$A_4$ lepton flavor model and modulus stabilization from $S_4$ modular symmetry

We study the modulus stabilization in an $A_4$ model whose $A_4$ flavor symmetry is originated from the $S_4$ modular symmetry. We can stabilize the modulus so that the $A_4$ invariant superpotential leads to the realistic lepton masses and mixing angles. We also discuss the phenomenological aspect of the present model as a consequence of the modulus stabilization.


Introduction
The origin of the flavor structure is one of important mysteries in particle physics. The recent development of the neutrino oscillation experiments provides us helpful information to investigate the flavor physics. The neutrino oscillation experiments have presented two large flavor mixing angles, which contrast with quark mixing angles. The T2K and NOνA strongly indicate the CP violation in the neutrino oscillation [1,2]. Thus, we are in the era to develop the flavor theory with facing the experimental data.
Geometrical symmetries of compact space in extra dimensional field theories and superstring theory can be origins of non-Abelian discrete flavor symmetries 1 . Torus compactification and orbifold compactification are simple compactifications. These compactifications have the modular symmetry SL(2, Z) as the geometrical symmetry. The shape of the torus is described by the modulus τ , and the modular group transforms the modulus non-trivially. The modular group SL(2, Z) has infinite order, but it includes finite subgroups such as Γ 2 S 3 , Γ 3 A 4 , Γ 4 S 4 and Γ 5 A 5 [24]. Furthermore, the modular group transforms zero-modes each other [25][26][27][28][29][30]. Thus, the modular symmetry is a sort of flavor symmetries. However, Yukawa couplings as well as other couplings are functions of the modulus, and those couplings also transform non-trivially under the modular symmetry.
In minimal model building, we do not need to introduce flavon fields to break flavor symmetries because flavor symmetries are broken when the value of τ is fixed. We can realize lepton and quark masses and mixing angles by choosing a proper value of the modulus τ as well as other parameters of models. It is important how to fix the value of τ , i.e. the modulus stabilization. The modulus value can be fixed as a minimum of scalar potential in supergravity theory. The modular invariant supergravity theory was studied [59] 2 . Indeed, the modulus stabilization was studied by assuming the SL(2, Z) modular invariance for the non-perturbative superpotential in supergravity theory [63,64] The purpose of this paper is to study the modulus stabilization and its phenomenological implications in Γ N flavor models. We consider the modulus stabilization by using the model in Ref. [52] as an illustrating model. Non-Abelian discrete symmetries can be anomalous [66]. (See also for anomalies of the modular symmetry in concrete models [67].) For example, S 4 can be anomalous and thus broken down to A 4 by anomalies. In the model of Ref. [52], the S 4 modular symmetry is imposed at the tree level and assumed to be broken to A 4 by anomalies.
In this paper, we study an A 4 invariant superpotential of the modulus τ to stabilize it at a supersymmetric minimum of the supergravity scalar potential. We discuss phenomenological aspects in our model. This paper is organized as follows. In section 2, we give a brief review on the modular symmetry and the S 4 anomaly. In section 3, we review on the A 4 flavor model in Ref. [52].
In section 4, we study the modulus stabilization in the A 4 model. In section 5, we study phenomenological aspects through the modulus stabilization in the A 4 model. Section 6 is devoted to our conclusion. Relevant representations of S 4 and A 4 groups are presented in Appendix A. We list the input data of neutrinos in Appendix B. In Appendix C, we show a scenario to induce the modulus potential.
2 Modular symmetry and S 4 anomaly

Modular symmetry
We give a brief review on the modular symmetry and modular forms. The torus compactification is the simplest compactification. The modulus τ of the torus transforms under the modular transformation as where a, b, c, d are integer with satisfying ad − bc = 1. This is the symmetry P SL(2, Z) = SL(2, Z)/Z 2 , which is denoted by Γ. The modular symmetry is generated by two elements, S and T : They satisfy the following algebraic relations, Furthermore, we define the congruence subgroups of level N as The quotient subgroups Γ N are given as Γ N ≡ Γ/Γ(N ), and these are finite for N = 2, 3, 4, 5, i.e. Γ 2 S 3 , Γ 3 A 4 , Γ 4 S 4 , Γ 5 A 5 . The algebraic relation T N = I is satisfied for Γ(N ) in addition to Eq.(3). We study the modular invariant supergravity theory. We use the unit that M P = 1 where M P denotes the reduced Planck scale. A typical Kähler potential of the modulus field τ is written as follows: The Kähler potential transforms under the modular symmetry as Supergravity theory can be written by G, where W denotes the superpotential in supergravity theory. We require that G is invariant under the modular transformation. The superpotential W therefore transforms as under the modular transformation. That is, the superpotential must be a holomorphic function of the modular weight −1. Chiral matter fields φ (I) with the modular weight −k I transform as under the modular symmetry, where ρ(γ) ij is a unitary matrix in Γ N . Their Kähler potential can be written as Moreover, the modular forms of weight k are the holomorphic functions of τ and transform as The modular forms of Γ(4) have been constructed by use of the Dedekind eta function, η(τ ), in Ref. [35].
where q = e 2πiτ . The modular forms of the weight 2 are written by where ω = e 2πi/3 and Y (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 τ ) = a 1 η (τ + 1/2) These five modular forms correspond to reducible representations of Γ 4 S 4 , and these are decomposed into the 2 and 3 representations under S 4 , The generators, S and T , are represented on the above modular forms, for 2, and for 3 . The modular forms of higher weights are obtained as the products of Y S 4 2 (τ ) and Y S 4 3 (τ ). See for other representations in Appendix A.

