Effective Field Theories

by Alexey A Petrov, Andrew E Blechman

Overview

This book is a broad-based text intended to help the growing student body interested in constructing and applying methods of effective field theory to solve problems in their research. It begins with a review of using symmetries to identify the relevant degrees of freedom in a problem, and then presents a variety of methods that can be used to construct various effective theories. A detailed discussion of canonical applications of effective field theory techniques with increasing complexity is given, including Fermi's weak interaction, heavy-quark effective theory, and soft-collinear effective theory. Applications of these techniques to study physics beyond the standard model, dark matter, and quantum and classical gravity are explored. Although most examples come from questions in high-energy physics, many of the methods can also be applied in condensed-matter settings. Appendices include various factoids from group theory and other topics that are used throughout the text, in an attempt to make the book self-contained.

Product Details

ISBN-13:	9789814434928
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/13/2016
Pages:	320
Product dimensions:	5.90(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface vii

1 Introduction 1

1.1 Wherefore EFT? 1

1.2 EFT vs MFT 3

1.3 An example from Newton 4

1.4 A theorem of Weinberg 6

1.5 Organization of the book 7

2 Symmetries 9

2.1 Introduction 9

2.2 Noether's Theorem 9

2.3 Examples of Noether currents 11

2.4 Gauged symmetries and Noether's procedure 13

2.5 Broken symmetries and Goldstone's Theorem 15

2.5.1 Nonrelativistic NG-bosons 18

2.6 The BEHGHK Mechanism of Anderson 20

2.6.1 A little history 21

2.6.2 An example 23

2.6.3 An interlude: superconductivity 25

2.7 CCWZ construction of EFT 27

2.8 Explicit breaking; spurion analysis 30

2.9 Anomalous symmetries 32

2.9.1 Anomalies in the path integral 33

2.9.2 The chiral anomaly and its consequences 35

2.9.3 't Hooft anomaly matching 37

2.10 Notes for further reading 38

3 Elementary Techniques 41

3.1 Canonical (engineering) dimensions 41

3.1.1 Dimensional analysis 41

3.1.2 Example: hydrogen atom 44

3.2 Dimensional transmutation 46

3.3 Callan-Symanzik Equation 49

3.4 The renormalization group 53

3.4.1 Engineering dimensions of Γ⁽ⁿ⁾ 53

3.4.2 Physics of the anomalous dimension 54

3.5 Renormalizability and Effective Field Theory 57

3.5.1 Dropping renormalizability 57

3.5.2 Matching 57

3.6 Subtraction schemes as part of EFT definition 58

3.7 Decoupling. Appelquist-Carrazone theorem in various schemes 60

3.8 Notes for further reading 63

4 Effective Field Theories of Type I 65

4.1 Introduction 65

4.2 Real physics: Euler-Heisenberg Lagrangian 70

4.3 Fermi Theory of Weak Interactions as an effective theory 71

4.4 Fermi theory to one loop: ΔS = 2 processes in EFT 73

4.4.1 Setting up the EFT approach 74

4.4.2 A more detailed calculation 76

4.5 QCD corrections in EFTs 78

4.5.1 Matching at one loop in QCD 79

4.5.2 Renormalization group improvement and EFTs 83

4.5.3 Complete basis. Penguin operators 85

4.6 Chiral perturbation theory 88

4.6.1 Goldstone bosons and their properties 90

4.6.2 Sources in chiral perturbation theory 93

4.6.3 Applications: Gell-Mann-Okubo relation 97

4.6.4 Power counting. Chiral loops and higher orders in _ΧPT 99

4.6.5 Naive dimensional analysis 101

4.6.6 Baryons and chiral perturbation theory 103

4.7 Notes for further reading 104

5 Effective Field Theories of Type II. Part A. 107

5.1 Introduction 107

5.2 Heavy Quark Effective Theory 108

5.2.1 Quantum mechanics of heavy particles 109

