Modulus $\tau$ linking leptonic CP violation to baryon asymmetry in $A_4$ modular invariant flavor model

We propose an $A_4$ modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus $\tau$. The value of the modulus $\tau$ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses at $3\sigma$ confidence level. The predictive Dirac CP phase $\delta_{CP}$ is in the ranges $[0^\circ,50^\circ]$, $[170^\circ,175^\circ]$ and $[280^\circ,360^\circ]$ for ${\rm Re}\, [\tau]<0$, and $[0^\circ,80^\circ]$, $[185^\circ,190^\circ]$ and $[310^\circ,360^\circ]$ for ${\rm Re}\,[\tau]>0$ at $3\sigma$ confidence level. The sum of three neutrino masses is predicted in $[60, 84]$ meV, and the effective mass for the $0\nu\beta\beta$ decay is in [0.003, 3] meV. On the other hand, there is no allowed region of the modulus $\tau$ for the inverted hierarchy of neutrino masses at $3\sigma$ confidence level. The modulus $\tau$ links the Dirac CP phase to the cosmological baryon asymmetry (BAU) via the leptogenesis. Due to the strong wash-out effect, the predictive baryon asymmetry $Y_B$ can be at most the same order of the observed value. Then, the lightest right-handed neutrino mass is restricted in the range of $M_1 =[1.5,\,6.5] \times 10^{13}$ GeV. We find the correlation between the predictive $Y_B$ and the Dirac CP phase $\delta_{CP}$. Only two predictive $\delta_{CP}$ ranges, $[5^\circ,40^\circ]$ (${\rm Re}\,[\tau]>0$) and $[320^\circ,355^\circ]$ (${\rm Re}\,[\tau]<0$) are consistent with the BAU.


Introduction
One interesting approach to the origin of flavor structure is to impose a flavor symmetry on a theory. The non-Abelian discrete groups are attractive ones to understand flavor structure of quarks and leptons. The S 3 flavor symmetry was studied to understand the large mixing angle [1] in the oscillation of atmospheric neutrinos [2] as well as discussing the Cabibbo angle [3,4]. For the last twenty years, the non-Abelian discrete symmetries of flavors have been developed [5][6][7][8][9][10][11][12][13][14], that is motivated by the precise observation of flavor mixing angles of leptons. Among them, the A 4 flavor symmetry provides a simple 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 clues of the A 4 flavor symmetry because of a lot of free parameters associated with scalar flavon fields.
An interesting 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 flavor symmetry, the prediction of the Dirac CP 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 [107][108][109][110][111][112]. The modular invariance has been also studied in the framework of the generalized CP symmetry [113,114]. It provided a significant scheme to predict Dirac and Majorana CP phases of leptons. A viable lepton model was proposed in the modular A 4 symmetry [80], in which the CP violation is realized by fixing τ , that is, the breaking of the modular symmetry. Afterward, the systematic search of the viable A 4 model was done [81].
The CP violation by the modulus τ raises a question. Is the leptonic CP violation linked to the baryon asymmetry of the universe (BAU)? The BAU is now measured very precisely by the cosmic microwave background radiation [115]. One of the most studied scenarios for baryogenesis is the canonical leptogenesis scenario [116], in which the decays of right-handed neutrinos can generate the lepton asymmetry that is partially converted into the baryon asymmetry via the sphaleron process [117]. The sign of the BAU is controlled by the CP violation pattern in the leptonic sector. In general, the sign of the BAU cannot be predicted uniquely even if the Dirac and Majorana CP phases are determined. This is because there exist additional phases associated with right-handed neutrinos which decouple from the low energy phenomena if right-handed neutrinos are sufficiently heavy. However, there are non-trivial relations between the properties of right-handed neutrinos and the low energy observables of neutrinos in the A 4 modular symmetry. Indeed, the modulus τ controls the CP phases of both the left-handed sector of neutrinos and the right-handed one in our scheme. Under these situations, it is interesting to investigate the sign and magnitude of the BAU.
In the framework of the modular symmetry, the BAU has been studied in A 4 model of leptons, where the source of CP violation is a complex parameter in the Dirac neutrino mass matrix in addition to the modulus τ [40]. In our work, the origin of the CP violation is only in modulus τ , therefore, it is also only the source of the leptogenesis. We present a modular A 4 invariant model with the CP symmetry, where 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 [118,119] and the Dirac CP phase of leptons, which is expected to be observed at T2K and NOνA experiments [120,121]. Then, we examine a link between the predictive Dirac CP phase and the BAU. The paper is organized as follows. In section 2, we give a brief review on the 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 lepton mixing and CP phases. In section 5, we give the framework of the leptogenesis in our model. In section 6, we discuss the link between the predictive Dirac CP phase and BAU numerically. Section 7 is devoted to the summary. In Appendices A and B, we give the tensor product of the A 4 group and the modular forms, respectively. In Appendix C, we show the definition of PMNS matrix elements and how to obtain the Dirac CP phase, the Majorana phases and the effective mass of the 0νββ decay. In Appendix D, alternative A 4 models and their results are presented. In Appendix E, we give the relevant formulae of the leptogenesis explicitly.

