Uncovering Quantum Field Theory and the Standard Model

Description

This textbook provides an accessible introduction to quantum field theory and the Standard Model of particle physics. It adopts a distinctive pedagogical approach with clear, intuitive explanations to complement the mathematical exposition. The book begins with basic principles of quantum field theory, relating them to quantum mechanics, classical field theory, and statistical mechanics, before building towards a detailed description of the Standard Model. Its concepts and components are introduced step by step, and their dynamical roles and interactions are gradually established. Advanced topics of current research are woven into the discussion and key chapters address physics beyond the Standard Model, covering subjects such as axions, technicolor, and Grand Unified Theories. This book is ideal for graduate courses and as a reference and inspiration for experienced researchers. Additional material is provided in appendices, while numerous end-of-chapter problems and quick questions reinforce the understanding and prepare students for their own research. Cover Half-title page Reviews Title page Copyright page Contents Preface Intention of this Book Subjects of this Book Structure of this Book Acknowledgments Notations and Conventions Glossary Part I Quantum Field Theory Ouverture: Concepts of Quantum Field Theory Point Particles versus Fields at the Classical Level Particles versus Waves in Quantum Theory Classical and Quantum Gauge Fields Ultraviolet Divergences, Regularization, and Renormalization Euclidean Quantum Field Theory versus Classical Statistical Mechanics 1 Basics of Quantum Field Theory 1.1 From Point Particle Mechanics to Classical Field Theory 1.2 Quantum Mechanical Path Integral 1.3 Path Integral in Euclidean Time 1.4 Spin Models in Classical Statistical Mechanics 1.5 Quantum Mechanics versus Classical Statistical Mechanics 1.6 TransferMatrix 1.7 Lattice Field Theory Exercises 2 Scalar Field Theory and Canonical Quantization 2.1 Scalar Fields 2.2 Noether Current 2.3 From the Lagrangian to the Hamilton Density 2.4 Commutation Relations for the Scalar Field Operators 2.5 Hamilton Operator in Scalar Field Theory 2.6 Vacuum State and Vacuum Energy 2.7 Cosmological Constant Problem 2.8 Particle States and their Energies and Statistics 2.9 Momentum Operator Exercises 3 From Particles toWavicles and Back 3.1 Model for Ions Forming a Crystal 3.2 Phonon Creation and Annihilation Operators 3.3 Quantum States of a Vibrating Crystal 3.4 Phonons as Wavicles 3.5 Explicit Breaking of Continuous Translation Symmetry 3.6 Debye Field Theory of the Vibrating Solid 3.7 FromWavicles Back to Particles 3.8 What is Space? Exercises 4 Perturbative Scalar Field Functional Integral in Dimensional Regularization 4.1 FromMinkowski to Euclidean Space–Time 4.2 Euclidean Propagator and Contraction Rule 4.3 Perturbative Expansion of the Functional Integral 4.4 Dimensional Regularization 4.5 2-Point Function to 1 Loop 4.6 Mass Renormalization 4.7 Connected, Disconnected, and 1-Particle Irreducible Diagrams 4.8 Feynman Rules for the λφ[sup(4)] Model 4.9 4-Point Function to 1 Loop 4.10 Dimensional Regularization of J(p[sup(2)] 4.11 Renormalization of the Coupling 4.12 