Book contents
- Frontmatter
- Contents
- Preface to the First Edition
- Preface to the Second Edition
- 0 Introduction
- 1 Classical fields, symmetries and their breaking
- 2 Path integral formulation of quantum field theory
- 3 Feynman rules for Yang–Mills theories
- 4 Introduction to the theory of renormalization
- 5 Quantum electrodynamics
- 6 Renormalization group
- 7 Scale invariance and operator product expansion
- 8 Quantum chromodynamics
- 9 Chiral symmetry; spontaneous symmetry breaking
- 10 Spontaneous and explicit global symmetry breaking
- 11 Higgs mechanism in gauge theories
- 12 Standard electroweak theory
- 13 Chiral anomalies
- 14 Effective lagrangians
- 15 Introduction to supersymmetry
- Appendix A Spinors and their properties
- Appendix B Feynman rules for QED and QCD and Feynman integrals
- Appendix C Feynman rules for the Standard Model
- Appendix D One-loop Feynman integrals
- Appendix E Elements of group theory
- References
- Index
13 - Chiral anomalies
Published online by Cambridge University Press: 13 January 2010
- Frontmatter
- Contents
- Preface to the First Edition
- Preface to the Second Edition
- 0 Introduction
- 1 Classical fields, symmetries and their breaking
- 2 Path integral formulation of quantum field theory
- 3 Feynman rules for Yang–Mills theories
- 4 Introduction to the theory of renormalization
- 5 Quantum electrodynamics
- 6 Renormalization group
- 7 Scale invariance and operator product expansion
- 8 Quantum chromodynamics
- 9 Chiral symmetry; spontaneous symmetry breaking
- 10 Spontaneous and explicit global symmetry breaking
- 11 Higgs mechanism in gauge theories
- 12 Standard electroweak theory
- 13 Chiral anomalies
- 14 Effective lagrangians
- 15 Introduction to supersymmetry
- Appendix A Spinors and their properties
- Appendix B Feynman rules for QED and QCD and Feynman integrals
- Appendix C Feynman rules for the Standard Model
- Appendix D One-loop Feynman integrals
- Appendix E Elements of group theory
- References
- Index
Summary
Triangle diagram and different renormalization conditions
Introduction
Anomalies have already been mentioned in this book on several occasions. In this chapter we systematically discuss the fermion anomaly in (3+1)-dimensional quantum field theory (Adler 1969, Bell & Jackiw 1969). Its existence can be traced back to the short-distance singularity structure of products of local operators.
To be specific let us consider QCD. As discussed in Chapter 9, apart from being invariant under gauge transformations in the colour space its lagrangian is also invariant under the global SU(N) × SU(N) × U(1) × UA(1) chiral group of transformations acting in the flavour space.† Fermions belong to a vector-like, i.e. real, representation of the gauge group and for N > 2 to a complex representation of the chiral group (see Appendix E). We know from Chapters 9 and 10 that the Noether currents corresponding to the global flavour symmetry, although external with respect to the strong interaction gauge group, acquire important dynamical sense. The axial non-abelian currents couple to Goldstone bosons (pseudoscalar mesons) and the left-handed chiral currents couple to the intermediate vector bosons, i.e. they are gauge currents of the weak interaction gauge group. Moreover, the conservation of the U(1) current corresponds to baryon number conservation whereas the conservation of the UA(1) current is a problem (the so-called UA(1) problem): it can be seen that the spontaneous breakdown of the SU(N) × SU(N) implies the same for the UA(1) but there is no good candidate for the corresponding Goldstone boson in the particle spectrum.
It is clear from the preceding discussion that Green's functions involving chiral or axial currents are of direct physical interest.
- Type
- Chapter
- Information
- Gauge Field Theories , pp. 457 - 494Publisher: Cambridge University PressPrint publication year: 2000