Abstract
The flavor structure of quarks and leptons and quark-lepton unification are studied in the framework of Pati-Salam models with A4 modular symmetry. The three generations of the left-handed and right-handed fermions are assigned to be triplet or singlets of A4. The light neutrino masses are generated through the type-I seesaw mechanism. We perform a systematic classification of Pati-Salam models according to the transformations of matter fields under the A4 modular symmetry, and the general form of the fermion mass matrix is given. We present four phenomenologically viable benchmark models which provide excellent descriptions of masses and flavor mixing of quarks and leptons, including neutrinos. In such models we find that the normal ordered neutrino mass spectrum is preferred over the inverted case, with neutrinoless double beta decay predicted to be too small to be observed by the next generation of experiments.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Particle Data collaboration, Review of Particle Physics, Prog. Theor. Exp. Phys. 2022 (2022) 083C01 [INSPIRE].
I. Esteban, M. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178 [arXiv:2007.14792] [INSPIRE].
S.F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340] [INSPIRE].
Z.-z. Xing, Flavor structures of charged fermions and massive neutrinos, Phys. Rep. 854 (2020) 1 [arXiv:1909.09610] [INSPIRE].
F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93 (2021) 015007 [arXiv:1912.06028] [INSPIRE].
G.-J. Ding and J.W.F. Valle, The symmetry approach to quark and lepton masses and mixing, arXiv:2402.16963 [INSPIRE].
S.F. King, Unified Models of Neutrinos, Flavour and CP Violation, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413] [INSPIRE].
F. Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire. . . : Guido Altarelli’s Legacy, A. Levy et al. eds., World Scientific (2019), pp. 227–266 [https://doi.org/10.1142/9789813238053_0012] [arXiv:1706.08749] [INSPIRE].
T. Kobayashi and M. Tanimoto, Modular flavor symmetric models, arXiv:2307.03384 [INSPIRE].
G.-J. Ding and S.F. King, Neutrino mass and mixing with modular symmetry, Rept. Prog. Phys. 87 (2024) 084201 [arXiv:2311.09282] [INSPIRE].
F.J. de Anda, S.F. King and E. Perdomo, SU(5) grand unified theory with A4 modular symmetry, Phys. Rev. D 101 (2020) 015028 [arXiv:1812.05620] [INSPIRE].
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi, Modular S3-invariant flavor model in SU(5) grand unified theory, Prog. Theor. Exp. Phys. 2020 (2020) 053B05 [arXiv:1906.10341] [INSPIRE].
X. Du and F. Wang, SUSY breaking constraints on modular flavor S3 invariant SU(5) GUT model, JHEP 02 (2021) 221 [arXiv:2012.01397] [INSPIRE].
Y. Zhao and H.-H. Zhang, Adjoint SU(5) GUT model with modular S4 symmetry, JHEP 03 (2021) 002 [arXiv:2101.02266] [INSPIRE].
P. Chen, G.-J. Ding and S.F. King, SU(5) GUTs with A4 modular symmetry, JHEP 04 (2021) 239 [arXiv:2101.12724] [INSPIRE].
S.F. King and Y.-L. Zhou, Twin modular S4 with SU(5) GUT, JHEP 04 (2021) 291 [arXiv:2103.02633] [INSPIRE].
G.-J. Ding, S.F. King and C.-Y. Yao, Modular S4 × SU(5) GUT, Phys. Rev. D 104 (2021) 055034 [arXiv:2103.16311] [INSPIRE].
Y. Abe, T. Higaki, J. Kawamura and T. Kobayashi, Fermion hierarchies in SU(5) grand unification from \( {\Gamma}_6^{\prime } \) modular flavor symmetry, JHEP 08 (2023) 097 [arXiv:2307.01419] [INSPIRE].
I. de Medeiros Varzielas, S.F. King and M. Levy, A modular SU(5) littlest seesaw, JHEP 05 (2024) 203 [arXiv:2309.15901] [INSPIRE].
G. Charalampous, S.F. King, G.K. Leontaris and Y.-L. Zhou, Flipped SU(5) with modular A4 symmetry, Phys. Rev. D 104 (2021) 115015 [arXiv:2109.11379] [INSPIRE].
X.K. Du and F. Wang, Flavor structures of quarks and leptons from flipped SU(5) GUT with A4 modular flavor symmetry, JHEP 01 (2023) 036 [arXiv:2209.08796] [INSPIRE].
G.-J. Ding, S.F. King and J.-N. Lu, SO(10) models with A4 modular symmetry, JHEP 11 (2021) 007 [arXiv:2108.09655] [INSPIRE].
G.-J. Ding, S.F. King, J.-N. Lu and B.-Y. Qu, Leptogenesis in SO(10) models with A4 modular symmetry, JHEP 10 (2022) 071 [arXiv:2206.14675] [INSPIRE].
J.C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10 (1974) 275 [Erratum ibid. 11 (1975) 703] [INSPIRE].
S.F. King and M. Malinsky, A4 family symmetry and quark-lepton unification, Phys. Lett. B 645 (2007) 351 [hep-ph/0610250] [INSPIRE].
R. de Adelhart Toorop, F. Bazzocchi and L. Merlo, The Interplay Between GUT and Flavour Symmetries in a Pati-Salam × S4 Model, JHEP 08 (2010) 001 [arXiv:1003.4502] [INSPIRE].
S.F. King, A model of quark and lepton mixing, JHEP 01 (2014) 119 [arXiv:1311.3295] [INSPIRE].
S.F. King, A to Z of Flavour with Pati-Salam, JHEP 08 (2014) 130 [arXiv:1406.7005] [INSPIRE].
