Neutrino mixing from finite modular groups

We study the lepton flavor models, whose flavor symmetries are finite subgroups of the modular group such as $S_3$ and $A_4$. In our models, couplings are also nontrivial representations of these groups and modular functions of the modulus. We study the possibilities that these models realize realistic values of neutrino masses and lepton mixing angles.


Introduction
One of the unsolved but important mysteries in particle physics is the mystery about the flavor structure of the quarks and leptons such as the generation number, mass hierarchy and mixing angles. Such a mystery would provide us with hints to explore physics beyond the standard model. Indeed, several types of scenarios have been proposed to understand quark and lepton masses and mixing angles as well as CP phases.
One of interesting ideas is to impose non-Abelian discrete flavor symmetries. Many models have been proposed imposing non-Abelain discrete flavor symmetries, e.g., S 3 , A 4 , S 4 and other various finite groups. (See for review [1][2][3][4].) In particular, the lepton sector has been intensively studied, because at least two of three lepton mixing angles are large compared with the quark mixing angles and their experimental results have been improved precisely. Recently, CP symmetry and its violation were also studied [5][6][7].
Superstring theory is a promising candidate for the unified theory of all interactions including gravity and matter fields such as quarks and leptons as well as Higgs fields. It is shown that superstring theory on certain compactifications leads to non-Abelian discrete flavor symmetries. For example, heterotic string theory on orbifolds can lead to D 4 , ∆(54), etc. [8].
On the other hand, string theory on tori and orbifolds has the modular symmetry. For example, modular symmetries were studied in heterotic orbifold models [17][18][19] and magnetized D-brane models [20][21][22]. In general, modular transformations act nontrivially on string modes and interchange massless modes such as quarks and leptons to each other. In this sense, modular symmetry is a non-Abelain discrete flavor symmetry. Furthermore, it is interesting that the modular symmetry includes S 3 , A 4 , S 4 , A 5 as its congruence subgroups, Γ(N). However, there is a difference between the modular symmetry and the usual flavor symmetries. 2 Coupling constants such as Yukawa couplings also transform nontrivially under the modular symmetry [17,[20][21][22], while coupling constants are invariant under the usual flavor symmetries, although flavon fields, which develop their vacuum expectation values (VEVs), transform nontrivially under flavor symmetries. Moreover, Yukawa couplings as well as higher order coupling constants are modular functions of moduli [20,[23][24][25].
By use of the above aspects, an interesting ansatz was proposed in Ref. [26], where Γ(3) ≃ A 4 was used and leptons were assigned to triplets and singlets of A 4 . Also coupling constants were assigned to A 4 triplet and singlets, which are modular functions. Then, neutrino masses and mixing angles were analyzed. Such an ansatz would be interesting in order to bridge a gap between underlying theory such as superstring theory and low-energy physics like neutrino phenomena. Our purpose in this paper is to study systematically the above approach. We study the A 4 model following Ref. [26]. Also, we extend our analysis to S 3 models. This paper is organized as follows. In section 2, we give a briel review on modular symmetry. In section 3, using experimental values, we write the neutrino mass matrix, which is convenient 1 In Ref. [11], relations between enhanced gauge symmetries and non-Abelian discrete flavor symmetries were studied.
2 See for their difference and relation [22].
to our analyses. In section 4, we study systematically the A 4 model following Ref. [26]. In section 5, we construct S 3 models in a way similar to the A 4 model and study them systematically. Section 6 is our conclusion and discussions. In appendix A, we briefly review modular functions and show modular functions corresponding to the A 4 triplet and the S 3 doublet.