Anomaly
A discrete symmetry can be anomalous like a continuous symmetry [66,[68][69][70]. Each element g in a non-Abelian discrete group satisfies g N = 1, that is, the Abelian Z N subgroup. If all of Abelian discrete subgroups in a non-Abelian discrete group are anomaly-free, the whole non-Abelian symmetry is anomaly-free [66]. Otherwise, the non-Abelian symmetry is anomalous, and anomalous subgroup is broken. Furthermore, each element g is represented by a matrix ρ(g). If det ρ(g) = 1, the corresponding Z N is always anomaly-free. On the other hand, if det ρ(g) = 1, the corresponding Z N symmetry can be anomalous [4,5,66]. In Refs. [4,5], it is shown explicitly which subgroups can be anomalous in non-Abelian discrete symmetries. The S 4 group is isomorphic to (Z 2 × Z 2 ) S 3 . The Z 2 symmetry of S 3 can be anomalous in S 4 . In general, the 2 and 3 representations as well as 1 have det ρ(g) = −1 while the 1 and 3 representations have det ρ(g) = 1. Indeed, ρ(S) and ρ(T ) for 2 as well as 3 and 1 have det(ρ(S)) = det(ρ(T )) = −1.
If the above Z 2 symmetry in S 4 is anomalous, S 4 is broken to A 4 by anomalies. In this case, S and T themselves are anomalous, butS = T 2 andT = ST are anomaly-free. These anomaly-free elements satisfy if we impose T 4 = I, that is, the A 4 algebra is realized. The modular forms of weight 2 for S 4 correspond to A 4 representations as follows: We have Note that these are not modular forms of Γ(3) becauseS = T 2 andT = ST do not generate SL(2, Z). We can also write S 4 singlet modular forms of weights 4 and 6 Both are trivial singlets 1 also under A 4 . These are useful for our study.

A 4 lepton model from S 4 modular symmetry
We briefly review on the A 4 lepton flavor model in Ref. [52]. Our A 4 flavor symmetry is originated from the S 4 modular symmetry by assuming that the S 4 symmetry is broken to A 4 by anomalies as mentioned in the previous section. The model in this paper is described in the supergravity basis where the superpotential has the modular weight −1. On the other hand, the model in Ref. [52] is a global supersymmetric model where the superpotential has the vanishing weight. Thus, we rearrange modular weights of chiral superfields. We assign the modular weight −1 to all of the left-handed and right-handed leptons and Higgs fields.
For the A 4 flavor symmetry, the left-handed lepton doublets, (L e , L µ , L τ ) T correspond to the A 4 triplet L 3 , and the right-handed charged leptons are assigned to the A 4 singlets of 1, 1 , 1 , i.e. e c 1 , µ c 1 , τ c 1 ; while the up-and down-sector Higgs fields, H u and H d , are assigned to the trivial singlet. The charge assignment of the fields and modular forms is summarized in Table  1.
The superpotential of the neutrino mass term is given by the Weinberg operator: where Λ is a cut-off scale; and parameters a and b are complex constants in general. The superpotential of the mass term of the charged leptons is described as where α, β and γ are taken to be real and positive without loss of generality. The superpotential w in the global supersymmetry basis is related to one in the supergravity basis by |w| 2 = e K |W | 2 , i.e. |w ν | 2 = |W ν | 2 /|τ −τ | and |w e | 2 = |W e | 2 /|τ −τ | 4 . For canonically normalized lepton fields, the Majorana neutrino mass matrix is written as follows: where while the charged lepton matrix is given as with The parameters α , β , γ are determined by the observed charged lepton masses and the value of τ . We take a and b to be real in order to present a simple viable model. We scan parameters in the following ranges: where the fundamental domain of Γ(4) is taken into account. The lower-cut 0.1 of Im[τ ] is artificial to keep the accurate numerical calculation. The upper-cut 2.8 is large enough to estimate the modular forms. We input the experimental data within 3 σ C.L.
[71] of three mixing angles in the lepton mixing matrix [72] in order to constrain the magnitudes of parameters. We also put the observed neutrino mass ratio ∆m 2 sol /∆m 2 atm and the cosmological bound for the neutrino masses m i < 0.12 [eV] [73,74]. There are two possible spectra of neutrinos masses m i , which are the normal hierarchy (NH), m 3 > m 2 > m 1 , and the inverted hierarchy (IH), m 2 > m 1 > m 3 . Figure 1 shows allowed regions for NH (Cyan) and IH(Red), respectively.

