This book has its origin in a long series of lectures given at the Institute for Theoretical Physics, Warsaw University. It is addressed to graduate students and to young research workers in theoretical physics who have some knowledge of quantum field theory in its canonical formulation, for instance at the level of two volumes by Bjorken & Drell (1964, 1965). The book is intended to be a relatively concise reference to some of the field theoretical tools used in contemporary research in the theory of fundamental interactions. It is a technical book and not easy reading. Physical problems are discussed only as illustrations of certain theoretical ideas and of computational methods. No attempt has been made to review systematically the present status of the theory of fundamental interactions.

Preface to the First Edition

Preface to the Second Edition

0  Introduction

0.1  Gauge invariance

0.2  Reasons for gauge theories of strong and electroweak interactions

QCD

Electroweak theory

1  Classical fields, symmetries and their breaking

1.1  The action, equations of motion, symmetries and conservation laws

Equations of motion

Global symmetries

Space-time transformations

Examples

1.2  Classical field equations

Scalar field theory and spontaneous breaking of global symmetries

Spinor fields

1.3 Gauge field theories

U (1) gauge symmetry

Non-abelian gauge symmetry

1.4  From classical to quantum fields (canonicalquantization)

Scalar fields

The Feynman propagator

Spinor fields

Symmetry transformations for quantum fields

1.5 Discrete symmetries

Space reflection

Time reversal

Charge conjugation

Summary and the CPT transformation

CP violation in the neutral K-I-system

Problems

2  Path integral formulation of quantum field theory

2.1  Path integrals in quantum mechanics

Transition matrix elements as path integrals

Matrix elements of position operators

2.2  Vacuum-to-vacuum transitions and the imaginary time formalism

General discussion

Harmonic oscillator

Euclidean Green's functions

2.3  Path integral formulation of quantum field theory

Green's functions as path integrals

Action quadratic in fields

Gaussian integration

2.4  Introduction to perturbation theory

Perturbation theory and the generating functional

Wick's theorem

An example: four-point Green's function in .4

Momentum space

2.5  Path integrals for fermions; Grassmann algebra

Anticommuting c-numbers

Dirac propagator

2.6  Generating functionals for Green's functions and proper vertices; effective

potential

Classification of Green's functions and generating functionals

Effective action

Spontaneous symmetry breaking and effective action

Effective potential

2.7  Green's functions and the scattering operator

Problems

3  Feynman rules for Yang-Mills theories

3.1  The Faddeev-Popov determinant

Gauge invariance and the path integral

Faddeev-Popov determinant

Examples

Non-covariant gauges

3.2  Feynman rules for QCD

Calculation of the Faddeev-Popov determinant

Feynman rules

3.3  Unitarity, ghosts, Becchi-Rouet-Stora transformation

Unitarity and ghosts

BRS and anti-BRS symmetry

Problems

4  Introduction to the theory of renormalization

4.1  Physical sense of renormalization and its arbitrariness

Bare and 'physical' quantities

Counterterms and the renormalization conditions

Arbitrariness of renormalization

Final remarks

4.2  Classification of the divergent diagrams

Structure of the UV divergences by momentum power counting

Classification of divergent diagrams

Necessary counterterms

4.3  4: low.order renormalization

Feynman rules including counterterms

Calculation of Fig. 4.8(b)

Comments on analytic continuation to n # 4 dimensions

Lowest order renormalization

4.4  Effective field theories

Problems

5  Quantum electrodynamics

5.1  Ward-Takahashi identities

General derivation by the functional technique

Examples

5.2  Lowest order QED radiative corrections by the dimensional regularization

technique

General introduction

Vacuum polarization

Electron self-energy correction

Electron self-energy: IR singularities regularized by photon mass

On-shell vertex correction

5.3  Massless QED

5.4  Dispersion calculation of O(ot) virtual corrections in massless QED, in

(4 q: e) dimensions

Self-energy calculation

Vertex calculation

5.5  Coulomb scattering and the IR problem

Corrections of order ot

IR problem to all orders in

Problems

6  Renormalization group

6.1  Renormalization group equation (RGE)

Derivation of the RGE

Solving the RGE

Green's functions for rescaled momenta

 RGE in QED

6.2  Calculation of the renormalization group functions/3, F, Fm

6.3  Fixed points; effective coupling constant

Fixed points

Effective coUpling constant

6.4  Renormalization scheme and gauge dependence of the RGE

parameters

Renormalization scheme dependence

Effective ot in QED

Gauge dependence of the fl-function

Problems

7  Scale invariance and operator product expansion

7.1  Scale invariance

Scale transformations

Dilatation current

Conformal transformations

7.2  Broken scale invariance

General discussion

Anomalous breaking of scale invariance

7.3  Dimensional transmutation

7.4  Operator product expansion (OPE)

Short distance expansion

Light-cone expansion

7.5  The relevance of the light-cone

Electron-positron annihilation

Deep inelastic hadron leptoproduction

Wilson coefficients and moments of the structure function

7.6  Renormalization group and OPE

Renormalization of local composite operators

RGE for Wilson coefficients

OPE beyond perturbation theory

7.7  OPE and effective field theories

Problems

8  Quantum chromodynamics

8.1  General introduction

Renormalization and BRS invariance; counterterms

Asymptotic freedom of QCD

The Slavnov-Taylor identities

8.2  The background field method

8.3  The structure of the vacuum in non-abelian gauge theories

Homotopy classes and topological vacua

Physical vacuum

O-vacuum and the functional integral formalism

8.4  Perturbative QCD and hard collisions

Parton picture

Factorization theorem

8.5  Deep inelastic electron-nucleon scattering in first order QCD (Feynman

gauge)