Generalized CP symmetry
The CP transformation is non-trivial if the non-Abelian discrete flavor symmetry G is set in the Yukawa sector of a Lagrangian [112,122]. 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 transformation of ψ(x) in the irreducible representation r of the discrete flavor symmetry G. This transformation is so-called the generalized CP transformation. If X r is the unit matrix, the CP transformation is the trivial one. This is the case for the continuous flavor symmetry [122]. 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 [124,125].
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 [125].

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 . Under the modular transformation of Eq. (4), chiral superfields ψ i (i denotes flavors) with weight −k transform as [126], 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 . 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 [127]. 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 [128][129][130], 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 These modular forms have been explicitly given [22] in the symmetric base of the A 4 generators S and T for the triplet representation (see Appendix A) in Appendix B.

CP transformation of the modulus τ
The CP transformation in the modular symmetry was given by using the generalized CP symmetry [113]. We summarize the discussion in Ref. [113] briefly. Consider the CP and modular transformation γ of the chiral superfield ψ(x) with weight −k 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. (12) 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. (13), 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. (14) turns to where n is an integer. The chain CP → S → CP −1 = γ (S) imposes no further 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 can define the CP transformation of the modulus τ as

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 was given in Ref. [113] 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 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 (k) r (τ ), Eq. (21) gives X * r X r = |κ| 2 1 r . Therefore, the matrix X r is a 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 basis 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 lepton mass matrix in A 4 modular symmetry
In this section, we propose the CP invariant lepton mass matrix for introducing the A 4 modular symmetry. The three generations of the left-handed lepton doublets are assigned to be an A 4 triplet L, and the right-handed charged leptons e c , µ c , and τ c are A 4 singlets 1, 1 , and 1 , respectively. The three generations of the right-handed neutrinos are also assigned to be an A 4 triplet N c . The weight of the superfields of left-handed leptons is fixed to be −1 as a standard. The weight of right-handed neutrinos is also taken to be −1 in order to give a Dirac neutrino mass matrix in terms of modular forms of weight 2. On the other hand, weights of the right-handed charged leptons e c , µ c , and τ c are put (k e , k µ , k τ ) in general. Weights of Higgs fields H u , H d are fixed to be 0. The representations and weights for MSSM fields and modular forms of weight k are summarized in Table 1. Since we construct the CP invariant lepton mass matrices with minimum number of parameters, we fix weights k e = −1, k µ = −3, k τ = −5 for right-handed charged leptons. Then, we need modular forms of weight 2, 4 and 6, Y Then, the A 4 invariant superpotential of the charged leptons, w E , by taking into account the modular weights is obtained as where α e , β e , γ e , and γ e are constant parameters. 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 (see Appendic A), we can choose X 3 = 1 and X 1 = X 1 = X 1 = 1 as discused in Eq. (23).
Taking (e L , µ L , τ L ) in the flavor base, the charged lepton mass matrix M E is simply written as: where g e = γ e /γ e , and v d is VEV of the neutral component of H d . The 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. (28) as: Let us discuss the neutrino sector. In Table 1, the A 4 invariant superpotential for the neutrino sector, w ν , is given as: where γ ν and γ ν are Yukawa couplings, and Λ denotes a right-handed Majorana neutrino mass scale. By putting v u for VEV of the neutral component of H u and taking (ν e , ν µ , ν τ ) for neutrinos, the Dirac neutrino mass matrix, M D , is obtained as where g D = γ ν /γ ν . On the other hand the right-handed Majorana neutrino mass matrix, M N is written as follows: By using the type-I seesaw mechanism, the effective neutrino mass matrix, M ν is obtained 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.
if g e and g D are taken to be real. Therefore, the source of the CP violation is only non-trivial Re[τ ] after breaking the modular symmetry. In the next section, we present a numerical analysis of the CP violation by fixing the modulus τ with real g e and g D .

