Spontaneous CP violation by modulus $\tau$ in $A_4$ model of lepton flavors

We discuss the modular $A_4$ invariant model of leptons combining with the generalized CP symmetry. In our model, both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus $\tau$. The source of the CP violation is a non-trivial value of ${\rm Re} [\tau]$ while other parameters of the model are real. The allowed region of $\tau$ is in very narrow one close to the fixed point $\tau=i$ for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase $\delta_{CP}$ is predicted clearly in $[98^\circ,\,110^\circ]$ and $[250^\circ,\,262^\circ]$ for NH at $3\,\sigma$ confidence level. On the other hand, $\delta_{CP}$ is in $[95^\circ,100^\circ]$ and $[260^\circ,\,265^\circ]$ for IH at $5\,\sigma$ confidence level. The predicted $\sum m_i$ is in $[82,\,102]$ meV for NH and $\sum m_i=[134,\,180]$ meV for IH. The effective mass $\langle m_{ee}\rangle$ for the $0\nu\beta\beta$ decay is predicted in $[12.5,\,20.5]$ meV and $[54,\,67]$ meV for NH and IH, respectively.


Introduction
The non-Abelian discrete symmetries are attractive ones to understand flavors of quarks and leptons. The S 3 flavor symmetry was a pioneer for the quark flavor mixing [1,2]. It was also discussed to understand the large mixing angle [3] in the oscillation of atmospheric neutrinos [4]. For the last twenty years, the non-Abelian discrete symmetries of flavors have been developed, that is motivated by the precise observation of flavor mixing angles of leptons [5][6][7][8][9][10][11][12][13][14]. Among them, the A 4 flavor model is an attractive one because the A 4 group is the minimal one including a triplet irreducible representation, which allows for a natural explanation of the existence of three families of quarks and leptons [15][16][17][18][19][20][21]. However, it is difficult to obtain clear predictions of the A 4 flavor symmetry because of a lot of free parameters associated with scalar flavon fields.
Recently, a new approach to the lepton flavor problem has been put forward based on the invariance under the modular transformation [22], where the model of the finite modular group Γ 3 A 4 has been presented. In this approach, fermion matrices are written in terms of modular forms which are holomorphic functions of the modulus τ . This work inspired further studies of the modular invariance approach to the lepton flavor problem.
In order to test the modular symmetry of flavors, the prediction of the CP violating Dirac phase is important. The CP transformation is non-trivial if the non-Abelian discrete flavor symmetry is set in the Yukawa sector of a Lagrangian. Then, we should discuss so called the generalized CP symmetry in the flavor space [87][88][89][90][91]. It can predict the CP violating phase [92]. The modular invariance has been also studied combining with the generalized CP symmetry in flavor theories [93,94]. It provides a powerful framework to predict CP violating phases of quarks and leptons.
In our work, we present the modular A 4 invariant model with the generalized CP symmetry. Both CP and modular symmetries are broken spontaneously by the vacuum expectation value (VEV) of the modulus τ . We discuss the phenomenological implication of this model, that is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing angles [95,96] and the CP violating Dirac phase of leptons, which is expected to be observed at T2K and NOνA experiments [97,98].
The paper is organized as follows. In section 2, we give a brief review on the generalized CP transformation in the modular symmetry. In section 3, we present the CP invariant lepton mass matrix in the A 4 modular symmetry. In section 4, we show the phenomenological implication of our model. Section 5 is devoted to the summary. In Appendix A, we present the tensor product of the A 4 group. In Appendix B, we show the modular forms for weight 2 and 4. In Appendix C, we show how to determine the coupling coefficients of the charged lepton sector. In Appendix D , we present how to obtain the Dirac CP phase, the Majorana phases and the effective mass of the 0νββ decay.
2 Generalized CP transformation in modular symmetry

