an introductory course of particle physics - palash b. pal

For graduate students unfamiliar with particle physics, An Introduc-tory Course of Particle Physics teaches the basic techniques and fundamental theories related to the subject. It gives students the competence to work out various properties of fundamental particles, such as scattering cross-section and lifetime. The book also gives a lucid summary of the main ideas involved.

In giving students a taste of fundamental interactions among ele-mentary particles, the author does not assume any prior knowledge of quantum field theory. He presents a brief introduction that sup-plies students with the necessary tools without seriously getting into the nitty-gritty of quantum field theory, and then explores advanced topics in detail. The book then discusses group theory, and in this case the author assumes that students are familiar with the basic definitions and properties of a group, and even SU(2) and its repre-sentations. With this foundation established, he goes on to discuss representations of continuous groups bigger than SU(2) in detail.

The material is presented at a level that MSc and PhD students can understand, with exercises throughout the text at points at which 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.

K22084

An Introductory course of

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PARTICLE PHYSICS

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PARTICLE PHYSICS

PAlAsh B. PAlsAhA InstItute of nucleAr PhysIcs

KolKAtA, IndIA

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

2015 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

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To

Lincoln Wolfenstein

Whose inspiration has guided menot only through my PhD thesis

but throughout my life

Contents

List of Figures xvii

List of Tables xxi

Preface xxiii

Notations xxvii

1 Scope of particle physics 11.1 What are elementary particles? . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Inventory of elementary fermions . . . . . . . . . . . . . . . . . . . . . . . 21.3 Which properties? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Fundamental interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 High energy physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Relativity and quantum theory . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Natural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Plan of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Relativistic kinematics 162.1 Lorentz transformation equations . . . . . . . . . . . . . . . . . . . . . . . 162.2 Vectors and tensors on spacetime . . . . . . . . . . . . . . . . . . . . . . . 192.3 Velocity, momentum and energy . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Invariances and conservation laws . . . . . . . . . . . . . . . . . . . . . . . 242.6 Kinematics of decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.1 Lifetime and time dilation . . . . . . . . . . . . . . . . . . . . . . . 252.6.2 Two-body decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.3 Three-body decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Kinematics of scattering processes . . . . . . . . . . . . . . . . . . . . . . . 312.7.1 Center-of-mass frame . . . . . . . . . . . . . . . . . . . . . . . . . 312.7.2 Fixed-target frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Symmetries and groups 353.1 The role of symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Examples and classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.1 Parameters and generators . . . . . . . . . . . . . . . . . . . . . . 423.4.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5.1 Matrix and differential representations . . . . . . . . . . . . . . . . 453.5.2 Irreducible representations . . . . . . . . . . . . . . . . . . . . . . 48

vii

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3.5.3 Kronecker product of representations . . . . . . . . . . . . . . . . 493.5.4 Decomposition under a subgroup . . . . . . . . . . . . . . . . . . . 51

3.6 Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.6.1 Generators and algebra . . . . . . . . . . . . . . . . . . . . . . . . 523.6.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.6.3 Extended Lorentz group and its representations . . . . . . . . . . 56

3.7 Poincare group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 A brisk tour of quantum field theory 614.1 Motivating quantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Plane wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.1 Scalar fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.3 Dirac fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4 Making Lorentz invariants with fields . . . . . . . . . . . . . . . . . . . . . 75

4.4.1 Invariants with scalar and vector fields . . . . . . . . . . . . . . . . 754.4.2 Invariants involving Dirac fields . . . . . . . . . . . . . . . . . . . 754.4.3 Final recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Lagrangians for free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.6 Noether currents and charges . . . . . . . . . . . . . . . . . . . . . . . . . 824.7 Quantum fields as operators . . . . . . . . . . . . . . . . . . . . . . . . . . 844.8 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.9 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.10 From Lagrangian to Feynman rules . . . . . . . . . . . . . . . . . . . . . . 90

4.10.1 External lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.10.2 Internal lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.10.3 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.10.4 Other factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.10.5 Feynman amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.11 Calculation of decay rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.11.1 General formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.11.2 Two-body decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.11.3 Three-body decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.12 Calculation of cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . 994.12.1 General formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.12.2 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.12.3 Mandelstam variables . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.13 Differential decay rates and cross-sections . . . . . . . . . . . . . . . . . . . 1054.13.1 Angular distribution in the CM frame . . . . . . . . . . . . . . . . 1054.13.2 Invariant form of angular distribution . . . . . . . . . . . . . . . . 1074.13.3 Angular distribution in FT frame . . . . . . . . . . . . . . . . . . 1084.13.4 Other differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.14 Feynman diagrams that do not represent physical amplitudes . . . . . . . 110

5 Quantum electrodynamics 1125.1 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.1.1 Global phase symmetry . . . . . . . . . . . . . . . . . . . . . . . . 1125.1.2 Local symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1.3 Charge conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Interaction vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3 Elastic scattering at second order . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.1 Electronelectron scattering . . . . . . . . . . . . . . . . . . . . . . 1195.3.2 Electronpositron scattering . . . . . . . . . . . . . . . . . . . . . 1245.3.3 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.4 Inelastic scattering at second order . . . . . . . . . . . . . . . . . . . . . . 1305.4.1 Pair creation and pair annihilation . . . . . . . . . . . . . . . . . . 130

Contents ix

5.4.2 Muon pair production . . . . . . . . . . . . . . . . . . . . . . . . . 1315.5 Scalar QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.6 Multi-photon states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.6.2 Two-photon states . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.6.3 Three-photon states . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.7 Higher-order effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.7.1 Electromagnetic form factors . . . . . . . . . . . . . . . . . . . . . 1385.7.2 Form factors in non-relativistic limit . . . . . . . . . . . . . . . . . 1415.7.3 Vertex function at one-loop . . . . . . . . . . . . . . . . . . . . . . 1455.7.4 Anomalous magnetic moment . . . . . . . . . . . . . . . . . . . . . 148

6 Parity and charge conjugation 1506.1 Discrete symmetries in classical electrodynamics . . . . . . . . . . . . . . . 1506.2 Parity transformation of fields . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.2.1 Parity invariance of a Lagrangian . . . . . . . . . . . . . . . . . . . 1516.2.2 Free scalar fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.2.3 Free photon field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.2.4 Free fermion fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.2.5 Interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.3.1 Nature of the transformation . . . . . . . . . . . . . . . . . . . . . 1586.3.2 Free bosonic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.3.3 Free fermion fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.3.4 Interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.4 Parity properties of particle states . . . . . . . . . . . . . . . . . . . . . . . 1646.4.1 Intrinsic parity for bosons . . . . . . . . . . . . . . . . . . . . . . . 1646.4.2 Intrinsic parity for fermions and antifermions . . . . . . . . . . . . 1656.4.3 Orbital parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.5 Charge conjugation properties of particle states . . . . . . . . . . . . . . . 1696.6 Multi-photon states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.7 Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.8 Parity assignment of different particles . . . . . . . . . . . . . . . . . . . . 1756.9 Signature of parity violation . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.9.1 Correlations in experiments . . . . . . . . . . . . . . . . . . . . . . 1776.9.2 Parity violating transitions . . . . . . . . . . . . . . . . . . . . . . 1806.9.3 Parity violating coupling with external fields . . . . . . . . . . . . 1816.9.4 Connection with field theory . . . . . . . . . . . . . . . . . . . . . 182

6.10 Consequences of charge conjugation symmetry . . . . . . . . . . . . . . . . 1846.11 CP symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7 Time-reversal and CPT symmetries 1887.1 Anti-unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.1.2 Rules for working with operators . . . . . . . . . . . . . . . . . . . 189

7.2 Time reversal transformation on fields . . . . . . . . . . . . . . . . . . . . 1917.2.1 Free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917.2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.3 CPT transformation on fields . . . . . . . . . . . . . . . . . . . . . . . . . 1947.4 CPT theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.5 Consequences of CPT symmetry . . . . . . . . . . . . . . . . . . . . . . . . 1967.6 Time reversal transformation on states . . . . . . . . . . . . . . . . . . . . 1987.7 Signature of time reversal violation . . . . . . . . . . . . . . . . . . . . . . 199

