Abstract
We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire. . . : Guido Altarelli’s Legacy, A. Levy, S. Forte and G. Ridolfi eds. (2019) DOI [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 S4 Symmetry, Nucl. Phys. B 939 (2019) 292 [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].
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].
H. Okada and M. Tanimoto, CP violation of quarks in A4 modular invariance, Phys. Lett. B 791 (2019) 54 [arXiv:1812.09677] [INSPIRE].
G.-J. Ding, S. F. King and X.-G. Liu, Neutrino mass and mixing with A5 modular symmetry, Phys. Rev. D 100 (2019) 115005 [arXiv:1903.12588] [INSPIRE].
P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, Generalised CP Symmetry in Modular-Invariant Models of Flavour, JHEP 07 (2019) 165 [arXiv:1905.11970] [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].
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, A4 lepton flavor model and modulus stabilization from S4 modular symmetry, Phys. Rev. D 100 (2019) 115045 [Erratum ibid. 101 (2020) 039904] [arXiv:1909.05139] [INSPIRE].
T. Asaka, Y. Heo, T. H. Tatsuishi and T. Yoshida, Modular A4 invariance and leptogenesis, JHEP 01 (2020) 144 [arXiv:1909.06520] [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].
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida, C P violation in modular invariant flavor models, Phys. Rev. D 101 (2020) 055046 [arXiv:1910.11553] [INSPIRE].
G.-J. Ding and F. Feruglio, Testing Moduli and Flavon Dynamics with Neutrino Oscillations, JHEP 06 (2020) 134 [arXiv:2003.13448] [INSPIRE].
X.-G. Liu, C.-Y. Yao, B.-Y. Qu and G.-J. Ding, Half-integral weight modular forms and application to neutrino mass models, Phys. Rev. D 102 (2020) 115035 [arXiv:2007.13706] [INSPIRE].
G.-J. Ding, F. Feruglio and X.-G. Liu, Automorphic Forms and Fermion Masses, JHEP 01 (2021) 037 [arXiv:2010.07952] [INSPIRE].
C.-Y. Yao, X.-G. Liu and G.-J. Ding, Fermion Masses and Mixing from Double Cover and Metaplectic Cover of A5 Modular Group, arXiv:2011.03501 [INSPIRE].
T. Kobayashi, S. Nagamoto and S. Uemura, Modular symmetry in magnetized/intersecting D-brane models, PTEP 2017 (2017) 023B02 [arXiv:1608.06129] [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].
T. Kobayashi and S. Tamba, Modular forms of finite modular subgroups from magnetized D-brane models, Phys. Rev. D 99 (2019) 046001 [arXiv:1811.11384] [INSPIRE].
Y. Kariyazono, T. Kobayashi, S. Takada, S. Tamba and H. Uchida, Modular symmetry anomaly in magnetic flux compactification, Phys. Rev. D 100 (2019) 045014 [arXiv:1904.07546] [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 scheme from ten-dimensional string theory — I. Basic results, Phys. Lett. B 808 (2020) 135615 [arXiv:2006.03059] [INSPIRE].
A. Baur, M. Kade, H. P. Nilles, S. Ramos-Sanchez and P. K. S. Vaudrevange, The eclectic flavor symmetry of the ℤ2 orbifold, JHEP 02 (2021) 018 [arXiv:2008.07534] [INSPIRE].
A. Baur, M. Kade, H. P. Nilles, S. Ramos-Sanchez and P. K. S. Vaudrevange, Siegel modular flavor group and CP from string theory, Phys. Lett. B 816 (2021) 136176 [arXiv:2012.09586] [INSPIRE].
H. Ohki, S. Uemura and R. Watanabe, Modular flavor symmetry on a magnetized torus, Phys. Rev. D 102 (2020) 085008 [arXiv:2003.04174] [INSPIRE].
S. Kikuchi, T. Kobayashi, S. Takada, T. H. Tatsuishi and H. Uchida, Revisiting modular symmetry in magnetized torus and orbifold compactifications, Phys. Rev. D 102 (2020) 105010 [arXiv:2005.12642] [INSPIRE].
S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada and H. Uchida, Modular symmetry by orbifolding magnetized T2 × T2: realization of double cover of ΓN, JHEP 11 (2020) 101 [arXiv:2007.06188] [INSPIRE].
K. Hoshiya, S. Kikuchi, T. Kobayashi, Y. Ogawa and H. Uchida, Classification of three-generation models by orbifolding magnetized T2 × T2, PTEP 2021 (2021) 033B05 [arXiv:2012.00751] [INSPIRE].