Numerical results of leptonic CP violation
We have presented the CP invariant lepton mass matrices in the A 4 modular symmetry. The tiny neutrino masses are given via type-I seesaw. The CP symmetry is broken spontaneously by VEV of the modulus τ . Thus, VEV of τ breaks the CP invariance as well as the modular invariance. The source of the CP violation is the real part of τ . Indeed, the spontaneous CP violation is realized by fixing τ . Then, the Dirac CP phase and Majorana phases are predicted clearly with reproducing observed lepton mixing angles and two neutrino mass squared differences. The predictive CP phases are possibly linked to the phase of the leptogenesis [116]. Our parameters are real ones α e , β e , γ e , γ e , γ ν , γ ν and Λ in addition to the complex τ . Observed input data are three charged lepton masses, three flavor mixing angles and two neutrino mass squared differences. Since γ ν and Λ appear only with the combination γ 2 ν /Λ in the neutrino mass matrix, the input data determine completely our parameters apart from error-bars of the experimental data. Therefore, the lepton mixing angles, the Dirac phase and Majorana phases are predicted in the restricted ranges.
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 [131,132]:   [133].
We also input the lepton mixing angles and neutrino mass parameters which are given by NuFit 5.0 in Table 2 [133]. In our analysis, the Dirac CP phase δ CP (see Appendix C) 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 PMNS matrix U PMNS [118,119] 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 C) 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 [115,134].
Let us discuss numerical results for NH of neutrino masses. We scan τ in the fundamental domain of SL(2, Z). The real parameters g e and g D are scanned in [−10, 10]. As a measure of good-fit, we adopt the sum of one-dimensional χ 2 functions for five accurately known observables ∆m 2 atm , ∆m 2 sol , sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 in NuFit 5.0 [133]. In addition, we employ Gaussian approximations for fitting m e , m µ and m τ .
In Fig. 1   Due to the rather broad range of Re [τ ], the predictive Dirac CP phase δ CP , which is defined in Appendix C, is not so restricted. In Fig. 2   In Fig. 3, we show a prediction of δ CP versus m i . It is also found different six predicted regions. The sum of neutrino masses m i is restricted in the narrow range [60,84] meV at χ 2 ≤ 3 (magenta). It is consistent with the cosmological bound 120 meV in the minimal cosmological model, ΛCDM + m i . [115,134].
We can calculate the effective mass m ee for the 0νββ decay by using the Dirac CP phase and Majorana phases as seen in Appendix C. The predicted m ee is at χ 2 ≤ 3. It is difficult to reach this value in the future experiments of the neutrinoless double beta decay. We have checked the correlation between three mixing angles and δ CP plane. The predicted δ CP is correlated weakly with sin 2 θ 23 in our model. It has the broadest ranges of [0 We show the best fit sample for NH in Table 3, where numerical values of parameters and output are listed. As a measure of goodness of fit, we show square root of the sum of one-dimensional χ 2 functions.
We have also scanned the parameter space for the case of IH of neutrino masses. We have found parameter sets which reproduce the observed masses and three mixing angles sin 2 θ 23 , sin 2 θ 12 , and sin 2 θ 13 at χ 2 ≤ 5. However, there is no parameter sets below χ 2 = 4.
The allowed region of τ is restricted in the narrow regions. As shown in Fig. 5 Table 3: Numerical values of parameters and observables at the best fit points of NH and IH.
In Fig. 6 [133]. On the other hand, the sum of neutrino masses m i is restricted in the narrow range [143,147] meV at χ 2 = 4-5. Therefore, the case of IH will be excluded by the improved cosmological bound in the near future. The predicted m ee is given as m ee = [35,36] meV.
We also present the best fit set of IH in Table 3, where values of relevant parameters are listed compared with the values of the NH case.  In our numerical calculations, we have not included the RGE effects in the lepton mixing angles. 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 ≤ 10 unless neutrino masses are almost degenerate [27].
We have presented CP invariant lepton mass matrices with minimum number of parameters by fixing weights (k e = −1, k µ = −3, k τ = −5) for right-handed charged leptons. Therefore, the charged lepton mass matrix is given by modular forms of weight 2, 4 and 6, Y and Y 3 . However, this choice for weights is not a unique one even if we consider the seesaw model with minimum number of parameters. We have examined alternative three choices: (k e = −1, k µ = −1, k τ = −5), (k e = −1, k µ = −1, k τ = −7) and (k e = −1, k µ = −3, k τ = −7). The correponding charged lepton mass matrices are presented in Appendix D. In those three models, we have also obtained successful numerical results, which are not so different from above ones. We present samples of parameter sets for each model in Appendix D for NH of neutrino masses. The cases of IH are omitted since their χ 2 are larger than 4. Our study of the leptogenesis is focused on the case of weights (k e = −1, k µ = −3, k τ = −5) in the next section.

