Lepton Flavor Mixing and CP Violation in the Minimal Type-(I+II) Seesaw Model with a Modular $A_4$ Symmetry

In this paper, we study the implications of the modular $A^{}_4$ flavor symmetry in constructing a supersymmetric minimal type-(I+II) seesaw model, in which only one right-handed neutrino and two Higgs triplets are introduced to account for the tiny neutrino masses, flavor mixing and CP violation. The right-handed neutrino as well as the Higgs triplets in this model are assigned into the trivial one-dimensional irreducible representation of the modular group $A^{}_{4}$. We show that the individual contributions to the neutrino masses from the right-handed neutrino and the Higgs triplet are comparable. We also find that the neutrino mass matrix can possess an approximate $\mu-\tau$ reflection symmetry for some specific values of free model parameters. Moreover, our model predicts relatively large masses of three light neutrinos, thus can be easily tested in future neutrino experiments.


Introduction
Neutrino oscillation experiments in the past two decades have provided us with the very solid evidence that neutrinos are massive and lepton flavor mixing indeed exists [1,2]. In order to generate tiny neutrino masses, one can extend the standard model (SM) by adding a few new particles and allowing for the lepton number violation, and then the tiny masses of light neutrinos can be attributed to the introduced heavy degrees of freedom. This is the so-called seesaw mechanism. For example, in the typical type-I seesaw mechanism [3][4][5][6], three right-handed neutrinos, which are singlets under the SU(2) L × U(1) Y gauge symmetry of the SM, are introduced and the smallness of light neutrino masses can thus be explained by the heavy mass scale of the right-handed neutrinos.
Another interesting realization of the seesaw mechanism is the type-II seesaw mechanism [7][8][9][10][11][12], in which an additional Higgs triplet under SU(2) L is added into the SM. Therefore, the gaugeinvariant Lagrangian relevant for lepton masses and flavor mixing can be written as where L and E R denote the left-handed lepton doublet and the right-handed charged-lepton singlet, H and ∆ are the Higgs doublet and triplet, respectively. Note that in Eq. (1.1), C L ≡ C L T with C = iγ 2 γ 0 being the charge-conjugation matrix has been defined. After the spontaneous symmetry breaking, we can obtain the charged-lepton and neutrino mass matrices as where µ, λ and λ ∆ are the coupling coefficients and M ∆ denotes the mass of the Higgs triplet.
Then the vev's v = µ 2 /(λ − 2λ 2 ∆ ) and v ∆ = λ ∆ v 2 /M ∆ can be derived from Eq. (1.2). The small value of v ∆ , which is suppressed by the large mass scale of M ∆ , can also explain the observed tiny neutrino masses. In addition, there is another possibility that the light neutrino masses are not originated from one single seesaw mechanism. For instance, one can consider a combination of the type-I and type-II seesaw mechanism, i.e., the type-(I+II) seesaw mechanism, in which both right-handed neutrinos and the Higgs triplet are introduced. In this case, the Lagrangian in Eq. (1.1) becomes where N R and M R denote the right-handed neutrinos and their Majorana mass matrix respectively. Note that in Eq. (1.3), H ≡ iσ 2 H * and N C R ≡ CN R T have been defined.
Although the seesaw model provides us with an elegant way to explain the tiny neutrino masses, it can not account for the flavor structures existing in the lepton mass matrices. As a consequence, the model is in general lacking of predictive power for lepton mass spectra, flavor mixing pattern and CP violation [13]. On this account, non-Abelian discrete flavor symmetries have been implemented in the seesaw model to explain the flavor mixing in recent literature, e.g., Refs. [14][15][16][17][18][19]. To be specific, one can first assume that the Lagrangian maintains an overall discrete flavor symmetry at some high-energy scale. Next a few of gauge-singlet scalar fields which are called flavons are introduced to break down the whole symmetry into distinct residual symmetries in the charged-lepton and neutrino sectors [20][21][22][23][24][25]. Then the flavor structures will be determined by the vev's of these flavons. However, the introduction of flavons will inevitably bring a large number of free parameters into the model and how to experimentally prove the existence of the flavons is also a tough problem.