8 Isospin 2018.1 Nuclear energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.2 Isospin symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

x Contents

8.2.1 Group structure of isospin transformations . . . . . . . . . . . . . 2028.2.2 Isospin representations . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.3 Pions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.4 Isospin relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.4.1 Forbidden processes . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.4.2 Relative strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 2088.4.3 Smushkevichs method . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.5 G-parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118.6 Generalized Pauli principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 2148.7 Isospin and quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.7.1 Nucleons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2198.7.2 Pions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.7.3 More hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.8 Pionnucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.9 Isospin breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248.10 Baryon number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

9 Discovering particles 2289.1 Discoveries of electron, proton and neutron . . . . . . . . . . . . . . . . . . 2289.2 New particles in cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.3 Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2319.4 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.5 Hadronic zoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2389.6 Detecting short-lived particles . . . . . . . . . . . . . . . . . . . . . . . . . 241

9.6.1 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2429.6.2 Reconstruction of events . . . . . . . . . . . . . . . . . . . . . . . . 245

9.7 Discovering leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2469.8 Overview of particle physics experiments . . . . . . . . . . . . . . . . . . . 249

9.8.1 Non-accelerator experiments . . . . . . . . . . . . . . . . . . . . . 2499.8.2 Accelerator experiments . . . . . . . . . . . . . . . . . . . . . . . . 250

10 SU(3) quark model 25410.1 Strange quark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25410.2 Hypercharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10.2.1 Hypercharge and charge of hadrons . . . . . . . . . . . . . . . . . 25510.2.2 Hypercharge and isospin . . . . . . . . . . . . . . . . . . . . . . . . 256

10.3 SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.3.1 Some general properties of SU(N) groups . . . . . . . . . . . . . . 25810.3.2 Fundamental representation . . . . . . . . . . . . . . . . . . . . . . 25810.3.3 Conjugate representation . . . . . . . . . . . . . . . . . . . . . . . 26010.3.4 Examples of other representations . . . . . . . . . . . . . . . . . . 26110.3.5 Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26310.3.6 Decompositions in a subgroup . . . . . . . . . . . . . . . . . . . . 267

10.4 Mesons from three flavors of quarks . . . . . . . . . . . . . . . . . . . . . . 27010.5 Baryons from three flavors of quarks . . . . . . . . . . . . . . . . . . . . . 27310.6 U -spin and V -spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27510.7 SU(3) breaking and mass relations . . . . . . . . . . . . . . . . . . . . . . . 27710.8 Electromagnetic properties in SU(3) . . . . . . . . . . . . . . . . . . . . . . 282

10.8.1 Fundamental interaction . . . . . . . . . . . . . . . . . . . . . . . . 28210.8.2 Baryon magnetic moments . . . . . . . . . . . . . . . . . . . . . . 28310.8.3 Electromagnetic mass differences . . . . . . . . . . . . . . . . . . . 288

10.9 Decays of hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.9.1 Decays of members of pseudoscalar meson octet . . . . . . . . . . 28910.9.2 Decays of members of baryon octet . . . . . . . . . . . . . . . . . . 29110.9.3 Decays of members of baryon decuplet . . . . . . . . . . . . . . . . 29210.9.4 Decays of members of vector meson nonet . . . . . . . . . . . . . . 292

10.10 Summary of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 293

Contents xi

10.11 Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29410.11.1 Symmetry of baryon states . . . . . . . . . . . . . . . . . . . . . . 29410.11.2 Generalized Pauli principle . . . . . . . . . . . . . . . . . . . . . . 296

11 Non-abelian gauge theories 29811.1 Local SU(N) invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29811.2 Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

11.2.1 Transformation properties of gauge fields . . . . . . . . . . . . . . 30111.2.2 Field-strength tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 30311.2.3 Pure gauge Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 30411.2.4 Mathematical interlude . . . . . . . . . . . . . . . . . . . . . . . . 30611.2.5 Free gauge Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 310

11.3 Self-interaction of gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . 31111.3.1 Quartic vertices of gauge bosons . . . . . . . . . . . . . . . . . . . 31111.3.2 Cubic vertices of gauge bosons . . . . . . . . . . . . . . . . . . . . 312

11.4 FadeevPopov ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31311.5 Interaction of gauge bosons with other particles . . . . . . . . . . . . . . . 315

12 Quantum chromodynamics 31812.1 SU(3) of color . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31812.2 Running parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

12.2.1 Why parameters run . . . . . . . . . . . . . . . . . . . . . . . . . . 32212.2.2 Running of QED coupling . . . . . . . . . . . . . . . . . . . . . . . 32712.2.3 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33012.2.4 Running of the QCD coupling . . . . . . . . . . . . . . . . . . . . 333

12.3 QCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33512.4 Perturbative QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

12.4.1 Cross-sections involving quarks and gluons . . . . . . . . . . . . . 33712.4.2 Jet production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

12.5 The 1/N expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34812.6 Lattice gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

12.6.1 Scope and basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35212.6.2 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35412.6.3 Quantum field theory on Euclidean spacetime . . . . . . . . . . . . 35612.6.4 Quarks and gluons on a lattice . . . . . . . . . . . . . . . . . . . . 36012.6.5 Lattice action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36112.6.6 Fermion doubling problem . . . . . . . . . . . . . . . . . . . . . . 36412.6.7 Path integral on a lattice . . . . . . . . . . . . . . . . . . . . . . . 36712.6.8 Correlation functions and physical observables . . . . . . . . . . . 36812.6.9 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

12.7 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37112.8 Asymptotic properties of color gauge fields . . . . . . . . . . . . . . . . . . 375

12.8.1 Dual of field-strength tensor . . . . . . . . . . . . . . . . . . . . . 37512.8.2 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . . . 37912.8.3 QCD vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

13 Structure of hadrons 38513.1 Electronproton elastic scattering . . . . . . . . . . . . . . . . . . . . . . . 38513.2 Deep inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38913.3 Structure functions and charge distribution . . . . . . . . . . . . . . . . . . 39313.4 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39413.5 Partons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39613.6 Parton distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . 39913.7 Parton distribution and cross-section . . . . . . . . . . . . . . . . . . . . . 40213.8 Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40413.9 Scale dependence of parton distribution . . . . . . . . . . . . . . . . . . . . 40513.10 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

xii Contents

13.11 Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

14 Fermi theory of weak interactions 41014.1 Four-fermion interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41014.2 Helicity and chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

14.2.1 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41314.2.2 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41614.2.3 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41714.2.4 Neutrino helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41814.2.5 Weyl fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

14.3 Fierz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42014.3.1 Scalar combinations . . . . . . . . . . . . . . . . . . . . . . . . . . 42114.3.2 Pseudoscalar combinations . . . . . . . . . . . . . . . . . . . . . . 42314.3.3 Invariant combinations with fields . . . . . . . . . . . . . . . . . . 424

14.4 Elastic neutrinoelectron scattering . . . . . . . . . . . . . . . . . . . . . . 42614.5 Inelastic neutrinoelectron scattering . . . . . . . . . . . . . . . . . . . . . 42914.6 Muon and tau decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43014.7 Parity violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43414.8 Problems with Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . . 43814.9 Intermediate vector bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

15 Spontaneous symmetry breaking 44315.1 Examples of spontaneous symmetry breaking . . . . . . . . . . . . . . . . . 443

15.1.1 Breaking a Z2 symmetry . . . . . . . . . . . . . . . . . . . . . . . 44315.1.2 Breaking a U(1) symmetry . . . . . . . . . . . . . . . . . . . . . . 44615.1.3 Breaking a non-abelian symmetry . . . . . . . . . . . . . . . . . . 448

15.2 Goldstone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44915.3 Interaction of Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . 45115.4 Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

15.4.1 Gauge boson mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 45415.4.2 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45515.4.3 Unitary gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45715.4.4 Renormalizablity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

16 Standard electroweak model with leptons 45916.1 Chiral fermions and internal symmetries . . . . . . . . . . . . . . . . . . . 45916.2 Leptons and the gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . 46016.3 Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

