arXiv subjects

Morimitsu Tanimoto

Publications and source records attributed to Morimitsu Tanimoto.

At least 37 records · Page 2Linked to original sources

arXiv2021

Electron EDM arising from modulus $τ$ in the supersymmetric modular invariant flavor models

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 $τ$. 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 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 $τ$, we examine the correlation between the electron EDM and the decay rate of the $μ\rightarrow e γ$ 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 $τ$ in the supersymmetric modular invariant theory of flavors.

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arXiv2021

Towards unification of quark and lepton flavors in $A_4$ modular invariance

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 $τ$ are consistent with observed CKM matrix elements. The one is close to $τ=i$ and the other is in the larger ${\rm Im }[τ]$. On the other hand, lepton mass matrices work well only 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 $δ_{CP}^\ell$ and $\sin^2θ_{23}$ 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 $\sin^2θ_{23}$ and $δ_{CP}^\ell$. The effective neutrino mass of the $0νββ$ decay is $\langle m_{ee}\rangle=15$--$31$\,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 $Z_2$ 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.

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arXiv2021

Modulus $τ$ 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 $τ$. The value of the modulus $τ$ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses at $3σ$ confidence level. The predictive Dirac CP phase $δ_{CP}$ is in the ranges $[0^\circ,50^\circ]$, $[170^\circ,175^\circ]$ and $[280^\circ,360^\circ]$ for ${\rm Re}\, [τ]<0$, and $[0^\circ,80^\circ]$, $[185^\circ,190^\circ]$ and $[310^\circ,360^\circ]$ for ${\rm Re}\,[τ]>0$ at $3σ$ confidence level. The sum of three neutrino masses is predicted in $[60, 84]$ meV, and the effective mass for the $0νββ$ decay is in [0.003, 3] meV. On the other hand, there is no allowed region of the modulus $τ$ for the inverted hierarchy of neutrino masses at $3σ$ confidence level. The modulus $τ$ 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 $δ_{CP}$. Only two predictive $δ_{CP}$ ranges, $[5^\circ,40^\circ]$ (${\rm Re}\,[τ]>0$) and $[320^\circ,355^\circ]$ (${\rm Re}\,[τ]<0$) are consistent with the BAU.

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arXiv2021

Soft supersymmetry breaking terms and lepton flavor violations in modular flavor models

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, $μ\rightarrow e + γ$ 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 $μ\rightarrow e + γ$ 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 $μ\to e + γ$ 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 $τ$ significantly. It decreases of one order at the large ${\rm Im}\,τ$. We also calculate the branching ratios of tauon decays to $e + γ$ and $μ+ γ$. Predicted ones are at most ${\cal O}(10^{-15})$, which are much below the current experimental bounds.

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arXiv2021

Spontaneous CP violation by modulus $τ$ 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 $τ$. The source of the CP violation is a non-trivial value of ${\rm Re} [τ]$ 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 $[98^\circ,\,110^\circ]$ and $[250^\circ,\,262^\circ]$ for NH at $3\,σ$ confidence level. On the other hand, $δ_{CP}$ is in $[95^\circ,100^\circ]$ and $[260^\circ,\,265^\circ]$ for IH at $5\,σ$ 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 $0νββ$ decay is predicted in $[12.5,\,20.5]$ meV and $[54,\,67]$ meV for NH and IH, respectively.

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arXiv2020

Modular invariant flavor model of $\rm A_4$ and hierarchical structures at nearby fixed points

In the modular invariant flavor model of $\rm A_4$, we study the hierarchical structure of lepton/quark flavors at nearby fixed points of $τ=i$ and $τ=ω$ 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 $τ= i \infty$ 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.

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arXiv2020

Quark and lepton flavors with common modulus $τ$ in $A_4$ modular symmetry

We study quark and lepton mass matrices with the common modulus $τ$ 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 $τ$ 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 $τ$ which is deviated a little bit from $τ=i$. 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 $τ$ close to $i$ for NH of neutrino masses. Allowed regions of $τ$ 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 $τ$ for quarks and leptons. The minimal sum of neutrino masses $\sum m_i$ is $140$meV at the common $τ$. 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.

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arXiv2020

$CP$ violation in modular invariant flavor models

We study the spontaneous $CP$ violation through the stabilization of the modulus $τ$ in modular invariant flavor models. The $CP$-invaraiant potentential has the minimum only at ${\rm Re}[τ] = 0$ 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.

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arXiv2020

Modular $S_3$ invariant flavor model in SU(5) GUT

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 $τ$. 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.

