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Progress of Theoretical Physics Supplement No. 183
Non-Abelian Discrete Symmetries in Particle Physics
By H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto
Recently, non-Abelian discrete groups have been playing an important role
in particle physics. In particular, several non-Abelian discrete
flavor symmetries have been applied in order to understand
the flavor structure of quarks and leptons. However, non-Abelian discrete
symmetries may not be familar to all of particle physicists (compared with
non-Abelian continous symmetries). In this article, we review pedagogically
non-Abelian discrete groups and show some applications for physical aspects.
This article includes a brief review on general aspects of group
theory in Section 2 and useful theorems in Appendix A.
Thus, readers could read it without any expected background knowledge
on group theory.
In the sections of concrete non-Abelian discrete groups, we focus on
the physically favored or expected groups:
SN , AN , T ', DN , QN ,
Σ(2N 2) , Δ(3N 2) ,
T7 ,
Σ(3N 3) and Δ(6N 2),
and summarize their conjugacy classes,
character tables, irreducible representations, and tensor products, which
are important to apply for physics.
To apply for physics, it is important to understand their breaking patterns
and anomalies. We show their possible breaking patterns
for all groups appearing here, and also review anomalies of general non-Abelian discrete groups. Then, by adapting them to the above concrete groups,
we give explicitly anomaly-free conditions.
As illustrating examples, we also review concrete models by using
A4, S4, and Δ(54) groups.
We study how to construct
quark- and lepton- mass matrices and derive their predictions
based on experiments.
Moreover, it is known that there exist some different types of tensor
products for one group. This implies that physical scenarios can be
changeable. Here we also show the relation between their tensor products
based on S4 and A4 groups in
Appendices B and C respectively.
We hope that this review will contribute to further progress of
particle physics with non-Abelian discrete groups and also be a useful
reference for researchers and graduate students who are interested in
this field.
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