Search

A model of the axion-less solution to the strong CP problem has been recently constructed based on the $\bf T^2/\mathbb{Z}_3$ orbifold in the six dimensional space-time. We extend, in this paper, the model to include the lepton sector with three right-handed neutrinos. A crucial point in the extended model is that we have only one CP-violating phase and thus the CP phase in the neutrino oscillation can be predicted by the CP phase in the CKM matrix. The CP phase $\delta_{CP}$ in the PMNS matrix is predicted as $\delta_{CP}\simeq 192^\circ - 195^\circ$ which can be tested in the near future experiments. We also discuss the effective mass $m_{\beta\beta}$ responsible for the neutrino-less double beta decay of nucleus, which is predicted as $m_{\beta\beta} \simeq 8-11$\,meV. We consider the leptogenesis and confirm that our model generates the right sign of the baryon asymmetry if the first family right-handed neutrino is the lightest among the heavy right-handed neutrinos., Comment: Abstract and Section 4 are changed

Phenomenological aspects of non-invertible symmetries, in particular the flavor structure of quarks and leptons, are studied. We start with a $\mathbb{Z}_M$ discrete symmetry and gauge $\mathbb{Z}_2$ so as to obtain a non-invertible symmetry. We study which Yukawa textures can be derived from the non-invertible symmetries. Various textures can be realized and some of them cannot be realized by a conventional symmetry. For example, the nearest neighbor interaction texture as well as other interesting textures of quarks and leptons are obtained., Comment: 14 pages

We have studied the leptonic EDM and the LFV decays relating with the recent data of anomalous magnetic moment of muon, $(g-2)_{\mu}$ in the leptonic dipole operator. We have adopted the successful $\Gamma_2$ modular invariant model by Meloni-Parriciatu as the flavor symmetry of leptons. Suppose the anomaly of $(g-2)_{\mu}$, $\Delta a_{\mu}$ to be evidence of New Physics (NP), we have related it with the anomalous magnetic moment of the electron $\Delta a_e$, the electron EDM $d_e$ and the $\mu\to e \gamma$ decay. We found that the NP contributions to $\Delta a_{e(\mu)}$ are proportional to the lepton masses squared likewise the naive scaling $\Delta a_\ell \propto m^2_\ell$. The experimental constraint of $|d_e|$ is much tight compared with the one from the branching ratio $\mathcal{B} (\mu \to e \gamma)$ in our framework. Supposing the phase of our model parameter $\delta_{\alpha}$ for the electron to be of order one, we have estimated the upper-bound of $\mathcal{B}(\mu \to e \gamma)$, which is at most $10^{-21}-10^{-20}$. If some model parameters are real, leptonic EDMs vanish since the CP phase of the modular form due to modulus $\tau$ does not contribute to the EDM. However, we can obtain $\mathcal{B} (\mu \to e \gamma)\simeq 10^{-13}$ with non-vanishing $d_e$ in a specific case. The imaginary part of a parameter can lead to $d_e$ in the next-to-leading contribution. The predicted electron EDM is below $10^{-32}$e\,cm, while $\mathcal{B} (\mu \to e \gamma)$ is close to the experimental upper-bound. The branching ratios of $\tau\to e\gamma$ and $\tau\to \mu\gamma$ are also discussed., Comment: 37 pages, 12 figures. Version accepted for publication in EPJC

We study the interplay of New Physics (NP) among the lepton magnetic moment, the lepton flavor violation (LFV) and the electron electric dipole moment (EDM) in light of recent data of the muon $(g-2)_\mu$. The NP is discussed in the leptonic dipole operator with the $U(2)$ flavor symmetry of the charged leptons, where possible CP violating phases of the three family space are taken into account. It is remarked that the third-family contributes significantly to the LFV decay, $\mu \to e\gamma$, and the electron EDM. The experimental upper-bound on $\mu \to e\gamma$ decay gives a severe constraint on the parameters of the flavor model. The predicted electron EDM is rather large due to the CP violating phases in the three family space. In addition, we also study $(g-2)_{e,\tau}$ of the electron and tauon, and EDMs of the muon and tauon as well as the $\tau \to e \gamma$ and $\tau \to \mu \gamma$ decays. The $\tau_R \to \mu_L \gamma$ decay is predicted to be close to the experimental upper-bound., Comment: 25 pages, 14 figures. Published version in EPJ-C; minor comments added, typos corrected