Recently, a new and attractive approach to solve the flavor mixing problem, which is based on the modular invariance, has been proposed in Ref. [26]. Within the framework of modular symmetries, the Yukawa couplings are regarded as the modular forms with even weights, which transform as the multiplets under some finite modular symmetry groups Γ N . For a given value of N , Γ N is isomorphic to the well-known non-Abelian discrete symmetry group, e.g., Γ 2 S 3 [27][28][29][30], Γ 3 A 4 [31][32][33][34][35][36][37][38][39][40], Γ 4 S 4 [41][42][43][44] and Γ 5 A 5 [45][46][47]. Modular forms are the functions of the modulus τ , and the modular symmetry is broken and the flavor mixing pattern is generated right after the value of τ is fixed. Therefore the flavon field is not necessary in the framework of modular symmetries. Except the references we have mentioned above, there are also plenty of works related to other interesting aspects of modular symmetries, such as the combination of modular symmetries and the CP symmetry [48,49], multiple modular symmetries [50,51], the double covering of modular groups [52], the A 4 symmetry from the modular S 4 symmetry [53,54], the modular residual symmetry [55,56], the unification of quark and lepton flavors with modular invariance [57], the realization of texture zeros via the modular symmetry [58,59] and the applications of modular symmetries on other types of seesaw models [60,61].
In this paper, we investigate the minimal supersymmetric type-(I+II) seesaw model, where only one right-handed neutrino and two Higgs triplets are introduced, with the modular A 4 symmetry and explore its implications for lepton mass spectra, flavor mixing pattern and CP violation. The pure type-II seesaw model with two additional Higgs triplets under the modular A 4 symmetry has already been investigated in Ref. [60]. In such a framework, usually one has to require a large number of free model parameters and higher weights of the modular forms in order to find the suitable parameter space consistent with current experimental data. However, in our minimal type-(I+II) seesaw model, only a few of free parameters are introduced and most of the modular forms involved in are with the lowest non-trivial weights. We show that our model predicts relatively large masses of three light neutrinos, and the individual contributions to the neutrino masses from the right-handed neutrino and the Higgs triplet turn out to be comparable. In addition, we also find that the neutrino mass matrix can possess an approximate µ−τ reflection symmetry [62] for some specific values of free model parameters.
The remaining part of this paper is organized as follows. In Sec. 2, a brief summary of the modular A 4 symmetry is given. The concrete type-(I+II) seesaw model with the modular A 4 symmetry is then proposed in Sec. 3. The low-energy phenomenology of lepton mass spectra, flavor mixing pattern and CP violation in our model are discussed in Sec. 4. Finally, we summarize our main conclusions in Sec. 5. Some properties of the modular A 4 symmetry group are presented in Appendix A.

Modular A 4 Symmetry
The basics of modular symmetries have been expounded in previous works (See, e.g., Ref. [26]). In this section, we shall only give a brief review on the modular symmetry.
In a supersymmetric theory, the action S keeps invariant under the modular transformation where γ is the element of the modular group Γ with a, b, c and d being integers satisfying ad − bc = 1 and τ is an arbitrary complex number in the upper complex plane. As a consequence, the Kähler potential K(τ, τ , χ, χ) with χ being the supermultiplet is invariant up to the Kähler transformatsion K(τ, τ , χ, χ) → K(τ, τ , χ, χ) + f (τ, χ) + f (τ , χ) 1 , where f (τ, χ) itself is invariant under the modular transformation. Meanwhile, the superpotential W(τ, χ) is invariant as well and can be expanded in terms of the supermultiplets as follows W(τ, χ) = n {I 1 ,...,I n } Y I 1 ...I n (τ )χ (I 1 ) · · · χ (I n ) , (2.2) where the coefficients Y I 1 ...I n (τ ) take the modular forms, transforming under the finite modular group Γ N ≡ Γ/Γ(N ) (with Γ(N ) being the principal congruence subgroup of Γ) as where the even integer k Y is the weight of Y I 1 ...I n (τ ) and ρ Y is the representation matrix of Γ N . In addition, k Y and ρ Y must satisfy k Y = k I 1 + · · · + k I N and ρ Y ⊗ ρ I 1 ⊗ · · · ⊗ ρ I N 1, respectively. For the symmetry group Γ 3 A 4 of our interest, there are three linearly independent modular forms of the lowest non-trivial weight k Y = 2, denoted as Y i (τ ) for i = 1, 2, 3, which form a triplet 3 under the modular A 4 symmetry transformations [26], namely, (2.4) The exact expressions of Y i (τ ) (for i = 1, 2, 3) are presented in Appendix A. Based on the modular forms Y i (τ ) of weight k Y = 2, one can construct the modular forms of higher weights, such as k Y = 4 and k Y = 6. For k Y = 4, there are totally five independent modular forms, which transform as 1, 1 and 3 under the A 4 symmetry [26,37,58], namely, where the argument τ of all the modular forms is suppressed. For k Y = 6, we have seven independent modular forms, whose assignments under the A 4 symmetry are as follows [26,37,58] (2.6)

The Minimal Type-(I+II) Seesaw Model
In this section, we are going to construct a minimal type-(I+II) seesaw model with the modular A 4 symmetry. To begin with, let us first make some general remarks on the model building.
