Modular A5 symmetry for flavour model building

128

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 expectation 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)/2, τ T S = (1 + i √ 3)/2, τ T = i∞ in the fundamental domain which are invariant under the modular transformations indicated. We then generalise these fixed points to τ f = γτ S , γτ ST , γτ T S and γτ T in the upper half complex plane, and show that it is sufficient to consider γ ∈ Γ N . 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 ≈ 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. *

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing

Ding

King

Liu

et al. 2019

128

“…The first few terms of the q-expansions of these modular forms can be found in Ref. [51]. Our numerical results have made use of q-expansions up to O(q 100 ), but the results are unchanged when using up to O(q 5 ).…”

Section: B Finite Modular Group γ 5 and Level 5 Modular Formsmentioning

confidence: 98%

“…Level 5 modular forms of weight 2 have been built in Ref. [51], making use of the Jacobi theta function:…”

Section: B Finite Modular Group γ 5 and Level 5 Modular Formsmentioning

confidence: 99%

“…However, the main advantage of the recent approach is that it can be implemented in a bottom-up perspective, relying on the group transformation properties of modular forms of given weight and level. Several models of lepton masses and mixing angles have been built at level 2 [42, 43], 3 [10, 44-47], 4 [48-50] and 5 [51,52]. Extensions to quarks [53,54] and to grand unified theories [55,56] have also been proposed.…”

mentioning

confidence: 99%

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Modular invariant models of lepton masses at levels 4 and 5

Criado

Feruglio

King

2020

111

We explore alternative descriptions of the charged lepton sector in modular invariant models of lepton masses and mixing angles. In addition to the modulus, the symmetry breaking sector of our models includes ordinary flavons. Neutrino mass terms depend only on the modulus and are tailored to minimize the number of free parameters. The charged lepton Yukawa couplings rely upon the flavons alone. We build modular invariant models at levels 4 and 5, where neutrino masses are described both in terms of the Weinberg operator or through a type I seesaw mechanism. At level 4, our models reproduce the hierarchy among electron, muon and tau masses by letting the weights play the role of Froggatt-Nielsen charges. At level 5, our setup allows the treatment of left and right handed charged leptons on the same footing. We have optimized the free parameters of our models in order to match the experimental data, obtaining a good degree of compatibility and predictions for the absolute neutrino masses and the CP violating phases. At a more fundamental level, the whole lepton sector could be correctly described by the simultaneous presence of several moduli. Our examples are meant to make a first step in this direction. arXiv:1908.11867v1 [hep-ph] 30 Aug 2019 Recently, modular invariance has been invoked as candidate flavour symmetry [10]. In its simplest implementation a unique complex field, the modulus, acts as symmetry breaking parameter, thus simplifying the vacuum alignment problem. Modular invariance, in the limit of exact supersymmetry, completely determines the Yukawa couplings, to any order of the expansion in powers of the modulus. Moreover, neutrino masses, mixing angles and phases are all related to each other and, in minimal models, depend only on a few parameters. The formalism has been extended to consistently include CP transformations [11] 1 and it can involve several moduli [16,17]. The idea that Yukawa couplings are determined by a set of moduli is clearly not new, and has been naturally realized in the context of string theory [18][19][20][21][22], in D-brane compactification [23][24][25][26][27][28][29], in magnetized extra dimensions [30][31][32], and in orbifold compactification [33][34][35][36]. Modular invariance has also been incorporated in early flavour models [37][38][39][40][41]. However, the main advantage of the recent approach is that it can be implemented in a bottom-up perspective, relying on the group transformation properties of modular forms of given weight and level. Several models of lepton masses and mixing angles have been built at level 2 [42, 43], 3 [10, 44-47], 4 [48-50] and 5 [51,52]. Extensions to quarks [53,54] and to grand unified theories [55,56] have also been proposed. In most of the existing constructions, there is a unique symmetry breaking parameter: the modulus itself. While this scenario is certainly appealing since it minimizes the symmetry breaking sector, it does not yet provide a convincing explanation of the charged lepton masses. The mass hierarchy is achieved by...

Symplectic modular symmetry in heterotic string vacua: flavor, CP, and R-symmetries

Ishiguro

Kobayashi

Otsuka

2022

We examine a common origin of four-dimensional flavor, CP, and U(1)R symmetries in the context of heterotic string theory with standard embedding. We find that flavor and U(1)R symmetries are unified into the Sp(2h + 2, ℂ) modular symmetries of Calabi-Yau threefolds with h being the number of moduli fields. Together with the $$ {\mathbb{Z}}_2^{\mathrm{CP}} $$ ℤ 2 CP CP symmetry, they are enhanced to GSp(2h + 2, ℂ) ≃ Sp(2h + 2, ℂ) ⋊ $$ {\mathbb{Z}}_2^{\mathrm{CP}} $$ ℤ 2 CP generalized symplectic modular symmetry. We exemplify the S3, S4, T′, S9 non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and ℤ2, S4 flavor symmetries on three-parameter examples of Calabi-Yau threefolds. Thus, non-trivial flavor symmetries appear in not only the exact orbifold limit but also a certain class of Calabi-Yau three-folds. These flavor symmetries are further enlarged to non-Abelian discrete groups by the CP symmetry.