241
THE STANDARD MODEL Y. Grossman and Y. Nir DRAFT as of November 17, 2016 1
THE STANDARD MODEL
Y. Grossman and Y. Nir
DRAFT as of November 17, 2016
1
Contents
1 Lagrangians 9
1.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Fermions and scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Discrete spacetime symmetries: C, P and T . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 C and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.2 CP violation and complex couplings . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Model building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Abelian symmetries 16
2.1 Global discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Global continuous symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Product groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Fermion masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Local symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.1 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Homework 27
3 QED 29
3.1 Defining QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 The interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Parameter counting and tests of QED . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 QED with more fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6.1 Two Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2
3.6.2 Accidental symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6.3 Even more fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Homework 34
4 Non-Abelian symmetries 37
4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Global symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Local symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Running coupling constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Appendices 43
4.A Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.A.1 Free massless scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.A.2 Free massless Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.A.3 Free massive Dirac fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Homework 49
5 QCD 53
5.1 Defining QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 The interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4.1 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.5 Accidental symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.6 Combining QCD with QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Homework 59
6 Spontaneous Symmetry Breaking 61
6.1 Global discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2 Global Abelian continuous symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Global non-Abelian continuous symmetries . . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Fermion masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.5 Local symmetries: the Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . 70
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Appendices 74
6.A The Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3
Homework 76
7 The Leptonic Standard Model 80
7.1 Defining the LSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2.1 Lkin and the gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2.2 Lψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2.3 LYuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2.4 Lφ and spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . 82
7.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3.1 Scalars: back to Lφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3.2 Vector bosons: back to Lkin(φ) . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3.3 Fermions: back to LYuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.4 The interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4.1 The Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4.2 QED: Electromagnetic interactions . . . . . . . . . . . . . . . . . . . . . . . 88
7.4.3 Neutral current weak interactions . . . . . . . . . . . . . . . . . . . . . . . . 89
7.4.4 Charged current weak interactions . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4.5 Gauge boson self-interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.5 Global symmetries and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5.1 The interaction basis and the mass basis . . . . . . . . . . . . . . . . . . . . 94
7.5.2 The LSM parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5.3 Accidental symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.5.4 Discrete symmetries: C, P and CP . . . . . . . . . . . . . . . . . . . . . . . 97
7.6 Low Energy Tests of the LSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.6.1 CC weak interactions: Quasi-elastic neutrino–electron scattering . . . . . . . 97
7.6.2 NC weak interactions: neutrino–electron scattering . . . . . . . . . . . . . . 98
7.6.3 Forward-backward asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Homework 100
8 The Standard Model 106
8.1 Defining the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2.1 Lkin and the gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4
8.2.2 Lψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.3 LYuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.4 Lφ and spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . 110
8.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3.1 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.4 The interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.4.1 Neutral current weak interactions . . . . . . . . . . . . . . . . . . . . . . . . 112
8.4.2 Charged current weak interactions . . . . . . . . . . . . . . . . . . . . . . . . 113
8.4.3 Interactions of the Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.5 Accidental symmetries and parameter counting . . . . . . . . . . . . . . . . . . . . 116
8.5.1 Accidental symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.5.2 Parameter counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.5.3 Parameter counting in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.5.4 The strong CP parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.6 The CKM parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.7 P, C and CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Appendices 124
8.A Unitarity Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Homework 126
9 Electroweak Precision Measurements 132
9.1 The SM beyond tree level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.2 Electroweak Precision Measurements (EWPM) . . . . . . . . . . . . . . . . . . . . . 133
9.3 The weak mixing angle, θW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9.3.1 θ within the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.4 Custodial symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9.5 Probing new physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.5.1 NR operators and the q2 expansion . . . . . . . . . . . . . . . . . . . . . . . 139
9.5.2 The four generation SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10 QCD at the IR 144
10.1 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.1.1 The quark model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5
10.1.2 Hadron masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.1.3 Hadron lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.1.4 Hadron quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.1.5 The approximate symmetries of QCD: light quarks . . . . . . . . . . . . . . 149
10.1.6 The approximate symmetries of QCD: heavy quarks . . . . . . . . . . . . . . 150
10.2 High energy QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.2.1 Quark hadron duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.2.2 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Appendices 153
10.A quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.B SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.C Names and QN for hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Homework 155
11 Flavor physics 156
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
11.2 Flavor changing neutral current (FCNC) processes . . . . . . . . . . . . . . . . . . . 157
11.3 Testing the CKM sector: The ρ− η plane . . . . . . . . . . . . . . . . . . . . . . . 160
11.4 New flavor physics: The hd − σd plane . . . . . . . . . . . . . . . . . . . . . . . . . 161
11.5 Non-renormalizable terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
11.6 CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Appendices 168
11.A Neutral meson mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
11.A.1 Flavor oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
11.A.2 Time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11.A.3 The SM calculation of M12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.B CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
11.B.1 CP violation in decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11.B.2 CP violation in mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.B.3 CP violation in interference of decays with and without mixing . . . . . . . . 179
Homework 181
12 Neutrinos 185
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.2 The νSM: The SM with d = 5 terms . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.2.1 The neutrino spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
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12.2.2 The scale of generation of neutrino masses . . . . . . . . . . . . . . . . . . . 186
12.2.3 The neutrino interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
12.2.4 Accidental symmetries and the lepton mixing parameters . . . . . . . . . . . 189
12.3 The NSM: The SM with singlet fermions . . . . . . . . . . . . . . . . . . . . . . . . 190
12.3.1 Defining the NSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
12.3.2 The NSM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
12.3.3 The NSM spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
12.3.4 The Ni interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
12.3.5 The case of mN v: Sterile neutrinos . . . . . . . . . . . . . . . . . . . . . 196
12.4 Probing neutrino masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.4.1 Neutrino oscillations in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.4.2 The MSW effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
12.4.3 Non-uniform density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
12.4.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Appendices 205
12.A Probing neutrino masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.A.1 Kinematic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.A.2 Neutrinoless double-beta (0ν2β) decay . . . . . . . . . . . . . . . . . . . . . 205
Homework 207
13 Connection to cosmology 210
13.1 Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
13.1.1 The baryon asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
13.1.2 Sakharov conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
13.1.3 The suppression of KM baryogenesis . . . . . . . . . . . . . . . . . . . . . . 213
13.1.4 Leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
13.2 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
13.2.1 Observational Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
13.2.2 Why not the neutrinos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
13.2.3 The relic density: back of the envelope estimate . . . . . . . . . . . . . . . . 218
13.2.4 Detecting WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13.2.5 DM@LHC? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13.2.6 Supersymmetry: neutralino dark matter . . . . . . . . . . . . . . . . . . . . 220
Appendices 222
7
A Lie Groups 223
A.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
A.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
A.3 Lie groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
A.4 Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.5 SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.6 Classification and Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
A.7 Naming representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.8 Combining representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Homework 238
8
Chapter 1
Lagrangians
Modern physics encodes the basic laws of Nature in the action S, and postulates the principle
of minimal action in its quantum interpretation. In Quantum Field Theory (QFT), the action is
an integral over spacetime of the “Lagrangian density” or Lagrangian, L, for short. For most of
our purposes, it is enough to consider the Lagrangian, rather than the action. In this section we
explain how particle physicists “construct” Lagrangians. Later in the book we discuss how they
determine the numerical values of the parameters that appear in the Lagrangian, and how they
test whether a Lagrangian provides a viable description of Nature.
The QFT equivalent of the generalized coordinates of classical mechanics are the fields. The
action is given by
S =∫d4x L[φi(x), ∂µφi(x)] , (1.1)
where d4x = dx0dx1dx2dx3 is the integration measure in four-dimensional Minkowski space. The
index i runs from 1 to the number of fields. Here we denote a generic field by φ(x). Later, we use
φ(x) for a scalar field, ψ(x) for a fermion field, and V (x) for a vector field.
The action S has units of ML2T−1 or, equivalently, h. In a natural unit system, where h = 1, S
is taken to be “dimensionless.” Then, in four dimensions, L has natural dimensions of L−4 = M4.
In general, we require the following properties for the Lagrangian:
(i) It is a function of the fields and their derivatives only, so as to ensure translational invariance.
(ii) It depends on the fields taken at one spacetime point xµ only, leading to a local field theory.
(iii) It is real, so that the total probability is conserved.
(iv) It is invariant under the Poincare group, that is under spacetime translations and Lorentz
transformations.
(v) It is an analytic function in the fields. This is not a general requirement, but it is common to
all field theories that are solved via perturbation theory. In all of these, we expand around
9
a minimum, and this expansion means that we consider a Lagrangian that is a polynomial
in the fields.
(vi) It is invariant under certain internal symmetry groups. The invariance of S (or of L) is
in correspondence with conserved quantities and reflects basic symmetries of the physical
system.
We often impose two additional requirements:
(vii) Naturalness: Every term in the Lagrangian that is not forbidden by a symmetry should
appear.
(viii) Renormalizability. A renormalizable Lagrangian contains only terms that are of dimension
less than or equal to four in the fields and their derivatives.
The requirement of renormalizability ensures that the Lagrangian contains at most two ∂µ oper-
ations, and leads to classical equations of motion that are no higher than second order derivatives.
If the full theory of Nature is described by QFT, its Lagrangian should indeed be renormaliz-
able. The theories that we consider, however, and, in particular, the Standard Model, are only
low energy effective theories, valid up to some energy scale Λ. Therefore, we must include also
non-renormalizable terms. These terms have coefficients with inverse mass dimensions, 1/Λn,
n = 1, 2, . . .. For most purposes, however, the renormalizable terms constitute the leading terms in
an expansion in E/Λ, where E is the energy scale of the physical processes under study. Therefore,
the renormalizable part of the Lagrangian is a good starting point for our study.
Properties (i)-(v) are not the subject of this book. You must be familiar with them from
your QFT course(s). We do, however, deal intensively with the other requirements. Actually, the
most important message that we would like to convey is the following: (Almost) all experimental
data for elementary particles and their interactions can be explained by the standard model of a
spontaneously broken SU(3)× SU(2)× U(1) gauge symmetry.1
Writing down a specific Lagrangian is the endpoint of the process known as “model building,”
and the starting point for a phenomenological interpretation and experimental testing. In this
book we explain both sides of this modern way of understanding high energy physics.
We next show a few examples of simple Lagrangians.
1.1 Scalars
The Lagrangian for a real scalar field φ is given by
LS =1
2∂µφ∂µφ−
m2
2φ2 − µ
2√
2φ3 − λ
4φ4 . (1.2)
1Actually, the great hope of the high-energy physics community is to prove this statement wrong, and to find
an even more fundamental theory.
10
The Lagrangian LS of Eq. (1.2) is the most general renormalizable L(φ) that can be written, so
it satisfies the naturalness principle. We emphasize the following points:
1. The term with derivatives is called the kinetic term. It is necessary if we want φ to be a
dynamical field, namely to be able to describe propagation in spacetime.
2. We work in the “canonically normalized” basis where the coefficient of the kinetic term is
1/2. (This is true for a real scalar field. For a complex scalar field, the canonically normalized
coefficient of the kinetic term is 1.)
3. We do not write a constant term since it does not enter the equation of motion for φ.
4. In principle we could write a linear term but it is not physical, that is, we can always redefine
the field such that the linear term vanishes.
5. The quadratic term (φ2) is a mass-squared term.
6. The trilinear (φ3) and quartic (φ4) terms describe interactions.
7. Terms with five or more scalar fields (φn, n ≥ 5) are non-renormalizable.
1.2 Fermions
The Lagrangian for a Dirac fermion field ψ is given by
LF = iψ∂/ψ −mψψ . (1.3)
The Lagrangian LF of Eq. (1.3) is the most general renormalizable L(ψ) that can be written, so it
satisfies the naturalness principle. (There is a subtlety involved in this statement. By saying that
the fermion in question is of the Dirac type, we are implicitly imposing a symmetry that forbids
Majorana mass terms. We discuss this issue later.) We treat ψ and ψ as independent fields. The
reason is that a fermion field is complex, and it is more convenient to deal with ψ and ψ than with
Re(ψ) and Im(ψ). We emphasize the following points:
1. The derivative term is the kinetic term. It is necessary if we want ψ to be a dynamical field.
2. We work in the canonically normalized basis where the coefficient of the kinetic term is 1.
3. Terms with an odd number of fermion fields violate Lorentz symmetry, and so they are
forbidden.
4. The quadratic term (ψψ) is a mass term.
5. Terms with four or more fermion fields are non-renormalizable.
11
1.3 Fermions and scalars
The renormalizable Lagrangian for a single Dirac fermion and a single real scalar field includes, in
addition to the terms written in Eqs. (1.2) and (1.3), the following term:
LYuk = −Y ψψφ. (1.4)
Such a term is called a Yukawa interaction and Y is the dimensionless Yukawa coupling. The most
general renormalizable Lagrangian for a real scalar field and a Dirac fermion is thus
L(φ, ψ) = LS + LF + LYuk, (1.5)
where LS is given in (1.2), LF is given in (1.3), and LYuk in (1.4).
1.4 Symmetries
We always seek deeper reasons for the laws of Nature that have been discovered. In modern
theories, these reasons are closely related to symmetries. The term symmetry refers to an invariance
of the equations that describe a physical system. The fact that a symmetry and an invariance are
related concepts is obvious enough — a smooth ball has a spherical symmetry and its appearance
is invariant under rotation.
Symmetries are built into physics as invariance properties of the Lagrangian. If we construct
our theories to encode various empirical facts and, in particular, the observed conservation laws,
then the equations turn out to exhibit certain invariance properties. For example, if we want the
theory to have energy conservation, then the Lagrangian must be invariant under time translations
(and therefore cannot depend explicitly on time). From this point of view, the conservation law is
the input and the symmetry is the output.
Conversely, if we take the symmetries to be the fundamental rules, then various observed
features of particles and their interactions are a necessary consequence of the symmetry principle.
In this sense, symmetries provide an explanation of these features. In modern particle physics
(and, in particular, in this book), we often take the latter point of view, in which symmetries are
the input and conservation laws are the output.
In the following we discuss the consequences of imposing a symmetry on a Lagrangian. This
is the starting point of model building in modern particle physics: One defines the basic symme-
tries and the particle content, and then obtains the predictions that follow from these imposed
symmetries.
We emphasize here that there are, however, symmetries that are not imposed and are called
accidental symmetries. They are outputs of the theory rather than external constraints. Accidental
symmetries arise due to the fact that we truncate our Lagrangian. In particular, the renormalizable
12
terms in the Lagrangian often have accidental symmetries that are broken by non-renormalizable
terms or by anomalies. Since we study mostly renormalizable Lagrangians, we will often encounter
accidental symmetries.
There are various types of symmetries. First, we distinguish between spacetime and inter-
nal symmetries. Spacetime symmetries include the Poincare group of translations, rotations and
boosts. They give us the energy–momentum and angular momentum conservation laws. As men-
tioned above, we always impose this symmetry. The list of possible spacetime symmetries includes,
in addition, space inversion (also called parity) P , time-reversal T , and charge conjugation C.
(While C is not truly a spacetime symmetry, the way it acts on fermions and the CPT theorem
make it simpler to include C in the same class of operators.) We discuss them briefly in Section 1.5.
Internal symmetries act on the fields, not directly on spacetime. In other words, they act in
mathematical spaces that are generated by the fields. This is the kind of symmetries that we
discuss in detail. In Chapter 2 we introduce Abelian symmetries. In Chapter 4 we introduce
non-Abelian symmetries.
1.5 Discrete spacetime symmetries: C, P and T
The discrete spacetime symmetries, C, P , and T , play an important role in our understanding of
Nature. Each of these three symmetries has been experimentally shown to be violated in Nature,
as discussed in detail further below. The CPT combination seems, however, to be an exact
symmetry of Nature. On the experimental side, no sign of CPT violation has been observed. On
the theoretical side, CPT must be conserved for any Lorentz invariant local field theory. Since we
only consider such theories, we assume that CPT holds. In this case, CP and T are equivalent.
Thus, we usually refer to CP .
1.5.1 C and P
We only consider C and P in theories that involve fermions. Under C, particles and antiparticles
are interchanged by conjugating all internal quantum numbers, e.g., reversing the sign of the
electromagnetic charge, Q → −Q. Under P , the handedness of space is reversed, ~x → −~x, and
the chirality of fermion fields is reversed, ψL ↔ ψR. For example, a LH electron e−L transforms
under C into a LH positron e+L , and under P into a RH electron e−R. (For a formal discussion see,
for example, Section 3.6 of Ref. [11].) The result that is important for our purposes is that C and
P are violated in any chiral theory. This is the case if there is a different number of LH and RH
fields, or if the LH and RH fermion fields transform in a different way under the symmetry group.
13
1.5.2 CP violation and complex couplings
The CP transformation combines charge conjugation C with parity P . For example, a LH electron
e−L transforms under CP into a RH positron, e+R. CP is a good symmetry if there is a basis where
all the parameters of the Lagrangian are real. We do not prove it here but provide a simple
intuitive explanation of this statement.
Consider a theory with a single scalar, φ, and two sets of N fermions, ψiL and ψiR (i = 1, 2, ..., N).
The Yukawa interactions are given by
−L = YijψLiφψRj + Y ∗ijψRjφ†ψLi, (1.6)
where we write the two hermitian conjugate terms explicitly. The CP transformation of the fields
is defined as follows:
φ→ φ†, ψLi → ψLi, ψRi → ψRi. (1.7)
Therefore, a CP transformation exchanges the operators
ψLiφψRj CP ←→ ψRjφ†ψLi, (1.8)
but leaves their coefficients, Yij and Y ∗ij , unchanged. This means that CP is a symmetry of L if
Yij = Y ∗ij .
In practice, things are more subtle, since one can define the CP transformation in a more
general way than Eq. (1.7):
φ→ eiθφ†, ψiL → eiθLiψiL, ψiR → eiθRiψiR, (1.9)
with θ, θLi, θRi convention-dependent phases. Then, there can be complex couplings, yet CP would
be a good symmetry. The correct statement is that CP is violated if, using all freedom to redefine
the phases of the fields, one cannot find any basis where all couplings are real.
Below we will encounter the idea of gauge interactions. We note that a theory with only gauge
interactions conserves CP as the coupling constants are real.
1.6 Model building
As stated above, writing a Lagrangian is the endpoint of “model building.” The starting point of
our way to build models is to provide as input the following two ingredients:
(i) The symmetry;
(ii) The transformation properties of the various scalar and fermion fields under the symmetry
operation.2
2We use interchangeably the terms ”charge,” ”quantum numbers,” and ”representation” for the transformation
properties. We later explain these terms in more detail.
14
Then we write down the most general renormalizable Lagrangian that depends on the scalar and
fermion fields and is invariant under the symmetry.
A Lagrangian defined in the above way has a finite number of parameters. For a theory with N
parameters, we need to perform N appropriate measurements such that additional measurements,
from the (N + 1)’th and on, test the theory. Note that in principle we never need to determine
the values of the parameters, and just use experimental input to make predictions. In practice,
however, it is usually useful to consider the first N measurements as those that determine the
values of these parameters.
As we go along this book, we repeat the process of model building several times. We will see how
QED (the theory of electromagnetic interactions), QCD (the theory of strong interactions), the
LSM (the theory of electroweak interactions among leptons), and the Standard Model itself, can
be understood in this modern way of thinking, starting from a postulate of symmetry principles.
15
Chapter 2
Abelian symmetries
2.1 Global discrete symmetries
We start with a simple example of imposing an internal discrete symmetry. Consider a real scalar
field φ. The most general Lagrangian we can write is given in Eq. (1.2). We now impose a
symmetry: We demand that L is invariant under φ→ −φ, namely
L(φ) = L(−φ) . (2.1)
L is invariant under this symmetry if µ = 0. Thus, by imposing the symmetry we force µ = 0:
The most general L(φ) that is invariant under φ→ −φ is then
L =1
2∂µφ∂µφ−
m2
2φ2 − λ
4φ4 . (2.2)
What conservation law corresponds to this symmetry? We note that the number of φ-particles
in a system that is described by the Lagrangian of Eq. (2.2) can change, but only by an even
number. Therefore, if we define φ-parity as (−1)n, where n is the number of φ-particles in the
system, this φ-parity is conserved. If we do not impose the symmetry and µ 6= 0, then the number
of particle can change by any integer and φ-parity is not conserved.
It is a useful exercise to describe the symmetry in terms of group theory. Here the relevant
group is Z2. It has two elements that we call even (+) and odd (−). The multiplication table is
very simple:
(+) · (+) = (−) · (−) = (+), (+) · (−) = (−) · (+) = (−). (2.3)
When we say that we impose a Z2 symmetry on L, with the Z2 transformation law φ→ −φ, what
we mean is that L belongs to the even representation of Z2, and φ belongs to the odd representation
of Z2. For L to be Z2-even, each term in L must be Z2-even. Since φ is Z2-odd, we can keep only
terms with even powers of φ. Then we can construct the most general L and it is given by Eq.
(2.2). Later on we use the language of group theory to deal with more complicated situations.
16
2.2 Global continuous symmetries
We now extend our discussion to global continuous symmetries. The idea is that we demand that
L is invariant under rotation in some internal space. While (some of) the fields are not invariant
under rotation in that space, the combinations that appear in the Lagrangian are.
The fact that we can rotate between fields should not come as a surprise. This is nothing
but the idea of generalized coordinates. Any linear combinations of the fields can be used as our
coordinates, not just the one we choose to start with.
Our first example is that of a complex scalar field, φ. A complex field has two degrees of
freedom (DoF). There are two useful ways to write down explicitly the two DoF. First, we can use
a Cartesian form:
φ ≡ 1√2
(φR + iφI) , (2.4)
with φR and φI real scalar fields. The most general renormalizable L(φR, φI) is given by
L =1
2δij∂
µφi∂µφj −m2ij
2φiφj −
µijk
2√
2φiφjφk −
λijk`4φiφjφkφ` , i, j, k, ` = R, I. (2.5)
We impose invariance of the Lagrangian under rotations in the complex plane:(φR
φI
)→ O
(φR
φI
), O =
(cos θ sin θ
− sin θ cos θ
), (2.6)
where θ is a number that does not depend on xµ. Imposing that L(φR, φI) is invariant under the
transformation of Eq. (2.6) forbids many terms and relates other:
L(φR, φI) =1
2∂µφR∂µφR +
1
2∂µφI∂µφI −
m2
2(φRφR + φIφI)−
λ
4
(φ4R + φ4
I + 2φ2Iφ
2R
). (2.7)
In the language of group theory, the imposed symmetry – rotations in a two dimensional real plane
– is called SO(2).
Second, we can formulate the transformation law directly in terms of the complex field φ:
φ→ eiθφ, φ† → e−iθφ†. (2.8)
Imposing that L(φ, φ†) is invariant under the transformation of Eq. (2.8) leads to
L(φ, φ†) = ∂µφ†∂µφ−m2φ†φ− λ(φ†φ)2. (2.9)
In the language of group theory, the imposed symmetry – rotations in a one dimensional complex
plane – is called U(1). (Mathematically SO(2) and U(1) are equivalent. The different names
represent the way we think about the underlying space.) It is easy to check that the Lagrangians
in Eqs. (2.7) and (2.9) are equivalent. Eq. (2.9) is, however, more compact. We emphasize the
following points regarding Eq. (2.9):
17
• All three terms that appear in this equation and, in particular, the mass term, do not
violate any internal symmetry. Thus, there is no way to forbid them by imposing an internal
symmetry.
• We would obtain the same result if we scale θ by any non-zero number. Explicitly, we would
obtain the same Lagrangian with a transformation law φ→ exp(iqθ)φ for any finite q. Since
q is arbitrary, we can choose it to be one, as we did. The situation is different when we have
more than one field, as we discuss next.
2.3 Charge
We are now ready to define charge. We are used to the notion of charge from the specific case
of electromagnetism. In electromagnetism, charge has two aspects: (i) It sets the strength of the
interaction of the fermions with the photon; (ii) It is a conserved quantity. Below we first deal
with the latter point, while the aspect of interaction strength will emerge only when we generalize
our discussion to local symmetries in Section 2.6. Charge conservation is related to symmetry. The
general relation between internal global continuous symmetries and conserved charges is known
as Noether’s theorem. Here we provide a simple example while the general case is explained in
Appendix 4.A.
Consider a theory with two complex scalar fields, φ1 and φ2. To each field φi we assign a real
number qi. We impose a symmetry under the simultaneous phase rotation of both fields:
φ1 → exp(iq1θ)φ1, φ2 → exp(iq2θ)φ2. (2.10)
The conjugate fields transform as follows:
φ†1 → exp(−iq1θ)φ†1, φ†2 → exp(−iq2θ)φ
†2. (2.11)
We say that qi is the charge of the field φi. The charge of the conjugate field φ†i is −qi. While we
can always set one of the charges to 1, we cannot do it for both. The ratio of charges, q2/q1, is
physical.
The charge qi is an input to model building: We assign charges to the fields and write down the
Lagrangian that is invariant under the above rotations. As a concrete example, consider a model
with two complex scalar fields of charges q1 = 1 and q2 = 3. Then the most general renormalizable
Lagrangian that is invariant under Eq. (2.10) is
L = ∂µφ†1∂µφ1 + ∂µφ†2∂µφ2 −m21φ†1φ1 −m2
2φ†2φ2
−λ11(φ†1φ1)2 − λ22(φ†2φ2)2 − λ12(φ†1φ1)(φ†2φ2)− (ηφ31φ†2 + h.c.). (2.12)
A few comments are in order:
18
• All the interactions that are allowed by the symmetry conserve the charge. This can be seen
formally (and most generally) by the Noether’s theorem. It can also be seen for our specific
example by inspecting the Lagrangian of Eq. (2.12) and observing that each term carries an
overall charge zero, and therefore corresponds to creation and annihilation of particles such
that the initial and final charges are equal.
• Since charge is related to the phase shift of a field, it can only be assigned to complex fields.
Conversely, real fields carry no charge.
• There are two terms that are often used in the Physics jargon instead of “charges”: “quantum
numbers” (QN’s) and “representations.” The use of the term QN’s might be confusing as in
quantum mechanics charge is assigned to a particle while QN is assigned to a state. In QFT,
however, we use QN to describe the charges that are assigned to a field. The use of the term
representations will become clear when we discuss non-Abelian symmetries.
2.4 Product groups
In our discussion above, we introduced the simplest continuous Abelian symmetry – U(1). We can
generalize this idea and impose a larger symmetry, [U(1)]N . This gives us another tool for model
building. Consider, for example, a model with two complex scalar fields, where we require invari-
ance of the Lagrangian under two independent rotations by phases θa and θb. Such a symmetry is
called U(1)a × U(1)b. The two fields transform as follows:
φ1 → exp[i(qa1θa + qb1θb)]φ1, φ2 → exp[i(qa2θa + qb2θb)]φ2. (2.13)
The rotation of the conjugate fields is done with the replacement qa,bi → −qa,bi .
Consider the assignment (qa1 , qb1) = (1, 0), and (qa2 , q
b2) = (0, 1). In such a case, the η term of
the Lagrangian of Eq. (2.12) is forbidden, and we have (repeated indices are summed over)
L = ∂µφ†i∂µφi −m2iφ†iφi − λij(φ
†iφi)(φ
†jφj) (2.14)
We see again how imposing a symmetry can be used to forbid terms in the Lagrangian.
The U(1)a × U(1)b symmetry can be defined in various ways: Instead of having θa and θb as
the independent rotation angles, we can use linear combinations of them. For example, we can
use θ± ≡ θa ± θb as the two independent angles. The corresponding charges are q±i = qai ± qbi and
the symmetry is denoted by U(1)+ × U(1)− symmetry. Below we see examples of when such a
redefinition is useful.
It is illuminating to ask if there is a way to obtain the Lagrangian of Eq. (2.14) by imposing
a single U(1) symmetry as in Eq. (2.10). The answer is in the affirmative. For example, we can
19
assign charges q1 = 1 and q2 = 4. Note that this choice allows a term of the form φ41φ†2. The reason
we do not write it is that it is a dimension-five term and therefore non-renormalizable.
This is in fact our first example of an accidental symmetry. The symmetry we impose is a
U(1) symmetry with certain charge assignments, and the resulting Lagrangian has a U(1)× U(1)
symmetry, that is, there is an extra U(1) symmetry that is accidental. In fact, the Lagrangian of
Eq. (2.14) would be the same for any U(1) symmetry with q1 and q2 positive integers, co-prime to
each other, and q1 + q2 ≥ 5. To each such assignment corresponds a different non-renormalizable
term, φq21 φ†q12 , that breaks the [U(1)]2 symmetry down to the U(1) that we imposed.
2.5 Fermion masses
The basic fermion fields are two component Weyl fermions that are generically denoted by ψL and
ψR, where L and R denote, respectively, left-handed and right-handed chirality. They are related
to a Dirac field by
ψR,L =1
2(1± γ5)ψ. (2.15)
Each of ψL and ψR has two degrees of freedom and is a complex field. We can define a phase
transformation of the Weyl fermions:
ψL → exp(iqLθ)ψL, ψR → exp(iqRθ)ψR. (2.16)
The conjugate fields transform as
ψL → exp(−iqLθ)ψL, ψR → exp(−iqRθ)ψR. (2.17)
To understand the consequences of imposing such a U(1) symmetry on the Lagrangian, re-
call that the combination of renormalizability and Lorentz invariance allows only terms with two
fermion fields (or no fermion fields at all). We here focus on the fermion mass terms. There are
two possible mass terms for fermions: Dirac and Majorana.
Dirac masses couple left- and right-handed fields,
LmD = mDψLψR + h.c., (2.18)
where mD is the Dirac mass.
Majorana masses couple a left-handed or a right-handed field to itself. Consider ψR, a right-
handed field. Defining
ψcR = C ψRT, (2.19)
where C is the charge conjugation matrix, a Majorana mass term reads
LmM =mM
2ψcR ψR + h.c., (2.20)
20
where mM is the Majorana mass. Note that ψR and ψcR transform in the same way under all
symmetries. In particular, if ψR → exp(iqRθ)ψR, then we have ψcR → exp(iqRθ)ψcR. Similar
expressions hold for left-handed fields.
There is a classification of symmetries applied to fermions that is relevant for our discussion.
Consider a theory with a single left-handed and a single right-handed fermion fields and an imposed
U(1) symmetry. The symmetry is chiral if qL 6= qR. The symmetry is vectorial if qL = qR. More
generally, if there are many fermion fields, the symmetry is vectorial if all left-handed fields and
all right-handed fields can be matched into pairs with the same charge, qLi = qRi for each i, and
chiral otherwise.
We emphasize the following points regarding Eqs. (2.18) and (2.20):
• Since ψL and ψR are different fields, there are four degrees of freedom with the same Dirac
mass, mD. In contrast, since only one Weyl fermion field is involved in a Majorana mass
term, there are only two degrees of freedom that have the same Majorana mass, mM .
• The relative factor of 1/2 between Majorana and Dirac mass terms is the analog of the similar
factor between the mass-squared terms for a real and complex scalar fields.
• Consider a theory with one or more U(1) symmetries. To allow a Dirac mass, the charges of
ψL and ψR under these symmetries must be opposite, which is the case when q(ψL) = q(ψR).
Thus, to have a Dirac mass term, the fermion has to be in a vector representation of the
symmetry group.
• Since the transformations of ψcR and ψR are the same, a fermion field can have a Majorana
mass only if it is neutral under all U(1) symmetries. In particular, as we discuss below, fields
that carry electric charges cannot acquire Majorana masses. If we include any non-Abelian
group (to be discussed later), a fermion field can have a Majorana mass only if it is in a real
representation of the symmetry group.
• When there are m left-handed fields and n right-handed fields with the same quantum num-
bers, the Dirac mass terms for these fields form an m× n general complex matrix mD:
(mD)ij(ψL)i(ψR)j + h.c.. (2.21)
The mass eigenstates are, for m ≤ n, m Dirac fermions and (n −m) massless right-handed
fermions or, for m ≥ n, n Dirac fermions and (m− n) massless left-handed fermions. In the
SM, as we discuss later, fermion fields are present in three copies with the same quantum
numbers, and the Dirac mass matrices are 3× 3.
• When there are m neutral left-handed fields and n neutral right-handed fields, the mass
terms form an (m+n)× (m+n) symmetric complex matrix Mψ. This matrix consists of an
21
Table 2.1: Dirac and Majorana masses
Dirac Majorana
# of degrees of freedom 4 2
Representation vector neutral
Mass matrix m× n, general (m+ n)× (m+ n), symmetric
SM fermions quarks, charged leptons neutrinos (?)
m×m block of Majorana mass terms for the left-handed fields, an n× n block of Majorana
mass terms for the right-handed fields, and an m× n block of Dirac mass terms:
(ψL ψcR )
(m
(m×m)ML m
(m×n)D
mT (n×m)D m
(n×n)MR
)(ψcL
ψR
). (2.22)
The (m + n) mass eigenstates are Majorana fermions. In the SM, neutrinos are the only
neutral fermions. If they have Majorana masses, then their mass matrix is 3× 3.
We summarize these differences between Dirac and Majorana masses in Table 2.1.
The main lesson that we can draw from these observations is the following: Charged fermions in
a chiral representation are massless. In other words, if we encounter massless fermions in Nature,
there is a way to explain their masslessness from symmetry principles.
2.6 Local symmetries
So far we discussed global symmetries, that is, symmetries where θ is independent of spacetime. In
this section we discuss local symmetries (also called gauge symmetries), that is, symmetries where
the transformation can be different in different space-time points θ(xµ). The implications of local
symmetries are far reaching. The symmetries that are imposed in defining the Standard Model
are all local symmetries.
We generalize Eq. (2.8) and define a local transformation of a complex scalar field:
φ(x)→ eiqθ(x)φ(x). (2.23)
(From here on, we do not write explicitly the transformation of the conjugate field, and we omit the
index µ from xµ.) Note that a global transformation is a special case of the local transformation.
Moreover, all terms in the Lagrangian that do not involve derivatives of fields and which are
invariant under a global symmetry are also invariant under the corresponding local symmetry.
This is, however, not the case for derivative terms. The local transformation of ∂µφ is given by
∂µφ→ ∂µ[eiqθ(x)φ] = eiqθ(x)∂µφ+ iqeiqθ(x)[∂µθ(x)]φ . (2.24)
22
Consequently, the kinetic term of a scalar in not invariant under the local symmetry:
∂µφ† ∂µφ→(∂µφ† − iq[∂µθ(x)]φ†
)(∂µφ+ iq[∂µθ(x)]φ) 6= ∂µφ†∂µφ. (2.25)
The kinetic term violates the local symmetry also in the fermionic case. We learn that in a theory
that includes only scalars and fermions, a local symmetry acting on these scalar and fermion fields
forbids the kinetic terms. A field without a kinetic term is not dynamical and cannot describe the
particles we observe in Nature. Can we have a theory of dynamical scalars and fermions that is
invariant under a local symmetry?
The answer is in the affirmative. To do that, we have to “correct” for the extra terms that arise
in the transformation of the kinetic terms. For the global symmetry case, L remains invariant since
φ and ∂µφ transform in the same way, that is, both are just multiplied by the phase factor exp(iqθ).
Then, we construct all the terms in L as products of φ and φ† or their derivatives. (Recall, φ and
φ† transform with opposite phases.) This procedure gives us an idea of how to solve the situation
for the local case. We should replace ∂µφ with a so-called “covariant” derivative Dµφ and require
that Dµ transforms in such a way that the transformation law for Dµφ is
Dµφ→ eiqθ(x)Dµφ. (2.26)
If we have a globally invariant L(∂µφ, φ), and the transformation law (2.26) applies, then L(Dµφ, φ)
is guaranteed to be locally invariant under the corresponding symmetry.
How do we find what Dµ is? Given the transformation law (2.23), and the desired transforma-
tion law (2.26), let us try
Dµ = ∂µ + igqAµ , (2.27)
where g is a dimensionless constant, and Aµ is a vector field with the following transformation law:
Aµ → Aµ −1
g∂µθ. (2.28)
We leave it as a homework to check that the covariant derivative of a field transforms in the same
way as the field, namely that Eqs. (2.27) and (2.28) indeed lead to (2.26). We thus achieved our
goal of obtaining a locally invariant Lagrangian with dynamical scalar and/or fermion fields. We
do so by taking a Lagrangian L that is invariant under the global symmetry, and replacing ∂µ with
Dµ. The “price” to pay is that we must add vector fields to the model.
The field Aµ is called a gauge field. The constant g is called the coupling constant (for reasons
that will become clear below). Following our principle that we include all the terms that are
allowed, we add a kinetic term for Aµ. To do that, we define the field strength F µν :
F µν = ∂µAν − ∂νAµ. (2.29)
The Lorentz invariant kinetic term for the gauge field is given by
LV = −1
4F µνFµν . (2.30)
23
The Lagrangian (2.30) is the most general renormalizable L(A) that is invariant under the local
symmetry.
Note the following points:
• Given the kinetic term, Aµ is a dynamical field and its excitations are physical particles. For
example, as we show later, the photon is associated with such excitations.
• We work in the canonically normalized basis where the coefficient of the kinetic term is 1/4.
• While a kinetic term is invariant under the local phase transformation, a mass-squared term
– 12m2AµAµ – is not. You will prove it in your homework. Here we just emphasize the
result: Local invariance implies massless gauge fields. Gauge bosons have only two degrees
of freedom.
• The gauge field does not couple to itself, that is, there is no term in the renormalizable
Lagrangian that involves more than two gauge fields. Specifically, a trilinear term is not
Lorentz invariant, while a quartic term violates the local symmetry.
• You may be puzzled by the fact that the transformation law is additive, and not multiplica-
tive, as is the case for the scalar and fermion fields. You should not be entirely surprised.
When we write the transformation law for a scalar field as φ → expiqθ φ, the phase of the
field is shifted, arg(φ)→ arg(φ) + iqθ. The Aµ field transforms like a phase.
• Consider Lagrangian terms involving only scalar and/or fermion fields and no derivatives of
fields. If such a term is invariant under the global symmetry then it is also invariant under the
corresponding local symmetry. This is not the case for the terms involving the gauge fields.
Indeed, a mass-squared term and a quartic interaction term are invariant under the global
U(1) but not under the local U(1). This is related to the fact that the U(1) transformation
law for scalars and fermions involves a multiplicative phase factor, while the one for gauge
fields involves an additive shift.
• The transformation of Eq. (2.28) is familiar from classical electromagnetism, where Aµ is
just the vector potential. In that context, the invariance of the vector potential under (2.28)
is often stated as the fact that only E and B are physical, while Aµ is redundant, and thus
we have an extra freedom that we called the gauge freedom. Here we reverse the logic: we
impose the gauge symmetry and the result is that the photon is massless and that only E
and B are physical.
• If a local symmetry decomposes into several commuting factors, each factor requires its
corresponding gauge field and has its own independent coupling constant. For example, if
the symmetry is a local U(1)a × U(1)b, we must include two massless gauge fields, Aaµ and
Abµ, and their transformation laws involve two independent coupling constants, ga and gb.
24
2.6.1 Charge
The notion of charge under a global symmetry was introduced in Eq. (2.10). The global symmetry
implies charge conservation. In this subsection we show that, in the case of a local symmetry, there
is an additional implication to the charge: it sets the strength of the interaction with the gauge
boson.
Consider a local U(1) where the expression for the covariant derivative of a field of charge q is
given by
Dµ = ∂µ + igqAµ . (2.31)
Take, for example, a fermion field with charge q under the local U(1). Its kinetic term is
iψD/ψ = iψ∂/ψ − gqψA/ψ. (2.32)
The second term is an interaction term between the fermion and the vector field. The strength
of the interaction is given by the gauge coupling constant g times the charge q. In particular, the
larger the charge, the stronger the coupling to the gauge boson.
Two comments are in order with regard to Eqs. (2.31) and (2.32):
1. Unlike the derivative ∂µ, the covariant derivative Dµ depends on the charge q of the field on
which it acts. It is different for fields of different charges.1
2. What appears in a Lagrangian is the combination gq, and not g and q separately. Indeed,
for the Abelian case, one can re-scale the coupling constant g and the charge q such that gq
remains the same, and the physics of the model is unaffected. When we have several fields
of different charges, one should re-scale all of them in the same way. The ratio between
different charges, say q2/q1, is physical and sets the relative strength of the interactions of
φ2 and φ1 with the gauge field Aµ. The situation is different in the non-Abelian case as we
discuss later.
We can summarize the situation as follows. To have a Lagrangian of dynamical scalar and/or
fermion fields that is invariant under a local U(1) symmetry requires the introduction of a vector
boson. The vector boson necessarily interacts with all scalars and fermions that are charged under
the symmetry. Conversely, the gauge boson does not couple to fields that are neutral (q = 0) under
the symmetry. The global symmetry implies charge conservation. The local symmetry implies, in
addition, that the charge sets the strength of the interaction with the corresponding gauge field.
1Perhaps it would have been helpful to denote a U(1)-related covariant derivative by Dµq , emphasizing the q-
dependence. It became, however, customary to keep the q-dependence implicit and write just Dµ and we comply
with this norm of the community.
25
2.7 Summary
We described the first steps in the process of model building for scalars and/or fermions. We need
to provide as input the following two ingredients:
(i) The symmetry;
(ii) The charges of the fermions and scalars.
Then we write down the most general renormalizable Lagrangian that depends on the scalar and
fermion fields and is invariant under the symmetry. If the imposed symmetry is local, corresponding
vector fields must be added.
The most general renormalizable Lagrangian with scalar, fermion and gauge fields can be
decomposed into
L = Lkin + Lψ + LYuk + Lφ. (2.33)
Here Lkin describes the free propagation in spacetime of all dynamical fields, as well as the gauge
interactions, Lψ gives the fermion mass terms, LYuk describes the Yukawa interactions, and Lφgives the scalar potential. In all the examples we will study, our task will be to find the specific
form of each of these four parts of the renormalizable Lagrangian.
In the SM, only local symmetries are imposed. Similarly, in most of the extensions of the SM,
only local and global discrete symmetries are imposed. While it is possible, in principle, to impose
also global continuous symmetries, this is rarely done in current model building. The reason for
that is twofold. First, there are arguments that suggest that continuous global symmetries are
always broken by gravitational effects and thus can only arise as accidental, rather than imposed
symmetries. Second, there is no obvious phenomenological motivation to impose such symmetries.
Thus, in all the models presented in this book that aim to describe Nature, global continuous
symmetries are not imposed.
In the next chapter we show how QED, the theory of electromagnetic interactions, can be
derived by starting from the above principles.
26
Homework
Question 2.1: Charges
Consider a system with three complex fields, φ1, φ3 and φ4, where a field φq carries charge q.
Write all the allowed interaction terms with three and four fields.
Question 2.2: Local symmetries
In order to “gauge” a symmetry we assume that the symmetry transformation parameter is
xµ-dependent, θ = θ(xµ), and the field transformation properties are:
φ(xµ)→ eiqθ(xµ)φ(xµ), Aµ(xµ)→ Aµ(xµ)− 1
g∂µθ(xµ), (2.34)
where g is a coupling constant and q is the charge of the field. We consider a complex field φ with
the following Lagrangian
L = |∂µφ|2 +m2|φ|2 + λ|φ|4 (2.35)
1. Explain why, for any terms that do not involve derivatives, the θ → θ(xµ) substitution has
no effect on the symmetry properties.
2. Show that the kinetic term is not invariant under a local transformation, φ(xµ)→ eiqθ(xµ)φ(xµ).
3. The covariant derivative is given by:
Dµφ = (∂µ + igqAµ)φ. (2.36)
Show that the covariant derivative of the field transforms in the same way as the field itself:
Dµφ→ eiqθ(xµ)Dµφ (2.37)
or equivalently:
Dµ → eiqθ(xµ)Dµe−iqθ(xµ). (2.38)
We can therefore replace the ordinary derivative with a covariant derivative to make the
Lagrangian (2.35) gauge invariant.
27
4. Show that a mass term for A, that is, m2AµAµ is not invariant under the transformation
properties of Eq. (2.34).
5. Write L that is invariant under the local transformation with the inclusion of A and its
kinetic term. Keep terms up to d = 4.
Question 2.3: Chiral symmetry
Consider a single free Dirac fermion
L = iψ ∂/ ψ , (2.39)
and the following transformations:
ψ → eiα ψ, ψ → eiαγ5 ψ . (2.40)
1. Show that the adjoint field transforms as
ψ → ψ e−iα, ψ → ψ eiαγ5 . (2.41)
2. Show that the ψ → eiα ψ transformation is vectorial: ψR,L → eiαψR,L.
3. Show that the ψ → eiαγ5 ψ transformation is chiral: ψR,L → e±iαψR,L.
4. Show that the Lagrangian (2.39) is invariant under both rotations of Eq. (2.40).
5. We now add a dirac mass term to the Lagrangian:
L = ψ[i∂/−m]ψ . (2.42)
Show that the presence of m 6= 0 breaks the chiral symmetry and conserves the vectorial
one. The above result is the source of the statement that we can use chiral symmetries to
forbid mass terms for fermions.
28
Chapter 3
QED
You are familiar with Quantum ElectroDynamics (QED) from your QFT courses, where the QED
Lagrangian is given and its implications are studied. In this Section we introduce QED using tools
of model building: We postulate a symmetry principle and derive the Lagrangian.
3.1 Defining QED
The simplest version of QED is defined as follows:
(i) The symmetry is a local
U(1)EM. (3.1)
(ii) There are two fermion fields,
eL(−1), eR(−1). (3.2)
The sub-indices L,R denote the chirality: left-handed and right-handed, respectively. The
number in parenthesis is the U(1)EM charge.
(iii) There are no scalars.
3.2 The Lagrangian
As discussed above, the most general renormalizable Lagrangian with scalar, fermion, and gauge
fields can be decomposed into
L = Lkin + Lψ + LYuk + Lφ. (3.3)
It is now our task to find the specific form of the Lagrangian made of the fermion fields eL and eR
subject to the U(1)EM gauge symmetry.
29
The imposed U(1)EM gauge symmetry requires that we include a single gauge boson, Aµ (of
charge q = 0) that we call the photon field. The corresponding field strength is given by
F µν = ∂µAν − ∂νAµ. (3.4)
The covariant derivative is
Dµ = ∂µ + ieqAµ, (3.5)
where e is the coupling constant and, specifically for the eL,R fields, q = −1.
Lkin includes the kinetic terms of all the fields:
Lkin = −1
4F µνFµν − ieLD/ eL − ieRD/ eR. (3.6)
Lψ includes a Dirac mass term for the electron fields:
Lψ = meeLeR + h.c. . (3.7)
Finally, since there are no scalar fields in the model,
LYuk = 0, Lφ = 0. (3.8)
3.3 The spectrum
The spectrum of the model that we defined above consists of a massive Dirac fermion of mass me,
and a massless gauge boson – the photon. We call the Dirac fermion “the electron,” and denote it
by e. (It is somewhat unfortunate that the QED coupling constant and the electron field are both
denoted by e. Which of the two options is meant should be clear from the context.)
We emphasize the following points:
• The electron is a Dirac field, which has four DoF.
• The reason we can write a mass term for the electron is that QED is vectorial, that is, the
eL and eR fields are assigned the same charge.
• The masslessness of the photon is a consequence of the gauge symmetry.
3.4 The interactions
Expanding Dµ, we obtain the photon–fermion interaction term:
Lint = eeA/e , (3.9)
30
where we used qe = −1. When the charge of the electron is set by convention to q = −1 (as we do
in our definition of the model), the coupling constant e is related to the fine structure constant as
α =e2
4π. (3.10)
We emphasize the following two points:
• The photon is not charged under QED (qA = 0) and consequently, at tree level, photons do
not interact with photons.
• The QED interaction is vector-like and, consequently, it conserves C, P and CP .
3.5 Parameter counting and tests of QED
The QED Lagrangian has two free parameters: the electron mass, me, and the coupling constant,
e or, equivalently, α ≡ e2/(4π). Both have been measured with impressive accuracy:
me = 0.510998928± 0.000000011 MeV,
α−1 = 137.035999074± 0.000000044. (3.11)
This model has no accidental symmetry.
The fact that the photon is massless is a prediction of QED, independent on the value of its
parameters. The massless photon field should generate a long term potential of the form e2q/r.
This is indeed the form of the Coulomb potential.
Given that the model has two parameters, once two appropriate experiments are carried out
to measure these parameters, one can make predictions for any other QED-related observable.
Examples can be found in several textbooks on QFT, for example, in Chapter 5 of [11].
3.6 QED with more fermions
3.6.1 Two Dirac fermions
We study a generalization of QED where we add a second fermion to the theory. We obtain
interesting lessons regarding symmetries. Our definition of the model is as follows:
(i) The symmetry is a local
U(1)EM. (3.12)
(ii) There are four fermions fields,
`iL(−1), `iR(−1) (i = 1, 2). (3.13)
31
(iii) There are no scalars.
The Lagrangian is a simple extension to the previous model. Since the model has no scalars,
we still have LYuk = Lφ = 0. The canonically-normalized kinetic terms for the fermions read
Lkin = −i`1LD/ `
1L − i`2
LD/ `2L − i`1
RD/ `1R − i`2
RD/ `2R. (3.14)
The fermion mass terms read
Lψ = (`1L `
2L)
(m11 m12
m21 m22
)(`1R
`2R
)+ h.c.. (3.15)
We can always, however, transform to a different basis for the fermion fields,(`1L
`2L
)→(eL
µL
)= VL
(`1L
`2L
),
(`1R
`2R
)→(eR
µR
)= VR
(`1R
`2R
), (3.16)
such that in the new basis the mass matrix is diagonal:
VL
(m11 m12
m21 m22
)V †R =
(me
mµ
). (3.17)
Since we work with canonical kinetic terms, the above rotation keeps the kinetic terms invariant.
The basis where the mass matrix is diagonal is called the mass basis. The fermion states in this
basis are called mass eigenstates. We call the mass eigenstates the electron e and the muon µ
where, by definition, the muon is heavier than the electron, mµ > me.
In the language of the e and µ Dirac fields, the full QED Lagrangian reads
Le,µQED = −1
4F µνFµν − ieD/ e− iµD/µ+meee+mµµµ. (3.18)
3.6.2 Accidental symmetries
An interesting feature of this model is that it exhibits an accidental symmetry. The imposed
symmetry is local U(1)EM symmetry. The Lagrangian is, however, symmetric under a global
U(1)e × U(1)µ symmetry. The conventional charge assigment under this symmetry is
e(+1, 0), µ(0,+1). (3.19)
The symmetry is manifest in Eq. (3.18), where there is no term that involves both e and µ fields.
Thus, an independent phase rotation of each of them represents a symmetry. The global U(1)EM
(which is automatically imposed when imposing local U(1)EM) is, however, part of U(1)e×U(1)µ.
When the phase rotation of the e and µ is the same, that is just the symmetry we imposed.
We can rephrase our findings in the following way. Equivalently to U(1)e × U(1)µ, we can use
U(1)EM × U(1)e−µ, (3.20)
32
with the charges
qEMe = qEM
µ = −1, qe−µe = −qe−µµ = +1. (3.21)
U(1)EM is the imposed symmetry, while U(1)e−µ is an accidental symmetry.
The experimental implication of the accidental symmetry is that QED conserves the muon and
electron numbers. In any QED process, the number of electrons minus the number of positrons is
conserved, and the number of muons minus the number of anti-muons is conserved. The symmetry
we impose – electric charge conservation – is the sum of the above two conservation laws. For
example, the decay process µ− → e−e+e− is allowed by the imposed U(1)EM but violates the
accidental U(1)e−µ symmetry.
The accidental symmetry is broken by higher dimension operators. For example, the dimension-
six term,
eLeReLµR, (3.22)
breaks the accidental symmetry (it carries U(1)e−µ charge of −2), but is still invariant under
the imposed local symmetry (its U(1)EM charge is zero). It allows the QED-forbidden process
µ− → e−e+e−.
So far no decay process that violates the U(1)e−µ symmetry has been observed. In neutrino
oscillation experiments, however, such violation has been observed. We discuss it in more detail
in Chapter 12.
3.6.3 Even more fields
We can add more fields to QED. As long as we add pairs of LH and RH fields with the same
charge, the situation is rather similar to what we described above. There is always a basis where
the mass terms are diagonal, and thus each pair of LH and RH fields can be combined into a Dirac
field that does not interact with the other fields. In this basis
LQED = −1
4F µνFµν − i
∑j
ψjD/ψj −mjψjψj, (3.23)
where j = 1, . . . , N runs over all the Dirac fields. This model has a [U(1)]N symmetry, which, in
this form, is manifest.
Note that not all fields necessarily carry the same U(1)EM charge. (The charge of each field is
implicit in the covariant derivative.) Indeed, the charged elementary fermions known to us come
in three different charges: The charged leptons (such as the electron and the muon) with q` = −1,
the up-type quarks of charge qu = +2/3, and the down-type quarks of charge qd = −1/3. There
are three Dirac fermions of each of these three types. The QED Lagrangian for these nine Dirac
fields has a global [U(1)]9 symmetry. In other words, the number of fermions minus the number
of anti-fermions of a given charge and a given mass is conserved.
33
Homework
Question 3.1: Light by light scattering
One of the well known properties of waves is the superposition principle that states that two
electromagnetic waves do not interact with each other. The superposition principle, however, is
only a classical approximation and it does not hold at the quantum level. In order to understand
this statement we discuss the cross section for the scattering process γγ → γγ.
1. Explain why at tree level (also referred to as the classical level)
σ(γγ → γγ) = 0. (3.24)
2. Within the framework of QED with only an electron field, draw a one loop diagram that
contributes to the above process.
3. To leading order, the cross section was calculated in the 1930s by Euler and Kockel. (For a
review, see, for example, [13].) In the center of mass frame and for photons of energy E, the
differential cross section is given by
dσ
dΩ=
139α4
(180πE)2
E8
m8e
(3 + cos2 θ
)2. (3.25)
Explain the α4 and the 1/m8e factors.
4. Calculate the total cross section.
5. The above result takes into account only an intermediate electron. Estimate the effect of
including the muon.
6. The cross section is numerically very small for visible light. Calculate the probability of
one light by light interaction when two laser beams of visible light (of 500 nm) cross each
others for one second. Assume that the beam cross section is 10−6 m2 and that the density
of photons is 1021 photons per second per m2.
7. At what photon energies do you expect the effect to become significant?
34
Question 3.2: A mirror world
Consider the following model:
1. The symmetry is a local U(1)EM × U(1)D.
2. The fields in the theory are four-component Dirac fermions: e(−1, 0) and d(0,−1).
Our notation is such that the first number is the charge under U(1)EM and the second number is
the charge under U(1)D. We denote the gauge bosons by Aµ and Cµ, respectively. We assume
that the coupling constants are equal, gEM = gD ≡ e.
1. Write down the covariant derivative, Dµ, for e and for d.
2. Draw the Feynman rules for this theory, i.e. draw all vertices from the interaction terms in
the Lagrangian. Be sure to label the fields.
