Modular $S_3$ symmetric radiative seesaw model

We propose a one-loop induced radiative seesaw model applying a modular $S_3$ flavor symmetry, which is known as the minimal non-Abelian discrete group. In this scenario, dark matter (DM) candidate is correlated with neutrinos and lepton flavor violations (LFVs). We show several predictions of mixings and phases satisfying LFVs, observed relic density, and neutrino oscillation data.

In this paper, we apply a S 3 modular symmetry to the lepton sector in a framework of Ma model [1], where S 3 is known as the minimal symmetry in non-Abelian discrete flavor symmetry. Here, we introduce two right-handed neutrinos that correspond to two singlets under S 3 and an isospin doublet inert boson in standard model (SM), both of which have nonzero charge of modular weight. In order to get a radiative seesaw model, we introduce additional Z 2 symmetry since the modular invariance is not sufficient to retain the radiative seesaw model. Therefore, Z 2 plays an role in assuring stability of DM. However, we realize a neutrino predictive model under one of the active neutrino masses is vanishing due to the two right-handed Majorana fermions, where the two kinds of fields originate from the fact that there are only two singlets under S 3 . 2 This is the first achievement in several series of modular flavor symmetry projects.
In our analysis, we show several predictions to the lepton sector, satisfying constraints of LFVs as well as neutrino oscillation data. Also, bosonic DM is favored compared to the 1 Several reviews are helpful to understand whole the ideas [34][35][36][37][38][39][40][41][42] for traditional applications and [43,44] for modular symmetries. 2 If we assign the right-handed Majorana fields as doublet under S 3 , we cannot reproduce the observed neutrino oscillation data because of few free parameters. fermionic one, since the interacting coupling between DM and the SM particles are too tiny to explain the observed relic density. 3 This paper is organized as follows. In Sec. II, we give our model set up under modular S 3 symmetry. Then, we discuss right-handed neutrino mass spectrum, lepton flavor violation (LFV), relic density of DM and generation of the active neutrino mass at one loop level.
Finally we conclude and discuss in Sec. IV.

II. MODEL
The modular groupΓ is the group of linear fractional transformation γ acting on the modulus τ , belonging to the upper-half complex plane as: which is isomorphic to P SL(2, Z) = SL(2, Z)/{I, −I} transformation. This modular transformation is generated by S and T , which satisfy the following algebraic relations, We introduce the series of groups Γ(N) (N = 1, 2, 3, . . . ) defined by isomorphic to S 3 , A 4 , S 4 and A 5 , respectively [2].