Renormalizability of Scalar Field Theories 4.13 Condition for Renormalizability Exercises 5 Renormalization Group 5.1 Locality and Hierarchies of Energy Scales 5.2 Renormalization Group Blocking and Fixed Points 5.3 Gaussian Fixed Points of Lattice Scalar Field Theory 5.4 Blocking from the Continuum to the Lattice 5.5 Perfect Lattice Actions on the Renormalized Trajectory 5.6 Wilson–Fisher Fixed Points 5.7 Renormalization of Scalar Field Theory in a Cut-off Regularization 5.8 Callan–Symanzik Equation 5.9 β-Function and Anomalous Dimension to 1 Loop 5.10 Running Coupling 5.11 Infrared and Ultraviolet Fixed Points Exercises 6 Quantization of the Free Electromagnetic Field 6.1 Vector Potential and Gauge Symmetry 6.2 From the Lagrangian to the Hamilton Density 6.3 Hamilton Operator for the Photon Field 6.4 Gauss Law 6.5 Vacuum and Photon States 6.6 Momentum Operator of the Electromagnetic Field 6.7 Angular Momentum Operator and Helicity of Photons 6.8 Planck’s Formula and the Cosmic Background Radiation 6.9 Gauge Fixing and Photon Propagator Exercises 7 Charged States in Scalar Quantum Electrodynamics 7.1 Complex Scalar Field with Global U(1) Symmetry 7.2 Scalar Quantum Electrodynamics 7.3 Charged Particles as Infraparticles 7.4 Superselection Sectors 7.5 Charged Particles in a Periodic Volume 7.6 C-periodic Boundary Conditions Exercises 8 Canonical Quantization of FreeWeyl, Dirac, and Majorana Fermions 8.1 MasslessWeyl Fermions 8.2 Momentum, Angular Momentum, and Helicity of Weyl Fermions 8.3 Fermion Number, Parity, and Charge Conjugation 8.4 Cosmic Background Radiation of Neutrinos 8.5 Massive Dirac Fermions 8.6 Massive Majorana Fermions 8.7 Massive Weyl Fermions 8.8 Redundant Particle Labels and the Pauli Principle as a “Gauss Law” 8.9 Can We Supersede Gauge Symmetry? Exercises 9 Fermionic Functional Integrals 9.1 Grassmann Algebra, Pfaffian, and Fermion Determinant 9.2 Dirac Equation 9.3 Weyl and Majorana Equations 9.4 Euclidean Fermionic Functional Integral 9.5 Euclidean Lorentz Group 9.6 Charge Conjugation, Parity, and Time Reversal for Weyl Fermions 9.7 C, P, and T Transformations of Dirac Fermions 9.8 CPT Invariance in Relativistic Quantum Field Theory 9.9 Connections between Spin and Statistics 9.10 Euclidean Time Transfer Matrix Exercises 10 Chiral Symmetry in the Continuum and on the Lattice 10.1 Chiral Symmetry in the Continuum 10.2 Lattice Fermion Doubling Problem 10.3 Nielsen–Ninomiya No-Go Theorem 10.4 Absence of Neutrinos on a Lattice 10.5 Wilson Fermions 10.6 Perfect Lattice Fermions and the Ginsparg–Wilson Relation 10.7 Overlap Fermions Exercises 11 Non-Abelian Gauge Fields 11.1 Non-Abelian Gauge Fields at the Classical Level 11.2 Gauge Fixing and Faddeev–Popov Ghosts 11.3 Becchi–Rouet–Stora–Tyutin Symmetry 11.4 Nilpotency and BRST Cohomology 11.5 Aharonov–Bohm Effect as an Analogue of BRST Cohomology 11.6 Lattice Gauge Theory 11.7 Canonical Quantization of Compact U(1) Lattice Gauge Theory 11.8 Canonical Quantization of Non-Abelian Lattice Gauge Theory 11.9 Functional Integral for Compact U(1) Lattice Gauge Theory 11.10 Functional Integral for Non-Abelian Lattice Gauge Theory Exercises Part II Construction of the Standard Model Intermezzo: Concepts of the Standard Model The Standard Model: A Non-Abelian Chiral Gauge