3. We define, as usual, Fµν to be the field strength of Aµ, and we denote by Cµν that of Cµ.
Consider the term CµνFµν . Is that term gauge invariant? Is it Lorentz invariant? Based on
the answers to these two question, can we write it in our Lagrangian?
We now assume that there is no CµνFµν term. In that case we can write L = LEM + LD and
the two sectors are completely decoupled. In particular, the process ee → dd is not allowed as it
connects the two sectors.
4. We now add a scalar field S(−1, 1). Write down the covariant derivative Dµ for it, write down
its most general coupling to the fermions (up to dimension four), and draw the Feynman
rules.
5. Now the process ee→ dd is allowed. Draw a tree level Feynman diagram for this process.
6. We assume that the mass of the scalar, MS is large, that is, MS E, where E is the center
of mass energy of the incoming electron. In this limit, how does σ(ee→ dd) scale with MS?
7. We now add another fermion field, b(0,−2). Write down its couplings to the gauge bosons,
the scalar, and other fermions, and draw the corresponding Feynman rules (again, up to
dimension four).
8. There is no tree level diagram for the ee → bb process, but there are one loop ones. Draw
one such loop diagram.
35
The model that we considered in this homework is representative of a class of models where, in
addition to a sector with the particles and interactions known to us, one adds a sector that is
either completely decoupled from the first sector, or coupled to it only via very heavy degrees of
freedom. The (almost) decoupled sector is often called “dark sector.” Thus, Cµ would be called a
“dark photon”, and its interactions can be termed as “dark QED.”
36
Chapter 4
Non-Abelian symmetries
4.1 Basics
In previous chapters we discussed Abelian symmetries, such as U(1). For this type of symmetries,
also known as commutative symmetries, the result of applying two symmetry transformations does
not depend on the order in which they are applied. In this chapter we discuss non-Abelian sym-
metries, such as SU(2) and SU(3). For this type of symmetries, also known as non-commutative
symmetries, the result of applying two transformations might depend on the order of applying
them. Within the Standard Model, and in models that extend it, both Abelian and non-Abelian
symmetries play an important role.
In order to extend our discussion to non-Abelian symmetries, we use the language of Lie groups.
A short review of Lie groups is provided in an Appendix, aimed for self-study. From here on we
use this formalism assuming that the reader is familiar with it. We now emphasize some of the
differences between transformation laws for Abelian and non-Abelian symmetries.
• Consider a U(1) symmetry and a field φ that has charge q 6= 0 under this symmetry. For
q 6= 0, φ is necessarily a complex field. The transformation law for this field is of the form
φ→ eiqθφ. (4.1)
Thus, for an Abelian symmetry, the transformation law is defined for each single field sep-
arately. A U(1) symmetry operation changes the phase of the field proportionally to the
charge.
• Consider an SU(M) symmetry and a field φ in a representation R of dimension N > 1 under
this symmetry. Here φ is a vector with N components, φi with i = 1, . . . , N . If R is a
complex representation, φ is a complex field, while if R is a real representation, φ is real.
The transformation law for this field is of the form
φi →(eiTaθa
)ijφj. (4.2)
37
Here i, j = 1, . . . , N and a = 1, . . . , (M2 − 1). The Ta’s are the generators of the SU(M)
algebra:
[Ta, Tb] = ifabcTc. (4.3)
For the field φ in the N -dimensional representation R the Ta’s are represented by N × Nmatrices. Thus, for a non-Abelian symmetry, the transformation law is defined for each
multiplet of fields separately. The SU(M) symmetry operations consist of rotations among
the various components within each multiplet.
If the symmetry group is not simple, we can consider an independent rotation by each simple
subgroup. Then, it is convenient to represent the field as a vector under each of the simple Lie
subgroups. When applying a transformation under one subgroup, it does not affect the other ones.
For example, consider an SU(3) × SU(2) group, and a field that is a triplet under SU(3) and a
doublet under SU(2). We can denote the field by φαi, where α = 1, 2, 3 is the SU(3)-triplet index,
and i = 1, 2 is the SU(2)-doublet index. We can write separately the transformation laws under
an SU(3) symmetry transformation and under an SU(2) symmetry transformation:
φαi →(e(i/2)λaθa
)αβφβi,
φαi →(e(i/2)τbθb
)ijφαj. (4.4)
Here 12λa are the eight 3 × 3 Gell-Mann matrices, which are the SU(3) generators in the triplet
representation, and 12τb are the three 2× 2 Pauli matrices, which are the SU(2) generators in the
doublet representation.
4.2 Global symmetries
For a Lagrangian that is invariant under a U(1) symmetry, each term in the Lagrangian must
consist of products of fields such that the sum of their charges is zero. For a Lagrangian that is
invariant under a non-Abelian symmetry, each term must consist of products of various represen-
tations that are contracted into a singlet of the symmetry group.
Consider, for example, two SU(2)-doublets, ρ and σ. Under SU(2), 2 × 2 = 1 + 3. Thus, the
product ρσ can be decomposed into a singlet and a triplet representations of SU(2). If we want to
construct the quadratic term ρσ, we must contract them into the singlet combination, εijρiσj. If
the theory includes also an SU(2)-triplet ∆, then we can write a term ρσ∆. Now, ρ and σ should
be contracted into a triplet, ρiσj − δijρkσk, and then this triplet combination can be contracted
with ∆ to form a singlet (3× 3 = 1 + 3 + 5). In what follows, we do not write these contractions
explicitly. It should be understood that the contraction is the one needed to make each of the
Lagrangian terms a singlet under all imposed symmetries. In the simplest cases there is only one
way to do so.
38
To understand what kind of symmetries can be imposed, consider a model that has N complex
scalar fields φi. The kinetic terms are given by
Lkin = ∂µφ∗i∂
µφi. (4.5)
The symmetry of Lkin is U(N) = SU(N)×U(1), under which the scalar fields transform as a single
N -plet of charge q. The imposed non-Abelian symmetry can only be a subgroup of the SU(N). If
we choose to impose the full SU(N)× U(1), we have
L = ∂µφ†∂µφ−m2φ†φ− λ(φ†φ)2. (4.6)
Here φ transforms as (N)q, the fundamental representation of SU(N) with charge q under the
U(1). What we mean by φ†φ is the contraction of (N)−q × (N)q into (1)0 of SU(N)× U(1).
As another example, consider a model that has N left-handed Weyl fermions ψLi and N right-
handed Weyl fermions ψRi. The kinetic terms are given by
Lkin = iψLi∂/ψLi + iψRi∂/ψRi. (4.7)
The symmetry of Lkin is SU(N)L×SU(N)R×U(1)L×U(1)R, under which the ψL fields transform
as (N, 1)qL,0, while the ψR fields transform as (1, N)0,qR . The imposed non-Abelian symmetry can
only be a subgroup of the SU(N)L×SU(N)R. If we choose to impose the full [SU(N)×U(1)]2, only
the kinetic terms are allowed. If, for example, we impose the vectorial subgroup [SU(N)×U(1)]V ,
under which the left-handed and right-handed fields transform in the same way, a Dirac mass is
allowed. We can now write the Lagrangian in terms of the Dirac field ψ, which transforms as (N)q
under [SU(N)× U(1)]V :
Lkin = iψ∂/ψ −mψψ. (4.8)
Let us introduce a more specific example. Consider a model where we impose an SU(3)
symmetry and introduce a single scalar triplet (3) denoted by φ. The conjugate field, φ†, transform
as an anti-triplet (3). To construct the (renormalizable part of the) Lagrangian we need to find all
possible combinations of two, three or four triplets and anti-triplets that can be contracted into
an SU(3) singlet. At dimension two in the fields there is a single such combination, 3 × 3. At
dimension three in the fields there are two such combinations, 3× 3× 3 and 3× 3× 3. The singlet
in these latter contractions is antisymmetric under the exchange of the (anti-)triplets. This implies
that we cannot form a singlet if we have a single (or even two) (anti-)triplets. The most general
Lagrangian is then
L = ∂µφ†∂µφ−m2φ†φ− λ(φ†φ)2. (4.9)
Let us now consider the case of an SU(3) symmetry and three scalar triplets φi(3), i = 1, 2, 3.
Now there are trilinear scalar couplings:
L = ∂µφ†∂µφ−m2
ijφiφ∗j − λijklφ
†iφjφ
†kφl − [cijkφiφjφk + h.c.]. (4.10)
The m2 and λ terms are totally symmetric in the i, j, k, l indices, while c is anti-symmetric. Thus,
there is only one independent cijk coupling: cijk = εijkc.
39
4.3 Local symmetries
Theories with non-Abelian local symmetries are also called Yang-Mills theories. There are several
similarities to the case of local Abelian symmetries:
• Terms that depend on scalar and fermion fields but not on their derivatives, and which
are invariant under the corresponding global symmetry, are also invariant under the local
symmetry.
• The kinetic terms are not invariant under the local symmetry.
• To achieve such invariance we must add gauge fields, and replace the derivative ∂µφ with a
covariant derivative Dµφ such that it transforms like the field φ.
The more complicated SU(N) transformation law for φ,
φ→ eiTaθa(x)φ, (4.11)
where Ta are the N2 − 1 generators of the SU(N) algebra, leads to a more complicated transfor-
mation law for Dµ:
Dµ → eiTaθa(x)Dµe−iTaθa(x). (4.12)
Since the Ta form the adjoint representation of SU(N), to achieve local invariance we need to
introduce gauge fields, Aµa , in the adjoint representation. The covariant derivative is given by
Dµ = ∂µ + igTaAµa , (4.13)
where g is a dimensionless parameter called the coupling constant. The index a runs from 1 to
N2 − 1. The transformation law for Aµa is given by
Aµa → Aµa − fabcθbAµc −1
g∂µθa. (4.14)
The fact that the non-Abelian gauge field is in the adjoint representation of the gauge group and,
in particular, that – unlike the Abelian case – it is not a singlet, has significant consequences: It
leads to self-interactions of the gauge fields, as we discuss below.
To promote Aµa to a dynamical field, we must introduce a kinetic term. We define F µνa via
[Dµ, Dν ] = igTaFµνa . (4.15)
Then
TaFµνa = ∂µTaA
νa − ∂νTaAµa + ig[Ta, Tb]A
µaA
νb . (4.16)
Using
tr(TaTb) = δab, (4.17)
40
we can rewrite Eq. (4.16) as follows:
F µνa = ∂µAνa − ∂νAµa − gfabcA
µbA
νc . (4.18)
Compared to the Abelian case, Eq. (2.29), the non-Abelian case, Eq. (4.18) has an extra term.
This term is the source of self-interactions of the gauge fields.
We are now ready to find the kinetic term of the non-Abelian gauge fields:
LV = −1
4F µνa Faµν . (4.19)
The Lagrangian (4.19) is the most general renormalizable L(A) that is invariant under the non-
Abelian local symmetry.
Note the following points:
• Given the kinetic term, Aµa is a dynamical field and its excitations are physical particles. For
example, as we show later, the gluon is associated with such excitations.
• We work in the canonically normalized basis where the coefficient of the kinetic term is 1/4.
• While a kinetic term is invariant under the gauge transformation, a mass-squared term –12m2AµaAaµ – is not. Local invariance under non-Abelian symmetry implies massless gauge
fields in the adjoint representation.
• The non-Abelian gauge fields couple to themselves. This can be see by replacing F µνa in Eq.
(4.19) with the explicit expression (4.18), which leads to interaction terms that are trilinear
and quartic in the gauge fields:
Lself−interactions = gfabc(∂µAaν)AµbA
νc −
1
4g2(fabcA
µbA
νc )(fadeAdµAeν). (4.20)
• If the symmetry group decomposes into several commuting factors, each factor has its own
gauge fields in the corresponding adjoint representation and an independent coupling con-
stant. For example, if the symmetry is SU(3) × SU(2) × U(1), we must introduce three
irreducible representations of gauge fields: (8, 1)0 representation, with coupling constant g3,
to make the Lagrangian invariant under the local SU(3); (1, 3)0, with coupling constant g2,
to achieve invariance under local SU(2); and (1, 1)0, with coupling constant g1, to achieve
invariance under local U(1).
The self-interactions of the non-Abelian gauge fields constitute a significant difference with the
Abelian case. As mentioned above, the source of this difference is the fact that the U(1) gauge
fields are neutral under the U(1), while the SU(N) gauge fields are in the adjoint (and not in the
singlet) representation of the SU(N).
41
4.4 Running coupling constants
In QFT, the coupling constants depend on the energy scale. In the Physics jargon, we say that
the coupling constants run. Here we do not discuss the theoretical background of this effect, and
assume that the reader is familiar with it from QFT classes. We just give some basic formulae and
mention some important consequence of the running of the coupling constants.
The running is given by the beta function:
β(g) =∂g
∂ log µ, (4.21)
where µ is the relevant energy scale. The beta function depends on the field content of the
theory. The leading order effects depend only on fields that are charged under the symmetry. The
fact mentioned above, that Abelian gauge fields are neutral under the Abelian symmetry, while
non-Abelian gauge fields are in the adjoint representation of the gauge symmetry, has important
implications on the running of the corresponding coupling constants.
For a local U(1) theory, with coupling constant g1, and with nf chiral fermion fields with charge
|q| = 1, we have
β(g1) =nfg
31
24π2. (4.22)
(This result is often quoted for the case of nf Dirac fermions, where the 24 is replaced by 12.)
For a local SU(N) theory, with coupling constant gN , and with nf fermions in the fundamental
representation and nf fermions in the anti-fundamental representation, we have
β(gN) =(
2nf3− 11N
3
)g3N
16π2. (4.23)
The important difference between the running effects of the Abelian (4.22) and non-Abelian
(4.23) cases is in the sign of the beta function. In the U(1) case, the beta function is always positive.
Consequently, the lower the relevant energy scale, the smaller the coupling. In the SU(N) case it
can assume either sign. In particular, if the number of fermions in not too large, nf < (11/2)N
(and in particular in the pure gauge case), the sign is negative. In this case, the lower the relevant
energy scale, the larger the coupling.
Reliable calculations of experimental observables are possible only in the perturbative regime.
If a coupling grows large, one loses the ability to make accurate predictions. The question of the
running is then closely related to the question of when we can make accurate predictions. For
U(1) theories, this is the case in low energy (so called the IR). For SU(N) theories, if there are
not too many fermions, this is the case at high energies (the UV), while in the IR perturbativity
is lost. We discuss the implications of this situation in the next chapter where we present QCD –
the theory of strong interactions – as a specific and important, example.
42
Appendix
4.A Noether’s theorem
Let φi(x) be a set of fields, i = 1, 2, . . . , N , on which the Lagrangian L(φ) depends. Consider an
infinitesimal change δφi in the fields. This is a symmetry if
L(φ+ δφ) = L(φ). (4.24)
Since L depends only on φ and ∂µφ, we have
δL(φ) = L(φ+ δφ)− L(φ) =δLδφj
δφj +δL
δ(∂µφj)δ(∂µφj). (4.25)
The relation between symmetries and conserved quantities is expressed by Noether’s theorem: To
every symmetry in the Lagrangian there corresponds a conserved current. To prove the theorem,
one uses the equation of motion:
∂µδL
δ(∂µφj)=δLδφj
. (4.26)
The condition for a symmetry is then
∂µ
[δL
δ(∂µφj)
]δφj +
δLδ(∂µφj)
δ(∂µφj) = ∂µ
[δL
δ(∂µφj)δφj
]= 0. (4.27)
Thus, the conserved current (∂µJµ = 0) is
Jµ =δL
δ(∂µφj)δφj. (4.28)
The conserved charge (Q = 0) is given by
Q =∫d3x J0(x). (4.29)
We are interested in unitary transformations,
φ→ φ′ = Uφ, UU † = 1. (4.30)
43
Here φ is a vector with N components, U is an N ×N unitary matrix, and 1 stands for the N ×Nunit matrix. The reason that we are interested in unitary transformation is that they keep the
canonical form of the kinetic terms. A unitary matrix can always be written as
U = eiεaTa
, T a† = T a, (4.31)
where εa are numbers and T a are N × N hermitian matrices. For infinitesimal transformation
(εa 1),
φ′ ≈ (1 + iεaTa)φ =⇒ δφ = iεaT
aφ. (4.32)
A global symmetry is defined by εa = const(x). For internal symmetry, δ(∂µφ) = ∂µ(δφ). Thus,
for an internal global symmetry,
δ(∂µφ) = iεaTa∂µφ. (4.33)
In the physics jargon, we say that ∂µφ transforms like φ. The conserved current is
Jaµ = iδL
δ(∂µφ)T aφ. (4.34)
The matrices T a form am algebra of the symmetry group,
[T a, T b] = ifabcT c. (4.35)
The charges that are associated with these symmetry also satisfy the algebra:
[Qa, Qb] = ifabcQc. (4.36)
Note that T a are N × N matrices, while Qa are operators in the Hilbert space where the theory
lives.
4.A.1 Free massless scalars
Consider N real, free, massless scalar fields φj:
L(φ) =1
2(∂µφ)(∂µφ). (4.37)
The index j is called flavor index, and here and in what follows the summation over it is implicit.
The theory is invariant under the group of orthogonal N × N matrices, which is the group of
rotations in an N -dimensional real vector space. This group is called SO(N). The generators T ajk
are the N(N − 1)/2 independent antisymmetric N ×N imaginary matrices, that is
δφ = iεaTaφ, (4.38)
with T a antisymmetric and imaginary. (It must be imaginary so that δφ is real.) For an internal
global symmetry, the spacetime and the internal symmetry group are unrelated and thus ∂µφ
transforms like φ, that is
δ(∂µφ) = iεaTa(∂µφ). (4.39)
44
Then,
δL =δLδφδφ+
δLδ(∂µφ)
δ(∂µφ) = 0 + (∂µφ) iεaTa(∂µφ) = 0, (4.40)
where in the last step we used the antisymmetry of T a in its flavor indices. The associated conserved
current is then
Jaµ = i∂µφTaφ. (4.41)
The SO(N) group that we have found is the largest possible symmetry for a Lagrangian
involving N real scalar fields. In general, mass and interaction terms will reduce the symmetry to
a subgroup of SO(N). In the presence of such SO(N) breaking terms, an SO(N) transformation
by a broken generator constitutes a change of basis, where the mass and interaction terms change
their form. The SO(N) groups have no important role in the SM. We mention SO(4) when we
discuss the Higgs mechanism. In a more advanced course, you may encounter SO(10) as a grand
unifying group.
4.A.2 Free massless Dirac fermions
Consider N free, massless, spin-12, four-component fermion fields ψj:
L(ψ) = iψ∂/ψ (4.42)
The ψ’s are necessarily complex because of the Dirac structure. The theory is invariant under the
group of unitary N × N matrices. This group is called U(N) = SU(N) × U(1). The generators
are the independent N2 Hermitian matrices:
δψ = iεaTaψ, (4.43)
where T a is a general Hermitian traceless matrix. The generators can be divided into N2 − 1
traceless generators that correspond to the SU(N) algebra, and a single U(1) generator. The
transformation law of ψ is as follows:
δψ = δ(ψ†γ0) = (iεaTaψ)†γ0 = ψ†(−i)ε∗aT a†γ0 = −iψ†γ0εaT
a = −iψεaT a, (4.44)
where we used the hermiticity of T a. The T a matrices and the γµ matrices commute because they
act on different spaces. We also need to derive the transformation property of the derivative. For
an internal symmetry,
δ∂/ψ = ∂/δψ. (4.45)
For an internal global symmetry (εa independent of x)
δ∂/ψ = iεaTa∂/ψ. (4.46)
We then have
δL = δψδLδψ
+ δ∂/ψδLδ∂/ψ
+δLδψδψ +
δLδ∂/ψ
δ∂/ψ . (4.47)
45
We use
δψδLδψ
= (−iψεaT a)(i∂/ψ) = ψεaTa∂/ψ,
δ∂/ψδLδ∂/ψ
= 0,
δLδψδψ = 0,
δLδ∂/ψ
δ∂/ψ = (iψ)(iεaTa∂/ψ) = −ψεaT a∂/ψ, (4.48)
and find that δL = 0. The corresponding conserved current is
Jaµ = ψγµTaψ. (4.49)
The charge associated with U(1),∫d3xψ†ψ, is the fermion number operator.
In fact, the symmetry of N free massless Dirac fermions is larger: [U(N)]2, rather than just a
single U(N). To understand this point, let us define the following projection operators:
P± =1
2(1± γ5). (4.50)
The four-component Dirac fermion can be decomposed into a left-handed (LH) and a right-handed
(RH) Weyl spinor fields, ψL and ψR:
ψL = P−ψ, ψR = P+ψ, ψL = ψP+, ψR = ψP−. (4.51)
The Lagrangian of Eq. (4.42) can be written as follows:
L(ψ) = iψLj∂/ψLj + iψRj∂/ψRj. (4.52)
It follows straightforwardly that this Lagrangian has a symmetry under independent rotations of
the left-chirality and right-chirality fields, namely
[SU(N)× U(1)]L × [SU(N)× U(1)]R. (4.53)
The conserved currents are
JaLµ = ψLγµTaLψL, JaRµ = ψRγµT
aRψR. (4.54)
The symmetry (4.53) is the largest possible symmetry group for a Lagrangian of N Dirac fermions.
In general, mass and interaction terms break the symmetry to a subgroup of (4.53). Transforma-
tions with broken [SU(N)× U(1)]L × [SU(N)× U(1)]R generators constitute a change of basis.
The ψL and ψR are eigenstates of the chirality operator γ5 (with eigenvalues −1 and +1,
respectively). For massless fields, they are also helicity eigenstates. To see that, consider a plane
46
wave traveling in the z direction, p0 = p3, and p1 = p2 = 0. The Dirac equation in momentum
space is p/ψ = 0, so p(γ0 − γ3)ψ = 0, or
γ0ψ = γ3ψ. (4.55)
The spin angular momentum in the z direction is
J3 = σ12/2 = iγ1γ2/2. (4.56)
Then
J3ψL =i
2γ1γ2ψL =
i
2γ0γ0γ1γ2ψL =
i
2γ0γ1γ2γ0ψL =
i
2γ0γ1γ2γ3ψL =
=1
2γ5ψL = −1
2ψL. (4.57)
We learn that ψL describes a massless particle with helicity −1/2. Similarly, ψR describes a
massless particle with helicity +1/2.
4.A.3 Free massive Dirac fermions
Consider N free spin-12, four-component fermion fields ψi with universal mass m:
L = iψ∂/ψ −mψψ = iψL∂/ψL + iψR∂/ψR −mψLψR −mψRψL, (4.58)
where the “flavor” index j is omitted. This Lagrangian is invariant under the symmetry in which
the LH and RH fields rotate together, U(N) = SU(N) × U(1). We learn that a universal mass
term breaks [U(N)]2 → U(N). This U(N) symmetry is actually the one identified in Eq. (4.43),
with the conserved current of Eq. (4.49).
For a general, non-universal, mass term the symmetry is smaller. By performing an [SU(N)×U(1)]L × [SU(N)× U(1)]R transformation, the mass matrix M is modified:
M → VLMV †R, (4.59)
where VL,R are unitary matrices. We can always choose a basis in this way where the mass matrix
is diagonal and real. In this basis,
L = iψL∂/ψL + iψR∂/ψR −mi(ψLiψRi + ψRiψLi) = ψi(i∂/−mi)ψi. (4.60)
The symmetry is [U(1)]N . The conserved currents are
Jµi = ψiγµψi, (4.61)
and the corresponding conserved charges are simply the fermion number for each fermion (mass
eigenstate) separately,
Qi =∫d3xψ†iψi. (4.62)
47
Within the SM, this is the case of lepton flavor symmetry, which ensures that the flavor of the
leptons (namely, e, µ and τ) is conserved. There is also a similar flavor symmetry of the quark
sector, which is respected by the strong and the EM forces but not by the weak force.
The Lagrangian of Eq. (4.60) is the most general renormalizable Lagrangian that includes only
Direc fermion fields [see Eq. (1.3)]. Thus, in the absence of Yukawa interactions, any renormalizable
Lagrangian that has N Dirac-fermions fields has an accidental [U(1)]N symmetry.
48
Homework
Question 4.1: Running and GUT
Add a quesion about running and GUT
Question 4.2: Scalar Lagrangians
Consider a system with eight real scalar fields φi (i = 1, . . . , 8).
1. Write the kinetic term for this system. What is the symmetry of the kinetic term?
2. To impose a U(1) symmetry under which the scalars are charged, we must group the scalars
into U(1) representations. That is, we combine two real scalar fields to form a complex scalar
field. How many complex scalar fields there are?
3. We assign the scalar fields charges 1, 4, 10, 16. Write the most general Lagrangian that
satisfies the requirements we listed in class.
4. The above Lagrangian has an accidental symmetry. What is that accidental symmetry?
Write non-renormalizable terms that break the accidental symmetry completely.
5. We now impose an SO(8) symmetry and assign the eight scalars into a fundamental of SO(8).
That is, they form a vector in eight dimensional real space. Write the most general La-
grangian in this case. Explain why the mass and the interaction terms must be proportional
to a unit matrix.
6. Let us consider only the kinetic and mass terms. When we imposed SO(8) the mass matrix
was proportional to the unit matrix. Now, instead, we take the mass matrix to be
m2 = diag(m21,m
21,m
21,m
21,m
22,m
22,m
22,m
22), m2
1 6= m22. (4.63)
What is the symmetry of this Lagrangian?
49
7. Last, we would like to impose a U(2) symmetry. For this, we think about the model as
having four complex fields and assign each two complex fields into an SU(2) doublet
H1 =1√2
(φ1 + iφ2
φ3 + iφ4
), H2 =
1√2
(φ5 + iφ6
φ7 + iφ8
). (4.64)
The U(1) charge of the two doublets can be different. There is one four-scalar interaction
term that we can write that breaks the SO(8) symmetry. Write this term.
Question 4.3: Discrete sub-symmetries
Consider
L = ψi[(i∂/−m)δij]ψj (4.65)
where the flavor indices i, j run from 1 to 2. This Lagrangian is manifestly invariant under
some discrete symmetries in flavor space. These symmetries, however, are part of the U(2) flavor
symmetry. For example, the symmetry rotation ψi → −ψi can be generated by the U(1) symmetry:
U = eiα with α = π. Find the U(2)=SU(2)×U(1) rotations for the following two symmetries:
(i) ψ1 → ψ1, ψ2 → −ψ2 (4.66)
(ii) ψ1 ↔ ψ2 .
Hints: You can write your answer as a series of several rotations. The following identity can also
be useful
eiασi = I cosα + iσi sinα (4.67)
Question 4.4: Flavor symmetries
Consider a system with 2 Dirac fermions. The most general Lagrangian is
L = ψi(i∂/δij −mij)ψj (4.68)
where the flavor indices i, j run from 1 to 2. The matrix mij has to be Hermitian and thus we
need four real parameters to describe it.
1. The flavor symmetry of this Lagrangian is U(1) × U(1). However, this symmetry is not
manifest. To make it manifest proceed as follows. Write
mdiag = UmU † (4.69)
such that mdiag is diagonal. Define ψ′ = Uψ and show that L in (4.68) is equal to
L = ψ′i[i∂/δij − (mdiag)ij]ψ′j (4.70)
50
2. In (4.70) the two U(1) symmetries are manifest. They are the independent rotations of each
flavor. One simple way to define them is
ψi → eiαiψi for i = 1, 2 (4.71)
Show that these two U(1) symmetries can also be written as eiα0 and eiα3σ3 .
3. When m is proportional to the unit matrix, the flavor symmetry is SU(2)×U(1). In the most
general case the symmetry is only U(1)×U(1). In the above parametrization the two broken
symmetries are those correspond to σ1 and σ2. Show by explicit calculation that they are
broken (it is enough to show it only for one of them).
4. The two broken symmetry generators can be used to generate U of (4.69). Namely, we can
write
U = eiα2σ2eiα1σ1 , (4.72)
such that α1 and α2 are given in terms of the parameters of mij. (We are usually saying that
we used the symmetries to perform basis rotations.) Find α1 and α2. Hint: it may be easier
to perform the two rotations one after the other. First use U1 = eiα1σ1 to make m real and
symmetric, and then use U2 = eiα2σ2 to diagonalize it.
Question 4.5: A system with fermion and scalars
Consider a system with N Dirac fermion fields and one real scalar field, φ. Write the fermions
as a vector in flavor space ψT = (ψ1, ψ2, . . . , ψN). We impose a U(1) symmetry such that the
fermions are charged under this U(1) and the scalar is not. The most general Lagrangian for this
system is
L = ψi[i∂/δij −mij + λijφ]ψj + LS , (4.73)
where i, j are flavor indices, λij are the Yukawa couplings and LS includes the kinetic, mass and
self interaction terms for the scalar field.
1. Generally, the flavor symmetry of (4.73) is U(1). Show that when λij ∝ mij the symmetry
is larger, U(1)N .
2. When we add terms to a Lagrangian that break some of the symmetries the following relation
holds: “The number of new physical parameters is equal to the number of total (physical and
unphysical) new parameters minus the number of broken generators.” Namely, we “use” the
broken generators to remove unphysical parameters. Here we can think about an original
model with only kinetic terms that has an SU(N)×U(1) flavor symmetry, and then the mass
and Yukawa terms were added. Show that the number of physical parameters in the theory
describes by (4.73) is N2 + 1.
51
3. In the N = 1 case you should find that there are two physical parameters, that you can think
of as m and λ. More generally, you see that we can always find a basis where m is diagonal
but λ is not. Show that for N = 2 we can find a basis where λ is real. For N = 3, however,
we cannot do it in the most general case. We can always write any complex number as a
real number and a phase. What is the minimal number of phases that we need in the N = 3
case?
4. Find a way to modify the theory such that mij = 0 but allow for λij 6= 0 in the Lagrangian
(4.73). Explain why this modification must involve a chiral symmetry.
52
Chapter 5
QCD
Quantum ChromoDynamics (QCD), the theory of strong interactions, is our first example of a
non-Abelian theory. The theory, however, has additional interesting aspects. In particular, at low
energy it is not perturbative, and the perturbative approach that we use cannot be applied. In
this chapter we define the theory at the UV in a perturbative way. We discuss the IR aspects in
Chapter 10.
5.1 Defining QCD
QCD is defined as follows:
(i) The symmetry is a local
SU(3)C . (5.1)
(ii) There are six left-handed and six right-handed quark fields,
QLi(3), QRi(3), i = 1, ..., 6. (5.2)
(iii) There are no scalars.
5.2 The Lagrangian
The most general renormalizable Lagrangian with fermion and gauge fields can be decomposed
into
L = Lkin + Lψ. (5.3)
It is now our task to find the specific form of the Lagrangian made of the fermion fields QLi and
QRi subject to the SU(3)C gauge symmetry.
53
The imposed SU(3)C gauge symmetry requires that we include a gauge field in the adjoint
representation of SU(3)C , Gµa , that we call the gluon field. The corresponding field strength is
given by
Gµνa = ∂µGν
a + ∂νGµa − gsfabcG
µbG
νc . (5.4)
Here, gs is the coupling constant, and fabc are the structure constants of SU(3). The covariant
derivative is
Dµ = ∂µ + igsGµaLa, (5.5)
where La, a = 1, . . . , 8, are the SU(3) generators, [La, Lb] = fabcLc. In the triplet representation,
La = 12λa, where λa are the eight 3 × 3 Gell-Mann matrices given explicitly in Eq. (A.19). The
covariant derivative is the same for all the fermions in the theory and is given by
Dµ = ∂µ +i
2gsG
µaλa. (5.6)
Lkin includes the kinetic terms of all the fields:
Lkin = −1
4Gµνa Gaµν − iQLiD/QLi − iQRiD/QRi. (5.7)
Lψ includes Dirac mass terms for the quarks:
Lψ = mQijQLiQRj + h.c. . (5.8)
We can always, however, make a unitary transformation of the fermion fields,
QLi → (VL)ijQLj, QRi → (VR)ijQRj, (5.9)
such that in the new basis the mass matrix is diagonal (this basis is called the mass basis):
VLmQV †R = diag(mu,md,ms,mc,mb,mt). (5.10)
The transformation (5.9) by unitary matrices preserves the canonical normalization of the kinetic
terms. We call the mass eigenstates the up u, down d, strange s, charm c, bottom b and top t
quarks where, by definition, mu < md < ms < mc < mb < mt.
In the language of the quark Dirac mass eigenstate fields, q ≡ (QL, QR)T , the full QCD La-
grangian reads
LQCD = −1
4Gµνa Gaµν − iqD/ q −mqqq, q = u, d, s, c, b, t. (5.11)
5.3 The spectrum
When discussing the spectrum of QCD, the energy scale at which it is probed makes a significant
difference. The reason is that the strong coupling constant (uniquely among the SM coupling
constants) becomes stronger at lower energy or, equivalently, the larger the distance between
54
(anti)quarks. This leads to confinement: quarks and antiquarks bind into color-singlet states.
Here we discuss the short distance spectrum. The IR spectrum is discussed further below.
At short distance, the spectrum of QCD consists of six massive Dirac quarks of mass mq in the
color-triplet representation, and a massless gluon, in the color-octet representation. We emphasize
the following points:
• Since each of the six quarks is a color-triplet Dirac fermion, it has twelve DoF.
• The reason that we can write mass terms for all six quarks is that QCD is vectorial, that is,
the QL and QR fields are both color-triplets.
• The masslessness of the gluon is a consequence of the gauge symmetry. Since it is a color-octet
gauge boson, it has sixteen DoF.
5.4 The interactions
Expanding Dµ, we obtain the quark–gluon interaction term:
Lint = −gs2qiλaG/aqi . (5.12)
Expanding Gµνa Gaµν we obtain the gluon self-interactions:
Lself−int = gsfabc(∂µGν
a)GµbG
νc + g2
s(fabcGµbG
νc )(fadeG
µdG
νe) . (5.13)
We emphasize the following points:
• Both the quark–gluon interaction and the gluon self-interaction depend on the same coupling
constant, gs.
• The quark–gluon interaction is vector-like, diagonal and universal.
• The strong interaction conserves P , T , and C and any combination of them.
The QCD Lagrangian has seven free parameters: the six masses, mq, and the coupling constant,
gs or, equivalently, αs ≡ g2s/(4π). The values of the parameters depend on the scale where they
are measured, and on the renormalization scheme we are using. We do not discuss these issues
here, and just quote the numbers from the PDG:
mu = 2.3+0.7−0.5 MeV, md = 4.8+0.5
−0.3 MeV, ms = 95± 5 MeV, (5.14)
mc = 1.275± 0.025 GeV, mb = 4.18± 0.03 GeV, mt = 160+5−4 GeV.
The value of the coupling constant is given by
αs(m2Z) = 0.1185± 0.0006, (5.15)
where mZ ∼ 91 GeV is the mass of the Z boson that we discuss in Chapter 7.
55
5.4.1 Confinement
We do not observe free quarks in Nature. To explain it one has to postulate that all asymptotic
states are singlets of SU(3)C . This is the confinement hypothesis: quarks, which are SU(3)C
triplets, must be confined within color-singlet bound states.
The confinement hypothesis is consistent with the way the strong coupling runs. The beta
function for the SU(N) coupling constant is given in Eq. (4.23). For QCD, where the gauge group
is SU(3) and the number of relevant quarks at µ > mt is six, we have
β(gs)(µ > mt) = − 7g3s
16π2< 0. (5.16)
For lower energy scales, where the number of relevant quarks is smaller, β(gs) becomes even more
negative, β(gs) = −(33 − 2nlightq )/(48π2), where nlight
q is the number of quarks with mq < µ. The
fact that the beta function is negative is very important. It implies that the strong coupling
constant becomes larger for lower energies. This behavior is what leads to confinement: when two
quarks are traveling away from each other, at some point it becomes energetically favorable to pop
up a quark-antiquark pair from the vacuum. The result is that we do not observe free quarks at
the IR. All quarks are confined into color-singlet objects.
The scale where the transition between the UV and IR description of QCD occurs is called
ΛQCD. Roughly speaking, it is the scale where the coupling constant become strong, gs ≈ 4π. It
occurs around ΛQCD ≈ 300 MeV (The value depends on the renormalization scheme). For most
practical purposes, however, the important question is at what energy scales we can treat QCD as
a perturbative theory. This is the case for process where q2 Λ2QCD.
5.5 Accidental symmetries
The kinetic term of QCD has a large accidental global symmetry, U(6)L×U(6)R. The mass terms
break it, in general, to [U(1)]6:
U(1)u × U(1)d × U(1)s × U(1)c × U(1)b × U(1)t. (5.17)
In the mass basis, Eq. (5.11), this symmetry is manifest.
The following comments are in order:
• The [U(1)]6 symmetry predicts that the quarks are stable.
• More generally, the accidental symmetry implies that quark flavor is conserved. For example,
uu→ cc is allowed, but, for example, uc→ cu is not.
• Both predictions are violated in Nature. In fact, the weak interaction breaks the [U(1)]6 sym-
metry down to a single U(1) called baryon number. This U(1)B is one where all quarks rotate
56
with the same phase. It is also called the diagonal U(1). This symmetry breaking by the
weak interactions allows the QCD-forbidden processes to be mediated by weak interactions.
• At and below the scale of present experiments, the weak interactions are considerably weaker
than the strong interactions. Thus, the QCD Lagrangian and its accidental [U(1)]6 symmetry
constitute a good approximation.
• At low energies, the QCD Lagrangian has even larger approximate accidental symmetries:
isospin, SU(3)-flavor and heavy quark symmetry. These are discussed later.
• Similarly to QED, also QCD conserves C, P and CP since it is a vectorial theory with masses
and gauge interaction only.
5.6 Combining QCD with QED
In this section we combine QCD, the local SU(3)C theory of strong interactions, with QED, the
local U(1)EM theory of the electromagnetic (EM) interactions. The model describes the quark
spectrum and interactions in Nature, neglecting the weak interactions.
The model is defined as follows:
(i) The symmetry is a local
SU(3)C × U(1)EM. (5.18)
(ii) There are six left-handed and six right-handed quark fields,
ULi(3)+2/3, URi(3)+2/3, DLi(3)−1/3, DRi(3)−1/3, i = 1, ..., 3. (5.19)
(iii) There are no scalars.
The resulting Lagrangian combines QED with QCD. The imposed symmetry requires that we
include the following nine gauge field DoF:
Gµa(8)0, Aµ(1)0. (5.20)
The covariant derivative acting on the fermions fields is given by
Dµ = ∂µ +i
2gsG
µaλa + ieqAµ, (5.21)
where q = +2/3(−1/3) for the Ui(Di) fields.
Lkin includes the kinetic terms of all the fields:
Lkin = −1
4Gµνa Gaµν −
1
4F µνFaµν − iULiD/ULi − iURiD/URi.− iDLiD/DLi − iDRiD/DRi. (5.22)
57
From the kinetic term we read off the gauge interactions. The gluon interactions are identical to
the pure QCD Lagrangian and the photon interactions are identical to the pure QED Lagrangian
(except that the quark charges are different from those of the electron). The mass terms are the
same as in the QCD Lagrangian. In terms of the Dirac mass eigenstate quark fields, they read
Lψ = −muuu−mccc−mttt−mddd−msss−mbbb, (5.23)
where u, c, t are the three q = +2/3 fields and d, s, b are the three q = −1/3 fields.
The resulting theory is not much different from just putting together LQCD +LQED. The global
symmetry of the kinetic terms is now [U(3)]4 (instead of the [U(6)]2 symmetry of the pure QCD
case), but the mass terms break it to the same symmetry of QCD, [U(1)]6.
We can also add the three charged leptons to this model. They are in the (1)−1 representation
of SU(3)C × U(1)EM. The terms that involve them are the same as in Eq. (3.18).
58
Homework
Question 5.1: β function
In class we discussed the idea that coupling constants run, that is, that higher order corrections
can be absorbed into the definition of the coupling constant. For example, consider the cross section
for e+e− → µ+µ−. In the CM frame and to leading order, this is given by (up to normalization
factors)
σ =α2
E4, (5.24)
where E is the energy of the electron. Higher order effects change this result. Most of the effect
can be absorbed into the running of α, that is
σ =α(µ)2
E4, (5.25)
where µ ∼ E is the energy scale in the problem, and α(µ) is a running coupling constant that
satifies the differential equation:∂α
∂ log(µ)= β(α), (5.26)
where the beta function can be calculated to the desired precision in perturbation theory. In QED
at one loop with only electrons in the loop we have
β(α) = Bα2, B =2
3π. (5.27)
1. Verify that the solution of the beta function equation is
1
α(µ1)=
1
α(µ2)+B log
(µ2
µ1
). (5.28)
2. Use α(me) ≈ 1/137.0 to calculate α(mZ) where mZ ∼ 91 GeV.
3. Find the Landau pole, that is, find µ where α(µ) → ∞. What can you say about this
infinity?
59
Measurements found that α(mZ) ≈ 1/128. The reason for the disagreement with your result above
is that there are other particles in the loop beside the electrons. The generalization of eq. (5.28) is
1
α(µ1)=
1
α(µ2)+B
∑i
Q2iN
iC log
(mi
µ1
). (5.29)
where the sum over i is the sum of all the charged particles with mass below µ1, i.e. we assume
µ2 ≤ mi < µ1. N iC = 1(3) for leptons (quarks), and Qi is the electric charge.
4. Give a physical argument why we only sum over particles with mass less than µ1.
5. Use the physical masses and charges of the known fermions
q = −1 : m` ∼ (0.5, 100, 1777) MeV,
q = 2/3 : mu ∼ (0.3, 1.4, 174) GeV,
q = −1/3 : md ∼ (0.3, 0.4, 4.2) GeV, (5.30)
and calculate α(mZ). How close is your number to the measured value? (Note that mu, md
and ms are much larger than what you find in PDG, because the correct quark masses to
use here are “valence quark masses”, which is not what the PDG gives.)
We now move to QCD where the beta function is more complicated
B = −(
11− 2nf3
)1
2π(5.31)
where nf is the number of quark flavors with masses below the relevant scale. Below we use the
input αs(mZ) ∼ 0.12
6. Note that the sign of the beta function changes sign depending on the number of flavors.
How many flavors are needed to change the sign of the beta function?
7. Sketch the shape of the function αs(µ) for µ between 1 and 104 GeV for (i) a theory with
fewer flavors, and (ii) a theory with more flavors than this critical value. Use log scale for µ.
8. Estimate ΛQCD, that is, the scale where αs = 1. For simplicity, you can neglect all quark
masses but that of the top.
9. The proton mass is very roughly mP ≈ 3ΛQCD. Can you tell if the mass of the proton would
be lighter or heavier if we did not have the third generation, assuming the same measured
value of αs(mZ)?
60
Chapter 6
Spontaneous Symmetry Breaking
The notion of broken symmetries may seem strange: In what sense is there a difference between
the case that we call “broken symmetry” and the case of not having the symmetry at all? The
idea of a broken symmetry is however meaningful in two scenarios:
• Explicit breaking of a symmetry by a small parameter. The Lagrangian includes terms
that break the symmetry, but these terms are characterized by a small parameter. The small
parameter can be either a small dimensionless coupling, or a small ratio between mass scales.
The symmetry is then approximate, and one can obtain selection rules for processes that are
forbidden in the symmetry limit.
• Spontaneous symmetry breaking (SSB). The Lagrangian is symmetric, but the vacuum state
is not. Even though with SSB there is no longer a conserved charge, the number of parameters
is the same as in the case of unbroken symmetry. In this sense, the predictive power of a
spontaneously broken symmetry is as strong as that of the unbroken symmetry.
In this chapter we introduce the idea of spontaneous symmetry breaking and analyze its conse-
quences.
SSB is based on the following ingredients. Symmetries of interactions are determined by the
symmetries of the Lagrangian. The states, however, do not have to obey these symmetries. Con-
sider, for example, the hydrogen atom. While the Lagrangian is invariant under rotations, an
eigenstate does not have to be. Any state with a finite m quantum number is not invariant under
rotation around the z-axis. This is a general case when we have degenerate states.
In perturbative QFT we always expand around the lowest energy state. This lowest state is
called the “vacuum” state. When the vacuum state is degenerate, the fact that we can expand
around any of the vacua, and the physics would be the same, is a consequence of the symmetry.
Yet, when we expand around a specific vacuum state out of the degenerate set of vacua, we expand
around a state that does not conserve the symmetry.
61
The name “spontaneously broken” indicates that there is no preference as to which of the states
is chosen. A very simple example is that of a hungry donkey. Consider a donkey that stands exactly
halfway between two stacks of hay. Symmetry tells us that it costs the same amount of energy to
go to either stack. Thus, the donkey cannot choose where to go and would not go anywhere! Yet,
in reality, the donkey would make an arbitrary choice and go to one of the stacks to eat. We say
that the donkey spontaneously breaks the symmetry between the two sides.
In previous chapters we encountered the predictive power of imposed symmetries. In this
chapter we show that spontaneously broken symmetries are no less predictive than exact ones,
though the predictions are different. While the symmetry is no longer manifest, in the sense that
processes that are forbidden in the symmetry limit may become allowed if it is spontaneously
broken, there are subtle relations between these ‘forbidden’ processes and the allowed ones which
reveal that the Lagrangian does have this symmetry. This is why a spontaneously broken symmetry
is also called a hidden symmetry.
6.1 Global discrete symmetries
Consider a model with a single real scalar field φ, where we impose φ-parity:
φ→ −φ. (6.1)
The Lagrangian reads
L =1
2(∂µφ)(∂µφ)− µ2
2φ2 − λ
4φ4. (6.2)
In particular, the symmetry forbids a φ3 term.
Hermiticity of L requires that µ2 and λ are real. The scalar potential should be physically
relevant, so we must have λ > 0. As for the µ2 term, we can have either µ2 > 0 or µ2 < 0. For
µ2 > 0 we have an ordinary φ4 theory, and µ2 is the mass-squared of φ. The case of interest for
our purposes is
µ2 < 0. (6.3)
The minimum of the scalar potential should satisfy
0 =∂V
∂φ= φ(µ2 + λφ2). (6.4)
Thus, the potential has two possible minima:
φ± = ±√−µ2
λ≡ ±v. (6.5)
The classical solution would be either φ+ or φ−. We say that φ acquires a vacuum expectation
value (VEV):
〈φ〉 ≡ 〈0|φ|0〉 6= 0. (6.6)
62
Perturbative calculations should involve expansions around the classical minimum. Since the
two solutions are physically equivalent, the physics cannot depend on our choice, but we must
make a choice. Let us choose — without loss of generality — to expand around φ+. We define a
field φ′ with a vanishing VEV:
φ′ = φ− v. (6.7)
In terms of φ′, the Lagrangian reads
L =1
2(∂µφ
′)(∂µφ′)− 1
2(2λv2)φ′2 − λvφ′3 − λ
4φ′4, (6.8)
where we used µ2 = −λv2 and discarded a constant term.
Let us emphasize several points:
(i) The Lagrangian (6.8) includes all possible terms for the real scalar field φ′. In particular,
it has no φ′-parity symmetry. Thus, the φ → −φ symmetry is hidden. It is spontaneously
broken by our choice of the ground state 〈φ〉 = +v.
(ii) Yet, the Lagrangian (6.8) is not the most general renormalizable L(φ′). While the general
L(φ′) depends on three independent parameters [see Eq. (1.2)], (6.8) depends on only two.
The two parameters can be chosen to be v and λ, or µ2 and λ. The first choice is the one we
made in writing Eq. (6.8). The second choice employs the same parameters of the original,
manifestly symmetric L(φ), see Eq. (6.2). It demonstrates that the SSB does not introduce
additional new parameters.
(iii) The coefficients of the quadratic and trilinear terms in (6.8) are different from those of the
quadratic and trilinear terms in Eq. (6.2). In contrast, the coefficients of the quartic terms
are the same. This is a general result: as long as we consider only the renormalizable terms,
SSB changes dimensionful parameters, but not dimensionless ones.
(iv) While the symmetry is manifest in Eq. (6.2), the phenomenological interpretation of this
model should start from Eq. (6.8). Specifically, the model describes a scalar particle of
mass-squared 2λv2 = −2µ2. This particle is an excitation of the φ′ field.
The fact that the three terms in the scalar potential — the mass term, the trilinear term and
the quartic term — depend on only two parameters means that there is a relation between the
three couplings. In terms of the parameters of the general Lagrangian (1.2), the relation is [note
that µ is here the coefficient of the trilinear term, different from µ of Eq. (6.2)]
µ2 = 4λm2. (6.9)
This relation is the clue that the symmetry is spontaneously, rather than explicitly, broken.
63
6.2 Global Abelian continuous symmetries
Consider a model with a complex scalar field φ, with q = 1 under an imposed U(1) symmetry, and
thus its tranformation is
φ→ eiθφ. (6.10)
The Lagrangian reads
L = (∂µφ†)(∂µφ)− µ2φ†φ− λ(φ†φ)2. (6.11)
Equivalently, we can rewrite the Lagrangian in terms of two real scalar fields, h and ξ, where
φ = (h+ iξ)/√
2, (6.12)
and impose an SO(2) symmetry,(h
ξ
)→(h
ξ
)′=
(cos θ sin θ
− sin θ cos θ
)(h
ξ
). (6.13)
The Lagrangian reads
L =1
2(∂µh)(∂µh) +
1
2(∂µξ)(∂
µξ)− µ2
2(h2 + ξ2)− λ
4(h2 + ξ2)2. (6.14)
The µ2 and λ parameters are real, and we must have λ > 0. We consider the case that µ2 < 0.
We define v2 = −µ2/λ. The scalar potential can be written (up to a constant term) as
V = λ
(φ†φ− v2
2
)2
. (6.15)
Thus, φ acquires a VEV:
2〈φ†φ〉 = 〈h2 + ξ2〉 = v2 = −µ2
λ. (6.16)
In the (h, ξ) plane, there is a circle of radius v that corresponds to minima of the potential. We
now have to choose a specific vacuum to expand around it. We choose the real component of φ to
carry the VEV (〈Im φ〉 = 0):
〈h〉 = v, 〈ξ〉 = 0. (6.17)
We define the real scalar fields
h′ = h− v, ξ′ = ξ (6.18)
with vanishing VEVs:
〈h′〉 = 〈ξ′〉 = 0. (6.19)
We obtain the Lagrangian in terms of h′ and ξ′:
L =1
2(∂µh
′)(∂µh′) +1
2(∂µξ
′)(∂µξ′)− λv2h′2 − λvh′(h′2 + ξ′2)− λ
4(h′2 + ξ′2)2. (6.20)
Note the following points:
64
1. The SO(2) symmetry is spontaneously broken. This can be seen from the presence of the
h′(h′2 + ξ′2) term.
2. The Lagrangian describes one massive scalar, h′, with m2 = 2λv2, and one massless scalar,
ξ′.
3. The Lagrangian of Eq. (6.20) is not the most general Lagrangian for two real scalar fields.
Many terms are missing, while others, that would have been independent in the general case,
are related. In particular, there are only two independent parameters, as for a Lagrangian
with an unbroken SO(2).
4. The quartic terms, with dimensionless couplings, are the same in (6.14) and (6.20). Only
dimensionful couplings are modified.
5. If the symmetry were not broken, it would be impossible distinguish the two components
of the complex scalar field. With the symmetry spontaneously broken, these two DoF are
distinguishable. For example, they have different masses.
6. We chose a basis by assigning the VEV to the real component of φ. This is an arbitrary
choice. We made it since it is convenient. The physics does not depend on this choice.
7. Since the symmetry is (spontaneously) broken, the Lagrangian is no longer invariant under
the transformation (6.13). This transformation can be viewed as fixing θ or a choice of basis
(where the VEV is real and positive).
8. A particle with negative mass-squared is called a tachyon and it travels faster than light. The
example above shows why tachyons do not appear in QFT. For the physical interpretation, we
have to expand around the minimum, and thus physical particles have positive mass-squared.
Fields with negative mass-squared are sometimes referred to as tachyonic fields.
One of the most interesting features of the model presented here is the existence of a mass-
less scalar field. This result is not particular to our specific model, but the result of a general
theorem: The spontaneous breaking of a continuous global symmetries is always accompanied by
the appearance of a massless scalars called Goldstone Bosons. We discuss it in more detail in
Appendix 6.A.
Here we only briefly describe the intuition of that theorem. We can get a SSB only when the
vacuum is degenerate, and in a case of a continuous symmetry, this degeneracy is also continuous.
In the case of a U(1) symmetry the shape of the potential is usually called “a Mexican hat.” In this
case when expanding around any point in the valley it can be seen that one direction is flat. A flat
direction in the potential correspond to a massless DoF. The Goldstone theorem is a generalization
of this simple picture. Fig. 6.1 demostrate this point.
65
Figure 6.1: The Mexcian hat potential. The masses of the two DoFs correspond to the second
derivative of the potential. We see that one of them (the pic on the left) is flat, while the other
one (on the right) is not flat. In the global case these two corresponds to the Goldstone boson and
the massive DoF. In the local case the flat direction is the one “eaten”by the gauge boson and the
massive oen is the Higgs boson. (The plots are taken from [9].)
6.3 Global non-Abelian continuous symmetries
The Non-Abelian case is similar to the Abelian one, with a significant different, that is, that the
symmetry may not be totaly broken. Since there are more than one generator, some of them may
not be broken by the vacuum. The exact pattern of symmetry breaking depends on the specific
potential and the represenations of the scalar fields.
A generator corresponds to a broken symmetry if the vacuum is not invariant under an operation
by the corresponding group elements. On the contrary, a generator corresponds to an unbroken
symmetry if the vacuum is invariant to an operation by the corresponding group elements. Given
the fact that the group elements is the exponet of the generator, we see that the condition can be
represent in terms of the generators as follows. We denote the the vacuum state by 〈φ〉. A broken
generator gives
Ta〈φ〉 6= 0 (6.21)
while an unbroken one has
Ta〈φ〉 = 0 (6.22)
More details are given in the homework.
We now give two examples where we discuss breaking of SU(2) by a triplet and by a doublet.
Breaking by a triple
Consider a model with an imposed SO(3) symmetry, and a scalar field, φ, that transform as a
triple the symmetry. This is one example where we cn get intution in the non-abelian case. What
we have here is a vector in a real 3d space. Once a vector is places in space the full 3d rotation
66
breaks into a rotation in a plane that is perpendicular to the vector. Here we show this inturion
explicitly.
Given the fact that φ is a triple it transform as
φ→ e(i/2)Laθaφ. (6.23)
where (La)bc = εabc are the triplet representation of the SO(3) algebra.
The Lagrangian reads
L = Lkin −µ2
2φ†φ− λ
4(φ†φ)2. (6.24)
We take µ2 < 0, and define v2 = −µ2/λ. Then φ acquires a VEV: |〈φ〉| = v. The triple φ has
three DoFs. We choose a basis and define the vev to be in the z direction
φ =
φ1
φ2
v + φ3
, (6.25)
such that 〈φi〉 = 0, i = 1, 2, 3.
The Lagrangian for the φi fields can be written as
L = Lkin + L2 + L3 + L4, (6.26)
where Ln includes terms that are n’th power in the φi fields. Let us comment on the significance
of each of these parts of the Lagrangian:
• Quadratic terms:
L2 = −λv2φ23. (6.27)
We learn that the model has one massive scalar, φ3, of mass-squared m23 = 2λv2, and
two massless scalars, m21 = m2
2 = 0. This is a manifestation of the Goldstone theorem.