• A criterion for the model building is that our model should be economical enough, which means that the number of free model parameters should be as small as possible. To be specific, we have eight low-energy observables, including three charged-lepton masses {m e , m µ , m τ }, two independent neutrino mass-squared differences {∆m 2 21 , ∆m 2 31 } in the normal mass ordering (NO) case where m 1 < m 2 < m 3 or {∆m 2 21 , ∆m 2 32 } in the inverted mass ordering (IO) case where m 3 < m 1 < m 2 and three mixing angles {θ 12 , θ 13 , θ 23 }. Therefore the number of free model parameters should be no more than eight in order to have predictive power for the other parameters, such as the CP-violating phases.
• As the modular symmetry is intrinsically working in the supersymmetric framework, we should introduce one chiral superfield N C 2 which contains the right-handed neutrino singlet and a pair of SU(2) L triplet Higgs superfields { ∆ 1 , ∆ 2 } with the hypercharges {+1, −1} defined as where ∆ ++ 1 ( ∆ −− 2 ), ∆ + 1 ( ∆ − 2 ) and ∆ 0 1 ( ∆ 0 2 ) denote the doubly-charged, singly-charged and neutral components of ∆ 1 ( ∆ 2 ), respectively. N C , ∆ 1 and ∆ 2 are all arranged to be the trivial singlet under the modular A 4 symmetry in our model for simplicity. Furthermore, the superfields for Higgs doublets { H u , H d } with the hypercharges {+1/2, −1/2} are also assigned into 1 under the modular A 4 symmetry. As a consequence, we do not need to change the remaining part of the MSSM irrelevant for leptonic favor mixing.
• The superfields for three lepton doublets { L 1 , L 2 , L 3 } are arranged as a triplet 3 under the A 4 symmetry, while the superfields for three charged-lepton singlets { E C 1 , E C 2 , E C 3 } should be assigned into three different singlets of A 4 (e.g., E C 1 ∼ 1, E C 2 ∼ 1 and E C 3 ∼ 1 ). Otherwise, it will be difficult to explain the strong mass hierarchy of three charged leptons, namely, m e m µ m τ .
• The modular forms relevant for lepton masses and flavor mixing can be exactly determined from the two identities k Y = k I 1 + · · · + k I N and ρ Y ⊗ ρ I 1 ⊗ · · · ⊗ ρ I N 1 after the weights and representations of the superfields are fixed. Note that since both the right-handed neutrino and Higgs triplets are introduced into our model, more terms will appear in the whole superpotential. Consequently, there remains less freedom for us to adjust the weights and representations under the modular A 4 symmetry of all the superfields as well as the modular forms. In Table 1, we show the charge assignments of the chiral superfields and the couplings under the SU(2) L gauge symmetry and the modular A 4 symmetry for our model, and the corresponding modular weights are listed in the last row. Note that k D = 4 and k R = 6 are the lowest weights which the modular forms f D and f R can take respectively under the premise that k Y = k I 1 + · · · + k I N should be satisfied in each superpotential.
