quantum field theory and critical phenomena

Quantum Field Theoryand

Critical Phenomena

J. Zinn-JustinCEN-Saclay

CLARENDON PRESS • OXFORD1989

Contents

1 Algebraic Preliminaries 11.1 The Gaussian Integral 11.2 Perturbation Theory . 31.3 Complex Structures 41.4 A Useful Identity 51.5 Functional Differentiation 61.6 Determinants of Operators 61.7 Differentiation in Grassmann Algebras 71.8 Integration in Grassmann Algebras 91.9 Gaussian Integrals with Grassmann Variables 13

Bibliographical Notes 172 Euclidean Path Integrals in Quantum Mechanics 18

2.1 General Remarks 182.2 Path Integrals: The General Idea 192.3 Hamiltonians of the Form p2/2m+ V (q,t) 202.4 More General Hamiltonians 232.5 Hamiltonians Quadratic in Momentum Variables 262.6 Explicit Evaluation of a Simple Path Integral: The Harmonic

Oscillator in an External Force 292.7 Perturbed Harmonic Oscillator 332.8 Correlation Functions 362.9 The 2-Point Function 38

2.10 Fermionic Path Integral . . . . < • . 382.11 Quantum Mechanics in Real Time 422.12 Calculation of the 5-Matrix 432.13 Application: The Eikonal Approximation 45

Bibliographical Notes 47Exercises 48

Appendix 51A2.1 The Spectrum of the 0 (2) Symmetric Rigid Rotator 51A2.2 The Spectrum of the O(N) Symmetric Rigid Rotator . . . . 52A2.3 Perturbation Theory in the Operator Formalism 55

3 Stochastic Differential Equations: Langevin, Fokker-PlanckEquations 57

3.1 The Langevin Equation 573.2 The Fokker-Planck Equation 58

xii Contents

3.3 Equilibrium Distribution 613.4 A Special Example 623.5 Path Integral Representation 643.6 A Simple Example: The Linear Langevin Equation 663.7 Stochastic Processes on Riemannian Manifolds 68

Bibliographical Notes . 71Exercises 72

Appendix Markov's Stochastic Processes: A Few Remarks 73A3.1 The Spectrum of the Transition Matrix 74A3.2 Detailed Balance 76A3.3 Continuous Time Limit 77A3.4 Construction of a Stochastic Process with a Given Equilibrium

Distribution 78A3.5 Discretized Langevin Equation 79

4 Functional Integrals in Field Theory 824.1 The Functional Integral 824.2 ' Perturbative Expansion of Functional Integrals 854.3 Wick's Theorem and Feynman Diagrams 884.4 Algebraic Identities and Functional Integrals 914.5 Field Theories with Fermions 98


Appendix Euch'dean Dirac Fermions and 7 Matrices 102A4.1 Dirac 7 Matrices l% 102A4.2 An Explicit Construction 103A4.3 Transformation of Spinors 104A4.4 The Matrix 7s 106A4.5 Reflections 107A4.6 Other Symmetries 107A4.7 Continuous Internal Symmetries 109A4.8 Trace of Products of 7 Matrices 110A4.9 The Fierz Transformation 112

A4.10 Connection between Spin and Statistics 1135 Generating Functionals of Correlation Functions. Loopwise

Expansion . 1145.1 Generating Functionals of Connected Correlation Functions:

Cluster Properties 1145.2 Proper Vertices 1175.3 Semiclassical or Loopwise Expansion 1215.4 Legendre Transformation and 1-Irreducibility 1275.5 Physical Interpretation of the 1PI Functional 129

Bibliographical Notes . . . » 130Exercises 130

Appendix 131A5.1 Higher Orders in the Loop Expansion 131

Contents . xiii

A5.2 Decay of Connected Correlation Functions 1336 D i v e r g e n c e s i n P e r t u r b a t i o n T h e o r y , P o w e r C o u n t i n g . . . . 1 3 5

