Narrow Results

The modular flavor symmetry is studied to explain the mass hierarchy of quarks and leptons. In this review, at first, we introduce briefly the framework of the modular symmetries and its residual symmetries, and then, present the general framework of generating the mass hierarchy close to the symmetric points of modulus without fine-tuning. Finally, we present a minimal ${\mathrm{\Gamma }}_3$ modular invariant model of quarks. We then describe how to obtain the hierarchical quark mass patterns in the vicinity of ${\tau }_0=i\infty$ and investigate the flavor structure of the quark mass matrices. The results of our study show that describing correctly the quark mass hierarchies without severe fine-tuning and the CP violation is remarkably challenging within the modular invariance approach to flavors.

In this paper, we construct a model with the help of modular symmetry in the framework of minimal inverse seesaw [ISS(2,3)]. We have used $\mathrm{\Gamma }(3)$ modular group which is isomorphic to non-Abelian discrete symmetry group $A_4$. In this group, there are three Yukawa modular forms of weight 2. In this model, we study neutrino masses and mixings for both normal and inverted hierarchies. Use of modular symmetry reduces the need for more number of extra flavons and their specific VEV alignments, as such, minimality of the model is maintained to a great extent. Along with $A_4$ symmetry group, we have used $Z_3$ to restrict certain interaction terms in the Lagrangian. Further, we calculate the effective mass to address the phenomena of neutrinoless double-beta decay ($0\nu \beta \beta$). The values of effective mass are found to lie within the bound ($m_{\mathrm{\text{eff}}}<0.165$eV) as predicted by different $0\nu \beta \beta$ experiments.

In this paper, we have realized the left–right symmetric model (LRSM) with modular symmetry. Most of the previous works on LRSM have been done considering discrete flavor symmetry, but this work has been carried out using $\mathrm{\Gamma }$(3) modular group which is isomorphic to nonabelian discrete symmetry group $A_4$. The advantage of using modular symmetry is the nonrequirement for the use of extra particles called “flavons”. In this model, the Yukawa couplings are expressed in terms of modular forms $(Y_1,Y_2,Y_3)$. In this work, we have studied minimal LRSM for both type-I and type-II dominances. Here, we have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino masses. The results obtained are well within the experimental limits for the desired values of sum of neutrino masses. We have also briefly analyzed the effects of the implications of modular symmetry on neutrinoless double beta decay with the new physics contributions and the correlation of lepton flavor violation and lightest neutrino mass within the framework of modular symmetric LRSM.

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.

In this paper, we discuss a minimal lepton flavor model with two modular $A_4$ symmetries, one acting on neutrinos and the other acting on charged leptons. Two corresponding moduli fields are restricted at stabilizers. To avoid vanishing or degenerate lepton masses, the stabilizer in the charged lepton sector always preserves a $Z_3$ symmetry, and that in the neutrino preserves a $Z_2$ symmetry. By scanning all viable stabilizers, a unique Trimaximal TM2 mixing is predicted. However, the prediction of the lightest neutrino mass and the mass parameter in neutrinoless double beta decay, as well as two Majorana phases, are different and classified into three cases.

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 τ = ω. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators.