Abstract
We study the phenomenological implications of the modular symmetry Γ(3) ≃ A4 of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus τ, which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with A4 symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses, the seesaw model is available by taking account of recent experimental data of neutrino oscillations and the cosmological bound of sum of neutrino masses. The predicted sin2θ23 is restricted to be larger than 0.54 and δCP = ±(50°-180°). Since the correlation of sin2θ23 and δCP is sharp, the prediction is testable in the future. It is remarkable that the effective mass mee of the neutrinoless double beta decay is around 22 meV while the sum of neutrino masses is predicted to be 145 meV. On the other hand, for the inverted hierarchy of neutrino masses, only the Dirac neutrino model is consistent with the experimental data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T2K collaboration, K. Abe et al., Measurement of neutrino and antineutrino oscillations by the T2K experiment including a new additional sample of ν e interactions at the far detector, Phys. Rev. D 96 (2017) 092006 [Erratum ibid. D 98 (2018) 019902] [arXiv:1707.01048] [INSPIRE].
W. Morgan, T2K Status, Results, and Plans, talk given at the XXVIII International Conference on Neutrino Physics and Astrophysics, Heidelberg, Germany, 4-9 June 2018, https://doi.org/10.5281/zenodo.1286751.
NOvA collaboration, P. Adamson et al., Constraints on Oscillation Parameters from ν e Appearance and ν μ Disappearance in NOvA, Phys. Rev. Lett. 118 (2017) 231801 [arXiv:1703.03328] [INSPIRE].
M. Sanchez, NOvA Results and Prospects?, talk given at the XXVIII International Conference on Neutrino Physics and Astrophysics, Heidelberg, Germany, 4-9 June 2018, https://doi.org/10.5281/zenodo.1286757.
G. Altarelli and F. Feruglio, Discrete Flavor Symmetries and Models of Neutrino Mixing, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211] [INSPIRE].
H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Non-Abelian Discrete Symmetries in Particle Physics, Prog. Theor. Phys. Suppl. 183 (2010) 1 [arXiv:1003.3552] [INSPIRE].
H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, An introduction to non-Abelian discrete symmetries for particle physicists, Springer, Lect. Notes Phys. 858 (2012) 1 [INSPIRE].
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, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, Neutrino Mass and Mixing: from Theory to Experiment, New J. Phys. 16 (2014) 045018 [arXiv:1402.4271] [INSPIRE].
E. Ma and G. Rajasekaran, Softly broken A 4 symmetry for nearly degenerate neutrino masses, Phys. Rev. D 64 (2001) 113012 [hep-ph/0106291] [INSPIRE].
K.S. Babu, E. Ma and J.W.F. Valle, Underlying A 4 symmetry for the neutrino mass matrix and the quark mixing matrix, Phys. Lett. B 552 (2003) 207 [hep-ph/0206292] [INSPIRE].
G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing from discrete symmetry in extra dimensions, Nucl. Phys. B 720 (2005) 64 [hep-ph/0504165] [INSPIRE].
G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A 4 and the modular symmetry, Nucl. Phys. B 741 (2006) 215 [hep-ph/0512103] [INSPIRE].
Y. Shimizu, M. Tanimoto and A. Watanabe, Breaking Tri-bimaximal Mixing and Large θ 13, Prog. Theor. Phys. 126 (2011) 81 [arXiv:1105.2929] [INSPIRE].
S.K. Kang, Y. Shimizu, K. Takagi, S. Takahashi and M. Tanimoto, Revisiting A 4 model for leptons in light of NuFIT 3.2, Prog. Theor. Exp. Phys. 2018 (2018) 083B01 [arXiv:1804.10468] [INSPIRE].
S.T. Petcov and A.V. Titov, Assessing the Viability of A 4 , S 4 and A 5 Flavour Symmetries for Description of Neutrino Mixing, Phys. Rev. D 97 (2018) 115045 [arXiv:1804.00182] [INSPIRE].
T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [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].
P. Ko, T. Kobayashi, J.-h. Park and S. Raby, String-derived D 4 flavor symmetry and phenomenological implications, Phys. Rev. D 76 (2007) 035005 [Erratum ibid. D 76 (2007) 059901] [arXiv:0704.2807] [INSPIRE].
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models, Nucl. Phys. B 820 (2009) 317 [arXiv:0904.2631] [INSPIRE].
M. Berasaluce-Gonzalez, P.G. Camara, F. Marchesano, D. Regalado and A.M. Uranga, Non-Abelian discrete gauge symmetries in 4d string models, JHEP 09 (2012) 059 [arXiv:1206.2383] [INSPIRE].
J. Lauer, J. Mas and H.P. Nilles, Duality and the Role of Nonperturbative Effects on the World Sheet, Phys. Lett. B 226 (1989) 251 [INSPIRE].