Theory Renormalizability of Non-Abelian Gauge Theories Triviality and Incorporation of Gravity Fundamental Standard Model Parameters Hierarchies of Scales and Approximate Global Symmetries Local and Global Symmetries Explicit versus Spontaneous Symmetry Breaking Anomalies in Local and Global Symmetries Power of Lattice Field Theory 12 Spontaneous Breakdown of Global Symmetries: From CondensedMatter to Higgs Bosons 12.1 Effective Scalar Fields for Cold CondensedMatter 12.2 Vacua in the λ|φ|[sup(4)] Model 12.3 Higgs Doublet Model 12.4 Goldstone Theorem 12.5 Mermin–Wagner–Hohenberg–Coleman Theorem 12.6 Low-Energy Effective Field Theory 12.7 Hierarchy Problem 12.8 Solving the Hierarchy Problem with Supersymmetry? 12.9 Is Nature Natural? 12.10 Triviality of the Standard Model 12.11 Electroweak Symmetry Restoration at High Temperature 12.12 Extended Model with Two Higgs Doublets Exercises 13 Local Symmetry and the Higgs Mechanism: From Superconductivity to Electroweak Gauge Bosons 13.1 Higgs Mechanism in Scalar Electrodynamics 13.2 Higgs Mechanism in the Electroweak Theory 13.3 Identification of the Electric Charge 13.4 Accidental Custodial Symmetry 13.5 Variants of the Standard Model with Modified Gauge Symmetry 13.6 Scalar Electrodynamics on the Lattice 13.7 SU(2)[sub(L)] Gauge–Higgs Model on the Lattice 13.8 Small Electroweak Unification 13.9 Electroweak Symmetry Breaking in an SU(3) Unified Theory Exercises 14 Gluons: From Confinement to Deconfinement 14.1 Gluons in the Continuum and on the Lattice 14.2 Quark Confinement and the Wegner–Wilson Loop 14.3 Character Expansion and Group Integration 14.4 Strong Coupling Limit of Lattice Yang–Mills Theory 14.5 Asymptotic Freedom and Natural Continuum Limit 14.6 How Strong is the Strong Force? 14.7 Roughening Transition 14.8 Systematic Low-Energy Effective String Theory 14.9 Lüscher Term as a Casimir Effect 14.10 Cosmological Constant Problem on the StringWorld-Sheet 14.11 Gluon Confinement and the Fredenhagen–Marcu Operator 14.12 Glueball Spectrum 14.13 Polyakov Loop and Center Symmetry 14.14 Deconfinement at High Temperatures 14.15 Exceptional Confinement and Deconfinement in G(2) Yang–Mills Theory Exercises 15 One Generation of Leptons and Quarks 15.1 Electron and Left-Handed Neutrino 15.2 CP and T Invariance of Gauge Interactions 15.3 Fixing the Lepton Weak Hypercharges 15.4 Triangle Gauge Anomalies in the Lepton Sector 15.5 Witten’s Global SU(2)[sub(L)] Gauge Anomaly in the Lepton Sector 15.6 Up and Down Quarks 15.7 Anomaly Cancellation between Leptons and Quarks 15.8 Electric Charges of Quarks and Baryons 15.9 Anomaly Matching 15.10 Right-Handed Neutrinos 15.11 Lepton and Baryon Number Anomalies 15.12 Gauge Anomaly-Free Technicolor Model Exercises 16 Fermion Masses 16.1 Electron and Down Quark Masses 16.2 Up Quark Mass 16.3 NeutrinoMass from a Dimension-5 Operator 16.4 Mass Hierarchies of Fermions 16.5 Neutrino Mass Term and Reconsideration of CP 16.6 Lepton and Baryon Number Violation by Higher-Dimensional Operators 16.7 Charge Quantization, FermionMasses, and Consistency with Gravity 16.8 Dirac andMajoranaMasses from Right-Handed Neutrino Fields 16.9 Seesaw Mass-by-Mixing Mechanism 16.10 Right-Handed Neutrinos and Electric Charge Quantization 16.11 Lepton–BaryonMixing for