We explore alternative descriptions of the charged lepton sector in modular invariant models of lepton masses and mixing angles. In addition to the modulus, the symmetry breaking sector of our models includes ordinary flavons. Neutrino mass terms depend only on the modulus and are tailored to minimize the number of free parameters. The charged lepton Yukawa couplings rely upon the flavons alone. We build modular invariant models at levels 4 and 5, where neutrino masses are described both in terms of the Weinberg operator or through a type I seesaw mechanism. At level 4, our models reproduce the hierarchy among electron, muon and tau masses by letting the weights play the role of Froggatt-Nielsen charges. At level 5, our setup allows the treatment of left and right handed charged leptons on the same footing. We have optimized the free parameters of our models in order to match the experimental data, obtaining a good degree of compatibility and predictions for the absolute neutrino masses and the $CP$ violating phases. At a more fundamental level, the whole lepton sector could be correctly described by the simultaneous presence of several moduli. Our examples are meant to make a first step in this direction.
We study a double covering of modular $A_4$ flavor symmetry in which we construct lepton models in cases of canonical seesaw and radiative seesaw models. Through $χ$ square numerical analysis, we show some predictions for the cases, depending on normal and inverted hierarchies.
We perform a comprehensive analysis of the homogeneous finite modular group $Γ'_4\equiv S'_4$ which is the double covering of $S_4$ group. The weight 1 modular forms of level 4 are constructed in terms of Dedekind eta function, and they transform as a triplet $\mathbf{\hat{3}'}$ of $S'_4$. The integral weight modular forms until weight 6 are built from the tensor products of weight 1 modular forms. We perform a systematical classification of $S'_4$ modular models for lepton masses and mixing with/without generalized CP, where the left-handed leptons are assigned to triplet of $S'_4$ and right-…
We propose a quark-lepton model via leptoquarks and modular $A_4$ symmetry. Since the neutrino mass is induced at one-loop level mediated by down quarks as well as leptoquarks, we have to explain lepton and quark masses and mixings with a single modulus $τ$. Here, we find predictions for lepton and quark sectors with unified modulus $τ$, and show several constraints originating from leptoquarks.
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.
We study models of lepton masses and mixing based on broken modular invariance. We consider invariance under the finite modular group $Γ_4 \simeq S_4$ and focus on the minimal scenario where the expectation value of the modulus is the only source of symmetry breaking, such that no flavons need to be introduced. After constructing a basis for the lowest weight modular forms, we build two minimal models, one of which successfully accommodates charged lepton masses and neutrino oscillation data, while predicting the values of the Dirac and Majorana CPV phases.
We propose a lepton model under modular $A_4$ and gauged $U(1)_R$ symmetries, in which the neutrino masses are induced at one-loop level. Thanks to the modular $A_4$ symmetry, we have several predictions on the lepton sector, especially, on fixed points of $τ=i,ω\equiv e^{2πi/3},i\times \infty$ each of which has remnant symmetry; $Z_2$ for $τ=i$ and $Z_3$ for $τ=ω, i\infty$. These points are favored by a string theory and phenomenologically interesting.
We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group $Γ_7$, which is isomorphic to $PSL(2,Z_{7})$, the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as $PSL_2(7)$ or $Σ(168)$. At weight 2, there are 26 linearly independent modular forms, organised into a tri…
We perform a systematical analysis of the $A_4$ modular models with generalized CP for the masses and flavor mixing of quarks and leptons, and the most general form of the quark and lepton mass matrices is given. The CP invariance requires all couplings real in the chosen basis and thus the vacuum expectation value of the modulus $τ$ uniquely breaks both the modular symmetry and CP symmetry. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find 20 models with 7 real free parameters that can accommodate the experimental data of…
We study quark and lepton masses and mixings in a double covering of modular $A_4$ flavor symmetry in which we search for common solution of a single modulus $τ$, applying chi-square numerical analysis under the as minimum framework as possible. We have found the common region of $τ$ within $5σ$ interval that is narrow in case of normal hierarchy, but not found in case of inverted hierarchy.