16.3.1 Gauge bosons and their masses . . . . . . . . . . . . . . . . . . . . 46216.3.2 Couplings of photon . . . . . . . . . . . . . . . . . . . . . . . . . . 46616.3.3 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

16.4 Gauge interaction of fermions . . . . . . . . . . . . . . . . . . . . . . . . . 46816.5 Yukawa sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47116.6 Connection with Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . 473

16.6.1 Charged-current induced processes . . . . . . . . . . . . . . . . . . 47316.6.2 Neutral-current induced processes . . . . . . . . . . . . . . . . . . 47516.6.3 Processes induced by both types of currents . . . . . . . . . . . . . 476

16.7 Forward-backward asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 478

17 Electroweak interaction of hadrons 48217.1 Quarks in standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . 48217.2 Gauge interaction of quarks . . . . . . . . . . . . . . . . . . . . . . . . . . 48517.3 CKM matrix and its parametrization . . . . . . . . . . . . . . . . . . . . . 48717.4 Yukawa interaction of quarks . . . . . . . . . . . . . . . . . . . . . . . . . . 48917.5 Leptonic decays of mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

17.5.1 Decay of charged pions . . . . . . . . . . . . . . . . . . . . . . . . 49317.5.2 Decay of charged kaons . . . . . . . . . . . . . . . . . . . . . . . . 496

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17.5.3 Related processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 49717.6 Spin and parity of hadronic currents . . . . . . . . . . . . . . . . . . . . . 49817.7 Selection rules for charged currents . . . . . . . . . . . . . . . . . . . . . . 50217.8 Semileptonic decays of mesons . . . . . . . . . . . . . . . . . . . . . . . . . 50317.9 Neutral kaons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

17.9.1 Eigenstates in the neutral kaon sector . . . . . . . . . . . . . . . . 50417.9.2 Decays of neutral kaons into pions . . . . . . . . . . . . . . . . . . 50817.9.3 Kaon oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50917.9.4 K0L-K

0S mass difference . . . . . . . . . . . . . . . . . . . . . . . . 510

17.9.5 Leptonic decays of neutral kaons . . . . . . . . . . . . . . . . . . . 51717.10 Processes involving baryons . . . . . . . . . . . . . . . . . . . . . . . . . . 519

18 Global symmetries of standard model 52318.1 Accidental symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52318.2 Approximate symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52718.3 Chiral symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

18.3.1 Symmetries of massless Dirac Lagrangian . . . . . . . . . . . . . . 52818.3.2 Sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53018.3.3 Currents and charges . . . . . . . . . . . . . . . . . . . . . . . . . 53218.3.4 Chiral symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . 53518.3.5 PCAC and soft pion theorem . . . . . . . . . . . . . . . . . . . . . 53618.3.6 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53918.3.7 Chiral Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

18.4 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54418.4.1 Failure of Noethers theorem . . . . . . . . . . . . . . . . . . . . . 54418.4.2 QCD anomaly of U(1)A . . . . . . . . . . . . . . . . . . . . . . . . 55118.4.3 Decay of neutral pions . . . . . . . . . . . . . . . . . . . . . . . . . 55218.4.4 Anomaly cancellation for gauge currents . . . . . . . . . . . . . . . 55518.4.5 Non-anomalous global symmetry . . . . . . . . . . . . . . . . . . . 557

19 Bosons of standard model 55819.1 Interactions among bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

19.1.1 Self-interaction among gauge bosons . . . . . . . . . . . . . . . . . 55819.1.2 Self-interaction among scalars . . . . . . . . . . . . . . . . . . . . . 56219.1.3 Gauge-scalar interactions . . . . . . . . . . . . . . . . . . . . . . . 563

19.2 Decay of gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56519.3 Scattering of gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 56819.4 Equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57019.5 Custodial symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57319.6 Loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

19.6.1 Oblique parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 57619.6.2 Evaluation of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

19.7 Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58319.7.1 Elusiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58319.7.2 Theoretical bounds on mass . . . . . . . . . . . . . . . . . . . . . . 58419.7.3 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58819.7.4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

20 Hadrons involving heavy quark flavors 59320.1 Charm quark and charmed hadrons . . . . . . . . . . . . . . . . . . . . . . 593

20.1.1 Prediction of charm quark . . . . . . . . . . . . . . . . . . . . . . . 59320.1.2 Discovery of charmonium . . . . . . . . . . . . . . . . . . . . . . . 59420.1.3 Flavor SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59720.1.4 Discovery of other charm-containing hadrons . . . . . . . . . . . . 60220.1.5 Decay of mesons with charm quark . . . . . . . . . . . . . . . . . . 60320.1.6 Decay of baryons with charm quark . . . . . . . . . . . . . . . . . 606

20.2 Bottom quark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

xiv Contents

20.2.1 Discovery of bottom-containing hadrons . . . . . . . . . . . . . . . 60720.2.2 Decay of bottom-containing hadrons . . . . . . . . . . . . . . . . . 607

20.3 Neutral mesonantimeson systems . . . . . . . . . . . . . . . . . . . . . . . 60820.4 Top quark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61020.5 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61220.6 Heavy quark effective theory . . . . . . . . . . . . . . . . . . . . . . . . . . 613

20.6.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61320.6.2 Hadron fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61520.6.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

21 CP violation 62021.1 CP violation and complex parameters . . . . . . . . . . . . . . . . . . . . . 62021.2 KobayashiMaskawa theory of CP violation . . . . . . . . . . . . . . . . . 62121.3 Rephasing invariant formulation . . . . . . . . . . . . . . . . . . . . . . . . 62321.4 CP-violating decays of kaons . . . . . . . . . . . . . . . . . . . . . . . . . . 625

21.4.1 Observables and experimental results . . . . . . . . . . . . . . . . 62521.4.2 Identifying sources of CP violation . . . . . . . . . . . . . . . . . . 62621.4.3 CP violation in semileptonic decays . . . . . . . . . . . . . . . . . 62821.4.4 CP violation in decays into two pions . . . . . . . . . . . . . . . . 62921.4.5 Standard model estimates . . . . . . . . . . . . . . . . . . . . . . . 640

21.5 Other signals of CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . 64621.5.1 Decays of mesons involving heavier quarks . . . . . . . . . . . . . 64621.5.2 CP-violating correlations . . . . . . . . . . . . . . . . . . . . . . . 65121.5.3 Electric dipole moment . . . . . . . . . . . . . . . . . . . . . . . . 651

21.6 Unitarity triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65321.7 CP violation and T violation . . . . . . . . . . . . . . . . . . . . . . . . . . 65521.8 Strong CP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

21.8.1 Effective -parameter . . . . . . . . . . . . . . . . . . . . . . . . . 65721.8.2 Physical effects of . . . . . . . . . . . . . . . . . . . . . . . . . . 65921.8.3 Can be irrelevant? . . . . . . . . . . . . . . . . . . . . . . . . . . 660

22 Neutrino mass and lepton mixing 66322.1 Simple extension of standard model . . . . . . . . . . . . . . . . . . . . . . 66322.2 Neutrino oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

22.2.1 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 66522.2.2 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . 66922.2.3 Matter effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67322.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

22.3 Majorana fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68022.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68022.3.2 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

22.4 Consequences of lepton mixing . . . . . . . . . . . . . . . . . . . . . . . . . 68522.4.1 Lepton flavor violation . . . . . . . . . . . . . . . . . . . . . . . . . 68522.4.2 CP violation in leptonic sector . . . . . . . . . . . . . . . . . . . . 689

22.5 Lepton number violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69122.5.1 Neutrinoless double beta decay . . . . . . . . . . . . . . . . . . . . 69122.5.2 Lepton to antilepton conversion . . . . . . . . . . . . . . . . . . . 694

22.6 Models of neutrino mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69422.6.1 Adding singlet neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 69422.6.2 Effective Lagrangian for neutrino mass . . . . . . . . . . . . . . . . 697

23 Beyond the standard model 70023.1 Shortcomings of standard model . . . . . . . . . . . . . . . . . . . . . . . . 70023.2 Leftright symmetric model . . . . . . . . . . . . . . . . . . . . . . . . . . 70223.3 Grand unified theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70423.4 Horizontal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70923.5 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Contents xv

