Related papers: Quarks at the modular $S_4$ cusp

We study $A_4 \times A_4 \times A_4$ modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points $\tau = i\infty$ and…

We study modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms, and modular forms in the vicinity of the modular fixed points become hierarchical depending on their residual charges.…

We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity…

We study the possibility to generate the quark mass hierarchies as well as the CKM quark mixing and CP violation without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the…

We investigate theories of flavour based on genus $g=2$ modular invariance and analyze how fermion mass hierarchies can be generated in this context, in the vicinity of invariant points or regions in moduli space where a residual symmetry…

We propose a quark flavor model based on modular $S'_4$ with a general CP symmetry. CP violation in the quark sector is entirely realized by the modulus $\langle \tau \rangle$. We show that the canonical normalization induced by the…

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…

We study in the bottom-up framework the possibility to generate the quark mass hierarchies without fine-tuning, the quark mixing and CP-violation (CPV) in a flavour model with $S^\prime_4$ modular symmetry having minimal number of…

We propose models to explain the hierarchies of the quark masses and mixing by utilizing the $S_4^\prime$ modular flavor symmetry. The hierarchy is realized by the modulus $\tau$ stabilized at $\mathrm{Im}\,\tau \gg 1$, where the residual…

We study the large mass hierarchy and CP violation in the modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms. Modular forms near the modular fixed points are approximately given by…

We formulate general conditions under which the strong CP problem is solved by spontaneous CP violation. Quark-mass matrix elements are polynomials in the CP-breaking order parameters, engineered such that their determinant is a real…

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…

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…

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…

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…

We have performed a systematical analysis of lepton and quark masses models based on $\Gamma_4\cong S_4$ modular symmetry with gCP symmetry. We have considered both cases that neutrinos are Majorana particles and Dirac particles. All…

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…

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…

Flavor mixing and the quark mass spectrum are intimately related. In view of the observed strong hierarchy of the quark and lepton masses and of the flavor mixing angles it is argued that the description of flavor mixing must take this into…

A ${\cal G}_{\cal F}=SU(3)_{Q}\times SU(3)_{u}\times SU(3)_{d}$ invariant scalar potential breaking spontaneously the quark flavor symmetry can explain the standard model flavor puzzle. The approximate alignment in flavor space of the…