Spontaneous symmetry breaking requires the appearance of massless Goldstone bosons in
correspondence to the broken generators. Since the SO(3) algebra has three generators and
it is brokn to SO(2) that has one generator our model must have two Goldstone bosons.
• Trilinear terms:
L3 = λvφ1φiφi. (6.28)
The fact that L3 6= 0 is a manifestation of the SO(3) breaking.
• The quartic terms:
L4 = −(λ/4)(φ21 + φ2
2 + φ23)2. (6.29)
This part of the Lagrangian has dimensionless couplings and therefore it is unchanged from
the symmetric form.
67
We can check that the unbroken generator is indeed rotation around the z axis. Write Lz
explicitly, we see that it anihhilate the vacuum0, 1, 0
−1, 0, 0
0, 0, 0
0
0
v
= 0 (6.30)
We elaborate on this in the homework.
Breaking by a complex doublet
Consider a model with an imposed SU(2) symmetry, and a complex scalar field, φ, that transform
as a doublet under the symmetry:
φ→ e(i/2)τaθaφ. (6.31)
The Lagrangian reads
L = Lkin − µ2φ†φ− λ(φ†φ)2. (6.32)
We take µ2 < 0, and define v2 = −µ2/λ. Then φ acquires a VEV: |〈φ〉| = v/√
2. The complex
doublet φ has four DoFs. We choose a basis and define
φ =1√2
(φ3 + iφ4
v + φ1 + iφ2
), (6.33)
such that 〈φi〉 = 0, i = 1, 2, 3, 4. The Lagrangian for the φi fields can be written as
L = Lkin + L2 + L3 + L4, (6.34)
where Ln includes terms that are n’th power in the φi fields. Let us comment on the significance
of each of these parts of the Lagrangian:
• Quadratic terms:
L2 = −λv2φ21. (6.35)
We learn that the model has one massive scalar, φ1, of mass-squared m21 = 2λv2, and three
massless scalars, m22,3,4 = 0. This is a manifestation of the Goldstone theorem. Spontaneous
symmetry breaking requires the appearance of massless Goldstone bosons in correspondence
to the broken generators. Since the SU(2) algebra has three generators, our model must
have three Goldstone bosons.
• Trilinear terms:
L3 = λvφ1φiφi. (6.36)
The fact that L3 6= 0 is a manifestation of the SU(2) breaking.
68
• The quartic terms:
L4 = −(λ/4)(φ21 + φ2
2 + φ23 + φ2
4)2. (6.37)
This part of the Lagrangian has dimensionless couplings and therefore it is unchanged from
the symmetric form.
The symmetry structure of this model is worth a further discussion. The complex doublet φ
has four real DoF. If we write φ of Eq. (6.31) in the form
φ =1√2
(φ3 + iφ4
φ1 + iφ2
), (6.38)
it becomes clear that the Lagrangian of Eq. (6.32) depends only on the combination (φ21 + φ2
2 +
φ23 + φ2
4) and consequently has an accidental symmetry such that the overall symmetry is SO(4).
The VEV of φ breaks SO(4) → SO(3). This can be seen from the fact that the Lagrangian of
Eqs. (6.34)–(6.37) depends on φ2,3,4 only in the combination (φ22 +φ2
3 +φ24). The SO(4) group has
six generators while SO(3) has three, hence the appearance of three Goldstone bosons.
6.4 Fermion masses
Spontaneous symmetry breaking can give masses to chiral fermions. We explain this statement by
an explicit example.
Consider a model with a U(1) symmetry. The field content consists of a left-handed fermion
ψL, a right-handed fermion ψR, and a complex scalar φ with the following U(1) charges:
q (ψL) = 1, q (ψR) = 2, q (φ) = 1. (6.39)
The most general Lagrangian we can write is
L = Lkin − µ2φ†φ− λ(φ†φ)2 − Y φψRψL + h.c.. (6.40)
Since the fermions are charged and chiral, we cannot write mass terms for them (Lψ = 0).
We take µ2 < 0, so that the scalar potential is the one given in Eq. (6.14), leading to a VEV
for φ: |〈φ〉| = v/√
2 6= 0. As above, we choose 〈Reφ〉 = v/√
2, 〈Imφ〉 = 0. We define the real
fields h and ξ in such a way that they have vanishing VEVS:
φ =h+ v + iξ√
2, (6.41)
Expanding around the chosen vacuum we find
L = Lkin − V (h, ξ)− Y v√2ψRψL −
Y√2
(h+ iξ)ψRψL + h.c. , (6.42)
69
where V (h, ξ) can be read off Eq. (6.20). We learn that ψL and ψR combined to form a Dirac
fermion with mass
mψ =Y v√
2. (6.43)
This is possible because the symmetry under which the fermion is chiral is completely broken. The
two real scalar fields, h and ξ, couple to the fermion with same Yukawa coupling, and the coupling
is proportional to the fermion mass.
In a more general case, the symmetry might be only partially broken, namely a subgroup of
the original group remains unbroken. In this case, a necessary condition for generating fermion
masses is that the fermion representation is vector-like under the unbroken subgroup.
6.5 Local symmetries: the Higgs mechanism
In this section we discuss spontaneous breaking of local symmetries. We demonstrate it by studying
a U(1) gauge symmetry. We will find out that breaking of a local symmetry results in mass terms
for the gauge bosons that correspond to the broken generators. It is a somewhat surprising result,
since the spontaneous breaking of a global symmetry gives massless Goldstone bosons. In the
case of a local symmetry, these would-be Goldstone bosons are “eaten” by the gauge bosons and
become the longitudinal components of the resulting massive vector-bosons.
Consider a theory similar to the one discussed in Section 6.2, where we have a single scalar
field that is charged under a U(1) symmetry. The difference is that here we impose a local U(1)
symmetry, that is
φ→ eiθ(x)φ. (6.44)
The Lagrangian is given by
L = (Dµφ)†(Dµφ)− 1
4FµνF
µν − µ2φ†φ− λ(φ†φ)2. (6.45)
The covariant derivative is given by
Dµφ = (∂µ + igAµ)φ , (6.46)
Aµ is the gauge field, Fµν is defined in Eq. (2.29), and g is the coupling constant.
We consider the case of µ2 < 0, leading to SSB via a VEV of φ:
〈φ〉 =v√2, v2 = −µ
2
λ. (6.47)
We choose the real component of φ to carry the VEV. We again write the complex scalar in terms
of two scalar fields with vanishing VEVs, 〈h〉 = 〈ξ〉 = 0 but, unlike the global case, it is convenient
to write the two DoFs as a phase, ξ(x) and a magnitude, h(x):
φ(x) = eiξ(x)/v v + h(x)√2
. (6.48)
70
Note that we normalized ξ(x) such that it has mass dimension one. To leading order Eq. (6.48) is
the same as Eq. (6.41). We usually refer to Eq. (6.48) as a non-linear realization and to Eq. (6.41)
as a linear realization.
When a symmetry is spontaneously broken, the Lagrangian is no longer invariant under the
broken symmetry transformation. Instead, the transformation constitutes a change of basis. We
can use this change of basis to our advantage, by choosing a basis that makes the physics of the
model more transparent. This is what we do here by choosing a specific gauge: θ(x) = −ξ(x)/v.
(It is fully legitimate to choose the phase to be related to a field.)
With this choice of gauge,
φ→ φ′ =1√2
(h+ v), Aµ → Vµ = Aµ +i
gv∂µξ. (6.49)
The Lagrangian in terms of h and Vµ reads
L = −1
4FµνF
µν +1
2(∂µh)(∂µh) +
1
2(g2v2)VµV
µ − 1
2(2λv2)h2 (6.50)
+g2
2VµV
µh(2v + h)− λvh3 − λ
4h4.
Note the following points:
1. The elementary particles of this model are a massive vector boson of mass-squared m2V =
(gv)2 and a massive scalar of mass-squared m2h = 2λv2.
2. The ξ field is “eaten” in order to give mass to the gauge boson. It was a conveniant choice
to make the phase to be the “eaten” DoF. The total number of degrees of freedom does
not change: instead of the scalar ξ, we have the longitudinal component of a massive vector
boson.
3. In the limit g → 0 we have mV → 0. This situation describes a massless gauge boson
and a massless scalar. We see that in that limit the longitudinal component is the massless
Goldstone boson as expected.
4. The field that acquires a VEV, in our case φ, is called the Brout-Englert-Higgs (BEH) field
or the Higgs field. The h scalar is called “a Higgs boson.”
5. The Lagrangian (6.45) depends on three parameters. They can be taken to be g, v and λ.
The Lagrangian (6.50) has two mass terms and four interaction terms which depend on the
same three parameters. Thus, the six relevant terms, which would be independent in the
absence of a symmetry, obey three relations among them. This is a sign of SSB.
6. The hV V coupling is proportional to the mass-squared of the vector boson.
7. The dimensionless V V hh and hhhh couplings are unchanged from the symmetric Lagrangian.
71
In the example above we consider the case of a SSB of a local U(1) symmetry. The basic
ingredients are, however, much more generic and apply also to non-Abelian symmetries and to
product groups. In fact, the SM incorporates SSB of a local SU(2) × U(1) symmetry. The
following lessons are generic to all cases of spontaneous breaking of a local symmetry:
1. Spontaneous symmetry breaking gives masses to the gauge bosons related to the broken
generators.
2. Gauge bosons related to an unbroken subgroup remain massless, because their masslessness
is protected by the symmetry.
3. The field that acquires a VEV (the BEH field) must be a scalar field. Otherwise its VEV
would break Lorentz invariance.
Spontaneous breaking of a local symmetry can give masses also to fermions, as is the case
for a spontaneously broken global symmetry. To see this, we add to our model fermions, with
q(ψR) − q(ψL) = q(φ), as in the model of Section 6.4. Working in the physical gauge, θ(x) =
−ξ(x)/v, we learn that Eq. (6.42) is modified to
L = Lkin − V (h)− Y v√2ψRψL −
Y√2hψRψL + h.c.. (6.51)
In the physical gauge, the ξ field does not appear explicitly anymore, as expected. The lon-
gitudinal component of the vector boson couples to the fermion, with a coupling strength that
is proportional to the fermion mass (∝ mf/v). The coupling of the transverse component is
proportional to the gauge coupling.
6.6 Summary
Symmetries in QFT have a strong predictive, or explanatory, power. The main consequences of
the various types of symmetries are summarized in Table 6.1.
To construct a model, we first define the following three ingredients:
(i) The symmetry;
(ii) The transformation properties of the fermions and scalars;
(iii) The pattern of spontaneous symmetry breaking (SSB).
Then we write the most general renormalizable Lagrangian that is invariant under the symmetry.
The renormalizable Lagrangian has a finite number of parameters that we need to determine
by experiment. In principle, for a theory with N independent parameters, we need to perform N
72
Table 6.1: Symmetries
Type Consequences
Spacetime Conservation of energy, momentum, angular momentum
Discrete Selection rules
Global (exact) Conserved charges
Global (spon. broken) Massless scalars
Local (exact) Interactions, massless spin-1 mediators
Local (spon. broken) Interactions, massive spin-1 mediators
appropriate measurements to extract the values of the parameters. Additional measurements test
the theory.
Strictly speaking, the pattern of the SSB is not an input, that is, it depends on the values of
the parameters. In all models that we discuss, the SSB pattern depends on the sign of µ2. Yet,
since the phenomenology is so different based on this choice, we prefer to quote it as an input
ingredient.
73
Appendix
6.A The Goldstone Theorem
The Goldstone Theorem states the following: The spontaneous breaking of a global continuous
symmetry is accompanied by massless scalars. Their number and quantum numbers equal those of
the broken generators.
Consider φj to be some multiplet of scalar fields with the Lagrangian
L(φ) =1
2(∂µφj)(∂
µφj)− V (φ) (6.52)
where L(φ) is invariant under some symmetry group as in Eq. (4.2)
φj → (exp [iT aθa])jk φk. (6.53)
We want to perturb around a minimum of the potential V (φ). We expect the φ field to have
a VEV, 〈φ〉, which minimizes V . The condition that 〈φ〉 is an extremum of V (φ) reads
V ′j∣∣∣φ=〈φ〉
= 0 ∀j, where V ′j ≡∂V
∂φj(6.54)
The condition for minimum at v is, in addition to (6.54), that the second derivative matrix at the
extremum
m2ij ≡
∂2V
∂φi∂φj
∣∣∣∣∣φ=〈φ〉
(6.55)
is a positive semidefinite matrix, that is, that all of its eigenvalues are nonnegative. Note that m2ij
is the scalar mass-squared matrix. We can see that by expanding V (φ) in a Taylor series in the
shifted fields φ′ = φ− 〈φ〉 and noting that the mass term is the one with two fields.
Now we check for the behavior of 〈φ〉 under the transformation (6.53). There are two cases. If
Ta〈φ〉 = 0 (6.56)
for all a, the symmetry is not broken. This is certainly what happens if 〈φ〉 = 0. It is also possible
that
Ta〈φ〉 6= 0 for some a. (6.57)
74
This is the case when Ta is spontaneous broken.
We focuse on the case where some generators of the original symmetry are spontaneously
broken while others are not. Note that the set of generators satisfying Eq. (6.56) is closed under
commutation becasue
Ta〈φ〉 = 0 & Tb〈φ〉 = 0 =⇒ [Ta, Tb] 〈φ〉 = 0 , (6.58)
and therfore they generate the unbroken subgroup of the original symmetry group.
Because V is invariant under Eq. (6.53), we have to leading order (that is, for θa 1)
V (φ+ δφ)− V (φ) = i∂V (φ)
∂φkθa(T
a)klφl = 0. (6.59)
If we differentiate with respect to φj, and set φ = 〈φ〉 we get
m2jk(T
a)klφl + Vk(〈φ〉)(T a)kj = 0. (6.60)
The second term drops out because we work around the minimum, see Eq. (6.54), and we obtain
m2jk(T
a)kl〈φ〉l = 0. (6.61)
For T a in the unbroken subgroup, T a〈φ〉 = 0 and Eq. (6.61) is trivially satisfied. But if T a〈φ〉 6= 0,
Eq. (6.61) requires that T a〈φ〉 is an eigenvector of m2 with eigenvalue zero. It corresponds to a
massless boson field given by
φj(Ta)jl〈φ〉l (6.62)
which is called a Goldstone boson.
75
Homework
Question 6.1: SSB with many scalars
Consider a Lagrangian for N interacting real scalar fields φi with i = 1..N ,
L =1
2∂µφi∂
µφi −1
2µ2∑i
(φ2i
)− 1
4λ
[∑i
(φ2i
)]2
, (6.63)
with µ2 < 0 and λ > 0. This Lagrangian is a generalization of Eq. (6.11). It is symmetric under
SO(N) rotation φi → Uijφj.
1. Show that L describes a massive field of mass√−2µ2 and N − 1 massless Goldstone bosons.
2. What is the unbroken symmetry group?
3. The Goldstone theorem states that the number of massless bosons is equal to the number of
broken generators. Show explicitly that this relation holds.
Question 6.2: Broken and unbroken symmetries
In this question we are going to elaborate on Eqs. (6.21) and (6.22) that stated that a unbroken
generator Ta annihilate the vacuum, Ta〈φ〉 = 0 while a broken one does not, Ta〈φ〉 6= 0. For this
consider the operation of a generator on the vacuum
〈φ′〉 = eiTaθ〈φ〉. (6.64)
1. Explain why the symmetry is unbroken if 〈φ′〉 = 〈φ〉 for any θ and that is broken if there is
a θ such that 〈φ′〉 6= 〈φ〉
2. Explain why the above implies that Ta〈φ〉 = 0 if Ta corresponds to an unbroken symmetry.
3. A familiar example is the case of a vector in 3d, that breaks the symmetry from an SO(3) to
SO(2) (or from SU(2) to U(1)) that is from a rotation is a 3D into rotation on the plane. The
unbroken symmetry is around the direction of the vector, or in the plane perpendicular to
76
it. Consider a case where we choose the normalized vector to be ~v = (0, 1, 0). Show that Lz
is still a symmetry while Lx and Ly are not. It is useful to recall the explicit representation
of Li for a vector.
Lx =
0 1 0
1 0 1
0 1 0
Ly =
0 i 0
−i 0 i
0 −i 0
Lz =
1 0 0
0 0 0
0 0 −1
(6.65)
4. Now consider a generic normalized vector, ~v = (a, b, c) such that a2 + b2 + c2 = 1. Show that
[(a− c)Lx − i(a+ c)Ly − 2bLz]~v = 0. (6.66)
The above shows that there is always one generator that is not broken, so indeed the unbroken
symmetry is SO(2).
5. While a 3 (aka vector) breaks SU(2) to U(1) a 2 (aka as spinor) breaks it to nothing. In order
to show it we need to prove that there is no way to annihilate a spinor with any combination
of the generators, which in this case are just the Pauli matrices. Consider a simple case
where the spinor is just ~s = (0, 1) and show that any normalized linear combination of the
Pauli matrices does not annihilate it. From that we learn that choosing a spinor the SU(2)
symmetry it totally broken.
Question 6.3: The sigma model
A classic example of a spontaneous symmetry breaking with Goldstone bosons is the so called
“σ-model”, which tries to explain pion-nucleon interactions.
We begin by putting the proton and neutron in a doublet of global SU(2) which we call isospin.
We further impose a chiral SU(2)L×SU(2)R symmetry such that the left-handed field transform
under SU(2)L and the right-handed field under SU(2)R. Then the most general L is
L = iψ∂/ψ, ψ =
(P
N
). (6.67)
The infinitesimal symmetry transformations are
δψL = iεaLTaψL, δψR = iεaRT
aψR. (6.68)
1. Show that L is invariant under these chiral symmetries. Rewrite them in the form
δψ = iεaT aψ, δψ = iγ5εa5T
aψ, (6.69)
and express εa and εa5 in term of εaL and εaR.
77
What you showed is that we can write the symmetry in a different basis. Instead of SU(2)L ×SU(2)R we can write SU(2)V × SU(2)A. (SU(2)V is also called “diagonal SU(2)”.)
2. A mass term for ψ breaks one of the SU(2) symmetries. Which one? Show it.
Instead of adding a mass term we introduce a scalar field, Σ, that transforms as doublet under
both SU(2)L and SU(2)R. Then the most general Lagrangian I can write is
L = Lkin − g(ψLΣψR + h.c.) + L(Σ†Σ) . (6.70)
In general Σ is a doublet under two SU(2) symmetries so naively it must have 8 real components.
Yet, recalling that SO(4)∼ SU(2)× SU(2) you should not be surprised that we can write it only
in term of 4 real scalar fields. We write
Σ = σ + iτaπa, (6.71)
such that τa are the Pauli matrices.
3. Show that the rotations of δΣ rotate these fields into each other, and write the rotations in
terms of σ and π and the ε and ε5 parameters. I will start you off: δσ = −εa5πa.
4. Rewrite the scalar-fermion couplings in terms of σ and π.
In order to give the nucleon mass we need to break the chiral symmetry spontaneously
SU(2)L × SU(2)R → SU(2) (6.72)
that is, we like to break the symmetry that forbid the nucleon mass. For that we like the Σ field
to acquires a vev. This can be done if we write its scalar potential as
1
4λ[σ2 + ~π2 − F 2
π
]2. (6.73)
Here Fπ is the so called “pion decay constant” and it is the only mass scale in the theory.
5. What are the minima of V ? Find a minimum when only σ but not π acquires a vev.
6. Rewrite L in terms of ψ, πa and s ≡ σ − Fπ. Note that all these fields are physical, that is,
have no vev. What are the masses of these new fields?
7. What is the nucleon-nucleon-pion interaction term? Show that it satisfied g = mN/Fπ.
(This relation is know as the Goldberger-Treiman relation and it satisfied in Nature to a
good accuracy.)
8. How many generators are broken? Does the Goldstone theorem hold?
78
9. In Nature the pions have small masses (small compared to the nucleon). That is, we like to
think about a small breaking of a symmetry. Which is this symmetry? Can you think about
a way to modify the model such that the pions end up with small masses? What other terms
have to be added once you decide to have a small breaking of the symmetry?
Question 6.4: The Higgs mechanism
We consider the model of section 6.5 with the Lagrangian of Eq. (6.45).
1. Show that to leading order in ξ/v and h/v, Eq. (6.48) is the same as Eq. (6.41), that is, show
that
eiξ(xµ)/v v + h(xµ)√2
≈ v + h(xµ) + iξ(xµ)√2
, (6.74)
2. Use Eqs. (6.48) and (6.49) to derive Eq. (6.50)
3. Draw the tree-level diagrams for the hh → hh scattering and write down the amplitude.
Note that there are two types of diagrams.
4. Calculate (up to numerical constants and phase space) the differential cross section in the
limit where E v and θ ∼ 1. Here E is the center of mass energy of the collision and θ is
the scattering angle such that θ = 0 is forward scattering.
5. Consider now the same theory but with µ2 > 0 and calculate (again, up to numerical con-
stants and phase space) the φφ → φφ differential cross section the limit where E µ and
θ ∼ 1. Explain the similarity to the result obtained in the previous item.
79
Chapter 7
The Leptonic Standard Model
7.1 Defining the LSM
The Leptonic Standard Model (LSM) incorporates the three aspects of imposed symmetries that
are discussed in previous chapters: Abelian symmetries, non-Abelian symmetries, and spontaneous
symmetry breaking. Moreover, the model is relevant to Nature. It accounts for the weak, electro-
magnetic and Yukawa interactions of the leptons. The LSM is part of the SM. The complete SM
adds quarks and strong interactions to the LSM.
In Section 6.6 we presented the three ingredients that are required to define a model. For the
LSM, these three ingredients are defined as follows:
(i) The symmetry is a local
SU(2)L × U(1)Y . (7.1)
(ii) The pattern of spontaneous symmetry breaking is as follows:
SU(2)L × U(1)Y → U(1)EM. (7.2)
(iii) There are three fermion generations, each consisting of two different representations:
LLi(2)−1/2, ERi(1)−1, i = 1, 2, 3. (7.3)
There is a single scalar multiplet:
φ(2)+1/2. (7.4)
We use the notation (N)Y such that N is the irreducible representation (irrep) under SU(2)L
and Y is the hypercharge (the charge under U(1)Y ). What we mean by Eq. (7.3) is that there are
nine Weyl fermion degrees of freedom that are grouped into three copies (“generations”) of the
same gauge representations. The three fermionic degrees of freedom in each generation form an
SU(2)-doublet (of hypercharge −1/2) and an SU(2)-singlet (of hypercharge −1).
80
It is now our task to find the specific form of the Lagrangian of Eq. (2.33) made ofthe LLi, ERi
[Eq. (7.3)] and φ [Eq. (7.4)] fields, subject to the gauge symmetry of Eq. (7.1) and leading to the
SSB of Eq. (7.2).
7.2 The Lagrangian
7.2.1 Lkin and the gauge symmetry
The gauge group is given in Eq. (7.1). It has four generators: three Ta’s that form the SU(2)
algebra
[Ta, Tb] = iεabcTc, (7.5)
where a, b, c = 1, 2, 3 and a single Y that correspond to the U(1) group, and thus does not form
an algebra. The generators of the SU(2) and the U(1) must commute as they belong to different
gauge groups
[Ta, Y ] = 0. (7.6)
There are two independent coupling constants in Lkin: there is a single g for all the SU(2) couplings
and a different one, g′, for the U(1) coupling. The SU(2) couplings must all be the same because
they mix with one another under SU(2) rotations. The U(1) coupling can be different from that
of the SU(2) because the generator Y never appears as a commutator of SU(2) generators.
The local symmetry requires four gauge boson DoFs, three in the adjoint representation of
SU(2) and one related to the U(1) symmetry:
W µa (3)0, Bµ(1)0. (7.7)
The corresponding field strengths are given by [see Eqs. (2.29) and (4.18)]
W µνa = ∂µW ν
a − ∂νW µa − gεabcW
µb W
νc , Bµν = ∂µBν − ∂νBµ. (7.8)
The covariant derivative is
Dµ = ∂µ + igW µa Ta + ig′BµY. (7.9)
We define Lkin to include the kinetic terms of all the fields:
Lkin = −1
4W µνa Waµν −
1
4BµνBµν − iLLiD/LLi − iERiD/ERi − (Dµφ)†(Dµφ). (7.10)
For the SU(2)L doublets Ta = σa/2 (σa are the Pauli matrices), while for the SU(2)L singlets,
Ta = 0. [For SU(2)L triplets, (Ta)bc = εabc, which has already been used in writing Eq. (7.8).]
Explicitly,
Dµφ =(∂µ +
i
2gW µ
a σa +i
2g′Bµ
)φ,
DµLL =(∂µ +
i
2gW µ
a σa −i
2g′Bµ
)LL,
DµER = (∂µ − ig′Bµ)ER. (7.11)
81
7.2.2 LψThere are no mass terms for the fermions in the LSM. We cannot write Dirac mass terms for the
fermions because they are assigned to chiral representations of the gauge symmetry. We cannot
write Majorana mass terms for the fermions because they all have Y 6= 0. Hence,
Lψ = 0. (7.12)
7.2.3 LYuk
The Yukawa part of the Lagrangian is given by
LYuk = Y eijLLiERj φ+ h.c., (7.13)
where i, j = 1, 2, 3 are flavor indices. The Yukawa matrix Y e is a general complex 3× 3 matrix of
dimensionless couplings. Without loss of generality, we can choose a basis where Y e is diagonal
and real (see the discussion in subsection 7.5.1):
Y e = diag(ye, yµ, yτ ). (7.14)
7.2.4 Lφ and spontaneous symmetry breaking
The Higgs potential, which leads to the spontaneous symmetry breaking, is given by
Lφ = −µ2φ†φ− λ(φ†φ
)2. (7.15)
The discussion follows the same lines as the U(1) and SU(2) models presented in Chapter 6. The
quartic coupling λ is dimensionless and real, and has to be positive for the potential to be bounded
from below. The quadratic coupling µ2 has mass dimension 2 and is real. If the gauge symmetry
is to be spontaneously broken, Eq. (7.2), we must take µ2 < 0. Defining
v2 = −µ2
λ, (7.16)
we can rewrite Eq. (7.15) as follows (up to a constant term):
−Lφ = λ
(φ†φ− v2
2
)2
. (7.17)
The scalar potential (7.17) implies that the scalar field acquires a VEV, |〈φ〉| = v/√
2. We choose
a gauge,
〈φ〉 =
(0
v/√
2
). (7.18)
82
This VEV breaks the SU(2)×U(1) symmetry down to a U(1) subgroup. The generator of U(1)EM,
Q, must be the generator of the unbroken subgroup. Thus we identify the unborken generator as
Q = T3 + Y, (7.19)
as this combination vanishes for the down component where we choose the vev to be.
Before we proceed, let us clarify a few points regarding our choice of having the VEV in the
direction of the T3 = −1/2 component of φ:
1. We could equally well choose the have the VEV in the direction of the T3 = +1/2 component.
In this case we would have Q = T3 − Y , and the physics would remain the same.
2. Let us write explicitly the two components of SU(2)L doublets:
LL1 =
(νeL
eL
), φ =
(φ+
φ0
). (7.20)
The charge under U(1)EM of the different components, q, is given by
q(νeL) = 0, q(eL) = −1, q(eR) = −1, q(φ+) = +1, q(φ0) = 0. (7.21)
3. If SU(2)L×U(1)Y were an exact symmetry of Nature, there would be no way of distinguishing
particles of different electric charges in the same SU(2)L multiplet. The SSB makes, for
example, νeL distinguishable from eL.
Let us denote the four real components of the scalar doublet as three phases, θa(x) (a = 1, 2, 3),
and one magnitude, h(x). We choose the three phases to be the three “would be” Goldstone bosons,
in a way that is similar to the case we discuss in Section 6.5. In our case the broken generators
are T1, T2, and T3 − Y , and thus we write
φ(x) = exp [(i/2) (σaθa(x)− Iθ3(x))]1√2
(0
v + h(x)
). (7.22)
The local SU(2)L × U(1)Y symmetry of the Lagrangian allows one to rotate away the explicit
dependence on the three θa(x). They represent the three would-be Goldstone bosons that are
eaten by the three gauge bosons that acquire masses as a result of the SSB. See the discussion in
Section 6.5. In this gauge,
φ(x) =1√2
(0
v + h(x)
). (7.23)
7.2.5 Summary
The renormalizable part of the Leptonic Standard Model Lagrangian is given by
LSM = − 1
4W µνa Waµν −
1
4BµνBµν − (Dµφ)†(Dµφ)− iLLiD/LLi − iERiD/ERi
+(Y eijLLiERj φ+ h.c.
)− µ2φ†φ− λ
(φ†φ
)2, (7.24)
where i, j = 1, 2, 3.
83
7.3 The Spectrum
7.3.1 Scalars: back to LφThe scalar sector contains one real scalar field that we denote by h. This is the Higgs boson of the
LSM. Its mass can be obtained by plugging (7.23) into (7.17), and is given by
mh =√
2λv. (7.25)
Experiment gives (PDG 2015)
mh = 125.09± 0.24 GeV. (7.26)
7.3.2 Vector bosons: back to Lkin(φ)
Since the symmetry that is related to three out of the four generators is spontaneously broken,
three of the four vector bosons acquire masses, while one remains massless. To see how this
happens, we examine (Dµ〈φ〉)†(Dµ〈φ〉). Using Eq. (7.11) for Dµφ, we obtain:
Dµ
(0
v/√
2
)=
i√8
(gW µa σa + g′Bµ)
(0
v
)=
i√8
(gW µ
3 + g′Bµ g(W µ1 − iW
µ2 )
g(W µ1 + iW µ
2 ) −gW µ3 + g′Bµ
)(0
v
). (7.27)
The mass terms for the vector bosons are thus given by (we omit Lorentz indices)
LMV=
1
8(0 v)
(gW3 + g′B g(W1 − iW2)
g(W1 + iW2) −gW3 + g′B
)(gW3 + g′B g(W1 − iW2)
g(W1 + iW2) −gW3 + g′B
)(0
v
). (7.28)
We define an angle θW via
tan θW ≡g′
g. (7.29)
We define four gauge boson states:
W± =1√2
(W1 ∓ iW2), Z0 = cos θWW3 − sin θWB, A0 = sin θWW3 + cos θWB. (7.30)
We see that the W± are charged under electromagnetism (hence the superscripts ±), while A0 and
Z0 are not. In terms of the vector boson fields of Eq. (7.30), we write Eq. (7.28) as follows:
LMV=
1
4g2v2W+W− +
1
8(g2 + g′2)v2Z0Z0. (7.31)
We learn that the four states of Eq. (7.30) are the mass eigenstates, with masses
m2W =
1
4g2v2, m2
Z =1
4(g2 + g′2)v2, m2
A = 0. (7.32)
(Recall that for a complex field φ with mass m the mass term is m2|φ|2 while for a real field it is
m2φ2/2.) Three points are worth emphasizing:
84
1. As anticipated, three vector boson acquire masses.
2. m2A = 0 is not a prediction, but rather a consistency check on our calculation.
3. The angle θW represents a rotation angle of the two neutral vector bosons from the interaction
basis, where fields have well-defined transformation properties under the full gauge symmetry,
(W3, B), into the mass basis for the vector bosons, (Z,A).
In Chapter 6 it is emphasized that SSB leads to relation between observables that would have
been independent in the absence of a symmetry. One such important relation involves the vector-
boson masses and their couplings:m2W
m2Z
=g2
g2 + g′2. (7.33)
It is conventional to express this relation in terms of θW , defined in Eq. (7.29):
ρ ≡ m2W
m2Z cos2 θW
= 1. (7.34)
This relation is testable. The left hand side of Eq. (7.33) can be derived from the measured
spectrum, and the right hand side from interaction rates. The ρ = 1 relation is a consequence of
the SSB by SU(2)-doublets. (See the homework for other possibilities.) It thus tests this specific
ingredient of the LSM.
The experimental values of the weak gauge boson masses are given by (PDG 2014)
mW = 80.385± 0.015 GeV; mZ = 91.1876± 0.0021 GeV. (7.35)
We can then use the ρ = 1 relation to determine sin2 θW :
mW
mZ
= 0.8815± 0.0002 =⇒ sin2 θW = 1− (mW/mZ)2 = 0.2229± 0.0004. (7.36)
Below we describe the determination of sin2 θW by various interaction rates. We will see that the
ρ = 1 is indeed realized in Nature (within experimental errors, and up to calculable quantum
corrections that we discuss in length in Chapter 9).
7.3.3 Fermions: back to LYuk
Here we see how the chiral fermions acquire masses via SSB. The Yukawa part of the Lagrangian
is given by Eq. (7.13). The SSB allows us to tell the upper and lower components of the doublet.
In the basis defined in Eq. (7.14), we denote these components as follows:
LL1 =
(νeL
eL
), LL2 =
(νµL
µL
), LL3 =
(ντL
τL
), (7.37)
where e, µ, τ are ordered by the size of ye,µ,τ (from smallest to largest). We also define
ER1 = eR, ER2 = µR, ER3 = τR, (7.38)
85
Eq. (7.21) tells us that the neutrinos (νeL, νµL, ντL) have charge zero, while the left-handed charged
leptons (eL, µL, τL) and the right handed leptons (eR, µR, τR) carry charge −1.
With φ0 acquiring a VEV, the Yukawa term has a piece that corresponds to the charged lepton
masses. These terms are the ones obtained by replacing φ by its VEV in Eq. (7.13), leading to
−yev√2eL eR −
yµv√2µL µR −
yτv√2τL τR + h.c.. (7.39)
namely
me =yev√
2, mµ =
yµv√2, mτ =
yτv√2. (7.40)
The crucial point is that while the leptons are in a chiral representation of the full gauge
group SU(2)L × U(1)Y , the charged leptons — e, µ, τ — are in a vectorial representation of the
subgroup that is not spontaneously broken, that is U(1)EM. This situation is the key to opening
the possibility of acquiring masses as a result of the SSB, as realized in Eq. (7.39).
The charged lepton masses have been measured:
me = 0.510998928(11) MeV, mµ = 105.6583715(35) MeV, mτ = 1776.82(16) MeV.
(7.41)
The neutrinos are massless in this model. There are no LR(2)−1/2 fields in the LSM, so there
are no Dirac mass terms for the neutrinos in the symmetry limit. There are no NR(1)0 fields
in the LSM, so the neutrinos cannot acquire Dirac mass as a result of the SSB. A-priori, since
the neutrinos have no charge under U(1)EM, the possibility of acquiring Majorana masses is not
closed. Yet, lepton number is an accidental symmetry of the theory (see Section 7.5.3) and thus
the neutrinos do not acquire Majorana masses from renormalizable terms.
In your homework you will find that the number of Higgs representations that can give the
gauge boson their masses is large, but only very few also give masses to the fermions.
7.3.4 Summary
We presented the details of the spectrum of the leptonic standard model. These are summarized
in the Table 7.1. All masses are proportional to the VEV of the scalar field, v. For the three
massive gauge bosons, and for the three charged leptons, this must be the case: In the absence
of spontaneous symmetry breaking, the former would be protected against acquiring a mass by
the gauge symmetry and the latter by their chiral nature. For the Higgs boson, the situation is
different, as a mass-squared term does not violate any symmetry. Here it is just a manifestation
of the fact that the LSM has a single dimensionful parameter, which can be taken to be v, and
therefore all masses must be proportional to this parameter.
86
Table 7.1: The LSM particles
particle spin Q mass (theo)
W± 1 ±1 12gv
Z0 1 0 12
√g2 + g′2v
A0 1 0 0
h 0 0√
2λv
e 1/2 −1 1√2yev
µ 1/2 −1 1√2yµv
τ 1/2 −1 1√2yτv
νe 1/2 0 0
νµ 1/2 0 0
ντ 1/2 0 0
7.4 The interactions
IN this Section, we obtain the interactions among the mass eigenstates of well-defined EM charge
of the LSM. The scalar potential of Eq. (7.15) leads to Higgs self-interactions. The Yukawa terms
of Eq. (7.13) lead to Higgs-mediated Yukawa interactions among the charged leptons. The kinetic
terms of Eq. (7.10) lead to three type of interactions mediated by vector bosons: The photon-
mediated electromagnetic interactions (QED), the Z-mediated weak interactions (neutral current
weak interactions), and the W±-mediated weak interactions (charged current weak interactions).
To obtain the latter three types of interactions, we need to rewrite the covariant derivative given
in Eq. (7.9) in terms of the vector boson mass eigenstates, A, Z and W± defined in Eq. (7.30):
Dµ = ∂µ + ig(W+µ T
+ +W−µ T
−) (7.42)
+i(g sin θW T3 + g′ cos θW Y )Aµ + i(g cos θW T3 − g′ sin θW Y )Zµ,
where T± = (T1 ∓ iT2)/√
2.
7.4.1 The Higgs boson
The kinetic, gauge-interaction, self-interaction and Yukawa interaction terms of h are given by
Lh =1
2∂µh∂
µh− 1
2m2hh
2 − m2h
2vh3 − m2
h
8v2h4
+ m2WW
−µ W
µ+
(2h
v+h2
v2
)+
1
2m2ZZµZ
µ
(2h
v+h2
v2
)
− h
v(me eL eR +mµ µL µR +mτ τL τR + h.c.) . (7.43)
87
We write Lh in a way that demonstrates that all of the Higgs couplings can be expressed in terms
of the masses of the particles to which it couples.
The Higgs mass is given in Eq. (7.25), mh =√
2λv. It determines its quartic self-coupling,
m2h
2v2= λ, (7.44)
which is unchanged from the quartic coupling in (7.15), and its trilinear self-coupling,
m2h
2v= λv, (7.45)
which arises as a consequence of the SSB. The Higgs coupling to the weak interaction gauge bosons
is proportional to their masses-squared. The dimensionless hhV V couplings,
m2W
v2=g2
4,
m2Z
2v2=g2 + g′2
8(7.46)
are unchanged from Eq. (7.10). The hV V couplings,
2m2W
v=g2v
2,
m2Z
v=
(g2 + g′2)v
4, (7.47)
arise as a consequence of the SSB.
There is neither an hAA nor hhAA coupling. One can understand the absence of these couplings
in two ways. First, the Higgs boson is electromagnetically neutral, so it should not couple to the
electromagnetic force carrier. Second, the photon is massless, so it should not couple to the Higgs
boson.
The Yukawa couplings of the Higgs bosons to the charged leptons are proportional to their
masses: the heavier the lepton, the stronger the coupling. Note that these couplings, m`/v =
y`/√
2, are unchanged from Eq. (7.13).
7.4.2 QED: Electromagnetic interactions
In this Section we study the photon interactions and show that we recover QED. Using Eq. (7.42)
and the definition of θW in Eq. (7.29), we find that the photon coupling is proportional to
(g sin θW T3 + g′ cos θW Y ) =gg′√g2 + g′2
(T3 + Y ). (7.48)
This is what we wanted! The coupling is proportional to T3 + Y , which we defined as Q, the
generator of U(1)EM. The photon coupling is conventionally defined as eQ. We learn that
g =e
sin θW, g′ =
e
cos θW. (7.49)
Thus, the electromagnetic interactions are described by the QED Lagrangian [see Eq. (3.18)],
which is now understood as the part of the LSM Lagrangian that involves the photon field, A and
the charged fermions:
LQED = −1
4FµνF
µν + eAµ`iγµ`i, (7.50)
88
where Fµν = ∂µAν − ∂νAµ. The `1,2,3 = e, µ, τ fields are the Dirac fermions with Q = −1 that are
formed from the T3 = −1/2 component of a left-handed lepton doublet and from a right-handed
lepton singlet, for example τ = (τL, τR)T .
The QED interactions are discussed in Chapter 3. Here we only emphasize again some impor-
tant features that arise from Eq. (7.50):
1. The photon couplings are vector-like: It couples to the left-handed and right-handed fields
in the same way.
2. Thus, electromagnetic interactions are parity conserving.
3. Diagonality. The photon couples to e+e−, µ+µ− and τ+τ−, but not to e±µ∓, e±τ∓ or
µ±τ∓ pairs. Thus, electromagnetic interactions do not change flavor. This is a result of the
unbroken local U(1)EM symmetry.
4. Universality: the couplings of the photon to the different generations are universal. This is
a result of the U(1)EM gauge invariance.
7.4.3 Neutral current weak interactions
In this Section we study the Z-boson interactions with fermions. Using Eq. (7.42) and the definition
of θW in Eq. (7.29), we find that the Z-boson coupling is proportional to
(g cos θW T3 − g′ sin θW Y ) =g
cos θW(T3 − sin2 θWQ). (7.51)
This leads to the following Zff interactions:
LZff =g
cos θW
[−(
1
2− sin2 θW
)`iLZ/ `iL + sin2 θW `iRZ/ `iR +
1
2νiLZ/ νiL
]. (7.52)
where `1,2,3 = e, µ, τ and ν1,2,3 = νe, νµ, ντ . Note that, unlike the photon, the Z couples to neutrinos.
Z-exchange gives rise to neutral current weak interactions (NCWI). Eq. (7.52) reveals some further
important features of the model:
1. The Z-boson couplings are chiral: It couples to left-handed and right-handed fields with
different strength.
2. Thus, the Z-interactions are parity violating.
3. Diagonality. The Z-boson couples to, for example, e+e−, µ+µ−, νeLνeL and νµLνµL, but not
to, for example, e±µ∓ and νeLνµL pairs. Consequently, there are no flavor changing neutral
currents (FCNCs). This can be thought of a result of an accidental U(1)3 symmetry of the
model, see Section 7.5.3.
89
4. Universality: the couplings of the Z-boson to the different generations within each of the
three sectors (νL, `L, `R) are universal. This is a result of a special feature of the LSM: all
fermions of given chirality and given charge come from the same SU(2)×U(1) representation.
The above points have been experimentally tested. For example, the branching ratios of the
Z-boson into charged lepton pairs,
BR(Z → e+e−) = (3.363± 0.004)% , (7.53)
BR(Z → µ+µ−) = (3.366± 0.007)% ,
BR(Z → τ+τ−) = (3.367± 0.008)% .
beautifully confirm universality:
Γ(µ+µ−)/Γ(e+e−) = 1.0009± 0.0028, (7.54)
Γ(τ+τ−)/Γ(e+e−) = 1.0019± 0.0032.
Diagonality is also tested by the following experimental searches:
BR(Z → e+µ−) < 1.7× 10−6, (7.55)
BR(Z → e+τ−) < 9.8× 10−6,
BR(Z → µ+τ−) < 1.2× 10−5.
The branching ratio of Z decays into invisible final states which, in our model, is interpreted
as the decay into final neutrinos, is measured to be
BR(Z → νν) = (20.00± 0.06)%. (7.56)
From Eq. (7.52) we obtain
BR(Z → `+`−)
BR(Z → ν`ν`)=
(1/2− sin2 θW )2 + sin4 θW1/4
= 1− 4 sin2 θW + 8 sin4 θW . (7.57)
We can thus extract sin2 θW from the experimental data, sin2 θW = 0.226, consistent with Eq. (7.36).
We discuss these decays in more detail in Chapter 9.
We also remark that we can write the coupling of the Z to fermions in terms of Dirac fermions.
We define
gL = T3 − sin2 θWQ, gR = − sin2 θWQ, (7.58)
gV = gL + gR = T3 − 2 sin2 θWQ, gA = gL − gR = T3
and we get tha for any fermion ψ the coupling is
LZψψ =g
cos θWψ(gV − gAγ5)Z/ψ, (7.59)
90
7.4.4 Charged current weak interactions
In this Section we study the W±-boson interactions with fermions. Using Eq. (7.42), the definition
of θW in Eq. (7.29), and the explicit form of the Ta matrices, we find the W -boson couplings to
fermion pairs are given by
LW = − g√2νiL W/
+`−iL + h.c. (7.60)
The interactions mediated by theW± vector-bosons are called charged current interactions (CCWI).
They are unique among the interactions of the LSM, as the fermion pairs to which the W -boson
couples consist of two different fermions, a neutrino and a charged lepton. This must be the case
as the W -bosons are charged, so they must change the identity of the particle with which they
interact.
Eq. (7.60) reveals some important features of the model:
1. Only left-handed particles take part in charged-current interactions.
2. Parity violation: a consequence of the previous feature is that the W -mediated interactions
violate parity.
3. Diagonality: the charged current interactions couple each charged lepton to a single neutrino,
and each neutrino to a single charged lepton.
4. Universality: the couplings of the W -boson to τ ντ , to µνµ and to eνe are equal. This is a
result of the local nature of the imposed SU(2): a global symmetry would have allowed an
independent coupling to each lepton pair.
All of these predictions have been experimentally tested. As an example of how well universality
works, consider the decay rates of the W -bosons to the three lepton pairs:
BR(W+ → e+νe) = (10.75± 0.13)× 10−2,
BR(W+ → µ+νµ) = (10.57± 0.15)× 10−2,
BR(W+ → τ+ντ ) = (11.25± 0.20)× 10−2. (7.61)
You must be impressed by the nice agreement!
The charged current interaction gives rise to all flavor changing weak decays. One example is
the µ− → e−νµνe decay. One can use this decay rate as yet another independent way to determine
sin2 θW from an interaction rate. The low-energy W -propagator is well approximated via a four
fermion coupling:g2
m2W − q2
≈ g2
m2W
=4πα
sin2 θWm2W
≡ 4√
2GF , (7.62)
where we used g2 = 4πα/ sin2 θW based on Eqs. (3.10) and (7.49) . The measured muon lifetime,
τµ = (2.197034± 0.000021)× 10−6 s, (7.63)
91
determines GF via
Γµ =1
τµ=G2Fm
5µ
192π3f(m2
e/m2µ)(1 + δRC), f(x) = 1− 8x+ 8x3 − x4 − 12x2 log x, (7.64)
where f(x) is the phase space function for a three body decay with two massless final particles
(it is normalized to 1 in the case when all final particles are massless) and δRC encodes radiative
corrections, and is known to O(α2). One gets:
GF = 1.16637(1)× 10−5 GeV−2. (7.65)
Using α of Eq. (??), mW of Eq. (7.35) and GF of Eq. (7.65), we obtain
sin2 θW = 0.215, (7.66)
in good agreement with Eq. (7.36). The difference between the two is accounted for by higher
order radiative corrections (we discuss them in Chapter 9).
Note that GF determines also the VEV. Using GF = g2/(4√
2m2W ) and m2
W = g2v2/4 we obtain
v = (√
2GF )−1/2 ≈ 246 GeV. (7.67)
7.4.5 Gauge boson self-interactions
The gauge boson self-interactions that are presently most relevant to experiments are the W+W−V
(V = Z,A) couplings which, in the LSM, have the following form:
LWWV = ie cot θW[(W+
µνW−µ −W−
µνW+µ)Zν +W+
µ W−ν Z
µν]
+ ie[(W+
µνW−µ −W−
µνW+µ)Aν +W+
µ W−ν A
µν]. (7.68)
Here Wµν = ∂µWν − ∂νWµ, Zµν = ∂µZν − ∂νZµ, and Aµν = ∂µAν − ∂νAµ. (Note that we usually
use Fµν instead of what we now denote as Aµν .)
The above interaction depends on only two parameters, e and θW . It is much more restrictive
than the most general one. Moreover, these parameters can be measured from other sectors of the
theory. Thus, it can be used to test the theory. For example, the most general CP invariant form
of the couplings is given by (see for example [12] and references therein)
LWWV = −igWWV
[gV1 (W+
µνW−µ −W−
µνW+µ)V ν + κVW
+µ W
−ν V
µν +
(λV /M2W )W+ν
µ W−ρν V µ
ρ −igV5 εµνρσ(W+µ ∂ρW
−ν −W−
ν ∂ρW+µ )Vσ
], (7.69)
where gWWA = e, gWWZ = e cot θW and, due to EM gauge invariance, 1− gγ1 = gγ5 = 0. (Note that
we use here γ interchangeably with A.) The LSM predicts the following values for the parameters:
gZ1 = κγ = κZ = 1, λγ = λZ = gZ5 = 0. (7.70)
92
Table 7.1: The LSM lepton interactions
interaction force carrier coupling range
electromagnetic (EM) γ eQ long
NC weak Z0 e(T3−s2WQ)
sW cWshort
CC weak W± g short
Yukawa h y` short
The experimental values are
gZ1 = 0.98± 0.02, κZ = 0.92± 0.07, λZ = −0.09± 0.07,
κγ = 0.97± 0.04, λγ = −0.03± 0.02, gZ5 = 0.07± 0.09, (7.71)
in very good agreement with the SM predictions.
Last we present the quartic gauge boson couplings within the SM
L4V = g2 cos2 θW(W+µ W
−ν Z
µZν −W+µ W
−µZνZν)
+g2(W+µ W
−ν A
µAν −W+µ W
−µAνAν)
+g2
2
(W+µ W
−ν
) (W+µW−ν −W+νW−µ
)+e2 cot θW
[(W+µ W
−ν
)(ZµAν − ZνAµ)− 2W+
µ W−µZνA
ν]. (7.72)
The experimental precision is not yet good enough to significantly probe them.
7.4.6 Summary
Leptons have four types of interactions. These interactions are summarized in Table 7.1.
The name weak interactions is somewhat misleading. In fact, the weak coupling g is larger
than the electromagnetic coupling e. The more important feature is that the weak interactions are
mediated by massive vector bosons, and consequently they are short range, while the electromag-
netic interactions are mediated by the massless photon and hence they are long range. It is the
short range of the weak interactions which makes the neutrinos, which do not have electromagnetic
interactions, very hard to detect.
93
7.5 Global symmetries and parameters
7.5.1 The interaction basis and the mass basis
The interaction basis is the one where all fields have well-defined transformation properties under
the imposed symmetries of the Lagrangian. In particular, in this basis, the gauge interactions are
universal.
If there are several fields with the same quantum numbers, then the interaction basis is not
unique. The kinetic and gauge terms are invariant under a global unitary transformation among
these fields. On the other hand, the Yukawa terms and the fermion mass terms are, in general, not
invariant under a unitary transformation among fermion fields with the same quantum numbers,
fi → U fjifi, while the Yukawa terms and scalar potential are, in general, not invariant under a
unitary transformation among scalar fields with the same quantum numbers, si → U sjisi. Thus, by
performing such transformations, we are changing the interaction basis.
In the LSM, there are three copies of (2)−1/2 fermions and three copies of (1)−1 fermions.
Transforming the first by a unitary transformation UL, and the latter by an independent unitary
transformation UR, the Yukawa matrix Y e is transformed into ULYeU †R. The matrix Y e is a 3× 3
complex matrix and thus has, in general, nine complex parameters. We can always find a bi-
unitary transformation that would make Y e real and diagonal, and thus depend on only three real
parameters:
Y e → ULYeU †R = Y e
diag = diag(ye, yµ, yτ ). (7.73)
Often one chooses a basis where the number of Lagrangian parameters is minimal, as is the case
with the diagonal basis of Eq. (7.73). One could work in any other interaction basis. However,
when calculating physical observables, only the eigenvalues of Y †e Ye would play a role. Using the
diagonal basis just provides a shortcut to this result.
The mass basis is the one where all fields have well defined transformation properties under the
symmetries that are not spontaneously broken and are mass eigenstates. The fields in this basis
correspond to the particles that are eigenstates of free propagation in spacetime. The Lagrangian
parameters in this basis correspond directly to physical observables.
For the LSM, the interaction eigenstates have well defined transformation properties under the
SU(2)L × U(1)Y symmetry:
Wa(3)0, B(1)0, LL1,2,3(2)−1/2, ER1,2,3(1)−1, φ(2)+1/2. (7.74)
The mass eigenstates have well defined electromagnetic charge and mass:
W±, Z0, A0, e−, µ−, τ−, νe, νµ, ντ , h0. (7.75)
The number of degrees of freedom is the same in both bases. To verify this statement one has
to take into account the following features:
94
1. Wa and B have only transverse components, while W± and Z0 have also a longitudinal one.
2. LL and ER are Weyl fermions, while e, µ, τ are Dirac fermions.
3. φ is a complex scalar, while h is a real one.
The three electromagnetically neutral neutrino states are, at the renormalizable level, massless
and, in particular, degenerate. Thus, there is freedom in choosing the mass basis for the neutrinos.
We choose the basis where the W± couplings to the charged lepton mass eigenstates are diagonal.
One could choose a different mass basis, related to the one we chose by a unitary transformation
of the three neutrino fields, νe
νµ
ντ
→ν1
ν2
ν3
= U †
νe
νµ
ντ
. (7.76)
Let us see how the decay rate of the W -boson into an electron and a neutrino is calculated in this
basis. Since the experiment does not distinguish between ν1, ν2, ν3, one has to sum over all three
species:
Γ(W+ → e+ν) =∑
i=1,2,3
Γ(W+ → e+νi) = Γ(W+ → e+νe)(|Ue1|2 + |Ue2|2 + |Ue3|2)
= Γ(W+ → e+νe). (7.77)
Thus, if the neutrinos are degenerate, the elements of the matrix U have no physical significance;
They cannot appear in any physical observable. Our choice of basis (νe, νµ, ντ ) provides a shortcut
to this result.
Later we will see that non-renormalizable terms provide the neutrinos with (non-degenerate)
masses, and then the mass basis becomes unique.
7.5.2 The LSM parameters
There are seven independent parameters in the LSM. This implies that, in principle, we need to
perform seven appropriate measurements and then we can make predictions for any other processes
involving the leptons and the Higgs boson that are mediated by the EM, weak or Yukawa inter-
actions. It is convenient to think of these experiments as measurements of the seven parameters.
There are various ways in which we can choose the seven independent parameters, for example,
g, g′, v, λ, ye, yµ, yτ . (7.78)
Another example would be mW , mZ , mh, me, mµ, mτ , and α. This example shows that by
measuring the spectrum of the LSM fermions and the fine structure constant, all other interaction
rates are predicted.
95
A good choice of parameters would be one where the experimental errors in their determination
are the smallest. As of now this set is the following:
α, GF , me, mµ, mτ , mZ , mh. (7.79)
By now, all seven parameters have been measured, with mh (or, equivalently, λ in the previous
list) the latest addition. In the following we use the above seven parameters to show a few more
examples of how the LSM has been tested.
7.5.3 Accidental symmetries
If we set the Yukawa couplings to zero, LYuk = 0, the LSM gains a large accidental global symmetry:
GglobalLSM (Y e = 0) = U(3)L × U(3)E = SU(3)L × SU(3)E × U(1)L × U(1)E. (7.80)
The (LL1, LL2, LL3) fields transform as (3, 1)qL,0 under this symmetry. The (ER1, ER2, ER3) fields
transform as (1, 3)0,qR under the symmetry. All other fields are singlets, (1, 1)0,0. Concerning the
U(1) factors, the choice of qL and qR is arbitrary (except that both must not equal zero). It is
customary to normalize these charges to +1.
