This lecture note provides a tutorial review of non-Abelian discrete groups and presents applications to particle physics where discrete symmetries constitute an important principle for model building. While Abelian discrete symmetries are often imposed in order to control couplings for particle physics―particularly model building beyond the standard model―non-Abelian discrete symmetries have been applied particularly to understand the three-generation flavor structure. The non-Abelian discrete symmetries are indeed considered to be the most attractive choice for a flavor sector: Model builders have tried to derive experimental values of quark and lepton masses, mixing angles and CP phases on the assumption of non-Abelian discrete flavor symmetries of quarks and leptons, yet lepton mixing has already been intensively discussed in this context as well. Possible origins of the non-Abelian discrete symmetry for flavors are another topic of interest, as they can arise from an underlying theory, e.g., the string theory or compactification via orbifolding as geometrical symmetries such as modular symmetries, thereby providing a possible bridge between the underlying theory and corresponding low-energy sector of particle physics.
The book offers explicit introduction to the group theoretical aspects of many concrete groups, and readers learn how to derive conjugacy classes, characters, representations, tensor products, and automorphisms for these groups (with a finite number) when algebraic relations are given, thereby enabling readers to apply this to other groups of interest. Further, CP symmetry and modular symmetry are also presented.
Author(s): Tatsuo Kobayashi
Series: Lecture Notes in Physics, 995
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Pages: 352
City: Berlin
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
2 Basics of Finite Groups
3 SN
3.1 S3
3.2 S4
4 AN
4.1 A4
4.2 A5
5 T'
6 DN
6.1 DN with N= Even
6.2 DN with N= Odd
6.3 D4
6.4 D5
7 QN
7.1 QN with N=4n
7.2 QN with N=4n+2
7.3 Q4
7.4 Q6
8 QD2N
8.1 Generic Aspects
8.2 QD16
9 Σ(2N2)
9.1 Generic Aspects
9.2 Σ(18)
9.3 Σ(32)
9.4 Σ(50)
10 Δ(3N2)
10.1 Δ(3N2) with N/3 neq Integer
10.2 Δ(3N2) with N/3 = Integer
10.3 Δ(27)
10.4 Δ(48)
10.4.1 Conjugacy Classes and Tensor Products
10.4.2 The Different Bases of Δ(48)
11 TN
11.1 Generic Aspects
11.2 T7
11.3 T13 Group Theory
11.4 T19 Group Theory
12 Σ(3N3)
12.1 Generic Aspects
12.2 Σ(81)
13 Δ(6N2)
13.1 Δ(6N2) with N/3= Integer
13.2 Δ(6N2) with N/3= Integer
13.3 Δ(54)
13.4 Δ(96)
13.4.1 Conjugacy Classes
13.4.2 Characters and Representations
13.4.3 Tensor Products
14 Subgroups and Decompositions of Multiplets
14.1 S3
14.2 S4
14.3 A4
14.4 A5
14.5 T'
14.6 General DN
14.7 D4
14.8 General QN
14.9 Q4
14.10 QD2N
14.11 General Σ(2N2)
14.12 Σ(32)
14.13 General Δ(3N2)
14.14 Δ(27)
14.15 General TN
14.16 T7
14.17 General Σ(3N3)
14.18 Σ(81)
14.19 General Δ(6N2)
14.20 Δ(54)
15 Finite Subgroups of Continuous Groups
15.1 Finite Subgroups of SO(3)
15.1.1 SO(3)toDN
15.1.2 SO(3)toA4
15.1.3 SO(3)toS4
15.2 Finite Subgroups of SU(2)
15.2.1 SU(2)toQN
15.2.2 SU(2)toT'
15.3 Finite Subgroups of SU(3)
15.3.1 SU(3)toA4
15.3.2 SU(3)toA5
15.3.3 SU(3)toS4
15.3.4 SU(3)toDN
15.3.5 SU(3)toT'
15.3.6 SU(3)toΔ(3N2)
15.3.7 SU(3)toΔ(6N2)
16 Modular Symmetry
17 Automorphism
17.1 Z3
17.2 Z2timesZ'2
17.3 A4
17.4 Σ(18)
17.5 Δ(27)
18 Anomalies
18.1 Generic Aspects
18.2 Explicit Calculations
18.3 Comments on Anomalies
19 Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models
19.1 Neutrino Flavor Mixing and Neutrino Mass Matrix
19.2 A4 Flavor Symmetry
19.2.1 Realizing the Tri-Bimaximal Mixing of Flavors
19.2.2 Breaking Tri-Bimaximal Mixing
19.3 S4 Flavor Model
19.4 Alternative Flavor Mixing
19.5 Modular A4 Invariance and Neutrino Mixing
19.6 Comments on Other Applications
19.7 Comment on Origins of Flavor Symmetries
20 Generalized CP Symmetry
20.1 CP Transformation in Flavor Space
20.2 CP Violating Phase and Its Group Theoretical Origin
20.3 Modular Symmetry with Generalized CP Symmetry
20.3.1 CP Transformation of the Modulus τ
20.3.2 CP Transformation of Modular Multiplets
20.4 CP Violation by Modulus τ in A4 Model of Leptons
A Useful Theorems
B Representations of S4 in Several Bases
B.1 The Basis I
B.2 The Basis II
B.3 The Basis III
B.4 The Basis IV
C Representations of A4 in Different Basis
D Representations of A5 in Different Basis
D.1 The Basis I
D.2 The Basis II
E Representations of T' in Different Basis
E.1 The Basis I
E.2 The Basis II
F Representations of Δ(96) in Different Basis
G Other Smaller Groups
G.1 Z 4 Z4
G.2 Z 8 Z2
G.3 (Z2timesZ4)Z2 (I)
G.4 (Z2timesZ4)Z2 (II)
G.5 Z3 Z8
G.6 (Z6timesZ2) Z2
G.7 Z9 Z3
H Generators of the Modular Group
I Modular Forms
