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We study quark and lepton mass matrices in the A4 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. The viable region of τ is close to τ=i. Lepton mass matrices also work well at nearby τ=i, which overlaps with the one of the quark sector, for the normal hierarchy of neutrino masses. In the common τ 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 δCPℓ and sin2θ23 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 sin2θ23 and δCPℓ. The effective neutrino mass of the 0νββ decay is ⟨mee⟩=15–31 meV. It is remarked that the modulus τ is fixed at nearby τ=i in the fundamental domain of SL(2, Z), which suggests the residual symmetry Z2 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.

A bstract 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 τ . The source of the CP violation is a non-trivial value of Re[ τ ] while other parameters of the model are real. The allowed region of τ is in very narrow one close to the fixed point τ = i for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase δ CP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δ CP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑ m i is in [82, 102] meV for NH and ∑ m i = [134, 180] meV for IH. The effective mass 〈 m ee 〉 for the 0 νββ decay is predicted in [12 . 5, 20 . 5] meV and [54 , 67] meV for NH and IH, respectively.

Abstract 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 τ. The source of the CP violation is a non-trivial value of Re[τ] while other parameters of the model are real. The allowed region of τ is in very narrow one close to the fixed point τ = i for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase δ CP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δ CP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑m i is in [82, 102] meV for NH and ∑m i = [134, 180] meV for IH. The effective mass 〈m ee 〉 for the 0νββ decay is predicted in [12.5, 20.5] meV and [54, 67] meV for NH and IH, respectively.

A bstract The electric dipole moment (EDM) of electron is studied in the supersymmetric A4 modular invariant theory of flavors with CP invariance. The CP symmetry of the lepton sector is broken by fixing the modulus τ . Lepton mass matrices are completely consistent with observed lepton masses and mixing angles in our model. In this framework, a fixed τ also causes the CP violation in the soft SUSY breaking terms. The electron 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 4–6 TeV depending on five cases of the lepton mass matrices. In order to see the effect of CP phase of the modulus τ , we examine the correlation between the electron EDM and the decay rate of the μ → eγ 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 branching ratio is approximately proportional to the square of | d e / e |. The SUSY mass scale will be constrained by the future sensitivity of the electron EDM, | d e / e | ≃ 10 − 30 cm. 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 τ in the supersymmetric modular invariant theory of flavors.

A bstract Phenomenological aspects of non-invertible symmetries, in particular the flavor structure of quarks and leptons, are studied. We start with a ℤ M discrete symmetry and gauge 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.

Abstract 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 δ CP and the mixing sin2 θ 23 depend on the sum of neutrino masses for NH.

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 ) μ . 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, μ → e γ , and the electron EDM. The experimental upper-bound on μ → e γ 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 , τ of the electron and tauon, and EDMs of the muon and tauon as well as the τ → e γ and τ → μ γ decays. The τ R → μ L γ decay is predicted to be close to the experimental upper-bound.

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 τ = ω in modular flavor symmetric models. We present models that the NNI forms of the quark mass matrices are simply realized at the fixed point τ = ω 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 τ = ω 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.

A bstract We study the coupling selection rules associated with non-group symmetries, i.e., ℤ 2 gauging of ℤ M symmetries. We clarify which Yukawa textures can be derived by our selection rules for M = 3 , 4, and 5, and obtain various textures including the nearest neighbor interaction type and its extension. Some of them cannot be realized by a conventional group-like symmetry. They lead to interesting phenomenology such as a solution to the strong CP problem without axion.

A bstract We study the phenomenological implications of the modular symmetry Γ(3) ≃ A 4 of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus τ , which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with A 4 symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses, the seesaw model is available by taking account of recent experimental data of neutrino oscillations and the cosmological bound of sum of neutrino masses. The predicted sin 2 θ 23 is restricted to be larger than 0 . 54 and δ CP = ±(50°-180°). Since the correlation of sin 2 θ 23 and δ CP is sharp, the prediction is testable in the future. It is remarkable that the effective mass m ee of the neutrinoless double beta decay is around 22 meV while the sum of neutrino masses is predicted to be 145 meV. On the other hand, for the inverted hierarchy of neutrino masses, only the Dirac neutrino model is consistent with the experimental data.