23.6 Higher dimensional theories . . . . . . . . . . . . . . . . . . . . . . . . . . 71223.7 String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

Appendix A Units and constants 717

Appendix B Short summary of particle properties 719

Appendix C Timeline of major advances in particle physics 724

Appendix D Properties of spacetime 727D.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727D.2 Levi-Civita symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

D.2.1 Contraction properties . . . . . . . . . . . . . . . . . . . . . . . . . 729D.2.2 Transformation properties . . . . . . . . . . . . . . . . . . . . . . . 729D.2.3 Spatial antisymmetric tensor . . . . . . . . . . . . . . . . . . . . . 731

Appendix E ClebschGordan co-efficients 732

Appendix F Dirac matrices and spinors 735F.1 Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

F.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735F.1.2 Associated matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 737F.1.3 Contraction formulas . . . . . . . . . . . . . . . . . . . . . . . . . 738F.1.4 Trace formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738F.1.5 Strings of Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . 741F.1.6 A property of the matrix C . . . . . . . . . . . . . . . . . . . . . . 742

F.2 Dirac spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743F.2.1 Plane wave solutions of Dirac equation . . . . . . . . . . . . . . . 743F.2.2 Normalization of spinors . . . . . . . . . . . . . . . . . . . . . . . . 745F.2.3 Conjugation properties . . . . . . . . . . . . . . . . . . . . . . . . 745F.2.4 Result of 0 multiplying the spinors . . . . . . . . . . . . . . . . . 746F.2.5 Result of 5 multiplying the spinors . . . . . . . . . . . . . . . . . 747F.2.6 Spin sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749F.2.7 Projection matrices on spinors . . . . . . . . . . . . . . . . . . . . 749

F.3 Bilinears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752F.3.1 Gordon identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752F.3.2 Squaring amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 753F.3.3 Non-relativistic reduction . . . . . . . . . . . . . . . . . . . . . . . 754

Appendix G Evaluation of loop integrals 756G.1 Introducing Feynman parameters . . . . . . . . . . . . . . . . . . . . . . . 756G.2 Reduction to invariant integrals . . . . . . . . . . . . . . . . . . . . . . . . 758G.3 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759G.4 Angular integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761G.5 Integration over the magnitude . . . . . . . . . . . . . . . . . . . . . . . . 761G.6 Divergent integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762

Appendix H Feynman rules for standard model 766H.1 External lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766H.2 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766H.3 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

H.3.1 Vertices in gauge sector . . . . . . . . . . . . . . . . . . . . . . . . 767H.3.2 Gauge bosons and Higgs bosons . . . . . . . . . . . . . . . . . . . 769H.3.3 Scalar self-interactions . . . . . . . . . . . . . . . . . . . . . . . . . 770H.3.4 Vertices involving fermions . . . . . . . . . . . . . . . . . . . . . . 770

Appendix I Books and other reviews 772

xvi Contents

Appendix J Answers to selected exercises 775

Index 777

List of Figures

2.1 Kinematically allowed region in a three-body decay . . . . . . . . . . . . . 302.2 Two-particle scattering in the CM frame . . . . . . . . . . . . . . . . . . . 322.3 Two-particle scattering in the FT frame . . . . . . . . . . . . . . . . . . . . 33

4.1 Interactions in diagrammatic form . . . . . . . . . . . . . . . . . . . . . . . 884.2 Two-particle elastic scattering from 3 interaction . . . . . . . . . . . . . . 894.3 Two-particle elastic scattering from 4 interaction . . . . . . . . . . . . . . 1004.4 More diagrams for elastic scattering of scalar particles . . . . . . . . . . . . 1104.5 4-point and 2-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1 Creation and annihilation of particles at the basic QED vertex . . . . . . . 1175.2 Feynman rule for the QED vertex. . . . . . . . . . . . . . . . . . . . . . . . 1185.3 Diagram for electronelectron scattering . . . . . . . . . . . . . . . . . . . . 1205.4 Diagrams for electronpositron scattering . . . . . . . . . . . . . . . . . . . 1255.5 Diagrams for Compton scattering . . . . . . . . . . . . . . . . . . . . . . . 1275.6 Diagram for muonantimuon production . . . . . . . . . . . . . . . . . . . . 1325.7 Feynman rules for scalar QED vertices. . . . . . . . . . . . . . . . . . . . . 1355.8 Effective vertex between a fermion and the photon . . . . . . . . . . . . . . 1395.9 One-loop correction to the vertex function in QED. . . . . . . . . . . . . . 1465.10 More one-loop corrections to the vertex function in QED. . . . . . . . . . . 148

6.1 Furrys theorem says that such a vertex is impossible . . . . . . . . . . . . 185

8.1 Low-lying states of the nuclei 7Li and 7Be . . . . . . . . . . . . . . . . . . . 2028.2 Basic vertices in the Yukawa theory. . . . . . . . . . . . . . . . . . . . . . . 2068.3 States of the dineutron, the deuteron and the diproton . . . . . . . . . . . . 215

9.1 Discovery of the positron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2319.2 The basic idea of a cyclotron. . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.3 Schematic diagram of a linear accelerator. . . . . . . . . . . . . . . . . . . . 2339.4 Detector surrounding the beam pipe . . . . . . . . . . . . . . . . . . . . . . 2369.5 Parts of a modern detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379.6 Bubble chamber tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2399.7 Cross-section near resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 244

10.1 Isospin and hypercharge of JP = 0 mesons . . . . . . . . . . . . . . . . . 25610.2 Isospin and hypercharge of JP = 1 mesons. . . . . . . . . . . . . . . . . . 25710.3 Pictorial meaning of Eq. (10.35) . . . . . . . . . . . . . . . . . . . . . . . . 26510.4 Young tableaux of the conjugate of a representation . . . . . . . . . . . . . 26710.5 Isospin and hypercharge of quarks. . . . . . . . . . . . . . . . . . . . . . . . 27110.6 The JP = 1/2+ baryon octet . . . . . . . . . . . . . . . . . . . . . . . . . . 27410.7 The JP = 3/2+ baryon decuplet . . . . . . . . . . . . . . . . . . . . . . . . . 275

11.1 Feynman rule for quartic interactions of gauge bosons . . . . . . . . . . . . 31211.2 Feynman rule for cubic interactions of gauge bosons. . . . . . . . . . . . . . 313

xvii

xviii List of Figures

11.3 Feynman rules involving ghosts. . . . . . . . . . . . . . . . . . . . . . . . . 31411.4 Gauge boson vertex with fermions. . . . . . . . . . . . . . . . . . . . . . . . 31611.5 Gauge boson vertices with scalars. . . . . . . . . . . . . . . . . . . . . . . . 317

12.1 Strong interaction between two baryons . . . . . . . . . . . . . . . . . . . . 32012.2 Pion exchange as an effective picture . . . . . . . . . . . . . . . . . . . . . . 32112.3 Diagrams for the two-point function . . . . . . . . . . . . . . . . . . . . . . 32512.4 Virtual electronpositron pairs around a bare electron . . . . . . . . . . . . 32912.5 Self-energy of gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 33112.6 Extra diagram for qg qg scattering . . . . . . . . . . . . . . . . . . . . . 34112.7 Gluongluon elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . 34212.8 Schematic diagram for two-jet production . . . . . . . . . . . . . . . . . . . 34412.9 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34512.10 Basic Feynman diagram responsible for three-jet events . . . . . . . . . . . 34612.11 The double-line notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34912.12 A non-planar diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35012.13 Two-point function and decay diagram for mesons . . . . . . . . . . . . . . 35112.14 Pictorial representations of basic links and plaquettes . . . . . . . . . . . . 36012.15 Going towards the continuum limit on the lattice . . . . . . . . . . . . . . . 37112.16 Magnetic lines of force in a superconductor . . . . . . . . . . . . . . . . . . 37212.17 First non-zero term for W (R, T ) . . . . . . . . . . . . . . . . . . . . . . . 374

13.1 Diagram for electronproton elastic scattering . . . . . . . . . . . . . . . . 38613.2 Electronproton inelastic scattering . . . . . . . . . . . . . . . . . . . . . . 39013.3 Ocean quarks and gluons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40113.4 Parton distribution functions for the proton . . . . . . . . . . . . . . . . . . 408

