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Quark masses and mixing from Modular $S'_4$ with Canonical Kähler Effects

Ivo de Medeiros Varzielas, Manuel Paiva

Abstract

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 $\langle τ\rangle$. We show that the canonical normalization induced by the Kähler metric plays a crucial role in reproducing the observed hierarchies, while maintaining coupling constants of order $\mathcal{O}(1)$. The minimal model achieves a great fit to the quark sector data, which we take as the PDG 2024 data extrapolated to the GUT scale.

Quark masses and mixing from Modular $S'_4$ with Canonical Kähler Effects

Abstract

We propose a quark flavor model based on modular with a general CP symmetry. CP violation in the quark sector is entirely realized by the modulus . We show that the canonical normalization induced by the Kähler metric plays a crucial role in reproducing the observed hierarchies, while maintaining coupling constants of order . The minimal model achieves a great fit to the quark sector data, which we take as the PDG 2024 data extrapolated to the GUT scale.

Paper Structure

This paper contains 12 sections, 30 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Framework
  3. Modular symmetry
  4. Modular-invariant supersymmetric theories
  5. Model
  6. Superfields and Superpotential
  7. Mass Matrices
  8. Numerical Analysis and Phenomenological Results
  9. Conclusions
  10. Modular Forms and Clebsch-Gordan Coefficients of S'4
  11. Tensor Products
  12. Explicit Modular Forms

Figures (3)

  • Figure 1: Projection in the $(\bar{\rho}, \bar{\eta})$ plane of the theoretical $1\sigma$ parameter space, as derived from a Gaussian sampling around the global $\chi^2$ minimum. The red marker denotes the exact global best-fit configuration of the $S'_4$ modular framework. The blue scatter points represent the theoretical $1\sigma$ region. The shaded background ellipses delineate the $1\sigma$, $2\sigma$, and $3\sigma$ experimentally allowed regions derived from the updated PDG 2024 global fit.
  • Figure 2: Topographical contour map of the global $\chi^2$ in the complex modulus plane.
  • Figure 3: The Jarlskog invariant $J_{CP}^q$ as a function of $\operatorname{Re}(\tau)$. The invariant vanishes when the vacuum aligns with the imaginary axis.