Fermions
Bosons where k is the so-called as the modular weight.
We discuss the modular symmetric theory without supersymmetry. In this paper, we fix the S 3 (N = 2) modular group. Under the modular transformation of Eq.(II.1), fields φ (I) transform as where −k I is the modular weight and ρ (I) (γ) denotes an unitary representation matrix of The kinetic terms of their scalar fields are written by which is invariant under the modular transformation. Also, the Lagrangian should be invariant under the modular symmetry. Here, we describe our scenario based on the Ma model, where field contents are exactly the same as the Ma model [1]. The S 3 representation and modular weight are given by Table I, while the ones of Yukawa couplings are given by Table II. Under these symmetries, one writes renormalizable Lagrangian as follows: whereη ≡ iσ 2 η * , σ 2 being second Pauli matrix.
The modular forms with the lowest weight 2; Y (2) 2 ≡ (y 1 , y 2 ), transforming as a doublet of S 3 is written in terms of Dedekind eta-function η(τ ) and its derivative [48]: Then, any couplings of higher weight are constructed by multiplication rules of S 3 , and one finds the following couplings: Higgs potential is given by 1 ||η| 2 (II.11) 1 ||η| 4 + λ Hη |Y which can be the same as the original potential of Ma model without loss of generality, because of additional free parameters. The point is that one does not have a term H † η due to absence of S 3 singlet with modular weight 2 that arises from the feature of modular symmetry.
The structure of Yukawa couplings are determined by the modular symmetry. Therefore, our model is more predictive than the standard Ma model. After the electroweak spontaneous symmetry breaking, the charged-lepton mass matrix is given by Then the charged-lepton mass eigenstate can be found by In our numerical analysis below, one can numerically fix the free parameters α ℓ , β ℓ , γ ℓ to fit the three charged-lepton masses after giving all the numerical values. Therefore, σ ℓ is an input parameter that is free.
The right-handed neutrino mass matrix is given by It suggests that right-handed neutrinos are diagonal with two degenerate masses for the second and third fields, and we define 1 . The Dirac Yukawa matrix is given by 2,2 ] T . Lepton flavor violations also arises from y D as [49,50]  which will be imposed in our numerical calculation.
Neutrino mass matrix is given at one-loop level by where m R(I) is a mass of the real (imaginary) component of η 0 . Then the neutrino mass 0.12 eV is given by the recent cosmological data [54]. Then, one finds Each of mixing is given in terms of the component of U M N S as follows: (II. 19) We provide the experimentally allowed ranges for neutrino mixings and mass difference squares at 3σ range [57] as follows: Also, the effective mass for the neutrinoless double beta decay is given by m ee = |D ν 1 cos 2 θ 12 cos 2 θ 13 + D ν 2 sin 2 θ 12 cos 2 θ 13 e iα 21 + D ν 3 sin 2 θ 13 e i(α 31 −2δ CP ) |, (II. 21) where its observed value could be measured by KamLAND-Zen in future [55].
To achieve numerical analysis, we derive several relations of the normalized neutrino mass matrix as follows:m where the last line is the first order approximation of the small mass difference between m 2 R and m 2 I ; m 2 R − m 2 I = ∆m 2 . 4 Then the normalized neutrino mass eigenvalues are given in terms of neutrino mass eigenvalues; diag(m 2 ν 1 ,m 2 ν 2 ,m 2 ν 3 ) = diag(m 2 ν 1 , m 2 ν 2 , m 2 ν 3 )/k 2 3 . It is found that k 2 3 is given by where normal hierarchy is assumed and ∆m 2 atm is the atmospheric neutrino mass difference square. Comparing Eq.(II.22) and Eq.(II.25), we find ∆m 2 is rewritten by the other parameters as follows: (II.24) The solar neutrino mass difference square is also found as In numerical analysis, this value should be within the experimental result, while ∆m 2 atm is expected to be input parameter.
DM is expected to be an imaginary component of inert scalar η; η I . In order to avoid the oblique parameters, we assume to be m η ± ≈ m I for simplicity. In this case, the mass of DM is uniquely fixed by the observed relic density which suggests it is within 534 ± 8.5 GeV [56], if the Yukawa coupling is not so large. In fact, tiny Yukawa couplings are requested by satisfying the data. Thus, we just work on the mass of η at this narrow range.

III. NUMERICAL ANALYSIS
Here, we show numerical analysis to satisfy all of the constraints that we discussed above, where we work on a basis that the neutrino mass ordering is normal hierarchy. 5 The range of absolute value of the five complex dimensionless parameters α ν , β ν , ρ ν , σ ν , σ ℓ are taken to be [0.01 − 1],while the mass parameters M 0 , M 1 are of the order [50,500] TeV. We have only two right handed neutrino, therefore m 1 =0 eV and α 21 =0 [deg]. 4 Advantage of this approximation is thatk α does not depend on ∆m. 5 We have checked that the inverted hierarchy is not favored in our model.    3. The lightest Majorana mass eigenstate is given by [2−9] TeV.

IV. CONCLUSION AND DISCUSSION
We have constructed a predictive lepton model with modular S 3 symmetry in framework of one-loop induced radiative seesaw model. The DM stability is naturally assured by Z 2 symmetry, and DM is correlated with neutrinos in a specific manner, where their interactions are determined by the S 3 symmetry that is known as the minimal group in non-Abelian discrete flavor symmetries. In our numerical analyses, we have highlighted several remarks as follows: 1. The Dirac phase and the Majorana phase are strongly correlated.
2. The typical region of modulus τ is found in narrow space as -0.