This book contains the material of a set of Lectures on The Standard Model and Beyond,which were delivered to the first year graduate students of the Department of Physics of Nanjing University, during the Fall of 2012.

0.1 PREFACE i

1 Introduction to Particle Physics 1

1.1 The natural System of Units 1

1.2 The Planck Natural system of units 4

1.3 Invariants in kinematics 5

1.4 Cross sections and Luminosities 7

      1.4.1 Luminosity for fixed target experiments 7

      1.4.2 Lmninosity for colliding beam experiments 8

1.5 Kinematics of particle decay 10

1.6 Invariants in the scattering of two particles 11

1.7 Transformation properties of non invariant quantities 14

2 Elements of Group Theory 17

2.1 Definitions 17

2.2 Matrix groups 18

      2.2.1 The exponential of a matrix 20

      2.2.2 Determination of the independent parameters 23

2.3 The structure constants 26

2.4 Representations 27

      2.4.1 The fundamental representation 27

      2.4.2 The adjoined representation 30

2.5 Homomorphism-isomorphism 32

2.6 Some further examples 33

2.7 The proper orthogonal groups O（3） and O（4） 35

2.8 Symmetries and conservation laws-Noether's theorem 39

3 The Dirac Theory 45

3.1 Preliminaries-The Klein Gordon equation 45

3.2 The Dirac equation 47

      3.2.1 The spinors u and v 49

      3.2.2 Projection operators 52

      3.2.3 Various representations of the Dirac Matrices 52

3.3 Interpretation of negative energy solutions 53

3.4 The notion of helicity 54

3.5 Charge conjugation for 4-spinors 57

3.6 Types of currents 58

4 The standard Model-Symmetry and particle content 61

4.1 The essential ingredients 61

4.2 The notion of a local symmetry 61

4.3 Non Abelian gauge transformation 63

      4.3.1 Transformations associated with a non Abelian group 64

      4.3.2 Some Examples 65

4.4 Gauge invariant Lagrangians 67

      4.4.1 Only gauge fields 67

      4.4.2 Lagrangian in the presence of scalar fields 69

      4.4.3 Lagrangian in the presence of fermion fields 70

4.5 The particle content of the SM 70

4.6 The electroweak Lagrangian 71

5 The Higgs Mechanism 73

5.1 The Higgs Mechanism in global gauge transformations 73

5.2 The Higgs Mechanism in gauge theories 79

5.3 Fundamental theorem of Higgs Mechanism 87

6 The Group SU（3）-Quantum Chromodynamics 89

6.1 Quantum chromodynamics QCD 89

6.2 The one gluon exchange potential 92

      6.2.1 Process involving only baryons 92

      6.2.2 processes involving the creation of a qq pair 93

      6.2.3 processes involving interaction of mesons 93

6.3 Low energy formalism 94

      6.3.1 The orbital part at the quark level 95

      6.3.2 The kinetic energy part 95

      6.3.3 The confining potential 95

      6.3.4 Fitting the strength of the confining potential 96

6.4 Matrix elements involving two quarks 97

      6.4.1 The confining potential 97

      6.4.2 The one gluon exchange potential 97

7 Fermion Masses and Currents 99

7.1 Fermion masses 99

7.2 The currents 103

      7.2.1 The EM current 105

      7.2.2 The charged currents 105

      7.2.3 The neutral currents 106

7.3 The contact interaction 108

7.4 Determination of the Standard Model Parameters 109

7.5 Summary 110

8 Rates and Cross Sections in Electroweak Theory 111

8.1 Brief review and formulae 111

8.2 Decay widths and cross section 112

      8.2.1 The decay of a vectcr boson 113

8.3 Muon-antimuon production in electron- positron colliders 117

8.4 neutrino electron scatterir 120

9 Supersymmetry for Pedestrians 125

9.1 Introduction 125

9.2 The particle content of MSSM 126

      9.2.1 gauge particles 126

      9.2.2 Fermions and s-fermions 127

      9.2.3 The Higgs content 127

9.3 The Higgs mechanism 127

9.4 The Fermion masses 130

9.5 Supersymmetry breaking 132

9.6 Some remarks about the particle spectrum 132

10 SU（5）-An Example of Grand Unification 137

10.1 Mathematical Introduction 137

10.2 The structure of the GUT SU（5） 137

10.3 The particle content 139

      10.3.1 The Fermions 139

      10.3.2 The gauge bosons 140

      10.3.3 The Higgs content 141

10.4 The Higgs Mechanism 141

10.5 The gauge boson masses 144

10.6 Gauge couplings 144

10.7 Baryon Asymmetry 146

11 A Brief Introduction to Cosmology 151

11.1 Cosmological Principles 151

11.2 The eXpanding Universe-Tie big bang scenario 151

      11.2.1 The receding of galaxies 152

      11.2.2 The background microwave radiation 155

      11.2.3 The abundance of primordial 4/2 He and other light elements in the Universe 156

11.3 Evolution of the Universe 157

11.4 The Cosmological Constant - Dark Energy 161

11.5 The standard cosmological model 164

11.6 The proper length and the horizon 165

      11.6.1 The proper length entering the prototype candles 166

      11.6.2 The horizon 167

11.7 temperature 168

      11.7.1 The relic neutrinos 170

11.8 The role of dark matter 174

      11.8.1 The rotational velocities 174

      11.8.2 Gravitational lensing 178

      11.8.3 The observation of bullet cluster 178

11.9 Dark energy 181

      11.9.1 Observations with standard candles 182

      11.9.2 The microwave background radiation -The earliest picture of the Universe 184

11.10 The WMAP and PLANCK Observations 188

12 Neutrino oscillations 193

12.1 Introduction 193

12.2 The formalism 194

12.3 Neutrino oscillation experiments 196

12.4 The absolute scale of the neutrino mass 203

12.5 Neutrinos as probes 209

13 C.P.T and all that 211

13.1 Space inversion P 211

      13.1.1 Parity conservation-parity violation 214

13.2 The helicity of the neutrinos 217

      13.2.1 The determination of the helicity of neutrinos 219

      13.2.2 Weak interaction and handedness 221

13.3 Charge Conjugation 221

      13.3.1 Charge conjugation for 4-spinors 223

      13.3.2 Transformation of currents 224

      13.3.3 Symmetry relation of the Dirac mass term 224

      13.3.4 Symmetry relation of the Majorana mass term 225

      13.3.5 The charge conjugation of composite systems 225

      13.3.6 How good a symmetry is the charge conjugation? 226

13.4 Time reversal 227

      13.4.1 Time reversal in classical mechanics 227

      13.4.2 Time reversal in quantum mechanics 229

      13.4.3 Premature evidence for T violation 231

13.5 CP the combined action of P and C symmetries 231

      13.5.1 The neutral Kaon system 233

      13.5.2 Strangeness oscillations 236

      13.5.3 Kaon regeneration 236

      13.5.4 CP Violation 239

      13.5.5 Evidence for T-violation 243

13.6 The combined symmetry CPT 244

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