Finite Modular Groups and Lepton Mixing (1112.1340v1)

Abstract: We study lepton mixing patterns which are derived from finite modular groups Gamma_N, requiring subgroups G_nu and G_e to be preserved in the neutrino and charged lepton sectors, respectively. We show that only six groups Gamma_N with N=3,4,5,7,8,16 are relevant. A comprehensive analysis is presented for G_e arbitrary and G_nu=Z2 x Z2, as demanded if neutrinos are Majorana particles. We discuss interesting patterns arising from both groups G_e and G_nu being arbitrary. Several of the most promising patterns are specific deviations from tri-bimaximal mixing, all predicting theta_13 non-zero as favoured by the latest experimental data. We also comment on prospects to extend this idea to the quark sector.

• The paper introduces finite modular groups as a novel framework to derive predictive lepton mixing patterns through symmetry breaking.
• It systematically examines groups with N=3,4,5,7,8,16, detailing how configurations in neutrino and charged lepton sectors yield distinct mixing matrices.
• The study highlights the potential for refined particle physics models that leverage discrete symmetries to explain fermion mass hierarchies and mixing angles.

Finite Modular Groups and Lepton Mixing

The paper of lepton mixing patterns inspired by finite modular groups presents an intriguing approach to understanding the unsolved problem of fermion mixing in particle physics, particularly the distinct discrepancy observed between quark and lepton mixing matrices. The paper investigates the use of finite modular groups, denoted $\Gamma_N$, as a foundation for deriving patterns in lepton mixing, with considerations extended to both the neutrino and charged lepton sectors.

Overview

Finite modular groups $\Gamma_N$ are explored with $N=3,4,5,7,8,16$, which are found to be the relevant configurations for the described methodologies. These groups offer discrete symmetry patterns that can potentially map the structures related to lepton mixing matrices. The research focuses on instances where the neutrinos are Majorana particles, necessitating the preservation of $Z_2 \times Z_2$ in the neutrino sector, and considers arbitrary choices for the subgroup $G_e$ in the charged lepton sector.

Key Findings and Methodology

The paper discusses different modular groups by addressing the possibility of breaking the flavour symmetry $G_f$ into subgroups $G_\nu$ and $G_e$. By embedding the left-handed leptons into three-dimensional irreducible representations of $G_f$, lepton mixing then emerges from the mismatch between the relative embedding of these subgroups into $G_f$. Notably, this research emphasizes specific configurations of $G_\nu=Z_2 \times Z_2$ and $G_e$ as various cyclic groups or their combinations, leading to highly predictive lepton mixing patterns.

Through a methodical exploration:

• $A_4$ unveils a unique mixing pattern predicting maximized values for both the solar and atmospheric mixing angles.
• $S_4$ offers tri-bimaximal and bimaximal mixing patterns, which diverge from empirical data mainly in terms of $\theta_{13}$ leading to considerations for extending these scenarios or seeking further perturbative corrections.
• $A_5$ and other higher order groups like $PSL(2,Z_7)$, $\Delta(96)$, and $\Delta(384)$ further expand on promising and diverse mixing pattern possibilities including those with non-zero $\theta_{13}$ favored by data.

Theoretical Implications and Future Exploration

The bold suggestion of leveraging finite modular groups emphasizes the symmetry-based origin of fermion masses and mixing. The research suggests that integrating finite modular groups within the existing theoretical framework can provoke novel approaches and applications of discrete symmetries, particularly in building models accommodating realistic lepton and even quark sectors.

Practical Applications

The insight from these modular groups could significantly influence the construction of new particle physics models that predict not only mixing matrices but also the possible hierarchy of masses. Such developments might lead to benchmarks for experimental testing of theoretical predictions, especially as the experimental precision in neutrino oscillation parameters improves.

Conclusion and Speculation on AI integration

While the paper provides a comprehensive and foundational approach to lepton mixing via finite modular groups, future extensions might contemplate applying similar constructs using various computational frameworks or would permit exploration of synthetic combinatorial methods in AI. Such interdisciplinary outreach may open avenues for automatically generating and validating predictive models based on configurational symmetries or enhancing machine learning algorithms with insights from group theoretical symmetries, potentially benefiting broader AI deployment in modeling complex systems.