14.1 Helicity of decay-products of a spinless particle . . . . . . . . . . . . . . . . 41514.2 Differential cross-section for neutrinoelectron scattering . . . . . . . . . . . 42814.3 Electron energy spectrum in muon decay. . . . . . . . . . . . . . . . . . . . 43314.4 Intermediate vector boson for weak interactions . . . . . . . . . . . . . . . . 441

15.1 Shape of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44415.2 Shape of the potential of a complex scalar field. . . . . . . . . . . . . . . . 44715.3 Diagrams for decay of H into two Goldstone bosons . . . . . . . . . . . . . 45315.4 Scattering of a Goldstone boson and another scalar particle. . . . . . . . . 454

16.1 Propagators in R gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46816.2 Feynman rules for fermion interactions with W and Z bosons . . . . . . . . 46916.3 Feynman rules for fermion interactions with scalar fields . . . . . . . . . . . 47216.4 Muon decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47316.5 Elastic -e scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47516.6 Elastic e-e scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47716.7 Diagram for muonantimuon production . . . . . . . . . . . . . . . . . . . . 478

17.1 Diagrams for fermionfermion scattering . . . . . . . . . . . . . . . . . . . . 48917.2 Fermion interactions with unphysical scalars . . . . . . . . . . . . . . . . . 49117.3 Fermion interaction with the Higgs boson . . . . . . . . . . . . . . . . . . . 49217.4 Quark-level Feynman diagram for charged pion decay . . . . . . . . . . . . 49317.5 K0- bK0 transition diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 51017.6 K0- bK0 box diagrams involving unphysical Higgs bosons . . . . . . . . . . . 51317.7 More diagrams for K0- bK0 transition . . . . . . . . . . . . . . . . . . . . . . 51717.8 K0L decay to

+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

17.9 Neutrinoquark scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52017.10 -p and b-p scattering cross-section . . . . . . . . . . . . . . . . . . . . . . 521

18.1 Triangle diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

19.1 Feynman rule for cubic interactions of gauge bosons. . . . . . . . . . . . . . 559

List of Figures xix

19.2 Feynman rules for quartic interactions of gauge bosons. . . . . . . . . . . . 56019.3 Feynman rules for cubic interactions of scalars . . . . . . . . . . . . . . . . 56319.4 Vertices involving two gauge bosons and one scalar boson . . . . . . . . . . 56519.5 Diagrams for W+W elastic scattering . . . . . . . . . . . . . . . . . . . . 56819.6 One-loop correction to fermionfermion scattering . . . . . . . . . . . . . . 57619.7 Self-energy diagram for gauge bosons . . . . . . . . . . . . . . . . . . . . . 58019.8 Higgs boson production at hadron colliders . . . . . . . . . . . . . . . . . . 58819.9 Higgs boson decay to two photons . . . . . . . . . . . . . . . . . . . . . . . 59019.10 Branching ratios for Higgs boson decay . . . . . . . . . . . . . . . . . . . . 591

20.1 The ratio R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59620.2 Mesons in SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59820.3 Spin-1/2 baryons in SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60020.4 Spin-3/2 baryons in SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60220.5 Explanation of the OZI rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 60420.6 One-loop diagrams for b s+ . . . . . . . . . . . . . . . . . . . . . . . . 608

21.1 Quark-level tree diagrams contributing to K0 2 . . . . . . . . . . . . . 64121.2 Gluonic corrections to tree diagrams contributing to K0 2 . . . . . . . 64121.3 Penguin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64321.4 Diagrams for photon coupling to quarks . . . . . . . . . . . . . . . . . . . . 65221.5 Unitarity triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65321.6 Schematic diagram for test of T violation . . . . . . . . . . . . . . . . . . . 656

22.1 Contours of equal oscillation probability . . . . . . . . . . . . . . . . . . . . 66822.2 KamLAND result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67222.3 Summary of solutions of neutrino oscillation experiments . . . . . . . . . . 67822.4 Neutrino mass hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67922.5 One-loop diagram for the process e+ . . . . . . . . . . . . . . . . . . 68522.6 One-loop diagram for the process 3e . . . . . . . . . . . . . . . . . . . 68822.7 More diagrams for 3e involving Majorana neutrinos . . . . . . . . . . . 69022.8 Quark-level diagram of 0 process . . . . . . . . . . . . . . . . . . . . . . 69222.9 Different versions of the seesaw mechanism . . . . . . . . . . . . . . . . . . 698

23.1 Baryon number violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

G.1 Contour for performing k0 integration. . . . . . . . . . . . . . . . . . . . . . 760

List of Tables

1.1 Elementary fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Elementary bosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Dimensions of important kinematic quantities in natural units. . . . . . . . 13

4.1 Feynman rules for external lines. . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1 Electromagnetic quantities under parity and time reversal . . . . . . . . . . 1576.2 Kinematical variables under parity and time reversal . . . . . . . . . . . . . 178

10.1 Magnetic moments of baryons in the octet . . . . . . . . . . . . . . . . . . 285

14.1 Feynman rules for external lines related to a left-chiral Weyl field. . . . . . 420

21.1 Time-dependent CP-violating effects in B meson decays . . . . . . . . . . . 65121.2 Taggings required for tests of different discrete symmetries . . . . . . . . . 657

22.1 Feynman rules for external Majorana fermion lines . . . . . . . . . . . . . . 683

23.1 Counting parameters of the standard model. . . . . . . . . . . . . . . . . . 700

A.1 Conversion factors in natural units . . . . . . . . . . . . . . . . . . . . . . . 717A.2 Physical constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

B.1 Properties of fundamental bosons . . . . . . . . . . . . . . . . . . . . . . . 719B.2 Properties of leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720B.3 Properties of various baryons . . . . . . . . . . . . . . . . . . . . . . . . . . 720B.4 Properties of various mesons . . . . . . . . . . . . . . . . . . . . . . . . . . 721B.5 Properties of quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

E.1 ClebschGordan co-efficients for 12 1

2. . . . . . . . . . . . . . . . . . . . 733

E.2 ClebschGordan co-efficients for 1 12

. . . . . . . . . . . . . . . . . . . . . 733E.3 ClebschGordan co-efficients for 1 1 . . . . . . . . . . . . . . . . . . . . . 734

xxi

Preface

Modern theory of particle interactions is based on gauge theories. Amongthe four fundamental interactions, all but the gravitational one have beenmoulded in the form of gauge theories. There are two important ingredientsin a gauge theory: first, it has to be a quantum field theory, and second, itmust have some internal symmetry that governs its dynamics.

In giving the reader a taste of fundamental interactions among elementaryparticles, I have not assumed any prior knowledge of quantum field theory.I have presented the necessary tools of quantum field theory in Ch. 4 of thisbook. Of course, such a brief introduction can never be complete, and manyresults have been stated without proof. I have taken care to ensure that thenumber of such results is kept to a minimum without seriously getting into thenitty-gritty of quantum field theory, and the nature of these results is moredidactic than conceptual. But if the reader wants to understand the reasonsbehind these formulas, he or she will have to supplement this book with abeginner-level textbook on quantum field theory.

I do not take the same attitude regarding some advanced topics of quantumfield theory. These have been dealt with in more detail. Such discussions startfrom Ch. 11 of the book and appear in different sections of the chapters thatfollow. In other words, I assume only a beginners knowledge of quantum fieldtheory, not an experts knowledge.

In regard to group theory, I have assumed a somewhat different level ofpreparation from the reader. I assumed that the reader is familiar with thebasic definition and properties of a group, and is even familiar with the groupSU(2) and its representations. This seemed like a fair assumption since thesetopics are covered by any textbook on quantum mechanics in the context ofangular momentum operators. For continuous groups bigger than SU(2), I donot assume any prior knowledge. Their representations have been discussedin detail in the text.