The Yukawa couplings break this symmetry into the following subgroup:
GglobalLSM = U(1)e × U(1)µ × U(1)τ . (7.81)
The U(1) factors are called electron number, muon number, and tau number, respectively. The
charges of νeL and e are (1, 0, 0), of νµL and µ are (0, 1, 0), and of ντL and τ are (0, 0, 1). It is
thus a subgroup of GglobalLSM and is conserved. Thus, electron number, muon number, tau number,
and total lepton number are accidental symmetries of the LSM. This situation allows, for example,
the muon decay mode µ− → e−νeνµ, but forbids µ− → e−γ and µ− → e−e+e−. Also scattering
processes such as e+e− → µ+µ− are allowed, but e+µ− → µ+e− is forbidden.
it is useful to also define total lepton number which is the sum of these three lepton flavor
numbers and it corresponds to a U(1)L symmetry. Clearly total lepton number is also an accidental
symmetry of the leptonic SM. The conservation of total lepton number explains why Majorana
masses for neutrinos are not allowed within the LSM.
These accidental symmetries are, however, all broken by nonrenormalizable terms of the form
(1/Λ)LLiLLjφφ. If the scale Λ is high enough, these breaking effects are very small. It means
that the “forbidden” processes mentioned above are expected to occur, but at very low rates. It
also implies that we expect very small Majorana masses. We discuss these points in detail in
Chapter 12.
Finally, let us point out that the breaking of the symmetry (7.80) into (7.81) is by the Yukawa
couplings — ye, yµ, yτ — which are small, ofO(10−6, 10−3, 10−2), respectively. Thus, the full [U(3)]2
remains an approximate symmetry of the LSM.
96
7.5.4 Discrete symmetries: C, P and CP
The LSM violates C and P as it is a chiral theory: there are more LH DoF than there are RH
ones. This implies C and P violation.
On the other hand, CP is conserved by the LSM. This can be seen by the fact that the seven
parameters of the models defined in Eqs. (7.78) or (7.79) can be chosen real. In fact, we explicitly
found a basis where this is the case.
Experimentally, P violation was demonstrated in many ways. One example is given by the
measurement of τ polarization Pτ (also denoted as Aτ ) in the Z → τ+τ− decay. It is given by
Pτ ≡σR − σLσR + σL
, (7.82)
where σR(σL) is the cross section of producing a RH (LH) tau in Z decay. In a parity invariant
theory, Pτ = 0. The LSM prediction at tree level can be read from the couplings of the Z to the
fermions in Eq. (7.52):
Pτ =(1/2 + s2
W )2 − (s2W )2
(1/2 + s2W )2 + (s2
W )2≈ 0.16, (7.83)
where we used s2W = 0.23. Experimentally [10],
Pτ = 0.143± 0.004 (7.84)
which corresponds to sin2 θW = 0.2320± 0.0005. The small deviations from other determinations
that we discuss in this section are mainly due to higher order corrections that we neglect.
7.6 Low Energy Tests of the LSM
Nowadays, experiments produce the W and Z bosons and measure their properties directly. It
is interesting to understand, however, how the SM was tested at the time before the energy in
experiments became high enough for such direct production. It is not only the historical aspect
that is interesting; It is also important to see how we can use low energy data to understand shorter
distances.
7.6.1 CC weak interactions: Quasi-elastic neutrino–electron scattering
Let us compare the charged current contributions to the two elastic scattering processes
νµe− → νeµ
−, νee− → νµµ
−. (7.85)
(These processes are sometimes called “inverse muon decays”.) Since these are flavor changing
processes, in the LSM the only contributions come from W exchange. We consider scattering with
97
a center-of-mass energy in the range m2µ s m2
W . In particular, we can consider the leptons
massless.
We define θ to be the angle between the incoming (anti)neutrino and the outgoing muon.
Then cos θ = 1 corresponds to backward scattering of the beam particle. For the νee− scattering,
νL and `L have positive and negative helicities, respectively. Thus, in the center of mass frame,
their spins are in the same direction. Therefore (Jz)i = +1. When the scattering is backwards, the
respective momenta of the antineutrinos and the charged leptons change to the opposite directions,
and so do their helicities: (Jz)f = −1. Therefore, backward ν` scattering is forbidden by angular
momentum conservation. In fact, the process νee→ νµµ proceeds entirely in a J = 1 state with net
helicity +1. That is, only one of the three states is allowed. In contrast, in νµe→ νeµ, backward
scattering has (JZ)i = (JZ)f = 0 and all helicity states are allowed. The full calculation yields, for
m2µ s m2
W , and working in the electron rest frame
dσ(νµe− → νeµ
−)
dΩ=G2F s
4π2,
dσ(νee− → νµµ
−)
dΩ=G2F s
16π2(1− cos θ)2, (7.86)
σ(νµe− → νeµ
−) =G2F s
π, σ(νee
− → νµµ−) =
G2F s
3π,
with s = 2meEν . In particular, the ratio of cross sections is predicted by the LSM. To leading
order it does not depend on any parameter:
σ(νµe− → νeµ
−)
σ(νee− → νµµ−)= 3. (7.87)
7.6.2 NC weak interactions: neutrino–electron scattering
There are several observables that can be used to test neutral currents interactions. The first
example is low energy elastic scattering:
νµe− → νµe
−, νµe− → νµe
−. (7.88)
Since the W -boson couples diagonally, it does not couple to a νµe− pair. Since neutrinos are
involved, these processes cannot be mediated by photons. Consequently, the νµe → νµe and
νµe− → νµe
− scattering processes are mediated purely by the Z-boson.
We can use the ratio
R ≡ σ(νµe→ νµe)
σ(νµe→ νµe)(7.89)
to fix sin2 θW . Defining
geL = −1/2 + sin2 θW , geR = sin2 θW , (7.90)
geV = geL + geR = −1/2 + 2 sin2 θW , geA = geL − geR = −1/2,
98
working in the electron rest frame the results read
σνµ =G2F s
2π
[(geL)2 +
1
3(geR)2
], σνµ =
G2F s
2π
[(geR)2 +
1
3(geL)2
]. (7.91)
with s = 2meEν .
The experimental result of the scattering rates give geA = −0.507 ± 0.014 and geV = −0.040 ±0.015 [?] (in the “Electroweak model and constraints on new physics” review) that leads to
sin2 θW = 0.230± 0.008. (7.92)
in good agrrment with other determinations.
7.6.3 Forward-backward asymmetry
We consider e+e− → µ+µ− scattering. This process is mediated by both QED interactions and
NC weak interactions. The former are vector-like contributions, and therefore conserve parity.
The latter are parity violating. The interference between the photon-mediated contribution and
the Z-mediated contribution leads to a forward-backward asymmetry, which is a manifestation of
parity violation.
The forward-backward asymmetry is defined as follows:
AFB =σF − σBσF + σB
, σF = 2π∫ 1
0d cos θ
dσ
d cos θ, σB = 2π
∫ 0
−1d cos θ
dσ
d cos θ. (7.93)
A detailed calculation gives, for m2µ s m2
Z ,
dσ
dΩ=α2
4s
[1 + cos2 θ − 4g2
A
c2W s
2W
s
m2Z
cos θ
], (7.94)
yielding
AFB(m2µ s m2
Z) = − 3g2A
2c2W s
2W
s
m2Z
. (7.95)
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Homework
Question 7.1: Some algebra
1. Starting from Eq. (7.28) and using the definitions of Eqs. (7.29) and (7.30) derive Eq. (7.31).
2. Starting from Eq. (7.42) and using the definition of θW in Eq. (7.29) and of Q in Eq. (7.19)
derive both sides of Eq. (7.51).
3. Using the gauge boson kinetic terms from Eq. (7.24), and the definitions in Eqs. (7.29), (7.30)
and (7.49) derive Eq. (7.68).
Question 7.2: Lepton universality
Here we consider muon and tau decays in the SM.
1. Find in the PDG the main decay mode of the muon. What is its width?
2. Find in the PDG the bound on the decay
Γ(µ→ eγ). (7.96)
What is the SM prediction to this mode? Give a short explanation not just a number.
3. Draw the tree level Feynman diagram for the leading muon decay.
4. We now move to tau decays. Based on lepton universality, what do you expect for the
following ratiosΓ(τ → eνν)
Γ(τ → µνν),
Γ(µ→ eνν)
Γ(τ → eνν). (7.97)
Compare your results with the PDG and explain any small deviation that you find compared
with your predictions.
100
Question 7.3: A modified leptonic SM
Consider a model where the gauge group is SU(2)L × U(1)Y broken to U(1)EM . we consider
one generation of leptons. We assume that the leptons and Higgs representations are
LL(3)−1, ER(2)−3/2, φ(2T + 1)1/2 , (7.98)
where T is undetermined.
1. Can we have a bare mass term for the leptons?
2. What are the possible values of T such that some of the leptons are massive after SSB?
Namely, what T can give a Yukawa term of the form LLER φ?
3. Explain why only T = 1/2 is consistent with the observed masses of the W and Z bosons.
From now on we then assume T = 1/2.
4. By considering the components of the fields and the Yukawa term, argue that the model
contains a massless q = 0 state (the neutrino), a massive, q = −1 Dirac state (the electron)
and a massive, q = −2 Dirac state that we did not observed in nature, that we denote as λ.
5. Show that mλ =√
2me. (Note: this question do require some amount of calculation. Thus,
if you get stuck move on to the next item.)
6. Write down the fermionic charged current interaction terms in the Lagrangian, that is, the
coupling of the W to the fermions. You are asked to keep track of factors of 2. Recall that
the spin-1/2 matrices are
1
2
(0 1
1 0
),
1
2
(0 −ii 0
),
1
2
(1 0
0 −1
). (7.99)
and the spin-1 matrices are
1√2
0 1 0
1 0 1
0 1 0
, 1√2
0 −i 0
i 0 −i0 i 0
,
1 0 0
0 0 0
0 0 −1
. (7.100)
7. Is the λ stable? Explain.
8. Calculate the ratioΓ(W− → eR
+λ−−R )
Γ(W− → eL+λ−−L ). (7.101)
Hint: make sure you are not doing a long calculation.
101
9. Is this model a candidate to replace the SM? Explain.
Question 7.4: ρ for a general Higgs
In the SM the Higgs transforms under SU(2)L × U(1)Y as (2)1/2. However, any scalar that is
charged under the gauge group and acquires a vev will break the SM gauge symmetry. We assume
that the Higgs potential is given by Eq. (7.17).
1. Consider a scalar φ that transforms as (2T + 1)Y . Since SU(2) is a non Abelian group,
2T + 1 has to be a positive integer, that is, T is a non negative half integer. Since U(1)
is Abelian, a priori Y can assume any real value. Yet, we like φ to be responsible for the
SU(2)L × U(1)Y → U(1)EM breaking where we define Q = T3 + Y . This definition restricts
the possible values for Y . Find these values.
2. Show that ρ, defined as
ρ ≡ m2W
m2Z cos2 θW
, (7.102)
is given by
ρ =T (T + 1)− Y 2
2Y 2. (7.103)
Hint: Recall that the 2T + 1 dim. representation of SU(2) is given by
T3 = diagT, T − 1, T − 2, . . . ,−T (7.104)
T1 =
0 a1 0 . . . 0
a1 0 a2 0...
0 a2. . . . . .
...... 0
. . . 0 a2T
0 . . . . . . a2T 0
T2 =
0 ia1 0 . . . 0
−ia1 0 ia2 0...
0 −ia2. . . . . .
...... 0
. . . 0 ia2T
0 . . . . . . −ia2T 0
where
ai =
√T (T + 1)− (T − i)(T − i+ 1)
2(7.105)
3. For T > 0 and Y = 0 one can see from Eq. (7.103) that ρ →∞ independent of T . Explain
this result using symmetry arguments.
4. Suppose that there exist several Higgs representations (i = 1, . . . , N) whose neutral members
acquire vevs vi. Find ρ in terms of vi, Ti and Yi.
5. Assume that, in addition to the usual Higgs doublet T = 1/2, Y = 1/2 with vev vW , there
exists one other multiplet Ti, Yi which acquires a much smaller vev vi. Find δρ ≡ ρ− 1 to
first order in (vi/vW )2.
102
6. Assume that experimentally −0.01 ≤ δρ ≤ +0.005. Find the constraint on (vi/vW )2 for the
following multiplets: (5)−1 and (4)3/2.
7. From Eq. (7.103) it is clear that ρ = 1 for all 3Y 2 = T (T+1) multiplets. Since experimentally
ρ is very close to 1, we assume that the SM Higgs is one of these multiplets. While from the
consideration of ρ alone there is no difference which multiplet we take, in the SM we do make
a choice and take T = 1/2 and Y = 1/2. What is the advantage of the SM Higgs compare
to the other possible choices?
Question 7.5: Left Right Symmetric (LRS) model for leptons
Assume the Left Right Symmetric (LRS) model for leptons: The gauge group is
GLR = SU(2)L × SU(2)R × U(1)X . (7.106)
The fermions transform as
LL(2, 1)−1, LR(1, 2)−1, (7.107)
where our notation is (NL, NR)qX such that NL (NR) is the representation under SU(2)L (SU(2)R)
and qX is the charge under U(1)X . The left handed leptons are denote as LL = (νL, e−L) and the
right handed leptons as LR = (νR, e−R). We use WL, WR and C to denote the seven gauge fields,
and gL, gR and gX to denote the three different coupling constants of the SU(2)L, SU(2)R and
U(1)X groups respectively. Note that we have extended both the gauge group and the lepton
content of the SM.
1. Write down the commutation relations between the various generators [the analogue of (7.5)].
2. Write down the covariant derivative [the analogue of (7.9)].
3. The LSM group SU(2)L×U(1)Y has to be included in the LRS group which is the case when
U(1)Y ⊂ SU(2)R × U(1)X . Find the linear combination of the LRS generators which gives
the SM generator Y . Then, find the linear combination of generators which gives the electric
charge Q [the analogue of (7.19)].
4. What are the SU(2)L × U(1)Y charges of the right handed neutrino, νR?
5. Write down explicitly the charged current interactions of the leptons [the analogue of (7.60)].
It is enough to write it for one generation only.
6. In the SM muon decay is mediates by WL. In the LRS model there is one more tree level
Feynman diagram that contributes to muon decay. Draw this new diagram.
103
7. Assuming that there is no mixing between WL and WR find the ratio of the new amplitude
to the SM amplitude in terms of the coupling constants and the masses of the gauge bosons.
8. The Bµ field of the SM must be a linear combination of the W µR3 and Cµ. The orthogonal
linear combination is called Z ′µ. Write the expression of Bµ and Z ′µ [the analogue of (7.30)].
Use θR to define the mixing angle.
9. Express gR and gX in terms of g′ and θR [the analogue of (7.49)].
10. Find the coupling of the Z ′ to the fermions in terms of T3R and Y [the analogue of (7.52)].
11. We now add a scalar, φ, to the model and demand that it couples to the fermions. (We need
it in order that after it acquires a vev the fermion will be massive.) Namely, we want that a
term of the form
LLφLR + h.c. (7.108)
will be allowed. Find the representation of the scalar under GLR. What are the electric
charges of the various components of the scalar field?
12. The first guess for the SSB sector may be to use the scalar that we know we must add,
namely the one that couples to the fermions. As you just found it is
Φ(2, 2)0 =
(Φ0
1 Φ+1
Φ−2 Φ02
)(7.109)
where the indecies are the electric charges and we explicitly wrote Φ in a matrix notation
such that the transformation law under SU(2)L × SU(2)R is Φ→ ULΦU †R. We now assume
that Φ acquires a vev
〈Φ〉 =
(k1 0
0 k2
)(7.110)
Explain why we require that only the neutral components of Φ acquire a vev. What is the
symmetry breaking pattern generated by these vevs?
13. This choice, however, does not give a realistic model. For example, the lightest charged
gauge boson is an equal mixture of WL and WR. (The calculation is somewhat lengthy, but
straighforward. You are encourage to do it and check the above statment yourself.) In order
to solve the problem we require a SSB pattern that will result in mWR mWL
. This is done
by adding two more scalars
∆L = (3, 1)2 ∆R = (1, 3)2 (7.111)
Write each of the scalars as a triplet (namely, as a vector with three components). What is
the hypercharge (Y ) and the electric charge (Q) of each component?
104
14. We assume that the neutral components of ∆L and ∆R acquire vevs, vL and vR, respectively.
What is the symmetry breaking pattern generated by each of these vevs? (Namely, assume
first that the only vev is vL and write down the SSB pattern. Then repeat the question
assuming that the only vev is vR.)
15. We assume the following hierarchy
vL ki vR . (7.112)
Now you are going to show why we assume these relations. Which of the four vevs (k1,
k2, vL and vR) affect the ρ = 1 relation? What bounds on the vevs can you deduce from
the experimental bound |ρ − 1| < 0.01 ? (You can neglect mixing between the WL and WR
bosons.)
16. Assuming the above relations, estimate the masses of the W±L and W±
R in terms of the
vevs and gauge couplings (do not worry about numbers and sub-leading corrections). What
bounds on the vevs can you deduce from the experimental bound on the right handed am-
plitude in muon decay |gVRR| < 0.033 ? (You can assume that gVLL = 1. Then, gVRR is nothing
but the ratio between the right handed and left handed muon decay amplitudes.)
105
Chapter 8
The Standard Model
8.1 Defining the Standard Model
The Standard Model (SM) is defined as follows:
(i) The symmetry is a local
SU(3)C × SU(2)L × U(1)Y . (8.1)
(ii) The pattern of spontaneous symmetry breaking is as follows:
SU(3)C × SU(2)L × U(1)Y → SU(3)C × U(1)EM (QEM = T3 + Y ). (8.2)
(iii) There are three fermion generations, each consisting of five different representations:
QLi(3, 2)+1/6, URi(3, 1)+2/3, DRi(3, 1)−1/3, LLi(1, 2)−1/2, ERi(1, 1)−1, i = 1, 2, 3.
(8.3)
There is a single scalar multiplet:
φ(1, 2)+1/2. (8.4)
We use the notation (A,B)Y such that A (B) is the irrep under SU(3)C (SU(2)L, and Y is the
hypercharge. The fermions that transform as triplets of SU(3)C are called quarks, while those
that transform as singlets of SU(3)C are called leptons.
8.2 The Lagrangian
As explained in previous chapters, the most general renormalizable Lagrangian with scalar and
fermion fields can be decomposed into
L = Lkin + Lψ + LYuk + Lφ. (8.5)
106
It is now our task to find the specific form of the Lagrangian made of the fermion fields QLi, URi,
DRi, LLi and ERi (8.3), and the scalar field φ (8.4), subject to the gauge symmetry (8.1) and
leading to the SSB of Eq. (8.2).
8.2.1 Lkin and the gauge symmetry
The gauge group is given in Eq. (8.1). It has twelve generators: eight La’s that form the SU(3)
algebra, three Tb’s that form the SU(2) algebra, and a single Y that correspond to the U(1) group:
[La, Lb] = ifabcLc, [Ta, Tb] = iεabcTc, [La, Tb] = [La, Y ] = [Tb, Y ] = 0. (8.6)
Thus there are three independent coupling constants in Lkin: gs related to the SU(3)C subgroup,
g related to the SU(2)L subgroup, and g′ related to the U(1)Y subgroup.
The local symmetry requires twelve gauge bosons degrees of freedom, eight in the adjoint
representation of SU(3)C , three in the adjoint representation of SU(2)L, and one related to the
U(1)Y symmetry:
Gµa(8, 1)0, W µ
a (1, 3)0, Bµ(1, 1)0. (8.7)
The corresponding field strengths are given by
Gµνa = ∂µGν
a − ∂νGµa − gsfabcG
µbG
νc ,
W µνa = ∂µW ν
a − ∂νW µa − gεabcW
µb W
νc ,
Bµν = ∂µBν − ∂νBµ. (8.8)
The covariant derivative is given by
Dµ = ∂µ + igsGµaLa + igW µ
b Tb + ig′Y Bµ. (8.9)
For the SU(3)C triplets La = 12λa (λa are the Gell-Mann matrices), while for the SU(3)C singlets,
La = 0. For the SU(2)L doublets Tb = 12σb (σb are the Pauli matrices), while for the SU(2)L
singlets, Tb = 0. For SU(3)C adjoints, (La)bc = fabc and for SU(2)L adjoints, (Ta)bc = εabc, which
have already been used in writing (8.8).
Lkin includes the kinetic terms of all the fields:
Lkin = − 1
4Gµνa Gaµν −
1
4W µνb Wbµν −
1
4BµνBµν
− iQLiD/QLi − iURiD/URi − iDRiD/DRi − iLLiD/LLi − iERiD/ERi− (Dµφ)†(Dµφ). (8.10)
Explicitly, the covariant derivatives acting on the various fermion fields are given by
DµQL =(∂µ +
i
2gsG
µaλa +
i
2gW µ
b σb +i
6g′Bµ
)QL,
107
DµUR =(∂µ +
i
2gsG
µaλa +
2i
3g′Bµ
)UR,
DµDR =(∂µ +
i
2gsG
µaλa −
i
3g′Bµ
)DR,
DµLL =(∂µ +
i
2gW µ
b σb −i
2g′Bµ
)LL,
DµER = (∂µ − ig′Bµ)ER. (8.11)
8.2.2 LψThere are no mass terms for the fermions of the SM,
Lψ = 0. (8.12)
In Chapter 7 we have seen that this is the case for leptons. Note that a larger symmetry means
stronger constraints, hence it is impossible that lepton masses would become allowed when the
gauge symmetry is extended to include SU(3)C . As concerns the quarks, we cannot write Dirac
mass terms because they are assigned to chiral representations of the SU(2)L × U(1)Y gauge
symmetry. We cannot write Majorana mass terms for the quarks because they all have Y 6= 0.
8.2.3 LYuk
The Yukawa part of the Lagrangian is given by
LYuk = Y uijQLiURj φ+ Y d
ijQLiDRj φ+ Y eijLLiERj φ+ h.c., (8.13)
where i, j = 1, 2, 3 are flavor indices, and φa = εabφ∗b . The Yukawa matrices Y u, Y d and Y e are
general complex 3× 3 matrices of dimensionless couplings.
For the leptons, the situation is the same as what we discussed in Chapter 7. For the quarks,
because there are two Yukawa matirices, the situation is different in a significant way.
We start with the leptons. Without loss of generality, we can use a bi-unitary transformation,
Y e → Ye = UeLYeU †eR, (8.14)
to change the basis to one where Y e is diagonal and real:
Y e = diag(ye, yµ, yτ ). (8.15)
In the basis defined in Eq. (8.15), we denote the components of the lepton SU(2)-doublets, and
the three lepton SU(2)-singlets, as follows:(νeL
eL
),
(νµL
µL
),
(ντL
τL
); eR, µR, τR, (8.16)
where e, µ, τ are ordered by the size of ye, yµ, yτ (from smallest to largest).
108
We now move to the quarks. Similarly, without loss of generality, we can use a bi-unitary
transformation,
Y u → Yu = VuLYuV †uR, (8.17)
to change the basis to one where Y u is diagonal and real:
Y u = diag(yu, yc, yt). (8.18)
In the basis defined in Eq. (8.18), we denote the components of the quark SU(2)-doublets, and
the quark up SU(2)-singlets, as follows:(uL
duL
),
(cL
dcL
),
(tL
dtL
); uR, cR, tR, (8.19)
where u, c, t are ordered by the size of yu, yc, yt (from smallest to largest).
We can use yet another bi-unitary transformation,
Y d → Yd = VdLYdV †dR, (8.20)
to change the basis to one where Y d is diagonal and real:
Y d = diag(yd, ys, yb). (8.21)
In the basis defined in Eq. (8.21), we denote the components of the quark SU(2)-doublets, and
the quark down SU(2)-singlets, as follows:(udL
dL
),
(usL
sL
),
(ubL
bL
); dR, sR, bR, (8.22)
where d, s, b are ordered by the size of yd, ys, yb (from smallest to largest).
If VuL 6= VdL, as is the general case, then the interaction basis defined by (8.18) is different
from the interaction basis defined by (8.21). In the former, Y d can be written as a unitary matrix
times a diagonal one,
Y u = Y u, Y d = V Y d. (8.23)
In the latter, Y u can be written as a unitary matrix times a diagonal one,
Y d = Y d, Y u = V †Y u. (8.24)
In either case, the matrix V is given by
V = VuLV†dL, (8.25)
where VuL and VdL are defined in Eqs. (8.17) and (8.20), respectively. Note that VuL, VuR, VdL and
VdR depend on the basis from which we start the diagonalization. The combination V = VuLV†dL,
however, does not. This is a hint that V is physical. Indeed, below we see that it plays a crucial
role in the charged current weak interactions.
109
8.2.4 Lφ and spontaneous symmetry breaking
The scalar field is a singlet of the SU(3)C group. Thus, the form of Lφ is the same as in the LSM,
Lφ = −µ2φ†φ− λ(φ†φ
)2. (8.26)
Choosing µ2 < 0 and λ > 0 leads, as in the LSM, to spontaneous symmetry breaking, with
|〈φ〉| = v/√
2 (v2 = −µ2/λ). Since φ is SU(3)C singlet, the SU(3)C subgroup remains unbroken,
and the pattern of spontaneous symmetry breaking is as required by Eq. (8.2).
The spontaneous breaking of SU(2)L×U(1)Y into U(1)EM allows us to distinguish the compo-
nents of the SU(2)L-doublet fermions fields by their electromagnetic charges. For the lepton fields
of Eq. (7.20), we presented the charges in Eq. (7.21): −1 for the T3 = −1/2 member and 0 for
the T3 = +1/2 member. Writing down the two components of the SU(2)L-doublet quark fields as
QL =
(UL
DL
), (8.27)
we have the following EM charges of the quark fields:
q(UL) = +2
3, q(DL) = −1
3, q(UR) = +
2
3, q(DR) = −1
3. (8.28)
In what follows, we often call the q = −1/3 quarks “down-type quarks,” and the q = +2/3 quarks
“up-type quarks.”
8.2.5 Summary
The renormalizable part of the Standard Model Lagrangian is given by
LSM = − 1
4Gµνa Gaµν −
1
4W µνb Wbµν −
1
4BµνBµν − (Dµφ)†(Dµφ)
− iQLiD/QLi − iURiD/URi − iDRiD/DRi − iLLiD/LLi − iERiD/ERi+(Y uijQLiURj φ+ Y d
ijQLiDRj φ+ Y eijLLiERj φ+ h.c.
)− λ
(φ†φ− v2/2
)2, (8.29)
where i, j = 1, 2, 3.
8.3 The Spectrum
8.3.1 Bosons
Given the spontaneous breaking of the SU(2)L × U(1)Y symmetry to the U(1)EM subgroup, the
spectrum of the electroweak gauge bosons remains the same as in the LSM: three massive vector
110
bosons, W± and Z0, and a massless photon, A0. Furthermore, since the breaking is induced by an
SU(2)L-doublet, the ρ ≡ m2W/(m
2Z cos2 θW ) = 1 relation holds.
The new ingredient is the existence of a gluon in the octet representation of SU(3)C . Since the
SU(3)C gauge symmetry remains unbroken, the gluon is massless.
As concerns scalars, the three would-be Goldstone bosons become the longitudinal components
of the three massive vector bosons. The fourth scalar degree of freedom is the Higgs boson h, a
real massive scalar field,
8.3.2 Fermions
Since the SM allows no bare mass terms for the fermions, their masses can only arise from the
Yukawa part of the Lagrangian, which is given in Eq. (8.13). Indeed, with 〈φ0〉 = v/√
2, Eq.
(8.13) has a piece that corresponds to charged lepton masses:
me =yev√
2, mµ =
yµv√2, mτ =
yτv√2, (8.30)
a piece that corresponds to up-type quark masses,
mu =yuv√
2, mc =
ycv√2, mt =
ytv√2, (8.31)
and a piece that corresponds to down-type quark masses,
md =ydv√
2, ms =
ysv√2, mb =
ybv√2. (8.32)
We conclude that all charged fermions acquire Dirac masses as a result of the spontaneous symme-
try breaking. The key to this feature is that, while the charged fermions are in chiral representations
of the full gauge group SU(3)C × SU(2)L × U(1)Y , they are in vector-like representations of the
SU(3)C × U(1)EM group:
• The LH and RH charged lepton fields, e, µ and τ , are in the (1)−1 representation.
• The LH and RH up-type quark fields, u, c and t, are in the (3)+2/3 representation.
• The LH and RH down-type quark fields, d, s and b, are in the (3)−1/3 representation.
On the other hand, as is the case discussed in Section 7.3.3 the neutrinos remain massless:
mνe = mνµ = mντ = 0. (8.33)
The experimental values of the charged fermion masses are
me = 0.510998910(13) MeV, mµ = 105.658367(4) MeV, mτ = 1776.82(16) MeV,
mu = 1.5− 3.1 MeV, mc = 1.29+0.05−0.11 GeV, mt = 172.9± 0.12 GeV,
md = 4.1− 5.7 MeV, ms = 100+30−20 MeV, mb = 4.9+0.18
−0.06 GeV, (8.34)
where the quark masses are given at a scale µ = 2 GeV. We discuss the how the masses of the
quarkes are dtermined in more details in Chapter 10.
111
Table 8.1: The SM particles
particle spin color Q mass [v]
W± 1 (1) ±1 12g
Z0 1 (1) 0 12
√g2 + g′2
A0 1 (1) 0 0
g 1 (8) 0 0
h 0 (1) 0√
2λ
e, µ, τ 1/2 (1) −1 ye,µ,τ/√
2
νe, νµ, ντ 1/2 (1) 0 0
u, c, t 1/2 (3) +2/3 yu,c,t/√
2
d, s, b 1/2 (3) −1/3 yd,s,b/√
2
8.3.3 Summary
The mass eigenstates of the SM, their SU(3)C × U(1)EM quantum numbers, and their masses in
units of the VEV v, are presented in Table 8.1. All masses are proportional to the VEV of the
scalar field, v. For the three massive gauge bosons, and for the fermions, this is expected: In the
absence of spontaneous symmetry breaking, the former would be protected from acquiring masses
by the gauge symmetry and the latter by their chiral nature. For the Higgs boson, the situation
is different, as a mass-squared term does not violate any symmetry.
8.4 The interactions
In this section, we discuss the interactions of the fermion and scalar mass eigenstates of the
Standard Model. The QED interactions of the leptons have been presented in Chapters 3 and
7. The electromagnetic interactions of the quarks are dictated by their charges, presented in Eq.
(8.28), and can be obtained along the lines explained in Section 3.6.3. The QCD interactions of
the quarks have been presented in Chapter 5. Here we focus on the weak and Yukawa interactions,
that is, the couplings of fermions to the W , Z and h bosons.
8.4.1 Neutral current weak interactions
The Z couplings to fermions can be written as follows:
LZ,fermions =e
sin θW cos θW(T3i − sin2 θWQi) ψiZ/ψi . (8.35)
112
Using the T3 and Y assignments of the various fermion fields, we find the following types of Z
couplings:
L =e
sW cW
[−(
1
2− s2
W
)eLZ/eL + s2
W eRZ/eR +1
2νeLZ/νeL (8.36)
+(
1
2− 2
3s2W
)uLZ/uL −
2
3s2W uRZ/uR −
(1
2− 1
3s2W
)dLZ/dL +
1
3s2W dRZ/dR
]+(e, νe, u, d→ µ, νµ, c, s) + (e, νe, u, d→ τ, ντ , t, b).
The Z couplings are chiral, parity-violating, diagonal and universal.
Omitting common factors, particularly, a factor of e2/(4s2W c
2W ), and phase-space factors, we
obtain the following predictions for the Z decays into a one-generation fermion-pair of each type:
Γ(Z → νν) ∝ 1,
Γ(Z → `¯) ∝ 1− 4s2W + 8s4
W ,
Γ(Z → uu) ∝ 3(
1− 8
3s2W +
32
9s4W
),
Γ(Z → dd) ∝ 3(
1− 4
3s2W +
8
9s4W
). (8.37)
Putting s2W = 0.225, we obtain
Γν : Γ` : Γu : Γd = 1 : 0.505 : 1.74 : 2.24. (8.38)
Experiments measure the following average branching ratio into a single generation of each fermion
species:
BR(Z → νν) = (6.67± 0.02)%,
BR(Z → `¯) = (3.37± 0.01)%,
BR(Z → uu) = (11.6± 0.6)%,
BR(Z → dd) = (15.6± 0.4)%, (8.39)
which, using central values, give
Γν : Γ` : Γu : Γd = 1 : 0.505 : 1.74 : 2.34, (8.40)
in very nice agreement with the predictions
8.4.2 Charged current weak interactions
We now study the couplings of the charged vector bosons, W±, to fermion pairs. For the lepton
mass eigenstates, things are simple, because there exists an interaction basis that is also a mass
basis. Thus, the W interactions must be universal also in the mass basis:
− g√2
(νeL W/
+e−L + νµL W/+µ−L + ντL W/
+τ−L + h.c.). (8.41)
113
As concerns quarks, things are more complicated, since there is no interaction basis that is also
a mass basis. In the interaction basis where the down quarks are mass eigenstates (8.22), the W
interactions have the following form:
− g√2
(udL W/
+dL + usL W/+sL + ubL W/
+bL + h.c.). (8.42)
The Yukawa matrices in this basis have the form (8.24), and in particular, for the up sector, we
have
LuYuk = (udL usL ubL)V †Y u
uR
cR
tR
, (8.43)
which tells us straightforwardly how to transform to the mass basis:uL
cL
tL
= V
udL
usL
ubL
. (8.44)
Using Eq. (8.44), we obtain the form of the W interactions (8.42) in the mass basis:
− g√2
(uL cL tL) V W/ +
dL
sL
bL
+ h.c.. (8.45)
Recalling that V = VuLV†dL is basis independent, you can easily convince yourself that we would
have obtained the same form starting from any arbitrary interaction basis.
Eq. (8.45) reveals some important features of the model:
1. Only left-handed particles take part in charged-current interactions. Consequently, parity is
violated by these interactions.
2. The W couplings to the quark mass eigenstates are neither universal nor diagonal. The
universality of gauge interactions is hidden in the unitarity of the matrix V .
The matrix V is called the CKM matrix.
Omitting common factors (particularly, a factor of g2/4) and phase-space factors, we obtain
the following predictions for the W decays:
Γ(W+ → `+ν`) ∝ 1,
Γ(W+ → uidj) ∝ 3|Vij|2 (i = 1, 2; j = 1, 2, 3). (8.46)
The top quark is not included because it is heavier than the W boson. Taking this fact into
account, and the CKM unitarity relations
|Vud|2 + |Vus|2 + |Vub|2 = |Vcd|2 + |Vcs|2 + |Vcb|2 = 1, (8.47)
114
we obtain
Γ(W → hadrons) ≈ 2Γ(W → leptons). (8.48)
Experimentally,
BR(W → leptons) = (32.40± 0.27)%, BR(W → hadrons) = (67.60± 0.27)%, (8.49)
which leads to
Γ(W → hadrons)/Γ(W → leptons) = 2.09± 0.01, (8.50)
in beautiful agreement with the SM prediction. The (hidden) universality within the quark sector
is tested by the prediction
Γ(W → uX) = Γ(W → cX) =1
2Γ(W → hadrons). (8.51)
Experimentally,
Γ(W → cX)/Γ(W → hadrons) = 0.49± 0.04. (8.52)
We discuss more aspects of the phenomenology related to the CKM matrix in Section 11.
8.4.3 Interactions of the Higgs boson
The Higgs boson has self-interactions, weak interactions, and Yukawa interactions:
Lh =1
2∂µh∂
µh− 1
2m2hh
2 − m2h
2vh3 − m2
h
8v2h4 (8.53)
+ m2WW
−µ W
µ+
(2h
v+h2
v2
)+
1
2m2ZZµZ
µ
(2h
v+h2
v2
)
− h
v(me eL eR +mµ µL µR +mτ τL τR
+mu uL uR +mc cL cR +mt tL tR +md dL dR +ms sL sR +mb bL bR + h.c.).
Note that the Higgs boson couples diagonally to the quark mass eigenstates. The reason for
this is that the Yukawa couplings determined both the masses and the Higgs couplings to the
fermions. Thus, in the mass basis the Yukawa interactions are also diagonal. Formally we can see
it as follows. Let us start from an arbitrary interaction basis:
hDLYdDR = hDL(V †dLVdL)Y d(V †dRVdR)DR
= h(DLV†dL)(VdLY
dV †dR)(VdRDR)
= h(dL sL bL)Y d(dR sR bR)T . (8.54)
We conclude that the Higgs couplings to the fermion mass eigenstates are diagonal, but not uni-
versal. Instead, they are proportional to the fermion masses: the heavier the fermion, the stronger
the coupling.
115
Thus, the Higgs boson decay is dominated by the heaviest particle which can be pair-produced
in the decay. For mh ∼ 125 GeV, this is the bottom quark. Indeed, the SM predicts the following
branching ratios for the leading decay modes:
BRbb : BRWW ∗ : BRgg : BRτ+τ− : BRZZ∗ : BRcc = 0.58 : 0.21 : 0.09 : 0.06 : 0.03 : 0.03. (8.55)
The following comments are in order with regard to Eq. (8.55):
1. From the six branching ratios, three (b, τ, c) stand for two-body tree-level decays. Thus, at
tree level, the respective branching ratios obey BRbb : BRτ+τ− : BRcc = 3m2b : m2
τ : 3m2c .
QCD radiative corrections are significant and suppress the two modes with the quark final
states (b, c) compared to one with the lepton final state (τ).
2. The WW ∗ and ZZ∗ modes stand for the three-body tree-level decays, where one of the vector
bosons is on-shell and the other off-shell.
3. The Higgs boson does not have a tree-level coupling to gluons since it carries no color (and
the gluons have no mass). The decay into final gluons proceeds via loop diagrams. The
dominant contribution comes from the top-quark loop.
4. Similarly, the Higgs decays into final two photons via loop diagrams with small (BRγγ ∼0.002), but observable, rate. The dominant contributions come from the W and the top-
quark loops which interfere destructively.
Experimentally, the decays into final ZZ∗, WW ∗ and γγ have been established, and there are
evidences for the τ+τ− one.
8.4.4 Summary
Within the SM, quarks have five types of interactions. These interactions are summarized in Table
8.1.
8.5 Accidental symmetries and parameter counting
8.5.1 Accidental symmetries
If we set the Yukawa couplings to zero, LYuk = 0, the SM gains a large accidental global symmetry:
GglobalSM (Y u,d,e = 0) = U(3)Q × U(3)U × U(3)D × U(3)L × U(3)E, (8.56)
where U(3)Q has (Q1, Q2, Q3) transforming as an SU(3)Q triplet, and all other fields singlets,
U(3)U has (U1, U2, U3) transforming as an SU(3)U triplet, and all other fields singlets, U(3)D has
116
Table 8.1: The SM quark interactions
interaction force carrier coupling range
electromagneric γ eQ long
Strong g gs long
NC weak Z0 e(T3−s2WQ)
sW cWshort
CC weak W± gV short
Yukawa h yq short
(D1, D2, D3) transforming as an SU(3)D triplet, and all other fields singlets, U(3)L has (L1, L2, L3)
transforming as an SU(3)L triplet, and all other fields singlets, and U(3)E has (E1, E2, E3) trans-
forming as an SU(3)E triplet, and all other fields singlets.
The Yukawa couplings break this symmetry into the following subgroup:
GglobalSM = U(1)B × U(1)e × U(1)µ × U(1)τ . (8.57)
Under U(1)B, all quarks (antiquarks) carry charge +1/3 (−1/3), while all other fields are neutral. It
explains why proton decay has not been observed. Possible proton decay modes, such as p→ π0e+
or p→ K+ν, are not forbidden by the SU(3)C ×U(1)EM symmetry. However, they violate U(1)B,
and therefore do not occur within the SM. The lesson here is quite general: The lightest particle
that carries a conserved charge is stable. The accidental U(1)B symmetry also explains why
neutron-antineutron oscillations have not been observed.
Note that U(1)B as well as each of the lepton numbers are anomalous. The combination
of B − L, however, is anomaly free. Due to the anomaly, baryon and lepton number violating
processes occur non-perturbatively. However, the non-perturbative effects obey ∆B = ∆L = 3n,
with n =integer, and thus do not lead to proton decay. Moreover, they are very small, and can be
neglected in almost all casses we study, and thus we do not discuss them any further.
The accidental symmetries of the renormalizable part of the SM Lagrangian also explain the
vanishing of neutrino masses. A Majorana mass term violates the accidental B − L symmetry by
two units. Thus, the symmetry prevents mass terms not only at tree level but also to all orders
in perturbation theory. Moreover, since the B − L symmetry is non-anomalous, Majorana mass
terms do not arise even at the non-perturbative level. We conclude that the renormalizable SM
gives the exact prediction:
mν = 0. (8.58)
117
8.5.2 Parameter counting
Before we discuss the SM parameters in detail, we explain the basics of identifying the number
of physical parameters. The Lagrangian written in a general interaction basis might include a
number of parameters that is larger than the number of physical parameters. This means that
when we express physical observables in terms of the Lagrangian parameters, only a subset of
these parameters (or combinations of them) will appear. It also means that there is a specific
basis where the non-physical parameters are identically zero. For the purpose of testing a model,
it is important to count and identify its physical parameters. In this subsection we explain how to
determine the number of physical parameters.
We start with a very simple example. Consider a hydrogen atom in a uniform magnetic field.
Before turning on the magnetic field, the hydrogen atom is invariant under spatial rotations, which
are described by the SO(3) group. Furthermore, there is an energy eigenvalue degeneracy of the
Hamiltonian: states with different angular momenta have the same energy. This degeneracy is a
consequence of the symmetry of the system.
When magnetic field is added to the system, we can define, without loss of generality, the
direction of the magnetic field. The common convention is to define the positive z direction to
be the direction of the magnetic field. Consider this choice more carefully. A generic uniform
magnetic field would be described by three real numbers: the three components of the magnetic
field (Bx, By, Bz). The magnetic field breaks the SO(3) symmetry of the hydrogen atom system
down to an SO(2) symmetry of rotations in the plane perpendicular to the magnetic field. The
one generator of the SO(2) symmetry is the only valid symmetry generator now; the remaining
two SO(3) generators in the orthogonal planes are broken. These broken symmetry generators
allow us to rotate the system such that the magnetic field points in the z direction:
OxzOyz(Bx, By, Bz) = (0, 0, B′z), (8.59)
where Oxz and Oyz are rotations in the xz and yz planes respectively. The two broken generators
were used to rotate away two unphysical parameters, leaving us with one physical parameter, the
magnitude of the magnetic field. We learn that, when turning on the magnetic field, all measurable
quantities in the system depend on only one new parameter, rather than the naıve three.
The results described above are more generally applicable. Particularly, they are useful in
studying the flavor physics of quantum field theories. Consider a gauge theory with matter content.
The kinetic and gauge terms (Lkin) have a certain global symmetry, Gf . In adding terms (Lψ +
Lφ +LYuk) that respect the imposed gauge symmetries, the global symmetry may be broken down
to a smaller symmetry group. In breaking the global symmetry, there is an added freedom to use
the broken Gf generators to change basis and, in particular, rotate away unphysical parameters,
as when a magnetic field is added to the hydrogen atom system.
We are interested in obtaining the number of parameters affecting physical measurements,
118
Nphys. In a general basis, the added terms depend on Ngeneral parameters. The global symmetry
of the entire model, Hf , has fewer generators than Gf . We call the difference in the number of
generators Nbroken. Then Nphys is given by
Nphys = Ngeneral −Nbroken. (8.60)
Furthermore, the rule in (8.60) applies separately to real parameters and to phases. A general
n × n complex matrix can be parameterized by n2 real parameters and n2 phases. Imposing
restrictions like Hermiticity or unitarity reduces the number of parameters required to describe
the matrix. A Hermitian matrix can be described by n(n + 1)/2 real parameters and n(n − 1)/2
phases. As for generators, the rules for unitary matrices are as follows. The generator of U(1) is,
clearly, a phase. For SU(N) the real parameters are associated with the SO(N) subgroup. Thus,
there are n(n− 1)/2 real parameters and n(n+ 1)/2 phases.
8.5.3 Parameter counting in the SM
The rule given by (8.60) can be applied to the standard model. Consider the quark sector of the
model. The kinetic term has a global symmetry
Gf = U(3)Q × U(3)U × U(3)D. (8.61)
A U(3) algebra has 9 generators (3 real and 6 imaginary), so the total number of generators of Gf
is 27. The Yukawa interactions defined in Eq. (8.13), Y F (F = u, d), are 3× 3 complex matrices,
which contain a total of 36 parameters (18 real parameters and 18 phases) in a general basis. These
parameters also break Gf down to baryon number:
U(3)Q × U(3)U × U(3)D → U(1)B. (8.62)
While U(3)3 has 27 generators, U(1)B has only one and thus Nbroken = 26. This broken symmetry
allows us to rotate away a large number of the parameters by moving to a more convenient basis.
Using (8.60), the number of physical parameters should be given by
Nphys = 36− 26 = 10. (8.63)
These parameters can be split into real parameters and phases. The three unitary matrices
generating the symmetry of the kinetic and gauge terms have a total of 9 real parameters and 18
phases. The symmetry is broken down to a symmetry with only one phase generator. Thus,
N(r)phys = 18− 9 = 9, N
(i)phys = 18− 17 = 1. (8.64)
Let us now identify these parameters. Of the 9 real parameters, 6 are the fermion masses and
three are the CKM matrix mixing angles. The one phase is the CP-violating phase of the CKM
mixing matrix. (In your homework you will count the number of parameters for different models.)
We thus conclude that the full SM has 18 parameters: 3 gauge couplings, 2 parameters of the
Higgs potential, the 3 lepton masses and the 10 parameters of the quark sector.
119
8.5.4 The strong CP parameter
The above counting of parameters is done at the classical level. Usually, when quantizing a
system, the number of parameters is not changed. Yet, there are exceptions that are related to
non-Abelian gauge groups. In the SM it turns out that there is one more renormalizable parameter
that is unphysical at the classical level but is physical at the quantum level. This parameter is
called θQCD:
LθQCD=θQCD
32π2εµνρσG
µνa G
ρσa . (8.65)
This term violates P and CP. In particular, it leads to an electric dipole moment (EDM) of the
neutron dn. The experimental upper bound on the EDM of the neutron,
dn < 2.9× 10−26 e cm, (8.66)
implies a very small value for θQCD, θQCD ∼< 10−9. The problem of why θQCD is so small is known as
the strong CP problem. We do not discuss it any further here. We just conclude that the number
of independent parameters in the quantum SM is thus 19: the 18 mentioned above and θQCD.
8.6 The CKM parameters
The Cabibbo-Kobayashi-Maskawa (CKM) matrix determines the strength of the couplings of the
W boson to quark-antiquark pairs,
LWqq = − g√2uLi γ
µ Vij dLj W+µ + h.c.. (8.67)
The form of the CKM matrix is not unique. First, there is freedom in defining V in that we
can permute between the various generations. This freedom is fixed by ordering the up quarks
and the down quarks by their masses, i.e. (u1, u2, u3) → (u, c, t) and (d1, d2, d3) → (d, s, b). The
elements of V are therefore written as follows:
V =
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
. (8.68)
Since the W -bosons couple to pairs that belong to different generations, V is often referred to as
the mixing matrix for quarks. Second, while a general unitary 3 × 3 matrix has six phases, we
concluded above that only one of these phases is physical. This implies that we can find bases
where V has a single phase. This physical phase is the Kobayashi-Maskawa phase that is usually
denoted by δKM.
The fact that there are only three real and one imaginary physical parameters in V can be made
manifest by choosing an explicit parametrization. For example, the standard parametrization, used
120
by the Particle Data Group (PDG) [?], is given by
V =
c12c13 s12c13 s13e
−iδ
−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e
iδ s23c13
s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e
iδ c23c13
, (8.69)
where cij ≡ cos θij and sij ≡ sin θij. The three sin θij are the three real mixing parameters while
δ is the Kobayashi-Maskawa phase. Another parametrization is the Wolfenstein parametrization
where the four mixing parameters are (λ,A, ρ, η) where η represents the CP violating phase. The
Wolfenstein parametrization is an expansion in the small parameter, λ = |Vus| ≈ 0.22. To O(λ3)
the parametrization is given by
V =
1− 1
2λ2 λ Aλ3(ρ− iη)
−λ 1− 12λ2 Aλ2
Aλ3(1− ρ− iη) −Aλ2 1
. (8.70)
Below we discuss in detail the measurements of the CKM parameters. Here we just mention that
the Wolfenstein parametrization provides a good approximation to the actual numerical values:
The CKM matrix is close to a unit matrix, with off-diagonal terms that are small. The order of
magnitude of each element can be read from the power of λ in the Wolfenstein parametrization.
Among the SM interactions, the W -mediated interactions are the only ones that are not diag-
onal. Consequently, all flavor changing processes depend on the CKM parameters. The fact that
there are only four independent CKM parameters, while the number of measured flavor changing
processes is much larger, allows for stringent tests of the CKM mechanism for flavor changing
processes. We emphasize, however, that there is an inherent difficulty in determining the CKM
parameters. While the SM Lagrangian has the quarks as its degrees of freedom, in Nature they
appear only within hadrons. Thus, for example, the Vcb matrix element affects the rates of b→ c`ν
decays, but what can be measured are hadronic processes such as B → D`ν decay. Can we re-
late the two processes? Our best opportunities to do so in a reliable way arise when we can use
approximate symmetries of QCD. An example of how isospin symmetry relates hadron decays to
the u→ d`ν transition is given in Appendix ??. Approximate symmetries such as isospin, SU(3)-
flavor and heavy quark symmetry are useful for semileptonic or leptonic decays, where the relevant
operators involve only two quarks. Thus, the most useful processes are those related to q → q′`ν
transitions:
• Processes related to u→ d`+ν transitions give |Vud| = 0.97425± 0.00022.
• Processes related to s→ u`−ν transitions give |Vus| = 0.2253± 0.0008.
• Processes related to c→ d`+ν or to νµ + d→ c+ µ− transitions give |Vcd| = 0.225± 0.008.
• Processes related to c→ s`+ν or to cs→ `+ν transitions give |Vcs| = 0.986± 0.016.
121
• Processes related to b→ c`−ν transitions give |Vcb| = 0.0411± 0.0013.
• Processes related to b→ u`−ν transitions give |Vub| = 0.0041± 0.0005.
There are two additional tree level processes that depend on the CKM parameters:
• Processes related to single top production in hadron colliders give |Vtb| = 1.02± 0.03.
• Processes related to b→ scu and b→ suc transitions give γ ≡ arg(−VudV
∗ub
VcdV∗cb
)= (68.0+8.0
−8.5)o.
All of these eight different classes of measurements are indeed consistent with the following ranges
of the CKM parameters:
λ = 0.2254± 0.0006, A = 0.81± 0.02, ρ = 0.12± 0.02, η = 0.354± 0.015. (8.71)
Additional information comes from measurements of loop processes, which are discussed in
Chapter 11. In particular, |Vts| and |Vtd| (as well as various additional relative phases between
CKM combinations) are measured in this way. The present status of our knowledge of the absolute
values of the various entries in the CKM matrix can be summarized as follows:
|V | =
0.97427± 0.00014 0.22536± 0.00061 (3.55± 0.15)× 10−3
0.22522± 0.00061 0.97343± 0.00015 0.0414± 0.0012
(8.86± 0.33)× 10−3 0.0405± 0.0012 0.99914± 0.00005
. (8.72)
8.7 P, C and CP
The single phase of the CKM matrix is the only source of CP violation within the SM. Various
parameterizations differ in the way that the freedom of phase rotation is used to leave a single
phase in V . One can define, however, a CP violating quantity in V that is independent of the
parametrization. This quantity, the Jarlskog invariant, JCKM, is defined through
Im(VijVklV∗ilV∗kj) = JCKM
3∑m,n=1
εikmεjln, (i, j, k, l = 1, 2, 3). (8.73)
In terms of the explicit parameterizations given above, we have
JCKM = c12c23c213s12s23s13 sin δ ≈ λ6A2η. (8.74)
While there is room for CP violation in the SM, so that we expect that indeed CP is violated,
this is not necessarily the case. A necessary and sufficient condition for CP violation in the quark
sector of the SM is given by
∆m2tc∆m
2tu∆m
2cu∆m
2bs∆m
2bd∆m
2sdJCKM 6= 0, (8.75)
where ∆m2ij ≡ m2
i −m2j . Eq. (8.75) puts the following requirements on the SM in order that CP
is violated:
122
1. Within each quark sector, there should be no mass degeneracy;
2. None of the three mixing angles should be zero or π/2;
3. The phase should be neither 0 nor π.
123
Appendix
8.A Unitarity Triangles
A very useful concept is that of the unitarity triangles. The unitarity of the CKM matrix leads to
various relations among the matrix elements, for example,∑i
VidV∗is = 0. (8.76)
There are six such relations and they require the sum of three complex quantities to vanish.
Therefore, they can be geometrically represented in the complex plane as a triangle and are called
“unitarity triangles”. It is a feature of the CKM matrix that all unitarity triangles have equal
areas. Moreover, the area of each unitarity triangle equals |JCKM|/2 while the sign of JCKM gives
the direction of the complex vectors around the triangles.
The triangle which corresponds to the relation
VudV∗ub + VcdV
∗cb + VtdV
∗tb = 0. (8.77)
has its three sides of roughly the same length. Furthermore, both the length of its sides and
its angles are experimentally accessible. For these reasons, the term “the unitarity triangle” is
reserved for Eq. (8.77). We further define the rescaled unitarity triangle. It is derived from (8.77)
by choosing a phase convention such that (VcdV∗cb) is real and dividing the lengths of all sides by
|VcdV ∗cb|. The rescaled unitarity triangle is similar to the unitarity triangle. Two vertices of the
rescaled unitarity triangle are fixed at (0,0) and (1,0). The coordinates of the remaining vertex
correspond to the Wolfenstein parameters (ρ, η). The unitarity triangle is shown in Fig. 8.A.1.
The lengths of the two complex sides are
Ru ≡∣∣∣∣VudVubVcdVcb
∣∣∣∣ =√ρ2 + η2, Rt ≡
∣∣∣∣VtdVtbVcdVcb
∣∣∣∣ =√
(1− ρ)2 + η2. (8.78)
The three angles of the unitarity triangle are defined as follows:
α ≡ arg
[− VtdV
∗tb
VudV ∗ub
], β ≡ arg
[−VcdV
∗cb
VtdV ∗tb
], γ ≡ arg
[−VudV
∗ub
VcdV ∗cb
]. (8.79)
They are physical quantities and can be independently measured, as we discuss below. Another
commonly used notation is φ1 = β, φ2 = α, and φ3 = γ. Note that in the standard parametrization
γ = δKM.
124
VudV∗ub
VcdV ∗cb
VtdV∗tb
VcdV ∗cb
(ρ, η)
α
βγ
(0, 0) (1, 0)
Figure 8.A.1: The unitarity triangle.
125
Homework
Question 8.1: Semi-leptonic decays and CKM
A semi-leptonic decay is one where in the final state we have both hadrons and leptons. Here
we consider semileptonic b decays.