6.1 The Problem of Divergences and Renormalization:The 4? Field Theory at One-Loop Order 136

6.2 Divergences in Perturbation Theory: General Analysis . . . 1406.3 Classification of Renormalizable Field Theories 1456.4 Power Counting for Operator Insertions 148

Bibliographical Notes 1507 Regularization Methods 151

7.1 Cut-Off and Pauli-Villars Regularization 1517.2 Lattice Regularization . 1557.3 Dimensional Regularization ". 158

Bibliographical Notes 164Appendix Schwinger's Proper Time Regularization 1658 Introduction to Renormalization Theory. Renormalization

Group Equations 1698.1 Power Counting. Renormalized Action 1698.2 One-Loop Divergences 1738.3 Divergences Beyond One-Loop: Skeleton Diagrams 1758.4 Bare and Renormalized Correlation Functions, Operator 0 2

Insertions 1798.5 Callan-Symanzik Equations 1818.6 Inductive Proof of Renormalizability 1848.7 The (<t>2<l>2) Correlation Function 1898.8 The Renormalized Action 1908.9 The Massless Theory 192

8.10 Homogeneous Renormalization Group (RG) Equations in aMassive Theory 196

8.11 Covariance of RG Functions 197Bibliographical Notes 198Exercises 200

Appendix 201A8.1 Super-renormalizable Field Theories • 201A8.2 Background Field Method and the S-Matrix . 204

9 Dimensional Regularization and Minimal Subtraction:Calculation of RG Functions 208

9.1 Bare and Renormalized Actions: Renormalization Group (RG)Functions 208

9.2 Dimensional Regularization: The Form of RenormalizationConstants 210

9.3 Minimal Subtraction Scheme 2119.4 The Massless Theory 2139.5 A Few Technical Remarks 2149.6 Two-Loop Calculation of Renormalization Constants and RG

Functions in the <j>* Field Theory 215

xiv Contents

9.7 Generalization to Several Component Fields 2219.8 Renormalization and RG Functions at One-Loop in a

Theory with Bosons and Fermions 225Bibliographical Notes 230Exercises 230

10 Renormalization of Composite Operators. Short DistanceExpansion 231

10.1 Renormalization of Operator Insertions 23110.2 Equations of Motion 23610.3 Short Distance Expansion (SDE) of Operator Products . . . 24010.4 Large Momentum Expansion of the SDE Coefficients . . . -. 24510.5 Callari-Symanzik Equation for the First Coefficient of the SDE 24610.6 SDE beyond Leading Order 24810.7 SDE of Products of Arbitrary Local Operators 24910.8 Light Cone Expansion (LCE) of Operator Products . . . . 250


11 Linearly Realized Symmetries and Renormalization 25211.1 Conventions and Notations 25211.2 Simple Algebraic Remarks 25311.3 Linearly Realized Global Symmetries 25411.4 Linear Symmetry Breaking 25811.5 Spontaneous Symmetry Breaking 26211.6 Quadratic Symmetry Breaking 26411.7 Application: Chiral Symmetry 270


Appendix Currents and Noether's Theorem 280A 11.1 Real Time Classical Field Theory ' 2 8 0A11.2 Euclidean Field Theory 281All.3 The Energy Momentum Tensor 282All.4 Energy Momentum Tensor and Euclidean Field Theory . . . 284A11.5 Dilatation and Conformal Invariance 28512 Non-Linearly Realized Symmetries: The Example of the

Non-Linear cr-Model . „. . ' 28812.1 The Non-Linear <r-Model: Definition 28812.2 Perturbation Theory 29012.3 Power Counting 29212.4 Lattice Regularization and Statistical Mechanics 29312.5 Perturbative Regularizations 29412.6 Infrared (IR) Divergences 29512.7 WT Identities 29612.8 Renormalization . 29912.9 Solution of the WT Identities 301