Keeping these assignments above in mind, now it is straightforward for us to write down the modular A 4 invariant superpotential W, which can be decomposed into three parts W = W l + W I + W II with where α 1 , α 2 and α 3 are three coupling coefficients in the charged-lepton sector which we can set to be real and positive without loss of generality while g 1 , g 2 and Λ are the coupling coefficients in the neutrino sector. In Eq. (3.2), the individual contributions to neutrino masses from the type-I and type-II seesaw mechanisms can be read from W I and W II , respectively. When the modulus parameter τ is fixed, the modular symmetry is broken down and the superpotential reads where λ l and λ D turn out to be the charged-lepton and Dirac neutrino Yukawa coupling matrices, respectively, λ R becomes the right-handed neutrino mass matrix and λ II is the neutrino mass matrix induced by the type-II seesaw.
On the other hand, the superpotential relevant for the couplings between the Higgs doublets and triplets, which is just the supersymmetric version of Eq. (1.2), can be written as [64] where µ, λ 1 and λ 2 are the coupling coefficients. After the supersymmetry breaking and the SU(2) L × U(1) Y gauge symmetry breakdown, all the Higgs fields get their own vev's and one can then obtain the lepton mass terms from Eq. (3.2). It has been indicated in Ref. [44] that there exists the following correspondence between the lepton mass matrices and the Yukawa coupling matrices in the MSSM framework under the left-right convention for the fermion mass terms where v d = v cos β and v u = v sin β are respectively the vev of the neutral scalar component field of H d to that of H u , with tan β ≡ v u /v d being their ratio, and v 1 = λ 2 v 2 u /M ∆ is the vev of the neutral scalar component field of ∆ 1 and can be derived from Eq. (3.4). Note that here we use" * " to denote the complex conjugation. Therefore, by using the product rules of the A 4 symmetry group collected in Appendix A, we can obtain the charged-lepton mass matrix 6) and the Dirac neutrino mass matrix Since only one right-handed neutrino is introduced, the Majorana mass matrix will degenerate to a complex number After applying the type-I seesaw mechanism M I ≈ −M D M −1 R M T D , we arrive at the neutrino mass matrix from the type-I seesaw Meanwhile, the neutrino mass matrix induced by the type-II seesaw can be expressed as (3.10) Then the whole effective neutrino mass matrix should be a combination of M I and M II . Since the overall phase of any lepton mass matrix is irrelevant for lepton masses and flavor mixing, one can take g 1 in Eq. (3.7) to be real and it is convenient to define a new complex parameter as 2λ 2 g 2 Λ/(3g 2 1 M ∆ ) ≡ g = ge iφ g with g = | g| and φ g ≡ arg( g). Therefore M ν can be written as From Eq. (3.11) we can find that each element in the mass matrix M II contains only one single modular form with a weight of 2, whereas every element in the matrix generated by M D M T D is a multiplication of two modular forms of weight 4, thus having a total modular weight of 8. However, M −1 R contributes another weight of −6, therefore the overall weight of M I is also 2, same as the weight of M II .

Low-energy Phenomenology
Next we discuss the low-energy phenomenology of our model. As can be seen from the previous section, there are totally eight free model parameters, which are a complex modulus τ (or equivalently two real parameters Re τ and Im τ ), three real parameters v d α 3 / √ 2, α 1 /α 3 and α 2 /α 3 in the charged-lepton sector and two complex parameters g and φ g as well as an overall factor v 2 u g 2 1 /(2Λ) in the neutrino sector. The number of free parameters is the same as that of the lowenergy observables. As a result, our model should be predictive. Then we proceed to explore the phenomenological implications for lepton mass spectra, flavor mixing and CP violation. We carry out a numerical analysis of our model and demonstrate that the predictions are consistent with the experimental data only in the NO case at the 1σ level. The main strategy for numerical analysis is analogous to what we have done in Ref. [44]. Here we list it as follows.