We review the modular flavor symmetric models of quarks and leptons focusing on our works. We present some flavor models of quarks and leptons by using finite modular groups and discuss the phenomenological implications. The modular flavor symmetry gives interesting phenomena at the fixed point of modulus. As a representative, we show the successful texture structure at the fixed point $\tau = \omega$. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators., Comment: 40 pages, 5 figures, the contribution to a special book dedicated to the memory of Professor Harald Fritzsch

We classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries $\Gamma_N$. Since all the couplings in supersymmetric theories of finite modular symmetries $\Gamma_N$ are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight $1/M$, we find $\mathbb{Z}_{2M}$ symmetries, including the generalized baryon and lepton parities, $R$-parity, $\mathbb{Z}_3$ baryon triality and $\mathbb{Z}_6$ proton hexality. Such $\mathbb{Z}_{2M}$ symmetries are enlarged to $\mathbb{Z}_{2M} \rtimes \mathbb{Z}_2^{\text{CP}}$ symmetries together with the CP transformation., Comment: 23 pagess, v2: published version

We study systematically derivation of the specific texture zeros, that is the nearest neighbor interaction (NNI) form of the quark mass matrices at the fixed point $\tau=\omega$ in modular flavor symmetric models. We present models that the NNI forms of the quark mass matrices are simply realized at the fixed point $\tau=\omega$ in the $A_4$ modular flavor symmetry by taking account multi-Higgs fields. Such texture zero structure originates from the $ST$ charge of the residual symmetry $Z_3$ of $SL(2,Z)$. The NNI form can be realized at the fixed point $\tau = \omega$ in $A_4$ and $S_4$ modular flavor models with two pairs of Higgs fields when we assign properly modular weights to Yukawa couplings and $A_4$ and $S_4$ representations to three generations of quarks. We need four pairs of Higgs fields to realize the NNI form in $A_5$ modular flavor models., Comment: 37 pages

We study the CP violation and the CP phase of quark mass matrices in modular flavor symmetric models. The CP symmetry remains at $\tau = e^{2 \pi i/3}$ by a combination of the $T$-symmetry of the modular symmetry. However, the $T$-symmetry breaking may lead to the CP violation at the fixed point $\tau = e^{2 \pi i/3}$. We study such a possibility in magnetized orbifold models as examples of modular flavor symmetric models. These models, in general, have more than one candidates for Higgs modes, while generic string compactifications also lead to several Higgs modes. These Higgs modes have different behaviors under the $T$-transformation. The light Higgs mode can be a linear combination of those modes so as to lead to realistic quark mass matrices. The CP phase of mass matrix does not appear in a certain case, which is determined by the $T$-transformation behavior. Deviation from it is important to realize the physical CP phase. We discuss an example leading to non-vanishing CP phase at the fixed point $\tau = e^{2 \pi i/3}$., Comment: 21 pages

We update and improve past efforts to predict the leptonic Dirac CP-violating phase with models that predict perturbatively modified tribimaximal or bimaximal mixing. Simple perturbations are applied to both mixing patterns in the form of rotations between two sectors. By translating these perturbed mixing matrices to the standard parameterization for the neutrino mixing matrix we derive relations between the Dirac CP-phase and the oscillation angles. We use these relations together with current experimental results to constrain the allowed range for the CP-phase and determine its probability density. Furthermore, we elaborate on the prospects for future experiments probing on the perturbations considered in this work. We present a model with $A_4$ modular symmetry that is consistent with one of the described perturbed scenarios and successfully predicts current oscillation parameter data., Comment: 24 pages, 4 figures