Modular transformation and its subgroup symmetries
In this section, we give a brief review on the modular symmetry and its congruence subgroups. Toroidal compactification is one of simple compactifications. The two-dimensional torus T 2 is obtained as T 2 = R 2 /Λ. Here, Λ is the two-dimensional lattice, which is spanned by two lattice vectors, α 1 = 2πR and α 2 = 2πRτ . We use the complex coordinate, where R is a real parameter, and τ is a complex modulus parameter.
The choice of the basis vectors has some ambiguity. The same lattice can be spanned by other bases, where a, b, c, d are integer with satisfying ad − bc = 1. That is the SL(2, Z) transformation. The modular parameter τ = α 2 /α 1 transforms as under (1). Both lattice bases (α 1 , α 2 ) and (α ′ 1 , α ′ 2 ), and both modular parameters, τ and τ ′ , lead to the same lattice Λ and the same T 2 . The modular transformation is generated by the S and T transformations, T : τ −→ τ + 1.
In addition, these satisfy the following algebraic relations, On top of that, when we impose T N = 1, the so-called congruence subgroups Γ(N) can be realized. The congruence subgroups Γ(N) are isomorphic to (even) permutation groups, e.g. Γ(2) ≃ S 3 , Γ(3) ≃ A 4 , Γ(4) ≃ S 4 , and Γ(5) ≃ A 5 . (For subgroups of the modular group, e.g., see Ref. [27].) String theory on T 2 as well as orbifolds T 2 /Z N has the modular symmetry. Furthermore, four-dimensional low-energy effective field theory on the compactification T 2 × X 4 as well as (T 2 /Z N ) × X 4 also has the modular symmetry, where X 4 is a four-dimensional compact space.
A set of chiral superfields φ (I) transform under the modular transformation (2) as a multiplet [28], where −k I is the so-called modular weight and ρ (I) denotes a representation matrix. Modular invariant kinetic terms expanded around a VEV of the modulus τ are written by Also, the superpotential should be invariant under the modular symmetry. That is, the superpotential should have vanishing modular weight in global supersymmetric models. Indeed, Yukawa coupling constants as well as higher-order couplings constants are modular functions of τ [20,[23][24][25]. In the framework of supergravity theory, the superpotential must be invariant up to the Kähler transformation [28]. That implies that the superpotential of supergravity models with the above kinetic term should have modular weight one. In sections 4 and 5, we consider the global supersymmetric models, and require that the superpotential has vanishing modular weight, although it is straightforward to arrange modular weights of chiral superfields for supergravity models. The Dedekind eta-function η(τ ) is one of famous modular functions, which is written by where q = e 2πiτ . The η(τ ) function behaves under S and T transformations as The former transformation implies that the η(τ ) function has the modular weight 1/2. The modular functions (Y 1 , Y 2 , Y 3 ) with weight 2, which behave as an A 4 triplet, are obtained as in Ref. [26], where ω = e 2πi/3 . (See Appendix A.) We can obtain the modular functions with weight 2, which behave as an S 3 doublet, by a similar technique. (See Appendix A.) We use the following expansions: In section 5, we fit Y 2 /Y 1 to experimental data. That is, we use the following expansion: Such an expansion would be valid for |24q| 0.1, i.e.

Neutrino mass matrix
Before studying the A 4 and S 3 models, here using experimental values of neutrino oscillations we write the neutrino mass matrix, which is convenient to our analyses in Sections 4 and 5.

Experimental values
Flavor eigenstates of neutrino (ν e , ν µ , ν τ ) are linear combinations of mass eigenstates (ν 1 , ν 2 , ν 3 ). Their mixing matrix U, i.e. the so-called PMNS matrix can be written by where c ij = cos θ ij and s ij = sin θ ij for mixing angles θ ij , δ CP is the Dirac CP phase, and α i are Majorana CP phases. The mass-squared differences are defined by where m i is the mass eigenvalue of ν i . We also define the ratio between the mass-squared differences as Experimental values with normal ordering (NO) and inverted ordering (IO) are shown in Ta

General setup of our models
In sections 4 and 5, we consider the models, where the charged lepton mass matrix is diagonal, following [26]. In such models, the neutrino mass matrix m ν is written as by use of the PMNS matrix. By using three column vectors where The matrices m up to the phase. Here, we have omitted to write explicitly some elements, because m ν is the symmetric matrix. These matrix elements have the following relations: where t ij = tan θ ij . Thus, realistic models should be consistent with these rules (23).
In the case of IO (m 3 ≪ m 1 ≃ m 2 ), the mass matrix m ν can be approximated by Since α 2 has not been determined by experiments, there is one more parameter in IO than in NO.

A model
In this section, we consider the model in Ref. [26] systematically. This model is the supersymmetric model, although we can construct a similar nonsupersymmetric model. This model has the flavor symmetry Γ(3) ≃ A 4 . We concentrate on the lepton sector. Table 2 shows the A 4 representations and k I of lepton and Higgs superfields, L i , e c Ri and H u,d . Recall that −k I is the modular weight. The superfield φ is a flavon field and the A 4 triplet. We arrange k I such that the charged lepton masses are not modular functions of τ and the flavon field does not appear in the Weinberg operator. For example, we take k φ = 3, k H u,d = 0, k L = 1 and k ei = −4 [26]. Then, the superpotential terms of the charged lepton sector can be written by, where the β i are constant coefficients. We assume that the flavon multiplet develops the VEV along the direction, φ = (u, 0, 0). Such a VEV as well as the VEV v d of the neutral component of H d leads to the diagonal charged lepton mass matrix, By choosing proper values of couplings β i , we can realize the experimental values of the charged lepton masses, m e,µ,τ .
On the other hand, we can write the Weinberg operator in the superpotential Since k L = 1 and k Hu = 0, the couplings Y = (Y 1 , Y 2 , Y 3 ) must be a modular form with modular weight 2 and an A 4 triplet. We use the modular functions (10). Then, we obtain the mass matrix of neutrinos written by where v u denotes the VEV of the neutral component of H u . Note that the charged lepton mass matrix is diagonal. Thus, mixing angles as well as the CP phases originated from the neutrino mass matrix. In the following subsections, we perform numerical analyses by using the mass matrix (28).