The idea of writing such a book grew when I taught a one-semester courseon particle physics for several years during the last decade at my home insti-tute, the Saha Institute of Nuclear Physics. However, the book contains muchmore than what could be covered in that one-semester course. Students enterour institute after completing their M.Sc. programs, so most of them alreadycome with some background of particle physics. Some preliminary material

xxiii

xxiv Preface

presented in the book can therefore be omitted for them, but I have includedit for the sake of completeness. The book also contains some advanced mate-rial that could not be covered in the course due to lack of time. I think finallyI have been able to cover the subject at a level where an M.Sc. student canstart reading the book and understand a good part of it, and also beginningPh.D. students can learn something from it. Anyone teaching a course fromthe book will have to probably choose from the topics, depending on the levelof students in the course. But the book is not only about teaching courses: Ihope established researchers will also find the book useful.

I have put in a lot of exercises for the benefit of the students. I haveplaced them not at the end of the chapters but within the text, in placeswhere I think performing the exercise would be most helpful for the students.Even if a student does not feel like solving a particular exercise while readingthrough the text, it is advised that the student reads the statement of theexercise, because some exercises complement the text in a useful way. Someexercises have been marked with a sign, signifying that they have in factbeen worked out elsewhere in the book, although I dont mention where.

It took me a lot of time to finish the book, much more than what I wouldhave liked. I am very fortunate that Lincoln Wolfenstein, who introducedme to the world of particle physics, has kindly read earlier drafts of severalchapters of the book and has made illuminating and helpful remarks. I amalso deeply indebted to a long list of students and colleagues, who have ei-ther helped me out when I was stuck, or suggested modifications to earlierdrafts, or read some chapters and made extensive comments. The list includesBiswajit Adhikary, Sunanda Banerjee, Bireswar Basu-Mallick, Gautam Bhat-tacharyya, Francisco Botella, Gustavo Branco, Sayan Chakrabarti, SomdebChakraborty, Subhasis Chattopadhyay, Abhishek Chowdhury, Dipankar Das,Asit Kumar De, Jadunath De, Amit Ghosh, Kumar Sankar Gupta, AvarothHarindranath, Aminul Islam, Anirban Kundu, Amitabha Lahiri, DebasishMajumdar, Parthasarathi Mitra, Kuntal Mondal, Santanu Mondal, SwagataMukherjee, Munshi Golam Mustafa, Jose F. Nieves, Shoili Pal, Jorge Romao,Shibaji Roy, Satyajit Saha, Santosh Samaddar, Subir Sarkar, Ashoke Sen,Joao Silva and I apologize to anyone whose name may have been omitted.I also received help and advice from Michael Barnett, Kunio Inoue, SergueyPetcov, Hitoshi Murayama, Abdelhak Djouadi, Sven-Olaf Moch, Kevin Mc-Farland, Heinrich Pas, Subir Sarkar, Horst Wahl and Graeme Watt regardingobtaining permissions for reproducing several illustrations used in the book.Publishers of various journals have also been very kind in granting permis-sions. I have acknowledged them in proper places. I cannot overstate myindebtedness to my home institute, the Saha Institute of Nuclear Physics, forproviding me with wonderful infrastructural facilities. For some extended pe-riod during the writing of this book, I was a visitor at the Instituto SuperiorTecnico in Lisbon, and I thank my hosts for their hospitality. Peripheral helpfrom Kamakshya Prasad Modak and Souvik Priyam Adhya is also appreci-ated. And finally, I want to thank the wonderful support staff at the CRC

Preface xxv

Press, who have helped me in a lot of ways in preparing the manuscript in itsfinal form.

Despite all the help I have received, I am sure that some mistakes musthave remained in the book. I am responsible for all of them. Since I havesubmitted a camera-ready version of the book to the publishers, I am evenresponsible for any typographic errors. I will appreciate if any reader pointsout any mistake and will keep a list on the internet. For now, the errata willbe located at http://www.saha.ac.in/theory/palashbaran.pal/books/partphys/errata.html. In case I have to change this web address, pleasesearch on the internet with palash+particle+course.

Palash B. Pal

Notations

This is not an exposition of all symbols used in the book. It is an explanationof why such a list could not be made.

It would have been nice if a one-to-one correspondence could be main-tained between the physical quantities and the symbols that represent them.Alas, it was not possible to do that without creating notations which were un-necessarily non-transparent, or uncomfortably different from accepted ones,or both. Take, for example, the lowercase letter s. It is used to representspin, for obvious reasons. But it is also used customarily to denote one of theMandelstam variables. One might think of using the uppercase letter for oneof the two, but then it will be confused with the symbol for the strangenessquantum number. Similarly, the letter C is used for charge conjugation aswell as for the charm quantum number. Maybe the worst confusion occurswith B, which can be the baryon number or the bottom quantum number, aswell as the symbol for the branching ratio of a decay channel.

In the end, I stopped worrying about these things by making two adjust-ments with my conscience. First, I assumed that the reader would be matureenough to understand the meaning of a symbol from its immediate context.Second, in some cases, I used different font shapes of the same letter, some-times with shades of gray, to stand for different things. Here are some (anddefinitely not all) examples of uses of this sort:

s Mandelstam variable

s Spin vector

Irrational number

Pion

Decay rate

Gamma function

In some cases, I have pointed out such shape differences within the text.If the reader carefully notices them, the differences should be apparent. Con-versely, if the reader feels that looking at font shapes is too much of a bother,at least he or she is no worse off than looking at identical symbols for differentquantities.

In a very few cases, I have used notation that is definitely non-standard.It is customary to denote antiparticles by a bar above the symbol for denotingthe corresponding particle, e.g., for antineutrinos. I never liked this ideabecause the overbar is used for constructing bilinears of fermion fields with

xxvii

xxviii Notations

definite transformation properties under the Lorentz group, e.g., denotesa scalar. So I have used hats to denote antiparticles. Thus, q stands forantiquark and for antineutrinos. I am not claiming that the hat has notbeen used for any other purpose. For example, unit 3-vectors have beendenoted by n. My only claim is that such other uses occur in very few places,so that they do not conflict with my use of symbols for antiparticles in a largeway.

There is a bound state of the charm quark and its antiquark that is referredto as J/ in the literature, and called j-psi. I do not like using the slash inthe name. Using the flexibilities of LATEX typesetting, I have superposed theshapes of the two letters involved to create the notation J. Some of myfriends find it outrageous. I obviously dont.

Chapter 1

Scope of particle physics

1.1 What are elementary particles?

Particle physics is the branch of physics which studies elementary particlesand their properties.

It is easy to clarify what we mean by elementary particles: these areparticles which do not have any substructure. In other words, these are objectswhich are not made up of smaller objects. Turning the tables around, wecan say that the elementary particles are the fundamental constituents of allobjects in the universe.

It is not so easy to tell which particles are elementary. It is possible tosay which particles are not elementary, if one knows of some experiment thatshows substructure of the object. But no experiment can guarantee that agiven object does not have any substructure. A new experiment might showsubstructure in an object that was earlier considered elementary. Atoms usedto be thought of as indivisible until a series of experiments performed in thelate 19th and early 20th century confirmed the existence of electrons and theatomic nucleus. It was soon realized that for all atoms except the lightest one,viz., hydrogen, the nucleus has a substructure. Protons and neutrons cameto be collectively called nucleons because they are the constituents of nuclei,and treated as elementary particles. A few decades later, new experimentsindicated that the nucleons themselves have substructure, so they cannot becalled elementary particles any longer. We now believe that they are madeup of quarks , which are elementary particles like the electrons.

Thus the list of elementary particles changes with time. There is no guar-antee that todays elementary particles would not turn out to be compositeobjects tomorrow. But this cannot also stop us from having a discussion onelementary particles as they are known today. If things change tomorrow, thediscussion will change, no doubt. We can only employ the best knowledge thatis available at the moment to decide which objects are elementary. Needlessto say, that will be our approach in this book. In 1.2 and 1.4, we are going

1

2 Chapter 1. Scope of particle physics

to provide an inventory of all particles which are believed to be elementary.We will discuss the properties of these particles in this book.