5.2.2 From quantum mechanics to field theory: HQET. Field redefinitions 111

5.2.3 Spin symmetry and its consequences 115

5.2.4 More symmetry: reparameterization invariance 117

5.2.5 HQET Green's functions. Radiative corrections 118

5.2.6 External currents and external states 122

5.3 Different degrees of freedom: heavy mesons 130

5.3.1 Heavy meson states. Tensor formalism 130

5.3.2 Leading-order Lagrangian 133

5.3.3 Subheading Lagrangians 134

5.3.4 Calculations with HH_ΧPT 135

5.4 Light baryons in heavy particle formalism 139

5.4.1 Leading-order Lagrangian 139

5.5 Notes for further reading 142

6 Effective Field Theories of Type-II. Part B. 145

6.1 Introduction. Non-relativistic QED and QCD 145

6.2 NRQCD Lagrangian at the scale m_Q 147

6.3 Going down: non-perturbative scales m_Qv and m_Qv² 151

6.3.1 Electromagnetic decays of the η_c 155

6.3.2 Inclusive decays of the η_c into light hadrons 156

6.3.3 Inclusive decays of the Χ_c into light hadrons 158

6.4 Going down: perturbativc scales m_Qv and m_Qv² pNRQCD 160

6.4.1 Example: heavy quarkonium potential 161

6.5 Different degrees of freedom: hadronic molecules 163

6.6 Notes for further reading 168

7 Effective Field Theories of Type-III. Fast Particles in Effective Theories 171

7.1 Infrared divergences 171

7.2 Soft-Collinear Effective Theory 174

7.2.1 Quantum mechanics of fast particles 174

7.2.2 SCET power counting 175

7.2.3 SCET action 178

7.3 Symmetries of SCET 182

7.3.1 Gauge invariance 182

7.3.2 Reparametrization invariance 184

7.4 Examples 185

7.4.1 B → X_s γ 185

7.4.2 B → D_π 188

7.4.3 Deep Inelastic Scattering 191

7.4.4 Jet production 197

7.5 Notes for further reading 199

8 Standard Model as an Effective Field Theory 203

8.1 Introduction 203

8.2 Standard model as the leading term in the EFT expansion 205

8.2.1 Dimcnsion-5 operators: fermion number violation 206

8.2.2 Dimension-6 operators: parameterizing new physics 208

8.2.3 Experimental tests and ohscrvables 209

8.3 BSM particles in EFT 211

8.3.1 Dark matter at colliders: effective operators 212

8.3.2 Mono-Higgs signatures of dark matter at LHC 213

8.4 Notes for further reading 217

9 Effective Field Theories of Gravity 219

9.1 Introduction 219

9.2 Review of general relativity 221

9.2.1 Geodesies and affine connection 221

9.2.2 General relativity and the weak field limit 222

9.2.3 Gravity sources: energy-momentum tensor 226

9.3 Building an effective field theory 227

9.3.1 Quantization. Feynman rules 228

9.3.2 Quantum EFT for gravity 231

9.3.3 Newtonian potential 233

9.3.4 Postscript 234

9.1 Classical EFT: NRGR 235

9.4.1 Setting up the problem 236

9.4.2 Gravition modes 238

9.4.3 Feynman rules 241

9.4.4 Gravitational radiation 242

9.4.5 Renormalization 245

9.5 Notes for further reading 247

10 Outlook 249

10.1 Super symmetry 249

10.2 Extra dimensions 250

10.3 Technicolor and compositeness 251

10.4 High-T_c superconductivity 252

Appendix A Review of Group Theory 253

A.1 General definitions 253

A.2 Continuous groups 255

A.3 Representation theory of Lie groups 258

A.4 Young Tableaux for SU(N) 262

A.5 Group theory coefficients 266

A.6 Notes for further reading 267

Appendix B Short Review of QED and QCD 269

B.1 Quantum electrodynamics 269

B.2 Quantum chromodynamics 270

B.2.1 QCD Lagrangian and Feynman rules 271

B.2.2 Symmetries of the QCD Lagrangian 277

Appendix C Useful Features of Dimensional Regularization 279

C.1 Overview of dimensional regularization 279

C.2 Useful formulas 280

C.3 Dimensional regularization vs other schemes 282

C.4 Advanced features: scaleless integrals 283

C.5 Advanced features: integration by parts 285

C.6 Advanced features: method of regions 287

Bibliography 291

Index 301