Generalized CP symmetry
Let us start with discussing the generalised CP symmetry [92,99]. The CP transformation is nontrivial if the non-Abelian discrete flavor symmetry G is set in the Yukawa sector of a Lagrangian. Let us consider the chiral superfields. The CP is a discrete symmetry which involves both Hermitian conjugation of a chiral superfield ψ(x) and inversion of spatial coordinates, where x P = (t, −x) and X r is a unitary transformations of ψ(x) in the irreducible representation r of the discrete flavor symmetry G. If X r is the unit matrix, the CP transformation is the trivial one. This is the case for the continuous flavor symmetry [99]. However, in the framework of the non-Abelian discrete family symmetry, non-trivial choices of X r are possible. The unbroken CP transformations of X r form the group H CP . Then, X r must be consistent with the flavor symmetry transformation, where ρ r (g) is the representation matrix for g in the irreducible representation r.
This equation defines the consistency condition, which has to be respected for consistent implementation of a generalized CP symmetry along with a flavor symmetry [101,102]. This chain CP → g → CP −1 maps the group element g onto g and preserves the flavor symmetry group structure. That is a homomorphism v(g) = g of G. Assuming the presence of faithful representations r, Eq. (3) defines a unique mapping of G to itself. In this case, v(g) is an automorphism of G [101]. It has been also shown that the full symmetry group is isomorphic to a semi-direct product of G and H CP , that is G H CP , where H CP Z CP 2 , is the group generated by the generalised CP transformation under the assumption of X r being a symmetric matrix [102].

Modular symmetry
The modular groupΓ is the group of linear fractional transformations 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 ), called principal congruence subgroups, where N is the level 1, 2, 3, . . . . These groups are defined by For N = 2, we defineΓ(2) ≡ Γ(2)/{I, −I}. Since the element −I does not belong to Γ(N ) for N > 2, we haveΓ(N ) = Γ(N ). The quotient groups defined as Γ N ≡Γ/Γ(N ) are finite modular groups. In these finite groups Γ N , T N = 1 is imposed. The groups Γ N with N = 2, 3, 4, 5 are isomorphic to S 3 , A 4 , S 4 and A 5 , respectively [23]. Modular forms f i (τ ) of weight k are the holomorphic functions of τ and transform as under the modular symmetry, where ρ(γ) ij is a unitary matrix under Γ N . Superstring theory on the torus T 2 or orbifold T 2 /Z N has the modular symmetry [103][104][105][106][107][108]. Its low energy effective field theory is described in terms of supergravity theory, and string-derived supergravity theory has also the modular symmetry. Under the modular transformation of Eq. (4), chiral superfields ψ i (i denotes flavors) transform as [109], We study global supersymmetric models, e.g., minimal supersymmetric extensions of the Standard Model (MSSM). The superpotential which is built from matter fields and modular forms is assumed to be modular invariant, i.e., to have a vanishing modular weight. For given modular forms this can be achieved by assigning appropriate weights to the matter superfields.
The kinetic terms are derived from a Kähler potential. The Kähler potential of chiral matter fields ψ i with the modular weight −k is given simply by where the superfield and its scalar component are denoted by the same letter, andτ = τ * after taking VEV of τ . Therefore, the canonical form of the kinetic terms is obtained by changing the normalization of parameters [28]. The general Kähler potential consistent with the modular symmetry possibly contains additional terms [110]. However, we consider only the simplest form of the Kähler potential. For Γ 3 A 4 , the dimension of the linear space M k (Γ(3)) of modular forms of weight k is k + 1 [111][112][113], i.e., there are three linearly independent modular forms of the lowest non-trivial weight 2, which form a triplet of the A 4 group, Y As shown in Appendix A, these modular forms have been explicitly obtained [22] in the symmetric base of the A 4 generators S and T for the triplet representation: where ω = exp(i 2 3 π) .