An elementary particle does not have to be subatomic, in the sense thatit does not have to be a constituent of atoms in some way or other. Navely,when we think about elementary particles, we have one of the two followingpropositions in our mind, or maybe some mixture of the two:

1. Elementary particles are constituents of larger conglomerates.

2. Elementary particles do not have any constituents.

It is important, therefore, to be absolutely clear at the outset that when wetalk about the field of elementary particle physics, we use the word ele-mentary only in the second sense. The existence of the first property mightbe crucial in detecting a particular elementary particle, but is not necessaryfor deciding whether a particular particle should be included in the list ofelementary particles. In other words, the capability of forming larger con-glomerates is a property that an elementary particle may or may not have.In fact, most elementary particles do not have this property for one reasonor other. For example, if the particle is very unstable, it would not have thechance of coming in contact with other particles to form a big, macroscopicchunk of matter. Also, not all particles can bind with others. One of thebest-known elementary particles, the photon, cannot bind to other particlesand is not a constituent of any bigger structure.

1.2 Inventory of elementary fermions

In 1897, the electron was discovered by J. J. Thomson, who suggested thatelectrons were constituents of atoms. Before that, it was believed that theatoms were indivisible. In search for all constituents of atoms, it was soondiscovered that the negative electric charge of the electrons is counterbalancedin atoms by the positive charge of the nucleus. The nucleus, though small,contains protons and neutrons, which are much heavier than the electrons.The electrons, protons and neutrons thus constitute atoms, a picture thatwas established finally by the discovery of the neutron in 1932 by JamesChadwick.

The electron, the proton and the neutron are all examples of fermions,which have an intrinsic angular momentum, or spin, that is a half-integralmultiple of the fundamental constant ~. In particular, all three of these havespin equal to 12~. The other alternative is to have spin in integral multiples of~, and particles carrying such spin are called bosons. In this section, we willconsider elementary fermions only.

The proton and the neutron are not elementary particles, as we know now.But the electron is. The electron thus goes back longer into history than anyother particle which is considered elementary even now.

1.2. Inventory of elementary fermions 3

With the first observation of the muon () in 1937 and the subsequentstudy of its properties, it was clear that there are other elementary particleswhich are very much similar in their properties to the electron, except thatthey are heavier. Rabi asked his famous question, Who ordered that? ex-pressing his surprise and disgust at the fact that Nature seems to be repeatingthings. We dont know the answer to Rabis question, but we know that therepetition does exist. Of course there are some differences in the propertiesof the muon and the electron. For example, the muon, being heavier, candecay into lighter particles, which the electron cannot. But, as this exampleshows, the apparent differences can all be traced to the difference in mass. Ata basic level, the electron and the muon have the same interactions. In mod-ern terminology, we call the muon a particle of the second generation. Muchlater, in 1975, another such particle has been discovered, which is called thetau particle (), or simply the tau. It belongs to the third generation.

In 1930, in order to explain the continuous spectrum of electrons in nuclearbeta decay, Pauli proposed that a neutral fermion was produced in such pro-cesses. Neutral fermions are collectively called neutrinos . Paulis conjecturewas proved in 1956, and thus the existence of one kind of neutrinos was con-firmed. In the modern terminology, these are called the electron-neutrinos(e)since they are related to beta decay, where an electron is emitted. In the 1960s,it was realized that the muon is also accompanied by its own neutrino, whichis now called the muon-neutrino (). The tau particle is also believed tohave its own neutrino which is different from e and , and is denoted by. The neutrinos, therefore, also conform to the generational structure.

These neutrinos, along with the electron, the muon and the tau, form aclass of elementary particles which are called leptons . There is another classof elementary particles which are called quarks . Unlike the leptons, no onehas seen quarks in their free state. Quarks always appear in bound states,examples of which are protons and neutrons that occur in atomic nuclei.Bound states involving quarks are called hadrons .

Properties of quarks must be inferred from the properties of hadrons.Six types of quarks are necessary to understand the properties of all knownhadrons. These quarks are given the names up (u), down (d), charm (c),strange (s), top (t) and bottom (b). They also seem to follow the gener-ational structure, in the sense that the properties of u, c and t are similar,and so are the properties of d, s and b. Thus the six quarks come in threegenerations. These, along with the three generations of leptons mentionedearlier, exhaust the list of elementary fermions as it is known today.

A cautionary remark should accompany this list. It is known that at leastany particle carrying a non-zero electric charge must have an antiparticlewhich carries an opposite charge. Neutrinos are electrically neutral. It isnot clear whether they have antiparticles which are different from themselves.But all other particles listed above are charged. The electron and its likesare negatively charged, so they have their own antiparticles as well. Theantiparticle of the electron is called the positron, whereas those for the muon


Table 1.1: Elementary fermions. All of them have spin- 12. The antiparticles have not

been included in the list.

Name Symbol

Lepto

ns

Electron eElectron-neutrino eMuon Muon-neutrino Tau Tau-neutrino

Quark

s

Up uDown dCharm cStrange sTop tBottom b

and the tau do not have any such special names: they are called just antimuonand antitau. All quarks have electric charge, so there must be six differenttypes of antiquarks. Their names are also quite mundane, antiup, antidownetc.

It may be a matter of taste to decide whether to count the antiparticlesas separate entries while preparing a list of elementary particles. On the onehand, it can be argued that any antiparticle is indeed different from the cor-responding particle, at least in the electric charge and often in other respectsas well, so they ought to be counted separately. On the other hand, it can beargued that the properties of any antiparticle follow from the properties of thecorresponding particle, and therefore the antiparticle is not an independentobject.

In compiling a list of all fundamental fermions in Table 1.1, we have takenthe latter viewpoint. The names appearing in this table, along with their an-tiparticles, complete the list of elementary fermions to the best of our knowl-edge today. There are also elementary particles which are bosons. We willget acquainted with them in 1.4.

1.3 Which properties?

We have announced our list of elementary fermions. Let us suppose that wewill also agree on our list of elementary bosons. Even after that, it remains tobe decided which properties of these particles should fall within the purviewof particle physics. It seems that if we successfully identify all elementaryparticles and know all of their properties, we will know their behavior in all

1.3. Which properties? 5

sorts of circumstances and in any sort of conglomeration. This would meanthat we would know the properties of any system. Thus, in principle, itmight seem that this branch of physics should cover all aspects of physics,and indeed everything else that there is to know. Following the properties ofsingle particles, we should be able to know everything about any system.

But this would be a worthless pursuit. We really cannot discuss allproperties of all systems starting from elementary particles. The problemsare twofold. First, if we have a large number of particles in a system, thereis enormous practical difficulty in discussing the properties of that systemstarting from those of individual particles. Often, while discussing a physicalprocess, it is beneficial to proceed by taking larger units comprised of manyparticles, whose identities are not disturbed in the process. For example, whilediscussing a football match, we talk about the players and the ball, treatingthem as units. It would not make much sense to talk about the atoms in theball, or the electrons in the ball.

The second kind of problem is that, in discussing large systems, one oftenencounters useful parameters which appear as co-operative effects, i.e., be-cause of the presence of a large number of particles. Things like temperatureand entropy cannot be defined for a single particle, yet they are undeniablyvery helpful concepts while discussing thermal properties of a system.

It is therefore customary to treat physical problems in a step-by-step man-ner, not going beyond the substructure more than is necessary to understanda particular problem. For the football match, we do not need to get intothe anatomy of the players. If a player is injured and has to be operatedupon, then we stop considering the player as a unit and discuss his or heranatomy. But even then, we think in terms of bones and organs rather thanmolecules and atoms. For discussing the properties of a system, it is usuallynot necessary to go beyond one level of substructure.

Turning the table around, we can say that the properties of any elementaryparticle are hardly discernible in systems which have more than one level ofstructure. This will be our guiding principle in determining the scope of thesubject. We will discuss elementary particles, and the next level of structuresformed by them.

For example, we will study interactions of the electron and its antipar-ticle, the positron. We can also discuss the positronium system, which is abound state of an electron and a positron. We will study protons, becausethey are the first level structures formed from quarks. In fact, the protonis just an example of the class of structures known, as mentioned earlier, ashadrons. There are many hadrons, which can be broadly divided into twoclasses. Hadrons which have half-integral spin, e.g., the proton and the neu-tron, fall into the class called baryons . Hadrons with integral spin, like thepion, are called mesons . In other words, baryons are fermions, mesons arebosons. We will encounter many kinds of hadrons and discuss their propertiesin this book.