Leptogenesis
The BAU at the present universe is measured very precisely by the cosmic microwave background radiation as [115]: at 3 σ confidence level, where Y B is defined by the ratio between the number density of baryon asymmetry n B and the entropy density s. One of the most attractive scenarios for baryogenesis is the canonical leptogenesis scenario [116] in which the decays of right-handed neutrinos can generate the lepton asymmetry that is partially converted into the baryon asymmetry via the sphaleron process [117]. The sign and magnitude of the BAU are predicted by the masses and Yukawa coupling constants of right-handed neutrinos. If their masses are hierarchical, the lightest one must be O(10 9 ) GeV [135] to explain the BAU. The sign of the BAU depends on the CP phase strucuture in the lepton mass matrices. In general, the sign of the BAU cannot be predicted uniquely even if the Dirac and Majorana CP phases are determined. This is because there exist generally one or more additional phases associated with right-handed neutrinos which decouple from the low energy phenomena even if right-handed neutrinos are sufficiently heavy. However, our predictive Dirac and Majorana CP phases are linked to the BAU because the CP violation is originated from only τ in our model with A 4 modular symmetry. Let us discuss the leptogenesis by decays of right-handed neutrinos in our model. Since the mass ratios of right-handed neutrinos are not so large, and then we have to include the effects of all three right-handed neutrinos to the leptogenesis. For simplicity, we assume that the reheating temperature of inflation is sufficiently higher than the mass of the heaviest right-handed neutrino and that the initial abundances of all right-handed neutrinos are zero. On the other hand, the mass degeneracy of the right-handed neutrinos is not so large, and so the resonant enhancement of the leptogenesis [136,137] does not occur. Thus, we shall use the formalism based on the Boltzmann equations to estimate the asymmetries. Moreover, as we show below, the required masses of right-handed neutrinos are O(10 13 ) GeV, and so we can apply the simple one-flavor approximation of the leptogenesis. Therefore, we neglect the so-called flavor effect [138][139][140][141][142][143][144][145]. Furthermore, to achieve successful leptogenesis via the decay of such heavy right-handed neutrinos, the reheating temperature must be higher than O(10 13 ) GeV. In the framework of supersymmetry (SUSY), such high reheating temperature can cause the overproduction of gravitinos, which is called the gravitino problem [146,147]. However, in our scenario, we assume that SUSY is broken at close to the Planck scale. In this situation, SUSY particles, including the gravitino, have masses around the Planck scale. Therefore, gravitino cannot be thermally produced after inflation. So the constraint on the reheating temperature due to the gravitino problem can be eliminated.
The flavor structure of our model appears in the CP asymmetry parameter ε I , which is: It is proportional to the imaginary part of Yukawa couplings as: Here, y ν y † ν is given by the Dirac neutrino mass matrix M D in the real diagonal base of the righthanded Majorana neutrino mass matrix M N as follows: where M D and M N are given in Eqs. (31) and (32), respectively, and M 1 , M 2 and M 3 are real. The Boltzmann equations are then solved numerically and the total lepton asymmetry Y L from the decays of right-handed neutrinos is estimated. The present baryon asymmetry can be estimated as Y B = −8/23Y L for the two Higgs doublets (see Appendix E).
In our model, the phases in the PMNS matrix and the high energy phases associated with righthanded neutrinos are originated in the modulus τ . In this situation, there may exist the correlations between the phases in the PMNS matrix and the yield of the BAU.
Since the best-fit point of the modulus τ is rather close to the fixed point τ = i for NH as seen in Table 3, we can calculate approximately the asymmetry parameter ε I of Eq.(41) in terms of a small complex parameter , which is defined as τ = i + in perturbation. This analytic calculation is possible due to the simple Dirac neutrino mass matrix at τ = i. It is found that the leading term of Im{(y ν y † ν ) Ij } 2 is given by Im [ 2 ]. On the other hand, it is almost impossible to give an approximate form of the CP phase δ CP or the CP violating measure J CP at low energy. Since the right-handed Majorana neutrino mass matrix M N gives one massless and two degenerated masses at τ = i in our model, its inverse is a singular one at τ = i. After seesaw, the left-handed Majorana mass matrix is unstable at nearby τ = i for perturbation. Indeed, we could not obtain a reliable analytic expression for J CP . Therefore, we study the correlations between the CP phase δ CP and the yield of the BAU in numerical calculations.