N[sub(c)] =1 Exercises 17 Several Generations and Flavor Physics of Quarks and Leptons 17.1 Electroweak versus Mass Eigenstates 17.2 Generation-Specific Lepton Numbers and Lepton Universality 17.3 Cabibbo–Kobayashi–Maskawa Quark Mixing Matrix 17.4 Flavor-Changing Neutral Currents and the GIM Mechanism 17.5 CP Violation with Neutral Kaons and B-Mesons 17.6 Pontecorvo–Maki–Nakagawa–Sakata Lepton Mixing Matrix 17.7 Neutrino Oscillations 17.8 Overview of Fundamental Standard Model Parameters 17.9 Low-Energy Theory Perspective on the Standard Model Physics Exercises Part III Strong Interaction 18 Quantum Chromodynamics 18.1 Deconstructing the Standard Model 18.2 Asymptotic Freedom 18.3 Structure of Chiral Symmetry 18.4 Dynamical Realization of Chiral Symmetry 18.5 Lattice QCD 18.6 Ginsparg–Wilson Relation and Lüscher’s Lattice Chiral Symmetry 18.7 Under-Appreciated Fermionic Hierarchy Problem 18.8 DomainWall Fermions and a Fifth Dimension of Space–Time Exercises 19 Topology of Gauge Fields 19.1 Adler–Bell–Jackiw Anomaly 19.2 Topological Charge 19.3 Topology of a Gauge Field on a Compact Manifold 19.4 SU(2) Instanton 19.5 θ-Vacuum States 19.6 Analogy with Energy Bands in a Periodic Crystal 19.7 Some Questions Related to θ 19.8 Atiyah–Singer Index Theorem 19.9 Zero-Mode of the SU(2) Instanton 19.10 Index Theorem on the Lattice Exercises 20 U(1)[sub(A)]-Problem 20.1 Nature of the Problem 20.2 QCD in the Large-N[sub(c)] Limit 20.3 Witten–Veneziano Formula for the ηˊ-Meson Mass 20.4 Topological Susceptibility from Lattice Gauge Theory Exercises 21 Spectrum of Light Baryons and Mesons 21.1 Isospin Symmetry 21.2 Nucleon and Δ-Isobar 21.3 Anti-Quarks and Mesons 21.4 Strange Hadrons 21.5 Gell-Mann–Okubo BaryonMass Formula 21.6 Meson Mixing 21.7 Hadron Spectrum from Lattice QCD 21.8 Hadrons for N[sub(c)] =5 Exercises 22 Partons and Hard Processes 22.1 Electron–Positron Annihilation into Hadrons 22.2 R-Ratio as Evidence for N[sub(c)] =3 22.3 Deep-Inelastic Electron–Nucleon Scattering 22.4 Deep-Inelastic Neutrino–Nucleon Scattering 22.5 Sum Rules Exercises 23 Chiral Perturbation Theory 23.1 Effective Theory for Pions, Kaons, and the η-Meson 23.2 Masses of Pseudo-Nambu–Goldstone Bosons 23.3 Low-Energy Effective Theory for Nambu–Goldstone Bosons and Photons 23.4 Electromagnetic Corrections to the Nambu–Goldstone Boson Masses 23.5 Effective Theory for Nucleons and Pions 23.6 QCD Contributions to the W- and Z-BosonMasses 23.7 Remarks about Technicolor 23.8 Hypothesis of Minimal Flavor Violation Exercises 24 Topology of Nambu–Goldstone Boson Fields 24.1 Skyrmions 24.2 Anomaly Matching for N[sub(f)] = 2 24.3 G-Parity and its Explicit Breaking 24.4 Electromagnetic Decay of the Neutral Pion 24.5 Evidence for N[sub(c)] = 3 from π[sup(0)] → γγ ? 24.6 Wess–Zumino–Novikov–Witten Term 24.7 Intrinsic Parity and Its Anomalous Breaking 24.8 Electromagnetic Interactions of Pions, Kaons, and η-Mesons 24.9 Electromagnetic Interactions of Nambu–Goldstone Bosons for N[sub(f)] ≥ 3 24.10 Can One See the Number of Colors? 