A.E. Cárcamo Hernández, S. Kovalenko, J.W.F. Valle and C.A. Vaquera-Araujo, Predictive Pati-Salam theory of fermion masses and mixing, JHEP 07 (2017) 118 [arXiv:1705.06320] [INSPIRE].
I. Antoniadis, G.K. Leontaris and J. Rizos, A Three generation SU(4) × O(4) string model, Phys. Lett. B 245 (1990) 161 [INSPIRE].
T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4d string models with a Pati-Salam symmetry: Orbifold grand unified theories from heterotic string compactification on a ℤ6 orbifold, Nucl. Phys. B 704 (2005) 3 [hep-ph/0409098] [INSPIRE].
T. Li, R. Sun and L. Wu, String-scale gauge coupling relations in the supersymmetric Pati-Salam models from intersecting D6-branes, JHEP 03 (2023) 210 [arXiv:2212.05875] [INSPIRE].
K.R. Dienes, New constraints on SO(10) model building from string theory, Nucl. Phys. B 488 (1997) 141 [hep-ph/9606467] [INSPIRE].
A. Baur, H.P. Nilles, A. Trautner and P.K.S. Vaudrevange, Unification of Flavor, CP, and Modular Symmetries, Phys. Lett. B 795 (2019) 7 [arXiv:1901.03251] [INSPIRE].
R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Finite Modular Groups and Lepton Mixing, Nucl. Phys. B 858 (2012) 437 [arXiv:1112.1340] [INSPIRE].
X.-G. Liu and G.-J. Ding, Neutrino Masses and Mixing from Double Covering of Finite Modular Groups, JHEP 08 (2019) 134 [arXiv:1907.01488] [INSPIRE].
M.-C. Chen, S. Ramos-Sánchez and M. Ratz, A note on the predictions of models with modular flavor symmetries, Phys. Lett. B 801 (2020) 135153 [arXiv:1909.06910] [INSPIRE].
F. Feruglio, V. Gherardi, A. Romanino and A. Titov, Modular invariant dynamics and fermion mass hierarchies around τ = i, JHEP 05 (2021) 242 [arXiv:2101.08718] [INSPIRE].
H.P. Nilles, S. Ramos-Sánchez and P.K.S. Vaudrevange, Eclectic Flavor Groups, JHEP 02 (2020) 045 [arXiv:2001.01736] [INSPIRE].
H.P. Nilles, S. Ramos-Sanchez and P.K.S. Vaudrevange, Lessons from eclectic flavor symmetries, Nucl. Phys. B 957 (2020) 115098 [arXiv:2004.05200] [INSPIRE].
H.P. Nilles, S. Ramos–Sánchez and P.K.S. Vaudrevange, Eclectic flavor scheme from ten-dimensional string theory. Part I. Basic results, Phys. Lett. B 808 (2020) 135615 [arXiv:2006.03059] [INSPIRE].
M.-C. Chen, V. Knapp-Perez, M. Ramos-Hamud, S. Ramos-Sanchez, M. Ratz and S. Shukla, Quasi-eclectic modular flavor symmetries, Phys. Lett. B 824 (2022) 136843 [arXiv:2108.02240] [INSPIRE].
G.-J. Ding, S.F. King, C.-C. Li, X.-G. Liu and J.-N. Lu, Neutrino mass and mixing models with eclectic flavor symmetry ∆(27) ⋊ T ′, JHEP 05 (2023) 144 [arXiv:2303.02071] [INSPIRE].
C.-C. Li and G.-J. Ding, Eclectic flavor group ∆(27) ⋊ S3 and lepton model building, JHEP 03 (2024) 054 [arXiv:2308.16901] [INSPIRE].
A. Melfo and G. Senjanovic, Minimal supersymmetric Pati-Salam theory: Determination of physical scales, Phys. Rev. D 68 (2003) 035013 [hep-ph/0302216] [INSPIRE].
G. Ross and M. Serna, Unification and fermion mass structure, Phys. Lett. B 664 (2008) 97 [arXiv:0704.1248] [INSPIRE].
Planck collaboration, Planck 2018 results. Part VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [Erratum ibid. 652 (2021) C4] [arXiv:1807.06209] [INSPIRE].
KamLAND-Zen collaboration, Search for the Majorana Nature of Neutrinos in the Inverted Mass Ordering Region with KamLAND-Zen, Phys. Rev. Lett. 130 (2023) 051801 [arXiv:2203.02139] [INSPIRE].
LEGEND collaboration, The Large Enriched Germanium Experiment for Neutrinoless ββ Decay: LEGEND-1000 Preconceptual Design Report, arXiv:2107.11462 [INSPIRE].
nEXO collaboration, nEXO: neutrinoless double beta decay search beyond 1028 year half-life sensitivity, J. Phys. G 49 (2022) 015104 [arXiv:2106.16243] [INSPIRE].
DUNE collaboration, Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF, arXiv:1512.06148 [INSPIRE].
Hyper-Kamiokande collaboration, Hyper-Kamiokande Design Report, arXiv:1805.04163 [INSPIRE].
Acknowledgments
GJD is supported by the National Natural Science Foundation of China under Grant Nos. 12375104, 11975224. JNL is supported by the Grants No. NSFC-12147110 and the China Post-doctoral Science Foundation under Grant No. 2021M70. SFK would like to thank CERN for its hospitality. SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation programme under Marie Skłodowska-Curie grant agreement HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2404.06520
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ding, GJ., Jiang, SY., King, S.F. et al. Pati-Salam models with A4 modular symmetry. J. High Energ. Phys. 2024, 134 (2024). https://doi.org/10.1007/JHEP08(2024)134
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2024)134