12.10 A Few Remarks 303

Contents xv

12.11 Renormalization of Dimensionless Operators andParametrization of the Sphere 305

12.12 The Linearized Representation 30612.13 IR Finiteness of O (N) Invariant Correlation Functions

in 2 Dimensions 307Bibliographical Notes 308

1 3 M o d e l s o n H o m o g e n e o u s S p a c e s i n T w o D i m e n s i o n s . . . . 3 0 913.1 Construction of the Model: Linear Coordinates 30913.2 Quantization and Perturbation Theory near Dimension 2 . . 31213.3 WT Identities 31213.4 The Field Renormaliza"tions . : 31413.5 Homogeneous Spaces: Arbitrary Parametrization 31513.6 Explicit Construction 32113.7 Generalizations 326

Bibliographical Notes 327Appendix Tensorial Analysis on Riemannian Manifolds 329A13.1 Tensors 329A13.2 Covariant Derivatives: Affine Connection . 330A13.3 The Curvature (Riemann) Tensor 332A13.4 Covariant Volume Elements 333A13.5 Homogeneous Spaces: Tensors and Lie Algebra 334A13.6 Manifolds Embedded in Euclidean Space 335A13.7 Geometrical Interpretation: Parallel Transport 336A13.8 Fermions, the Vielbein 34014 Symmetric Spaces: Non-Local Conservation Laws,

Renormalization Group 34314.1 The Classical Action 34414.2 Quantization and Perturbation Theory 34614.3 Coupling Constant RG Function at One-Loop Order . . . . 34714.4 One-Loop /^-Function and Background Field Method . . . . 350

Bibliographical Notes 352Appendix Symmetric Spaces: Classification 353A14.1 Definition 353A14.2 A Basic Property 354A14.3 Classification of Symmetric Spaces 355A14.4 Expression of the Metric Tensor for Symmetric Spaces . . . 35715 Slavnov-Taylor and BRS Symmetry. Stochastic Field Equations 358

15.1 Slavnov-Taylor (ST) Symmetry 35815.2 BRS Symmetry 36015.3 Grassmann Coordinates 36215.4 Relation between ST and BRS Symmetry 36315.5 BRS Symmetry and Group Manifolds 36415.6 Stochastic Equations 36515.7 Consequences of BRS Symmetry: General Discussion . . . . 36715.8 Gradient Equations 369

xvi Contents

15.9 Extension of Grassmannian Symmetries 37015.10 Application: Stochastic Field Equations 372

Bibliographical Notes 3761 6 R e n o r m a l i z a t i o n a n d S t o c h a s t i c D y n a m i c a l E q u a t i o n s . . . . 3 7 8

16.1 Langevin and Fokker-Planck Equations 37816.2 Time-Dependent Correlation Functions 38016.3 ST Symmetry and Equilibrium Correlation Functions . . . 38216.4 Renormalization and BRS Symmetry 38516.5 The Purely Dissipative Langevin Equation 38616.6 Stochastic Field Equations on Group Manifolds: The Langevin

Equation for Chiral Fields 39016.7 The Langevin Equation on Riemannian Manifolds 39316.8 Equilibrium Correlation Functions 396

Bibliographical Notes 398Appendix Renormalization Constants and RG Functions at Two-Loops:

Supersymmetric Perturbation Theory 399A16.1 The (<t>2)2 Field Theory 399A16.2 Perturbative Calculation of t]u (g) 400A16.3 The Non-Linear <r-Model 40217 Abelian Gauge Theories 406

17.1 The Massive Vector Field . 40617.2 Local Abelian Symmetry 40817.3 The Massless Gauge Field , . 41117.4 Perturbation Theory, Regularization 41117.5 WT Identities, Renormalization 41517.6 Gauge Dependence 41717.7 Physical Observables and Massless Gauge Field 42117.8 The Abelian Higgs Mechanism 42117.9 Quantization of the Higgs Mechanism 423

17.10 Physical Observables. Unitarity of the S-Matrix 42517.11 Massless Gauge Field: Hamiltonian Formalism 42617.12 More General Gauges 429