Baryon asymmetry
Let us then show the results of the BAU by right-handed neutrinos in our model by using parameter sets of section 4 at χ 2 ≤ 3 for NH of neutrino masses. At first, we discuss the sign of the BAU produced by right-handed neutrinos in the model. The sign of the BAU is determined by the τ and the sign of the real parameter g D as shown in Fig. 7. In order to obtain the observed positive Y B (orange points), the region of (Re[τ ] < 0, g D < 0) or (Re[τ ] > 0, g D > 0) is required. Indeed, we can see this situation in Fig. 9, where the positive (orange) and negative (blue) signs of Y B are shown in the g D -δ CP plane. The positive and negative regions of Y B are clearly separated.  Next, we discuss the magnitude of the BAU yield. The yield of the BAU depends on the masses M i and Yukawa coupling constants of right-handed neutrinos. These parameters are highly restricted due to the symmetry in our model. First, the allowed range of the mass ratios of right-handed neutrinos is shown in Fig. 10. It is found that M 3 /M 2 decreases on the curve in the range of [1.1, 1.6] depending on M 2 /M 1 = [1. 6, 7.6]. Those mass ratios suggest that all three right-handed neutrinos should be taken into account in the calculation of the leptogenesis.
We find that the yield can be at most the same order of the observed value of the BAU in Eq. (39). This is because the model predicts a relatively large value of the effective neutrino mass of the leptogenesism 1 which is defined asm 1 = (y ν y ν † ) 11 v u 2 /M 1 . We show the predictive Y B in thẽ m 1 -M 1 plane in Fig. 11, where four predictive ranges of Y B are discriminated by colors. We find numericallym 1 40 meV orm 1 60 meV, and then the strong wash-out effect is inevitable. In order to obtain the observed BAU,m 1 = [60, 61] meV and M 1 = [1.5, 6.5] × 10 13 GeV are required as seen in Fig. 11. It is an important consequence that the lightest right-handed neutrino mass should be in the restricted range. Thus, the absolute values of right-handed neutrino masses can be determined from the BAU.
We show predictive Y B versus M 1 in Fig. 12, where M 1 is taken to be O(10 13 ) GeV. The predictive Y B is rather broad in this range of M 1 . Especially, it expands maximally at M 1 = 3.36 × 10 13 GeV. That is because the larger M 1 is, the more the wash-out effect of the ∆L = 2 processes is important. Thereby, the lightest right-handed neutrino mass is restricted to the specific range M 1 = [1.5, 6.5] × 10 13 GeV.   In order to see which parameter causes the predictive broad Y B of Fig. 12, we show the M 2 /M 1 dependence of Y B at M 1 = 3.36×10 13 GeV in Fig. 13. It is clearly found that the predictive magnitude of Y B depends on M 2 /M 1 crucially in addition to the magnitude of M 1 . If M 2 /M 1 is fixed in [1.6, 2], the predictive Y B is in the narrow range, which is consistent with the observed one. However, the present neutrino oscillation data still allow the range M 2 /M 1 = [1.6, 7.6] because of the broad τ region in Fig. 1.
Finally, we present Y B versus δ CP at M 1 = 3.36 × 10 13 GeV to see the correlation between the predictive Y B and the low energy CP violating measure δ CP in Fig. 14. As seen in Fig. 2, there are six regions for the predictive δ CP versus Re [τ ]. Among them, only two regions are available to reproduce the observed BAU. As seen in Fig. 14 [66,84] meV, which is read from the output in Fig. 3. In conclusion, the precise determination of the Dirac CP phase δ CP and the sum of neutrino masses m i will test our model in the future. We have discussed the leptogenesis in the model of weights (k e = −1, k µ = −3, k τ = −5) for right-handed charged leptons. Actually, we have also examined the case of other alternative weights in Appendix D. For the case of (k e = −1, k µ = −1, k τ = −5), the predictive Y B is marginal to reproduce the observed one. On the other hand, the CP asymmetry parameter ε I of Eq. (40) is much smaller than the above result in the cases of (k e = −1, k µ = −1, k τ = −7) and (k e = −1, k µ = −3, k τ = −7). Thus, the leptogenesis provides a crucial test to select the favorable models of leptons.

