Abstract
We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold 𝕋2/ℤN, with a single torus defined by one complex coordinate z and a single modulus field τ, playing the role of a flavon transforming under a finite modular symmetry. We then consider 10d orbifolds with three factorizable tori, each defined by one complex coordinate zi and involving the three moduli fields τ1, τ2, τ3 transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form (𝕋2)3/(ℤN × ℤM), and list those which have fixed values of the moduli fields (up to an integer). The key advantage of such 10d orbifold models over 4d models is that the values of the moduli are not completely free but are constrained by geometry and symmetry. To illustrate the approach we discuss a 10d modular seesaw model with \( {S}_4^3 \) modular symmetry based on (𝕋2)3/(ℤ4 × ℤ2) where τ1 = i, τ2 = i + 2 are constrained by the orbifold, while τ3 = ω is determined by imposing a further remnant S4 flavour symmetry, leading to a highly predictive example in the class CSD(n) with n = 1 − \( \sqrt{6} \).
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S.F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340] [INSPIRE].
S.F. King, Unified Models of Neutrinos, Flavour and CP Violation, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413] [INSPIRE].
S. Ferrara, D. Lust, A.D. Shapere and S. Theisen, Modular Invariance in Supersymmetric Field Theories, Phys. Lett. B 225 (1989) 363 [INSPIRE].
S. Ferrara, D. Lust and S. Theisen, Target Space Modular Invariance and Low-Energy Couplings in Orbifold Compactifications, Phys. Lett. B 233 (1989) 147 [INSPIRE].
K. Ishiguro, T. Kobayashi and H. Otsuka, Symplectic modular symmetry in heterotic string vacua: flavor, CP, and R-symmetries, JHEP 01 (2022) 020 [arXiv:2107.00487] [INSPIRE].
D. Cremades, L.E. Ibanez and F. Marchesano, Computing Yukawa couplings from magnetized extra dimensions, JHEP 05 (2004) 079 [hep-th/0404229] [INSPIRE].
K. Ishiguro, T. Kobayashi and H. Otsuka, Landscape of Modular Symmetric Flavor Models, JHEP 03 (2021) 161 [arXiv:2011.09154] [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), p. 227–266 [https://doi.org/10.1142/9789813238053_0012] [arXiv:1706.08749] [INSPIRE].
F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93 (2021) 015007 [arXiv:1912.06028] [INSPIRE].
J.T. Penedo and S.T. Petcov, Lepton Masses and Mixing from Modular S4 Symmetry, Nucl. Phys. B 939 (2019) 292 [arXiv:1806.11040] [INSPIRE].
P.P. Novichkov, J.T. Penedo and S.T. Petcov, Double cover of modular S4 for flavour model building, Nucl. Phys. B 963 (2021) 115301 [arXiv:2006.03058] [INSPIRE].
X.-G. Liu, C.-Y. Yao and G.-J. Ding, Modular invariant quark and lepton models in double covering of S4 modular group, Phys. Rev. D 103 (2021) 056013 [arXiv:2006.10722] [INSPIRE].
F. Feruglio, The irresistible call of τ = i, arXiv:2211.00659 [INSPIRE].
I. de Medeiros Varzielas, S.F. King and Y.-L. Zhou, Multiple modular symmetries as the origin of flavor, Phys. Rev. D 101 (2020) 055033 [arXiv:1906.02208] [INSPIRE].
S.F. King and Y.-L. Zhou, Trimaximal TM1 mixing with two modular S4 groups, Phys. Rev. D 101 (2020) 015001 [arXiv:1908.02770] [INSPIRE].
I. de Medeiros Varzielas and J. Lourenço, Two A4 modular symmetries for Tri-Maximal 2 mixing, Nucl. Phys. B 979 (2022) 115793 [arXiv:2107.04042] [INSPIRE].
I. de Medeiros Varzielas and J. Lourenço, Two A5 modular symmetries for Golden Ratio 2 mixing, Nucl. Phys. B 984 (2022) 115974 [arXiv:2206.14869] [INSPIRE].
M.R. Devi, Retrieving texture zeros in 3 + 1 active-sterile neutrino framework under the action of A4 modular-invariants, arXiv:2303.04900 [INSPIRE].
I. de Medeiros Varzielas, S.F. King and M. Levy, Littlest modular seesaw, JHEP 02 (2023) 143 [arXiv:2211.00654] [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].
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 — I. Basic results, Phys. Lett. B 808 (2020) 135615 [arXiv:2006.03059] [INSPIRE].
A. Baur et al., The eclectic flavor symmetry of the Z2 orbifold, JHEP 02 (2021) 018 [arXiv:2008.07534] [INSPIRE].
G.-J. Ding et al., Neutrino mass and mixing models with eclectic flavor symmetry ∆(27) ⋊ T′, JHEP 05 (2023) 144 [arXiv:2303.02071] [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].
M. Fischer, M. Ratz, J. Torrado and P.K.S. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 01 (2013) 084 [arXiv:1209.3906] [INSPIRE].
P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Modular S4 models of lepton masses and mixing, JHEP 04 (2019) 005 [arXiv:1811.04933] [INSPIRE].
C. Hagedorn, S.F. King and C. Luhn, A SUSY GUT of Flavour with S4 × SU(5) to NLO, JHEP 06 (2010) 048 [arXiv:1003.4249] [INSPIRE].
G.-J. Ding, S.F. King, C. Luhn and A.J. Stuart, Spontaneous CP violation from vacuum alignment in S4 models of leptons, JHEP 05 (2013) 084 [arXiv:1303.6180] [INSPIRE].
G.-J. Ding, S.F. King, X.-G. Liu and J.-N. Lu, Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing, JHEP 12 (2019) 030 [arXiv:1910.03460] [INSPIRE].
F.J. de Anda and S.F. King, An S4 × SU(5) SUSY GUT of flavour in 6d, JHEP 07 (2018) 057 [arXiv:1803.04978] [INSPIRE].
G. Altarelli, F. Feruglio and Y. Lin, Tri-bimaximal neutrino mixing from orbifolding, Nucl. Phys. B 775 (2007) 31 [hep-ph/0610165] [INSPIRE].
A. Adulpravitchai, A. Blum and M. Lindner, Non-Abelian Discrete Flavor Symmetries from T2/Z(N) Orbifolds, JHEP 07 (2009) 053 [arXiv:0906.0468] [INSPIRE].
A. Adulpravitchai and M.A. Schmidt, Flavored Orbifold GUT — an SO(10) × S4 model, JHEP 01 (2011) 106 [arXiv:1001.3172] [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].
M.-C. Chen et al., Quasi-eclectic modular flavor symmetries, Phys. Lett. B 824 (2022) 136843 [arXiv:2108.02240] [INSPIRE].
I. Esteban et al., The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178 [arXiv:2007.14792] [INSPIRE].
NuFit webpage, http://www.nu-fit.org.
L. Nilse, Classification of 1D and 2D orbifolds, AIP Conf. Proc. 903 (2007) 411 [hep-ph/0601015] [INSPIRE].
Acknowledgments
SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation programme under Marie Sklodowska-Curie grant agreement HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). SFK would also like to thank CERN for its hospitality.
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de Anda, F.J., King, S.F. Modular flavour symmetry and orbifolds. J. High Energ. Phys. 2023, 122 (2023). https://doi.org/10.1007/JHEP06(2023)122
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DOI: https://doi.org/10.1007/JHEP06(2023)122