CP transformation of the modulus τ
The CP transformation in the modular symmetry was given by using the generalized CP symmetry [93]. We summarize the discussion in Ref. [93] briefly. Consider the CP and modular transformation γ of the chiral superfield ψ(x) assigned to an irreducible unitary representation r of Γ N . The chain CP → γ → CP −1 = γ ∈Γ is expressed as: where τ CP −1 is the operation of CP −1 on τ . The result of this chain transformation should be equivalent to a modular transformation γ which maps ψ(x) to (c τ + d ) −k ρ r (γ )ψ(x). Therefore, one obtains Since X r , ρ r and ρ r are independent of τ , the overall coefficient on the right-hand side of Eq. (13) has to be a constant (complex) for non-zero weight k: where |λ| = 1 due to the unitarity of ρ r and ρ r . The values of λ, c and d depend on γ.
Taking γ = S (c = 1, d = 0) , and denoting c (S) = C, d (S) = D while keeping λ(S) = λ, we find τ = (λτ * CP −1 − D)/C from Eq. (14), and consequently, Let us act with chain CP → T → CP −1 on the mudular τ itself: The resulting transformation has to be a modular transformation, therefore λ/C is an integer. Since |λ| = 1, we find |C| = 1 and λ = ±1. After choosing the sign of C as C = ∓1 so that Im[τ CP ] > 0, the CP transformation of Eq. (15) turns to where n is an integer. The chain CP → S → CP −1 = γ (S) imposes no furher restrictions on τ CP . It is always possible to redefine the CP transformation in such a way that n = 0 by using the freedom of T transformation. Therefore, we define that the modulus τ transforms under CP as without loss of generality.
The same transformation of τ was also derived from the higher dimensional theories [94]. The four-dimensional CP symmetry can be embedded into (4+d) dimensions as higher dimensional proper Lorentz symmetry with positive determinant. That is, one can combine the four-dimensional CP transformation and d-dimensional transformation with negative determinant so as to obtain (4 + d) dimensional proper Lorentz transformation. For example in six-dimensional theory, we denote the two extra coordinates by a complex coordinate z. The four-dimensional CP symmetry with z → z * or z → −z * is a six-dimensional proper Lorentz symmetry. Note that z = x + τ y, where x and y are real coordinates. The latter transformation z → −z * maps the upper half plane Im[τ ] > 0 to the same half plane. Hence, we consider the transformation z → −z * (τ → −τ * ) as the CP symmetry.

CP transformation of modular multiplets
Chiral superfields and modular forms transform in Eqs. (8) and (9), respectively, under a modular transformation. Chiral superfields also transform in Eq. (1) under the CP transformation. The CP transformation of modular forms were given in Ref. [93] as follows. Define a modular multiplet of the irreducible representation r of Γ N with weight k as Y (k) r (τ ), which is transformed as: under the CP transformation. The complex conjugated CP transformed modular forms Y r (τ ) under a modular transformation, namely: where u(γ) ≡ CP γCP −1 . Using the consistency condition of Eq. (3), we obtain Therefore, if there exist a unique modular multiplet at a level N , weight k and representation r, which is satisfied for N = 2-5 with weight 2, we can express the modular form Y (k) r (τ ) as: where κ is a proportional coefficient. Since Y (22) gives X * r X r = |κ| 2 1 r . Therefore, the matrix X r is symmetric one, and κ = e iφ is a phase, which can be absorbed in the normalization of modular forms. In conclusion, the CP transformation of modular forms is given as: It is also emphasized that X r = 1 r satisfies the consistency condition Eq. (3) in a basis that generators of S and T of Γ N are represented by symmetric matrices because of ρ * . The CP transformations of chiral superfields and modular multiplets are summalized as follows: where X r = 1 r can be taken in the base of symmetric generators of S and T . We use this CP transformation of modular forms to construct the CP invariant mass matrices in the next section.

CP invariant mass matrix in A 4 modular symmetry
Let us discuss the CP invariant lepton mass matrix in the framework of the A 4 modular symmetry.
We assign the A 4 representation and weight for superfields of leptons in Table 1, where the three left-handed lepton doublets compose a A 4 triplet L, and the right-handed charged leptons e c , µ c and τ c are A 4 singlets. The weights of the superfields of left-handed leptons and right-handed charged leptons are −2 and 0, respectively. Then, the simple lepton mass matrices for charged leptons and neutrinos are obtained [75]. The superpotential of the charged lepton mass term is given in terms of modular forms of weight 2, Y 3 . It is given as: where L is the left-handed A 4 triplet leptons. We can take real for α e , β e and γ e . Under CP, the superfields transform as: and we can take η d = 1 without loss of generality. Since the representations of S and T are symmetric as seen in Eq. (11), we can choose X 3 = 1 and X 1 = X 1 = X 1 = 1.
Taking (e L , µ L , τ L ) in the flavor base, the charged lepton mass matrix M E is simply written as: where v d is VEV of the neutral component of H d , and coefficients α e , β e and γ e are taken to be real without loss of generality. Under CP transformation, the mass matrix M E is transformed following from Eq. (24) as: Let us discuss the neutrino mass matrix. Suppose neutrinos to be Majorana particles. By using the Weinberg operator, the superpotential of the neutrino mass term, w ν is given as: where Λ is a relevant cutoff scale. Since the left-handed lepton doublet has weight −2, the superpotential is given in terms of modular forms of weight 4, Y 3 , Y 1 and Y 1 . By putting v u for VEV of the neutral component of H u and using the tensor products of A 4 in Appendix A, we have where Y 3 , Y 1 and Y 1 are given in Eq. (47) of Appendix B, and g ν 1 , g ν 2 are complex parameters in general. The neutrino mass matrix is written as follows: which is the same one in Ref. [75]. Under CP transformation, the mass matrix M ν is transformed following from Eq. (24) as: In a CP conserving modular invariant theory, both CP and modular symmetries are broken spontaneously by VEV of the modulus τ . However, there exist certain values of τ which conserve CP while breaking the modular symmetry. Obviously, this is the case if τ is left invariant by CP, i.e.
which leads to g ν 1 and g ν 2 being real. Since parameters α e , β e , γ e are also real, the source of the CP violation is only non-trivial Re[τ ] after breaking the modular symmetry. In the next section, we present numerical analysis of the CP violation by investigating the value of the modulus τ .

