An Introductory Course of Particle Physics

806 Pages·2014·5.822 MB·English

by  Pal, Palash B

Physics
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For graduate students unfamiliar with particle physics, An Introduc-
A A
tory Course of Particle Physics teaches the basic techniques and
n A I c
fundamental theories related to the subject. It gives students the R n ntroductory ourse of
I
competence to work out various properties of fundamental particles,
n
T PARTICLE
such as scattering cross-section and lifetime. The book also gives a t
r
lucid summary of the main ideas involved. I
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In giving students a taste of fundamental interactions among ele-
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mentary particles, the author does not assume any prior knowledge c PHYSICS
E
of quantum field theory. He presents a brief introduction that sup- t
o
plies students with the necessary tools without seriously getting into
P r
the nitty-gritty of quantum field theory, and then explores advanced y
topics in detail. The book then discusses group theory, and in this H c
case the author assumes that students are familiar with the basic
Y o P B. P
definitions and properties of a group, and even SU(2) and its repre- u AlAsh Al
sentations. With this foundation established, he goes on to discuss S r
s
representations of continuous groups bigger than SU(2) in detail. I e
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o
The material is presented at a level that MSc and PhD students can
f
understand, with exercises throughout the text at points at which S
performing the exercises would be most beneficial. Anyone teaching
a one-semester course will probably have to choose from the topics
covered, because this text also contains advanced material that
might not be covered within a semester due to lack of time. Thus it
provides a teaching tool with the flexibility to customize the course
to suit your needs.
PP
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K22084
K22084_cover.indd 1
A I c
n ntroductory ourse of
PARTICLE
PHYSICS
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
A I c
n ntroductory ourse of
PARTICLE
PHYSICS
P B. P
AlAsh Al
sAhA InstItute of nucleAr PhysIcs
KolKAtA, IndIA
CRC Press
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To
Lincoln Wolfenstein
Whose inspiration has guided me
not only through my PhD thesis
but throughout my life
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
List of Figures xvii
List of Tables xxi
Preface xxiii
Notations xxvii
1 Scope of particle physics 1
1.1 Whatareelementaryparticles? . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Inventory ofelementaryfermions . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Whichproperties? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Fundamental interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Highenergyphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Relativityandquantum theory . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Naturalunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Planofthebook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Relativistickinematics 16
2.1 Lorentztransformationequations . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Vectors andtensorsonspacetime . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Velocity,momentum andenergy . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Invariances andconservation laws . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Kinematicsofdecays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6.1 Lifetimeandtimedilation. . . . . . . . . . . . . . . . . . . . . . . 25
2.6.2 Two-bodydecays. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.3 Three-bodydecays . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Kinematicsofscatteringprocesses . . . . . . . . . . . . . . . . . . . . . . . 31
2.7.1 Center-of-massframe . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7.2 Fixed-targetframe . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Symmetries and groups 35
3.1 Theroleofsymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Grouptheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Examplesandclassification. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Parametersandgenerators . . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5.1 Matrixanddifferentialrepresentations . . . . . . . . . . . . . . . . 45
3.5.2 Irreduciblerepresentations . . . . . . . . . . . . . . . . . . . . . . 48
vii
viii Contents
3.5.3 Kronecker productofrepresentations . . . . . . . . . . . . . . . . 49
3.5.4 Decompositionunderasubgroup . . . . . . . . . . . . . . . . . . . 51
3.6 Lorentzgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.1 Generators andalgebra . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6.3 Extended Lorentzgroupanditsrepresentations . . . . . . . . . . 56
3.7 Poincar´egroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 A brisk tour of quantum field theory 61
4.1 Motivatingquantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Planewavesolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Scalarfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Diracfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 MakingLorentzinvariantswithfields . . . . . . . . . . . . . . . . . . . . . 75
4.4.1 Invariants withscalarandvector fields. . . . . . . . . . . . . . . . 75
4.4.2 Invariants involvingDiracfields . . . . . . . . . . . . . . . . . . . 75
4.4.3 Finalrecipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Lagrangians forfreefields . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Noether currentsandcharges . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Quantum fieldsasoperators . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.9 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.10 FromLagrangiantoFeynmanrules . . . . . . . . . . . . . . . . . . . . . . 90