24.11 Techni-Baryons, Techni-Skyrmions, and Topological Dark Matter Exercises Part IV Selected Topics beyond the Standard Model 25 Strong CP-Problem 25.1 Rotating θ into the Mass Matrix 25.2 θ-Angle in Chiral Perturbation Theory 25.3 θ-Angle at Large Nsub(c)] 25.4 Peccei–Quinn Symmetry 25.5 U(1)[sub(PQ)] Symmetry Breaking and the Axion 25.6 Astrophysical and Cosmological Axion Effects 25.7 Elimination of the Weak SU(2)[sub(L)] Vacuum-Angle 25.8 Is there an Electromagnetic CP-Problem? Exercises 26 Grand Unified Theories 26.1 Minimal SU(5) Model 26.2 Fermion Multiplets 26.3 Lepton–Quark Transitions and Proton Decay 26.4 Baryon Asymmetry in the Universe 26.5 Thermal Baryon Number Violation in the Standard Model 26.6 Topological Excitations as Cosmic Relics 26.7 ’t Hooft–Polyakov Monopole and Callan–Rubakov Effect 26.8 Dirac–Schwinger–Zwanziger Dyon Quantization Condition 26.9 Julia–Zee Dyon andWitten Effect 26.10 Fermion Masses and the Hierarchy Problem 26.11 Spin(10) Structure 26.12 Neutrino Masses in the Spin(10) GUT 26.13 Small Unification with SU(3), G(2), Spin(6), or Spin(7) 26.14 Grand or not so Grand Unification? Exercises Finale Appendix A Highlights in the Development of Particle Physics A.1 Development of Experimental High-Energy Physics A.2 Development of Quantum Field Theory and the Standard Model Appendix B Units, Hierarchies, and Fundamental Parameters B.1 Man-Made versus Natural Units B.2 Energy Scales and Particle Masses Appendix C Structure of Minkowski Space–Time C.1 Lorentz Transformations C.2 Gradient as a 4-Vector and d’Alembert Operator Appendix D Relativistic Formulation of Classical Electrodynamics D.1 Current and Vector Potential D.2 Field Strength Tensor D.3 Maxwell Equations D.4 Space–Time Scalars from Field Strength Tensors D.5 Transformation of Electromagnetic Fields D.6 Action and Euler–Lagrange Equation D.7 Energy–Momentum Tensor Appendix E From the Galilei to the Poincaré Algebra E.1 Galilei Algebra E.2 Poincaré and Lorentz Algebras Appendix F Lie Groups and Lie Algebras F.1 Definition of a Lie Algebra F.2 Simple and Semi-Simple Lie Algebras F.3 Representations of Lie Algebras F.4 Lie Algebra so(3) and its Representations F.5 Unitary Group SU(2) versus Orthogonal Group SO(3) F.6 Unitary Group SU(n) and its Algebra su(n) F.7 Group SU(3) and its Algebra su(3) F.8 Permutation Group S[sub(N)] F.9 su(n) Representations and Young Tableaux F.10 Tensor Product of su(n) Representations F.11 Tensor Product of {3} and {[sub(bar)3} in su(3) F.12 Orthogonal Group SO(n) and its Algebra so(n) F.13 Symplectic Group Sp(n) and its Algebra sp(n) F.14 Exceptional Group G(2) and its Algebra g(2) F.15 Graphical Method for Tensor Product Reduction Appendix G Homotopy Groups and Topology G.1 Maps from S[sup(d)] to S[sup(n)] G.2 Topological Charge in 2-d Abelian Gauge Theory G.3 Homotopy Groups of Lie GroupManifolds Appendix H Monte Carlo Method H.1 Concept of a Markov Chain H.2 Detailed Balance H.3 Ergodicity and its Implications H.4 Convergence to the Stationary Distribution H.5 Metropolis Algorithm H.6 Error Analysis H.7 Critical Slowing Down H.8 Supercritical Slowing Down and Sign Problems H.9 Complexity Classes and the Severity of Sign Problems H.10 Quantum Computation and Simulation of Real-Time Evolution Appendix I Phase Transitions and Critical Phenomena I.1 Phase Transitions and Critical Points I.2 Critical Exponents I.3 Universal Critical Behavior I.4 Scaling Hypothesis I.5 Critical Exponents and Scaling Laws: An Overview References Author Index Subject Index