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 i…
We present a comprehensive analysis of neutrino mass and lepton mixing in theories with $A_4$ modular symmetry, where the only flavon field is the single modulus field $τ$, and all masses and Yukawa couplings are modular forms. Similar to previous analyses, we discuss all the simplest neutrino sectors arising from both the Weinberg operator and the type I seesaw mechanism, with lepton doublets and right-handed neutrinos assumed to be triplets of $A_4$. Unlike previous analyses, we allow right-handed charged leptons to transform as all combinations of $\mathbf{1}$, $\mathbf{1}'$ and $\mathbf{1}…
We study the lepton flavor violation (LFV), the leptonic magnetic moments $(g-2)_{μ,\,e} $ and the electric dipole moment (EDM) of the electron in the Standard-Model Effective Field Theory with the $Γ_N$ 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, $Γ_3$ for the flavor symmetry, and discuss the dipole operators at nearby fixed point $τ=i$, where observed lepton masses and mixing angles are well reproduced. Suppose the a…
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.
The even weight modular forms of level $N$ can be arranged into the common irreducible representations of the inhomogeneous finite modular group $Γ_N$ and the homogeneous finite modular group $Γ'_N$ which is the double covering of $Γ_N$, and the odd weight modular forms of level $N$ transform in the new representations of $Γ'_N$. We find that the above structure of modular forms can naturally generate texture zeros of the fermion mass matrices if we properly assign the representations and weights of the matter fields under the modular group. We perform a comprehensive analysis for the $Γ'_3\co…
We propose to generate the flavor structures of the Standard Model plus neutrinos from flipped SU(5) GUT with $A_4$ modular flavor symmetry. Possible way to assign different moduli values for quarks and leptons in modular GUT scheme is discussed. We propose to reduce the multiple modular symmetries to a single modular symmetry in the low energy effective theory with proper boundary conditions. We classify all possible scenarios in this scheme according to the assignments of the modular $A_4$ representations for matter superfields and give the expressions of the quark and lepton mass matrices p…
本文研究了由有限模群Gamma_N导出的轻子混合斑图，它们要求子群G_nu和G_e分别保持在中微子和带电轻子扇区中。我们证明了只有6个群Gamma_N（N= 3，4，5，7，8，16）是相关的.本文对任意G_e和G_nu= Z ~ 2 × Z ~ 2的情况作了全面的分析。我们讨论了由群G_e和G_nu都是任意群所引起的有趣的模式。几个最有前途的模式是具体的偏差，从三双峰混合，所有预测θ_13非零的青睐，最新的实验数据。我们还评论的前景，将这一想法扩展到夸克部门。
In this paper, we investigate the double covering of modular $Γ^{}_5 \simeq A^{}_5$ group and derive all the modular forms of weight one for the first time. The modular forms of higher weights are also explicitly given by decomposing the direct products of weight-one forms. For the double covering group $Γ^\prime_5 \simeq A^\prime_5$, there exist two inequivalent two-dimensional irreducible representations, into which we can assign two right-handed neutrino singlets in the minimal seesaw model. Two concrete models with such a salient feature have been constructed to successfully explain lepton…
Sum rules in the lepton sector provide an extremely valuable tool to classify flavour models in terms of relations between neutrino masses and mixing parameters testable in a plethora of experiments. In this manuscript we identify new leptonic sum rules arising in models with modular symmetries with residual symmetries. These models simultaneously present neutrino mass sum rules, involving masses and Majorana phases, and mixing sum rules, connecting the mixing angles and the Dirac CP-violating phase. The simultaneous appearance of both types of sum rules leads to some non-trivial interplay, fo…
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 elemen…
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 neu…
We have performed a systematical analysis of lepton and quark masses models based on $Γ_4\cong S_4$ modular symmetry with gCP symmetry. We have considered both cases that neutrinos are Majorana particles and Dirac particles. All possible nontrivial representation assignments of matter fields are considered, and the most general form of fermion mass matrices are given. The phenomenologically viable models with the lowest number of free parameters together with the results of fit are presented. We find out nine lepton models with seven real free parameters including the real and imaginary parts …
We propose a new mechanism to explain neutrino masses with lepton number conservation, in which the Dirac neutrino masses are generated at the two-loop level involving a dark matter candidate. In this model, branching ratios of lepton flavor violating decays of the Higgs boson can be much larger than those of lepton flavor violating decays of charged leptons. If lepton flavor violating decays of the Higgs boson are observed at future collider experiments without detecting lepton flavor violating decays of charged leptons, most of the models previously proposed for tiny neutrino masses are excl…
In this paper, we study the implications of the modular $A^{}_4$ flavor symmetry in constructing a supersymmetric minimal type-(I+II) seesaw model, in which only one right-handed neutrino and two Higgs triplets are introduced to account for the tiny neutrino masses, flavor mixing and CP violation. The right-handed neutrino as well as the Higgs triplets in this model are assigned into the trivial one-dimensional irreducible representation of the modular group $A^{}_{4}$. We show that the individual contributions to the neutrino masses from the right-handed neutrino and the Higgs triplet are com…
We consider two Ansatze for the neutrino masses and mixings in which the permutation symmetry is implemented in various orders. We discuss the possible see-saw mechanisms and the charged lepton masses for the two cases in the presence of a Higgs triplet and three Higgs doublets.