Summary
We have presented A 4 modular invariant flavor models of leptons with the CP invariance. The origin of the CP violation is only in the modulus τ . Both CP and modular symmetries are broken spontaneously by the VEV of the modulus τ . We have discussed the phenomenological implication of this model, that is flavor mixing angles and CP violating phases.
We have found allowed region of τ which is consistent with the observed lepton mixing angles and lepton masses for NH at χ 2 ≤ 3. The CP violating Dirac phase δ CP is predicted in [0 The predicted m i is in [60,84] meV. By using the predicted Dirac phase and the Majorana phases, we have obtained the effective mass m ee for the 0νββ decay, which is in [0.003, 3] meV.
We have also studied the case of IH of neutrino masses. There are no parameter sets which reproduce the observed masses and three mixing angles below χ 2 = 4. but we have found parameter sets at χ 2 = 4-5.
Our CP invariant lepton mass matrices have minimum number of parameters, eight, apart from the overall scale by putting weights (k e = −1, k µ = −3, k τ = −5) for right-handed charged leptons. However, this choice for weights is not a unique one. We have examined alternative three choices: In those three models, we have also obtained successful numerical results.
The modulus τ links the Dirac CP phase to the baryon asymmetry. We have studied the leptogenesis in our model with NH of neutrino masses. The sign of the BAU is determined by the signs of both Re[τ ] and g D . In order to obtain the observed positive Y B , (Re[τ ] < 0, g D < 0) or (Re[τ ] > 0, g D > 0) is required. Due to the strong wash-out effect, the yield can be at most the same order of the observed value of the BAU. Then, the lightest right-handed neutrino is in the restricted range M 1 = [1.5, 6.5] × 10 13 GeV. In addition, the predictive Y B also depends on M 2 /M 1 crucially.
We have found the correlation between the predictive Y B and the low energy CP violating measure δ CP . Among six regions of the predictive δ CP versus Re Thus, our scheme of the modulus τ linking the Dirac CP phase to the baryon asymmetry gives rise to the idea for an important test of the phase relevant to leptogenesis.

A Tensor product of A 4 group
We take the generators of A 4 group for the triplet in the symmetric base as follows: 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 [128][129][130], 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. We use the base of the generators S and T in Eq. (43) for the triplet representation. Then, the modular forms of weight 2 (k = 2) transforming as a triplet of A 4 , Y 3 (τ ) = (Y 1 (τ ), Y 2 (τ ), Y 3 (τ )) T , can be written in terms of η(τ ) and its derivative [22]: The overall coefficient in Eq. (47) 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 using the A 4 tensor products of Appendix A in terms of the modular forms with weight 2, Y where c ij and s ij denote cos θ ij and sin θ ij , respectively. The rephasing invariant CP violating measure of leptons [148,149] is defined by the PMNS matrix elements U αi . It is written in terms of the mixing angles and the CP violating phase as: 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 We can calculate δ CP , α 21 and α 31 with these relations by taking account of 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 ) .