Structure functions and Born approximation

Deep inelastic quark structure functions in the first order in the strong

coupling constant

Final result for the quark structure functions

Hadron structure functions; probabilistic interpretation

8,6  Light-cone variables, light-like gauge

8.7  Beyond the one-loop approximation

Comments on the IR problem in QCD

Problems

9  Chiral symmetry; spontaneous symmetry breaking

9.1  Chiral symmetry of the QCD lagrangian

9.2  Hypothesis of spontaneous chiral symmetry breaking in strong interactions

9.3  Phenomenological chirally symmetric model of the strong interactions

(model)

9.4  Goldstone bosons as eigenvectors of the mass matrix and poles of Green's

functions in theories with elementary scalars

Goldstone bosons as eigenvectors of the mass matrix

General proof of Goldstone's theorem

9.5  Patterns of spontaneous symmetry breaking

9.6  Goldstone bosons in QCD

10  Spontaneous and explicitglobal symmetry breaking

10.1 Internal symmetries and Ward identities

Preliminaries

Ward identities from the path integral

Comparison with the operator language

Ward identities and short-distance singularities of the operator

products

Renormalization of currents

10.2 Quark masses and chiral perturbation theory

Simple approach

Approach based on use of the Ward identity

10.3 Dashen's theorems

Formulation of Dashen's theorems

Dashen's conditions and global symmetry broken by weak gauge interactions

10.4: Electromagnetic rr+-rr mass difference and spectral function sum

rules

Electromagnetic rr+-:r mass difference from Dashen's formula

Spectral function sum rules

Results

11  Brout-Englert-Higgs mechanism in gauge theories

11.1 Brout-Englert-Higgs mechanism

1112 Spontaneous gauge symmetry breaking by radiative corrections

11.3 - Dynamical breaking of gauge symmetries and vacuum alignment

Dynamical breaking of gauge symmetry

Examples

Problems

12  Standard electroweak theory

12.1 The lagrangian

12.2 Electroweak currents and physical gauge boson fields

12.3 Fermion masses and mixing

12.4 Phenomenology of the tree level lagrangian

Effective four-fermion interactions

Z couplings

12.5 Beyond tree level

Renormalization and counterterms

Corrections to gauge boson propagators

Fermion self-energies

Running or(#) in the electroweak theory

Muon decay in the one-loop approximation

Corrections to the Z partial decay widths

12.6 Effective low energy theory for electroweak processes

QED as the effective low energy theory

12.7 Flavour changing neutral-current processes

QCD corrections to CP violation in the neutral kaon system

Problems

13  Chirai anomalies

13.1 Triangle diagram and different renormalization conditions

Introduction

Calculation of the triangle amplitude

Different renormalization constraints for the triangle amplitude

Important comments

13.2 Some physical consequences of the chiral anomalies

Chiral invariance in spinor electrodynamics

Chiral anomaly for the axial U(1) current in QCD; UA(1) problem

Anomaly cancellation in the SU(2) U(1) electroweak theory

Anomaly-free models

13.3 Anomalies and the path integral

Introduction

Abelian anomaly

Non-abelian anomaly and gauge invariance

Consistent and covariant anomaly

13.4 Anomalies from the path integral in Euclidean space

Introduction

Abelian anomaly with Dirac fermions

Non-abelian anomaly and chiral fermions

Problems

14  Effective lagrangians

14.1 Non-linear realization of the symmetry group

Non-linear a-model

Effective lagrangian in the a (x) basis

Matrix representation for Goldstone boson fields

14.2 Effective lagrangians and anomalies

Abelian anomaly

The Wess-Zumino term

Problems

15  Introduction to supersymmetry

15.1 Introduction

15.2 The supersymmetry algebra

15.3 Simple consequences of the supersymmetry algebra

15.4 Superspace and superfields for N = 1 supersymmetry

Superspace

Superfields

15.5 Supersymmetric lagrangian; Wess-Zumino model

15.6 Supersymmetry breaking

15.7 Supergraphs and the non-renormalization theorem

Appendix A: Spinors and their properties

Lorentz transformations and two-dimensional representations of the

group SL(2, C)

Solutions of the free Weyl and Dirac equations and their properties

Parity

Time reversal

Charge conjugation

Appendix B: Feynman rules for QED and QCD and Feynman integrals

Feynman rules for the .dP4 theory

Feynman rules for QED

Feynman rules for QCD

Dirac algebra in n dimensions

Feynman parameters

Feynman integrals in n dimensions

Gaussian integrals

>,-parameter integrals

Feynman integrals in light-like gauge n A = 0, n2 = 0

Convention for the logarithm

Spence functions

Appendix C: Feynman rules for the Standard Model

Propagators of fermions

Propagators of the gauge bosons

Propagators of the Higgs and Goldstone bosons

Propagators of the ghost fields

Mixed propagators (only counterterms exist)

+' Gaug interactions of fermions

Yukawa interactions of fermions

Gauge interactions of the gauge bosons

Self-interactions of the Higgs and Goldstone bosons

Gauge interactions of the Higgs and Goldstone bosons

Gauge interactions of the ghost fields

Interactions of ghosts with Higgs and Goldstone bosons

Appendix D: One-loop Feynman integrals

Two-point functions

Three- and four-point functions

General expressions for the one-loop vector boson self-energies

Appendix E: Elements of group theory

Definitions

Transformation of operators

Complex and real representations

Traces

model

References

Index