S. Kikuchi, T. Kobayashi and H. Uchida, Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds, arXiv:2101.00826 [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].
D. Cremades, L. E. Ibáñez and F. Marchesano, Computing Yukawa couplings from magnetized extra dimensions, JHEP 05 (2004) 079 [hep-th/0404229] [INSPIRE].
W. Buchmüller, M. Dierigl, E. Dudas and J. Schweizer, Effective field theory for magnetic compactifications, JHEP 04 (2017) 052 [arXiv:1611.03798] [INSPIRE].
D. M. Ghilencea and H. M. Lee, Wilson lines and UV sensitivity in magnetic compactifications, JHEP 06 (2017) 039 [arXiv:1703.10418] [INSPIRE].
W. Buchmüller, M. Dierigl and E. Dudas, Flux compactifications and naturalness, JHEP 08 (2018) 151 [arXiv:1804.07497] [INSPIRE].
T. Hirose and N. Maru, Cancellation of One-loop Corrections to Scalar Masses in Yang-Mills Theory with Flux Compactification, JHEP 08 (2019) 054 [arXiv:1904.06028] [INSPIRE].
D. Mumford, Tata lectures on theta I, Birkhäuser ed., Springer, Boston U.S.A. (1983).
U. Dudley, Elementary number theory: Second edition, Dover Books on Mathematics, Dover Publications, Dover U.K. (2012).
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Magnetic flux, Wilson line and orbifold, Phys. Rev. D 80 (2009) 126006 [arXiv:0907.5274] [INSPIRE].
J. Bruinier, G. van der Geer, G. Harder and D. Zagier, 1-2-3 of modular forms, Springer, Berlin Germany (2008).
The GAP Group, GAP — Groups, Algorithms, and Programming, version 4.11.0 (2020) https://www.gap-system.org/Releases/4.11.0.html.
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
V. S. Kaplunovsky and J. Louis, Model independent analysis of soft terms in effective supergravity and in string theory, Phys. Lett. B 306 (1993) 269 [hep-th/9303040] [INSPIRE].
P. Ramond, Group theory: A physicist’s survey, Cambridge University Press (2010).
R. N. Mohapatra and G. Senjanovic, Neutrino Mass and Spontaneous Parity Nonconservation, Phys. Rev. Lett. 44 (1980) 912.
T. W. B. Kibble, G. Lazarides and Q. Shafi, Walls Bounded by Strings, Phys. Rev. D 26 (1982) 435 [INSPIRE].
D. Chang, R. N. Mohapatra and M. K. Parida, Decoupling Parity and SU(2)-R Breaking Scales: A New Approach to Left-Right Symmetric Models, Phys. Rev. Lett. 52 (1984) 1072 [INSPIRE].
S. Biermann, A. Mütter, E. Parr, M. Ratz and P. K. S. Vaudrevange, Discrete remnants of orbifolding, Phys. Rev. D 100 (2019) 066030 [arXiv:1906.10276] [INSPIRE].
P. Di Vecchia, A. Liccardo, R. Marotta and F. Pezzella, Kähler Metrics and Yukawa Couplings in Magnetized Brane Models, JHEP 03 (2009) 029 [arXiv:0810.5509] [INSPIRE].
S. A. Abel and A. W. Owen, N point amplitudes in intersecting brane models, Nucl. Phys. B 682 (2004) 183 [hep-th/0310257] [INSPIRE].
L. E. Ibáñez and A. M. Uranga, String theory and particle physics: An introduction to string phenomenology, Cambridge University Press (2012).
L. E. Ibáñez and D. Lüst, Duality anomaly cancellation, minimal string unification and the effective low-energy Lagrangian of 4-D strings, Nucl. Phys. B 382 (1992) 305 [hep-th/9202046] [INSPIRE].
K. R. Dienes, Modular invariance, finiteness, and misaligned supersymmetry: New constraints on the numbers of physical string states, Nucl. Phys. B 429 (1994) 533 [hep-th/9402006] [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998), pp. 402.
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: 2102.11286
Address from September 2021: Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, U.S.A. (Víctor Knapp-Pérez)
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
Almumin, Y., Chen, MC., Knapp-Pérez, V. et al. Metaplectic flavor symmetries from magnetized tori. J. High Energ. Phys. 2021, 78 (2021). https://doi.org/10.1007/JHEP05(2021)078
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)078