Numerical results of leptonic CP violation
We have presented the CP invariant lepton mass matrices in the A 4 modular symmetry. These mass matrices are the same ones in Ref. [75] except for parameters g ν 1 and g ν 2 being real. If the CP violation will be confirmed at the experiments of neutrino oscillations, the CP symmetry should be broken spontaneously by VEV of the modulus τ . Thus, VEV of τ breaks the CP symmetry as well as the modular invariance. The source of the CP violation is only the real part of τ . This situation is different from the previous work in Ref. [75], where imaginary parts of g ν 1 and g ν 2 also break the CP symmetry explicitly. Our phenomenological concern is whether the spontaneous CP violation is realized due to the value of τ , which is consistent with observed lepton mixing angles and neutrino masses. If this is the case, the CP violating Dirac phase and Majorana phases are predicted clearly under the fixed value of τ .
Parameter ratios α e /γ e and β e /γ e are given in terms of charged lepton masses and τ as shown in Appendix C. Therefore, the lepton mixing angles, the Dirac phase and Majorana phases are given by our model parameters g ν 1 and g ν 2 in addition to the value of τ . As the input charged lepton masses, we take Yukawa couplings of charged leptons at the GUT scale 2 × 10 16 GeV, where tan β = 5 is taken as a bench mark [114,115]: y e = (1.97 ± 0.024) × 10 −6 , y µ = (4.16 ± 0.050) × 10 −4 , y τ = (7.07 ± 0.073) × 10 −3 ,  The best fit ±1 σ of neutrino parameters from NuFIT 5.0 for NH and IH [116].
We also input the lepton mixing angles and neutrino mass parameters which are given by NuFit 5.0 in Table 2 [116]. In our analysis, δ CP is output because its observed range is too wide at 3 σ confidence level. We investigate two possible cases of neutrino 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 . Neutrino masses and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U PMNS [95,96] are obtained by diagonalizing M † E M E and M † ν M ν . We also investigate the effective mass for the 0νββ decay, m ee (see Appendix D) and the sum of three neutrino masses m i since it is constrained by the recent cosmological data, which is the upper-bound m i ≤ 120 meV obtained at the 95% confidence level [117,118].