Normal ordering in A 4 model
Here, we study the NO case. In the case of NO, there are constraints (23) for the realistic mass matrix. By using the equations in the second line of (23), we obtain This is the theoretical prediction in this model. By putting experimental values with ±3σ, the left-and right-hand sides of (29) become t 2 13 e 2iδ CP ∼ 0.02, respectively. The theoretical prediction (29) is inconsistent with experimental data (30). Thus, this model does not reproduce the experimental results for NO.

Inverted ordering in A 4 model
Here, we study the IO case. The mass matrix in this model m model ν (28) obeys the following relations: Figure 1 shows a plot of Re F 1 . The shaded region shows the 3σ deviation of s ij and δ CP . Figure 2 shows plots of Re F i , i = 2, 3. Since a realistic model must satisfy F i = 0 at the same α 2 , this model does not reproduce the experimental results within the 3σ range for IO.

S 3 model
In this section, we construct the models with the flavor symmetry Γ(2) ≃ S 3 in a way similar to the A 4 model, and study them systematically.
(1, 0) 2 3 Table 3: S 3 representations and k I in the S 3 models. Table 3 shows the S 3 representations and k I of lepton and Higgs superfields. The φ (1) and φ (2) fields are flavon fields, and φ (1) and φ (2) are S 3 singlet and doublet, respectively. In order to distinguish e Ra and e R b , we assign k I different from each other. For such a purpose, we can impose an additional symmetry, e.g. Z 2 . We assign k I such that we can realize the diagonal charged lepton mass matrix similar to the A 4 model. Indeed, the superpotential terms in the charged lepton sector can be written by where the β i are constant coefficients. We assume that the flavon fields develop their VEVs as where a, b, c, d ∈ C are constant coefficients. Y (1) and Y (2) are modular forms with modular weight 2, and Y (1) and Y (2) are S 3 singlet and doublet 3 , respectively. Note that since 1 ′ in 2 × 2 = 1 + 1 ′ + 2 is antisymmetric, L (2) L (2) 1 ′ = 0. We denote Y (1) = Y and Y (2) = (Y 1 , Y 2 ). There are 6 ways to assign 3 generations of lepton L i to S 3 singlet L (1) and doublet L (2) . The replacement of L (2) = (L i , L j ) → (L j , L i ) corresponds to the replacement of Y 1 ↔ Y 2 and less affects analysis. Therefore, we study 3 models shown in table (4). The fifth column shows the neutrino mass matrix in each model. Since parameters b and c always appear in bY and cY , respectively, we rewrite B ≡ bY and C ≡ cY .

Normal ordering in S 3 models
Here, we study the NO case in the three S 3 models. In the model 1, the neutrino mass matrix is written by Since the charged lepton mass matrix is diagonal, the PMNS matrix is determined only by the neutrino mass matrix. The consistency conditions (23) are written by Model L (1) Experimental values lead to t 2 13 = O(10 −2 ) and 3s 2 23 = O(1). Hence this model is inconsistent with experiments. Note that since this result (41) does not depend on Y i , we obtain the same result by replacing Y 1 ↔ Y 2 , that is, L (2) = (L 1 , L 2 ) → (L 2 , L 1 ).
In the models 2 and 3, we can analyze in the same way as in the model 1. A prediction of the model 2 is e 2iδ CP t 2 13 = 3c 2 23 , and a prediction of the model 3 is Both of these predictions are inconsistent with experiments. Hence, these models are not realistic.

Inverted ordering in S 3 models
Here, we study the IO case in the three S 3 models.

