Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing
Abstract In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup Γ N of the modular symmetry, broken by the vacuum expec- tation value (VEV) of a modulus field τ. If the VEV of the modulus τ takes some special value, a residual subgroup of Γ N would be...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-12-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP12(2019)030 |
| _version_ | 1819315464482848768 |
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| author | Gui-Jun Ding Stephen F. King Xiang-Gan Liu Jun-Nan Lu |
| author_facet | Gui-Jun Ding Stephen F. King Xiang-Gan Liu Jun-Nan Lu |
| author_sort | Gui-Jun Ding |
| collection | DOAJ |
| description | Abstract In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup Γ N of the modular symmetry, broken by the vacuum expec- tation value (VEV) of a modulus field τ. If the VEV of the modulus τ takes some special value, a residual subgroup of Γ N would be preserved. We derive the fixed points τ S = i, τ ST = (−1 + i 3 $$ \sqrt{3} $$ )/2, τ TS = (1 + i 3 $$ \sqrt{3} $$ )/2, τ T = i∞ in the fundamental domain which are in-variant under the modular transformations indicated. We then generalise these fixed points to τ f = γτ S , γτ ST , γτ TS and γτ T in the upper half complex plane, and show that it is suffi-cient to consider γ ∈ ΓN. Focussing on level N = 4, corresponding to the flavour group S 4, 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 $$ \sqrt{6} $$ ≈ 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 A 4 flavour symmetry. |
| first_indexed | 2024-12-24T10:00:31Z |
| format | Article |
| id | doaj.art-d3a31bb81a5e4524be47f4d108ce5fb5 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-24T10:00:31Z |
| publishDate | 2019-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-d3a31bb81a5e4524be47f4d108ce5fb52022-12-21T17:01:06ZengSpringerOpenJournal of High Energy Physics1029-84792019-12-0120191213510.1007/JHEP12(2019)030Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixingGui-Jun Ding0Stephen F. King1Xiang-Gan Liu2Jun-Nan Lu3Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of ChinaPhysics and Astronomy, University of SouthamptonInterdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of ChinaInterdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of ChinaAbstract In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup Γ N of the modular symmetry, broken by the vacuum expec- tation value (VEV) of a modulus field τ. If the VEV of the modulus τ takes some special value, a residual subgroup of Γ N would be preserved. We derive the fixed points τ S = i, τ ST = (−1 + i 3 $$ \sqrt{3} $$ )/2, τ TS = (1 + i 3 $$ \sqrt{3} $$ )/2, τ T = i∞ in the fundamental domain which are in-variant under the modular transformations indicated. We then generalise these fixed points to τ f = γτ S , γτ ST , γτ TS and γτ T in the upper half complex plane, and show that it is suffi-cient to consider γ ∈ ΓN. Focussing on level N = 4, corresponding to the flavour group S 4, 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 $$ \sqrt{6} $$ ≈ 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 A 4 flavour symmetry.https://doi.org/10.1007/JHEP12(2019)030CP violationDiscrete SymmetriesNeutrino Physics |
| spellingShingle | Gui-Jun Ding Stephen F. King Xiang-Gan Liu Jun-Nan Lu Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing Journal of High Energy Physics CP violation Discrete Symmetries Neutrino Physics |
| title | Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing |
| title_full | Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing |
| title_fullStr | Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing |
| title_full_unstemmed | Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing |
| title_short | Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing |
| title_sort | modular s 4 and a 4 symmetries and their fixed points new predictive examples of lepton mixing |
| topic | CP violation Discrete Symmetries Neutrino Physics |
| url | https://doi.org/10.1007/JHEP12(2019)030 |
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