粒子物理学家用非阿贝尔离散对称导论=aN lntroduction to Non Abelian Discrete Symmetries for Particle PhysicistsPDF电子书下载

1 Introduction 1

References 3

2 Basics of Finite Groups 13

References 20

3 SN 21

3.1 S3 21

3.1.1 Conjugacy Classes 21

3.1.2 Characters and Representations 22

3.1.3 Tensor Products 22

3.2 S4 25

3.2.1 Conjugacy Classes 27

3.2.2 Characters and Representations 27

3.2.3 Tensor Products 29

References 30

4 AN 31

4.1 A4 31

4.2 A5 34

4.2.1 Conjugacy Classes 35

4.2.2 Characters and Representations 35

4.2.3 Tensor Products 37

References 41

5 T′ 43

5.1 Conjugacy Classes 43

5.2 Characters and Representations 44

5.3 Tensor Products 47

6 DN 51

6.1 DN with N Even 51

6.1.1 Conjugacy Classes 52

6.1.2 Characters and Representations 52

6.1.3 Tensor Products 54

6.2 DN with N Odd 56

6.2.1 Conjugacy Classes 56

6.2.2 Characters and Representations 56

6.2.3 Tensor Products 57

6.3 D4 58

6.4 D5 59

7 QN 61

7.1 QN with N=4n 61

7.1.1 Conjugacy Classes 62

7.1.2 Characters and Representations 62

7.1.3 Tensor Products 62

7.2 QN with N=4n+2 64

7.2.1 Conjugacy Classes 64

7.2.2 Characters and Representations 64

7.2.3 Tensor Products 65

7.3 Q4 66

7.4 Q6 67

8 QD2N 69

8.1 Generic Aspects 69

8.1.1 Conjugacy Classes 70

8.1.2 Characters and Representations 70

8.1.3 Tensor Products 71

8.2 QD16 72

9 ∑(2N2) 75

9.1 Generic Aspects 75

9.1.1 Conjugacy Classes 75

9.1.2 Characters and Representations 76

9.1.3 Tensor Products 77

9.2 ∑(18) 78

9.3 ∑(32) 80

9.4 ∑(50) 84

10 △(3N2) 87

10.1 △(3N2) with N/3≠Integer 87

10.1.1 Conjugacy Classes 88

10.1.2 Characters and Representations 89

10.1.3 Tensor Products 89

10.2 △(3N2) with N/3 Integer 91

10.2.1 Conjugacy Classes 91

10.2.2 Characters and Representations 92

10.2.3 Tensor Products 93

10.3 △(27) 94

References 95

11 TN 97

11.1 Generic Aspects 97

11.1.1 Conjugacy Classes 98

11.1.2 Characters and Representations 99

11.1.3 Tensor Products 99

11.2 T7 100

11.3 T13 102

11.4 T19 104

References 108

12 ∑(3N3) 109

12.1 Generic Aspects 109

12.1.1 Conjugacy Classes 110

12.1.2 Characters and Representations 111

12.1.3 Tensor Products 112

12.2 ∑(81) 113

References 121

13 △(6N2) 123

13.1 △(6N2)with N/3≠Integer 123

13.1.1 Conjugacy Classes 123

13.1.2 Characters and Representations 126

13.1.3 Tensor Products 128

13.2 △(6N2) with N/3 Integer 131

13.2.1 Conjugacy Classes 131

13.2.2 Characters and Representations 133

13.2.3 Tensor Products 134

13.3 △(54) 138

13.3.1 Conjugacy Classes 138

13.3.2 Characters and Representations 139

13.3.3 Tensor Products 141

References 145

14 Subgroups and Decompositions of Multiplets 147

14.1 S3 147

14.1.1 S3→Z3 148

14.1.2 S3→Z2 148

14.2 S4 149

14.2.1 S4→S3 150

14.2.2 S4→A4 151

14.2.3 S4→∑(8) 151

14.3 A4 152

14.3.1 A4→Z3 152

14.3.2 A4→Z2×Z2 153

14.4 A5 153

14.4.1 A5→A4 153

14.4.2 A5→D5 153