We discuss a minimal flavour model with twin modular symmetries, leading to trimaximal TM$_1$ lepton mixing in which the first column of the tri-bimaximal lepton mixing matrix is preserved. The model involves two modular $S_4$ groups, one acting in the neutrino sector, associated with a modulus field value $τ_{SU}$ with residual $Z^{SU}_2$ symmetry, and one acting in the charged lepton sector, associated with a modulus field value $τ_{T}$ with residual $Z^{T}_3$ symmetry. Apart from the predictions of TM$_1$ mixing, the model leads to a new neutrino mass sum rule which implies lower bounds on …
We study one-loop induced radiative seesaw model applying a modular $A_4$ flavor symmetry, in which the neutrino mass matrix is achieved by two different Yukawa couplings one of which also contributes to positive value of muon anomalous magnetic moment as well as lepton flavor violations. Thanks to the specific mass matrix via $A_4$ symmetry and its modular weight, we find several predictions for lepton sector through our numerical analysis.
We propose a predictive lepton model under a modular $S_4$ symmetry, where the neutrino mass matrix arises from a radiative seesaw at one-loop level. The tree-level mass matrix is forbidden by well-assigned modular weights, which also play an important role in stabilizing dark matter candidate due to a remnant $Z_2$ symmetry even after breaking the modular symmetry. Supposing three families of the Majorana neutrinos, right-handed charged-leptons and left-handed charged-leptons to be embedded respectively into singlet, doublet, and triplet under $S_4$, we obtain the predictive mass matrices in …
We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level $N$ can be arranged into irreducible representations of the homogeneous finite modular group $Γ'_N$ which is the double covering of $Γ_N$. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of $Γ'_3 \cong T'$. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on $T'$ modular symmetry.
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 doub…
Instead of right-handed neutrino singlets, the standard model is extended to include lepton triplets $(Σ^+, Σ^0, Σ^-)$. Each quark and lepton family may now transform under an anomaly-free $U(1)_X$ gauge symmetry, known already for many years. A new sequential application is presented, using just the one Higgs doublet of the standard model, together with two $U(1)_X$ Higgs singlets. The resulting structure has hierarchical quark and lepton masses, as well as a viable seesaw neutrino mass matrix.
We make an investigation of modular $Γ^{\prime}_5 \simeq A^{\prime}_5$ group in inverse seesaw framework. Modular symmetry is advantageous because it reduces the usage of extra scalar fields significantly. Moreover, the Yukawa couplings are expressed in terms of Dedekind eta functions, which also have a $q$ expansion form, utilized to achieve numerical simplicity. Our proposed model includes six heavy fermion superfields i.e., $\mathcal{N}_{Ri}$, $\mathcal{S}_{Li}$ and a weighton. The study of neutrino phenomenology becomes simplified and effective by the usage of $A^\prime_5$ modular symmetry…
We discuss the $SU(5)$ grand unified extension of flavour models with multiple modular symmetries. The proposed model involves two modular $S_4$ groups, one acting in the charged fermion sector, associated with a modulus field value $τ_T$ with residual $Z_3^T$ symmetry, and one acting in the right-handed neutrino sector, associated with another modulus field value $τ_{SU}$ with residual $Z_2^{SU}$ symmetry. Quark and lepton mass hierarchies are naturally generated with the help of weightons, which are SM singlet fields, where their non-zero modular weights play the role of Froggatt-Nielsen cha…
