Non-Abelian Discrete Symmetries in Particle Physics (1003.3552v2)

Abstract: We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed them explicitly for $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $\Sigma(2N^2)$, $\Delta(3N^2)$, $T_7$, $\Sigma(3N^3)$ and $\Delta(6N^2)$, which have been applied for model building in the particle physics. We also present typical flavor models by using $A_4$, $S_4$, and $\Delta (54)$ groups. Breaking patterns of discrete groups and decompositions of multiplets are important for applications of the non-Abelian discrete symmetry. We discuss these breaking patterns of the non-Abelian discrete group, which are a powerful tool for model buildings. We also review briefly about anomalies of non-Abelian discrete symmetries by using the path integral approach.

• The paper presents a comprehensive framework for using non-Abelian discrete symmetries, such as Sₙ and Δ(6N²), to constrain flavor models and address anomalies.
• It systematically applies group-theoretical analyses, detailing conjugacy classes, character tables, and tensor products to align theoretical predictions with quark-lepton mixing data.
• The study highlights practical implications for CP violation, neutrino oscillations, and dark matter models, supporting future research in unified particle physics theories.

Overview of Non-Abelian Discrete Symmetries in Particle Physics

The focus of this paper is on non-Abelian discrete symmetries within the context of particle physics, particularly their applications in model building to analyze flavor physics. Symmetries are pivotal in theoretical physics due to their ability to unify multiple phenomena and constrain model parameters like Yukawa couplings, impacting our understanding of particle interactions. The paper explores various non-Abelian discrete groups such as $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $\Sigma(2N^2)$, $\Delta(3N^2)$, $T_7$, $\Sigma(3N^3)$, and $\Delta(6N^2)$, outlining their group-theoretical aspects—conjugacy classes, characters, representations, and tensor products—thus demonstrating their utility in constructing models that seek to mirror the experimental data related to quark and lepton masses and mixing angles.

Key Numerical Components and Concepts

The paper systematically presents numerical results and theoretical frameworks for numerous discrete groups:

• Symmetric and Alternating Groups ($S_N$, $A_N$): These groups allow permutations of $N$ objects and their even-subgroup equivalents. For example, $S_3$ consists of six elements and has three conjugacy classes, leading to three irreducible representations, including a doublet.
• Dihedral and Binary Dihedral Groups ($D_N$, $Q_N$): Representing symmetries of regular polygons, these groups are structured as semi-direct products of cyclic groups, with $D_N$ accommodating non-commutative relationships between rotations and reflections.
• $\Sigma(2N^2)$ and $\Delta(3N^2)$ Groups: These complex structures are indicative of the richness of discrete symmetry applications. For instance, $\Sigma(2N^2)$ is isomorphic to $(Z_N \times Z_N')\rtimes Z_2$, and the paper systematically defines their conjugacy classes and character tables, as shown in the $\Sigma(18)$ section with 18 elements and 9 conjugacy classes.
• $\Delta(6N^2)$ and Associated Groups: The $\Delta(6N^2)$ group extends ideas from smaller group constructs, building upon the non-Abelian behavior of flavor group symmetries relevant to three-generation particle models.

Implications and Applications

The implications are significant for flavor physics—non-Abelian discrete symmetries are widely considered for theories addressing the flavor problem, characterized by unexplained patterns in particle masses and their interactions. These symmetries can guide phenomenology beyond the Standard Model, promising natural explanations for observed mixing angles and hierarchies. The toolset developed through this extensive group-theoretical review offers robust foundations for exploring mechanisms for CP violation, neutrino oscillations, and even insights into dark matter models.

Anomalies and Model Building

A unique focus of the paper involves discussing anomalies within non-Abelian symmetries using path integrals, offering conditions for anomaly cancellation which are vital for maintaining valid symmetries at the quantum level. Anomaly considerations are crucial for ensuring the consistency of gauge theories extended by discrete symmetries.

Future Directions

Theoretical and experimental advancements in particle physics suggest that discrete non-Abelian symmetries will continue to play an instrumental role. As more complexity in flavor physics emerges, extending symmetry considerations into higher dimensions or incorporating them within string theory and GUTs could provide deeper insights. Accordingly, future research directions might involve leveraging these symmetries within experimental constraints, exploring new auxiliary symmetries, and further bridging gaps between low-energy phenomenological models and overarching unification theories.

In summary, this paper provides an extensive analysis of non-Abelian discrete symmetries, offering comprehensive mathematical and phenomenological frameworks useful for constructing viable models in the field of particle physics. The results indicate strong potential for these symmetries in explaining existing phenomenological data and exploring new physics.