Case of normal hierarchy of neutrino masses
Let us discuss numerical results for NH of neutrino masses. The ratios α e /γ e and β e /γ e are given after fixing charged lepton masses and τ as shown in Appendix C. However, in practice, we scan α e /γ e and β e /γ e to obtain the observed charged lepton mass ratio and include them in χ 2 fit as well as three mixing angles and ∆m 2 atm /∆m 2 sol . We have already studied the lepton mass matrices in Eqs. (27) and (31) phenomenologically at the nearby fixed points of the modulus because the spontaneous CP violation in Type IIB string theory is possibly realized at nearby fixed points, where the moduli stabilization is performed in a controlled way [119,120]. There are two fixed points in the fundamental domain of P SL(2, Z), τ = i and τ = ω. Indeed, the viable τ of our lepton mass matrices is found around τ = i [75].
Based on this result of Ref. [75], we scan τ around i while neutrino couplings g ν 1 and g ν 2 are scanned in the real space of [−10, 10]. As a measure of good-fit, we adopt the sum of one-dimensional χ 2 function for four accurately known dimensionless observables ∆m 2 atm /∆m 2 sol , sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 in NuFit 5.0 [116]. In addition, we employ Gaussian approximations for fitting m e /m τ and m µ /m τ by using the data of PDG [121].
In Fig. 1 we show the allowed region on the Re [τ ] -Im [τ ] plane, where three mixing angles and ∆m 2 atm /∆m 2 sol are consistent with observed ones. The green, yellow and red regions correspond to 2σ, 3σ and 5σ confidence levels, respectively.  The allowed region of g ν 1 and g ν 2 , which are real parameters, for NH. Colors denote same ones in Fig. 1.
The allowed region of τ is restricted in the narrow regions. This result is contrast to the previous one in Ref. [75], where non-trivial phases of g ν 1 and g ν 2 enlarged the allowed region of τ . The predicted range of τ is in Re [τ ] = ±[0.073, 0.083] and Im [τ ] = [1.006, 1.014] at 3 σ confidence level (yellow), which are close to the fixed point τ = i.
The allowed region of g ν 1 and g ν 2 is also shown in Fig. 2 [82,102] meV for 3 σ confidence level (yellow). The minimal cosmological model, ΛCDM + m i , provides the upper-bound m i < 120 meV [117,118]. Thus, our predicted sum of neutrino masses is consistent with the cosmological bound 120 meV.
In Fig. 4, we show the allowed region on the sin 2 θ 23 -m i plane. Since m i depends on the value of sin 2 θ 23 significantly, the crucial test of our prediction will be available in the near future.  We can calculate the effective mass m ee for the 0νββ decay by using the Dirac phase and Majorana phases as seen in Appendix D. We show the predicted value of m ee versus sin 2 θ 23 as seen in Fig. 6. The predicted m ee is in [12.5, 20.5] meV for 3 σ confidence level (yellow). The prediction of m ee 20 meV will be testable in the future experiments of the neutrinoless double beta decay.

Case of inverted hierarchy of neutrino masses
We discuss the case of IH of neutrino masses. In Fig. 7, we show the allowed region on the Re [τ ] -Im [τ ] plane, where the red region corresponds to 5 σ confidence level like in Fig. 1. However, there are no green and yellow regions of 2 σ and 3 σ confidence levels.  The allowed region of g ν 1 and g ν 2 is also shown in Fig. 8, where g ν 1 is restricted in the narrow range of [−1.20, −1.15] while g ν 2 is rather large as in [4.8, 9.6] for 5 σ.  m i < 120 meV [117,118], however, it becomes weaker when the data are analysed in the context of extended cosmological models [121]. The predicted sum of neutrino masses of IH may be still consistent with the cosmological bound.
We show the allowed region on the m i -sin 2 θ 23 plane in Fig. 10. The precise measurement of sin 2 θ 23 will provide a severe test for our prediction since sin 2 θ 23 > 0.55 is obtained for IH.
In Fig. 11 We also show the predicted value of m ee versus sin 2 θ 23 as seen in Fig. 12. The predicted m ee is in [54,67] meV for 5 σ confidence level.    Table 3: Numerical values of parameters and observables at the sample points of NH and IH.

Parameter samples of NH and IH
We show the numerical result of two samples for NH and IH, respectively. In Table 3, parameters and outputs of our calculations are presented for both NH and IH. We also present the mixing matrices of charged leptons U E and neutrinos U ν for the samples of Table 3. For NH, those are: which are given in the diagonal base of the generator S in order to see the hierarchical structure of flavor mixing [75]. The PMNS mixing matrix is given as U PMNS = U † E U ν . The diagonal base of S is obtained by using the following unitary matrix: which leads to V S S V † S = diag (1, −1, −1) [75]. Then, the charged lepton and neutrino mass matrices are transformed as which are also given in the diagonal base of the generator S. For both NH and IH, the mixing matrix of charged leptons U E is hierarchical one, on the other hand, two large mixing angles of 1-2 and 2-3 flavors appear in the neutrino mixing matrix U ν .
In our numerical calculations, we have not included the RGE effects in the lepton mixing angles and neutrino mass ratio ∆m 2 sol /∆m 2 atm . We suppose that those corrections are very small between the electroweak and GUT scales. This assumption is justified well in the case of tan β ≤ 5 unless neutrino masses are almost degenerate [27].

