High Energy Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"35011","DOI":"10.1103\/PhysRevD.105.035011","article-title":"Simplified textures of the seesaw model for trimaximal neutrino mixing","volume":"105","author":"Zhao","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"115541","DOI":"10.1016\/j.nuclphysb.2021.115541","article-title":"Texture of one equality in neutrino mass matrix","volume":"971","author":"Ismael","year":"2021","journal-title":"Nucl Phys B"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2369","DOI":"10.1103\/PhysRevD.17.2369","article-title":"Neutrino oscillations in matter","volume":"17","author":"Wolfenestein","year":"1978","journal-title":"Phys. Rev. D"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Fukugita, M., and Yanagida, T. (2003). Physics of Neutrinos and Applications to Astrophysics, Springer.","DOI":"10.1007\/978-3-662-05119-1"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"53005","DOI":"10.1103\/PhysRevD.70.053005","article-title":"Leptonic CP violation phases using an ansatz for the neutrino mass matrix and application to leptogenesis","volume":"70","author":"Nasri","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"93011","DOI":"10.1103\/PhysRevD.71.093011","article-title":"Rephasing invariant parametrization of flavor mixing matrices","volume":"71","author":"Kuo","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1007\/JHEP11(2010)080","article-title":"Rephasing invariance and neutrino mixing","volume":"11","author":"Chiu","year":"2010","journal-title":"J. High Energy Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"2579","DOI":"10.1140\/epjc\/s10052-013-2579-3","article-title":"Properties of the neutrino mixing matrix","volume":"73","author":"Chiu","year":"2013","journal-title":"Eur. Phys. J. C"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"53006","DOI":"10.1103\/PhysRevD.86.053006","article-title":"Generalized Friedberg-Lee model for CP violation in neutrino physics","volume":"86","author":"Razzaghi","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1103\/PhysRevLett.55.1039","article-title":"Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal Nonconservation","volume":"55","author":"Jarlskog","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"A6","DOI":"10.1051\/0004-6361\/201833910","article-title":"Planck 2018 results-VI. Cosmological parameters","volume":"641","author":"Aghanim","year":"2020","journal-title":"Astron. Astrophys."},{"key":"ref_63","unstructured":"Gando, Y. (2018). et al. [KamLAND-Zen Collaboration]. Neutrinoless double beta decay search with liquid scintillator experiments. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2410\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:17:40Z","timestamp":1760145460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,14]]},"references-count":63,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112410"],"URL":"https:\/\/doi.org\/10.3390\/sym14112410","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,14]]}}}