We study the lepton flavor violation (LFV), the leptonic magnetic moments $(g-2)_{\mu,\,e} $ and the electric dipole moment (EDM) of the electron in the Standard-Model Effective Field Theory with the $\Gamma_N$ modular flavor symmetry. We employ the stringy Ansatz on coupling structure that 4-point couplings of matter fields are written by a product of 3-point couplings of matter fields. We take the level 3 finite modular group, $\Gamma_3$ for the flavor symmetry, and discuss the dipole operators at nearby fixed point $\tau=i$, where observed lepton masses and mixing angles are well reproduced. Suppose the anomaly of the anomalous magnetic moment of the muon to be evidence of the new physics (NP), we have related it with $(g-2)_{e}$, LFV decays, and the electron EDM. It is found that the NP contribution to $(g-2)_{e}$ is proportional to the lepton masses squared likewise the naive scaling. We also discuss the correlations among the LFV processes $\mu\to e\gamma$, $\tau\to \mu\gamma$ and $\tau\to e\gamma$, which are testable in the future. The electron EDM requires the tiny imaginary part of the relevant Wilson coefficient in the basis of real positive charged lepton masses, which is related to the $\mu\to e\gamma$ transition in our framework., Comment: 33 pages, references added

The International Linear Collider (ILC) is on the table now as a new global energy-frontier accelerator laboratory taking data in the 2030s. The ILC addresses key questions for our current understanding of particle physics. It is based on a proven accelerator technology. Its experiments will challenge the Standard Model of particle physics and will provide a new window to look beyond it. This document brings the story of the ILC up to date, emphasizing its strong physics motivation, its readiness for construction, and the opportunity it presents to the US and the global particle physics community., Comment: 356 pages, Large pdf file (40 MB) submitted to Snowmass 2021; v2 references to Snowmass contributions added, additional authors; v3 references added, some updates, additional authors

We study a scenario to derive four-dimensional modular flavor symmetric models from higher dimensional theory by assuming the compactification consistent with the modular symmetry. In our scenario, wavefunctions in extra dimensional compact space are modular forms. That leads to constraints on combinations between modular weights and $\Gamma_N$ ($\Gamma_N')$ representations of matter fields. We also present illustrating examples., Comment: 21 pages, v2: published version

We study the modular symmetric standard-model effective field theory. We employ the stringy Ansatz on coupling structure that 4-point couplings $y^{(4)}$ of matter fields are written by a product of 3-point couplings $y^{(3)}$ of matter fields, i.e., $y^{(4)} = y^{(3)}y^{(3)}$. In this framework, we discuss the flavor structure of bilinear fermion operators and 4-fermion operators, where the holomorphic and anti-holomorphic modular forms appear. From the Ansatz, the $A_4$ modular-invariant semileptonic four-fermion operator $[\,\bar E_R \Gamma E_R][\,\bar D_R \Gamma D_R\,]$ does not lead to the flavor changing (FC) processes since this operator would be constructed in terms of gauge couplings $g$ as $y^{(3)} \sim g$. The chirality flipped bilinear operator $[\,\bar D_R \Gamma D_L\,]$ also does not lead FC if the mediated mode corresponds to the Higgs boson $H_d$. In this case, the flavor structure of this operator is the exactly same as the mass matrix. On the other hand, if the flavor structure of the operator is not the exactly same as the mass matrix, the situation would change drastically. Then, we obtain the non-trivial relations of FC transitions at nearby fixed points $\tau=i,\,\omega\,, i\infty$, which are testable in the future. As an application, we discuss the relations of the lepton flavor violation processes $ \mu\to e\gamma,\, \tau\to \mu\gamma$ and $\tau\to e\gamma$ at nearby $\tau_e=i$, where the successful lepton mass matrix was obtained. We also study the flavor changing 4-quark operators in the $A_4$ modular symmetry of quarks. As a result, the minimal flavor violation could be realized by taking relevant specific parameter sets of order one., Comment: 48 pages, 14 tables, revised discussion in Sec.4, several tables modified

Minimalistic Dirac radiative neutrino mass model based on modular symmetry is proposed. We predict maximum number of observables possible including neutrino mass splittings, neutrino mass scale, lepton mixing angles, and Dirac phases in the leptonic sector with minimum number of input parameters possible. Model is capable of accommodating multicomponent dark matter, thanks to the $\emph{R}-$parity and accidental scotogenic $\mathbb{Z}_2$ discrete symmetry. Furthermore, even-though neutrinos are Dirac in our model, matter-antimatter asymmetry of the Universe is achieved via neutrinogenesis mechanism. Phenomenology of the dark sector including various dark matter candidates is briefly discussed., Comment: 24 pages, 6 figures, 4 tables