• First of all, the modulus parameter τ is randomly generated in the right-hand part of the fundamental domain G, which is defined as This domain can be identified by using the basic properties of the modular forms as clearly explained in Ref. [42]. One can also notice that if the replacement τ → −τ * is made in Eq. (A.8), the modular forms Y i (τ ) will change to their complex-conjugate counterparts, i.e., Y i (−τ * ) = Y * i (τ ). If we further replace g with g * in the neutrino sector, all the lepton mass matrices will become their complex-conjugate counterparts. Under such a transformation, In the charged-lepton sector, once we randomly choose the values of {Re τ, Im τ }, the parameters v d α 3 / √ 2, α 1 /α 3 and α 2 /α 3 can be calculated from the following identities where we take m e = 0.511 MeV, m µ = 105.7 MeV and m τ = 1776.86 MeV for the observed charged-lepton masses [2]. Notice that Eqs. (4.2)-(4.4) have multiple solutions, corresponding to the different hierarchies of α 1 , α 2 and α 3 . Later we will see there exist two kinds of hierarchies (α 1 α 3 α 2 and α 1 α 2 α 3 ) which can lead to the realistic mixing pattern, with different predictions to the value of θ 23 . So far all the parameters in M l have been determined. It is then easy to diagonalize the charged-lepton mass matrix via Region B Region A Figure 1: Allowed ranges of the model parameters {Re τ, Im τ } and {g, φ g } in the NO case, where the 1σ (yellow dots) and 3σ (red dots) ranges of neutrino mixing parameters and mass-squared differences from the global-fit analysis of neutrino oscillation data have been input [65]. The bestfit values are indicated by the black stars. In the left panel, the horizontal dashed line separates the parameter space of {Re τ, Im τ } into two regions: Region A and Region B.
• Next the values of the other two parameters g ∈ (0, 10] and φ g ∈ [0 • , 360 • ) are randomly generated. Therefore, the effective neutrino mass matrix M ν is determined up to the overall scale parameter v 2 u g 2 1 /(2Λ). We introduce a ratio r defined as r ≡ ∆m 2 21 /∆m 2 31 in the NO case or r ≡ ∆m 2 21 /|∆m 2 32 | in the IO case which is irrelevant to this overall scale parameter, and this ratio can help us restrict the values of Re τ , Im τ , g and φ g . The overall parameter where c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 13, 23) have been defined and δ, ρ and σ are the Dirac and two Majorana CP-violating phases, respectively.
As we have mentioned before, the predictions of our model are consistent with the experimental data only in the NO case at the 1σ level. The allowed parameter space of {Re τ, Im τ }, {g, φ g } and {α 1 /α 3 , α 2 /α 3 } has been shown in Figs. 1-2, where the 1σ (3σ) range is denoted by the yellow (red) dots. As one can see from the left panel of Fig. 1, almost all the range [0, 0.5] of Re τ is allowed at the 3σ level while the value of Im τ is restricted to be larger than 1.75. Note that here we artificially cut off the parameter space of Im τ at Im τ = 4, since in the range where   Figure 2: Allowed ranges of two ratios {α 1 /α 3 , α 2 /α 3 } in the charged-lepton sector for the NO case, where the 1σ (yellow dots) and 3σ (red dots) ranges of neutrino mixing parameters and masssquared differences from the global-fit analysis of neutrino oscillation data have been input [65]. The best-fit value is indicated by the black star. The left panel corresponds to the hierarchy α 1 α 3 α 2 where only the 3σ range is allowed and the right panel is related to α 1 α 2 α 3 .
Im τ > 4, we find that the predicted values of mixing angles and CP-violating phases tend to be stable and the sum of three light neutrino masses m ν = m 1 + m 2 + m 3 > 2 eV, which has already been far away from the favored region of the latest Planck observations [66], thus being out of our interest. Actually we can separate the parameter space of {Re τ, Im τ } into two regions depending on the values of Im τ , to be specific, Region A with 1.75 < Im τ < 2.07 and Region B with 2.07 < Im τ < 4. An important feature to distinguish these two regions is that only the hierarchy α 1 α 3 α 2 is permitted in Region A while both the hierarchies α 1 α 3 α 2 and α 1 α 2 α 3 are allowed in Region B. The reason for this fact will be discussed in detail later. On the other hand, in Region A the value of Im τ can only change in a narrow region. However Re τ can vary in a wide range, from 0.04 to 0.5. On the contrary, in Region B the value of Re τ is about 0.03 while Im τ can reach very large values. The constraints on g and φ g within the 3σ level are 0.82 < g < 0.92 and 1.92 • < φ g < 21.8 • respectively, as can be seen from the right panel of Fig. 1. The value of g measures the individual contributions to the neutrino masses from the type-I and type-II seesaw mechanisms, and g ∼ 1 means that their individual contributions are comparable to each other.