Bibliographical Notes 433Appendix RG Equations. Calculations at One-Loop Order . . . . 435A17.1 The RG Equations . . .„ 435A17.2 The One-Loop /3-Function 43618 Non-Abelian Gauge Theories 440

18.1 Geometrical Construction 44018.2 The Gauge Invariant Action 44318.3 The Faddeev-Popov Quantization 44418.4 Hamiltonian Formalism 44618:5 BRS Symmetry 44818.6 Perturbation Theory, Regularization 44918.7 The Non-Abelian Higgs Mechanism 45118.8 The Standard Model of Weak Electromagnetic Interactions . 456

•Contents xvii

18.9 Quantum Chromodynamics (QCD) 462Bibliographical Notes 468

Appendix Anomalies 470A18.1 The Abelian Anomaly 470A18.2 Non-Abelian Anomaly 478A18.3 Physical Consequences 480A18.4 Wess-Zumino Consistency Conditions 48219 Renormalization of Gauge Theories: General Formalism . . . 483

19.1 Notations and Group Structure 48319.2 Quantization 48419.3 BRS Symmetry 48619.4 WT Identities . . 48919.5 Renormalization 49119.6 Solution of WT Identities: General Considerations 49219.7 The Renormalized Action 49419.8 Gauge Independence . 499.

Bibliographical Notes 50020 Critical Phenomena: General Considerations 502

20.1 Introduction 50220.2 Phase Transitions and Transfer Matrix 50420.3 The Infinite Transverse Size Limit: Ising-Like Systems . . . 50720.4 Order Parameter and Cluster Properties 51020.5 Stochastic Processes and Phase Transitions 51220.6 Example of Continuous Symmetries 513

Bibliographical Notes 51521 Mean Field Theory for Ferromagnetic Systems 516

21.1 Ising-like Ferromagnetic Systems 51621.2 Mean Field Expansion . 51721.3 Mean Field Approximation 51921.4 Universality within the Mean Field Approximation 52221.5 Beyond the Mean Field Approximation 52721.6 Power Counting and the Role of Dimension Four 52921.7 Tricritical Points 531


Appendix Mean Field Theory: General Formalism 533A21.1 Mean Field Theory 533A21.2 Mean Field Expansion 536A21.3 High Temperature Series and Mean Field 537A21.4 Low Temperature and Mean Field 53822 General Renormalization Group Analysis. The Critical Theory

near Dimension Four 54022.1 The Abstract Renormalization Group: General Formulation . 54122.2 The Gaussian Fixed Point 545

xviii Contents

22.3 The Critical Theory near Dimension Four:The Effective Field Theory 547

22.4 Renormalization Group Equations for the Critical Theory:The e-Expansion 549

22.5 Solution of the RG Equations 552Bibliographical Notes 555Exercises 556

23 Scaling Behaviour in the Critical Domain 55723.1 Strong Scaling above Tc: The Renormalized Theory . . . . 55723.2 Critical Correlation Functions with <f>2 (x) Insertions . . . . 56223.3 Expansion around the Critical Theory 56423.4 Scaling Laws above Tc 56523.5 Correlation Functions with <j>2 Insertions 56723.6 Scaling Laws in a Magnetic Field: The Equation of State . . 56823.7 Correlation Functions below Tc 57123.8 The AT-Vector Model 57223.9 Asymptotic Expansion of the 2-Point Function in the

Critical Domain 578Bibliographical Notes 580

Appendix The Scaling Behaviour of the Specific Heat for a = 0 . . 58224 Corrections to Scaling Behaviour 584

24.1 Corrections to Scaling below Four Dimensions 58424.2 Logarithmic Corrections at the Upper Critical Dimension . . 58624.3 Irrelevant Operators and the Question of Universality . . . 58924.4 Corrections to Scaling Coming from Irrelevant Operators . . 59224.5 Fixed Point in Hamiltonian Space and Improved Actions . . 59324.6 Application: Uniaxial Systems with Strong Dipolar Forces . 594