The electric dipole moment (EDM) of electron is studied in the supersymmetric $\rm A_4$ modular invariant theory of flavors with CP invariance. The CP symmetry of the lepton sector is broken by fixing the modulus $\tau$. Lepton mass matrices are completely consistent with observed lepton masses and mixing angles in our model. In this framework, a fixed $\tau$ also causes the CP violation in the soft SUSY breaking terms. The elecrton EDM arises from the CP non-conserved soft SUSY breaking terms. The experimental upper bound of the electron EDM excludes the SUSY mass scale below $2-6$ TeV for five cases of the lepton mass matrices. In order to see the effect of CP phase of the modulus $\tau$, we examine the correlation between the electron EDM and the decay rate of the $\mu \rightarrow e \gamma$ decay, which is also predicted by the soft SUSY breaking terms. The correlations are clearly predicted in contrast to models of the conventional flavor symmetry. The SUSY mass scale will be constrained by the future sensitivity of the electron EDM, $|d_e/e| \simeq 10^{-30}$. Indeed, it could probe the SUSY mass range of $10-20$ TeV in our model. Thus, the electron EDM provides a severe test of the CP violation via the modulus $\tau$ in the supersymmetric modular invariant theory of flavors., Comment: 33 pages, 8 figures, 4 tables; v3: corrections for equations, discussion added, figures and tables changed, conclusions unchanged. arXiv admin note: text overlap with arXiv:2105.14292

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., Comment: 30 pages, 14 figures, 7 tables, minor corrections

We study the soft supersymmetry (SUSY) breaking terms due to the modulus F-term in the modular flavor models of leptons. It is found that the soft SUSY breaking terms are constrained by the modular forms, and specific patterns are derived. Those phenomenological implications are discussed in such as the lepton flavor violation, $\mu \rightarrow e + \gamma$ decay. In order to examine numerically, two modular flavor $A_4$ models are taken. The SUSY breaking scale is significantly constrained by inputting the observed upper bound of the $\mu \rightarrow e + \gamma$ decay. The SUSY mass scale is larger than around $8$ TeV and $5$ TeV for the two $A_4$ models, respectively. Therefore, the current experimental upper bound for the $\mu \to e + \gamma$ decay corresponds to the new physics of the SUSY particle at the $5$ -- $10$ TeV scale in the modular flavor models. The predicted branching ratio depends on a modulus $\tau$ significantly. It decreases of one order at the large ${\rm Im}\,\tau$. We also calculate the branching ratios of tauon decays to $e + \gamma$ and $\mu + \gamma$. Predicted ones are at most ${\cal O}(10^{-15})$, which are much below the current experimental bounds., Comment: 16 pages, 3 figures, accepted version

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., Comment: 24 pages, 12 figures, 3 tables. arXiv admin note: text overlap with arXiv:2009.14242

In the modular invariant flavor model of $\rm A_4$, we study the hierarchical structure of lepton/quark flavors at nearby fixed points of $\tau=i$ and $\tau=\omega$ of the modulus, which are in the fundamental domain of $\rm PSL(2,\mathbb{Z})$. These fixed points correspond to the residual symmetries $\mathbb{Z}_2^{S}=\{I, S \}$ and $\mathbb{Z}_3^{S}=\{ I, ST, (ST)^2 \}$ of $\rm A_4$, where $S$ and $T$ are generators of the $A_4$ group. The infinite $\tau= i \infty$ also preserves the residual symmetry of the subgroup $\mathbb{Z}^T_3=\{ I,T,T^2 \}$ of $\rm A_4$. We study typical two-type mass matrices for charged leptons and quarks in terms of modular forms of weights $2$, $4$ and $6$ while the neutrino mass matrix with the modular forms of weight $4$ through the Weinberg operator. Linear modular forms are obtained approximately by performing Taylor expansion of modular forms around fixed points. By using them, the flavor structure of the lepton and quark mass matrices are examined at nearby fixed points. The hierarchical structure of these mass matrices is clearly shown in the diagonal base of $S$, $T$ and $ST$. The observed PMNS and CKM mixing matrices can be reproduced at nearby fixed points in some cases of mass matrices. By scanning model parameters numerically at nearby fixed points, our discussion are confirmed for both the normal hierarchy and inverted one of neutrino masses. Predictions are given for the sum of neutrino masses and the CP violating Dirac phase of leptons at each nearby fixed point., Comment: 53 pages, 10 figures, 6 tables; version accepted for publication in Physical Review D