To determine the model parameters from neutrino oscillation data and describe how well the model is consistent with observations, we construct the χ 2 -function by regarding the bestfit values q bf j of the oscillation parameters q j ∈ {sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , ∆m 2 21 , ∆m 2 31 } from the global analysis in Ref. [65] as experimental measurements, namely, where p i ∈ {Re τ, Im τ, g, φ g } stand for the free model parameters and q j (p i ) denote the model predictions for observables with σ j being the symmetrized 1σ uncertainties from the global-fit analysis, which has already been given in Table 2. Since the current constraint on δ from the global-fit results is rather weak, we will not include the information of δ in the χ 2 -function. The m ν > 0.6 eV Figure 3: The relation between the sum of light neutrino masses m ν = m 1 + m 2 + m 3 and Im τ for the NO case, where the 1σ (yellow dots) and 3σ (red dots) ranges of neutrino mixing parameters and mass-squared differences from the global-fit analysis of neutrino oscillation data have been input [65]. The best-fit value is indicated by the black star. The gray shaded region denotes the range m ν > 0.6 eV which is excluded by the latest Planck observations [66]. allowed ranges of model parameters can be obtained by following the standard χ 2 -fit approach. Based on the χ 2 -fit analysis, we find that the minimum χ 2 min = 0.232 is obtained in the NO case with the following best-fit values of the model parameters which together with the charged-lepton masses m α (for α = e, µ, τ ) lead to v d α 3 / √ 2 = 1.77 GeV, α 1 /α 3 = 2.88 × 10 −3 and α 2 /α 3 = 5.96 × 10 −2 . Once the model parameters are known, we can get the constraints on the observables q j and the predictions for the CP-violating phases δ, ρ and σ, as well as the effective neutrino mass m β for beta decays and m ββ for neutrinoless double-beta decays, which will be presented in Table 3.
The relation between m ν and Im τ is presented in Fig. 3, from which we can find that the value of m ν is tightly related to Im τ . To be specific, as the value of Im τ increases, m ν will also increase. This can be understood in the following way. Since Im τ in our model is larger than 1.75, |q| = e −2πIm τ < 4 × 10 −3 in the Fourier expansions of modular forms Y i (τ ) (for i = 1, 2, 3) in Eq. (A.8) is a small parameter. Therefore we can retain only the leading order terms in the expansions of Y i (τ ) and Eq. (A.8) will change to where t ≡ −6q 1/3 = −6e 2πiτ /3 has been defined. Then the neutrino mass matrix M ν can be expressed in an approximate form up to O(t 3 ) All the elements except (M ν ) 11 , (M ν ) 23 and (M ν ) 32 in the right-hand side of Eq. (4.9) are suppressed by the higher order terms of t. As the value of Im τ becomes larger, (M ν ) 11 , (M ν ) 23 and (M ν ) 32 will dominant the eigenvalues of M ν . Given the parameter space of g, we can find that the modulus of (M ν ) 11 is close to that of (M ν ) 23 (Note that (M ν ) 23 = (M ν ) 32 exactly holds due to the nature of the Majorana mass matrix) especially when Im τ is large enough, implying a quasi-degeneracy among m 1 , m 2 and m 3 . The high degeneracy of three neutrino masses requires a large m ν , which is why the value of m ν increases with the rise of Im τ . Fig. 3 also indicates that the minimal value of m ν predicted in our model is 0.16 eV, which has already exceeded the upper bound on the sum of neutrino masses m ν < 0.12 eV from the Planck observations [66]. However this upper bound is cosmological model dependent and obtained by combining other experimental data such as the baryon acoustic oscillation (BAO), the gravitational lensing of galaxies and the high multipole TT, TE and EE polarization spectra. If only the BAO data and the cosmic microwave background (CMB) lensing reconstruction power spectrum are taken into consideration, the restriction to m ν can be liberalized to m ν < 0.6 eV [66]. Therefore, our model can still be compatible with the Planck observations in the disfavored region where 0.12 eV < m ν < 0.6 eV. In addition, since our model predicts relatively large values of three neutrino masses, it can be easily tested in future neutrino experiments.