Appendix General RG Equations 601A24.1 A General Equivalence 601A24.2 Large Momentum Mode Partial Integration and RG Equations 60225 Calculation of Universal Quantities 604

25.1 The e-Expansion 60525.2 The Perturbative Expansion at Fixed Dimension 61325.3 The Series Summation . ". 61525.4 Numerical Estimates of Critical Exponents 61825.5 Comparison with Lattice Model Estimates 62025.6 Critical Exponents from Experiments 62125.7 Amplitude Ratios 622

Bibliographical Notes 623Appendix Non-Magnetic Statistical Models and the (4>2) Field

Theory . 626A25.1 Statistics of Self-Repelling Chains, Approximations . . . . 626A25.2 Equivalence with the <f Field Theory 628

Contents xix

A25.3 RG Approach to SAW and Statistics of Polymers 629A25.4 Liquid-Vapour Phase Transition and Field Theory 6312 6 T h e (<f>2)2 F i e l d T h e o r y i n t h e L a r g e N L i m i t . . . . . . . . 6 3 5

26.1 The Large N Limit 63626.2 The 1/N Expansion 64026.3 Calculations and Results 64126.4 Generalizations 644


27 Ferromagnetic Order at Low Temperature: the Non-Linearer-Model 646

27.1 The [<£2(x)]2 Field Theory at Low Temperature 64727.2 The Non-Linear a-Model: RG Equations 64927.3 Discussion of the RG Equations 65127.4 Results beyond One-Loop ; 65427.5 The Dimension Two 65627.6 The Large N Limit 65727.7 Generalizations 659

Bibliographical Notes 66028 The 0(2) Non-Linear a-Model 661

28.1 The Spin Correlation Functions 66228.2 Correlation Functions in a Field 66428.3 The Coulomb Gas in 2 Dimensions 66728.4 0(2) Non-Linear <r-Model and Coulomb Gas 67128.5 The Critical 2-point Function in the 0(2) Model 67228.6 The Massive Thirring Model 67428.7 A Two-Fermion Model 677

Bibliographical Notes 678Appendix Algebraic Identities in 2D Massless Fermion Models . . . 679A28.1 Sine-Gordon and Thirring Models 679A28.2 The 2-Fermion Model 68029 Critical Dynamics 682

29.1 Introduction 68229.2 Renormalization and RG Equations near 4 Dimensions:

The Purely Dissipative Case 68429.3 RG Equations near 2 Dimensions: The Purely Dissipative Case 68829.4 Conserved Order Parameter (Model B) 69029.5 Relaxational Model with Energy Conservation (Model C) . . 69129.6 A Non-Relaxational Model (Model E) 694

Bibliographical Notes 6983 0 F i e l d T h e o r y i n a F i n i t e G e o m e t r y : F i n i t e S i z e S c a l i n g . . . 6 9 9

30.1 Renormalization Group in Finite Geometries 69930.2 Perturbation Theory in Finite Geometries: Periodic Boundary

Conditions 70230.3 The Periodic Hypercube 704

xx Contents

30.4 The Cylindrical Geometry 70930.5 Finite Size Effects in the Non-Linear (7-Model . 71330.fr The Free Energy a t Fixed Order Parameter Average . . . . 71930.7 Discrete Symmetries and Finite Size Effects 72030.8 Finite Size Effects and Dynamics 723

Bibliographical Notes 72931 Critical Properties of Gauge Theories 730

31.1 Gauge Invariance on the Lattice 73031.2 The Pure Gauge Action. The Partition Function 73231.3 Low Temperature Analysis 73331.4 Wilson's Loop and Confinement 73531.5 Mean Field Theory 742

Bibliographical Notes 745Appendix Gauge Theory and Confinement in Two Dimensions . . . 747A31.1 Lattice Gauge Theories 747A31.2 The Schwinger Model 74832 Large Momentum Behaviour in Field Theory 753