14.4.3 A5→S3？D3 154

14.5 T′ 154

14.5.1 T′→Z6 154

14.5.2 T′→Z4 155

14.5.3 T′→Q4 155

14.6 General DN 155

14.6.1 DN→Z2 156

14.6.2 DN→ZN 157

14.6.3 DN→DM 157

14.7 D4 158

14.7.1 D4→Z4 158

14.7.2 D4→Z2×Z2 159

14.7.3 D4→Z2 159

14.8 General QN 159

14.8.1 QN→Z4 160

14.8.2 QN→ZN 161

14.8.3 QN→QM 161

14.9 Q4 162

14.9.1 Q4→Z4 162

14.10 QD2N 162

14.10.1 QD2N→Z2 163

14.10.2 QD2N→ZN 163

14.10.3 QD2N→DN/2 163

14.11 General∑(2N2) 164

14.11.1 ∑(2N2)→Z2N 164

14.11.2 ∑(2N2)→ZN×ZN 164

14.11.3 ∑(2N2)→DN 165

14.11.4 ∑(2N2)→QN 166

14.11.5 ∑(2N2)→∑(2M2) 166

14.12 ∑(32) 167

14.13 General △(3N2) 168

14.13.1 △(3N2)→Z3 169

14.13.2 △(3N2)→ZN×ZN 169

14.13.3 △(3N2)→TN 170

14.13.4 △(3N2)→△(3M2) 170

14.14 △(27) 172

14.14.1 △(27)→Z3 172

14.14.2 △(27)→Z3×Z3 172

14.15 General TN 173

14.15.1 TN→Z3 173

14.15.2 TN→ZN 173

14.16 T7 174

14.16.1 T7→Z3 174

14.16.2 T7→Z7 175

14.17 General ∑(3N3) 175

14.17.1 ∑(3N2)→ZN×ZN×ZN 175

14.17.2 ∑(3N3)→△(3N2) 175

14.17.3 ∑(3N3)→∑(3M3) 176

14.18 ∑(81) 176

14.18.1 ∑(81)→Z3×Z3×Z3 177

14.18.2 ∑(81)→△(27) 177

14.19 General△(6N2) 178

14.19.1 △(6N2)→∑(2N2) 179

14.19.2 △(6N2)→△(3N2) 180

14.19.3 △(6N2)→△(6M2) 180

14.20 △(54) 181

14.20.1 △(54)→S3×Z3 182

14.20.2 △(54)→∑(18) 182

14.20.3 △(54)→△(27) 183

15 Anomalies 185

15.1 Generic Aspects 185

15.2 Explicit Calculations 189

15.2.1 53 189

15.2.2 S4 190

15.2.3 A4 190

15.2.4 A5 191

15.2.5 T′ 192

15.2.6 DN (N Even) 193

15.2.7 DN (N Odd) 194

15.2.8 QN(N=4n) 194

15.2.9 QN(N=4n+2) 195

15.2.10 QD2N 196

15.2.11 ∑(2N2) 197

15.2.12 △(3N2)(N/3≠Integer) 198

15.2.13 △(3N2)(N/3 Integer) 199

15.2.14 TN 200

15.2.15 ∑(3N3) 201

15.2.16 △(6N2)(N/3≠Integer) 202

15.2.17 △(6N2)(N/3 Integer) 203

15.3 Comments on Anomalies 203

References 204

16 Non-Abellan Discrete Symmetry in Quark/Lepton Flavor Models 205

16.1 Neutrino Flavor Mixing and Neutrino Mass Matrix 205

16.2 A4 Flavor Symmetry 207

16.2.1 Realizing Tri-Bimaximal Mixing of Flavors 207

16.2.2 Breaking Tri-Bimaximal Mixing 209

16.3 S4 Flavor Model 211

16.4 Alternative Flavor Mixing 219

16.5 Comments on Other Applications 222

16.6 Comment on Origins of Flavor Symmetries 223

References 224

Appendix A Useful Theorems 229

References 235

Appendix B Representations of S4 in Different Bases 237

B.1 Basis Ⅰ 237

B.2 Basis Ⅱ 238

B.3 Basis Ⅲ 240

B.4 Basis Ⅳ 242

References 244

Appendix C Representations of A4 in Different Bases 245

C.1 Basis Ⅰ 245

C.2 Basis Ⅱ 245

References 246

Appendix D Representations of A5 in Different Bases 247

D.1 Basis Ⅰ 247

D.2 Basis Ⅱ 253

References 259

Appendix E Representations of T'in Different Bases 261

E.1 Basis Ⅰ 262

E.2 Basis Ⅱ 263

References 264

Appendix F Other Smaller Groups 265

F.1 Z4？Z4 265

F.2 Z8？Z2 268

F.3 (Z2×Z4)？Z2(Ⅰ) 270

F.4 (Z2×Z4)？Z2(Ⅱ) 272

F.5 Z3？Z8 275

F.6 (Z6×Z2)？Z2 277

F.7 Z9？Z3 281

References 283

Index 285