Model 1
In the model 1, the mass matrix is written by From this matrix, we find and hence This is a prediction of this model. First, we define a function with m ν in (24), and search a set of the values of (α 2 , s ij , δ CP ) satisfying F 1 = 0 within the 3σ experimental range. Here, we treat α 2 as a free parameter. Figure 3 shows a plot of the on a complex plane. The blue curve is an one-parametric plot of the F 1 (α 2 ) with α 2 ∈ [0, 2π) and the best-fit values of s ij and δ CP . The blue shaded region is a two-parametric plot of the F 1 (α 2 ; δ CP ) with α ∈ [0, 2π), δ CP ∈ (3σ-range), and the best-fit values of s ij . The origin At this stage, numerically expressed mass matrix m ν has degenerate eigenvalues m 2 1 = m 2 2 , and therefore m ν does not reproduce the mixing angle s 2 12 and mass-squared differences.
We can resolve the degeneracy m 2 1 = m 2 2 by changing parameters slightly. That is, we modify the parameter B → B + ǫ B to reproduce the s 2 12 and mass-squared differences by resolving the degeneracy. When the ǫ B = 0.030 + 0.026i, observables are given by and Majorana phases are given by These observables are consistent with experimental results up to 3σ, and the values of Majorana phases are predictions of the model 1.
Finally, we fit the values of Y 1 and Y 2 in (48) with the modular functions in (11). Since the overall coefficient of Y i (τ ) is not fixed in (89), we fit the value of Y 2 /Y 1 by Such a ratio can be obtained for τ = 0.505 + 0.781i. This value is sufficiently large compared with Eq. (14). When we change the generation assignment as L (2) = (L 1 , L 2 ) → (L 2 , L 1 ), Y 1 and Y 2 are replaced each other. Thus, we fit the values of Y 1 and Y 2 by and we get the solution τ = −0.507 + 0.960i.

Model 2
In the model 2, we can analyze with the same way as in the model 1.
The model 2 predicts We define and this is shown in Figure 4 By solving this equation, we obtain the solution τ = −0.487 + 0.714i. When we change the generation assignment as L (2) = (L 1 , L 3 ) → (L 3 , L 1 ), we find and we obtain the solution τ = 0.480 + 1.052i,

Model 3
The model 3 predicts We define and this is shown in Figure 5 similar to the model 1. The condition F 2 = 0 is satisfied at α 2 /π ≃ 0.14 and δ CP /π ≃ 1.53. An approximate solution of m model We can tune B → B + ǫ B with ǫ B = 0.001 + 0.081i, and we obtain and These observables are consistent with experimental results up to 1σ except for the δ CP . The value of δ CP is out of 3σ deviation. The values of Majorana phases are predictions of the model 3. In this model, we can also tune C or d instead of B. Finally, we fit the values of Y i by By solving this equation, we obtain the solution τ = 1.000 + 1.289i. When we change the generation assignment as L (2) = (L 2 , L 3 ) → (L 3 , L 2 ), we get This equation has no solution.

Conclusion
We have studied neutrino mixing in the models with A 4 and S 3 discrete flavor symmetries. In our models, couplings are also nontrivial representations under the discrete flavor symmetries, and they are modular functions. In the A 4 model, following [26], we assigned the three generations of leptons to the triplet of A 4 . In this case, the form of the neutrino mass matrix is strongly restricted as (28), and there are no realistic solution. In the S 3 models, we assigned the three generations of leptons to a singlet and a doublet of S 3 . In these cases, there are five model parameters except for an overall coefficient in Table 4. It may be easier to fit the experimental data by increasing the number of parameters. However, neutrino mass matrices in our models have restricted forms, and are written by modular functions. Thus, it is nontrivial to realize the experimental values by many parameters. Indeed, there are no solution in the case of normal ordering. In the case of inverted ordering, we can reconstruct experimental results except for the δ CP within the 3σ-range in all three models. Additionally, we can fit the mass matrix by using modular functions in all three models. Also, we have predictions on the Majorana CP phases. It would be interesting to study other assignments of leptons in A 4 and S 3 models such that the charged lepton mass matrix is not diagonal and depend on modular functions of τ . Also, it would be interesting to extend our analyses to other congruence subgroups, e.g. Γ(4) ≃ S 4 and Γ(5) ≃ A 5 . and the following transformations under S, Using them, we can construct the modular functions with weight 2 by with α + β + γ + δ = 0 because of Eq.(69). These functions transform under S and T as Now let us construct an A 4 triplet by the modular functions Y (α, β, γ, δ|τ ). We use the (3 × 3) matrix presentations of S and T as where ω = e 2πi/3 . They satisfy (ρ(S)) 2 = I, (ρ(S)ρ(T )) 3 = I, (ρ(T )) 3 = I, that is, Γ(3) ≃ A 4 . Using these matrices and Y (α, β, γ, δ|τ ), we search an A 4 triplet, which satisfy, Their solutions are written by where we set c = i/(2π). They can be expanded as Y 1 (τ ) = 1 + 12q + 36q 2 + 12q 3 + · · · , Y 2 (τ ) = −6q 1/3 (1 + 7q + 8q 2 + · · · ), (79) Y 3 (τ ) = −18q 2/3 (1 + 2q + 5q 2 + · · · ).