32.1 Large Momentum Behaviour in the (<j>2)2 Field Theory . . . 75332.2 The General </>* Field Theory 75832.3 Theories with Scalar Bosons and Fermions 76032.4 Gauge Theories 76232.5 Applications: The Theory of Strong Interactions 765

Bibliographical Notes 76833 Instantons in Quantum Mechanics: The Anharmonic Oscillator 770

33.1 Introduction 77033.2 The Anharmonic Oscillator for Negative Coupling:

Preliminary Considerations 77133.3 A Simple Integral 773

.33.4 Quantum Mechanics: The Saddle Point 77433.5 Instanton Contribution at Leading Order 776


Appendix The Calculation of the Jacobian 78334 Quantum Mechanics: Generalization 785

34.1 The Instanton Contribution 78534.2 Calculation of the Determinant: The Shifting Method . . . 78734.3 The Large 0 Limit 793

Bibliographical Notes 794Exercises . . • 794

Appendix The WKB Method and Instantons 795A34.1 The Classical Equations of Motion 795A34.2 T h e W K B M « t h o d 797A34.3 The Average Action in Pa th Integrals 80035 Unstable Vacua in Field Theory 802

35.1 The <pA Field Theory 802

Contents xxi

35.2 General Scalar Field Theories 80835.3 The Decay of the False Vacuum: Cosmological Interpretation. 811


Appendix Sobolev Inequalities 81436 Degenerate Classical Minima and Instantons 817

36.1 The Double-Well Potential 81736.2 The Periodic Cosine Potential 82036.3 Instantons in Stable Boson Field Theories: General Remarks 82336.4 Instantons in CP{N - 1) Models 82536.5 Instantons in the SU(2) Gauge Theory 826

Bibliographical Notes 830Appendix 831A36.1 Trace Formula for Periodic Potentials 831A36.2 Instantons and Stochastic Differential Equations 83237 Perturbation Theory at Large Orders and Instantons. *

The Summation Problem 83537.1 Quantum Mechanics 83537.2 Field Theory 83837.3 Divergent Series, Borel Summability, Summation Methods . 84037.4 Large Order Behaviour and Borel Summability 84137.5 Practical Summation Methods 843

Bibliographical Notes 847Appendix 849A37.1 Large Order Behaviour for Simple Integrals 849A37.2 Non-Loopwise Expansions 850A37.3 Large Coupling Behaviour of the Anharmonic Oscillator . . 850A37-4 Linear Differential Approximants 85138 The <̂ 4 Field Theory in Dimension Four 853

38.1 The Euclidean Equation of Motion 85338.2 Integration around the Saddle Point 85538.3 The Jacobian 85638.4 The Determinant 85738.5 The Coupling Constant Renormalization 85938.6 The Imaginary Part of the n-Point Function 86138.7 Large Order Behaviour 86238.8 The Problem of Renormalons 86338.9 The Massive Theory 866


Appendix 868A38.1 Instantons and RG Equations 868A38.2 Conformal Invariance 86939 Fermions and Large Order Behaviour 871

39.1 Example of a Yukawa-like Field Theory 871

xxii Contents

39.2 Evaluation of the Fermion Determinant for Large Fields . . 87239.3 The Large Order Behaviour 87539.4 The Case of QED 876

Bibliographical Notes 87740 Multi-lnstantons in Quantum Mechanics 878

40.1 The Role of Multi-lnstantons 87840.2 The Double-Well Potential 87940.3 The Periodic Cosine Potential 88940.4 General Potentials with Degenerate Minima 89240.5 The O (u) Symmetric Anharmonic Oscillator 89640.6 A General Conjecture 899


Appendix 902A40.1 The Determinant 902A40.2 The Instanton Interaction 903A40.3 Constraints 905A40.4 A Simple Example of Non-Borel Summability 906Index 909

quantum field theory and critical phenomena

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