1. Draw the tree-level diagram for b→ ueν and estimate the diagram
2. Estimate the ratioΓ(b→ ueν)
Γ(b→ ceν)(8.80)
as a function of some CKM matrix element. You can negelect the mass of the electron and
use the phase space function for a 1→ 3 decay with two massless final states. It is given in
Eq. (7.64) and we repeat it here
f(x) = 1− 8x+ 8x3 − x4 − 12x2 log x, xf ≡ (mf/mb)2. (8.81)
For the quark masses usemc
mb
≈ 0.3,mu
mb
≈ 0. (8.82)
3. Based on the following experimental data, estimate the relevant CKM matrix element ratio
Γ(b→ u`ν)
Γ(b→ c`ν)≈ 2× 10−2 . (8.83)
Question 8.2: Exotic light quarks
We consider a model with the gauge symmetry SU(3)C×SU(2)L×U(1)Y spontaneously broken
by a single Higgs doublet into SU(3)C × U(1)EM . The lepton sector is as in the SM. The quark
sector, however, differs from the SM one as it consists of three quark flavors, that is, we do not
have the c, b and t quarks. The quark representations are non-standard. Of the left handed quarks,
QL = (uL, dL) form a doublet of SU(2)L while sL is a singlet. All the right handed quarks are
singlets. All color representations and electric charges are the same as in the standard model.
126
1. Write down (a) the gauge interactions of the quarks with the charged W bosons (before
SSB); (b) the Yukawa interactions (before SSB); (c) the bare mass terms (before SSB); (d)
the mass terms after SSB.
2. Show that there are four physical flavor parameters in this model. How many are real and
how many imaginary? Is there CP violation in this model? Separate the parameters into
masses, mixing angles and phases.
3. Are there photon and gluons FCNC’s? Support your answer by an argument based on
symmetries.
4. Write down the gauge interactions of the quarks with the Z boson in both the interaction
basis and the mass basis. (You do not have to rewrite terms that do not change when you
rotate to the mass basis. Write only the terms that are modified by the rotation to the mass
basis.) Are there generally tree level Z exchange FCNC’s?
5. We assume that the masses of the particles and the value of the Cabibbo angle are as found
in Nature and that the leptons are described by the SM. Then, in this model we can have
process like KL → µ+µ−. Estimate its rate (normalized it to K+ → µ+ν).
6. Explain why this result practically ruled out the model.
Question 8.3: Two Higgs doublet model
Consider the two Higgs doublet model (2HDM) extension of the SM. In this model we add a
Higgs doublet to the SM fields. Namely, instead of the one Higgs field of the SM we now have two,
denoted by φ1 and φ2. For simplicity you can work with two generations when the third generation
is not explicitly needed.
1. Write down (in a matrix notation) the most general Yukawa potential of the quarks.
2. Carry out the diagonalization procedure for such a model. Show that the Z couplings are
still flavor diagonal.
3. In general, however, there are FCNCs in this model mediated by the Higgs bosons. To show
that, write the Higgs fields as Re(φi) = vi + hi where i = 1, 2 and vi 6= 0 is the vev of φi,
and define tan β = v2/v1. Then, write down the Higgs–fermion interaction terms in the mass
basis. Assuming that there is no mixing between the Higgs fields, you should find a non
diagonal Higgs fermion interaction terms.
127
4. Since there are FCNCs in this model processes like b → s`+`− can proceed at tree level.
Assume that tan β ∼ 1 and mHi ∼ mW and give a very rough estimate of the ratio
Γ(b→ sµ+µ−)
Γ(b→ cµ−ν)(8.84)
(For the numerical values use mb = 4.3 GeV and Vcb = 0.04.)
5. The current upper bound on this ratio is 5 × 10−6. Can we already probe this model using
this ratio?
6. Can you find a symmetry that will forbid the Higgs exchange FCNCs? In particular, try to
find a symmetry that will couple φ1 only to the up type quarks, and φ2 to the down type
quarks.
Question 8.4: τ decays
1. Almost always, the muon decays via µ− → e−νµνe. Draw the tree level Feynman diagram
for this process.
2. The tau lepton is relatively heavy, and can decay purely leptonically and semi-leptoniclly
(that is, also to hadrons). List the possible tree level decay processes of the tau lepton in
terms of leptons and quarks. (Note that while the charm is lighter than the tau, what we
care about is the mass of the lightest charmed, the D meson, is heavier than the tau.)
3. In terms of CKM elements, estimate the branching ratio of each mode.
4. Within the SM, the tau could not decay by τ− → µ−γ. Explain why.
5. We now move to the neutrino sector. We add three right handed neutrinos to the SM
N iR(1, 1)0, i = e, µ, τ. (8.85)
Now the neutrinos can have Dirac masses just like the other fermions of the SM.1 Write down
the terms in L which give neutrino masses. Write them both before and after the electroweak
SSB.
6. How many physical parameters are needed now in order to describe the lepton sector? Sep-
arate them into masses, mixing angles and phases.
1Here we ignore the possibility of lepton number violating mass, the so called Majorana mass. If you never heard
about Majorana mass, then just ignore this footnote. If you do know what it is, then assume you cannot write it.
Majorana masses can be forbidden by imposing lepton number as a global symmetry.
128
7. In this model, τ− → µ−γ is possible. Still, it is not allowed at tree level. Explain why.
8. We assume that there is no degeneracy in the neutrino sector. We denote the neutrino mass
eigenstates by νi with i = 1, 2, 3 where
m3 > m2 > m1. (8.86)
Draw the leading one loop diagram(s) for τ− → µ−γ.
9. We assume that m3 m2 m1 and we take m3 = 10−1 eV. In terms of the neutrino
masses, estimate the branching ratio of τ− → µ−γ. In case you need more information about
the neutrino parameters, just assume some values for them and explain your assumptions.
10. Consider a model with only two right-handed neutrino fields. How many physical parameters
are needed now in order to describe the lepton sector? Separate them into masses, mixing
angles and phases.
Question 8.5: W decays
In this question we study decays of the W boson. In all items you should neglect the effect of
the fermion masses when they are small compared to MW (namely, all the fermions but for the
top quark). The W → µ+νµ width at tree level is given by
Γµ ≡ Γ(W → µ+νµ) =g2mW
48π=GFm
3W
6√
2π≈ 227 MeV. (8.87)
1. What are the predicted widths for Γ(W → e+νe) and Γ(W → τ+ντ )?
2. Write down all the hadronic decays of the W in terms of quarks, that is, W → qq′, and
estimate their widths in terms of Γµ, the CKM elements and the number of colors, Nc = 3.
3. What is roughly ΓW , the total width of the W? Write your answer in GeV, to a precision of
two decimal points.
4. Experimentally ΓW = 2.12(4) GeV . Explain the difference between your result and the
experimental measurement.
Question 8.6: LeptoQuarks
In this question we study some properties of scalar LeptoQuarks (LQs). A LQ is a hypotettical
field which couples to a quark and a lepton. For example, F , which is a doublet of SU(2)L, couples
as
λFQEij QiLE
jR F, (8.88)
129
where SU(3)C and SU(2)L indices are omitted. Here i and j are generation indices. While λFQEij
above is in the flavor basis, in the following you can assume that the rotation to the mass basis
is small and can be neglected. Recall that the representations of the SM fermions under the SM
gauge group are
QL(3, 2)1/6, UR(3, 1)2/3, DR(3, 1)−1/3, LL(1, 2)−1/2, ER(1, 1)−1. (8.89)
1. What is the representation of F under the SM group?
2. What is the electric charge of each of the two components in F?
3. We assume that F does not acquire a VEV, that is, 〈F 〉 = 0. Shortly describe the phe-
nomenological problems of 〈F 〉 6= 0.
4. What are the baryon and lepton numbers of F? (Recall that we define the baryon number
of the proton to be +1, so the baryon number of the quarks is 1/3. We define the lepton
number of the electron to be +1.)
5. F contributes to the quark level decay b → sµ+e−. Draw the tree-level Feynman diagram
for this decay.
6. Next we find a bound on mF . (Here we neglect the splitting between the two components
of F .) For this we would like to compare the rate of the LQ mediated b → sµ+e− decay to
that of the W mediated one, b→ ce−ν. Draw the tree-level diagram for b→ ce−ν.
7. Estimate the ratioΓ(b→ sµ+e−)
Γ(b→ ce−ν). (8.90)
Express the ratio in terms of λFQEij , mF , g, mW , and the CKM matrix elements. You should
assume that mF mb and that the hadronic matrix elements are similar in the two decays
and thus cancel in the ratio.
8. We now assume that λFQEij ∼ g for all i and j. Using the experimental data
BR(b→ ce−ν) ∼ 10−1, BR(b→ sµ+e−) ∼< 10−5, (8.91)
estimate a lower bound on mF .
9. F couples to another combination of a quark and a lepton. Write it down. You can omit
SU(3)C and SU(2)L indices.
10. Besides F there is one more scalar doublet LQ which we denote by G. Find the representation
of G and write down its couplings (the equivalent of (8.88)). You can omit SU(3) and SU(2)
indices.
130
11. Now we move to other types of scalar LQs. There are several of them, but here we consider
only an SU(2)L singlet, R, that couples to SM fermions as
λRDEDREcRR, (8.92)
where the flavor indices are omitted. Recall that ψcR is a left handed field with opposite
charges compared to ψR. In particular, EcR is a left handed field which transforms as (1, 1)1
and carries lepton number of −1.
12. What is the SM representation of R?
13. R also couples to pairs of quarks. For example, we can have
λRUU U cRURR. (8.93)
Show that this term is indeed a singlet under the SM gauge group. Recall that U cR is a left
handed field which transforms as (3, 1)−2/3. As for SU(3) algebra, recall that 3 × 3 = 6 + 3
and 3× 3 = 1 + 8.
14. R breaks both baryon and lepton numbers. That is, the lepton and baryon numbers as-
signment of R in (8.92) is inconsistent with that in (8.93). Show that this is indeed the
case.
15. R can mediate proton decay. Draw a Feynman diagram for the decay
P+ → π0µ+, (8.94)
and give a rough estimate of it in terms of the coupling and mR. Be sure to write the flavor
dependence of the couplings. (You can assume mR 1 GeV .)
16. Experimental data tell us that the proton lifetime is longer than about 1033 years. Taking
all the couplings of R to be of order one, estimate a lower bound on mR. Express your result
in units of GeV. Recall that h = 1 = 6.58× 10−22 MeV s.
131
Chapter 9
Electroweak Precision Measurements
9.1 The SM beyond tree level
The SM is not a full theory of Nature. It is only a low energy effective theory, valid below some
scale Λ mZ . Then, the SM Lagrangian should be extended to include all non-renormalizable
terms, suppressed by powers of Λ:
L = LSM +1
ΛOd=5 +
1
Λ2Od=6 + · · · , (9.1)
where Od=n represents operators that are products of SM fields, transforming as singlets under the
SM gauge group, of overall dimension n in the fields. For physics at an energy scale E well below
Λ, the effects of operators of dimension n > 4 are suppressed by (E/Λ)n−4. Thus, in general, the
higher the dimension of an operator, the smaller its effect at low energies.
In previous sections, we studied the gauge sector of the SM at tree level and with only renor-
malizable terms. We can classify the effects of including loop corrections and nonrenormalizable
terms into three broad categories:
1. Forbidden processes: Various processes are forbidden by the accidental symmetries of the
Standard Model. Nonrenormalizable terms (but not loop corrections!) can break these
accidental symmetries and allow the forbidden processes to occur. Examples include neutrino
masses and proton decay.
2. Rare processes: Various processes are not allowed at tree level. These effects can often be
related to accidental symmetries that hold within a particular sector of, but not in the entire,
SM. Here both loop corrections and nonrenormalizable terms can contribute. Examples
include flavor changing neutral current (FCNC) processes.
3. Tree level processes: Often tree level processes in a particular sector depend on a small subset
of the SM parameters. This situation leads to relations among different processes within this
132
sector. These relations are violated by both loop effects and nonrenormalizable terms. Here,
precision measurements and precision theory calculations are needed to observe these small
effects. Examples include electroweak precision measurements (EWPM).
As concerns the last two types of effects, where loop corrections and nonrenormalizable terms
may both contribute, their use in phenomenology can be divided to two eras. Before all the SM
particles have been directly discovered and all the SM parameters measured, one could assume
the validity of the renormalizable SM and indirectly measure the properties of the yet unobserved
SM particles. Indeed, the charm quark, the top quark and the Higgs boson masses were predicted
in this way. Once all the SM particles have been observed and the parameters measured directly,
the loop corrections can be quantitatively determined, and effects of nonrenormalizable terms can
be unambiguously probed. Thus, at present, all three classes of processes serve to search for new
physics.
9.2 Electroweak Precision Measurements (EWPM)
Consider a situation where a class of processes is described at tree level by only one sector of a
theory, and where this sector depends on only a small number of parameters. If the number of
observables is larger than the number of parameters, relations among the observables are predicted.
At the quantum level, however, the tree level relations are violated, and the processes depend on
all parameters of the theory. In some cases, the tree level predictions follow from a symmetry
which is respected by the relevant sector but not by other sectors of the theory. Violations of such
symmetry-based relations are particularly sensitive to loop effects. These features of quantum
theories are taken advantage of in the program of EWPM.
At tree level, all (flavor diagonal) electroweak processes depend on only three parameters of the
renormalizable SM Lagrangian. In the language of the SM Lagrangian, the three parameters are
g, g′ and v. It is convenient for our purposes to work with the combinations of these parameters
that are best measured: α, mZ and GF . The number of relevant observables is much larger than
three. Thus, at tree level, a large number of relations among these observables are predicted.
These predictions are, however, violated by SM loop effects, and possibly by nonrenormalizable
operators that are generated by beyond SM (BSM) physics.
The full SM has eighteen parameters. The EWPM allow us to probe some of the additional
fifteen parameters through their modification of the tree level relations. Eleven of the fifteen
parameters (eight of the Yukawa couplings and the three CKM mixing angles) are small, and
consequently have negligible effects on deviations from the tree level relations. The four large
parameters are the Kobayashi-Maskawa (KM) phase, the strong coupling constant, the Higgs self-
coupling and the top Yukawa coupling. The KM phase has negligible effects on flavor diagonal
processes. As concerns the strong coupling constant, its universality and the fact that the elec-
133
troweak vector bosons do not couple directly to gluons combine to make its effect on the relevant
parameters very small. Thus, in practice, there are only two SM parameters that have significant
effects on the EWPM: mt/v and mh/v. In the past, when these masses had not yet been directly
measured, the EWPM were used to determine their values. Now, that the top quark and the Higgs
boson have been discovered and their masses are known from direct measurements, the EWPM
are used to probe nonrenormalizable operators, that is, BSM physics.
As concerns the experimental aspects of the EWPM program, the relevant processes can be
divided to two classes: low-energy and high-energy. The “low-energy” observables involve processes
with a characteristic energy scale well below mW and mZ , so that the intermediate W -boson or
Z-boson are far off-shell. The “high-energy” observables are measured in processes where the W -
boson or the Z-boson are on-shell. The low-energy EWPM include measurements of GF and α,
as well as data from neutrino scattering, deep inelastic scattering (DIS), atomic parity violation
(APV) and low energy e+e− scattering. The high-energy EWPM include measurements of the
masses, the total widths and partial decay widths of the W and Z bosons. A summary of the
current data on the relevant observables is given in Fig. ???. : Need to include the fig In the
appendix we discuss some examples of these observables in detail and derive their dependence on
the SM parameters.
9.3 The weak mixing angle, θW
As an example of the way EWPM can be used, we consider three definitions of the weak angle. Each
definition involves a different set of observables. At tree level, all three definitions are equivalent,
and correspond to
tan θtree ≡g′
g. (9.2)
At the one loop level, however, they differ.
(i) Definition in terms of α, GF and mZ :
sin2 2θ0 ≡4πα(mZ)√
2GFm2Z
. (9.3)
Quantitatively, θ0 is defined in terms of the best measured observables and thus has the
smallest experimental uncertainties.
(ii) Definition in terms of mW and mZ :
sin2 θW ≡ 1− m2W
m2Z
. (9.4)
This definition is based on the tree level relation ρ = 1 discussed earlier.
134
(iii) Definition in terms of gV and gA:
sin2 θi∗ ≡giA − giV
2Qi
, (9.5)
where i is a flavor index that is not summed, and where giV,A = giL ± giR refer to the Z
couplings to fermions in Eq. (7.59),
LZψψ ∝ ψiγµ(giV − giAγ5
)ψiZµ. (9.6)
In principle we have here nine different definitions of θ∗, one for each charged fermion type.
(The Zνν coupling does not involve θW .)
For any given model, one can compute all relevant corrections as functions of the model parame-
ters. For pedagogical and practical purposes we would like, however, to work in a simple framework
that represents a large class of models. To do so, we make the following working assumption:
1. The only significant effects are in the electroweak gauge boson propagators. These effects
are called oblique corrections.
We note that this assumption implies that all θi∗ are equal, that is, the relevant one loop effects
are flavor universal. In some models the assumption that all corrections are oblique does not hold,
yet the corrections are flavor universal.
We denote the oblique corrections to the electroweak gauge boson propagators by ΠAB(q2).
The propagators PAB are defined as follows:
PAB(q2) =−i
q2 −m2A
[δAB +
−iΠAB(q2)
q2 −m2B
]. (9.7)
Taking charge conservation into account, we learn that there are four ΠAB’s that do not vanish.
We can choose the set ΠWW , ΠZZ , Πγγ and ΠγZ(= ΠZγ), corresponding to the mass eigenstates.
Alternatively, we can use Π+−, Π33, Π00 and Π30, corresponding to the interaction eigenstates.
The transition between the two sets is straightforward.
At q2 = m2A, the relevant Π can be identified as correction to the mass of the corresponding
gauge boson. Thus, gauge invariance (specifically, m2γ = 0) guarantees that
Πγγ(q2 = 0) = ΠγZ(q2 = 0) = 0. (9.8)
To proceed further, we expand the ΠAB’s that are relevant for low energy observables, ΠAB(q2 m2W ), in q2, and make one more working assumption:
2. Terms of order (q2)2 and higher can be neglected in low energy observables:
ΠAB(q2 m2W ) = ΠAB(0) + q2Π′AB(0), Π′(q2) ≡ dΠ(q2)
dq2. (9.9)
135
How do the various corrections to the propagators affect the three differently-defined θ’s? We
define ∆ sin2 θA ≡ sin2 θA − sin2 θtree and obtain [Peskin]:
(i) Corrections to θ0 [See (21.131) of Peskin]:
∆ sin2 θ0 =sin2 2θ
4 cos 2θ
[Π′γγ(0) +
ΠWW (0)
m2W
− ΠZZ(m2Z)
m2Z
]. (9.10)
The three terms correspond to corrections to α, GF and mZ , respectively.
(ii) Corrections to θW [See (21.128) of Peskin]:
∆ sin2 θW = − 1
m2Z
[ΠWW (m2
W )− m2W
m2Z
ΠZZ(m2Z)
]. (9.11)
The two terms correspond to corrections to the tree level heavy gauge boson masses,
∆m2W = ΠWW (m2
W ), ∆m2Z = ΠZZ(m2
Z). (9.12)
(iii) Corrections to θ∗ [See (21.127) of Peskin]:
∆ sin2 θ∗ = − sin θ cos θΠγZ(m2
Z)
m2Z
. (9.13)
The corrections arise from the mixing between the off-shell photon and the Z-boson.
We learn that the loop effects are different for the three definitions. Once we extract the values of
sin2 θ from each of the three sets of observables, we can probe these effects.
9.3.1 θ within the SM
Our analysis so far applies to all models with only oblique corrections. In this subsection we discuss
the specific case of the SM. We make the following approximation:
3. We consider only one loop diagrams involving the top and bottom quarks.
Within the SM, these diagrams provide the leading oblique corrections. We thus need to calculate
diagrams of the general form
Plot of the diagram (9.14)
The relevant diagrams will be added to this text, but for now you can just find them in Fig. 21.12
of Peskin’s book. We do not reproduce the details of the calculation here, but we make one
comment regarding the finiteness of the results. Naively, each diagram is quadratically divergent.
Ward identities prevent, however, the quadratic divergences, leaving only logarithmic ones. The
logarithmic divergences appear in each of the ∆ sin2 θ, but they cancel in the differences between
136
any two observable quantities. The final results (see Eq. (21.157) of Peskin) are the following (we
approximate mb = 0):
sin2 θ0 − sin2 θ∗ =3α
16π cos2 2θ
m2t
m2Z
,
sin2 θW − sin2 θ∗ =−3α
16π sin2 θ
m2t
m2Z
. (9.15)
The factor of α/(16π) is typical of electroweak one-loop effects. The factor of 3 is the color factor
of the quarks in the loop. The θ dependence is different in the two cases, reflecting the specific
combination of observables involved in each case. The factor of m2t/m
2Z deserves a more detailed
discussion, which we now turn to.
Naively, quadratic dependence on the top mass is puzzling since it seems to violate the so-called
decoupling theorem. This theorem states that the effect of heavy states on low energy observables
must go to zero as their mass goes to infinity. The intuition behind this theorem is straightforward.
The heavier a state is, the smaller its effects (when off-shell) become. This can be understood based
on the uncertainty principle, or on second order perturbation theory, or simply by considering the
form of propagators in QFT. Why doesn’t this theorem apply to the top contribution to EWPM?
The solution to the puzzle lies in the fact that the SM quarks acquire their masses from the Higgs
mechanism. Consequently, their Yukawa couplings are proportional to their masses. The heavier
the top, the stronger its Yukawa coupling becomes. Indeed, the top-related loop corrections to
EWPM depend on the top coupling to the longitudinal W and Z, which is its Yukawa coupling.
The quadratic dependence on the top mass reflects the proportionality of the loop corrections to
the top Yukawa coupling, and not to its mass. (In fact, the mt →∞ limit cannot be taken, because
perturbation theory does not hold anymore.)
Eqs. (9.15) demonstrate the main point of this section: One can use the observables of the
EWPM to determine parameters outside the pure electroweak sector. Using the measured values of
the observables on the left-hand side of these equations, one can, first, determine mt and, second,
test the SM by examining whether the ranges allowed for mt from the two sets of observables
overlap.
Additional oblique corrections of interest come from loop diagrams with internal Higgs boson.
At one loop level, the mh-dependence is logarithmic (and thus the EWPM are less sensitive to
mh than they are to mt). This result is known as the screening theorem. We do not discuss it in
detail here. Two loop contributions are proportional to m2h, but are small because of the extra loop
factor. We note, however, that the precision of the EWPM was good enough to have sensitivity
to the Higgs-related oblique corrections and provided an allowed range for the Higgs mass well
before it was actually measured. The present allowed range from EWPM (removing all direct
measurements of the Higgs mass, production and decay rates) is [pdg] 60 GeV ≤ mh ≤ 127 GeV
at the 90% CL.
137
9.4 Custodial symmetry
The SM Higgs potential has an accidental symmetry. This so-called “custodial symmetry” predicts
tree level relations among various observables. The fact that the custodial symmetry is not a
symmetry of the full SM makes EWPM sensitive to the symmetry breaking SM parameters. This
is the topic of this subsection. The fact that the symmetry is accidental makes EWPM sensitive
to nonrenormalizable terms that violate the symmetry. This is one of the topics of the next
subsection.
Consider the SM Higgs potential:
V = µ2|φ|2 + λ|φ|4. (9.16)
Since φ is a complex, SU(2)L-doublet scalar field, it has four degrees of freedom:
φ =
(φ3 + iφ4
φ1 + iφ2
). (9.17)
The scalar potential, when written in terms of these four components, depends only on the com-
bination φ21 + φ2
2 + φ23 + φ2
4, and thus has manifestly an SO(4) symmetry. At the algebra level,
SO(4) ∼ SU(2) × SU(2). Out of the six generators, four are also generators of the gauge group
SU(2)L × U(1)Y . The two extra generators are then related to an accidental symmetry of the
scalar sector of the SM.
The VEV of the Higgs field breaks three of the generators, leaving (within the pure Higgs
sector) an unbroken SU(2) symmetry. This symmetry is called the custodial symmetry. Under
this symmetry, the (W1,W2,W3) DoF transform as a triplet. Consequently, the mass terms induced
by the spontaneous symmetry breaking are equal for these three DOFs.
The most general mass matrix in the (W1,W2,W3, B) basis, that is consistent with U(1)EM
gauge invariance, is given by m2W
m2W
m2Zc
2W m2
ZcW sW
m2ZcW sW m2
Zs2W
. (9.18)
The custodial symmetry requires that the top three diagonal terms are equal and thus thatm2Zc
2W =
m2W , namely the ρ = 1 relation.
The custodial symmetry holds at tree level for models with any number of scalar doublets and
singlets. It is however not a symmetry of the full SM. In particular, it is broken by the Yukawa
coupling since YD 6= YU . The strongest breaking parameter is then m2t − m2
b . This breaking is
communicated to the Higgs sector by loop effects, resulting in violation of the predictions that
follow from the custodial symmetry. This is the reason that the leading correction to the ρ = 1
relation is proportional to m2t −m2
b , see Eq. (9.15).
138
The custodial symmetry is just an accidental symmetry of the Higgs sector. It is broken
by nonrenormalizable operators of dimension six (and higher). We present these dimension six
operators in the next subsection.
9.5 Probing new physics
Within the SM, the EWPM program is sensitive at tree level to three input parameters and at the
loop level to a few more. Since we have more observables than relevant SM parameters, EWPM
can be used to test the SM. So far, no significant deviation from the SM was found. Furthermore,
as we already mentioned, all the relevant SM parameters are now directly measured, and therefore
the data can be use to constrain BSM physics.
As concerns the probing of new physics with EWPM, one can go in either of two ways. First,
one can consider a specific model, calculate the new contributions to the observables, and constrain
the BSM parameters by comparing to the experimental results (just as we did for the SM itself).
We demonstrate this by a brief discussion of the four generation extension of the Standard Model
in subsection 9.5.2. Second, one can consider the effects of nonrenormalizable terms without
committing to a specific model. This is what we do in some detail in subsection 9.5.1. Under
reasonable assumptions, to be specified below, there is only a small number of dimension six terms
that affect the EWPM. Here we discuss these operators and how they are constrained by EWPM.
9.5.1 NR operators and the q2 expansion
In general there are many operators that affect the EWPM. The number of operators with poten-
tially significant effects is however much smaller in a large class of models that fulfill the following
three conditions:
1. The scale of the new physics is much higher than the electroweak breaking scale, Λ mW .
(This condition holds, by definition, for new physics whose effects can be represented by
nonrenormalizable terms.)
2. The new physics generates only oblique corrections to the relevant observables.
3. There are no contributions from new heavy gauge bosons.
(The analysis below applies to a broader class of models than implied by the third condition
[barbieri], but we assume this stronger condition for the sake of simplicity.)
The contributions to oblique corrections from nonrenormalizable terms can come from tree-level
diagrams or from loop diagrams. We consider here only tree level contributions. Consequently, our
analysis concerns only operators that involve exactly two electroweak gauge fields and no fermions.
In addition to the two electroweak gauge fields, the operators can have derivatives and Higgs fields.
139
Tree level contributions come from replacing the Higgs fields in these operators with their VEV.
The leading contributions (for Λ v) come from dimension-six operators.
It turns out that there are only four dimension-six terms that contribute to the oblique correc-
tions [5, 3, 6, 4]:
Lo.c. =1
Λ2(cWBOWB + cHHOHH + cBBOBB + cWWOWW ) , (9.19)
where the cXY ’s are dimensionless coefficients, and the operators are defined as follows (note that
in the literature, a variety of normalizations are used):
OWB = (H†τaH)W aµνBµν →
1
2v2W 3
µνBµν ,
OHH = |H†DµH|2 →1
16v4(gW 3
µ − g′Bµ)2,
OBB = (∂ρBµν)2,
OWW = (DρWaµν)
2. (9.20)
The symmetry properties of these operators are important. In particular, the way they can be
probed by the EWPM depends on whether their contributions to the oblique corrections violate
both the electroweak gauge symmetry and the custodial symmetry, or just the electroweak symme-
try, or neither. (All operators that contribute to oblique corrections at tree level belong to one of
these three classes. In particular, no such operators break the custodial symmetry without break-
ing the electroweak gauge symmetry.) Among the four operators, OHH breaks both symmetries,
while OWB breaks the gauge symmetry but respects the custodial symmetry. The two remaining
operators, OBB and OWW , respect both symmetries.
Intuitively, symmetry breaking effects are easiest to probe. This intuition goes well with the
actual situation: If the new physics breaks the custodial symmetry with O(1) parameters, then
the lower bound on the scale Λ coming from OHH would be the strongest. (It corrects the ρ = 1
relation.) Conversely, if we assume that the new physics respects the custodial symmetry, so that
cHH = 0, then the bound coming from OWB is the strongest.
In addition to the expansion in inverse powers of Λ, we can expand in powers of q2. As we
will see below, this expansion leads to considerable simplification of the analysis. Since we deal
with oblique corrections, we expand the vacuum polarization amplitudes ΠAB(q2) with AB =
W+W−,W3W3, BB,W3B:
ΠAB(q2) = ΠAB(0) + q2Π′AB(0) +(q2)2
2Π′′AB(0) + · · · . (9.21)
What is the relation between the q2 and the 1/Λ expansions? First we note that the number of
derivatives in any operator is related to the order in the q2 expansion. Since Π is dimension two,
and the only dimensionful parameter in the SM is v, we can write
Π(0) = v2(1 + v2/Λ2 + · · ·),
140
Π′(0) = 1 + v2/Λ2 + · · · ,
Π′′(0) =
1
Λ2(1 + v2/Λ2 + · · ·),
Π(n)(0) =1
Λ2n−2(1 + v2/Λ2 + ...). (9.22)
Since we neglect terms of dimension higher than six, we can truncate the q2 expansion at order
(q2)2: Terms of order (q2)3 and higher are suppressed by at least 1/Λ4. One more observation from
the above is that the dimension-six operators affect Π(0), Π′(0) and Π′′(0). Specifically, examining
Eq. (9.20), we straightforwardly learn that OWB contributes to Π′W3B(0), OHH contributes to
ΠW3W3(0), ΠBB(0) and ΠW3B(0), OBB contributes to Π′′BB(0), and OWW contributes to Π′′W3W3(0)
and Π′′W+W−(0).
To be clear, the q2 expansion of Eq. (9.9) applies to any model but only to low energy
observables (q2 m2W ). The q2 expansion of Eq. (9.21) applies to all observables but only
to new physics models that are characterized by a scale that is much higher than the electroweak
scale, (Λ v).
S, T and U
To study the contributions of dimension-six terms to oblique corrections, we need to keep terms
up to Π′′
in the q2 expansion. Thus, the four functions, ΠAB(q2), are replaced by twelve numbers,
ΠAB(0), Π′AB(0), and Π′′AB(0). Out of these twelve numbers, two combinations vanish due to the
U(1)EM gauge invariance, Πγγ(0) = ΠγZ(0) = 0. Three other combinations are fixed by the three
tree-level SM parameters. The remaining seven parameters are then related to observables, and
can be fitted. The seven parameters are usually called S, T, U, V,W,X, Y . From the theoretical
side, one can obtain the contributions of the four dimension-six terms to these parameters and
finally constrain their coefficients in the Lagrangian.
In order to provide a simple demonstration of the relation to observables, we provisionally
truncate the expansion at order q2. This leaves only three parameters that neither vanish nor are
fixed by the tree level parameters. These are the S, T and U parameters defined as follows:
αT =ΠWW (0)
m2W
− ΠZZ(0)
m2Z
, (9.23)
αS
4 sin2 2θ= Π′ZZ(0)− 2 cos2 2θ
sin2 2θΠ′Zγ(0)− Π′γγ(0), (9.24)
αU
4 sin2 θ= Π′WW (0)− cos2 θΠ′ZZ(0)− sin 2θΠ′γZ(0)− sin2 θΠ′γγ(0). (9.25)
A few comments are in order:
1. These S, T and U parameters receive one loop contributions in the SM and possibly also
BSM contributions. One can subtract the SM values and redefine them such that within the
SM they vanish. With this new definition, a non-zero value would be a sign of new physics.
The values presented below correspond to this definition.
141
S
0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5
T
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
68%, 95%, 99% CL fit contours, U free
=173 GeV)t
=126 GeV, mH: Mref
(SM
SM Prediction 0.4 GeV± = 125.7 HM
0.94 GeV± = 173.18 tm
SM Prediction
[100,1000] GeV∈ H
with M
HM
G fitter SMB
Se
p 1
2
Figure 9.1: The current bounds on S and T taken from [8]
2. S, T and U are pure numbers. Their scaling by α−1 relative to ΠAB (or Π′AB) means that
they roughly reflect the allowed size of new physics contributions compared to the SM one
loop contributions.
3. The U parameter rarely provides a significant constraint. The reason is that U arises from
a dimension-eight operator, while S and T arise from dimension-six operators.
4. Any new physics that respects the custodial symmetry gives T = U = 0, while S may
be different from zero. More generally, T and U are proportional to custodial symmetry
breaking parameters.
The EWPM determine the allowed ranges for these parameters. The strongest bounds arise
from S and T but there are experimental correlations between them that have to be taken into
account, see Fig. 9.1. The current experimental bounds read [?]
S = +0.05± 0.11,
T = +0.09± 0.13,
U = +0.01± 0.11, (9.26)
and, in particular, S ≤ 0.14 and T ≤ 0.20 at 95% CL.
The contributions of the dimension-six operators to S and T are given by [4]
S =2 sin 2θ
α
v2
Λ2cWB,
142
T = − 1
2α
v2
Λ2cHH . (9.27)
We note that T is proportional to cHH , which is the only custodial symmetry breaking parameter
among these nonrenormalizable terms. Using the experimental upper bounds of Eq. (9.26) as
reference points gives
Λ2
cWB
> 9.7(
0.14
S
)TeV2,
Λ2
cHH> 4.4
(0.20
T
)TeV2. (9.28)
9.5.2 The four generation SM
The four generation SM is excluded by measurements of the Higgs production and decay rates.
Yet, it provides a simple example of constraining a specific new physics model by the EWPM. We
assume that the fourth generation fermions are heavier than the top quark. We further assume no
flavor mixing between the fourth generation and the three known ones. To leading order in the
masses of the new fermions one finds [7]
S =1
2π
[1− 1
3log
(m2u4
m2d4
)]+
1
6π
[1 + log
(m2ν4
m2e4
)],
T =1
2π sin2(2θW )
(m2u4−m2
d4
m2Z
)+
1
6π sin2(2θW )
(m2ν4−m2
e4
m2Z
), (9.29)
where mf4 are the masses of the fourth generation fermions. We learn that T is related to the mass
splitting between t4 and b4 and between ν4 and e4, which are the custodial symmetry breaking
parameters in this model. In the case of a degenerate quark doublet and a degenerate lepton
doublet, we have T = 0 and S = 2/(3π). In a way, the S parameter “counts” the number of extra
generations. The current measurement of S by itself excludes an extra degenerate generation (so
that T = U = 0) at the 7σ level. Yet, when the correlation with T is taken into account, the
bounds on the extra generations become much less severe.
143
Chapter 10
QCD at the IR
In Chapter 5 we presented the high energy effects of QCD, where the theory is perturbative. In
this chapter we discuss some low energy aspects of QCD.
10.1 Spectroscopy
10.1.1 The quark model
As we discussed in Chapter 5 QCD is strongly coupled at the IR, and consequently there is no
perturbative expansion in terms of the fundamental DoF, quarks and gluons. What is observed are
bound states which we call hadrons, which are SU(3)C-singlets. What we know from experiments
about hadrons are their masses, lifetimes, spins and electric charges.
Note, however, that the fact that there is no reliable perturbation theory in terms of quarks
and gluons does not imples that there is no way to solve QCD. For example Lattice QCD is a
framework that aimed at solving QCD from first principle, but it is not perturbative. Chiral
perturbation theory is a systematic expansion that works at low energy, but it used hadrons as
its DoF, and does not have quarks and gluons as its DoF. These two examples provide us with
a lot of understanding of QCD at low energy, but we will not elaborate of them. Instead, in the
following we discuss the quark model.
We use the quark model to understand the behavior of QCD. The idea is that all bound
states are color singlet combinations of quarks and anti-quarks. In particular, we assume that the
quantum numbers of the hadrons are dictated by the quantum numbers of the constituent quarks
and antiquarks. This is clearly a model, as it assumes the minimum quark content of the hadron,
while for strongly coupled theories the wave function of the hadron is unavoidably more complex.
Yet, the model works surprisingly well, and gives us a lot of insights into the IR properties of QCD.
It is difficult, however, to estimate the related errors. In this Section, we focus on the spectroscopy
of the hadrons. Yet, there are also properties of the dynamics that can be described in more
144
detailed models.
In the quark model, the simplest hadrons belong to one of three types:
• Mesons, which have quark-antiquark constituents, M = qq
• Baryons, which have three quark constituents, B = qqq;
• Antibaryons, which have three antiquark constituents, B = qqq.
The lightest mesons are the pions:
π+ = ud, π0 =1√2
(uu− dd), π− = du. (10.1)
The lightest baryons are the proton and the neutron:
p = uud, n = udd. (10.2)
Mesons carry no baryon number, baryons carry baryon number that we normalize to B = +1, and
anti-baryons carry baryon number B = −1. Note that the letter B is used to denote a baryon
and baryon number (as well a certain type of mesons). Which of these options is meant should be
clear from the context.
The fact that QCD is confining results in an infinite spectrum of bound states. Their names
and Quantum Numbers (QNs) are given in Appendix 10.C. For mesons the notation is similar
to that of the hydrogen atom, as it is a two body system. We have the total spin, J , the spin
combination of the spin of the quark and anti-quark, S, that can be zero or one, and the orbital
angular momentum, L.
The quark model can have states that involve more than two or three quarks and anti-quarks,
that are referred to as exotics. These include tetraquarks (qqqq) and pentaquarks (qqqqq). While
not usually described as part of the quark model, there are also states involving gluons, in particular
the glueballs made of gluons only. Some of these states have recently been discovered. We do not
elaborate further on them.
10.1.2 Hadron masses
If we could solve QCD we would be able to calculate its IR spectrum. Since, however, QCD is a
strongly coupled theory we cannot do it. Yet, we can make several qualitative statements.
The masses of the hadrons come from two sources: the masses of the constituent quarks and
the binding energy coming from QCD interactions. Let us compare the situation to that of the
hydrogen atom. The mass of a hydrogen atom comes from the masses of its constituent proton
and electron plus a potential energy V = −13.6 eV. Indeed, it can be experimentally checked
that the mass of the hydrogen atom is 13.6 eV less than the sum of the masses of the proton and
145
electron. To make this statement it is crucial that we are able to physically pull a proton and
electron asymptotically apart and measure their masses independent of their mutual electric field.
For hadrons the situation is more complicated. Confinement implies that we cannot pull
individual quarks apart and measure their masses independently. (We discuss the issues of quark
masses in mode details in Appendix 10.A.) If we were able to extract the quark masses at high
scales, where QCD is weak, then we could run these masses down to the IR. The running of the
masses to low energies involves, however, the interactions, which is precisely what we wanted to
separate from our mass measurement. Let us also remark that, unlike atoms, hadrons can have
masses larger than the sum of the masses of their constituent quarks. In other words, the potential
energy can be positive. This, of course, is a signature of confinement.
As concerns quark masses, we split the six quarks into two groups: Light quarks — u, d, s
— which have masses (even though renormalization-scheme dependent) much smaller than ΛQCD,
and heavy quarks — c, b, t — with masses much larger than ΛQCD. The problem we just described
is particularly relevant for the light u, d and s quark masses, , since we probe them at energies
much larger than their pole masses (to the extent that a pole mass is at all well defined). The
bottom line is that these quark masses are regularization-scheme dependent. To understand what
is meant by a light quark mass, the regularization scheme that is used must be spelled out. Because
of these complications, it is at times useful to define “constituent quark masses.” The idea is that
we include the binding energy into the mass. In that way, we can define the mass of the u and the
d quark to be a third of the mass of the proton and the neutron. While this definition is somewhat
more intuitive, it does not relate to a fundamental parameter in our Lagrangian. It is almost a
restatement of the definition of ΛQCD.
The situation with the heavy c and b quark masses is different, as the mass of hadrons with
heavy quarks arises mainly from the heavy quark mass. The binding energy is a small correction
of relative order ΛQCD/mc,b.
As concerns the top quark, its mass is so heavy that its decay width from the weak interaction
is larger that the width of the QCD resonance: Γt ≈ 1.4 GeV ΛQCD. This situation is often
described by saying that the top “does not form hadrons.” There are subtleties, however, in this
statement. Because the top decays so quickly, we can never identify the hadron that it forms.
Arguably it is even an ill-posed question to identify the top hadron within the very short time
scale. In principle, though, we can theoretically “turn off” the weak interaction and calculate a
spectrum of top hadrons. This is a very strongly coupled problem that is intractable by known
techniques, but in principle one can calculate the spectrum of top hadrons when the weak force is
neglected.
146
10.1.3 Hadron lifetimes
With some abuse of language, we classify hadrons as either stable particles or unstable resonances.
In this context, stability is not meant in the absolute sense. (In fact, the only truly stable hadron
is the proton.) We can define what we mean by stability in two ways: experiment-related and
theory-related. Experimentally, we can determine the lifetime (1) by directly measuring its decay
width or (2) by measuring a displaced vertex, which is possible if the particle is sufficiently long-
lived and have high velocity. We call particles “stable” if their lifetimes is large enough to be
measured by the latter method (or even larger, so that it escapes the detector before decaying).
Theoretically, a more rigorous definition is that a stable particle is one that does not decay through
QCD interactions. These are the hadrons whose decays necessarily violate the accidental [U(1)]6
symmetry of QCD, see Eq. (5.17). Thus, we define a stable particle as one that either is really
stable or decays only through weak interactions. Resonances are the particles which are not stable.
In fact, some resonances have a width that is so broad (of the order of their mass) that it becomes
very hard to determine whether or not it is a bound state at all. Recall that when we do QFT
we work with states in the asymptotic past and future. When a decay width of a particle is of
the order of its mass, the notion of an asymptotic state becomes ill defined. In the PDG, particles
whose name contains their mass in parenthesis, e.g. ρ(770), are resonances while those that do not
are stable.
As an example of a resonance, consider the ρ meson. This is a bound state of up and down
quark and anti-quark. The dominant contribution to its mass comes from QCD. We thus expect
the mass to be of the order of ΛQCD, which is hundreds of MeV. Indeed, mρ ≈ 775 MeV. The ρ
decay via QCD, almost always to two pions and its width is very large, Γρ ≈ 150 MeV, just a
factor of five smaller than its mass. As an example of a stable particle, consider the charged π
meson, which is also a bound state of up and down quark and anti-quark. Its mass is 140 MeV
and it decays via the weak interaction to leptons and its width is of order 10−8 eV, clearly much
smaller than its mass. The ratio Γρ/Γπ ∼ 1016 demonstrates the difference between a resonance
and a stable particle, and explains why the weakly decaying hadrons are called stable.
The stable mesons are the lightest JP = 0− mesons that are charged under the [U(1)]6 sym-
metry, and they play an important role in the investigation of flavor physics. Their list is given in
Table 10.1. Note that the four neutral meson pairs with well defined U(1)s × U(1)c × U(1)b QNs
are not mass eigenstates. In fact, each pair mixes into a pair of mesons of well defined masses and
decay widths: (KS, KL), (DH , DL), (BH , BL) and (BsH , BsL). Here, (PH , PL) (for P = D,B,BS)
are distinguished by mass (Heavy and Light), while (KS, KL) are distinguished by lifetime (Short-
lived and Long-lived). The mass splitting within each pair is tiny (see Table 11.1). The width
difference within each of the D, B and Bs pairs is tiny, but for the K pair it is very large.
When discussing the different between stable hadrons and resonance, tritium – a hydrogen
isotope with two neutrons – provides a useful analogy. The excited 2P tritium state emits a
147
Table 10.1: List of the weakly decaying JP = 0− mesons with their quark decomposition, U(1)s×U(1)c×U(1)b charge, mass and lifetime. ∗For the neutral K-meson mass eigenstates, KS and KL,
the respective lifetimes are τS = 9.0× 10−11 s and τL = 5.1× 10−8 s.
Meson qq′ (S,C,B) Mass [GeV] τ [s]
π± ud, ud (0, 0, 0) 0.140 2.6× 10−8
K± us, us (±1, 0, 0) 0.494 1.2× 10−8
K0, K0 ds, ds (±1, 0, 0) 0.498 ∗
D± cd, cd (0,±1, 0) 1.87 1.0× 10−15
D0, D0 cu, cu (0,±1, 0) 1.87 4.1× 10−16
D±s cs, cs (±1,±1, 0) 1.97 5.0× 10−15
B± ub, ub (0, 0,±1) 5.28 1.6× 10−12
B0, B0 db, db (0, 0,±1) 5.28 1.5× 10−12
Bs, Bs sb, sb (∓1, 0,±1) 5.37 1.5× 10−12
B±c cb, cb (0,±1,±1) 6.28 5.1× 10−13
photon to decay to the 1S ground state at a time scale of the order of a nanosecond. The 1S state
decays into helium-3 with a lifetime of the order of 18 years. One can similarly imagine an excited
(heavy) B resonance decaying rapidly to the B0 by emitting a pion and then the B0 decaying via
the weak interaction on a much longer time scale. One should be comfortable thinking about the
plethora of hadrons in analogy to excited hydrogen atoms in basic quantum mechanics. In fact,
let us turn to one of the basic features of hydrogen in quantum mechanics: the quantum numbers
that describe a state.
10.1.4 Hadron quantum numbers
Each hadron can be identified by its mass, and has a well defined width. In addition, the hadrons
are characterized by three kinds of hadronic quantum numbers:
• Exact QNs. These are the electromagnetic charge, Q and the spin, J .
• QNs that are respected by QCD and QED but broken by the weak interaction.
These are the charges under the global [U(1)]6 symmetry, often referred to as flavor QNs. In
addition for masosn we can define the discrete parity P , and for neutral mesons also charge
conjugation C. We define parity as
P = (−1)L+1 , (10.3)
148
where L is the relative orbital angular momentum between the quark and anti-quark. The
reason that the L = 0 state is parity odd has to do with the fact that the quarks are fermions,
and thus for L = 0 without they change sign under parity. As concerns charge conjugation,
charged states are not eigenstates of C, while neutral ones are.
• Approximate QNs. The approximate symmetries of the QCD interactions are isospin,
SU(3)-flavor, and heavy quark symmetry. They are discussed in Sections 10.1.5 and 10.1.6.
10.1.5 The approximate symmetries of QCD: light quarks
Consider the QCD Lagrangian for the up and the down quarks [see Eq. (5.11)]:
LQCD = −1
4Gµνa Gaµν − iqD/ q −mqqq, q = u, d. (10.4)
Since mu 6= md, this Lagrangian has a [U(1)]2 flavor symmetry. If, however, we had mu = md,
the flavor symmetry would be U(2) = SU(2)× U(1), such that the U(1) symmetry is just baryon
number where the up and down quarks transform in the same way. The SU(2) part is called
isospin symmetry. Under isospin the up and down quarks form a doublet:
Q =
(u
d
), Q =
(d
u
). (10.5)
The up and down quarks are much lighter than ΛQCD. In particular, md−mu ΛQCD. Given
that, the isospin symmetry is an approximate symmetry of the QCD interactions, broken by a
small parameter:
εI ≡md −mu
ΛQCD
∼ 10−2, (10.6)
One might wonder why the QCD scale enters here. In general, what we have to compare the mass
difference to is the relevant energy in the event. Due to confinement the energy is at least of order
the QCD scale, so the the symmetry breaking parameter may be smaller, but not larger than εI .
In addition to εI , the isospin symmetry is broken by QED due to the different charges of the
up and down quarks. The size of this breaking is of order α ∼ 0.01, which is of similar order as εI .
The weak interaction also break isospin. Yet, in most implications the weak interaction is small
enough that we can negelect it.
What are the implications of this approximate symmetry? The symmetry implies that the
hadron mass eigenstates can be assigned into well defined representations of isospin, where the
hadrons within an isospin multiplet are approximately degenerate. This statement is based on a
symmetry of the QCD Lagrangian and not on any quark model. We use the quark model to assign
the mass eigenstates into quark representations.
Consider, for example, the baryons. The two lightest baryons are the neutron and the proton
which are almost degenerate: mp = 938.272 MeV and mn = 939.565 MeV. It is natural to assume
149
that they form an isospin doublet. At somewhat higher scale, m∆ ≈ 1.23 GeV, there are the four
nearly degenerate ∆ states with charges Q = −1, 0,+1,+2, which fit nicely into an isospin quartet.
Within the quark model a baryon is made of three quarks. Thus, in term of isospin, the bartuons
that are form from u ad d quarks are a product of three isopsin doublets. This producs gives
1
2× 1
2× 1
2=
1
2+
3
2. (10.7)
(The other isospin-half in the combination is antisymmetric and thus vanishes for identical dou-
blets.) For the nucleons, the quark assignment is given in Eq. (10.2), while that of ∆ is given
by
∆− = ddd, ∆0 = ddu, ∆+ = duu, ∆++ = uuu. (10.8)
The fact that each multiplet is almost degenerate is a result of the approximate symmetry. The
difference of O(ΛQCD) between the masses of the nucleon doublet and the ∆ quartet is due to the
fact the QCD interaction that gives the binding energy is very different for different representations
of the isospin group.
The situation with the mesons is similar. Among the JP = 0− mesons, we identify the three
pions, which are quasi-degenerate (mπ± = 139.57 MeV, mπ0 = 134.98 MeV), as an isospin-triplet,
while the heavier η meson (mη = 547.86 MeV) is an isospin singlet. Indeed, the combination of
two isospin doublets gives1
2× 1
2= 0 + 1. (10.9)
The quark model assignment of the pions is given in Eq. (10.1), while that of η is given by
η0 =uu+ dd√
2. (10.10)
10.1.6 The approximate symmetries of QCD: heavy quarks
10.2 High energy QCD
In high energy QCD processes, both long and short distance physics are needed to explain the
observations. In this section we discuss a few aspects of it.
10.2.1 Quark hadron duality
the way we set up QCD, theoretical calculations are done in terms of quarks and gluons. Yet,
what we deal with experimentaly are hadrons. The question is then how can we connect the two
aspects of QCD, that is, how to use the experumental data to probe the theory.
The answer is given by the assumption of “quark-hadron duality.” The duality states that
inclusive hadronic observables at high energies, when integrated over large enough energt range,
150
can be described by the calculation in terms of quarks and gluons. (The generic name for the
quark or gluon in such a process is parton.) This duality is used in many cases with much success,
for example, in e+e− annihilation, deep inelastic scattering, τ decays, and semileptonic B decays.
Despite this success, the notion of the quark-hadron duality remained vague. In praticular,
there is no formal critiria of when ot can be used. Moreover, it is not clear how we can estimate
deviations from the duality results. Yet, the intuition is clear
To demonstrate the idea, consider the ratio of scattering cross sections (for s m2Z)
R(E) ≡∫ E
0 σ(e+e− → hadrons)∫ E0 σ(e+e− → µ+µ−)
. (10.11)
Quark hadron duality implies that it is given to leading order by
R(E) ≡∫ E
0 σ(e+e− → qq)∫ E0 σ(e+e− → µ+µ−)
≈ 3∑q
Q2q, (10.12)
where Qq is the EM charge of the quark q, the sum goes over all the quarks that are lighter than
E, and the approximation sign is due to the fact that we neglect the phase space factor, the weak
interaction effects, and higher orders in QCD.
The situation is more complicated when the initial state involves hadrons, for example, ep or
pp collisions. Then it is not clear what is the initial state in terms of quarks and gluons. The way
to go is to define a “parton distribution function” (PDF) that describes the distributions of the
quarks, antiquarks and gluons inside the initial hadrons as functions of the involved energy. The
PDF is a non-perturbative quantity that cannot be calculated from first principles. We determine
it by measuring it in one process and then using it in another. Once we have it, the calculation
is done on the same principle as discussed before. The calculation is done in terms of the partons
(quarks and gluons) and compared with the integrated measurements over hadrons.
10.2.2 Jets
When we smash particles together at a collider, the hard scattering process can be described by
the UV part of QCD. For example, we can think about e+e− → qq or e+e− → gg processes. Yet,
confinement tells us that we cannot see the two quarks going back to back as we see the muons in
the case of e+e− → µ+µ−. What we find is that qq pairs coming from the vacuum combine with
the original partons in order to form color-neutral final states.
The number of qq pairs that are created depends on the relative energy between the original
quarks. The higher the energy is, the more qq pairs are likely to be created. Most of the hadrons
that are produced are traveling roughly in the same direction as the original partons. The collection
of hadrons related to a single original parton comes under the name of a jet.
While there is no first principle way to relate jets to the UV properties of QCD, the fact that
jets form gives us ways to probe QCD at short distance. To a good approximation, a process like
e+e− → qq is searched for as e+e− → 2j (here j represents a jet), while e+e− → qqg as e+e− → 3j.
151
While to leading order what we care about is the fact that we have a jet, we can further study
the properties of the jet. Some of the properties depend on the parton that originates the jet. For
example, one can, statistically speaking, tell a quark jet from a gluon jet. Other properties are
universal. For example, the most likely mesons to be produced in the jet are pions since they are
the lightest. There are also not many baryons in a jet. We can understand this fact by the need
to pop up two pairs of qq and the combinatorics is such that the three qq pairs are more likely to
end up in three mesons than in two baryons.
152
Appendix
10.A quark masses
10.B SU(3)
10.C Names and QN for hadrons
The pseudoscalar (JPC = 0−+) mesons include the π, η, η′, K, D, and B mesons. The first three
are called ‘unflavored’ by the PDG, meaning that they contain no heavy flavor. The η and η′
contain some admixture of ss, but this has no net strangeness quantum number. The K, D, and
B are flavored mesons: The kaons (K) all have net strangeness, the D mesons have net charm,
and the B mesons have net beauty.
The vector mesons (JPC = 1−−) include the isospin-triplet ρ(770), and the ω(782) and φ(1020)
which are isospin singlets. The flavored vector mesons are indicated by stars relative to the
pseudoscalars: K∗, D∗, B∗.
The scalar (JPC = 0++) mesons are formed by quark configurations with orbital angular
momentum ` = 1 and spin s = 1 such that the net spin is zero: a0, a1, a2, . . .. The vector 1+−
mesons are formed by quark configurations with ` = 1 and s = 0: b0, b1, . . ..
The baryons with three light quarks (u, d combinations) include the isospin doublet of nucleons
N = p, n, and the isospin quartet of ∆ = ∆++,∆+,∆0,∆−.
The baryons with two u and/or d quarks include the isospin singlets Λ,Λc,Λb, and the isospin
triplets Σ,Σc,Σb. The isospin comes only from the light quarks. These particles are named
according to their heavy quark. No index implies an s, while a subscript c or b indicates charmed
or beautiful baryons.
Baryons with just one u or d quark are isospin doublets and are called Ξ,Ξc,Ξb (which is too
difficult for particle physicists to pronounce consistently so we tend to call them ‘cascades’). These
are also named according to their heavy quarks with s implied by no index. Thus we would name
a baryon with one light quark, a strange quark, and a charm quark the Ξc. Whether the light
quark is u or d is implied by the charge. For example, the Ξ+ must be a uus combination.