J. Lauer, J. Mas and H.P. Nilles, Twisted sector representations of discrete background symmetries for two-dimensional orbifolds, Nucl. Phys. B 351 (1991) 353 [INSPIRE].
W. Lerche, D. Lüst and N.P. Warner, Duality Symmetries in N = 2 Landau-Ginzburg Models, Phys. Lett. B 231 (1989) 417 [INSPIRE].
S. Ferrara, D. Lüst and S. Theisen, Target Space Modular Invariance and Low-Energy Couplings in Orbifold Compactifications, Phys. Lett. B 233 (1989) 147 [INSPIRE].
D. Cremades, L.E. Ibáñez and F. Marchesano, Computing Yukawa couplings from magnetized extra dimensions, JHEP 05 (2004) 079 [hep-th/0404229] [INSPIRE].
T. Kobayashi and S. Nagamoto, Zero-modes on orbifolds: magnetized orbifold models by modular transformation, Phys. Rev. D 96 (2017) 096011 [arXiv:1709.09784] [INSPIRE].
T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba and T.H. Tatsuishi, Modular symmetry and non-Abelian discrete flavor symmetries in string compactification, Phys. Rev. D 97 (2018) 116002 [arXiv:1804.06644] [INSPIRE].
F. Feruglio, Are neutrino masses modular forms?, arXiv:1706.08749 [INSPIRE].
T. Kobayashi, K. Tanaka and T.H. Tatsuishi, Neutrino mixing from finite modular groups, Phys. Rev. D 98 (2018) 016004 [arXiv:1803.10391] [INSPIRE].
J.T. Penedo and S.T. Petcov, Lepton Masses and Mixing from Modular S 4 Symmetry, arXiv:1806.11040 [INSPIRE].
J.C. Criado and F. Feruglio, Modular Invariance Faces Precision Neutrino Data, SciPost Phys. 5 (2018) 042 [arXiv:1807.01125] [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].
S. Ferrara, D. Lüst, A.D. Shapere and S. Theisen, Modular Invariance in Supersymmetric Field Theories, Phys. Lett. B 225 (1989) 363 [INSPIRE].
R.C. Gunning, Lectures on Modular Forms, Princeton University Press, Princeton NJ U.S.A. (1962).
NuFIT 3.2, (2018) http://www.nu-fit.org/.
E. Giusarma, M. Gerbino, O. Mena, S. Vagnozzi, S. Ho and K. Freese, Improvement of cosmological neutrino mass bounds, Phys. Rev. D 94 (2016) 083522 [arXiv:1605.04320] [INSPIRE].
S. Vagnozzi et al., Unveiling ν secrets with cosmological data: neutrino masses and mass hierarchy, Phys. Rev. D 96 (2017) 123503 [arXiv:1701.08172] [INSPIRE].
Planck collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [INSPIRE].
C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP-violation, Phys. Rev. Lett. 55 (1985) 1039 [INSPIRE].
Z. Maki, M. Nakagawa and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28 (1962) 870 [INSPIRE].
B. Pontecorvo, Neutrino Experiments and the Problem of Conservation of Leptonic Charge, Sov. Phys. JETP 26 (1968) 984 [Zh. Eksp. Teor. Fiz. 53 (1967) 1717] [INSPIRE].
S.M. Bilenky, S. Pascoli and S.T. Petcov, Majorana neutrinos, neutrino mass spectrum, CP-violation and neutrinoless double beta decay. 1. The Three neutrino mixing case, Phys. Rev. D 64 (2001) 053010 [hep-ph/0102265] [INSPIRE].
J.F. Nieves and P.B. Pal, Minimal Rephasing Invariant CP Violating Parameters With Dirac and Majorana Fermions, Phys. Rev. D 36 (1987) 315 [INSPIRE].
J.F. Nieves and P.B. Pal, Rephasing invariant CP-violating parameters with Majorana neutrinos, Phys. Rev. D 64 (2001) 076005 [hep-ph/0105305] [INSPIRE].
J.A. Aguilar-Saavedra and G.C. Branco, Unitarity triangles and geometrical description of CP-violation with Majorana neutrinos, Phys. Rev. D 62 (2000) 096009 [hep-ph/0007025] [INSPIRE].
I. Girardi, S.T. Petcov and A.V. Titov, Predictions for the Majorana CP-violation Phases in the Neutrino Mixing Matrix and Neutrinoless Double Beta Decay, Nucl. Phys. B 911 (2016) 754 [arXiv:1605.04172] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1808.03012
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kobayashi, T., Omoto, N., Shimizu, Y. et al. Modular A4 invariance and neutrino mixing. J. High Energ. Phys. 2018, 196 (2018). https://doi.org/10.1007/JHEP11(2018)196
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2018)196