Lepton Masses and Mixing from Modular $S_4$ Symmetry (1806.11040v2)

Abstract: We study models of lepton masses and mixing based on broken modular invariance. We consider invariance under the finite modular group $\Gamma_4 \simeq S_4$ 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.

An Analysis of Lepton Masses and Mixing from Modular $S_4$ Symmetry

The examination of the lepton sector through the lens of modular $S_4$ symmetry presents a distinctive approach to the flavor puzzle in particle physics. The paper by Penedo and Petcov outlines a framework wherein the lepton masses and mixing angles emerge from a broken modular symmetry, specifically using the finite modular group $\Gamma_4 \simeq S_4$. This paper circumvents the conventional introduction of flavon fields by relying solely on the expectation value of a modulus as the symmetry-breaking mechanism.

Key Contributions and Models

The authors construct two minimal models of lepton masses invoking the $S_4$ symmetry, where the finite modular group's invariance guides the structure of mass matrices. Central to this framework is the idea that the lowest weight modular forms serve as the basis for building such models. Focusing on the cases where the modular weight of the lepton doublets is set to 1 and 2, they explore the implications of these settings on lepton phenomenology.

In these models, neutrino masses are examined under the assumption that they originate from the Weinberg operator in a supersymmetric setting. The use of a finite subgroup of the modular group dictates mass ratios and mixing patterns, aligning predictions with available experimental data, notably without introducing supplemental fields or symmetry constraints.

Numerical Outcomes and Analysis

The paper's results demonstrate varying degrees of success across the two models. While the model with a lepton doublet weight of 1 struggles to align predictions with experimental values, the model with a doublet weight of 2 exhibits promising alignment with data from neutrino oscillation parameters. Such success highlights the ability of modular symmetry, specifically $\Gamma_4$, to address flavor issues in a minimal framework, dispensing with the complications added by multiple flavons and complex scalar potentials.

The authors note specific parameter choices leading to predictions for the Dirac CP violating phase and Majorana phases, offering testable outcomes for further experimental investigations. In the successful model, the effective Majorana mass relevant to neutrinoless double-beta decay is predicted near the sensitivity of upcoming experiments, marking a significant achievement for the model's validity.

Theoretical and Practical Implications

The theoretical implications are profound, as the research highlights modular symmetry as a viable path towards understanding fermion masses and mixing. This approach situates itself as an alternative to prior flavor models, presenting predictions grounded in symmetry principles without introducing excessive free parameters.

Practically, the work motivates experimentalists to investigate beyond confirmed parameters and seek signals of modular symmetry's fingerprints. The specific prediction for the effective Majorana mass makes future neutrinoless double-beta decay experiments crucial in testing the validity of these models.

Future Directions

The paper opens avenues for further exploration, including examining other finite subgroups of modular groups or extending the framework to include quarks. Additionally, integrating this formalism with grand unified theories could provide a more comprehensive picture of particle physics under modular symmetries.

In conclusion, this paper enriches the landscape of flavor physics, offering a modular symmetry-based explanation for lepton masses and mixing. Its minimalistic nature, coupled with successful data alignment, suggests that $S_4$ modular symmetry could represent a foundational aspect of particle physics, meriting deeper exploration and experimental interrogation.