Finally, there baryons with no u and d quarks. The most famous of these is Ω, composed of
153
three strange quarks. This is the particle that Ne’eman and Gell-mann predicted on the basis of
the decouplet of SU(3). They wrote down its charge, quark content, and mass a few years before
its discovery.
154
Homework
Question 10.1: Using the PDG
Read the quark model review and the light meson summery table from the PDG.
1. What are the component quarks of the D+ meson? What is its mass?
2. What are the component quarks of the Λ baryon? What is its spin?
3. What is the lifetime (in sec) and width (in eV) of the B+ meson?
4. Explain what P , C, J , I and G are. For each of these Quantum Numbers (QNs) indicate if
they are (i) exact in Nature; (ii) exact in QCD; or (iii) approximately conserved in QCD.
(Exact in QCD means to consider QCD gauge interactions and quark masses, approximate
means consider only QCD gauge interactions.)
5. Find the masses and the above mentioned QNs of the π, η, ρ and ω.
6. Find the Branching Ratios (BRs) of the ρ, η and ω to two and three pions.
7. You can see that the η does not decay to two pions, the ω decay rate to two pions is very
small, while the ρ decay almost always to two pions. Based on the QNs we mention above,
explain these results.
155
Chapter 11
Flavor physics
11.1 Introduction
The effects of non-renormalizable terms might be observed in rare processes, where the contribution
from the renormalizable SM is highly suppressed. The prime example of such processes are flavor
changing neutral current processes. In this section, we explain what these processes are, describe
the phenomenological constraints on deviations from the SM predictions, and extract lower bounds
on the scale that suppresses dimension-six terms that contribute to these processes.
The term “flavors” is used to describe several copies of the same gauge representation. Within
the Standard Model, each of the four different types (namely different SU(3)C ×U(1)EM represen-
tations) of fermionic particles comes in three flavors:
• Up-type quarks in the (3)+2/3 representation: u, c, t;
• Down-type quarks in the (3)−1/3 representation: d, s, b;
• Charged leptons in the (1)−1 representation: e, µ, τ ;
• Neutrinos in the (1)0 representation: ν1, ν2, ν3.
The term “flavor physics” refers to interactions that distinguish between flavors. Within the
SM, these are the W -mediated weak interactions and the Yukawa interactions. The term “flavor
parameters” refers to parameters that carry flavor indices. Within the SM, there are 13 flavor
parameters: the 9 charged fermion masses and the four CKM parameters. The term “flavor
universal” refers to interactions with couplings (or to parameters) that are proportional to a
unit matrix in flavor space. Within the SM, the strong, electromagnetic, and Z-mediated weak
interactions are flavor-universal. The term “flavor diagonal” refers to interactions with couplings
(or to parameters) that are diagonal, but not necessarily universal, in flavor space. Within the SM,
the Yukawa interactions are flavor-diagonal in the mass basis. Note that flavor universal couplings
are universral in any basis. Flavor diagonal that are not universal are diagonal only in one specific
156
Table 11.1: Measurements related to neutral meson mixing
Sector CP-conserving CP-violating
sd ∆mK/mK = 7.0× 10−15 εK = 2.3× 10−3
cu ∆mD/mD = 8.7× 10−15 AΓ/yCP ∼< 0.2
bd ∆mB/mB = 6.3× 10−14 SψK = +0.67± 0.02
bs ∆mBs/mBs = 2.1× 10−12 Sψφ = −0.04± 0.09
basis. In this section we only discuss flavor in th quark sector. (We will discuss the lepton sector
in Section 12.2.)
A central role in testing the CKM sector of the SM is played by flavor changing processes.
The term “flavor-changing” refers to processes where the initial and final flavor-numbers (that
is, the number of particles of a certain flavor minus the number of anti-particles of the same
flavor) are different. In “flavor changing charged current” (FCCC) processes, both up-type and
down-type flavors are involved. Examples are K− → µ−νµ which corresponds, at the quark
level, to su → µ−νµ transition, and B− → ψK− (b → ccs transition). Within the Standard
Model, these processes are mediated by the W -bosons and occur at tree level. In “flavor changing
neutral current” (FCNC) processes, either up-type or down-type flavors but not both are involved.
Examples of FCNC decays include K− → π−µ+µ− (sd → µ+µ− transition), and B− → φK−
(b→ sss transition). Within the Standard Model, these processes do not occur at tree level, and
are strongly suppressed.
11.2 Flavor changing neutral current (FCNC) processes
A very useful class of FCNC processes is that of neutral meson mixing. Nature provides us with
four pairs of CP -conjugate neutral mesons: K0−K0, B0−B0, B0s −B0
s, and D0−D0. Mixing in
this context refers to a transition such as K0 → K0 (sd→ ds).1 The experimental results for CP
conserving and CP violating observables related to neutral meson mixing (mass splittings and CP
asymmetries in tree level decays, respectively) are given in Table 11.1.
FCNC processes are highly suppressed within the SM. Our aim in this section is to explain the
suppression factors that affect FCNC within the SM.
(a) Loop suppression. The W -boson cannot mediate FCNC processes at tree level, since
it couples to up-down pairs, or to neutrino-charged lepton pairs. Obviously, only neutral bosons
1These transitions involve four-quark operators. When calculating the matrix elements of these operators be-
tween meson-antimeson states, approximate symmetries of QCD are of no help. Instead, one uses lattice calculations
to relate, for example, the B0 → B0 transition to the corresponding quark process, bd→ db.
157
can mediate FCNC at tree level. The SM has four neutral bosons: the gluon, the photon, the
Z-boson and the Higgs-boson. As concerns the massless gauge bosons, the gluon and the photon,
their couplings are flavor-universal and, in particular, flavor-diagonal. This is guaranteed by gauge
invariance. The universality of the kinetic terms in the canonical basis requires universality of the
gauge couplings related to the unbroken symmetries. Hence neither the gluon nor the photon can
mediate flavor changing processes at tree level. This result is therefore generic to any extension of
the SM that respect the unbroken gauge symmetries of the SM.
The situation concerning the Z-boson and the Higgs-boson is more complicated. In fact, the
diagonality of their tree-level couplings is a consequence of special features of the SM, and can be
violated with new physics.
The Z-boson, similarly to the W -boson, does not correspond to an unbroken gauge symmetry
(as manifest in the fact that it is massive). Hence, there is no fundamental symmetry principle
that forbids flavor changing couplings. Yet, as mentioned in Section 8.4.1, in the SM the couplings
are uuniveral and diagonal. The key point is the following. For each sector of mass eigenstates,
characterized by spin, SU(3)C representation and U(1)EM charge, there are two possibilities:
1. All mass eigenstates in this sector originate from interaction eigenstates in the same SU(2)L×U(1)Y representation.
2. The mass eigenstates in this sector mix interaction eigenstates of different SU(2)L × U(1)Y
representations (but, of course, with the same T3 + Y ).
Let us examine the Z couplings in the interaction basis in the subspace of all states that mix
within a given sector of mass eigenstates:
1. In the first class, the Z couplings in this subspace are universal, namely they are proportional
to the unit matrix (times T3−Q sin2 θW of the relevant interaction eigenstates). The rotation
to the mass basis maintains the universality: VfM × 1× V †fM = 1 (f = u, d, e; M = L,R).
2. In the second class, the Z couplings are only “block-universal”. In each sub-block i of mi
interaction eigenstates that have the same (T3)i, they are proportional to the mi ×mi unit
matrix, but the overall factor of (T3)i−Q sin2 θW is different between the sub-blocks. In this
case, the rotation to the mass basis, VfM × diag[(T3)1−Qs2W ]1m1 , [(T3)2−Qs2
W ]1m2 , . . .×V †fM , does not maintain the universality, nor even the diagonality.
The special feature of the SM fermions is that they belong to the first class: All fermion mass
eigenstates in a given SU(3)C × U(1)EM representation come from the same SU(3)C × SU(2)L ×U(1)Y representation. For example, all the left-handed up quark mass eigenstates, which are in
the (3)+2/3 representation, come from interaction eigenstates in the (3, 2)+1/6 representation. This
is the reason that the SM predicts universal Z couplings to fermions. If, for example, Nature had
also left-handed quarks in the (3, 1)+2/3 representation, then the Z couplings in the left-handed up
158
sector would be non-universal and the Z could mediate FCNC. In your homework, you will work
out an explicit example.
The Yukawa couplings of the Higgs boson are not universal. In fact, in the interaction basis,
they are given by completely general 3 × 3 matrices. Yet, as explained in Section 8.4.3, in the
fermion mass basis they are diagonal. The reason is that the fermion mass matrix is proportional
to the corresponding Yukawa matrix. Consequently, the mass matrix and the Yukawa matrix are
simultaneously diagonalized. The special features of the SM in this regard are the following:
1. All the SM fermions are chiral, and therefore there are no bare mass terms.
2. The scalar sector has a single Higgs doublet.
In contrast, either of the following possible extensions would lead to flavor changing Higgs cou-
plings:
1. There are quarks or leptons in vector-like representations, and thus there are bare mass
terms.
2. There is more than one SU(2)L-doublet scalar.
It is interesting to note, however, that not all multi Higgs doublet models lead to flavor changing
Higgs couplings. If all the fermions of a given sector couple to one and the same doublet, then the
Higgs couplings in that sector would still be diagonal. For example, in a model with two Higgs
doublets, φ1 and φ2, and Yukawa terms of the form
LYuk = Y uijQLiURj φ2 + Y d
ijQLiDRj φ1 + Y eijLLiERj φ1 + h.c., (11.1)
the Higgs couplings are flavor diagonal. In the physics jargon, we say that such models have natural
flavor conservation (NFC).
We conclude that within the SM, all FCNC processes are loop suppressed. However, in exten-
sions of the SM, FCNC can appear at the tree level, mediated by the Z boson or by the Higgs
boson or by new massive bosons.
(b) CKM suppression. Obviously, all flavor changing processes are proportional to off-
diagonal entries in the CKM matrix. A quick look at the absolute values of the off-diagonal
entries of the CKM matrix, Eq. (8.72), reveals that they are small. A rough estimate of the
CKM suppression can be acquired by counting powers of λ in the Wolfenstein parametrization,
Eq. (8.70): |Vus| and |Vcd| are suppressed by λ, |Vcb| and |Vts| by λ2, |Vub| and |Vtd| by λ3.
For example, the amplitude for b→ sγ decay comes from penguin diagrams, dominated by the
intermediate top quark, and suppressed by |VtbVts| ∼ λ2. As another example, the B0−B0 mixing
amplitude comes from box diagrams, dominated by intermediate top quarks, and suppressed by
|VtbVtd|2 ∼ λ6.
159
(c) GIM suppression. If all quarks in a given sector were degenerate, then there would
be no flavor changing W -couplings. A consequence of this fact is that FCNC in the down (up)
sector are proportional to mass-squared differences between the quarks of the up (down) sector.
For FCNC processes that involve only quarks of the first two generations, this leads to a strong
suppression factor related to the light quark masses, and known as Glashow-Iliopoulos-Maiani
(GIM) suppression.
Let us take as an example ∆mK , the mass splitting between the two neutral K-mesons. (A more
detailed discussion of neutral meson mixing can be found in Appendix 11.A.) We have ∆mK =
2|MKK |, where MKK corresponds to the K0 → K0 transition and comes from box diagrams.
The top contribution is CKM-suppressed compared to the contributions from intermediate up and
charm, so we consider only the latter:
MKK '∑
i,j=u,c
G2F
16π2〈K0|(dLγµsL)2|K0〉(VisV ∗idVjsV ∗jd)× F (xi, xj), (11.2)
where xi = m2i /m
2W . If we had mu = mc, the amplitude would be proportional to (VusV
∗ud+VcsV
∗cd)
2,
which vanishes in the two generation limit. We conclude that ∆mK ∝ (m2c −m2
u)/m2W , which is
the GIM suppression factor.
For the B0 − B0 and Bs − Bs mixing amplitudes, the top-mediated contribution is not CKM
suppressed compared to the lighter generations. The mass ratio m2t/m
2W enhances, rather than
suppresses, the top contribution. Consequently, the MBB amplitude is dominated by the top
contribution:
MBB 'G2Fm
2t
16π2〈B0|(dLγµbL)2|B0〉(VtbV ∗td)2 × F
(m2W
m2t
). (11.3)
Before we turn to discuss the present situation, we should mention that historically, FCNC
have served important role in predicting the existence of SM particles before they were directly
discovered, and in predicting their masses:
• The smallness of Γ(KL→µ+µ−)Γ(K+→µ+ν)
led to predicting a fourth (the charm) quark;
• The size of ∆mK led to a successful prediction of the charm mass;
• The measurement of εK led to predicting the third generation;
• The size of ∆mB led to a successful prediction of the top mass.
11.3 CP violation
There are two main reasons for the interest in CP violation:
• CP asymmetries provide some of the theoretically cleanest probes of flavor physics. The
reason for that is that CP is a good symmetry of the strong interactions. Consequently, for
some hadronic decays, QCD-related uncertainties cancel out in the CP asymmetries.
160
• There is a cosmological puzzle related to CP violation. The baryon asymmetry of the Uni-
verse is a CP violating observable, and it is many orders of magnitude larger than the SM
prediction. Hence, there must exist new sources of CP violation beyond the single phase of
the CKM matrix.
In this section, based on the fact that CP violation is related to complex parameters of the
Lagrangian, we prove that CP violation in a two generation SM is impossible, while CP violation
in a three generation SM requires a long list of conditions on its flavor parameters in order to
occur. The formalism and SM calculation of CP asymmetries is presented in Appendix 11.B. The
cosmological puzzle is presented in Section 13.1.
In Section 1.5.2, we argued that CP is violated if, using all freedom to redefine the phases of
the fields, one cannot find any basis where all the couplings in the Lagrangian are real. Let us
examine the situation in the mass basis of the SM. The couplings of the gluons, the photon and
the Z-boson are all real, as are the two parameters of the scalar potential. As concerns the fermion
mass terms and the weak gauge interactions, the relevant CP transformation laws are
ψiψj → ψjψi, ψiγµW+
µ (1− γ5)ψj → ψjγµW−
µ (1− γ5)ψi. (11.4)
Thus the mass terms and CC weak interaction terms are CP invariant if all the masses and couplings
are real. We can always choose the masses to be real. Then, let us focus on the couplings of W±
to quarks:
− g√2
(Vijuiγ
µW+µ (1− γ5)dj + V ∗ij djγ
µW−µ (1− γ5)ui
). (11.5)
The CP operation exchanges the two terms, except that Vij and V ∗ij are not interchanged. Thus
CP is a good symmetry only if there is a mass basis and choice of phase convention where all
couplings and masses are real.
In Section 8.7 we discussed CP violation in SM3. We reached the conclusion that CP is violated
if and only if fourteen different conditions are fulfilled, see Eq. (8.75) and the discussion that follows
it. These conditions can also be written as a single requirement on the quark mass matrices in the
interaction basis:
XCP ≡ Im
det[MdM
†d ,MuM
†u
]6= 0 ⇔ CP violation. (11.6)
This is a convention independent condition.
To appreciate the subtleties that allow CP violation in the SM, it is helpful to consider a two
generation Standard Model, SM2. This model is similar to the one defined in Section 8.1, which
in this section will be referred to as SM3, except that there are two, rather than three fermion
generations. Many features of SM2 are similar to SM3, but there is one important difference: CP is
a good symmetry of SM2, but not of SM3. To see how this difference comes about, let us examine
the accidental symmetries of SM2. We follow here the line of analysis of SM3 in Section 8.5.
161
If we set the Yukawa couplings to zero, LYuk = 0, SM2 gains an accidental global symmetry:
GglobalSM2 (Y u,d,e = 0) = U(2)Q × U(2)U × U(2)D × U(2)L × U(2)E, (11.7)
where the two generations of each gauge representation are a doublet of the corresponding U(2).
The Yukawa couplings break this symmetry into the subgroup
GglobalSM2 = U(1)B × U(1)e × U(1)µ. (11.8)
A-priori, the Yukawa terms depend on three 2×2 complex matrices, namely 12R+12I parameters.
The global symmetry breaking, [U(2)]5 → [U(1)]3, implies that we can remove 5× (1R+3I)−3I =
5R + 12I parameters. Thus the number of physical flavor parameters is 7 real parameters and no
imaginary parameter. The real parameters can be identified as two charged lepton masses, four
quark masses, and the single real mixing angle, sin θc = |Vus|.The important conclusion for our purposes is that all imaginary couplings can be removed
from SM2, and CP is an accidental symmetry of the model. Thus, the existence of (at least) three
generations is necessary to allow CP violation within the SM.
11.4 FCNC and CPV: Examples
11.5 Testing the CKM sector: The ρ− η plane
According to the SM, all quark flavor changing processes depend on only four independent CKM
parameters, that can be chosen to be those of the Wolfenstein parametrization (8.70): λ, A, ρ and
η. The number of flavor changing processes that can be measured is much larger, thus providing
a stringent test of the CKM picture of flavor physics.
The values of λ and A are known rather accurately from, respectively, K → π`ν and B → Xc`ν
decays:
λ = 0.2254± 0.0007, A = 0.811+0.022−0.012. (11.9)
Then, one can express all the relevant observables as functions of the two remaining parameters, ρ
and η, and check whether there is a range in the ρ−η plane that is consistent with all measurements.
The list of observables includes the following:
• The rates of inclusive and exclusive charmless semileptonic B decays depend on |Vub|2 ∝ρ2 + η2;
• The CP asymmetry in B → ψKS, SψKS = sin 2β = 2η(1−ρ)(1−ρ)2+η2 ;
• The rates of various B → DK decays depend on the phase γ, where eiγ = ρ+iη√ρ2+η2
;
• The rates of various B → ππ, ρπ, ρρ decays depend on the phase α = π − β − γ;
162
γ
γ
α
α
dm∆
Kε
Kε
sm∆ & dm∆
ubV
βsin 2
(excl. at CL > 0.95)
< 0βsol. w/ cos 2
exc
luded a
t CL >
0.9
5
α
βγ
ρ
1.0 0.5 0.0 0.5 1.0 1.5 2.0
η
1.5
1.0
0.5
0.0
0.5
1.0
1.5
excluded area has CL > 0.95
FPCP 13
CKMf i t t e r
Figure 11.1: Allowed region in the ρ, η plane. Superimposed are the individual constraints from
charmless semileptonic B decays (|Vub|), mass differences in the B0 (∆md) and Bs (∆ms) neutral
meson systems, and CP violation in K → ππ (εK), B → ψK (sin 2β), B → ππ, ρπ, ρρ (α), and
B → DK (γ).
• The ratio between the mass splittings in the neutral B and Bs systems is sensitive to
|Vtd/Vts|2 = λ2[(1− ρ)2 + η2];
• The CP violation in K → ππ decays, εK , depends in a complicated way on ρ and η.
The resulting constraints are shown in Fig. 11.1.
The consistency of the various constraints is impressive. In particular, the following ranges for
ρ and η can account for all the measurements:
ρ = +0.131+0.026−0.013, η = +0.345± 0.014. (11.10)
Given the consistency of the measurements with the renormalizable SM, and the fact that all
the SM parameters are known, one can use the upper bounds on possible deviations from the SM
predictions to set upper bounds on the size of non-renormalizable terms.
163
11.6 New flavor physics: The hd − σd plane
We now aim to go beyond testing the self-consistency of the CKM picture of flavor physics and
CP violation. Based on experimental information, we can actually prove that the KM phase is
different from zero and that, moreover, it is the dominant source of all observed CP violation. We
can also quantify how much room is left for new physics in this regard. In proving that the KM
mechanism is at work, we assume that charged-current tree-level processes are dominated by the
W -mediated SM diagrams. This is a very plausible assumption. It is difficult to construct a model
where new physics competes with the SM in flavor changing charged current processes, and does
not violate the constraints from flavor changing neutral current processes.
Thus we can use all tree level processes and fit them to ρ and η, as we did before. The list of
such processes includes the following:
1. Charmless semileptonic B-decays, b→ u`ν.
2. B → DK decays, which go through the quark transitions b→ cus and b→ ucs.
3. B → ρρ decays (and, similarly, B → ππ and B → ρπ decays) go through the quark transition
b → uud. With an isospin analysis, one can determine the relative phase between the tree
decay amplitude and the mixing amplitude. By incorporating the measurement of SψKS , one
can subtract the phase from the mixing amplitude, finally providing a measurement of the
angle γ.
In addition, we can use loop processes, but then we must allow for new physics contributions,
in addition to the (ρ, η)-dependent SM contributions. Of course, if each such measurement adds
a separate mode-dependent parameter, then we do not gain anything by using this information.
However, there is a number of observables where the only relevant loop process is B0−B0 mixing.
Within the SM, the B0−B0 mixing amplitude, MBB, is a function of ρ and η. We can parameterize
the most general modification of the SM prediction ofMBB in terms of two parameters, r2d signifying
the change in magnitude, and 2θd signifying the change in phase:
MBB = r2d e
2iθd MSMBB(ρ, η). (11.11)
The list of relevant observables includes SψKS , ∆mB and ASL, the CP asymmetry in semileptonic
B decays:
SψKS = sin(2β + 2θd),
∆mB = r2d(∆mB)SM,
ASL =(
∆ΓB∆mB
)SM sin 2θdr2d
+ (ASL)SM cos 2θdr2d
. (11.12)
164
An alternative way to present the data is to use the hd, σd parametrization,
r2de
2iθd = 1 + hde2iσd . (11.13)
While the rd, θd parameters give the relation between the full mixing amplitude and the SM one,
and are convenient to apply to the measurements, the hd, σd parameters give the relation between
the new physics and SM contributions, and are more convenient in testing theoretical models:
MNPBB = hde
2iσdMSMBB . (11.14)
Thus, we fit six observables to four parameters. The results of such fit, projected on the ρ− ηplane, can be seen in Fig. 11.1(a). It is clear that η 6= 0 is well established, proving that the
Kobayashi-Maskawa mechanism of CP violation is at work.
Figure 11.1: Constraints in the (a) ρ− η plane, and (b) hd − σd plane, assuming that NP contri-
butions to tree level processes are negligible.
γ
γ
)α(γ
)α(γ
ubV
ubV) & α(γ & γ
α
βγ
ρ
1.0 0.5 0.0 0.5 1.0 1.5 2.0
η
1.5
1.0
0.5
0.0
0.5
1.0
1.5
excluded area has CL > 0.95
2013
CKMf i t t e r
dh0.0 0.1 0.2 0.3 0.4 0.5
dσ
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0pvalue
excluded area has CL > 0.95
2013
CKMf i t t e r
In order to test model independently whether the SM dominates the observed CP violation,
and to put an upper bound on the new physics contribution to B0−B0 mixing, we need to project
the results of the fit on the hd − σd plane. If we find that hd 1, then the SM dominance in the
B0 −B0mixing amplitude will be established. If hd ∼> 1 for σd ∼ 0, π but hd 1 otherwise, then
the SM dominance in the observed CP violation will be established. The constraints in the hd−σdplane are shown in Fig. 11.1(b). We can make the following two statements:
1. A new physics contribution to B0 −B0mixing amplitude that carries a phase that is signif-
icantly different from the KM phase is constrained to lie below the 10-20% level.
2. A new physics contribution to the B0 −B0mixing amplitude which is aligned with the KM
phase lie below the the 30-40% level.
165
One can reformulate these statements as follows:
1. The KM mechanism dominates CP violation in B0 −B0mixing.
2. The CKM mechanism is a major player in B0 −B0mixing.
11.7 Non-renormalizable terms
Given that the SM is only an effective low energy theory, one should consider the effects of
non-renormalizable terms. As concerns quark flavor physics, consider, for example, the follow-
ing dimension-six set of operators:
L∆F=2NP =
∑i 6=j
zijΛ2
(QLiγµQLj)2, (11.15)
where the zij are dimensionless couplings. The consistency of the experimental results with the
SM predictions for neutral meson mixing, allows us to impose the condition |MNPPP | < |M
SMPP | for
P = K,B,Bs, which implies that
Λ >4.4 TeV
|V ∗tiVtj|/|zij|1/2∼
1.3× 104 TeV × |zsd|1/2
5.1× 102 TeV × |zbd|1/2
1.1× 101 TeV × |zbs|1/2(11.16)
A more detailed list of the bounds derived from the ∆F = 2 observables in Table 11.1 is given
in Table 11.1. The bounds refer to two representative sets of dimension-six operators: (i) left-left
operators, that are also present in the SM, and (ii) operators with different chirality, where the
bounds are strongest because of larger hadronic matrix elements.
The first lesson that we draw from these bounds on Λ is that new physics can contribute to
FCNC at a level comparable to the SM contributions even if it takes place at a scale that is six
orders of magnitude above the electroweak scale. A second lesson is that if the new physics has
a generic flavor structure, that is zij = O(1), then its scale must be above 104 − 105 TeV (or, if
the leading contributions involve electroweak loops, above 103 − 104 TeV). If indeed Λ TeV ,
it means that we have misinterpreted the hints from the fine-tuning problem and the dark matter
puzzle.
A different lesson can be drawn from the bounds on zij. It could be that the scale of new physics
is of order TeV, but its flavor structure is far from generic. Specifically, if new particles at the TeV
scale couple to the SM fermions, then there are two ways in which their contributions to FCNC
processes, such as neutral meson mixing, can be suppressed: degeneracy and alignment. Either of
these principles, or a combination of both, signifies non-generic structure.
166
Table 11.1: Lower bounds on the scale of new physics Λ, in units of TeV, for |zij| = 1 and for
Imzij ∼ 1, and upper bounds on zij, assuming Λ = 1 TeV.
Operator Λ [TeV] CPC Λ [TeV] CPV |zij| Im(zij) Observables
(sLγµdL)2 9.8× 102 1.6× 104 9.0× 10−7 3.4× 10−9 ∆mK ; εK
(sRdL)(sLdR) 1.8× 104 3.2× 105 6.9× 10−9 2.6× 10−11 ∆mK ; εK
(cLγµuL)2 1.2× 103 2.9× 103 5.6× 10−7 1.0× 10−7 ∆mD; AΓ
(cRuL)(cLuR) 6.2× 103 1.5× 104 5.7× 10−8 1.1× 10−8 ∆mD; AΓ
(bLγµdL)2 5.1× 102 9.3× 102 3.3× 10−6 1.0× 10−6 ∆mB; SψK
(bRdL)(bLdR) 1.9× 103 3.6× 103 5.6× 10−7 1.7× 10−7 ∆mB; SψK
(bLγµsL)2 1.1× 102 1.1× 102 7.6× 10−5 7.6× 10−5 ∆mBs
(bRsL)(bLsR) 3.7× 102 3.7× 102 1.3× 10−5 1.3× 10−5 ∆mBs
167
Appendix
11.A Neutral meson mixing
Neutral meson mixing is an FCNC process. Within the SM, it provides indirect measurements of
CKM parameters. Beyond the SM, it probes very high energy scales. In this Appendix, we present
the formalism that is used to investigate these processes, explain how the time evolution of the
neutral meson system depends on the tiny mass splitting between the two quasi-degenerate mass
eigenstates, and present the SM expression for this mass splitting.
11.A.1 Flavor oscillations
There are four neutral meson-pairs where mixing can occur: K0 − K0, D0 − D0, B0 − B0, and
B0s −B0
s.2
Consider a neutral meson P (P = K,D,B or Bs). Initially (t = 0) it is a superposition of P 0
and P 0:
|ψP (0)〉 = a(0)|P 0〉+ b(0)|P 0〉 . (11.17)
It evolves in time, and acquires components that correspond to all possible decay final states
f1, f2, . . .:|ψP (t)〉 = a(t)|P 0〉+ b(t)|P 0〉+ c1(t)|f1〉+ c2(t)|f2〉+ · · · . (11.18)
Our interest lies in obtaining only a(t) and b(t). For this aim, one can use a simplified formalism,
where the full Hamiltonian is replaced with a 2× 2 effective Hamiltonian H that is not Hermitian.
The non-Hermiticity is related to the possibility of decays, which makes the P 0, P 0 system an
open one. The complex matrix H can be written in terms of Hermitian matrices M and Γ as
H = M − i
2Γ . (11.19)
The matrices M and Γ are associated with (P 0, P 0)↔ (P 0, P 0) transitions via off-shell (dispersive)
and on-shell (absorptive) intermediate states, respectively. Diagonal elements of M and Γ are
associated with the flavor-conserving transitions P 0 → P 0 and P 0 → P 0. The CPT symmetry
2You may be wondering why there are only four such systems. If you do not wonder and do not know the answer,
then you should wonder. You will answer this question in your homework.
168
implies that M11 = M22 and Γ11 = Γ22. The off-diagonal elements are associated with the flavor
changing transitions P 0 ↔ P 0.
Before we proceed, let us clarify a semantic issue. The effective Hamiltonian H and, similarly,
its Hermitian part M , is a combination of operators. What we need for our purposes is its matrix
element between specific meson states. With some abuse of language, we denote by Mij both
the operator and its matrix element. Model independently, the diagonal matrix elements fulfill
M11 = M22 = m and Γ11 = Γ22 = Γ. The off-diagonal elements are those of interest to us. When
we refer to a specific meson system, we will use MPP for the matrix element 〈P 0|M12|P 0〉.In all cases (P = K,D,B,Bs), H is not a diagonal matrix. Thus, the states that have well
defined masses and decay widths are not P 0 and P 0, but rather the eigenvectors of H. We denote
the light and heavy eigenstates by PL and PH with masses mH > mL. (Another possible choice,
which is standard for K mesons, is to define the mass eigenstates according to their lifetimes. We
denote the short-lived and long-lived eigenstates by KS and KL with decay widths ΓS > ΓL. The
KL meson is experimentally found to be the heavier state.) The eigenstates of H are given by
|PL,H〉 = p|P 0〉 ± q|P 0〉, (11.20)
where (q
p
)2
=M∗
12 − (i/2)Γ∗12
M12 − (i/2)Γ12
, (11.21)
and with the normalization |p|2 + |q|2 = 1. Since H is not Hermitian, the eigenstates need not be
orthogonal to each other.
The masses and decay-widths are given by the real and imaginary parts of the eigenvalues,
respectively. The average mass and the average width are given by
m ≡ mH +mL
2, Γ ≡ ΓH + ΓL
2. (11.22)
The mass difference ∆m and the width difference ∆Γ are defined as follows:
∆m ≡ mH −mL, ∆Γ ≡ ΓH − ΓL. (11.23)
Here ∆m is positive by definition, while the sign of ∆Γ is to be determined experimentally. (Al-
ternatively, one can use the states defined by their lifetimes to have ∆Γ ≡ ΓS − ΓL positive by
definition.) It is useful to define dimensionless ratios x and y:
x ≡ ∆m
Γ, y ≡ ∆Γ
2Γ. (11.24)
We also define
θ = arg(M12Γ∗12). (11.25)
Solving the eigenvalue equation gives
(∆m)2 − 1
4(∆Γ)2 = 4|M12|2 − |Γ12|2, ∆m∆Γ = 4Re(M12Γ∗12). (11.26)
169
We move on to study the time evolution of a neutral meson. For simplicity, we assume CP
conservation. In Section 11.B we study CP violation, and there we relax this assumption. Many
important points can, however, be understood in the simplified case where CP is conserved. If CP
is a good symmetry of H then Γ12/M12 is real, leading to
|q/p| = 1 . (11.27)
It follows that the mass eigenstates are also CP eigenstates, and are orthogonal to each other,
〈PH |PL〉 = |p|2−|q|2 = 0. The phase of q/p is convention dependent, and not a physical observable.
As concerns the mass and decay widths, Eq. (11.26) simplifies to
∆m = 2|M12|, |∆Γ| = 2|Γ12|. (11.28)
Let us denote the time-evolved state of an initial state |P 〉 by |P (t)〉. For mass eigenstates, the
time evolution is simple, |PL,H(t)〉 = e−iEL,H t|PL,H〉. But the time evolution of |P 0(t)〉 and |P 0(t)〉is more complicated:
|P 0(t)〉 = cos(
∆E t
2
)|P 0〉+ i sin
(∆E t
2
)|P 0〉 ,
|P 0(t)〉 = cos(
∆E t
2
)|P 0〉+ i sin
(∆E t
2
)|P 0〉 . (11.29)
Since flavor is not conserved, the probability P to measure a specific flavor, that is P 0 or P 0,
oscillates in time:
P(P 0 → P 0)[t] =∣∣∣〈P 0(t)|P 0〉
∣∣∣2 =1 + cos(∆Et)
2,
P(P 0 → P 0)[t] =∣∣∣〈P 0(t)|P 0〉
∣∣∣2 =1− cos(∆Et)
2. (11.30)
Thus, neutral meson mixing, M12 6= 0, leads to flavor oscillations.
In the meson rest frame, ∆E = ∆m and t = τ , the proper time. Thus, ∆m sets the frequency
of the flavor oscillations. This is a very interesting result:
• On the theoretical side, ∆m is related to FCNC transitions: the quark transitions that
correspond to K0 −K0, D0 −D0, B0 −B0, and B0s −B0
s mixing are, respectively, sd→ sd,
uc → uc, bd → bd, and bs → bs. Thus, ∆m for each of the four systems gives an indirect
measurement of CKM parameters and can probe new physics.
• On the experimental side, we learn that by measuring the oscillation frequency we can
determine the mass splitting between the two mass eigenstates. One way this can be done
is by measuring the flavor of the meson both at production and decay. It is not trivial to
measure the flavor at both ends, and we do not explain here how it is done, but you are
encouraged to think and learn about it.
170
11.A.2 Time scales
There are various time scales involved in meson mixing, and understanding the hierarchy (or lack
of hierarchy) between them leads to insights and simplifications.
The first important time scale is the oscillation period. As can be seen from Eq. (11.30), the
oscillation time scale is given by ∆m.3
To understand which other time scales are relevant, we need to introduce the notion of “flavor
tagging.” The flavor eigenstates P 0 and P 0 have a well defined flavor content. For example, B0
(B0) is a bd (bd) bound state. The term ‘flavor tagging’ is used, in the physicists jargon, to the
experimental determination of whether a neutral P meson is in a P 0 or P 0 state. Flavor tagging is
provided to us by Nature, when the meson decays into a flavor-specific final state, namely a state
that can come from either P 0 or P 0 state, but not from both.4 Semi-leptonic decays are very good
flavor tags. Take, for example, semileptonic b (anti)quark decays:
b→ cµ−ν, b→ cµ+ν. (11.31)
Thus, the charge of the lepton tells us the flavor: µ+ comes from a B0 (or B+) decay, while µ−
comes from a B0 (or B−) decay. Of course, before the meson decays it could be in a superposition
of B0 and a B0. The decay acts as a quantum measurement. In the case of semileptonic decay, it
acts as a measurement of flavor vs. anti-flavor.
Thus, a second relevant time scale is that of flavor tagging. Since the flavor is tagged when the
meson decays, the relevant time scale is determined by the decay width, Γ. We can then use the
dimensionless quantity x [defined in Eq. (11.24)] to understand the possible hierarchies between
these two time scales:
1. x 1 (“slow oscillations”): The meson decays before it has time to oscillate, and thus
flavor is conserved to good approximation. Putting cos(∆mt) ≈ 1 in Eq. (11.30), we obtain
P(P 0 → P 0) ≈ 1 and P(P 0 → P 0) → 0. A measurement of ∆m is challenging, but
experiments can provide a useful upper bound even before the required precision for an
actual measurement is achieved. This case is relevant for the D system.
2. x 1 (“fast oscillation”): The meson oscillates many times before decaying, and thus the
oscillating term practically averages out to zero. Putting cos(∆mt) ≈ 0 in Eq. (11.30),
we obtain P(P → P ) ≈ P(P → P ) ≈ 1/2. A measurement of ∆m is challenging, but
experiments can provide a useful lower bound even before the required precision for an
actual measurement is achieved. This case is relevant for the Bs system.
3The time scale is, of course, 1/∆m. Physicists know, however, how to match dimensions. We thus interchange
between time and energy freely, counting on the reader to understand what we mean.4Final states that are common to the decays of both P and P are also very useful in flavor physics and, in
particular, to the study of CP violation. They will be discussed in Section 11.B.
171
3. x ∼ 1: The oscillation and decay times are roughly the same. The meson has time to oscillate
and the oscillations do not average out. This is the case where it is experimentally easiest to
measure ∆m. This case is relevant to both the K and the B systems. We emphasize that
the physics processes that determine Γ and ∆m are unrelated, so there is no reason to expect
x ∼ 1. Yet, amazingly, Nature has been kind enough to choose flavor parameters such that
x ∼ 1 in two out of the four neutral meson systems.
Thus, flavor oscillations give us sensitivity to mass differences of the order of the width, which
are much smaller than the mass itself. In fact, we have been able to measure mass differences
that are 14 orders of magnitude smaller than the corresponding masses. It is due to the quantum
mechanical nature of the oscillation that such high precision can be achieved.
In some cases there is one more time scale: ∆Γ. In such cases, we have one more relevant
dimensionless parameter y ≡ ∆Γ/(2Γ). Note that y is bounded, |y| ≤ 1. (This is in contrast to x
which has no upper bound.) Thus, we can talk about several cases depending on the values of y
and x.
1. |y| 1 and y x. In this case the width difference is irrelevant. This is the case for the
B0 system.
2. y ∼ x. In this case the width different is as important as the oscillation. This is the case in
the D system where y 1 and for the K system with y ∼ 1.
3. |y| ∼ 1 and y x. In this case the oscillation averages out and the width difference can
be observed simply as a difference in the lifetimes of the two mass eigenstates. This case is
relevant to the Bs system, where y ∼ 0.1.
There are few other limits (like y x) that are not realized in the four meson systems. Yet, they
might be realized in some other systems yet to be discovered.
To conclude this subsection, we present in Table 11.A.1 the experimental data on meson mixing.
Note that in all cases (including the K meson system) we define x and y as in Eqs. (11.23) and
(11.24). Note that for the B0 system, there is only an upper bound on y.
11.A.3 The SM calculation of M12
We now explain how the theoretical calculation of the mixing parameters is done. Our focus is on
∆m. We present the SM calculation, but the tools that we develop can be used in a large class of
models.
For the sake of concreteness, we discuss in this section the neutral B meson system. The oper-
ator M12 is given, within the SM, by CSM(dLγµbL)(dLγµbL), where CSM is the Wilson coefficient.
The matrix element is given by
MBB =CSM
2mB
〈B0|(dLγµbL)(dLγµbL)|B0〉. (11.32)
172
Table 11.A.1: Neutral meson mixing parameters
P m [GeV] Γ [GeV] x y
K0 0.498 3.68× 10−15 0.945 −0.997
D0 1.86 1.60× 10−10 0.0048± 0.0017 +0.014± 0.002
B0 5.28 4.33× 10−13 0.775± 0.006 −0.0075± 0.0090
Bs 5.37 4.34× 10−13 26.82± 0.23 −0.061± 0.008
The mass splitting is given by
∆mB = 2|MBB|, (11.33)
so that, within the SM, we have
∆mB = −1
3mBBBf
2BCSM, (11.34)
where we parameterized the hadronic matrix element as 〈B0|(dLγµbL)(dLγµbL)|B0〉 = −1
3m2BBBf
2B
(lattice calculations give√BfB ≈ 0.22 GeV).
Our task is then to calculate CSM. Since the operator in Eq. (11.32) is an FCNC operator,
within the SM it cannot be generated at tree level. The one loop diagrams that generate it are
called “box diagrams”. They are displayed in Fig. 11.A.1. The calculation of the box diagrams
gives, to a good approximation,
MBB =G2F
12π2mBm
2W (BBf
2B)S0(xt)(VtbV
∗td)
2, (11.35)
where xt = m2t/m
2W . A few comments are in order:
1. The box diagrams have two W -boson propagators, which yield the G2F factor.
2. The box diagrams have two up-type quark (i and j) propagators, yielding six different com-
binations: ij = uu, cc, tt, uc, ut, ct. Each such diagram depends on a different combination
of CKM elements and quark masses, (VibV∗id)(VjbV
∗jd)F (m2
i /m2W ,m
2j/m
2W ).
3. The unitarity of the CKM matrix implies that any (mi,mj)-independent terms vanish.
4. The three CKM combinations V ∗idVib are comparable in size. (They are all cubic in the
Wolfenstein parameter λ.)
5. The six kinematic functions F (m2i /m
2W ,m
2j/m
2W ) are very different in size. In particular,
S0(xt) = F (xt, xt) (where xt = m2t/m
2W ) is the largest.
173
b d
d b
ui
uj
Figure 11.A.1: A box diagram that generate an operators that can lead to B ↔ B transition.
6. The conclusion of the last two statements is that the dominant contribution comes from the
box diagram with two top-quark propagators. In the physicists’ jargon we say that MBB is
dominated by the top-quark.
The function S0(xt) is quadratically sensitive to mt. Similar to the EWPM, this non-decoupling
effect is related to the fact that the larger the top mass, the stronger its Yukawa coupling. When
∆mB was first measured, the top quark has not yet been discovered, and one could use Eq.
(11.35) to predict (correctly!) the top mass. At present, when the top mass is known (yielding
S0(xt) ≈ 2.36), Eq. (11.35) serves to constrain the CKM combination |VtbV ∗td|.Let us comment on the calculation of MPP in the other meson systems:
1. As concerns ∆mK , due to the CKM structure, it is dominated by the charm quark in the loop.
Consequently, ∆mK is GIM suppressed by a factor of m2c/m
2W . The lightness of the charm
quark implies also considerably larger theoretical uncertainties in the calculation compared
to ∆mB.
2. As concerns ∆mD, due to the CKM structure, the contributions involving the bottom quark
are suppressed. The calculation of the box diagrams with intermediate down and strange
quarks is not a good approximation to ∆mD.
3. As concerns ∆mBs , the calculation goes along very similar lines to that of ∆mB. In the ratio
∆mB/∆mBs , much of the uncertainty in the calculation of the hadronic matrix elements
cancels out, providing an excellent measurement of |Vtd/Vts|.
Finally, let us mention the calculation of ΓPP . An estimate of it can be made by calculating
the on-shell part of the box diagram. Yet, since the intermediate quarks are light and on-shell,
QCD effects are important, and the theoretical uncertainties in the calculation of ΓPP are large.
11.B CP violation
To date, CP violation has been observed in about thirty different decay modes. It has not been
observed in baryon decays, nor in the leptonic sector, nor in flavor diagonal processes, such as
174
electric dipole moments. We thus present in this Appendix the formalism and the SM calculation
of CP asymmetries in meson decays.
The experimental observation of CP violation is challenging for several reasons:
1. In order that there will be a CP asymmetry in a decay process, the presence of so-called
“strong phases”, which are CP conserving phases arising from intermediate on-shell particles,
is needed. These phases might be small (or vanish) and suppress the CP asymmetry (or make
it vanish).
2. CPT implies that the total width of a particle and its anti-particle are the same. Thus, any
CP violation in one channel must be compensated by CP violation with an opposite sign in
other channels. Consequently, CP violation is suppressed in inclusive measurements.
3. Within the SM, CP violation arises only when all three generations are involved. With the
smallness of the CKM mixing angles, this means that either the CP asymmetries are small,
or they appear in modes with small branching ratios.
CP violation in meson decays is an interference effect. In neutral meson decays the phe-
nomenology of CP violation is particularly rich thanks to the fact that meson mixing, as described
in Appendix 11.A, can contribute to the CP violating interference effects. One distinguishes three
types of CP violation in meson decays, depending on which amplitudes interfere:
1. In decay: The interference is between two decay amplitudes.
2. In mixing: The interference is between the absorptive and dispersive mixing amplitudes.
3. In interference of decays with and without mixing: The interference is between the direct
decay amplitude and a first-mix-then-decay amplitude.
The formalism and the SM calculation of the neutral meson mixing amplitude was presented
in Appendix 11.A. Before we proceed to discuss in more detail each of these three types of CP
violation, we present our notations, and some physics ingredients, concerning the decay amplitudes.
We do so for the specific case of B-meson decays, but our discussion applies to all meson decays.
We denote the amplitude of B → f decay by Af , and the amplitude of the CP conjugate
process, B → f , by Af . There are two types of phases that may appear in these decay amplitudes.
First, complex parameters in any Lagrangian term that contributes to Af appear in a complex
conjugate form in Af . In other words, CP violating phases change sign between Af and Af . In the
SM, these phases appear only in the couplings of the W±-bosons, hence the CP violating phases are
called “weak phases.” Second, phases can appear in decay amplitudes even when the Lagrangian
is real. They arise from contributions of intermediate on-shell states. These CP conserving phases
appear with the same sign in Af and Af . In meson decays, such rescattering is usually driven by
strong interactions, hence the CP conserving phases are called “strong phases.”
175
It is useful to factorize each contribution ai to Af into three parts: the magnitude |ai|, the
weak phase φi, and the strong phase δi. If there are two such contributions, Af = a1 +a2, we write
Af = |a1|ei(δ1+φ1) + |a2|ei(δ2+φ2),
Af = |a1|ei(δ1−φ1) + |a2|ei(δ2−φ2). (11.36)
It is further useful to define
φf ≡ φ2 − φ1, δf ≡ δ2 − δ1, rf ≡ |a2/a1|. (11.37)
Similarly, for neutral meson decays, it is useful to write
M12 = |M12|eiφM , Γ12 = |Γ12|eiφΓ . (11.38)
Each of the phases appearing in Eqs. (11.36) and (11.38) is convention dependent, but combinations
such as δ1 − δ2, φ1 − φ2, and φM − φΓ are physical.
To discuss the modifications to the time evolution, which was presented in Eq. (11.29) for the
CP conserving case, it is convenient to define another complex parameter,
λf ≡ (q/p)(Af/Af ). (11.39)
The time evolution of a B0 and B0 mesons is given by
|B0(t)〉 = g+(t) |B0〉 − (q/p) g−(t)|B0〉,
|B0(t)〉 = g+(t) |B0〉 − (p/q) g−(t)|B0〉 , (11.40)
where
g±(t) ≡ 1
2
(e−imH t−
12
ΓH t ± e−imLt−12
ΓLt). (11.41)
We define τ ≡ Γt. The time-dependent decay rate Γ(B0 → f)[t] (Γ(B0 → f)[t]) gives the
probability for an initially pure B0 (B0) meson to decay at time t to a final state f :
Γ(B0 → f)[t] = |Af |2e−τ
(cosh yτ + cosxτ) + |λf |2(cosh yτ − cosxτ)
−2Re [λf (sinh yτ + i sinxτ)],
Γ(B → f)[t] = |Af |2e−τ
(cosh yτ + cosxτ) + |λf |−2(cosh yτ − cosxτ)
−2Re[λ−1f (sinh yτ + i sinxτ)
] . (11.42)
11.B.1 CP violation in decay
CP violation in decay corresponds to
|Af/Af | 6= 1. (11.43)
176
In charged meson decays, this is the only possible contribution to the CP asymmetry:
Af± ≡Γ(B− → f−)− Γ(B+ → f+)
Γ(B− → f−) + Γ(B+ → f+)=|Af−/Af+|2 − 1
|Af−/Af+|2 + 1. (11.44)
Using Eq. (11.36), we obtain for r 1
Af± = 2rf sinφf sin δf . (11.45)
This result shows explicitly that we need two decay amplitudes, that is, rf 6= 0, with different
weak phases, φf 6= 0, π and different strong phases δf 6= 0, π.
A few comments are in order:
1. In order to have a large CP asymmetry, we need each of the three factors in (11.45) to be
large.
2. A similar expression holds for the contribution of CP violation in decay in neutral meson
decays. In this case there are, however, additional contributions.
3. Another complication with regard to neutral meson decays is that it is not always possible
to tell the flavor of the decaying meson, that is, if it is B0 or B0. This can be a problem or
a virtue.
4. In general the strong phase is not calculable since it is related to QCD. This is not a problem
if the aim is just to demonstrate CP violation, but it is if we want to extract the weak
parameter φf . In some cases, however, the phase can be independently measured, eliminating
this particular source of theoretical uncertainty.
D → K+K−
We give here an example of the SM contribution to CP violation in decay in the D → K+K− mode.
This decay proceeds via the quark transition c → ssu. Within the SM, there are contributions
from both tree (t) and penguin (pq, where q = d, s, b is the quark in the loop) diagrams. Factoring
out the CKM dependence, we have
AK+K− = (V ∗csVus)tKK +∑
q=d,s,b
(V ∗cqVuq)pqKK . (11.46)
Using CKM unitarity, AK+K− can be written in terms of just two CKM combinations:
AK+K− = (V ∗csVus)TKK + (V ∗cbVub)PbKK , (11.47)
where TKK = tKK + psKK − pdKK and P bKK = pbKK − pdKK . CP violating phases appear only in the
CKM elements, so thatAK+K−
AK+K−=
(V ∗csVus)TKK + (V ∗cbVub)PbKK
(VcsV ∗us)TKK + (VcbV ∗ub)PbKK
. (11.48)
177
Due to CKM suppression and loop suppression, we expect the P bKK-related contribution to be
much smaller than the TKK-related contribution, and thus the contribution from CP violation in
decay to the CP asymmetry is given by
AdK+K− ≈ −2Im(P bKK
TKK
)|V ∗cbVub||V ∗csVus|
sin γ, (11.49)
where γ is defined in Eq. (8.79). The super-index d on AdK+K− denotes that we include here only
the contribution from CP violation in decay.
The CKM parameters are known, and generate a suppression factor of O(10−3). The factor of
Im(P bKK/TKK) depends on the relative size of the penguin and tree contributions, as well as the
relative strong phase. Both ingredients arise from QCD dynamics at the scale of mD. At present,
there is no rigorous way to calculate this factor. Thus, one cannot use a measurement of AK+K−
to extract, for example, the value of the CP violating phase γ.
11.B.2 CP violation in mixing
CP violation in mixing corresponds to
|q/p| 6= 1 . (11.50)
In decays into flavor specific final states (Af = 0 and, consequently, λf = 0), and, in particular,
semileptonic neutral meson decays, this is the only source of CP violation:5
ASL(t) ≡ Γ[B0(t)→ `+X]− Γ[B0(t)→ `−X]
Γ[B0(t)→ `+X] + Γ[B0(t)→ `−X]=
1− |q/p|4
1 + |q/p|4. (11.51)
Using Eq. (11.21), we obtain for |Γ12/M12| 1,
ASL = − |Γ12/M12| sin(φM − φΓ). (11.52)
A few comments are in order:
1. Eq. (11.51) implies that this asymmetry of time-dependent decay rates is actually time
independent.
2. The calculation of |Γ12/M12| is difficult, since it depends on low-energy QCD effects. Hence,
it would be difficult in general to extract the value of the CP violating phase φM − φΓ from
a measurement of ASL.
5This statement holds within the SM where, to lowest order in GF , |A`+X | = |A`−X | and A`−X = A`+X = 0.
178
K → `νπ
We give here an example of the SM contribution to CP violation in K0−K0 mixing. It is measured
via the semileptonic asymmetry which is defined as follows:
δL ≡Γ(KL → `+ν`π
−)− Γ(KL → `−ν`π+)
Γ(KL → `+ν`π−) + Γ(KL → `−ν`π+)=
1− |q/p|2
1 + |q/p|2. (11.53)
This asymmetry is somewhat different from the one defined in Eq. (11.51), in that the decaying
meson is the neutral mass eigenstate, rather than the flavor eigenstate. Hence also the different
dependence on |q/p|. The experimental value is δL = (3.32± 0.06)× 10−3.
Here one can overcome the difficulty of calculating |Γ12| by taking into account the experimental
result that ∆ΓK/∆mK ≈ −2, and that, given that the CP violating effects are experimentally
determined to be small, ∆ΓK/∆mK ' |ΓKK/MKK |. Then one obtains
Re(εK) =1
4(1− |q/p|2) ' Im(MKK)
2∆mK
, (11.54)
where we use connect the commonly used CP violating parameter εK to our notations. Thus, to
find Re(εK) we need to obtain the SM contribution to MKK . Similarly to the neutral B system,
this contribution comes from box diagrams with intermediate up-type quarks, leading to
MKK =G2Fm
2W
12π2mK(BKf
2K)[S0(xc)(VcsV
∗cd)
2 + S0(xt)(VtsV∗td)
2 + S0(xc, xt)(VcsV∗cdVtsV
∗td)].
(11.55)
where xc = m2c/m
2W . In contrast to the case of MBB (11.35), in the neutral K system, MKK is
dominated by the charm quark. The reason is that, of the three relevant CKM combinations,
the top-related one is highly suppressed: |V ∗tdVts| ∼ λ5 compared to |V ∗cdVcs| ' λ. Thus, ∆mK is
dominated by the charm quark. We used this fact when writing down Eq. (11.2). The pure charm
contribution to Im(M12) is, however, highly suppressed, and the top quark is dominant in Re(εK).
11.B.3 CP violation in interference of decays with and without mixing
CP violation in interference of decays with and without mixing corresponds to
Im(λf )
1 + |λf |26= 0. (11.56)
It can be extracted from the CP asymmetry in decays into final CP eigenstates:
AfCP (t) ≡ Γ[B0(t)→ fCP ]− Γ[B0(t)→ fCP ]
Γ[B0(t)→ fCP ] + Γ[B0(t)→ fCP ]= Im(λfCP ) sin(∆m t). (11.57)
The last equality holds when the effects of CP violation in decay are negligible, |AfCP /AfCP | ' 1,
and the effects of CP violation in mixing are small, |q/p| ' 1. In this case, λfCP is a pure phase.
179
Using Eq. (11.39), we obtain for |Γ12/M12| 1,
Im(λfCP ) = Im(M∗
12
|M12|AfCPAfCP
)= − sin(φM + 2φ1). (11.58)
The phase φM is defined in Eq. (11.38), while the phase φ1 is defined in Eq. (11.36), and we
assume that a2 can be neglected.
B → ψKS
We give here an example of the SM contribution to CP violation in the interference of decays with
and without mixing in the B → ψKS mode. This is often called “the golden mode” with regard
to CP violation as its theoretical calculation is uniquely clean of hadronic uncertainties. In fact,
the CP asymmetry can be translated into a value of sin 2β [β is defined in Eq. (8.79)] with a
theoretical uncertainty smaller than one percent.
For the neutral B meson system, |ΓBB/MBB| 1 holds. From Eq. (11.35) we obtain
M∗BB
|MBB|=V ∗tbVtdVtbV ∗td
. (11.59)
The B → ψK decay proceeds via a b→ ccs transition:
AψK = (V ∗cbVcs)TψK + (V ∗ubVus)PuψK . (11.60)
The second term is CKM and loop suppressed, and can be safely neglected. Since B0 decays into
ψK0 while B0 decays into ψK0, an additional phase from K0−K0 mixing, (V ∗cdVcs)/(VcdV∗cs), enters
the calculation of AψKS/AψKS :AψKSAψKS
= −VcbV∗cd
V ∗cbVcd. (11.61)
Combining Eq. (11.59) and Eq. (11.61), we obtain
λψKS = −e−2iβ =⇒ Im(λψKS) = sin 2β. (11.62)
This demonstrate the power of CP asymmetries in measuring CKM parameters. The experimental
measurement of Im(λψKS) translates directly into the value of a CKM parameter, β, without any
hadronic parameters. A crucial role is played by the CP symmetry of the strong interactions.
