Introduction to Effective Field Theory: Thinking Effectively about Hierarchies of Scale

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Copyright Contents List of Illustrations List of Tables Preface Acknowledgements Part I Theoretical Framework 1 DecouplingandHierarchiesofScale 1.1 An Illustrative Toy Model ♦ 1.1.1 Semiclassical Spectrum 1.1.2 Scattering 1.1.3 The Low-Energy Limit 1.2 The Simplicity of the Low-Energy Limit ♦ 1.2.1 Low-Energy Effective Actions 1.2.2 Why It Works 1.2.3 Symmetries: Linear vs Nonlinear Realization 1.3 Summary Exercises 2 EffectiveActions 2.1 Generating Functionals – A Review ♥ 2.1.1 Connected Correlations 2.1.2 The 1PI (or Quantum) Action ♠ 2.2 The High-Energy/Low-Energy Split ♦ 2.2.1 Projecting onto Low-Energy States 2.2.2 Generators of Low-Energy Correlations ♠ 2.2.3 The 1LPI Action 2.3 The Wilson action ♦ 2.3.1 Definitions 2.4 Dimensional Analysis and Scaling ♦ 2.4.1 Dimensional Analysis 2.4.2 Scaling 2.5 Redundant Interactions ♦ 2.6 Summary Exercises 3 PowerCountingandMatching 3.1 Loops, Cutoffs and the Exact RG ♠ v 3.1.1 Low-Energy Amplitudes 3.1.2 Power Counting Using Cutoffs 3.1.3 The Exact Renormalization Group 3.1.4 Rationale behind Renormalization ♦ 3.2 Power Counting and Dimensional Regularization ♦ 3.2.1 EFTs in Dimensional Regularization 3.2.2 Matching vs Integrating Out 3.2.3 Power Counting Using Dimensional Regularization 3.2.4 Power Counting with Fermions 3.3 The Big Picture ♦ 3.3.1 Low-Energy Theorems 3.3.2 The Effective-Action Logic ♦ 3.4 Summary Exercises 4 Symmetries 4.1 Symmetries in Field Theory ♥ 4.1.1 Unbroken Continuous Symmetries 4.1.2 Spontaneous Symmetry Breaking 4.2 Linear vs Nonlinear Realizations ♦ 4.2.1 Linearly Realized Symmetries 4.2.2 Nonlinearly Realized Symmetries 4.2.3 Gauge Symmetries 4.3 Anomaly Matching ♠ 4.3.1 Anomalies♥ 4.3.2 Anomalies and EFTs 4.4 Summary Exercises 5 Boundaries 5.1 ‘Induced’ Boundary Conditions 5.2 The Low-Energy Perspective 5.3 Dynamical Boundary Degrees of Freedom 5.4 Summary Exercises 6 Time-DependentSystems 6.1 Sample Time-Dependent Backgrounds ♦ 6.1.1 View from the EFT 6.2 EFTs and Background Solutions ♦ 6.2.1 Adiabatic Equivalence of EFT and Full Evolution 6.2.2 Initial Data and Higher-Derivative Instabilities ♣ 6.3 Fluctuations about Evolving Backgrounds ♠ 6.3.1 Symmetries in an Evolving Background 6.3.2 Counting Goldstone States and Currents ♣ 6.4 Summary Exercises Part II Relativistic Applications 7 ConceptualIssues(RelativisticSystems) 7.1 The Fermi Theory of Weak Interactions ♦ 7.1.1 Properties of the W Boson 7.1.2 Weak Decays 7.2 Quantum Electrodynamics 7.2.1 Integrating Out the Electron 7.2.2 E ≫ me and Large Logs ♣ 7.2.3 Muons and the Decoupling Subtraction Scheme ♠ 7.2.4 Gauge/Goldstone Equivalence Theorems 7.3 Photons, Gravitons and Neutrinos 7.3.1 Renormalizable Interactions ♦ 7.3.2 Strength of Non-renormalizable Interactions ♦ 7.3.3 Neutrino-Photon Interactions ♣ 7.4 Boundary Effects 7.4.1 Surfaces between Media 7.4.2 Casimir Energies ♠ 7.5 Summary Exercises 8 QCDandChiralPerturbationTheory 8.1 Quantum Chromodynamics ♠ 8.1.1 Quarks and Hadrons 8.1.2 Asymptotic Freedom 8.1.3 Symmetries and Their Realizations 8.2 Chiral Perturbation Theory 8.2.1 Nonlinear Realization ♦ 8.2.2 Soft-Pion Theorems ♠ 8.2.3 Including Baryons 8.2.4 Loops and Logs ♦ 8.3 Summary Exercises 9 TheStandardModelasanEffectiveTheory 9.1 Particle Content