A salient feature of our model is that the predicted value of θ 23 shows a strong dependence on the free model parameters especially Im τ , as can be seen from the left panel of Fig. 4, where the red and blue curves correspond to the hierarchy α 1 α 3 α 2 and α 1 α 2 α 3 , respectively. Some useful remarks are as follows.
• There are two branches in the allowed range of {Im τ, θ 23 }, depending on which hierarchy of α 1 , α 2 and α 3 is taken into consideration. In the hierarchy α 1 α 3 α 2 , θ 23 is located in the first octant where θ 23 < 45 • while in the hierarchy α 1 α 2 α 3 , θ 23 is in the second octant, which is preferred by the latest global-fit analysis within 1σ level. If further long = U With the expression of P in Eq. (4.13), we can express U e3 and U (2) µ3 by using the elements of U (1) as (4.14) So finally we arrive at the expression of sin 2 θ (2) 23 If Im τ is sufficiently large, all the higher order terms of t can be neglected, and we will arrive at sin 2 θ (1) 23 ≈ 90 • . However when Im τ is not large enough, we should take the modification from higher order terms of t into consideration, especially the term of O(t). The numerical calculation shows that Re [t * U 51.3 • , which is exactly the upper bound of the 3σ range from the global-fit results of θ 23 . Therefore, if Im τ < 2.07, we could not find the proper value of θ (2) 23 located in the 3σ range any more. That is why only the hierarchy α 1 α 3 α 2 is allowed in Region A.
• The asymptotic behavior of θ 23 and δ when Im τ is extremely large deserves some more discussion. As can be seen from Fig. 4, θ 23 ≈ 45 • and δ ≈ 90 • or 270 • hold excellently when Im τ ∼ 4. The distinctive values of θ 23 and δ imply that the neutrino mass matrix M ν might possess an approximate µ − τ reflection symmetry [62], which means or equivalently |U µi | = |U τ i | (for i = 1, 2, 3). Actually as we have mentioned before, except the (M ν ) 11 , (M ν ) 23 and (M ν ) 32 , all the other elements in Eq. (4.9) are vanishing in the limit where Im τ tends to infinity. Under this limit it is easy to identify Eq. (4.17) is satisfied and M ν can be regarded as having a trivial µ − τ reflection symmetry. However, the non-zero value of t is required to slightly break down this symmetry and generate non-trivial values of the mass-squared differences as well as other mixing angles to fit the experiment data.
In order to illustrate this approximate µ − τ reflection symmetry indeed exists, we give a specific example of the numerical result for |U ν | as  Figure 5: In the left panel, the correlation between m β and m 1 is shown for the NO case where the 1σ (yellow dots) and 3σ (red dots) ranges of neutrino mixing parameters and mass-squared differences from the global-fit analysis of neutrino oscillation data have been input [65]. While the right panel is for the correlation between m ββ and m 1 . The gray shaded regions represent the range m ν > 0.6 eV which is excluded by the latest Planck observations [66] while the light gray shaded region in the right panel denotes the upper bound on m ββ from the KamLAND-Zen experiment [71]. The pink (purple) boundary in the right panel is obtained by using the 1σ (3σ) ranges of {θ 12 , θ 13 } and {∆m 2 21 , ∆m 2 31 } from the global-fit analysis.
• As can be seen in Eq. (4.14), the elements U e2 and U e3 in the lepton mixing matrix U keep invariant under the conversion from one hierarchy to another. Therefore different from θ 23 , the values of θ 12 and θ 13 do not depend on the hierarchies of α 1 , α 2 and α 3 .
On the other hand, once the neutrino mass spectrum and the mixing parameters are known, we can predict the effective neutrino mass for beta decays, In the case where three neutrino masses are quasi-degenerate, m β is approximately proportional to m 1 , as can be seen from the left panel of Fig. 5. The latest result from the KATRIN experiment, where the electron energy spectrum from tritium beta decays is precisely measured, indicates m β < 1.1 eV at the 90% confidence level [69,70]. With more data accumulated in KATRIN, the upper bound will be improved to m β < 0.2 eV. Then it will for sure provide us with some clues to test whether the corresponding parameter space in our model is still consistent with the experiment data or not. Furthermore, three light neutrinos are Majorana particles in the seesaw model, indicating that the neutrinoless double-beta decays of some even-even heavy nuclei could take place. The effective The right panel of Fig. 5 shows that the predicted values of m ββ in our model have already reached the upper bound from the KamLAND-Zen experiment [71], m upper ββ = 0.061 -0.165 eV, which is currently the best experimental constraint on m ββ . Hence our model is quite testable and can be easily ruled out in the next-generation neutrinoless double-beta decay experiments [72].