Modulus stabilization
We study the modulus stabilization in the A 4 symmetric model where the S 4 modular symmetry is assumed to be broken by anomalies. For the modulus stabilization, we need a modulusdependent superpotential W (τ ) which may be induced by non-perturbative effects. Such superpotential W (τ ) must have the modular weight −1 for the modular invariance. However, there is no modular form of odd weights for Γ(4). We need some mechanism to generate the superpotential term for modulus stabilization.
Here, we assume that the following superpotential, where we assumed that Λ d has the modular weight 3. This modulus superpotential may be induced from the condensation QQ = 0 in the hidden sector by strong dynamics such as supersymmetric QCD, and Λ The scalar potential in supergravity theory is written by using K in Eq. (5) and W in Eq.(29) as where with K τ = ∂K/∂τ and W τ = ∂W/∂τ . We analyze the minimum of the above scalar potential V by examining the stationary condition, ∂V /∂τ = 0. If there is a solution in the following equation, we have ∂V /∂τ = 0. Such a solution is a candidate for the potential minimum and corresponds to a supersymmetric minimum. However, the above scalar potential has no proper supersymmetric minimum. For the slice of Re(τ ) = 0, the value of |A(τ )| ≡ |D τ W |/Λ d is shown in Figure 2 for larger values of Im[τ ]. The value |D τ W | becomes to vanish as Im[τ ] → ∞. Similarly, |D τ W | becomes to vanish as Im[τ ] → 0, because Im[τ ] → 0 and ∞ are related to each other by the S transformation. The minimum corresponds to Im[τ ] → 0 and ∞. There is no supersymmetric minimum for a finite value of τ . On the other hand, the scalar potential has non-supersymmetric minima as shown in Figure  3. The minima correspond to τ = 1.54i + n, where n is integer. Unfortunately, these minima do not lead to realistic lepton mass matrices.
d ) 2 at these minima. We need to uplift the vacuum energy by other supersymmetry breaking effects in order to realize almost vanishing vacuum energy V ≈ 0. Such uplifting effects may not shift significantly the stabilized value τ = 1.54i + n because the modulus mass squared is large compared with the negative vacuum energy V ∼ −0.5 × (Λ Figure 3: A contour map of the scalar potential for W in Eq. (29). The potential minima correspond to τ = 1.54i + n, where n is integer.
Alternatively, we assume the following superpotential, where we assumed that Λ (−5) d has the modular weight −5. However, the corresponding scalar potential has no proper supersymmetric minimum. Fig. 4 shows the corresponding scalar potential. Its minima correspond to τ = 1.55i+n/2, where n is odd. Unfortunately, these values also do not lead to realistic lepton mass matrices.(See Figure 1 2 , and the modulus mass squared m 2 τ ∼ 400 × (Λ (−5) d ) 2 at these minima. The effects from uplifting the vacuum energy to V ≈ 0 on the stabilized value τ = 1.55i + n is small because the modulus mass squared is large compared with the negative vacuum energy V ∼ −2 × (Λ We can use the modular form Y (6) (τ ) instead of Y (4) (τ ) in Eqs. (29) and (33). When we replace Y (4) (τ ) in Eq. (29) by Y (6) (τ ), the corresponding scalar potential has the minimum at τ = 1.68i + 1/2. On the other hand, when we replace Y (4) (τ ) in Eq. (33) by Y (6) (τ ), the corresponding scalar potential has the minimum at τ = 1.69i. Unfortunately, these values of τ are not proper to realize the lepton masses and mixing angles.
Thus, we can stabilize the modulus, but its values are not realistic when the superpotential includes a single modular form. We need more terms to stabilize the modulus at a proper value. For example, we assume the following superpotential, Figure 4: A contour map of the scalar potential for W in Eq. (33). The potential minima correspond to τ = 1.55i + n/2, where n is odd.
where Λ (5) d is assumed to have the modular weight 5. Here, we define ρ = Λ d . This superpotential always has a supersymmetric minimum for a finite value of ρ. We focus on such a supersymmetric minimum.
For smaller values of τ , the Kähler potential of Eq.(5) may have corrections. Thus, we restrict ourselves to the case with τ = O(1). That is, we study the A, B and C regions in Figure 1. We can choose a proper value of ρ such that τ is fixed to be a value in the A, B and C regions through Eq. (32).    Similarly, we can use the following superpotential: by assuming that non-perturbative effects generate it and Λ . Then, similarly we can study the modulus stabilization by using this superpotential. Again, we analyze the supersymmetric condition Eq. (32). We can find values of the modulus τ , which satisfy the supersymmetric condition Eq.    As results, we can stabilize the modulus τ at realistic values in three regions A, B, C by using both the superpotential terms, (34) and (35) with proper values of the parameters, ρ and ρ . In the next section, we study phenomenological aspects of these three regions following the modulus stabilization by both the superpotential terms, (34) and (35).