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arXiv2020

$A_4$ lepton flavor model and modulus stabilization from $S_4$ modular symmetry

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.

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arXiv2020

New $A_4$ lepton flavor model from $S_4$ modular symmetry

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 $\sin^2θ_{23}$ depend on the sum of neutrino masses for NH.

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arXiv2019

Finite modular subgroups for fermion mass matrices and baryon/lepton number violation

We study a flavor model that the quark sector has the $S_3$ modular symmetry,while the lepton sector has the $A_4$ modular symmetry. Our model leads to characteristic quark mass matrices which are consistent with experimental data of quark masses, mixing angles and the CP violating phase. The lepton sector is also consistent with the experimental data of neutrino oscillations. We also study baryon and lepton number violations in our flavor model.

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arXiv2019

Sign of CP Violating Phase in Quarks and Leptons

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.

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arXiv2019

CP violation of quarks in $A_4$ modular invariance

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 $τ$. By inputting six quark masses and three CKM mixing angles, we can predict the CP violation phase $δ$ and the Jarlskog invariant $J_{CP}$. The predicted ranges of $δ$ 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.

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arXiv2018

Modular $A_4$ invariance and neutrino mixing

We study the phenomenological implications of the modular symmetry $Γ(3) \simeq A_4$ 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 $\sin^2θ_{23}$ is restricted to be larger than $0.54$ and $δ_{CP}=\pm (50^{\circ}\mbox{--}180^{\circ})$. Since the correlation of $\sin^2θ_{23}$ 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.

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arXiv2018

Towards the minimal seesaw model for the prediction of neutrino CP violation

We discuss the minimal seesaw model for the Dirac CP violating phase of the lepton mixing matrix. We introduce two right-handed Majorana neutrinos and obtain several textures of the tri-maximal lepton mixing matrices. Moreover, we discuss the observed baryon asymmetry of the universe through the leptogenesis mechanism. As the result, we obtain the specific model which predicts the negative sign of maximal Dirac CP violating phase and normal hierarchy of neutrino masses.

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arXiv2018

Revisiting $A_4$ model for leptons in light of NuFIT 3.2

We revisit the $A_4$ model for leptons in light of new result of NuFIT 3.2. We introduce a new flavon $η$ transforming as $A_4$ singlet $1'$ or $1"$ which couples to both charged leptons and neutrinos in next-leading order operators. The model consists of the five parameters: the lightest neutrino mass $m_1$, the vacuum expectation value of $η$ and three CP violating phases after inputting the experimental values of $Δm_{\rm atm}^2$ and $Δm_{\rm sol}^2$. The model with the $1"$ singlet flavon gives the prediction of $\sin^2 θ_{12}$ around the best fit of NuFIT 3.2 while keeping near the maximal mixing of $θ_{23}$. Inputting the experimental mixing angles with the $1\,σ$ error-bar, the Dirac CP violating phase is clearly predicted to be $|δ_\text{CP}|=50^\circ- 120^\circ$, which will be tested by the precise observed value in the future. In order to get the best fit value $\sin^2θ_{23}=0.538$, the sum of three neutrino masses is predicted to be larger than $90\,$meV. The cosmological observation for the sum of neutrino masses will also provide a crucial test of our predictions. It is remarked that the model is consistent with the experimental data only for the normal hierarchy of neutrino masses.

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arXiv2017

Towards the minimal seesaw model via CP violation of neutrinos

We study the minimal seesaw model, where two right-handed Majorana neutrinos are introduced, focusing on the CP violating phase. In addition, we take the trimaximal mixing pattern for the neutrino flavor where the charged lepton mass matrix is diagonal. Thanks to this symmetric framework, the $3\times 2$ Dirac neutrino mass matrix is given in terms of a few parameters. Numerical studies reveal that the observation of the CP violating phase can determine the flavor structure of the Dirac neutrino mass matrix in the minimal seesaw model. In particular, new minimal Dirac neutrino mass matrices are proposed in the case of $\rm TM_1$, which is derived by the additional 2-3 family mixing to the tri-bimaximal mixing basis in the normal hierarchy of neutrino masses. Our analyses include the Littlest seesaw model by King {\it et al.}, which is one of the specific one in our results. Furthermore, it is remarked that our $3\times 2$ Dirac neutrino mass matrix is reproduced by introducing gauge singlet flavons with the specific alignments of the VEV's. These alignments suggest the residual symmetry of $S_4$ group.

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