As a summary of this section, we list the best-fit values, together with the 1σ and 3σ ranges of all the free model parameters and observables in our model in Table 3.

Summary
The modular symmetry is a very attractive and interesting way to understand lepton flavor mixing. In this paper, we consider the application of the modular A 4 symmetry to the supersymmetric minimal type-(I+II) seesaw model, where only one right-handed neutrino and two Higgs triplets are introduced beyond the particle content of the SM. We successfully construct a model to account for lepton mass spectra and the flavor mixing, which is consistent with current neutrino oscillation data in the NO case.
In order to keep our model simple and economical enough, we assign the right-handed neutrino, two Higgs doublets and two Higgs triplets to be the trivial A 4 singlet 1, and implement the minimal set for the weights of modular forms (k e , k µ , k τ , k ∆ , k D , k R ) = (2, 2, 2, 2, 4, 6) under the premise that the sum of weights in each superpotential should be vanishing. We construct the mass matrices in both the charged-lepton and neutrino sectors under such a setup of weights. After performing the numerical analysis, we find out that our model is consistent with the latest global-fit results of neutrino oscillation data at the 1σ level only in the NO case and the individual contributions to the neutrino masses from the right-handed neutrino and the Higgs triplet are comparable. The allowed parameter space of the model parameters, namely, the modulus parameter τ = Re τ +iIm τ , three real parameters v d α 3 / √ 2, α 1 /α 3 and α 2 /α 3 in the charged-lepton sector, together with the coupling coefficient g = ge iφ g in the neutrino sector has been obtained. Moreover, we also give the constrained regions of three light neutrino masses {m 1 , m 2 , m 3 }, three neutrino mixing angles {θ 12 , θ 13 , θ 23 } and three CP-violating phases {δ, ρ, σ}, as well as the predictions for the effective neutrino masses m β in beta decays and m ββ in neutrinoless double-beta decays.
An interesting feature of our model is that the octant of θ 23 strongly depends on which hierarchy of α 1 , α 2 and α 3 is taken into consideration. To be specific, the hierarchy α 1 α 3 α 2 corresponds to the first octant of θ 23 while the hierarchy α 1 α 2 α 3 corresponds to the second octant of θ 23 . Furthermore, when the value of Im τ is sufficiently large, the neutrino mass matrix M ν will possess an approximate µ − τ reflection symmetry, indicating θ 23 ≈ 45 • and δ ≈ 90 • or 270 • . While the small parameter t ≡ −6q 1/3 = −6e 2πiτ /3 slightly breaks down this symmetry and generate realistic values for the mass-squared differences and other mixing angles. Besides, since our model predicts relatively large values of m ν = m 1 + m 2 + m 3 , m β and m ββ , it is very likely to be tested in the further neutrino experiments.
We stress that the hybrid seesaw models where not only one kind of seesaw mechanism is involved may lead to some new scenarios about flavor mixing and CP violation, and it deserves more attention to discuss the applications of the modular symmetry in such kinds of models. It is also interesting to study the type-(I+II) seesaw model with other kinds of finite modular symmetries. We hope to come back to these issues in the future works.

A The Γ 3 A 4 Symmetry Group
The permutation symmetry group A 4 has twelve elements and four irreducible representations, which are denoted as 1, 1 , 1 and 3. In the present work, we choose the complex basis which is used in Ref. [26] for the representation matrices of two generators S and T , namely, In this basis, we can explicitly write down the decomposition rules of the Kronecker products of any two A 4 multiplets.
As has been mentioned in Sec. 2, there exist three linearly independent modular forms of the lowest non-trivial weight k Y = 2, denoted as Y i (τ ) for i = 1, 2, 3. They transform as a triplet 3 under the A 4 symmetry, namely,