Phenomenological aspects of leptons
In this section, we discuss phenomenological results derived from the mass matrices of charged leptons and neutrinos for three regions A, B, C of the modulus in Fig.1, respectively.

Region of A
Let us present numerical results in the region A of the modulus τ . The parameter ρ which realizes the potential minimum for the superpotential (34) is shown in the Re[ρ]-Im[ρ] plane of Fig. 5, while Fig. 8 shows ρ for the potential minimum for the superpotential (35). In this case, NH is only available.
At first, we show a correlation between δ CP and sin 2 θ 23 in Fig. 11.  Table 2.

Region of B
We discuss numerical results in the region B of the modulus τ . The parameter ρ which realizes the potential minimum for the superpotential (34)   while Fig. 9 shows ρ for the potential minimum for the superpotential (35). In this case, NH of neutrino masses is only available. We show the correlation between δ CP and sin 2 θ 23 in Fig. 15 The neutrino mass dependence for sin 2 θ 23 is almost same as the result of region A, as seen in Fig. 17. The effective mass of the 0νββ decay m ee is also same as the result of region A, as seen in Fig. 18. We summarize the prediction for m ee and m i in Table 2.

Region of C
Finally, we present numerical discussions in the region C of the modulus τ . The parameter ρ which realizes the potential minimum for the superpotential (34) is shown in the Re[ρ]-Im[ρ] plane of Fig. 7, while Fig. 10 shows ρ for the potential minimum for the superpotential (35). In this case, IH is only available. We show the correlation between δ CP and sin 2 θ 23 in Fig. 19. We can see the weak sin 2 θ 23 dependence for δ CP . The predicted δ CP is in the range of ± . The effective mass of the 0νββ decay m ee is presented in Fig. 22. The prediction is in the range of 18-34 [meV]. We summarize the prediction for m ee and m i in Table 2. Figure 19: The predicted region on sin 2 θ 23 -δ CP plane in C for IH. Vertical red lines denote 3σ bound of observed data.

Conclusion
We have studied the modulus stabilization and its phenomenological aspects in the A 4 flavor model, where the A 4 flavor symmetry is originated from the S 4 modular symmetry. We can stabilize the modulus by a superpotential with a single modular form, but its modulus value is not favorable in lepton masses and mixing angles in the A 4 flavor model. If we assume two modular forms in the superpotential, we can stabilize the modulus at favorable values by using the parameter ρ as well as ρ . Proper values of ρ and ρ are of O(0.1 − 1). Thus, contributions due to two modular forms are important in our model. By choosing a proper value of ρ as well as ρ in the superpotential, we can stabilize the value of τ in our scalar potential such that one can realize the lepton masses and its mixing angles.
We have presented the neutrino phenomenology in the three different regions of τ (A,B,C) where modulus stabilization is realized. The CP violating phase of leptons, δ CP is distinctly predicted in three regions of τ . It is also emphasized that IH of neutrino masses is reproduced in only C region. The sum of neutrino masses is predicted in the restricted range for A, B and C respectively. The cosmological observation of it will provide a crucial test of our model. The effective mass of the 0νββ decay m ee is also predicted. The future experiments can probe our model since our prediction includes m ee = 22-24 [meV] [75]. Thus, our model realizes the modulus stabilization where the successful phenomenological results are obtained.
In this base, the multiplication rule of the A 4 triplet is More details are shown in the review [4,5].

C Modulus potential
We give a scenario on a plausible mechanism to induce the modulus superpotential. We assume a hidden sector, e.g. supersymmetric QCD which has the SU (N ) gauge symmetry and N f flavors of chiral matter fields, Q i andQ j , with fundamental and anti-fundamental representations. When N = N f , the mesons M = Q iQ j and the baryon B = ε i 1 ···i N Q i 1 · · · Q i N as well as anti-baryon condensate [79]. If the superpotential at the tree level has the mass term, W = m i i Q iQ j , the above condensation leads to the term W = m M . Suppose that the M has the modular weight −k − 1. The mass parameter must be a modular form of the weight k since the modular invariance requires that m has the modular weight k. The following superpotential may be induced W = cY (k) (τ ) M .
If there is another hidden sector to condensate, we may realize the superpotential where we assume that another meson fields M has the modular weight −k − 1. Furthermore, the condensation of the baryon B may induce another term with a different modular weight.