Summary
The modular invariant A 4 model of lepton flavors has been studied combining with the generalized CP symmetry. In our model, both CP and modular symmetries are broken spontaneously by VEV of the modulus τ . The source of the CP violation is a non-trivial value of Re[τ ] while parameters of neutrinos g ν 1 and g ν 2 are real. We have found allowed region of τ close to the fixed point τ = i, which is consistent with the observed lepton mixing angles and lepton masses for NH at 2 σ confidence level. The CP violating Dirac phase δ CP is predicted clearly in [98 • , 110 • ] and [250 • , 262 • ] at 3 σ confidence level. The predicted m i is in [82,102] meV with 3 σ confidence level. There is also allowed region of τ close to the fixed point τ = i for IH at 5 σ confidence level. The predicted δ CP is in [95 • , 100 • ] and [260 • , 265 • ] at 5 σ confidence level. The sum of neutrino masses is predicted in m i = [134, 180] meV. By using the predicted Dirac phase and the Majorana phases, we have obtained the effective mass m ee for the 0νββ decay, which are in [12.5, 20.5] meV for NH at 3 σ confidence level and in [54,67] meV for IH at 5 σ confidence level. Since KamLAND-Zen experiment [122] presented the upper bound on the effective Majorana mass as m ee < 61-165 meV by using a variety of nuclear matrix element calculations, the prediction of [54,67] meV for IH will be tested in the near future. Furthermore, the prediction of m ee 20 meV for NH will be also testable in the future experiments of the neutrinoless double beta decay.
In our model, the modulus τ dominates the CP violation. Therefore, the determination of τ is the most important work. Although we have constrained τ by observables of leptons phenomenologically, one also should pay attention to the recent theoretical work of the moduli stabilization from the viewpoint of modular flavor symmetries [123]. The study of modulus τ is interesting to reveal the flavor theory in both theoretical and phenomenological aspects.
where ω = e i 2 3 π for a triplet. In this base, the multiplication rule is where More details are shown in the review [6,7].

B Modular forms in A 4 symmetry
For Γ 3 A 4 , the dimension of the linear space M k (Γ(3)) of modular forms of weight k is k + 1 [111][112][113], i.e., there are three linearly independent modular forms of the lowest non-trivial weight 2. These forms have been explicitly obtained [22] in terms of the Dedekind eta-function η(τ ): where η(τ ) is a so called modular form of weight 1/2. In what follows we will use the following base of the A 4 generators S and T in the triplet representation: where ω = exp(i 2 3 π) . The modular forms of weight 2 (k = 2) transforming as a triplet of A 4 , Y ) T , can be written in terms of η(τ ) and its derivative [22]: The overall coefficient in Eq. (44) is one possible choice. It cannot be uniquely determined. The triplet modular forms of weight 2 have the following q-expansions: They satisfy also the constraint [22]: The modular forms of the higher weight, k, can be obtained by the A 4 tensor products of the modular forms with weight 2, Y 3 (τ ), as given in Appendix A. For weight 4, that is k = 4, there are five modular forms by the tensor product of 3 ⊗ 3 as: (47) where Y whereα e ≡ α e /γ e andβ e ≡ β e /γ e . Denoting charged lepton masses m 1 = m e , m 2 = m µ and m 3 = m τ , we have three equations as: where χ ≡ m 2 1 m 2 2 + m 2 2 m 2 3 + m 2 3 m 2 1 . The coefficients C e 1 , C e 2 and C e 3 depend only on Y i (τ )'s, where Y i (τ )'s are determined if the value of modulus τ is fixed. Those are given explicitly as follows: Then, we obtain two equations which describeα e andβ e in terms of masses and τ : where we redefine the parametersα 2 e +β 2 e = s andα 2 eβ 2 e = t. After fixing charged lepton masses and τ , we obtain s and t numerically. They are related as follows:

D Majorana and Dirac phases and m ee in 0νββ decay
Supposing neutrinos to be Majorana particles, the PMNS matrix U PMNS [95,96] is parametrized in terms of the three mixing angles θ ij (i, j = 1, 2, 3; i < j), one CP violating Dirac phase δ CP and two Majorana phases α 21 , α 31 as follows: where c ij and s ij denote cos θ ij and sin θ ij , respectively. The rephasing invariant CP violating measure of leptons [124,125] is defined by the PMNS matrix elements U αi . It is written in terms of the mixing angles and the CP violating phase as: J CP = Im U e1 U µ2 U * e2 U * µ1 = s 23 c 23 s 12 c 12 s 13 c 2 13 sin δ CP , where U αi denotes the each component of the PMNS matrix. There are also other invariants I 1 and I 2 associated with Majorana phases In terms of this parametrization, the effective mass for the 0νββ decay is given as follows: m ee = m 1 c 2 12 c 2 13 + m 2 s 2 12 c 2 13 e iα 21 + m 3 s 2 13 e i(α 31 −2δ CP ) .