and Symmetries♥ 9.1.1 The Lagrangian 9.1.2 Anomaly Cancellation ♣ 9.2 Non-renormalizable Interactions 9.2.1 Dimension-Five Interactions 9.2.2 Dimension-Six Interactions 9.3 Naturalness Issues♠ 9.3.1 Technical and ’t Hooft Naturalness ♦ 9.3.2 The Electroweak Hierarchy Problem 9.3.3 The Cosmological Constant Problem 9.4 Summary Exercises 10 GeneralRelativityasanEffectiveTheory 10.1 Domain of Semi-Classical Gravity ♦ 10.2 Time-Dependence and Cosmology ♠ 10.2.1 Semiclassical Perturbation Theory 10.2.2 Slow-Roll Suppression 10.3 Turtles All the Way Down? ♣ 10.3.1 String Theory 10.3.2 Extra Dimensions 10.4 Summary Exercises PartIII NonrelativisticApplications Part III Nonrelativistic Applications 11 ConceptualIssues(NonrelativisticSystems) 11.1 Integrating Out Antiparticles ♦ 11.2 Nonrelativistic Scaling ♦ 11.2.1 Spinless Fields 11.2.2 Spin-Half Fields 11.3 Coupling to Electromagnetic Fields ♠ 11.3.1 Scaling 11.3.2 Power Counting 11.4 Summary Exercises 12 ElectrodynamicsofNonrelativisticParticles 12.1 Schr ¨ odinger from Wilson ♦ 12.1.1 Leading Electromagnetic Interactions 12.1.2 Matching 12.1.3 Thomson Scattering 12.2 Multiple Particle Species ♠ 12.2.1 Atoms and the Coulomb Potential 12.2.2 Dipole Approximation 12.2.3 HQET 12.2.4 Particle-Antiparticle Systems 12.3 Neutral Systems 12.3.1 Polarizability and Rayleigh Scattering 12.3.2 Multipole Moments 12.4 Summary Exercises 13 First-QuantizedMethods 13.1 Effective Theories for Lumps ♦ 13.1.1 Collective Coordinates ♥ 13.1.2 Nonlinearly Realized Poincare´ Symmetry ♣ 13.1.3 Other Localized Degrees of Freedom 13.2 Point-Particle EFTs 13.2.1 Electromagnetic Couplings 13.2.2 Gravitational Couplings 13.2.3 Boundary Conditions I 13.2.4 Thomson Scattering Revisited 13.3 PPEFT and Central Forces ♠ 13.3.1 Boundary Conditions II 13.3.2 Contact Interaction 13.3.3 Inverse-Square Potentials: Fall to the Centre 13.3.4 Nuclear Effects in Atoms 13.4 Summary Exercises PartIV Many-BodyApplications 14 GoldstoneBosonsAgain 14.1 Magnons ♦ 14.1.1 Antiferromagnetism 14.1.2 Ferromagnetism 14.1.3 Physical Applications 14.2 Low-Energy Superconductors ♠ 14.2.1 Implications of the Goldstone Mode 14.2.2 Landau–Ginzburg Theory 14.3 Phonons ♣ 14.3.1 Goldstone Counting Revisited 14.3.2 Effective Action 14.3.3 Perfect Fluids 14.4 Summary Exercises 15 DegenerateSystems 15.1 Fermi Liquids ♦ 15.1.1 EFT Near a Fermi Surface 15.1.2 Irrelevance of Fermion Self-Interactions 15.1.3 Marginal Interactions 15.2 Superconductivity and Fermion Pairing ♠ 15.2.1 Phonon Scaling 15.2.2 Phonon-Coulomb Competition 15.3 Quantum Hall Systems ♣ 15.3.1 Hall and Ohmic Conductivity 15.3.2 Integer Quantum Hall Systems 15.3.3 Fractional Quantum Hall Systems 15.4 Summary Exercises 16 EFTsandOpenSystems 16.1 Thermal Fluids 16.1.1 Statistical Framework♥ 16.1.2 Evolution through Conservation 16.2 Open Systems 16.2.1 Density Matrices ♥ 16.2.2 Reduced Time Evolution♦ 16.3 Mean Fields and Fluctuations 16.3.1 The Mean/Fluctuation Split♦ 16.3.2 Neutrinos in Matter 16.3.3 Photons: Mean-Field Evolution♠ 16.3.4 Photons: Scattering and Fluctuations♠ 16.3.5 Domain of Validity of Mean-Field Theory 16.4 Late Times and Perturbation Theory ♣ 16.4.1 Late-Time Resummation 16.4.2 Master Equations 16.5 Summary Exercises AppendixA ConventionsandUnits AppendixB MomentumEigenstatesandScattering AppendixC QuantumFieldTheory:ACartoon AppendixD FurtherReading References Index