But a hydrogen atom will not fall within the purview of particle physics,because that is a bound state of the electron and the proton, and the latteritself is a bound state of elementary quarks. Even nuclei are not includedin discussions of particle physics, because they are bound states of protonsand neutrons, neither of which is elementary. We will make exception to thisrule only in a few places in the book when deuterium, tritium and heliumnuclei will make their appearances in our discussions. However, even in suchplaces, the aim will not be so much to understand the properties of thosebound states, but rather to illustrate some general principles that apply toelementary particles as well.

To summarize then, we will study the properties of elementary particlesas well as the first level of structures formed by them. There are two kindsof properties associated with any system. The first kind can be called staticproperties, which the system possesses even if it is left alone. The second kindof properties can be classified as dynamical, and involve the interactions ofthe system with other systems. For example, the mass, the spin, the chargeand the magnetic moment of a particle are static properties. On the otherhand, if we ask how an electron behaves when it collides with a pion, we areessentially inquiring about a dynamic property of the electron.

In practice, the difference between these two kinds of properties is lesspronounced than what their definitions might suggest. The reason is thateven a static property is revealed to us only through interactions. Even if weagree that the electron has a charge when it is left alone, we do not know aboutit until the electron undergoes some sort of interaction with other things, oreven with another electron. Thus, whether static or dynamic, a propertyof a particle is known through interactions. It is therefore good to take aninventory of interactions between particles.

1.4 Fundamental interactions

There are four kinds of fundamental interactions, as far as we know. Theoldest known one is the gravitational interaction, known since the time ofNewton in the 17th century. In the 19th century, electricity and magnetismwere unified into the electromagnetic theory, and this is the second kind ofinteraction that we recognize to be fundamental. In the early part of the20th century, with the discovery of the atomic nucleus, it was necessary tointroduce two more kinds of fundamental interactions. One of them is thestrong interaction, required to explain the stability of the atomic nuclei despitethe fact that the protons in the nuclei exert repulsive Coulomb forces on oneanother. The other is the weak interaction, needed to explain the phenomenonof beta radioactivity, in which an electron or a positron comes out of a nucleus.

In this book, gravitational interaction will not be discussed at all. Thereason is twofold. First, the gravitational interaction between elementaryparticles is much feebler than all other interactions. For example, the grav-

1.4. Fundamental interactions 7

Table 1.2: Elementary bosons.

Spin Name Number Symbol

1

Photon 1 W bosons 2 W+,W

Z boson 1 ZGluons 8 g

0 Higgs boson 1 H

itational potential between two protons is roughly 1039 times weaker thanthe Coulomb potential between them. Thus, gravitational interactions canbe neglected to a very good approximation. The second reason is that thereis no universally accepted quantum theory of gravitation. In this book, wewill discuss the other three kinds of interactions, using quantum theory as thetheoretical basis.

In quantum theory, interactions are also described in terms of exchangeof particles. For example, electromagnetic interactions are described by themediation of the photons, which are seen as the quanta of the electromagneticfield. Photons have spin 1 (in units of ~) and are therefore bosons. Otherinteractions are also mediated by spin-1 bosons. The weak interactions aremediated by the particles called W+ and W which have electric chargesshown as the superscripts and an uncharged boson Z. The strong interactionsare mediated by eight particles, which are collectively called gluons. Like thequarks, the gluons have also never been seen as free particles. Their existenceis inferred from theoretical calculations regarding the properties of hadrons.

In addition, the standard theory of electroweak interactions postulates thatthere is a spinless boson. It is called the Higgs boson. Since the inception ofthe standard model, there were strong theoretical reasons to believe that itexisted, although there were alternatives to the standard model which avoidthis prediction. In 2012, an announcement was made about the existence ofa boson that conforms with the Higgs boson of the standard model, althoughthere is not enough data to deduce various properties of the particle andtherefore to confirm that it is indeed the Higgs boson proposed in the standardmodel. We will assume that it is so and stick to the standard model, accordingto which the elementary bosons are those listed in Table 1.2.

Unlike the list of fermions, the names of all antiparticles have been includedin the list of bosons. The reason is that many of the elementary bosons aretheir own antiparticles. The photon, for example, is its own antiparticle. So isthe Z boson, and the Higgs boson according to the standard model. The W+

and the W are antiparticles of each other. Each gluon has its antiparticleincluded in the set of eight.


1.5 High energy physics

The branch of physics that is called particle physics is also called high energyphysics . The two names suggest that the study of the most minute objects issomehow related to the use of highest energies.

It is easy to argue that there has to be a connection between these twoextremes. If we shine an object with some kind of waves and want to seesubstructures of the object at a length scale smaller than a certain value ,we need waves whose wavelength is smaller than, or at most comparable to,:

. . (1.1)

The relation between the wave and particle natures is provided by the de-Broglie relation:

p =2~

, (1.2)

where p denotes the momentum of the particles that correspond to the waves,or in other words are the quanta of the waves. Eq. (1.1) can then be writtenas

p &~

. (1.3)

This is the crucial equation. It tells us that, in order to study length scales ororder , we must use probes whose quanta should have a minimum momentumwhose magnitude is inversely proportional to . Thus, in order to know aboutsmaller and smaller structures, we need to use more and more energetic probes.

We can look at the question in another way. Suppose we have some ob-jects whose lengths are of order , and we want to know whether they aremade up of some building blocks. Well, if there are such building blocks, orparticles, they are no doubt confined in some manner within a distance oforder of , which defines the size of the supposedly composite object. By theuncertainty principle, we conclude that the momentum uncertainty of theseobjects satisfies the relation

p &~

, . (1.4)

Now, if the momentum uncertainty is that big, the momentum itself must atleast be of that order. This sets up a minimum value for the kinetic energy ofthe particles that constitute the composite object. Since kinetic energy is amonotonically increasing function of momentum, it follows that the expressionfor the minimum possible kinetic energy should be of the form , where is positive, and its value depends on the relation connecting momentum andkinetic energy. For example, for non-relativistic particles the kinetic energy is

1.5. High energy physics 9

a quadratic function of the momentum, so that = 2. For massless particles, = 1.

Obviously, the particles must also possess an overall attractive potentialenergy. In order that the composite system is bound, the total energy mustbe negative, i.e., the magnitude of the potential energy has to be larger thanthat of the kinetic energy. However, the two magnitudes do not differ byhuge factors, as exemplified in Ex. 1.1 below. Thus, the magnitude of thetotal energy, called the binding energy of the composite object, is usuallycomparable with the individual contributions of the kinetic and the potentialenergies. We need to overcome this binding energy if we want to break upthe object and see its constituent particles. For this, we need probes whichare more energetic than the the binding energy. The minimum energy of theprobes will thus have to be proportional to , which shows that we needhigher energies in order to gain access to smaller length scales.

2 Exercise 1.1 Consider the electron in the hydrogen atom. Treatingit as a classical system and using Newtons laws of motion, showthat the magnitude of its potential energy is twice that of the kineticenergy.

2 Exercise 1.2 Take the size of a typical atom to be of order 108 cm.Use the arguments above to show that its binding energy has to be atleast of order of a few eV. [Note : 1 eV = 1.6 1012 erg.]

2 Exercise 1.3 For nuclei which have typical dimensions of order1013 cm, show that the binding energy has to be at least of the orderof tens of MeVs.

2 Exercise 1.4 Electrons have been bombarded at energies of the orderof 100 GeV, and those experiments have not revealed any substructureof the electron. Even if the electron has any substructure, argue thatits size must be smaller than of order 1016 cm.

Indeed, the history of discovery of smaller and smaller substructures isintimately connected to the ability of obtaining and controlling probes ofhigher and higher energies. In the second half of the 19th century, there wereimmense advancements in the science and technology of electricity, whichmade it possible to produce moderately high voltages. Such voltages wereapplied to electrodes to impart energies of the order of tens of eVs to atomicelectrons and strip them out of the atoms, activities which led to the discoveryof the electron. Radioactivity was discovered at the turn of the century. Soonafter, Rutherford realized that the products of radioactive de

an introductory course of particle physics - palash b. pal

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