Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing
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Springer
Received: October 16, 2019 Accepted: November 16, 2019 Published: December 4, 2019
Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing
a
Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, No. 96 JinZhai Road, Hefei, Anhui 230026, China b Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, U.K.
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup ΓN of the modular symmetry, broken by the vacuum expectation value (VEV) of a modulus field τ . If the VEV of the modulus τ takes some special value, a residual subgroup of ΓN would be preserved. We derive the fixed points τS = i, √ √ τST = (−1 + i 3)/2, τT S = (1 + i 3)/2, τT = i∞ in the fundamental domain which are invariant under the modular transformations indicated. We then generalise these fixed points to τf = γτS , γτST , γτT S and γτT in the upper half complex plane, and show that it is sufficient to consider γ ∈ ΓN . Focussing on level N = 4, corresponding to the flavour group S4 , we consider all the resulting triplet modular forms at these fixed points up to weight 6. We then apply the results to lepton mixing, with different residual subgroups in the charged lepton sector and each of the right-handed neutrinos sectors. In the minimal case of two right-handed neutrinos, we find three phenomenologically viable cases in which the light neutrino mass matrix only depends on three free parameters, and the lepton mixing takes the trimaximal TM1 pattern for two examples. One of these cases corresponds to a new √ Littlest Modular Seesaw based on CSD(n) with n = 1 + 6 ≈ 3.45, intermediate between CSD(3) and CSD(4). Finally, we generalize the results to examples with three right-handed neutrinos, also considering the level N = 3 case, corresponding to A4 flavour symmetry. Keywords: CP violation, Discrete Symmetries, Neutrino Physics ArXiv ePrint: 1910.03460
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)030
JHEP12(2019)030
Gui-Jun Ding,a Stephen F. King,b Xiang-Gan Liua and Jun-Nan Lua
Contents 1 Introduction
1
2 Modular symmetry and modular form multiplets of level N = 4 2.1 Modular forms of level 4
4 5 8 8 12 15
4 New predictive examples of lepton mixing 4.1 Examples of N = 4 4.1.1 Ge = Z3ST 2 4.1.2 Ge = Z4ST 4.2 Tri-direct modular model with three right-handed neutrinos
18 19 19 20 23
5 Fixed points and tri-direct modular model for N = 3
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6 Conclusion
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A Group theory of Γ4 ∼ = S4
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1
Introduction
It is well-known that there are huge mass hierarchies among the quarks and leptons, and the quark mixing angles are small while the lepton sector has two large mixing angles θ12 , θ23 and one small mixing angles θ13 which is of the same order of magnitude as the quark Cabibbo mixing angle [1]. The origin
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