We study quark and lepton mass matrices with the common modulus $\tau$ in the $A_4$ modular symmetry. The viable quark mass matrices are composed of modular forms of weights $2$, $4$ and $6$. It is remarked that the modulus $\tau$ is close to $i$, which is a fixed point in the fundamental region of SL$(2,Z)$, and the CP symmetry is not violated. Indeed, the observed CP violation is reproduced at $\tau$ which is deviated a little bit from $\tau=i$. The charged lepton mass matrix is also given by using modular forms of weights $2$, $4$ and $6$, where five cases have been examined. The neutrino mass matrix is generated in terms of the modular forms of weight $4$ through the Weinberg operator. Lepton mass matrices are also consistent with the observed mixing angles at $\tau$ close to $i$ for NH of neutrino masses. Allowed regions of $\tau$ of quarks and leptons overlap each other for all cases of the charged lepton mass matrix. However, the sum of neutrino masses is crucial to test the common $\tau$ for quarks and leptons. The minimal sum of neutrino masses $\sum m_i$ is $140$meV at the common $\tau$. The inverted hierarchy of neutrino masses is unfavorable in our framework. It is emphasized that our result suggests the residual symmetry $\mathbb{Z}_2^{S}=\{ I, S \}$ in the quark and lepton mass matrices., Comment: 24 pages, 7 tables, 18 figures. arXiv admin note: substantial text overlap with arXiv:1905.13421

We study the spontaneous $CP$ violation through the stabilization of the modulus $\tau$ in modular invariant flavor models. The $CP$-invaraiant potentential has the minimum only at ${\rm Re}[\tau] = 0$ or 1/2. From this prediction, we study $CP$ violation in modular invariant flavor models. The physical $CP$ phase is vanishing. The important point for the $CP$ conservation is the $T$ transformation in the modular symmetry. One needs the violation of $T$ symmetry to realize the spontaneous $CP$ violation., Comment: 19 pages, 2 figures

We study the modulus stabilization in an $A_4$ model whose $A_4$ flavor symmetry is originated from the $S_4$ modular symmetry. We can stabilize the modulus so that the $A_4$ invariant superpotential leads to the realistic lepton masses and mixing angles. We also discuss the phenomenological aspect of the present model as a consequence of the modulus stabilization., Comment: 23 pages, 22 figures, 3 tables. A revised version following the erratum to be published in PRD. arXiv admin note: text overlap with arXiv:1907.09141

We study a flavor model with $A_4$ symmetry which originates from $S_4$ modular group. In $S_4$ symmetry, $Z_2$ subgroup can be anomalous, and then $S_4$ can be violated to $A_4$. Starting with a $S_4$ symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only $A_4$ symmetry when $Z_2$ in $S_4$ is anomalous. We obtain modular forms of two singlets and a triplet representations of $A_4$ by decomposing $S_4$ modular forms into $A_4$ representations. We propose a new $A_4$ flavor model of leptons by using those $A_4$ modular forms. We succeed in constructing a viable neutrino mass matrix through the Weinberg operator for both normal hierarchy (NH) and inverted hierarchy (IH) of neutrino masses. Our predictions of the CP violating Dirac phase $\delta_{CP}$ and the mixing $\sin^2\theta_{23}$ depend on the sum of neutrino masses for NH., Comment: 17 pages, 6 figures, 2 tables. The numerical discussions are revised. A published version in JHEP