The size and the phase of the amplitude TψK cannot be calculated, but it is the same in the CP
conjugate amplitudes AψKS and AψKS and therefore cancels out when their ratio is taken.
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Homework
Question 11.1: The four mesonsIt is now time to come back to the question of why there are only four meson pairs that are relevant
to flavor oscillations. Explain why the following systems are irrelevant to flavor oscillations:
1. B+ −B−.
2. s− d. Conisder a case where, say, a charm decay into a superpostion of d and s quark.
3. T − T (a T is a meson made out of a t and a u quarks.)
4. K∗ −K∗ oscillation.
Hint: The last three cases all have to do with time scales. In principle there are oscillations in
these systems, but they are irrelevant.
Question 11.2: Kaons
Here we study some properties of the kaon system. We did not talk about it at all. You have
to go back and recall (or learn) how kaons decay, and combine that with what we discussed in the
lecture.
1. Because τL τS we have yK ≈ 1. What is the reason for that?
2. In a hypothetical world where we could change the mass of the kaon without changing any
other masses, how would the value of yK change if we made mK smaller or larger.
Question 11.3: Mixing beyond the SM
Consider a model without a top quark, in which the first two generations are as in the SM,
while the left–handed bottom (bL) and the right–handed bottom (bR) are SU(2) singlets.
1. Draw a tree-level diagram that contributes to B − B mixing in this model.
181
2. Is there a tree-level diagram that contributes to K − K mixing?
3. Is there a tree-level diagram that contributes to D − D mixing?
Question 11.4: Condition for CP violation
Using Eq. (11.36), show that in order to observe CP violation, Γ(B → f) 6= Γ(B → f), we
need two amplitudes with different weak and strong phases.
Question 11.5: Mixing formalism
In this question, you are asked to develop the general formalism of meson mixing.
1. Show that the mass and width differences are given by
4(∆m)2 − (∆Γ)2 = 4(4|M12|2 − |Γ12|2), ∆m∆Γ = 4Re(M12Γ∗12), (11.63)
and that ∣∣∣∣∣qp∣∣∣∣∣ =
∣∣∣∣∣∆m− i∆Γ/2
2M12 − iΓ12
∣∣∣∣∣ . (11.64)
2. When CP is a good symmetry all mass eigenstates must also be CP eigenstates. Show that
CP invariance requires ∣∣∣∣∣qp∣∣∣∣∣ = 1. (11.65)
3. In the limit Γ12 M12 show that
∆m = 2|M12|, ∆Γ = 2|Γ12| cos θ,
∣∣∣∣∣qp∣∣∣∣∣ = 1. (11.66)
4. Derive Eqs. (11.42).
5. Derive Eq. (11.57).
6. Show that when ∆Γ = 0 and |q/p| = 1
Γ(B → X`−ν)[t] = e−Γt sin2(∆mt/2), Γ(B → X`+ν)[t] = e−Γt cos2(∆mt/2). (11.67)
Question 11.6: B → π+π− and CP violation
One of the interesting decays to consider is B → ππ. Here we only briefly discuss it.
182
1. First assume that there is only tree level decay amplitude (that is, neglect penguin ampli-
tudes). Draw the Feynman diagram of the amplitude, paying special attention to its CKM
dependence.
2. In that case, which angle of the unitarity triangle is the time dependent CP asymmetry,
Eq. (11.57), sensitive to?
3. Can you estimate the error introduced by neglecting the penguin amplitude? (Note that one
can use isospin to reduce this error. Again, you are encouraged to read about it in one of
the reviews.)
Question 11.7: B decays and CP violation
Consider the decays B0 → ψKS and B0 → φKS. Unless explicitly noted, we always work
within the framework of the standard model.
1. B0 → ψKS is a tree-level process. Write down the underlying quark decay. Draw the
tree level diagram. What is the CKM dependence of this diagram? In the Wolfenstein
parametrization, what is the weak phase of this diagram?
2. Write down the underlying quark decay for B0 → φKS. Explain why there is no tree level
diagram for B0 → φKS.
3. The leading one loop diagram for B0 → φKS is a gluonic penguin diagram. As we have
discussed, there are several diagrams and only their sum is finite. Draw a representative
diagram with an internal top quark. What is the CKM dependence of the diagram? In the
Wolfenstein parametrization, what is the weak phase of the diagram?
4. Next we consider the time dependent CP asymmetries. We define as usual
λf ≡AfAf
q
p, Af ≡ A(B0 → f), Af ≡ A(B0 → f). (11.68)
In our case we neglect subleading diagrams and then we have |λ| = 1 and thus
af ≡Γ(B0(t)→ f)− Γ(B0(t)→ f)
Γ(B0(t)→ f) + Γ(B0(t)→ f)= −Imλf sin(∆mB t) (11.69)
Both aψKS and aφKS measure the same angle of the unitarity triangle. That is, in both cases,
Imλf = sin 2x where x is one of the angles of the unitarity triangle. What is x? Explain.
5. Experimentally,
ImλψKS = 0.68(2), ImλφKS = 0.59(14). (11.70)
Comment about these two results. In particular, do you think these two results are in
disagreement?
183
6. Assume that in the future we will find
ImλψKS = 0.68(1), ImλφKS = 0.55(3). (11.71)
That is, that the two results are not the same. Below are three possible “solutions”. For
each solution explain if you think it could work or not. If you think it can work, show how.
If you think it cannot, explain why.
(a) There are standard model corrections that we neglected.
(b) There is a new contribution to B0− B0 mixing with a weak phase that is different from
the SM one.
(c) There is a new contribution to the gluonic penguin with a weak phase that is different
from the SM one.
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Chapter 12
Neutrinos
12.1 Introduction
In the SM, the neutrinos are exactly massless. Experiments, however, established that neutrinos
have masses. While the individual neutrino mass eigenvalues are not known, two mass-squared
differences are inferred from experiments:
∆m221 ≡ m2
2 −m21 = (7.5± 0.2)× 10−5 eV2,
∆m232 ≡ m2
3 −m22 = ±(2.3± 0.1)× 10−3 eV2. (12.1)
We discuss the way these ranges were obtained in section 12.4. This is a clear experimental
indication of physics beyond the SM.
The SM prediction that the neutrinos are massless is related to the lepton number symmetry.
The SM prediction that the neutrinos do not mix is related to the lepton flavor symmetry. Similar
to other predictions that depend on accidental symmetries of the SM, these predictions are violated
in generic extensions of the SM. In Chapter 9 we saw that d = 6 terms violate the approximate
custodial symmetry of the scalar sector of the SM, and consequently are constrained by EWP
measurements. In Chapter 11 we saw that d = 6 terms violate the approximate flavor symmetries
of the SM, and consequently are constrained by measurements of FCNCs processes. In this chapter
we will see that d = 5 terms violate the accidental lepton number and lepton flavor symmetries of
the SM, and consequently are probed by measurements of neutrino masses and mixing.
12.2 The νSM: The SM with d = 5 terms
In this section we study a model that we call the νSM. It is the SM extended to include the most
general d = 5 terms.
There is a single class of dimension five terms that depend on SM fields and obey the SM
symmetries. These terms involve two SU(2)-doublet lepton fields and two SU(2)-doublet scalar
185
fields:
LνSM = LSM +Zνij
ΛφφLiLj, (12.2)
where Zν is a symmetric and complex 3 × 3 matrix of dimensionless couplings, and Λ is a high
mass scale, Λ v.
12.2.1 The neutrino spectrum
With φ0 acquiring a VEV, 〈φ0〉 = v/√
2, LνSM in Eq. (12.2) has a piece that corresponds to a
Majorana mass matrix for the neutrinos:
LνSM,mass =1
2(mν)ijνiνj, (mν)ij =
v2
ΛZνij. (12.3)
The matrix mν can be diagonalized by a unitary transformation:
VνLmνVTνL = mν = diag(m1,m2,m3). (12.4)
Majorana mass matrices are always symmetric. While the diagonalization of a general mass matrix
M involves a general bi-unitary transformation, Mdiag = VLMV †R, for a symmetric mass matrix
the diagonalization is by a unitary matrix and its transpose, as in Eq. (12.4).
We denote the corresponding neutrino mass eigenstates by ν1, ν2, ν3. If we choose a basis
where the mixing angles are all not larger than π/4, there are six possible mass ordering. Yet,
based on the experimental results (as we explain below) there are only two possibilities that are
consistent with the data. The convention here is that the states ν1 and ν2 are the ones separated by
the smaller mass-squared difference, with m2 > m1. The state ν3 is the one whose mass-squared
difference from the other two is the largest. It is not yet known experimentally whether it is
heavier (‘normal hierarchy’) or lighter (‘inverted hierarchy’) than the other two. This convention
is in one-to-one correspondence with the way that the experimental results are presented in Eq.
(12.1): |∆m232| > ∆m2
21 > 0.
12.2.2 The scale of generation of neutrino masses
In this section we explain the implications of the measured neutrino masses for the scale Λ where
these masses are generated. As long as experiments probe only the low energy effective theory,
what is measured is the combination Zν/Λ. Thus, there is an ambiguity in the definition of Λ and
Zν . The separation of the coefficient of a d = 5 term to a dimensionless coupling and a scale is
meaningful when we discuss a full high energy theory which generates the effective term. What we
refer to as the scale of a non-renormalizable term is Λ/Zν (or, in case that Zν is a matrix, as in Eq.
(12.2), Λ/Zνmax, where Zν
max is the largest eigenvalue of Zν). Note, however, that the combination
of a measurement of Λ/Zν and the assumption that Zν is generated by perturbative physics and
therefore Zν∼< 1 translates into an upper bound on Λ.
186
The measurements of the neutrino mass-squared differences, Eq. (12.1), do not tell us the
individual masses of the neutrinos, though they provide a lower bound on two mass eigenvalues:
There is at least one neutrino mass heavier than√|∆m2
32|,
mheaviest ≥√|∆m2
32| ' 0.05 eV, (12.5)
and there is at least one additional mass heavier than√
∆m221 ∼ 0.009 eV. There is, however,
additional information from experiments and cosmology which provides an upper bound on the
absolute mass scale of the neutrinos of order 1 eV. We discuss the experimental upper bounds in
Appendix 12.A.
The effective low energy Lagrangian of Eq. (12.2) where, by definition, Λ v, predicts that
the neutrino masses are much lighter than the weak scale:
m1,2,3 ∼ v2/Λ v. (12.6)
The fact that experiments find that the neutrinos are indeed much lighter than the W mass, makes
the notion that neutrino masses are generated by d = 5 terms very plausible.
In fact, all fermions of the SM except for the top quark are light relative to mW . The lightness of
charged fermions is related to the smallness of the corresponding Yukawa couplings. The question
of why Yukawa couplings are small may find an answer in a more fundamental theory, beyond the
SM. The neutrinos, however, are not only much lighter than mW , but also lighter by at least six
orders of magnitude than all charged fermions. This extreme lightness of the neutrinos is explained
if their masses are generated by d = 5 terms.
Clearly, the SM cannot be a valid theory above the Planck scale, Λ ∼< MPl. We thus expect
that mi ∼> v2/MPl ∼ 10−5 eV. A more relevant scale might be the scale of Grand Unified Theories
(GUTs). In GUTs, the GSM = SU(3)C ×SU(2)L×U(1)Y gauge group of the SM is assumed to be
a subgroup of a unifying group, such as SU(5), which is spontaneously broken to GSM at a scale
ΛGUT = O(1016 GeV). If the d = 5 terms are generated at ΛGUT, then we expect mν ∼ 10−2 eV.
Conversely, an experimental lower bound on neutrino masses provides an upper bound on the
scale of relevant new physics. Using the lower bound of Eq. (12.5) and the relation of Eq. (12.3),
we conclude that the SM cannot be a valid theory above the scale
Λ ∼<v2
mν
∼ 1015 GeV. (12.7)
This proves that the SM cannot be valid up to the Planck scale. Furthermore, this upper bound
is intriguingly close to the GUT scale.
12.2.3 The neutrino interactions
The addition of the dimension-five terms leads to significant changes in the phenomenology of the
lepton sector. The modifications can be understood by re-writing the neutrino-related terms in
187
Table 12.1: The neutrino interactions
interaction force carrier coupling
NC weak Z0 e/(2sW cW )
CC weak W± gU/√
2
Yukawa h 2m/v
the mass basis. The renormalizable SM gives
LSM,ν = iνα∂/να −g
2cWναZ/να −
g√2
(`LαW/
−να + h.c.), (12.8)
where α = e, µ, τ . (The Lagrangian (12.8) describes massless neutrinos, and consequently the basis
(νe, νµ, ντ ) serves as both an interaction basis and a mass basis.) The Lagrangian of Eq. (12.2)
gives
LνSM,ν = iνi∂/νi −g
2cWνiZ/νi −
g√2
(`LαW/
−Uαiνi + h.c.)
+miνiνi +2mi
vhνiνi +
mi
v2hhνiνi. (12.9)
Here α = e, µ, τ denotes only the charged lepton mass eigenstates, while i = 1, 2, 3 denotes the
neutrino mass eigenstates. The neutrino mass parameters m1,2,3 are real, and the mixing matrix
U is unitary. Starting from an arbitrary interaction basis, the matrix U is given by
U = VeLV†νL. (12.10)
While each of VeL and VνL is basis-dependent, the combination VeLV†νL is not. Explicitly we write
it as
U =
Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3
. (12.11)
The most significant changes from (12.8) to (12.9) concerning neutrino interactions are the
following:
• The leptonic charged current interactions are neither universal nor diagonal. Instead, they
involve the mixing matrix U .
• The Higgs boson has Yukawa couplings to neutrinos. These couplings break lepton num-
ber. The size of the Yukawa couplings is, however, tiny, of order mi/v ∼ 10−13, leading to
unobservably small branching ratio for h→ νν.
The νSM-neutrinos thus have three types of interactions, mediated by massive bosons. These
interactions are summarized in Table 12.1.
188
12.2.4 Accidental symmetries and the lepton mixing parameters
The dimension-five terms in Eq. (12.2) break the U(1)e × U(1)µ × U(1)τ accidental symmetry of
the SM. With the addition of only d = 5 terms, all that remains of the GglobalSM symmetry of the
SM [see Eq. (8.57)] is baryon number symmetry:
GglobalνSM = U(1)B. (12.12)
This symmetry is, however, anomalous and broken by non-perturbative effects. In addition, it is
broken by dimension-six terms.
The counting of flavor parameters in the quark sector remains unchanged: six quark masses
and four mixing parameters, of which one is imaginary. How many physical flavor parameters
are involved in the lepton sector? The Lagrangian of Eq. (12.2) involves the 3 × 3 matrix Y e (9
real and 9 imaginary parameters), and the symmetric 3 × 3 matrix Zν (6 real and 6 imaginary
parameters). The kinetic and gauge terns have a U(3)L×U(3)E accidental global symmetry, that is
completely broken by the Y e and Zν terms. Thus, the number of physical lepton flavor parameters
is (15R + 15I)− 2× (3R + 6I) = 9R + 3I . Six of the real parameters are the three charged lepton
masses me,µ,τ and the three neutrino masses m1,2,3. We conclude that the 3× 3 unitary matrix U
depends on three real mixing angles and three phases.
Why does the lepton mixing matrix U depend on three phases, while the quark mixing matrix V
depends on only a single phase? The reason for this difference lies in the fact that the Lagrangian of
Eq. (12.2) leads to Majorana masses for neutrinos. Consequently, there is no freedom in changing
the mass basis by redefining the neutrino phases, as such redefinition will introduce phases into
the neutrino mass terms. While redefinitions of the six quark fields allowed us to remove five non-
physical phases from V , redefinitions of the three charged lepton fields allows us to remove only
three non-physical phases from U . The two additional physical phases in U are called “Majorana
phases,” since they appear as a result of the (assumed) Majorana nature of neutrinos. They affect
lepton number violating processes.
A convenient parametrization of U is the following:
U =
c12c13 s12c13 s13e
−iδ
−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e
iδ s23c13
s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e
iδ c23c13
× diag(1, eiα1 , eiα2), (12.13)
where α1,2 are the Majorana phases, sij ≡ sin θij and cij ≡ cos θij. We describe the experimental
determination of the lepton mixing parameters in Section 12.4.
The present status of our knowledge of the absolute values of the various entries in the lepton
mixing matrix can be summarized as follows (we quote here the 3σ ranges):
|U | =
0.80− 0.85 0.51− 0.58 0.14− 0.16
0.22− 0.52 0.44− 0.70 0.61− 0.79
0.25− 0.53 0.46− 0.71 0.59− 0.78
. (12.14)
189
When working in the mass basis, the formalisms of quark and lepton flavor mixing are very
similar. The difference between these two phenomena arises due to the way neutrino experiments
are done. While quarks and charged leptons are identified as mass eigenstates, neutrinos are
identified as interaction eigenstates. Explicitly, they are identified as νe or νµ or ντ according to
whether they produce in the detector an e or µ or τ lepton, respectively.
There are several ways to think about fermion mixing. The basic point is that there are three
relevant matrices in each of the quark and lepton sectors – two mass matrices and one matrix
for the W -couplings – and at most two can be simultaneously diagonal. For quarks, mixing is
best understood in the mass basis. The W couplings to quark mass eigenstates are not diagonal,
namely mu and md are diagonal but V is not. For leptons, it is often more convenient to work in
an interaction basis which is also the charged lepton mass basis. Here me and the W couplings
are diagonal but mν is not.
Note also that the indices of the two matrices are reversed. In V the first index corresponds to
the T3 = +1/2 component of the doublet and the second one to the T3 = −1/2 component. In U
it is the other way around.
12.3 The NSM: The SM with singlet fermions
In Section 12.2 we introduce the νSM and show that the addition of non-renormalizable, dimension-
five terms to the SM Lagrangian gives neutrinos masses and, moreover, explains why they are much
lighter than the charged fermions. Non-renormalizable terms must arise from a more fundamental
theory. One uses the term “UV-completion” for a full high energy theory that leads to the effective
theory. For the full high energy theory we again write only renormalizable terms.
In this section we provide an example of a full high energy theory that generates at low energy
the dimension-five terms. (In your homework you will work out one more example.) This extension
amounts to adding heavy gauge-singlet fermions to the SM. The way these singlets generate masses
to the neutrinos is called the seesaw mechanism. The reason for this name, as will become clear
when we analyze the model, is that the heavier the singlet fermions, the lighter the neutrinos.
The Lagrangian Ld=5 of Eq. (12.30) can come not only from the NSM, but also from other
high-energy theories. In particular, there are three types of seesaw models, called type I, II, and
III, which differ by the type of heavy fields that one adds to the SM:
• Type I: (1, 1)0 fermion fields. (This is the NSM we discuss below.)
• Type II: (1, 3)−1 scalar fields. (An example is provided by the LRS model in your homework.)
• Type III: (1, 3)−1 fermion fields. (You will work out this model in your homework.)
All three types of seesaw models predict that the light neutrinos have Majorana masses, and that
their mass scale is inversely proportional to the mass scale of the new heavy particles (hence the
190
name “seesaw models”) and, in particular, much lower than the electroweak breaking scale. These
kind of models arises in various extensions of the SM, such as SO(10) grand unified theories (GUT),
and left-right symmetric (LRS) models.
12.3.1 Defining the NSM
The NSM is defined as follows:
(i) The symmetry is a local
SU(3)C × SU(2)L × U(1)Y . (12.15)
(ii) There are three fermion generations (i = 1, 2, 3), each consisting of six different representa-
tions:
QLi(3, 2)+1/6, URi(3, 1)+2/3, DRi(3, 1)−2/3, LLi(1, 2)−1/2, ERi(1, 1)−1, NRi(1, 1)0.
(12.16)
There is a single scalar multiplet:
φ(1, 2)+1/2. (12.17)
(iii) The pattern of spontaneous symmetry breaking is as follows:
SU(3)C × SU(2)L × U(1)Y → SU(3)C × U(1)EM (QEM = T3 + Y ). (12.18)
12.3.2 The NSM Lagrangian
The NSM has the same gauge group, the same scalar content, and the same pattern of spontaneous
symmetry breaking as the SM. In the fermion sector, all the SM representations are included. The
only difference is the addition of the fermionic NRi fields. Since the imposed symmetry is the same,
all the terms that appear in the SM Lagrangian appear also in the NSM Lagrangian. The NSM
Lagrangian has, however, several additional terms. These are all the terms that involve the NRi
fields. We can write:
LNSM = LSM + LN . (12.19)
Our task now is to find the specific form of LN . We note the following points in this regard:
1. Given that the NR fields are singlets of the gauge group, we have DµNR = ∂µNR.
2. Since the NR fields carry no conserved charge, they can have Majorana mass terms.
3. The combination LLNR transforms as (1, 2)+1/2 under the gauge group, and can thus have a
Yukawa coupling to the scalar doublet.
191
We thus obtain the most general form for the renormalizable terms in LN :
LN = iNRi∂/NRi −(
1
2MN
ij NRiNRj + Y νijLLiφNRj + h.c.
). (12.20)
Here MN is a symmetric 3 × 3 complex matrix, with entries of mass dimension 1, and Y ν is a
general 3× 3 complex matrix of dimensionless Yukawa couplings.
12.3.3 The NSM spectrum
As concerns the spectrum of this theory, clearly the bosonic spectrum remains unchanged from
the SM. As concerns the fermions, we note that, since the NR fields are singlets of the full gauge
group, they are also singlets of the unbroken subgroup, namely they transform as (1)0 under
SU(3)C×U(1)EM. This means that also the spectrum of the charged fermions (quarks and charged
leptons) remains unchanged from the SM.
As concerns the neutrinos (the νL components of the SU(2)-doublet leptons and the NR fields),
taking into account the spontaneous symmetry breaking, we find the following mass terms in LN :
LN ,mass = −1
2MN
ij NRiNRj −Y νijv√2νLiNRj + h.c.. (12.21)
This gives a 6× 6 neutrino mass matrix, that can be decomposed into four 3× 3 blocks as follows
[see Eq. (??)]: :
Mν =
(0 mD
mTD MN
), mD =
vY ν
√2. (12.22)
To obtain the six neutrino mass eigenstates, we need to diagonalize Mν .
Note that, unlike the case for the charged leptons and the quarks, where there are three mass
eigenstates, there are six neutrino mass eigenstates. The reason is that charged fermions have Dirac
masses, and each mass eigenstate has four DoF. The neutrinos have Majorana masses, where each
mass eigenstate has only two DoF. The total number of DoF is thus same in each sector.
Before we analyze the neutrino spectrum of the three generation case, we gain some intuition
by analyzing the simpler, one generation model, where we have only one copy of the L and N
fields. Without loss of generality, we choose a basis where Mν of Eq. (12.22) is a symmetric real
2 × 2 matrix. We further make an important assumption, inspired by both phenomenology and
theoretical model building: We assume that the MN is much larger than the electroweak breaking
scale, MN v. To leading order in mD/MN , we obtain the following mass eigenvalues and mixing
angle:
m1 =m2D
MN, m2 = MN , sin θ =
mD
MN. (12.23)
These results demonstrate several important points about the see-saw mechanism:
1. The mixing angle between the two states is very small. The light state is almost a pure
SU(2)-doublet ν, while the heavy one is almost a pure SU(2)-singlet N .
192
2. The light mass is inversely proportional to heavy one. This is the reason why the mechanism
that generates masses for the light neutrinos via their Yukawa couplings to heavy neutrinos
is called “the see-saw mechanism.”
3. The NSM constitutes a possible UV-completion of the νSM. Within the NSM, the unspecified
scale of the νSM, Λ, is interpreted as the mass scale of the heavy neutrino, MN .
We now return to the three generation case. We can always use a unitary transformation to
bring MN of Eq. (12.22) to a diagonal and real form:
MN → UTNM
NUN = MN = diag(M1,M2,M3). (12.24)
Unlike the SM, which has a single dimensionful parameter, v, the NSM has four dimensionful
parameters: v,M1,M2,M3. We assume that the Majorana masses of the heavy states are much
higher than the weak scale:
M1,2,3 v. (12.25)
We define mN to be the mass scale of the heavy neutrinos. Then, we can perform the diagonal-
ization of Mν to leading order in v/mN . First, we use the unitary matrix K,
K =
(1 mDM
−1N
−mDM−1N 1
), (12.26)
where we omitted terms of order v2/m2N , to block-diagonalize Mν :
KMνKT =
(−mT
DM−1N mD 0
0 MN
), (12.27)
The lower-right block is already diagonalized. The upper-left block,
mν = mTDM
−1N mD, (12.28)
can be diagonalized by a further unitary transformation:
V TνLmνVνL = mν = diag(m1,m2,m3). (12.29)
We thus learn the following points:
1. There are three heavy Majorana neutrinos of masses M1,M2,M3. We call these states
N1, N2, N3. These mass eigenstates are approximately SU(2)-singlet states, but have a small,
O(v/mN), SU(2)-doublet component. The masses are, by assumption, much larger than the
electroweak scale.
2. There are three light neutrinos of masses m1,m2,m3 of order v2/mN . We call these states
ν1, ν2, ν3. These mass eigenstates are approximately SU(2)-doublet states, but have a small,
O(v/mN), SU(2)-singlet component. The masses are much smaller than the electroweak
scale.
193
Table 12.1: The NSM particles
particle spin color Q mass
W± 1 (1) ±1 12gv
Z0 1 (1) 0 12
√g2 + g′2v
A0 1 (1) 0 0
g 1 (8) 0 0
h 0 (1) 0√
2λv
e, µ, τ 1/2 (1) −1 ye,µ,τv/√
2
ν1, ν2, ν2 1/2 (1) 0 m1,2,3
N1, N2, N2 1/2 (1) 0 M1,2,3
u, c, t 1/2 (3) +2/3 yu,c,tv/√
2
d, s, b 1/2 (3) −1/3 yd,s,bv/√
2
The details of the spectrum of the NSM are summarized in Table 12.1. There are three different
mass scales:
• The masses of all bosons and of the charged fermions are of order v.
• The masses of the (approximately) singlet neutrinos are heavy, of order mN .
• The masses of the (approximately) doublet neutrinos are light, of order v2/mN .
Furthermore, the heavier the gauge-singlet neutrinos, the lighter the SU(2)L-doublet neutrinos.
If the singlet neutrinos are very heavy, then they cannot be produced directly in experiments.
(Given that they are gauge-singlets, it would be difficult to produce them even if it were kinemat-
ically possible to do so.) They can thus be integrated out from the theory. This would leave the
SM as the effective low energy theory, with non-renormalizable terms suppressed by mN , the mass
scale of the heavy neutrinos. The dimension-five terms are
Ld=5 =Zνij
ΛφφLiLj, (12.30)
where
Zνij/Λ =
[Y ν(MN)−1Y νT
]ij. (12.31)
Thus, the leading terms in the low energy effective theory of the NSM are those of Eq. (12.2). We
learn that the NSM is indeed a possible UV completion of the νSM.
We end this subsection with an explanation why we choose to have three NR fields in our NSM.
• With a single NR field, the matrix Zν has two zero eigenvalues. Thus the model predicts
that two of the νi’s are massless and is therefore excluded.
194
• With two NR fields, the matrix Zν has a single zero eigenvalue. Thus the model predicts a
single massless νi and is phenomenologically viable. If experiments prove that the neutrino
spectrum is quasi-degenerate, the model will be excluded.
• With three NR fields, the matrix Zν is a general symmetric 3 × 3 matrix of complex, di-
mensionless couplings. Thus the model can accommodate any light neutrino spectrum and
mixing.
• The low energy effective theory of NSM models with more than three NR fields is the same
as that of the model with three NR fields.
Thus, three is the minimal number required to generate at low energy the most general LνSM. This
is the reason that we define the NSM in this way.
12.3.4 The Ni interactions
The interaction eigenstates NR are gauge-singlets. Consequently, they do not have any gauge
interactions. The only type of interaction that they do have is Yukawa interaction, as described
by the Lagrangian LN of Eq. (12.20). Unlike the Yukawa interactions of the SM, in the NSM the
Higgs boson has off-diagonal couplings. In particular, it couples the heavy Ni states to the light νj
states. As explained in Section 11.2, the special SM features which lead to diagonality are, first,
that for a given fermion sector all fermions are chiral and therefore there are no bare mass terms
and, second, that the scalar sector has a single Higgs doublet. In the NSM, the first condition is
violated, and the Higgs boson has neutrino flavor changing couplings.
The heavy mass eigenstates Ni have (in addition to the dominant NR component) a small
component, of order v/mN , of νL, the SU(2)L doublets. This means that the Ni fields have also
weak interactions. The leptonic charged current interaction depends on a 3 × 6 lepton mixing
matrix, where the three rows refer to the three charged lepton mass eigenstates (e, µ, τ), and
the six columns refer to the six neutrino mass eigenstates (ν1, ν2, ν3, N1, N2, N3). The neutral
current interaction of neutrinos is neither universal nor even diagonal. As explained in Section
11.2, universality of the Z couplings holds only if all fermions of given chirality and given color
and charge come from the same SU(2) × U(1) representation. In the NSM, neutrinos come from
two different types of SU(2) × U(1) representations, (2)−1/2 and (1)0, and therefore there is no
universality in the couplings of the Z-boson to neutrinos. The NSM demonstrates then that the
absence of tree-level FCNCs is a rather special feature of the SM, that is violated in general by
new physics.
The Yukawa and weak interactions of the Ni states imply that they are unstable, and decay
to a light lepton (charged or neutral) and a boson (h, Z, or W ). By assumption, however, the Ni
particles are heavy, and thus cannot be produced in experiments, and their interactions cannot be
195
directly probed. We thus do not study their interactions any further here. We note, however, that
the Yukawa interactions of the Ni particles might be the source of the baryon asymmetry of the
Universe, a scenario that is known by the name of leptogenesis. See Section 13.1.4 for a discussion.
12.3.5 The case of mN v: Sterile neutrinos
So far we worked under the assumption of Eq. (12.25), that is, mN v. Here we comment on the
consequences of an opposite hierarchy, mN v. Note that in this case the extreme lightness of
the neutrinos is accounted for by tiny Yukawa couplings, yν ∼< 10−12, rather than by a high scale
of new physics.
Let us first discuss the special case of MN = 0. In order to satisfy the Naturalness principle
discussed in Section 1, setting MN to zero requires that we postulate a global symmetry – lepton
number – in addition to the gauge symmetry of the SM. Lepton number is an anomalous symmetry,
but U(1)B−L is non-anomalous and achieves the same renormalizable Lagrangian, namely LNSM
with MN = 0.
With MN = 0, the neutrinos are Dirac particles, similarly to the charged fermions. This should
be the case, because in this model the neutrinos carry a conserved charged (L, or B−L). One may
think that this is the simplest way for neutrinos to acquire their masses. However, extending SM
not only by adding matter fields but also by imposing a global symmetry is a much more dramatic
modification than just adding matter fields and considering the most general Lagrangian, as is the
case for the NSM. Furthermore, in the NSM, the lightness of the doublet neutrinos is explained by
a new high scale of physics, which is well motivated. In contrast, the lightness of Dirac neutrinos
requires that their dimensionless Yukawa couplings are set to be tiny by hand.
For finite mN but with mN v, we have to consider two cases. First, if mN ∼ yνv, then
there are six very light Majorana mass eigenstates. Light states that are dominantly electroweak
singlets are called sterile neutrinos, while those that are dominantly electroweak doublets are called
active neutrinos. Sterile neutrinos can significantly change the neutrino phenomenology, but so far
there is no conclusive evidence for their existence. Second, if mN yνv, the six Majorana mass
eigenstates divide into three pairs of pseudo-Dirac neutrinos. Each pair is quasi-degenerate, with
average mass of order yνv and splitting of order mN .
The question of whether the neutrinos are Majorana or Dirac fermions is not yet experimentally
decided. If neutrinoless double beta decay is observed (see discussion in Section 12.A.2), it will
prove their Majorana nature.
196
12.4 Probing neutrino masses
12.4.1 Neutrino oscillations in vacuum
In experiments, neutrinos are produced and detected by charged current weak interactions. Thus,
the states that are relevant to production and detection are the SU(2)L-doublet partners of the
charged lepton mass eigenstates, namely νe, νµ, ντ . On the other hand, the eigenstates of free
propagation in spacetime are the mass eigenstates, ν1, ν2, ν3. In general, the interaction eigenstates
are different from the mass eigenstates:
|να〉 = U∗αi|νi〉 (α = e, µ, τ ; i = 1, 2, 3). (12.32)
Consequently, flavor is not conserved during propagation in spacetime and, in general, we may
produce να but detect νβ 6= να.
The probability Pαβ of producing neutrinos of flavor α and detecting neutrinos of flavor β is
calculable in terms of
• The neutrino energy E;
• The distance between source and detector L;
• The mass squared difference ∆m2ij ≡ m2
i −m2j ;
• The parameters – mixing angles and the Dirac phase – of the mixing matrix U ;
Starting from Eq. (12.32), we can write the expression for the time evolved |να(t)〉 (where |να(0)〉 ≡|να〉):
|να(t)〉 = U∗αi|νi(t)〉, (12.33)
where
|νi(t)〉 = e−iEit|νi(0)〉. (12.34)
In all cases of interest, the neutrinos are relativistic, and we then approximate
Ei = pi +m2i /(2Ei). (12.35)
Thus, the probability of a state that is produced as να to be detected as νβ is given by
Pαβ = |〈νβ|να(t)〉|2 . (12.36)
Explicit calculation (see your homework) gives
Pαβ = δαβ − 4∑j>i
Re(UαiU
∗βiU
∗αjUβj
)sin2
(∆m2
ijL
4E
)+ 2
∑j>i
Im(UαiU
∗βiU
∗αjUβj
)sin
(∆m2
ijL
2E
).
(12.37)
197
If we apply this calculation to the two generation case, where there is a single mixing angle
(and no relevant phase) and a single mass-squared difference,
U =
(cos θ sin θ
− sin θ cos θ
), ∆m2 = m2
2 −m21, (12.38)
we obtain, for α 6= β,
Pαβ = sin2 2θ sin2 x, x ≡ ∆m2L
4E. (12.39)
We learn that the time evolution of neutrinos that are produced in a flavor eigenstate exhibits
oscillations (as a function of time or, equivalently, distance) between the different flavor eigenstates.
This phenomenon is known as “neutrino oscillations.”
The expression (12.39) depends on two parameters that are related to the experimental design,
E and L, and two that are parameters of the Lagrangian, ∆m2 and θ. To be sensitive to the
Lagrangian parameters, one has to design the experiment appropriately:
∆m2L/E 1 Pαβ → 0,
∆m2L/E ∼ 1 Pαβ sensitive to ∆m2 and θ,
∆m2L/E 1 Pαβ → 12
sin2 2θ. (12.40)
We learn that to allow observation of neutrino oscillations, Nature needs to provide sin2 2θ that is
not too small. To get an intuition it is useful to write
x ≈ 1.27
(∆m2
eV2
)(L
km
)(GeV
E
). (12.41)
To probe small ∆m2, we need experiments with large L/E. Indeed, given natural neutrino sources
as well as neutrinos produced in reactors and accelerators, we can probe a rather large range of
∆m2; see the list in Table 12.1.
It is interesting to understand the differences between neutrino oscillations and neutral meson
oscillations (discussed in Section 11.A):
1. Mesons decay while the neutrinos are stable (at least on the time scale of experiments).
This brings the meson decay width into the analysis of the time evolution of mesons, see
Eq. (11.42). This is also the reason that in meson mixing the two Hamiltonian eigenstates
may be non-orthogonal.
2. Neutrino oscillations depend on the mixing angle. In neutral meson mixing we consider
particle-antiparticle oscillation. In this case, CPT requires that the two diagonal elements
in the mass matrix are equal and thus the mixing angle is maximal at π/4.
3. Consider the argument of the time dependent oscillation. Comparing Eq. (11.30) to Eq.
(12.39) we see that it is ∆mt for mesons and ∆m2t/(2E) for neutrinos. This apparent
198
Table 12.1: Neutrino oscillation experiments. The column ∆m2 gives the range to which the
corresponding class of experiments is sensitive. The physics of solar neutrinos, and the separation
to vacuum (VO) and matter (MSW) effects are explained in Section 12.4.2.
Source E[MeV] L[km] ∆m2[eV2]
Solar (VO) 1 108 =⇒ 10−11 − 10−9
SB Reactor 1 0.1− 1 =⇒ 10−3 − 10−2
LB Reactor 1 102 =⇒ 10−5 − 10−3
Atmospheric 103 101−4 =⇒ 10−5 − 1
SB accelerator 103−4 0.1 =⇒ ∼> 0.1
LB accelerator 104 102−3 =⇒ 10−3 − 10−2
Source n0[cm−3] r0[cm] ∆m2[eV2]
Solar (MSW) 6× 1025 7× 109 =⇒ 10−9 − 10−5
difference, however, is nothing but the effect of relativistic time dilation. Using, for the
quasi-degenerate mesons ∆m = ∆m2/(2m), and for the ultra-relativistic neutrinos t = τ/γ
(with τ the proper time) and γ = E/m, the dependencies on time and mass become the
same.
12.4.2 The MSW effect
To describe the propagation of neutrinos through matter, modifications to the oscillation formalism
are necessary. The smallness of the cross sections makes most effects of neutrino scattering off
medium negligible. This is, however, not the case for forward scattering, where there is no energy or
momentum exchange between the neutrinos and the medium. The effect of forward scattering is to
induce effective masses for the neutrinos (similar to the effect of medium on photon propagation).
The resulting modification to Pαβ of Eq. (12.39) can be very dramatic. This is known as the
Mikheyev-Smirnov-Wolfenstein (MSW) effect.
For the current measurements of neutrino flavor transitions, the relevant effects come from
neutrino propagation in the Sun or in Earth. In both cases, matter consists of electrons, protons
and neutrons. In particular, there are neither muons, nor tau-leptons nor anti-leptons in the
medium.
All neutrinos have the same (universal) neutral current interactions. In contrast, in matter
that has electrons but neither muons nor tau-leptons, only νe has charged current interactions
with matter. The effective potential induced by the charged current interactions of νe is given by
VC =√
2GFne ≈ 7.6ne
np + nn
(ρ
1014g/cm3
)eV , (12.42)
199
where ni (i = e, p, n) stands for the number density, and ρ is the mass density. For example,
at the solar core, ρ ∼ 100 g/cm3, which gives rise to VC ∼ 10−12 eV, while at the Earth core,
ρ ∼ 10 g/cm3, which gives rise to VC ∼ 10−13 eV.
Current data indicates that mν ∼> 10−3 eV, and thus mν VC . One may then naively think
that matter effects are irrelevant. Matter effects, however, arise from vector interactions while
masses are scalar operators. Consequently, the right comparison to make is between m2ν and EVC ,
where E is the neutrino energy. Since E mν , matter effects can be important. To see how this
enhancement of matter effects arises, consider a uniform, unpolarized medium at rest and a one
generation model. In this case, the four-vector effective interaction is given by Vµ = (VC , 0, 0, 0).
Due to VC the vacuum dispersion relation of the neutrino, pµpµ = m2, is modified as follows:
(pµ − Vµ)(pµ − V µ) = m2 ⇒ E ≈ p+ VC +m2
2p, (12.43)
where the approximation holds for ultra-relativistic neutrinos, E ≈ p m. Writing the dispersion
relation as E2 = p2 +m2m, we learn that the effective mass-squared in matter, m2
m, is given by
m2m = m2 + Ae, Ae ≡ 2EVC . (12.44)
It is the vector nature of the weak interaction that makes the matter effects practically relevant.
We analyze the matter effects in a two neutrino framework. In vacuum, in the mass basis
(ν1, ν2), the Hamiltonian can be written as
H = p+
( m21
2Em2
2
2E
). (12.45)
In the interaction basis (νe, νa), where νa is a combination of νµ and ντ , we have
H = p+m2
1 +m22
4E+
(−∆m2
4Ecos 2θ ∆m2
4Esin 2θ
∆m2
4Esin 2θ ∆m2
4Ecos 2θ
). (12.46)
In matter that contains only electrons, protons and neutrons, the Hamiltonian in the interaction
basis is modified from its vacuum form of Eq. (12.46):
H = p+ Va +m2
1 +m22
4E+
1
4E
(2Ae −∆m2 cos 2θ ∆m2 sin 2θ
∆m2 sin 2θ ∆m2 cos 2θ
). (12.47)
Omitting the part in the Hamiltonian that is proportional to the unit matrix in flavor space (which
plays no role in the oscillations), we see that the effective mass-squared difference and mixing angle
in matter are given by
∆m2m =
√(∆m2 cos 2θ − Ae)2 + (∆m2 sin 2θ)2, (12.48)
tan 2θm =∆m2 sin 2θ
∆m2 cos 2θ − Ae,
200
where the subindex m stands for matter.
The oscillation probability in matter with constant ne is simply obtained from Eq. (12.39) by
replacing ∆m2 and θ with ∆m2m and θm:
Pαβ = sin2 2θm sin2 xm , xm =∆m2
mL
4E. (12.49)
The following points are worth mentioning regarding Eq. (12.49):
1. The vacuum result is reproduced for A = 0, as it should.
2. Vacuum mixing is needed in order to get mixing in matter.
3. For ∆m2 cos 2θ |A|, the matter effect is a small perturbation to the vacuum result.
4. For ∆m2 cos 2θ |A|, the neutrino mass is a small perturbation to the matter effect. In
that case the oscillations are highly suppressed since the effective mixing angle is very small.
5. For ∆m2 cos 2θ = A, the mixing is maximal, namely it is on resonance.
12.4.3 Non-uniform density
When the matter density is not constant there are further modifications to the oscillation formal-
ism. Density variation results in changing the effective neutrino masses and their mixing angles.
Then, the flavor composition of the neutrinos along their path is a function of the medium density
profile.
At any point r on the neutrino path we care about the change of the potential. Given that the
potential is linear in the density, we define the derivative of the density
n′(r) =dn(r)
dr. (12.50)
For constant density, n′ = 0, the flavor conversion probability is controlled by the effective masses
and mixing angles. For varying density, n′ 6= 0, there are extra parameters that affect the flavor
conversion probability. Most important is the adiabatic parameter
Q(r) =∆m2 sin2 2θ
E cos 2θ
n(r)
n′(r). (12.51)
In the adiabatic limit, Q 1, the density variation is slow. In this case the constant density
formalism can be applied locally. Of particular interest is the case of large L where we can average
the oscillation and just think of propagation of mass eigenstates. In the adiabatic limit transition
between effective mass eigenstates is highly suppressed, and thus flavor transition is governed only
by the way the mass eigenstate evolves.
201
In the non-adiabatic limit, Q < 1, the density variation is fast. Then, transition between
effective mass eigenstates is possible, and the constant density formalism cannot be used. Both
limits can be of interest in reality, but here we concentrate on the adiabatic case.
To gain insight into the importance of the variation we consider the mixing angle that is function
of location, θm = θm(ne(x)):
tan 2θm(x) =∆m2 sin 2θ
∆m2 cos 2θ − 2√
2GFne(x)E. (12.52)
In particular, as ne(x) decreases, so does θm(x). Defining
nRe =∆m2 cos 2θ
2√
2GFE, (12.53)
we have
ne nRe =⇒ θm ≈ π/2,
ne = nRe =⇒ θm = π/4,
ne = 0 =⇒ θm = θ. (12.54)
We conclude that, for a small θ, νm2 propagating along a decreasing ne is mostly νe above nRe and
mostly νa for ne below nRe .
We now describe the characteristics of νe production and propagation in the Sun. The electron
density in the Sun can be parameterized as ne(x) ≈ 2n0 exp(−x/r0), where the relevant parameters
are given in Table 12.1. Consider the case where nprode nRe . Then, according to Eq. (12.54),
we have at the production point ν = νm2 (θm = π/2). Further assume that the propagation is
adiabatic at ne ∼ nRe . Then, at the resonance point we still have ν = νm2 (θm = π/4). Finally,
as the neutrino arrives to the surface of the Sun, it is still νm2 , but now, according to Eq. (12.54),
we have θm = θ, and the neutrino is simply the heavy mass eigenstate. Being a mass eigenstate,
it does not oscillate along its propagation to Earth. We conclude that for solar electron neutrinos
with energy in the range
E ∆m2
GFnprode
, (12.55)
and where the adiabatic condition is satisfied, the probability of being detected as νe is given by
PMSWee = |〈νe|ν2〉|2 = |Ue2|2 = sin2 θ. (12.56)
It is highly sensitive to θ and provides a way to probe small mixing angles.
On the other hand, for solar neutrinos with energy in the range
E ∆m2 cos 2θ
GFnprode
, (12.57)
202
namely nprode nRe , the produced state is ν = sin θ νm2 + cos θ νm1 . Approaching the surface of
the Sun, ν = sin θ ν2 + cos θ ν1 and Pee(R) = 1. Along the propagation to Earth, the neutrino is
subject to vacuum oscillations, with the final result [see Eq. (12.40)]
PVOee = 1− 1
2sin2 2θ. (12.58)
The comparison between Eqs. (12.56) and (12.58) demonstrates the importance of matter effects.
A few final remarks are in order:
1. Note that PMSWee < 1
2is possible, while PVO
ee > 12. For solar neutrinos, the transition between
those subject to the MSW effect, Eq. (12.56), and those subject to vacuum oscillations, Eq.
(12.58), occurs at E ∼ MeV.
2. Matter effects are sensitive to the sign of ∆m2, as manifest in Eq. (12.48). This is why we
know the sign of ∆m221 but not of ∆m2
32.
3. Examining Table 12.1, we conclude that, if sin θ ∼ 1, neutrino masses in the entire theoreti-
cally interesting range, 10−11 eV2∼< ∆m2
∼< eV2, could be discovered. For 10−2∼< sin θ 1,
neutrino masses could still be discovered via the adiabatic MSW effect for ∆m2 ∼ 10−5 eV2.
12.4.4 Experimental results
Neutrino flavor transitions have been observed for solar, atmospheric, reactor and accelerator
neutrinos. Five flavor parameters – two mass-squared differences and the three mixing angles –
have been measured:
∆m221 = (7.5± 0.2)× 10−5 eV2,
|∆m232| = (2.46± 0.05)× 10−3 eV2,
sin2 θ12 = 0.30± 0.01,
sin2 θ23 = 0.45± 0.03,
sin2 θ13 = 0.022± 0.001. (12.59)
All the results in the neutrino sector so far are consistent with the νSM. The following param-
eters are still not experimentally determined:
• The absolute mass scale of the neutrinos is still unknown. On one extreme, they could
be quasi-degenerate and as heavy as parts of eV. On the other extreme, they could be
hierarchical, with the lightest possibly massless.
• It is not known whether the spectrum has normal or inverted hierarchy.
• None of the three phases has been measured.
203
While the results can be accommodated in the νSM, there are other ways to explain the data.
The following questions are of interest as further tests of the idea that the νSM is the correct low
energy description of the neutrino sector:
• Are the neutrinos Dirac or Majorana fermions?
• Are there sterile neutrinos, that is, other light states that mix with the active neutrinos?
• Are there dimension six operators that significantly affect the neutrino interactions?
204
Appendix
12.A Probing neutrino masses
The positive evidences for neutrino masses only measure ∆m2. Here we discuss attempts to
measure the mass itself. As of now, these measurements give only upper bounds. Specifically, we
discuss kinematic tests for neutrino masses and neutrinoless double beta decay. We do not discuss
here astrophysical and cosmological probes of neutrino masses; We mention these in section 13.1.
12.A.1 Kinematic tests
In decays that produce neutrinos, the decay spectra are sensitive to neutrino masses. For example,
in the π → µν decay, the muon momentum is fixed (up to tiny width effects) by the masses of the
pion, the muon and the neutrino. To first order in m2ν/m
2π, the muon momentum in the pion rest
frame is given by
|~p | = 1
2mπ
(m2π −m2
µ −m2π +m2
µ
m2π −m2
µ
m2ν
). (12.60)
Since the correction to the massless neutrino limit is proportional to m2ν , the kinematic tests are
not very sensitive to small neutrino masses. The current best bounds obtained using kinematic
tests are the following [10]:
mν < 18.2 MeV from τ → 5π + ν ,
mν < 190 KeV from π → µν ,
mν < 2 eV from 3H→3He + e+ ν . (12.61)
The combination of oscillation experiments, which are sensitive to the neutrino mass-squared
differences, and kinematic tests, which are sensitive to the neutrino masses themselves, implies
that all three neutrino masses are lighter than 2 eV.
12.A.2 Neutrinoless double-beta (0ν2β) decay
Neutrino Majorana masses violate lepton number by two units. Therefore, if neutrinos have
Majorana masses we expect that there are also ∆L = 2 processes. The smallness of the neutrino
205
masses indicates that such processes have very small rates. Therefore, the only practical way to
look for ∆L = 2 processes is in places where the lepton number conserving ones are forbidden or
highly suppressed. Neutrinoless double-beta (0ν2β) decay, where the single beta decay is forbidden,
is such a process. An example for such processes is
3276Ge→ 34
76 Se + 2e− . (12.62)
The only physical background to 0ν2β decay is from double-beta decay with two neutrinos.
The 0ν2β decays are sensitive to the following combination of neutrino parameters:
mββ =3∑i=1
miU2ei. (12.63)
The best bound derived from 0ν2β decay is mββ < 0.34 eV [10]. We emphasize the following
points:
1. If the neutrinos are Dirac particles, lepton number is conserved, and their masses do not
contribute to 0ν2β decays.
2. The 0ν2β decay rate depends not only on m2ββ but also on some nuclear matrix elements.
Those matrix elements introduce theoretical uncertainties in extracting mββ from the signal,
or in deriving an upper bound on mββ if no signal is observed.
3. The 0ν2β decay is sensitive to other ∆L = 2 operators, and not only to the neutrino Majorana
masses. Thus, the relation between the 0ν2β decay rate and the neutrino mass is model
dependent.
206
Homework
Question 12.1: The νSM
Here you are asked to fill up some of the detailed omitted in the main text.
1. Restore the SU(2) indices that are implicit in Eq. (12.2).
2. Show that Eq. (12.3) arises from Eq. (12.2)Eq. (12.3) when the Higgs field is replaced by its
a VEV.
Question 12.2: 2× 2 matrices and the see-saw mechanism
Consider a 2× 2 Hermitian matrix:
M =
(a c
c∗ b
), (12.64)
with a and b real.
1. Show that the Eigenvalues are
λ1,2 =TrM ±
√(TrM)2 − 4 detM
2. (12.65)
2. Prove that
(TrM)2 ≥ 4 detM. (12.66)
This is needed in order to ensure that the Eigenvalues are real.
3. Next we move to study real symmetric matrices, that is, we assume that c is real. In that
case the matrix M can be diagonalized by an orthogonal matrix O:
O =
(cos θ sin θ
− sin θ cos θ
). (12.67)
We call θ the mixing angle. Show that
θ =1
2tan−1
(2c
b− a
). (12.68)
207
4. The seesaw mechanism is a special case of the above where a = 0 and c b. Show that, to
leading order in c/b, the eigenvalues and mixing angles are given by
λ1 =c2
b, λ2 = b, θ =
c
b, (12.69)
that is, confirming Eq. (12.23).
Question 12.3: Type III see-saw
Consider a one generation SM where we add one fermion field, N(1, 3)0. This model is called
the type III see-saw model and here you are asked to show how it generates the light neutrino
mass.
1. Write the new terms in L that involve N . Denote them by LNKin, LNψ , and LNYuk. Denote the
Majorana mass term by M . Write the covariant derivative explicitly. In the Yukawa term,
make sure you use correctly φ or φ.
2. The Higgs field acquires a VEV: 〈φ0〉 = v/√
2. Assume that M v. Show that the neutrino
acquires a mass, mν = cv2/M . Find out what c is.
3. Both the neutral and the charged components of N mix with the SM fields. The mixing
generates a mass splitting between the charged and neutral components of N . Estimate the
size of the splitting.
4. The mass splitting you found above is rather small and, in fact, negligible compared to those
generated by one loop effects. Discuss where these one loop effects come from, and estimate
their size.
Question 12.4: Neutrino oscillations
Here you are asked to derive some of the basic formulas of neutrino oscillations.
1. Derive Eq. (12.37). It is useful to recall that
Pαβ = |〈νβ|να(t)〉|2 =∑i
|〈νβ|νi〉〈νi|να(t)〉|2 . (12.70)
2. Derive expressions for the difference between the T -conjugate processes,
∆T ≡ P (νe → νµ)− P (νµ → νe), (12.71)
208
and for the difference between the CP -conjugate processes,
∆CP ≡ P (νe → νµ)− P (νe → νµ). (12.72)
Show that in the CP limit, which you can take as the case where U is real, ∆T = ∆CP = 0.
3. Due to CPT we know that ∆T −∆CP = 0. Check that this is indeed the case.
4. Consider a case where the production state is νe and the detection is done via elastic scat-
tering. In that case the detected neutrino is some superposition of flavor eigenstates which
we define as νd. Find P (νe → νd) as a function of distance. For simplicity assume only two
generations.
Question 12.5: Matter effects
1. Derive Eqs. (12.48) and (12.49).
2. Consider the case that xm is small. Show that to leading order in xm, matter effects cancel
in the oscillation probability, that is, you recover the vacuum result.
Question 12.6: Solar neutrinos
While by now ruled out, in the past one possible solution for the solar neutrino problem was
the so-called “vacuum oscillation.” The idea is that the mass difference is very small, and then
matter effects can be neglected, and the only relevant effects is the oscillation of the neutrinos
while traveling from the Sun to Earth.
1. Estimate ∆m2 which is needed for such a solution to work.
2. One prediction of this idea is seasonal variation. Explain how this can be observed and
estimate the magnitude of the effect.
209
Chapter 13
Connection to cosmology
13.1 Baryogenesis
13.1.1 The baryon asymmetry
Observations indicate that the number of baryons in the Universe is unequal to the number of
antibaryons. To the best of our understanding, all the structures that we see in the Universe –
stars, galaxies, and clusters – consist of matter (baryons and electrons) and there is no antimatter
(antibaryons and positrons) in appreciable quantities. Since various considerations suggest that
the Universe has started from a state with equal numbers of baryons and antibaryons, the observed
baryon asymmetry must have been generated dynamically, a scenario that is known by the name
of baryogenesis.
The baryon asymmetry of the Universe is expressed in the literature in various ways:
ηB ≡nB − nB
nγ
∣∣∣∣∣0
= (6.21± 0.16)× 10−10,
YB ≡nB − nB
s
∣∣∣∣0
= (8.75± 0.23)× 10−11, (13.1)
where nB, nB, nγ and s are the number densities of, respectively, baryons, antibaryons, photons
and entropy, a subscript 0 implies “at present time.” One can also present the asymmetry in terms
of the baryonic fraction of the critical energy density ΩB ≡ (ρB/ρcrit):
ΩBh2 = 0.0219± 0.0007, (13.2)
where h ≡ H0/(100 km s−1 Mpc−1) = 0.70± 0.01 is the present Hubble parameter.