D Alternative A 4 modular models
We present three alternative charged lepton mass matrices by putting different weights for (k e , k µ , k τ ), which are consistent with observed mixing angles and masses. Neutrino mass matrix is the same one in section 3.
D.1 Case of (k e = −1, k µ = −1, k τ = −5) for charged leptons The assignment of MSSM fields and modular forms are given in Table 4. Table 4: Representations and weights k for MSSM fields and modular forms of weights 2 and 6.
The A 4 invariant superpotential of the charged leptons, w E , by taking into account the modular weights is obtained as By using g e = γ e /γ e , the charged lepton mass matrix M E is simply written as: D.2 Case of (k e = −1, k µ = −1, k τ = −7) for charged leptons The assignment of MSSM fields and modular forms are given in Table 5. For k = 8, there are 9 modular forms by the tensor products of A 4 as: The A 4 invariant superpotential of the charged leptons, w E , by taking into account the modular weights is obtained as By using g e = γ e /γ e , the charged lepton mass matrix M E is simply written as: D.3 Case of (k e = −1, k µ = −3, k τ = −7) for charged leptons The assignment of MSSM fields and modular forms are given in Table 6. The A 4 invariant superpotential of the charged leptons, w E , by taking into account the modular weights is obtained as By using g e = γ e /γ e , the charged lepton mass matrix M E is simply written as:

D.4 Sample parameters of alternative models
In Table 7, we show parameters and output of our calculations in above three cases of (k e , k µ , k τ ) for NH.
(k e , k µ , k τ )  Table 7: Numerical values of parameters and observables at sample points of NH.

E Formulae of the leptogenesis
We solve the Boltzmann equations for right-handed neutrinos number densities n N I and the lepton asymmetry density n L as: N + 2γ tI + 2γ W I + γ W I + γ (2) where z = M 1 /T . Here we define the yields as Y N I = n N I /s and Y L = n L /s, with the entropy density of the universe s. The superscript "eq" denotes its equilibrium value. We apply the Boltzmann approximation and the yield for a massless particle with one degree of freedom in equilibrium is given by Y eq = 45/(2π 4 g * s ), with g * s = 110.75. The flavor summed CP asymmetry at the decay of the right-handed neutrino N I is given as where f V (x) and f S (x) are the contributions from vertex and self-energy corrections, respectively.
In the case of the standard model (SM) with right-handed neutrinos, they are given as The reaction density for the N I decay is given by where z = M 1 /T , a I = (M I /M 1 ) 2 , and K 1 (x) is the modified Bessel function of the second kind. Note that y ν is the Yukawa coupling matrix of neutrinos in the base where both the mass matrices of charged leptons and right-handed neutrinos are diagonalized. The reaction density for the scattering process A + B → C + D is expressed as where m A and m B are masses of the initial particles andσ(s) denotes the reduced cross section for the process. The expressions of the reduced cross sections for the ∆L = 1 processes induced through top Yukawa intraction, the ∆L = 2 scattering processes and the annihilation processes of right-handed neutrinos are found in Ref. [150]. The reduced cross section for LH u → LH u process which is correctly subtracted N I on-shell contribution is [151] σ x + a I a I where D I = [(x − a I ) 2 + a I c I ]/(x − a I ) with c I = (Γ N I /M 1 ) 2 , in which Γ N I is the total decay rate of right-handed neutrino N I . The explicit form of reduced cross sections for ∆L = 1 processes through the SU (2) L SM gauge interaction are found in Refs. [136,151], σ (2) W I (x) = 3g 2 2 (y ν y ν † ) II 8πx(x − a I ) 2a I x ln x − a I + a H a H + (x 2 + a 2 I ) ln σ W I (x) = 3g 2 2 (y ν y ν † ) II a I 16πx 2 x 2 − 4a I x + 3a 2 I a I + 4(x − a I ) ln x − a I + a H a H − x(4a H − a W )(x − a I ) a H (x − a I + m H ) .
Hereσ (1) W I ,σ W I andσ (3) W I correspond to the reduced cross sections of the processes N I L → H u W , N I W → LH u and N I H u → LW , respectively. We have used a X = m 2 X /M 2 1 where m X with X = L, H u , W, B are thermal masses of lepton doublets, up-type Higgs, SU (2) L gauge bosons and U (1) Y gauge boson, respectively. The reaction densities for the ∆L = 1 processes through U (1) Y gauge interactionσ (i) BI are obtained by replacing a W with a B and 3 2 g 2 2 with 1 4 g 2 Y inσ (i) W I . For the more accurate estimation of the baryon asymmetry, we have taken into account the one-loop RGE evolutions of couplings and the renormalization scale is taken as µ = 2πT .