4.10.1 Externallines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10.2 Internal lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.10.3 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.10.4 Otherfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.10.5 Feynmanamplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.11 Calculationofdecayrates . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.11.1 Generalformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.11.2 Two-bodydecays. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.11.3 Three-bodydecays . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.12 Calculationofcross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.12.1 Generalformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.12.2 Illustrativeexample . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.12.3 Mandelstamvariables . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.13 Differentialdecayratesandcross-sections. . . . . . . . . . . . . . . . . . . 105
4.13.1 AngulardistributionintheCMframe . . . . . . . . . . . . . . . . 105
4.13.2 Invariantformofangulardistribution . . . . . . . . . . . . . . . . 107
4.13.3 AngulardistributioninFTframe . . . . . . . . . . . . . . . . . . 108
4.13.4 Otherdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.14 Feynmandiagramsthatdonotrepresentphysicalamplitudes . . . . . . . 110
5 Quantum electrodynamics 112
5.1 Gaugeinvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.1 Globalphasesymmetry . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.2 Localsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.3 Chargeconservation . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Interaction vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Elasticscatteringatsecondorder . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.1 Electron–electron scattering. . . . . . . . . . . . . . . . . . . . . . 119
5.3.2 Electron–positronscattering . . . . . . . . . . . . . . . . . . . . . 124
5.3.3 Comptonscattering . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 Inelasticscatteringatsecondorder . . . . . . . . . . . . . . . . . . . . . . 130
5.4.1 Paircreationandpairannihilation . . . . . . . . . . . . . . . . . . 130
Contents ix
5.4.2 Muonpairproduction . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5 ScalarQED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.6 Multi-photonstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.6.2 Two-photonstates . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.6.3 Three-photonstates . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7 Higher-ordereffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7.1 Electromagnetic formfactors . . . . . . . . . . . . . . . . . . . . . 138
5.7.2 Formfactorsinnon-relativisticlimit . . . . . . . . . . . . . . . . . 141
5.7.3 Vertexfunctionatone-loop . . . . . . . . . . . . . . . . . . . . . . 145
5.7.4 Anomalousmagneticmoment. . . . . . . . . . . . . . . . . . . . . 148
6 Parity and charge conjugation 150
6.1 Discretesymmetriesinclassicalelectrodynamics . . . . . . . . . . . . . . . 150
6.2 Paritytransformationoffields . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2.1 ParityinvarianceofaLagrangian. . . . . . . . . . . . . . . . . . . 151
6.2.2 Freescalarfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.3 Freephotonfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2.4 Freefermionfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.5 Interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Chargeconjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3.1 Natureofthetransformation . . . . . . . . . . . . . . . . . . . . . 158
6.3.2 Freebosonicfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3.3 Freefermionfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3.4 Interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.4 Paritypropertiesofparticlestates . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.1 Intrinsicparityforbosons . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.2 Intrinsicparityforfermionsandantifermions . . . . . . . . . . . . 165
6.4.3 Orbitalparity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.5 Chargeconjugation propertiesofparticlestates . . . . . . . . . . . . . . . 169
6.6 Multi-photonstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.7 Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.8 Parityassignmentofdifferentparticles . . . . . . . . . . . . . . . . . . . . 175
6.9 Signatureofparityviolation . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.9.1 Correlationsinexperiments . . . . . . . . . . . . . . . . . . . . . . 177
6.9.2 Parityviolatingtransitions . . . . . . . . . . . . . . . . . . . . . . 180
6.9.3 Parityviolatingcouplingwithexternal fields . . . . . . . . . . . . 181
6.9.4 Connection withfieldtheory . . . . . . . . . . . . . . . . . . . . . 182
6.10 Consequences ofchargeconjugationsymmetry . . . . . . . . . . . . . . . . 184
6.11 CPsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 Time-reversal and CPT symmetries 188
7.1 Anti-unitaryoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.1.2 Rulesforworkingwithoperators . . . . . . . . . . . . . . . . . . . 189
7.2 Timereversaltransformationonfields . . . . . . . . . . . . . . . . . . . . 191
7.2.1 Freefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.3 CPTtransformationonfields . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.4 CPTtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.5 Consequences ofCPTsymmetry . . . . . . . . . . . . . . . . . . . . . . . . 196
7.6 Timereversaltransformationonstates . . . . . . . . . . . . . . . . . . . . 198
7.7 Signatureoftimereversalviolation . . . . . . . . . . . . . . . . . . . . . . 199
8 Isospin 201
8.1 Nuclearenergylevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.2 Isospinsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202