We present a flavor model with the $S_3$ modular invariance in the framework of SU(5) GUT. The $S_3$ modular forms of weights $2$ and $4$ give the quark and lepton mass matrices with a common complex parameter, the modulus $\tau$. The GUT relation of down-type quarks and charged leptons is imposed by the VEV of adjoint 24-dimensional Higgs multiplet in addition to the VEVs of $5$ and $\bar 5$ Higgs multiples of SU(5). The observed CKM and PMNS mixing parameters as well as the mass eigenvalues are reproduced properly. We discuss the leptonic CP phase and the effective mass of the neutrinoless double beta decay with the sum of neutrino masses., Comment: 16 pages, 5 figures and 1 table. Numerical data have been revised

We study quark and lepton mass matrices in the $A_4$ modular symmetry towards the unification of the quark and lepton flavors. We adopt modular forms of weights $2$ and $6$ for quarks and charged leptons, while we use modular forms of weight $4$ for the neutrino mass matrix which is generated by the Weinberg operator. We obtain the successful quark mass matrices, in which the down-type quark mass matrix is constructed by modular forms of weight $2$, but the up-type quark mass matrix is constructed by modular forms of weight $6$. Two regions of $\tau$ are consistent with observed CKM matrix elements. The one is close to $\tau=i$ and the other is in the larger ${\rm Im }[\tau]$. On the other hand, lepton mass matrices work well only at nearby $\tau=i$, which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common $\tau$ region for quarks and leptons, the predicted sum of neutrino masses is $87$--$120$meV taking account of its cosmological bound. Since both the Dirac CP phase $\delta_{CP}^\ell$ and $\sin^2\theta_{23}$ are correlated with the sum of neutrino masses, improving its cosmological bound provides crucial tests for our scheme as well as the precise measurement of $\sin^2\theta_{23}$ and $\delta_{CP}^\ell$. The effective neutrino mass of the $0\nu\beta\beta$ decay is $\langle m_{ee}\rangle=15$--$31$\,meV. It is remarked that the modulus $\tau$ is fixed at nearby $\tau=i$ in the fundamental domain of SL$(2,Z)$, which suggests the residual symmetry $Z_2$ in the quark and lepton mass matrices. The inverted hierarchy of neutrino masses is excluded by the cosmological bound of the sum of neutrino masses., Comment: 19 pages, 6 tables, 9 figures; the quark and lepton sectors have been significantly improved, and final results have been also changed

We discuss the relation between the CP violation of the quark mixing and that of the lepton mixing by investigating a CP violating observable, the Jarlskog invariant, as well as the CP violating Dirac phase. The down-type quark mass matrix with three zeros is given in terms of the minimal number of parameters, while the up-type quark mass matrix is diagonal. These quark mass matrices leading to the successful CKM mixing angles and CP violation are embedded in both the Pati--Salam and SU(5) models. The leptonic Jarlskog invariant $J_{CP}^l$ (as well as CP violating Dirac phase) is examined for two cases: the neutrino mass matrix is diagonal or non-diagonal, where no additional CP violating phase is introduced apart from the Majorana phases. In the case of the diagonal neutrino mass matrix, the favorable sign of the leptonic CP violation is obtained, however, the magnitude of $J_{CP}^l$ is at most ${\cal O}(10^{-4})$, which is too small compared with the expected value from the observation $-0.02$. In the case of the non-diagonal neutrino mass matrix where the tri-bimaximal mixing pattern is taken, we obtain the successful $J_{CP}^l$ up to its sign., Comment: 25 pages, 4 tables, some references are added(v.2), minor revisions(v.3)

We discuss the quark mass matrices in the $A_4$ modular symmetry, where the $A_4$ triplet of Higgs is introduced for each up-quark and down-quark sectors, respectively. The model has six real parameters and two complex parameters in addition to the modulus $\tau$. By inputting six quark masses and three CKM mixing angles, we can predict the CP violation phase $\delta$ and the Jarlskog invariant $J_{CP}$. The predicted ranges of $\delta$ and $J_{CP}$ are consistent with the observed values. The absolute value of $V_{ub}$ is smaller than $0.0043$, while $V_{cb}$ is lager than $0.0436$. In conclusion, our quark mass matrices with the $A_4$ modular symmetry can reproduce the CKM mixing matrix completely with observed quark masses., Comment: 15 pages, 1 table, 3 figures, accepted version for publication in Physics Letters B