The value of the baryon asymmetry of the Universe is inferred in two independent ways. The
first way is via big bang nucleosynthesis. This chapter in cosmology predicts the abundances of
the light elements, D, 3He, 4He, and 7Li. These predictions depend on a single parameter, which is
ηB. The abundances of D and 3He are very sensitive to ηB. The second way is from measurements
210
of the cosmic microwave background radiation. A larger ηB would enhance the odd peaks in the
spectrum. The fact that the two determinations agree gives much confidence in the value of the
baryon asymmetry. A consistent theory of baryogenesis should thus explain nB ≈ 10−9nγ and
nB = 0.
13.1.2 Sakharov conditions
Three conditions that are required to dynamically generate a baryon asymmetry were formulated
by Sakharov:
• Baryon number violation: This condition is required in order to evolve from an initial state
with ηB = 0 to a state with ηB 6= 0.
• C and CP violation: If either C or CP were conserved, then processes involving baryons
would proceed at precisely the same rate as the C- or CP-conjugate processes involving
antibaryons, with the overall effect that no baryon asymmetry is generated.
• Out of equilibrium dynamics: In chemical equilibrium, there are no asymmetries in quantum
numbers that are not conserved (such as B, by the first condition).
These necessary ingredients are all present in the Standard Model. Quantitatively, however,
the SM fails to explain the observed asymmetry:
• Baryon number is violated in the SM, and the resulting baryon number violating processes are
fast in the early Universe. The violation is due to the triangle anomaly, and leads to processes
that involve nine left-handed quarks (three of each generation) and three left-handed leptons
(one from each generation), see Fig. 13.1.1 At zero temperature, the amplitude of the
baryon number violating processes is proportional to e−8π2/g2, which is too small to have
any observable effect. At high temperatures (T TEWPT, where TEWPT is the temperature
where the electroweak phase transition starts to take place, of order 100 GeV), however,
these transitions become unsuppressed, Γ ∝ 250α5wT , and are thus faster than the expansion
rate of the Universe for TEWPT ∼< T ∼< 1012 GeV.
• The weak interactions of the SM violate C maximally and violate CP via the Kobayashi-
Maskawa mechanism. As argued in Section 13.1.3, the KM mechanism introduces a suppres-
sion factor of order 10−20 into the SM contribution to the baryon asymmetry. Since there
are practically no kinematic enhancement factors in the thermal bath, it is impossible to
generate ηB ∼ 10−9 with such a small amount of CP violation. Consequently, baryogenesis
implies that there must exist new sources of CP violation, beyond the KM phase of the SM.
1A selection rule is obeyed, ∆B = ∆L = ±3n, preserving proton stability.
211
Le
Lµ
Lτ
Q3
Q2
Q1
Figure 13.1: A schematic presentation of a B + L violating process in the SM.
• Within the Standard Model, departure from thermal equilibrium occurs at the electroweak
phase transition (EWPT). Here, the non-equilibrium condition is provided by the interac-
tions of particles with the bubble wall, as it sweeps through the plasma. The experimental
measurement of mh ∼ 126 GeV implies, however, that this transition is not strongly first
order, as required for successful baryogenesis (see Fig. 13.1.2). Thus, a different kind of
departure from thermal equilibrium is required from new physics or a modification to the
electroweak phase transition.
V
φ
Tc
T>Tc
T<Tc
φc
V
φ
T>Tc
T<Tc
Tc
Figure 13.2: The evolution of the Standard Model scalar potential with decreasing temperature
for (left) mh < 70 GeV and (right) mh > 70 GeV.
212
We learn that baryogenesis requires new physics that extends the SM in at least two ways. It
must introduce new sources of CP violation, and it must either provide a departure from thermal
equilibrium in addition to the EWPT or modify the EWPT.
Among the proposed scenarios for possibly successful baryogenesis we should mention GUT
baryogenesis, leptogenesis, electroweak baryogenesis, and the Affleck-Dine mechanism. We describe
leptogenesis in more detail in Section 13.1.4.
13.1.3 The suppression of KM baryogenesis
As explained in Section 11.6, the three generation SM violates CP if XCP 6= 0. The baryon
asymmetry of the Universe is a CP violating observable. As such, it is proportional to XCP . More
precisely, it is proportional to XCP/T12c , where Tc ∼ 100 GeV is the critical temperature of the
electroweak phase transition. When one puts the measured values of the quark masses and CKM
parameters, one obtains that XCP ∼ 10−20, and thus the KM mechanism cannot account for a
baryon asymmetry as large as O(10−10).
One may wonder why the suppression by XCP does not apply to all CP asymmetries measured
in experiments. After all, there are CP asymmetries such as Sππ that are experimentally of order
one and theoretically known to be suppressed by the KM phase (sin 2α) but by none of the mixing
angles or small quark mass-squared differences of XCP . The answer provides some insights as
to how the KM mechanism operates. As concerns the mixing angles, they often cancel in the
CP asymmetries which are ratios of CP violating to CP conserving rates. The physics behind
the mass factors in Eq. (11.14) is that, in order to exhibit CP violation, a process has to “go
through” all three flavors of each quark type, and “sense” that their masses are different from each
other. Sometimes, the experiment does that for us. For example, when experimenters measure
the CP asymmetry in B → ππ, they already distinguish the bottom, up, and down masses from
the others (by identifying the B and π mass eigenstates) and thus ‘get rid’ of the corresponding
mass factors. What remains is the (m2t −m2
c) factor, This factor does appear in ∆mB and, indeed,
if this factor were zero, the CP asymmetry, which is really Sππ sin(∆mBt), would vanish. In
contrast, baryogenesis is a flavor-blind process (it sums over all flavors), and is suppressed by all
six mass-squared factors of Eq. (??).
The important conclusion of the failure of the KM mechanism to account for the baryon
asymmetry is the following: There must exist sources of CP violation beyond the KM phase of
the SM.
13.1.4 Leptogenesis
The addition of the NRi fields, with the Yukawa (Y ν) and mass (MN) terms of Eq. (12.20), is
motivated by the seesaw mechanism for light neutrino masses. The addition of these terms implies,
213
however, an additional intriguing consequence: The physics of the singlet fermions is likely to play
a role in dynamically generating a lepton asymmetry in the Universe. The reason that leptogenesis
is qualitatively almost unavoidable once the seesaw mechanism is invoked is that the Sakharov
conditions, described in Appendix 13.1.2, are (likely to be) fulfilled:
• Lepton number violation: The Lagrangian terms (12.20) violate L because lepton number
cannot be consistently assigned to the NRi fields in the presence of Y ν and MN . If L(NR) = 1,
then Y ν respects L but MN violates it by two units. If L(NR) = 0, then MN respects L but
Y ν violates it by one unit. (Remember that the fact that the SM interactions violate B + L
implies that the requirement for baryogenesis from new physics is B − L violation, and not
necessarily B violation.)
• CP violation: Since there are irremovable phase in Y ν (once the `−L,R andN fields are rephased
to make Y e and MN real), the Lagrangian terms (12.20) provide new sources of CP violation.
• Departure from thermal equilibrium: The interactions of the Ni are only of the Yukawa type.
If the Y ν couplings are small enough, these interactions can be slower than the expansion
rate of the Universe, in which case the singlet fermions will decay out of equilibrium.
Thus, in the presence of the seesaw terms, leptogenesis is qualitatively almost unavoidable, and the
question of whether it can successfully explain the observed baryon asymmetry is a quantitative
one.
We consider leptogenesis via the decays of N1, the lightest of the singlet fermions Ni. When
the decay is into a single flavor α, N1 → Lαφ or Lαφ†, the baryon asymmetry can be written as
follows:
YB =
(135ζ(3)
4π4g∗
)× Csphal × ηeff × εCP. (13.3)
The first factor is the equilibrium N1 number density divided by the entropy density at temperature
T M1. It is of O(4× 10−3) when the number of relativistic degrees of freedom g∗ is taken as in
the SM, gSM∗ = 106.75. The other three factors on the right hand side of Eq. (13.3) represent the
following physics aspects:
1. εCP is the CP asymmetry in N1 decays. For every 1/εCP N1 decays, there is one more L than
there are L’s.
2. ηeff is the efficiency factor. Inverse decay, other “washout” processes, and inefficiency in N1
production, reduce the asymmetry by 0 ≤ ηeff ≤ 1. In particular, ηeff = 0 is the limit of N1
in perfect equilibrium, so no asymmetry is generated.
3. Csphal describes further dilution of the asymmetry due to fast processes which redistribute
the asymmetry that was produced in lepton doublets among other particle species. These
include gauge, Yukawa, and B + L violating non-perturbative effects.
214
Lα
LβLβ
LαLα
φ φφ
φφ
N1
N1
N1
N2 N
2
Figure 13.3: The diagrams contributing to the CP asymmetry ε.
These three factors can be calculated, with εCP and ηeff depending on the Lagrangian parame-
ters. The final result can be written (with some simplifying assumptions) as
YB ∼ 10−3 10−3 eV
mεCP, (13.4)
where the diagrams of Fig. 13.3 give (we define xj ≡M2j /M
21 )
εCP =1
8π
1
(Y ν†Y ν)11
∑j
Im[
(Y ν†Y ν)1j
]2√xj
[1
1− xj+ 1− (1 + xj) ln
(1 + xjxj
)], (13.5)
and
m =(Y ν†Y ν)11v
2
M1
. (13.6)
The plausible range for m is the one suggested by the range of hierarchical light neutrino
masses, 10−3 − 10−1 eV, so we expect a rather mild washout effect, ηeff ∼> 0.01. Then, to account
for YB ∼ 10−10, we need |εCP| ∼> 10−5 − 10−6. Using Eq. (13.5), we learn that this condition
roughly implies, for the seesaw parameters,
M1
M2
Im[(Y ν†Y ν)212]
(Y ν†Y ν)11∼> 10−4 − 10−5, (13.7)
which is quite natural.
We learn that leptogenesis is attractive not only because all the required features are qualita-
tively present, but also because the quantitative requirements are plausibly satisfied. In particular,
m ∼ 0.01 eV, as suggested by the light neutrino masses, is optimal for thermal leptogenesis as
it leads to effective production of N1’s in the early Universe and only mild washout effects. Fur-
thermore, the required CP asymmetry can be achieved in large parts of the seesaw parameter
space.
13.2 Dark Matter
Roughly 20% of the energy density of the Universe consists of neutral weakly interacting non-
baryonic matter, ‘dark matter’. The picture of structure formation by growth of fluctuations in
weakly interacting matter explains the elements of structure in the Universe from the fluctuations
215
in the cosmic microwave background down almost to the scale of galaxies. However, from the point
of view of particle physics, we have no idea what dark matter is made of. We know, however, that
the following conditions must be satisfied for the DM:
• Stable on cosmological time scales;
• Very weakly interacting;
• Have the right relic density.
The possibilities range in mass from axions (mass 10−5 eV) to primordial black holes (mass
10−5 M). One of the most attractive classes of dark matter candidates are WIMPs (weakly
interacting massive particles). They are heavy, neutral, weakly interacting particles with interac-
tion cross sections nevertheless large enough that they were in thermal equilibrium for some period
in the early universe.
13.2.1 Observational Evidence
Assuming that galaxies are in virial equilibrium, one can relate the mass at a given distance r from
the center of a galaxy to its rotational velocity:
v2 ∝ GNM(r)
r. (13.8)
The rotational velocity v is measured by observing 21 cm emission lines in HI regions (neutral
hydrogen) beyond the point where most of the light in the galaxy ceases. If the bulk of the mass is
associated with light, then beyond the point where most of the light stops, M should be constant
and v2 ∝ 1/r. This is not the case. The rotation curves are approximately flat, i.e. v ∼ constant
outside the core of the galaxy. In our own galaxy, v ' 240 km/s at the location of our solar system,
with little change out to the largest observable radius. This implies the existence of a dark halo,
with mass density ρ(r) ∝ 1/r2, i.e. M(r) ∝ r. At some point, ρ has to fall of faster, in order to
keep the total mass of the galaxy finite, but we do not know at what radius this happens. This
leads to a lower bound on the DM mass density, ΩDM ∼> 0.1.
The observation of clusters of galaxies tends to give somewhat larger values, ΩDM ' 0.2. These
observations include (i) measurements of the peculiar velocities of galaxies in the cluster, which
are a measure of their potential energy if the cluster is virialized; (ii) measurements of the X-ray
temperature of hot gas in the cluster, which again correlates with the gravitational potential felt by
the gas; and, most directly, (iii) studies of the (weak) gravitational lensing of background galaxies
on the cluster.
A particularly compelling example involves the bullet cluster which recently (on cosmological
time scales) passed through another cluster. As a result, the hot gas forming most of the baryonic
mass of the cluster was shocked and decelerated, whereas the galaxies in the clusters proceeded
216
on ballistic trajectories. Gravitational lensing shows that most of the total mass also moved
ballistically, indicating that DM self-interactions are weak.
The currently most accurate, if somewhat indirect, determination of ΩDM comes from global
fits of cosmological parameters to a variety of observations. For example, using measurements
of the anisotropy of the cosmic microwave background (CMB) and of the spatial distribution of
galaxies, the density of cold, non-baryonic matter is found to be [PDG]
Ωnbmh2 = 0.119± 0.002, (13.9)
where h is the Hubble constant in units of 100 km/(s Mpc). Some part of the baryonic matter
density [PDG],
Ωbh2 = 0.0223± 0.0002, (13.10)
may well contribute to (baryonic) DM, e.g. MACHOs or cold molecular gas clouds.
The DM density in the neighborhood of our solar system is of considerable interest. The most
recent estimates, based on a detailed model of our galaxy, and constrained by a host of observables
including the galactic rotation curve, finds [PDG]
ρlocalDM ∼ 0.4
GeV
cm3. (13.11)
13.2.2 Why not the neutrinos?
The SM does have dark matter particles among its list of elementary particles. These are the
neutrinos, which carry neither color charge, nor electromagnetic charge. The lightest of these (if
not all three) is stable. Can it constitute the dark matter of the Universe? The answer is negative.
Due to the weak interactions, neutrinos have been in thermal equilibrium at high enough
temperature. They decouple at a temperature TD, when the expansion rate of the Universe becomes
larger than their interaction rate. The cross section is given by σ ' G2FT
2 and the neutrino number
density is n ' T 3. Thus Γint = nσv ' G2FT
5 and
Γint
H' G2
FT5
T 2/mPl
'(
T
1 MeV
)3
. (13.12)
Below 1 MeV, the neutrino temperature Tν scales like a−1. Shortly after neutrino decoupling, the
temperature drops below the mass of the electron, and the entropy in e+e− pairs is transferred
to the photons, but not to the decoupled neutrinos. For T ∼> me, the particle species in thermal
equilibrium with photons include the photon (gγ = 2) and e+e− pairs (ge = 4), which give g =
2 + 78× 4 = 11/2. For T me, only the photons are in equilibrium, g = gγ = 2. For particles in
thermal equilibrium with the photons, g(aT )3 remains constant. Therefore, the value of aT after
the e± annihilation must be larger than before e± annihilation by a factor of the third root of the
217
ratio of g before (= 11/2) and after (= 2). Thus, aTγ increases by (11/4)1/3 while aTν remains
constant. Consequently, the present ratio of temperatures is
T
Tν=(
11
4
)1/3
= 1.40 =⇒ Tν = 1.95 K. (13.13)
The present number density of each flavor of neutrinos is
[n(νi)]0 =3
11[nγ]0 ≈ 110/cm3. (13.14)
Their contribution to the mass density of the universe is then
Ων =
∑imi[n(νi)]0
ρc∼ 10−3, (13.15)
where we took∑imi ∼ 0.05 eV.
Another major problem with the idea that neutrinos could constitute a major component in
the dark matter is that analyses of structure formation in the Universe indicate that most DM
should be “cold”, i.e., should have been non-relativistic at the onset of galaxy formation (when
there was a galactic mass inside the causal horizon).
13.2.3 The relic density: back of the envelope estimate
Let us study what happens to WIMP dark matter particle χ of mass m when the temperature T
cools below m. Annihilations with cross section σ(χχ → SM particles) try to maintain thermal
equilibrium, nχ ∝ exp(−m/T ). But they fail at T ∼< m, when nχ is so small that the collision rate
Γ experienced by χ becomes smaller than the expansion rate h:
Γ ∼ nχσ ∼< H ∼ T 2/mPl. (13.16)
Annihilations become ineffective, leaving the following out-of-equilibrium relic abundance of dark
matter particles:nχnγ∼ m2/(mPlσ)
m3∼ 1
mPlσm, (13.17)
i.e.,ρχ(T )
ργ(T )∼ m
T
nχnγ∼ 1
mPlσT. (13.18)
Inserting the observed DM density, ρχ ∼ ργ at T ∼ T0, and a typical cross section σ ∼ g4/m2,
gives
m ∼√T0mPl ∼ TeV, (13.19)
for a WIMP with g ∼ 1. More generally, to account for Ωih2 ∼ 0.1, we need
〈σv〉 ∼ 0.9 pb ∼ 1
108 GeV2 , (13.20)
218
which is a typical weak interaction rate. It is remarkable that the estimate points to a mass scale
where we expect for other reasons to find new physics beyond the standard model. The fact that
the cross section required to obtain the correct relic abundance is of the order of weak interaction
cross section has been termed “the WIMP miracle.”
13.2.4 Detecting WIMPs
The WIMP miracle not only provides a model-independent motivation for dark matter at the weak
scale, but it also has strong implications for how dark matter might be detected. For WIMPs χ to
have the observed relic density, they must annihilate to other particles. Assuming that these other
particles are SM particles, the necessity of χχ → SM SM interactions suggests three promising
strategies for dark matter detection:
• Indirect detection: if dark matter annihilated in the early Universe, it must also annihilate
now through χχ→ SM SM, and the annihilation products may be detected.
• Direct detection: dark matter can scatter off normal matter through χ SM→ χ SM interac-
tions, depositing energy that may be observed in sensitive, low background detectors.
• Particle colliders: dark matter may be produced at particle colliders through SM SM → χχ.
Such events are undetectable, but are typically accompanied by related production mecha-
nisms, such as SM SM → χχ + SM, where “SM” denotes one or more standard model
particles. These events are observable and provide signatures of dark matter at colliders.
It is important to note that the WIMP miracle not only implies that such dark matter interac-
tions must exist, it also implies that the dark matter-SM interactions must be efficient. Although
WIMPs may not be a significant amount of dark matter, they certainly cannot have an energy
density more than ΩDM. Cosmology therefore provides lower bounds on interaction rates. This
fact provides highly motivated targets for a diverse array of experimental searches that may be
able to detect WIMPs and constrain their properties.
13.2.5 DM@LHC?
Our current understanding of the weak interaction is that it arises from a gauge theory of the
group SU(2) × U(1) that is spontaneously broken at the hundred-GeV scale. An astronomer
might note this as a remarkable coincidence. A particle theorist might go further. There are
many possible, and competing, models of weak interaction symmetry breaking. In any of these
models, it is possible to add a discrete symmetry that makes the lightest newly introduced particle
stable. Generically, this particle is heavy and neutral and meets the definition of a WIMP that
we have given above. In many cases, the discrete symmetry in question is actually required for
219
the phenomenological viability of the model or arises naturally from its geometry. For example, in
models of supersymmetry, imposing a discrete symmetry called R-parity is the most straightforward
way to eliminate dangerous baryon-number violating interactions. Thus, as particle theorists, we
are almost justified in saying that the problem of electroweak symmetry breaking predicts the
existence of WIMP dark matter.
There is a further assumption that, when added to the properties of a WIMP just listed,
has dramatic consequences for experiments. Models of electroweak symmetry breaking typically
contain new heavy particles with QCD color. These appear as partners of the quarks to provide
new physics associated with the generation of the large top mass. In supersymmetry, and in many
other models, electroweak symmetry breaking appears as a result of radiative corrections due to
these particles, enhanced by the large coupling of the Higgs boson to the top quark. Thus, we
would like to add to the structure of WIMP models the assumption that there exists a new particle
that carries the conserved discrete symmetry and couples to QCD. This particle should have a mass
of the same order of magnitude as the WIMP, below TeV.
Any particle with these properties will be pair-produced at the CERN LHC with a cross section
of tens of pb. The particle will decay to quark or gluon jets and an imbalance of measured momen-
tum. These ‘missing energy’ events are well known to be a signature of models of supersymmetry.
In fact, they should be seen in any model (subject to the assumption just given) that contains a
WIMP dark matter candidate.
13.2.6 Supersymmetry: neutralino dark matter
In models of supersymmetry, the role of the WIMP χ is taken by the lightest ‘neutralino’ - a
mixture of the superpartners of γ and Z (‘gauginos’) and the superpartners of the neutral Higgs
bosons (‘higgsinos’). Depending on the spectrum and couplings of the superpatrtners, several
different reactions can dominate the process of neutralino pair annihilation.
The simplest possibility is that neutralinos annihilate to standard model fermions by exchanging
their scalar superpartners. Sleptons are typically lighter than squarks, so the dominant reactions
are χχ→ `+`−. It turns out, however, that this reaction is less important than one might expect
over most of the supersymmetry parameter space. Because neutralinos are Majorana particles,
they annihilate in the S-wave only in a configuration of total spin 0. However, light fermions are
naturally produced in a spin-1 configuration, and the spin-0 state is helicity suppressed by a factor
(m`/mχ)2. The dominant annihilation is then in the P-wave. Since the relic density is determined
as a temperature at which the neutralinos are non-relativistic, the annihilation cross-section is
suppressed and the prediction for the relic density is typically too large. To obtain values of the
relic density that agree with the observed value, we need light sleptons, with masses below 200
GeV.
Neutralinos can also annihilate to Standard Model vector bosons. A pure U(1) gaugino (‘bino’)
220
cannot annihilate to W+W− or Z0Z0. However, these annihilation channels open up if the gaugino
contains an admixture of SU(2) gaugino (‘wino’) or higgsino content. The annihilation cross
sections to vector bosons are large, so only a relatively small mixing is needed.
The annihilation to third generation fermions can be enhanced by a resonance close to threshold.
In particular, if the mass of the CP odd Higgs boson A0 is close to 2mχ, the resonance produced
by this particle can enhance the S-wave amplitude for neutralino annihilation to bb and τ+τ−.
If other superparticles are close in mass to the neutralino, these particles can have significant
densities when the neutralions decouple, and their annihilation cross sections can also contribute
to the determination of the relic density through a co-annihilation process. If the sleptons are only
slightly heavier than the neutralino, thereaction ˜χ → γ` and ˜ → `` can proceed in the S-wave
and dominate the annihilation Co-annihilation with W+ partners (‘charginos’) and with the top
squarks can also be important in some regions of the MSSM parameter space.
A common feature of all four mechanisms is that the annihilation cross section depends strongly
both on the masses of the lightest supersymmetric particles and on the mixing angles that relate
the original gaugino and higgsino states to the neutralino mass eigenstates. Both sets of parameters
must be fixed in order to obtain a precise prediction for the relic density.
221
Appendix
222
Appendix A
Lie Groups
A crucial role in model building is played by symmetries. You are already familiar with symmetries
and with some of their consequences. For example, Nature seems to have the symmetry of the
Lorentz group which implies conservation of energy, momentum and angular momentum. In order
to understand the interplay between symmetries and interactions, we need a mathematical tool
called Lie groups. These are the groups that describe all continuous symmetries.
In the following we only give definitions and quote statements without proving them. There
are many texts about Lie group where the statements we make below are proven. Three that are
very useful for particle physics purposes are the book by Howard Georgi (“Lie Algebras in particle
physics”), the book by Robert Cahn (“Semi-simple Lie algebras and their representations”) and
the physics report by Richard Slansky (“Group Theory for Unified Model Building”, Phys. Rept.
79 (1981) 1).
A.1 Groups
We start by presenting a series of definitions.
Definition: A group G is a set xi (finite or infinite), with a multiplication law ·, subject to the
following four requirements:
• Closure:
xi · xj ∈ G ∀ xi, xj. (A.1)
• Associativity:
xi · (xj · xk) = (xi · xj) · xk ∀ xi, xj, xk. (A.2)
• There is an Identity element I (or e) such that
I · xi = xi · I = xi ∀ xi. (A.3)
223
• Each element has an inverse element x−1i :
xi · x−1i = x−1
i · xi = I ∀ xi. (A.4)
Definition: A group is Abelian if all its elements commute:
xi · xj = xj · xi ∀ xi, xj. (A.5)
A non-Abelian group is a group that is not Abelian, that is, at least one pair of elements does not
commute.
Let us give a few examples:
• Z2, also known as parity, is a group with two elements, I and P , such that I is the identity
and P−1 = P . This completely specifies the multiplication table. This group is finite and
Abelian.
• ZN , with N=integer, is a generalization of Z2. It contains N elements labeled from zero
until N − 1. The multiplication law is the same as addition modulo N : xi · xj = x(i+j)mod N .
The identity element is x0, and the inverse element is given by x−1i = xN−i. This group is
also finite and Abelian.
• Multiplication of positive numbers. It is an infinite Abelian group. The identity is the
number one and the multiplication law is just a standard multiplication.
• S3, the group that describes permutation of 3 elements. It contains 6 elements. This group
is non-Abelian. In your homework you will find for yourself the 6 elements and their multi-
plication table.
A.2 Representations
One of the most important aspects of group theory that is relevant to physics is related to repre-
sentation theory, and that is what we discuss next.
Definition: A representation is a realization of the multiplication law among matrices.
Definition: Two representations are equivalent if they are related by a similarity transformation.
Definition: A representation is reducible if it is equivalent to a representation that is block
diagonal.
Definition: An irreducible representation (irrep) is a representation that is not reducible.
Definition: An irrep that contains matrices of size n× n is said to be of dimension n.
Statement: Any reducible representation can be written as a direct sum of irreps, e.g. D =
D1 +D2.
224
Statement: The dimension of all irreps of an Abelian group is one. For non-Abelian groups
there is at least one irrep that has dimension larger than one.
Statement: Any finite group has a finite number of irreps Ri. If N is the number of elements
in the group, the irreps satisfy ∑Ri
[dim(Ri)]2 = N. (A.6)
Infinite groups have infinite number of irreps.
Statement: For any group there exists a trivial representation such that all the matrices are
just the number 1. This representation is also called the singlet representation. As we see later, it
is of particular importance for us.
Let us give some examples for the statements that we made here.
• Z2: Its trivial irrep is I = 1, P = 1. The other irrep is I = 1, P = −1. Clearly these two
irreps satisfy Eq. (A.6).
• ZN : An example of a non-trivial irrep is xk = exp(i2πk/N).
• S3: In your homework you will work out its properties.
The groups that we are interested in are transformation groups of physical systems. Such
transformations are associated with unitary operators in the Hilbert space. We often describe the
elements of the group by the way that they transform physical states. When we refer to represen-
tations of the group, we mean either the appropriate set of unitary operators, or, equivalently, by
the matrices that operate on the vector states of the Hilbert space.
A.3 Lie groups and Lie Algebras
While finite groups are very important, the ones that are most relevant to particle physics and,
in particular, to the Standard Model, are infinite groups, in particular continuous groups, that is
of cardinality ℵ1. These groups are called Lie groups. They give us formal ways to talk about
rotations is any real or abstract space. The different groups corresponds to rotations in different
spaces.
Definition: A Lie group is an infinite group whose elements are labeled by a finite set of N
continuous real parameters α`, and whose multiplication law depends smoothly on the α`’s. The
number N is called the dimension of the group.
Different groups have different N . Yet, the dimension of the group does not uniquely defined
it. We discuss below the classifications of groups.
Statement: An Abelian Lie group has N = 1. A non-Abelian Lie group has N > 1.
The first example is a group we denote by U(1). It represents addition of real numbers modulo
2π, that is, rotation on a circle. Such a group has an infinite number of elements that are labeled
225
by a single continuous parameter α. We can write the group elements as M = exp(iα). We can
also represent it by M = exp(2iα) or, more generally, as M = exp(iXα) with X real. Each X
generates an irrep of the group.
We are mainly interested in compact Lie groups. We do not define this term formally here,
but we can use the U(1) example to give an intuitive explanation of what it means. A group of
adding with a modulo is compact, while just adding (without the modulo) would be non-compact.
In the first, if you repeat the same addition a number of times, you may return to your starting
point, while in the latter this would never happen. In other words, in a compact Lie group, the
parameters have a finite range, while in a non-compact group, their range is infinite. (Do not
confuse that with the number of elements, which is infinite in either case.) Another example is
rotations and boosts: Rotations represent a compact group while boosts do not.
Statement: The elements of any compact Lie group can be written as
M = exp(iα`X`) (A.7)
such that X` are specific Hermitian matrices. (We use the standard summation convention, that
is α`X` ≡∑` α`X`.)
Definition: The X` that are called the generators of the group.
Let us perform some algebra before we turn to our next definition. Consider two elements of a
group, A and B, such that in A only αa 6= 0, and in B only αb 6= 0 and, furthermore, αa = αb = λ:
A ≡ exp(iλXa), B ≡ exp(iλXb). (A.8)
Since A and B are in the group, each of them has an inverse. Thus also
C = BAB−1A−1 ≡ exp(iβcXc) (A.9)
is in the group. Let us take λ to be a small parameter and expand around the identity. Clearly, if
λ is small, also all the βc are small. Keeping the leading order terms, we get
C = exp(iβcXc) ≈ I + iβcXc, C = BAB−1A−1 ≈ I + λ2[Xa, Xb]. (A.10)
In the λ→ 0 limit, we have
[Xa, Xb] = iβcλ2Xc. (A.11)
The combinations
fabc ≡ λ−2βc (A.12)
is independent of λ. Furthermore, while λ and βc are infinitesimal, the fabc-constants do not
diverge. This brings us to a new set of definitions.
Definition: fabc are called the structure constants of the group.
226
Definition: The commutation relations [see Eq. (A.11)]
[Xa, Xb] = ifabcXc, (A.13)
constitute the algebra of the Lie group.
Note the following points regarding the Lie Algebra:
• The algebra defines the local properties of the group but not its global properties. Usually,
this is all we care about.
• The Algebra is closed under the commutation operator.
• Similar to our discussion of groups, one can define representations of the algebra, that is,
matrix representations of X`. In particular, each representation has its own dimension. (Do
not confuse the dimension of the representation with the dimension of the group.)
• The generators satisfy the Jacoby identity
[Xa, [Xb, Xc]] + [Xb, [Xc, Xa]] + [Xc, [Xa, Xb]] = 0. (A.14)
• For each algebra there is the trivial (singlet) representation which is X` = 0 for all `. The
trivial representation of the algebra generates the trivial representation of the group.
• Since an Abelian Lie group has only one generator, its algebra is always trivial. Thus, the
algebra of U(1) is the only Abelian Lie algebra.
• Non-Abelian Lie groups have non-trivial algebras.
• The generators of the Non-Abelian Lie groups are traceless.
The example of SU(2) algebra is well-known from QM courses:
[Xa, Xb] = iεabcXc. (A.15)
Here εabc are the structure constants of the SU(2) group. Usually, in QM, X is called L, S, or J .
The SU(2) group represents non-trivial rotations in a two-dimensional complex space. Its algebra
is the same as the algebra of the SO(3) group, which represents rotations in the three-dimensional
real space.
We should explain what we mean when we say that “the group represents rotations in a space.”
The QM example makes it clear. Consider a finite Hilbert space of, say, a particle with spin S. The
matrices that rotate the direction of the spin are written in terms of exponent of the Si operators.
For a spin-half particle, the Si operators are written in terms of the Pauli matrices. For particles
with spin different from 1/2, the Si operators will be written in terms of different matrices. We learn
227
that the group represents rotations in some space, while the various representations correspond to
different objects that can “live” in that space.
There are three important irreps that have special names. The first one is the trivial – or
singlet – representation that we already mentioned. Its importance stems from the fact that it
corresponds to something that is symmetric under rotations. While that might sound confusing it
is really trivial. Rotation of a singlet does not change its representation. Rotation of a spin half
does change its representation.
The second important irrep is the fundamental representation. This is the smallest non-trivial
irrep. For SU(2), this is the spinor, or spin half, representation. An important property of the
fundamental representation is that it can be used to get all other representations. We return to
this point later. Here we just remind you that this statement is well familiar from QM. One can
get spin-1 by combining two spin-1/2, and you can get spin-3/2 by combining three spin-1/2. Any
Lie group has a fundamental irrep.
The third important irrep is the Adjoint representation. It is made out of the structure constants
themselves. Think of a matrix representation of the generators. Each entry, T cij is labelled by three
indices. One is the c index of the generator itself, that runs from 1 to N , such that N depends
on the group. The other two indices, i and j, are the matrix indices that run from 1 to the
dimension of the representation. One can show that each Lie group has one representation where
the dimension of the representation is the same as the dimension of the group. This representation
is obtained by defining
(Xc)ab ≡ −ifabc. (A.16)
In other words, the structure constants themselves satisfy the algebra of their own group. In
SU(2), the Adjoint representation is that of spin-1. In your homework you will check for yourself
that the εijk are just the set of the three 3× 3 representations of spin 1.
Before closing this section we remarks about subalgebras and simple groups.
Definition: A subalgebra M is a set of generators that are closed under commutation.
Definition: Consider an algebra L with two subalgebra L1 and L2 such that for any X ∈ L1
and Y ∈ L2, [X, Y ] = 0. The algebra L is not simple and it can be written as a direct product:
L = L1 × L2.
Definition: A simple Lie algebra is an algebra that cannot be written as a direct product.
Definition: A semi-simple Lie algebra is an algebra without a U(1) factor.
Since any algebra can be written as a direct product of simple Lie algebras, we can think
about each of the simple algebras separately. A useful example is that of the U(2) group. A U(2)
transformation corresponds to a rotation in two-dimensional complex space. This group is not
semi-simple:
U(2) = SU(2)× U(1). (A.17)
Think, for example, about the rotation of a spinor. It can be separated into two: The trivial
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rotation is just a U(1) transformation, that is, a phase multiplication of the spinor. The non-
trivial rotation is the SU(2) transformation, that is, an internal rotation between the two spin
components.
A.4 Roots and Weights
Here we move to discuss properties of the algebra and the representations. From this point on
we only consider irreps, and thus we do not distinguish anymore between a representation and an
irrep.
Definition: The Cartan subalgebra is the largest subset of generators whose matrix represen-
tations can all be diagonalized at once. Obviously, these generators all commute with each other
and thus they constitute a subalgebra.
Definition: The number of generators in the Cartan subalgebra is called the rank of the algebra.
Let us consider a few examples. Since the U(1) algebra has only a single generator, it is of
rank one. SU(2) is also rank one. You can make one of its three generators, say Sz, diagonal, but
not two of them simultaneously. SU(3) is rank two. We later elaborate on SU(3) in much more
detail. (We have to, because the Standard Model has an SU(3) symmetry.)
Our next step is to introduce the terms roots and weights. We do that via an example. Consider
the SU(2) algebra. It has three generators. We usually choose S3 to be in the Cartan subalgebra,
and we can combine the two other generators, S1 and S2, to a raising and a lowering operator,
S± = S1 ± iS2. Any representation can be defined by the eigenvalues under the operation of the
generators in the Cartan subalgebra, in this case S3. For example, for the spin-1/2 representation,
the eigenvalues are −1/2 and +1/2; For the spin-1 representation, the eiganvalues are −1, 0, and
+1. Under the operation of the raising (S+) and lowering (S−) generators, we “move” from one
eigenstate of S3 to another. For example, for a spin-1 representation, we have S−|1〉 ∝ |0〉.Let us now consider a general Lie group of rank n. Any representation is characterized by the
possible eigenvalues of its eigenstates under the operation of the Cartan subalgebra: |e1, e2..., en〉.Statement: We can assemble all the operators that are not in the Cartan subalgebra into
“lowering” and “raising” operators. That is, when they act on an eigenstate they either move it
to another eigenstate or annihilate it.
Definition: The weight vectors (or simply weights) of a representation are the possible eigen-
values of the generators in the Cartan subalgebra.
Definition: The roots of the algebra are the various ways in which the generators move a state
between the possible weights.
Statement: The weights completely describe the representation.
Statement: The roots completely describe the algebra.
Statement: The weights of the adjoint representations are the roots of the Lie algebra.
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Note that both roots and weights live in an n-dimensional vector space, where n is the rank
of the group. The number of roots is the dimension of the group. The number of weights is the
dimension of the representation.
Let us return to our SU(2) example. The vector space of roots and weights is one-dimensional.
The three roots are −1, 0,+1. The trivial representation has only one weight, zero; The funda-
mental has two, ±1/2; The adjoint has three, 0,±1; and so on. You can also see that the weights
of the adjoint irrep are the roots of the algebra.
A.5 SU(3)
In this section we discuss the SU(3) group. It is more complicated than SU(2) and it allows us to
demonstrate few aspects of Lie groups that cannot be demonstrated with SU(2). Of course, it is
also important since it is relevant to physics.
SU(3) is a generalization of SU(2). It may be useful to think about it as rotations in three-
dimensional complex space. Similar to SU(2), the full symmetry of this rotations is called U(3),
and it can be written as a direct product of simple groups, U(3) = SU(3) × U(1). The SU(3)
algebra has eight generators. (You can see it by recalling that rotation in a complex space is
done by unitary matrices, and any unitary matrix can be written with Hermitian matrix in the
exponent. There are nine independent Hermitian 3 × 3 matrices. They can be separated to a
unit matrix, which corresponds to the U(1) part, and eight traceless matrices, which correspond
to the SU(3) part.) We we go on and study SU(3) without proving that it is related to the above
intuitive picture of rotation in three dimensional complex space.
Similar to the use of the Pauli matrices for the fundamental representation of SU(2), the
fundamental representation of SU(3) is usually written in terms of the Gell-Mann matrices,
Xa = λa/2, (A.18)
with
λ1 =
0 1 0
1 0 0
0 0 0
, λ2 =
0 −i 0
i 0 0
0 0 0
,
λ3 =
1 0 0
0 −1 0
0 0 0
, λ4 =
0 0 1
0 0 0
1 0 0
,
λ5 =
0 0 −i0 0 0
i 0 0
, λ6 =
0 0 0
0 0 1
0 1 0
,
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λ7 =
0 0 0
0 0 −i0 i 0
, λ8 =1√3
1 0 0
0 1 0
0 0 −2
. (A.19)
We would like to emphasize the following points:
1. The Gell-Mann matrices are traceless, as they should.
2. There are three SU(2) subalgebras. One of them is manifest and it is given by λ1, λ2 and
λ3. Can you find the other two?
3. It is manifest that SU(3) is of rank two: λ3 and λ8 are in the Cartan subalgebra.
Having explicit expressions of fundamental representation in our disposal, we can draw the
weight diagram. In order to do so, let us recall how we do it for the fundamental (spinor) rep-
resentation of SU(2). We have two basis vectors (spin-up and spin-down); we apply Sz on them
and obtain the two weights, +1/2 and −1/2. Here we follow the same steps. We take the three
vectors,
(1, 0, 0)T , (0, 1, 0)T , (0, 0, 1)T , (A.20)
and apply to them the two generators in the Cartan subalgebra, X3 and X8. We find the three
weights (+
1
2,+
1
2√
3
),
(−1
2,+
1
2√
3
),
(0,− 1√
3
). (A.21)
We can plot this in a weight diagram in the X3 −X8 plane. Please do it.
Once we have the weights we can get the roots. They are just the combination of generators
that move us between the weights. Clearly, the two roots that are in the Cartan are at the origin.
The other six are those that move us between the three weights. We find that they are(±1
2,±√
3
2
), (±1, 0) . (A.22)
Again, it is a good idea to plot it. This root diagram is also the weight diagram of the Adjoint
representation. In terms of the Gell-Mann matrices, we can see that the raising and lowering
generators are proportional to
I± =1
2(λ1 ± iλ2) V± =
1
2(λ4 ± iλ5) U± =
1
2(λ6 ± iλ7). (A.23)
The names I, U , and V are, at this point, just names. Later on we will see that they are related
to some specific SU(3) symmetry.
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A.6 Classification and Dynkin diagrams
The SU(3) example allows us to obtain more formal results. In the case of SU(2), it is clear what
are the raising and lowering operators. The generalization to groups with higher rank is as follows.
Definition: A positive (negative) root is a root whose first non-zero component is positive
(negative). A raising (lowering) operator correspond to a positive (negative) root.
Definition: A simple root is a positive root that is not the sum of other positive roots.
Statement: Every rank-k algebra has k simple roots. Which ones they are is a matter of
convention, but their relative lengths and angles are fixed.
In fact, it can be shown that the simple roots fully describe the algebra. It can be further
shown that there are only four possible angles and corresponding relative lengths between simple
roots:angle 90 120 135 150
relative length N/A 1 : 1 1 :√
2 1 :√
3.(A.24)
The above rules can be visualized using Dynkin diagrams. Each simple root is described by a
circle. The angle between two roots is described by the number of lines connecting the circles:
i i90 i i120 i y135 i y150
(A.25)
where the solid circle in a link represent the largest root.
There are seven classes of Lie groups. Four classes are infinite and three classes, called the
exceptional groups, have each only a finite number of Lie groups. Below you can find all the sets.
The number of circles is the rank of the group. Note that different names for the infinite groups
are used in the physics and mathematics communities. Below we give both names, but we use only
the physics names from now on.
i i . . . i iiSO(2k) [Dk]
y y . . . y iSO(2k + 1) [Ck]
i i . . . i ySp(2k) [Bk]
i i . . . i iSU(k + 1) [Ak]
(A.26)
i i i i iiE6
232
i i i i i iiE7
i i i i i i iiE8
i i y yF4
i yG2 (A.27)
Consider, for example, SU(3). The two simple roots are equal in length and have an angle of
120 between them. Thus, the Dynkin diagram is just h h.
Dynkin diagrams provide a very good tool to tell us also about what are the subalgebras of a
given algebra. We do not describe the procedure in detail here, and you are encouraged to read
it for yourself in one of the books. One simple point to make is that removing a simple root
always corresponds to a subalgebra. For example, removing simple roots you can see the following
breaking pattern:
E6 → SO(10)→ SU(5)→ SU(3)× SU(2). (A.28)
You may find such a breaking pattern in the context of Grand Unified Theories (GUTs).
Finally, we would like to mention that the algebras of some small groups are identical
SU(2) ' SO(3) ' Sp(2), SU(4) ' SO(6), SO(4) ' SU(2)× SU(2), SO(5) ' Sp(4).
(A.29)
A.7 Naming representations
We are now back to discuss representations. How do we name an irrep? In the context of SU(2),
which is rank one, there are three different ways to do so.
(i) We denote an irrep by its highest weight. For example, spin-0 denotes the singlet repre-
sentation, spin-1/2 refers to the fundamental representation, where the highest weight is 1/2, and
spin-1 refers to the adjoint representation, where the highest weight is 1.
(ii) We can define the irrep according to the dimension of the representation-matrices, which
is also the number of weights. Then the singlet representation is denoted by 1, the fundamental
by 2, and the adjoint by 3.
(iii) We can name the representation by the number of times we can apply S− to the highest
weight without annihilating it. In this notation, the singlet is denoted as (0), the fundamental as
(1), and the adjoint as (2).
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Before we proceed, let us explain in more detail what we mean by “annihilating the state”. Let
us examine the weight diagram. In SU(2), which is rank-one, this is a one dimensional diagram.
For example, for the fundamental representation, it has two entries, at +1/2 and −1/2. We now
take the highest weight (in our example, +1/2), and move away from it by applying the root that
corresponds to the lowering operator, −1. When we apply it once, we move to the lowest weight,
−1/2. When we apply it once more, we move out of the weight diagram, and thus “annihilate the
state”. Thus, for the spin-1/2 representation, we can apply the root corresponding to S− once to
the highest weight before moving out of the weight diagram, and — in the naming scheme (iii) —
we call the representation (1).
We are now ready to generalize this to general Lie algebras. Either of the methods (ii) and
(iii) are used. Method (ii) is straightforward, but somewhat problematic as there could be several
different representations with the same dimension. We give an example of such a situation later.
In scheme (iii), the notation used for a specific state is unambiguous. To use it, we must order
the simple roots in a well-defined (even if arbitrary) order. Then we have a unique highest weight.
We denote a representation of a rank-k algebra as a k-tuple, such that the first entry is the maximal
number of times that we can apply the first simple root on the highest weight before the state is
annihilated, the second entry refers to the maximal number of times that we can apply the second
simple root on the highest weight before annihilation, and so on. Take again SU(3) as an example.
We order the Cartan subalgebra as X3, X8 and the two simple roots as
S1 =
(+
1
2,+
√3
2
), S2 =
(+
1
2,−√
3
2
). (A.30)
Consider the fundamental representation where we chose the highest weight to be(1/2, 1/(2
√3)).
Subtracting S1 twice or subtracting S2 once from the highest weight would annihilate it. Thus
the fundamental representation is denoted by (1, 0). You can work out the case of the adjoint
representation and find that it should be denoted as (1, 1). In fact, it can be shown that any pair
of non-negative integers forms a different irrep. (For SU(2) with the naming scheme (iii), any
non-negative integer defines a different irrep.)
From now on we limit our discussion mostly to SU(N).
Statement: For any group the singlet irrep is (0, 0, ..., 0).
Statement: For any SU(N) algebra, the fundamental representation is (1, 0, 0, ..., 0).
Statement: For any SU(N ≥ 3) algebra, the adjoint representation is (1, 0, 0, ..., 1).
Definition: For any SU(N), the conjugate representation is the one where the order of the
k-tuple is reversed.
For example, (0, 1) is the conjugate of the fundamental representation, which is usually called
the anti-fundamental representation. An irrep and its conjugate have the same dimension. In
the naming scheme (ii), they are called m and m. Note that some representations are self–
conjugate, e.g., the adjoint representation, such representation are also called real representations.
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Representations that are not self conjugate are also called complex representations. Also note that
all irreps of SU(2) are real.
We now return to the notion that the groups that we are dealing with are transformation
groups of physical states. These physical states are often just particles. For example, when we
talk about the SU(2) group that is related to the spin transformations, the physical system that is
being transformed is often that of a single particle with well-defined spin. In this context we often
abuse the language by saying that the particle is, for example, in the spin-1/2 representation of
SU(2). What we mean is that, as a state in the Hilbert space, it transforms by the spin operator
in the 1/2 representation of SU(2). Similarly, when we say that the proton and the neutron form
a doublet of isospin-SU(2), we mean that we represent p by the vector-state (1, 0)T and n by
the vector-state (0, 1)T , so that the appropriate representation of the isospin generators is by the
2× 2 Pauli matrices. In other words, we loosely speak on “particles in a representation” when we
mean “the representation of the group generators acting on the vector states that describe these
particles.”
How many particles there are in a given irrep? Here, again, we consider only SU(N) and state
the results.
• Consider an (α) representation of SU(2). It has
N = α + 1, (A.31)
particles. The singlet (0), fundamental (1) and adjoint (2) representations have, respectively,
1, 2, and 3 particles.
• Consider an (α, β) representation of SU(3). It has
N = (α + 1)(β + 1)α + β + 2
2(A.32)
particles. The singlet (0, 0), fundamental (1, 0) and adjoint (1, 1) representations have, re-
spectively, 1, 3, and 8 particles.
• Consider an (α, β, γ) representation of SU(4). It has
N = (α + 1)(β + 1)(γ + 1)α + β + 2
2
β + γ + 2
2
α + β + γ + 3
3(A.33)
particles. The singlet (0, 0, 0), fundamental (1, 0, 0) and adjoint (1, 0, 1) representations have,
respectively, 1, 4, and 15 particles. Note that there is no α+γ+2 factor. Only a consecutive
sequence of the label integers appears in any factor.
• The generalization to any SU(N) is straightforward. It is easy to see that the fundamental
of SU(N) has N particles and the adjoint has N2 − 1 particles.
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In SU(2), the number of particles in a representation is unique. In a general Lie group, however,
the case may be different. Yet, it is often used to identify irreps. For example, in SU(3) we usually
call the fundamental 3, and the adjoint 8. For the anti-fundamental we use 3. In cases where there
are several irreps with the same number of particles we often use a prime to distinguish them. For
example, both (4, 0) and (2, 1) contain 15 particles. We denote them by 15 and 15′ respectively.
Last, we remark on how the count goes in terms of subgroup irreps. For example, any SU(3)
irreps can be wirtten in terms of its SU(2) subgroup ireps. Two usefull decompositions are 3 = 2+1
and 8 = 3 + 2 + 2 + 1, where the irreps on the left are that of SU(3) and on the right of SU(2).
You can decude them from the weight diagrams of the SU(3) irreps simply by moving only in one
of the SU(2) subgorup direction on the diagrams, for exmaple, only moving in the X3 direction.
While we do not elaborate further here, we only mention that there is a way to decompose any
irrep of a bigger group as a sum irreps of its subgroups.
A.8 Combining representations
When we study spin, we learn how to combine SU(2) representations. The canonical example is
to combine two spin-1/2 to generate a singlet (spin-0) and a triplet (spin-1). We often write it
as 1/2 × 1/2 = 0 + 1. There is a similar method to combine representations for any Lie group.
The basic idea is, just like in SU(2), that we need to find all the possible ways to combine the
indices and then assign it to the various irreps. That way we know what irreps are in the product
representation and the corresponding Clebsch-Gordan (GB) coefficients.
Here, however, we do not explain how to construct the product representation. The reason
is that often all we want to know is what irreps appear in the product representation, without
the need to get all the CG coefficients. (In particular, many times all we care about is how to
generate the singlet.) There is a simple way to do just this for a general SU(N). This method
is called Young Tableaux, or Young Diagrams. The details of the method are well explained in
several places, for example, in the PDG. In the homework you are asked to learn how to use it.
Here we give few examples for combining irreps in SU(3) that we will use when we discuss the
SM. Using naming scheme (ii) we have
3× 3 = 1 + 8, 3× 3 = 3 + 6, 3× 6 = 10 + 8. (A.34)
From this we can conclude that
3× 3× 3 = 10 + 8 + 8 + 1. (A.35)
Note that the number of particles on both sides are equal, as they should. Of particular interest
to us is that 3× 3 and 3× 3× 3 contain the singlet irrep.
We we combine identical irreps the result have well defined symmetry properties under exchange
of the two irreps, that is, they are even or odd under an exchange. For example, when combining
236
two spin halves, the singlet is odd while the triplet is even under the exchange. For SU(2) the rule
is simple, the highest irrep is symmetric and then going down it alternate. For other groups it is
more complicated and we do not discuss it here. When it is important we may add a subscript
to denote the symmetry property. For example, in S(2) we may write 2 × 2 = 1a + 3s and
3× 3 = 5s + 3a + 1s. In the example above for SU(3) we may write
3× 3 = 3a + 6s, 3× 3× 3 = 10s + 8m + 8m + 1a, (A.36)
where 8m refer to a mixed symmetry.
The symmetry properties are important when the two irreps that we are combining are identical.
For example, the antisymmetry combination of two vectors in real three dimensional space is the
cross product. Being antisymmetry we know it identically vanishes for two identical vectors,
~a×~a = 0. This result is a general one, and applies to all fully antisymmetric combinations in any
group.
Combining representations plays a very important role in physics. In particular, we would like
to find a combination of representations that combined into the singlets one, as such a combination
is invariant under rotation in the relevant space.
237
Homework
Question A.1: S3
In this question we study the group S3. It is the smallest finite non-Abelian group. You can
think about it as all possible permutation of three elements. The group has 6 elements. Thinking
about the permutations we see that we get the following representation of the group:
() =
1 0 0
0 1 0
0 0 1
(12) =
0 1 0
1 0 0
0 0 1
(13) =
0 0 1
0 1 0
1 0 0
(23) =
1 0 0
0 0 1
0 1 0
(123) =
0 1 0
0 0 1
1 0 0
(321) =
0 0 1
1 0 0
0 1 0
(A.37)
The names are instructive. For example, (12) represents exchanging the first and second elements.
(123) and (321) are cyclic permutation to the right or left.
1. Write explicitly the 6× 6 multiplication table for the group.
2. Show that the group is non-Abelian. Hint, it is enough to find one example.
3. Z3 is a sub group of S3. Find three generators that correspond to a Z3.
4. In class we mentioned the following theorem for finite groups
∑Ri
[dim(Ri)]2 = N, (A.38)
where N is the number of elements in the group and Ri are all the irreps. Based on this,
proof that the representation in Eq. (A.37) is reducible. Then, write it explicitly in a (1 + 2)
block diagonal representation. (Hint: find a vector which is an eigenvector of all the above
matrices.)
238
5. In the last item you found a two dimensional and a one dimensional representations of S3.
Based on (A.38) you know that there is only one more representation and that it is one
dimensional. Find it.
Question A.2: Lie algebras
Consider two general elements of a Lie groups,
A ≡ exp(iλXa), B ≡ exp(iλXb). (A.39)
where Xi is a generator. We think about λ as a small parameter. Then, consider a third element
C = BAB−1A−1 ≡ exp(iβcXc). (A.40)
Expand C in powers of λ and show that at lowest order you get the Lie algebra
[Xa, Xb] = ifabcXc, fabc ≡βcλ2. (A.41)
Question A.3: SU(3)
1. The three Gell–Mann matrices, aλ1, aλ2 and aλ3 satisfy an SU(2) algebra, where a is a
constant. What is a?
2. Does this fact mean that SU(3) is not a simple Lie group?
3. Draw the root diagram of SU(3).
4. There are two other independent combinations of Gell–Mann matrices that satisfy SU(2)
algebras. What are they? Hint: Look at the root diagram.
Question A.4: Dynkin diagrams
1. Draw the Dynkin diagram of SO(10).
2. What is the rank of SO(10)?
3. How many generators there are for SO(10)? (We did not proof a general formula for the
number of generators for SO(N). It should be simple for you to find such a formula using
your understanding of rotations in real N -dimensional spaces.)
4. Based on the Dynkin diagram show that SO(10) has the following subalgebras
SO(8), SU(5), SU(4)× SU(2), SU(3)× SU(2)× SU(2). (A.42)
In each case show which simple root you can remove from the SO(10) Dynkin diagram.
239
Question A.5: Representations
Here we practice finding the number of degrees of freedom in a given irrep.
1. In SU(5), how many particles there are in the (1, 1, 0, 0) irrep?
2. In SU(3) how many particles there are in the following irreps
(3, 0), (2, 2). (A.43)
3. Consider the (3, 0) irrep of SU(3). Draw its weight diagram and from it decompose it into
its SU(2) irreps.
Question A.6: Combining irreps
Here we are going to practice the use of Young Tableaux. The details of the method can be
found in the PDG (there is a link in the website of the course). Study the algorithm and do the
following calculations. Make sure you check that the number of particles on both sides is the same.
Write your answer both in the k-tuple notation and the number notation. For example, in SU(3)
you should write
(1, 0)× (0, 1) = (0, 0) + (1, 1), 3× 3 = 1 + 8. (A.44)
1. In SU(3) calculate
3× 3, 3× 8, 10× 8. (A.45)
2. Given that the quarks are SU(3)C triplets, 3, the anti-quarks are 3 and the gluons are color
octets, 8, which of the following could be an observable bound state?
qq, qq, qg, gg, qqg, qqq. (A.46)
Note that an observable bound state must be a color singlet.
3. Find what is 5 and 10 in SU(5) in a k-tuple notation.
4. Calculate 10× 10 in SU(5).
240
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