*Kazuo Fujikawa and Hiroshi Suzuki*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529132
- eISBN:
- 9780191712821
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529132.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers a possible intuitive explanation of quantum breaking of symmetries as well as the subjects which are not discussed in the main chapters, such as the descent formula for gauge ...
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This chapter considers a possible intuitive explanation of quantum breaking of symmetries as well as the subjects which are not discussed in the main chapters, such as the descent formula for gauge anomalies and the global SU(2) anomaly. The book has concentrated on the rather classical and basic aspects of quantum anomalies, which can be explicitly calculated by an elementary method in the path integral. Advanced subjects such as the anomaly cancellation in superstring theory and supersymmetric theory in general are not discussed in detail, but several references to these subjects are given.Less

This chapter considers a possible intuitive explanation of quantum breaking of symmetries as well as the subjects which are not discussed in the main chapters, such as the descent formula for gauge anomalies and the global *SU(2)* anomaly. The book has concentrated on the rather classical and basic aspects of quantum anomalies, which can be explicitly calculated by an elementary method in the path integral. Advanced subjects such as the anomaly cancellation in superstring theory and supersymmetric theory in general are not discussed in detail, but several references to these subjects are given.

*Jean Zinn-Justin*

- Published in print:
- 2019
- Published Online:
- August 2019
- ISBN:
- 9780198787754
- eISBN:
- 9780191829840
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198787754.003.0017
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Chapter 17 exhibits various examples where classical symmetries cannot be transferred to quantum theories. The obstructions are characterized by anomalies. The examples involve chiral symmetries ...
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Chapter 17 exhibits various examples where classical symmetries cannot be transferred to quantum theories. The obstructions are characterized by anomalies. The examples involve chiral symmetries combined with currents or gauge symmetries, leading to chiral anomalies. In particular, anomalies lead to obstruction in the construction of theories. In particular, the structure of the Standard Model of particle physics is constrained by the requirement of anomaly cancellation. Other applications, like the relation between electromagnetic pi0 decay and the axial anomaly, are described. Anomalies are related to the Dirac operator index, leading to relations between anomaly and topology. To prove anomaly cancellation beyond perturbation theory, one can use lattice regularization. However, the definition of lattice chiral transformations is non–trivial. It is based on solutions of the Ginsparg–Wilson relation.Less

Chapter 17 exhibits various examples where classical symmetries cannot be transferred to quantum theories. The obstructions are characterized by anomalies. The examples involve chiral symmetries combined with currents or gauge symmetries, leading to chiral anomalies. In particular, anomalies lead to obstruction in the construction of theories. In particular, the structure of the Standard Model of particle physics is constrained by the requirement of anomaly cancellation. Other applications, like the relation between electromagnetic pi0 decay and the axial anomaly, are described. Anomalies are related to the Dirac operator index, leading to relations between anomaly and topology. To prove anomaly cancellation beyond perturbation theory, one can use lattice regularization. However, the definition of lattice chiral transformations is non–trivial. It is based on solutions of the Ginsparg–Wilson relation.

*Michael Kachelriess*

- Published in print:
- 2017
- Published Online:
- February 2018
- ISBN:
- 9780198802877
- eISBN:
- 9780191841330
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802877.003.0017
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The axial anomaly is derived both from the non-invariance of the path-integral measure under UA(1) transformations and calculations of specific triangle diagrams. It is demonstrated that the ...
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The axial anomaly is derived both from the non-invariance of the path-integral measure under UA(1) transformations and calculations of specific triangle diagrams. It is demonstrated that the anomalous terms are cancelled in the electroweak sector of the standard model, if the electric charge of all fermions adds up to zero. The CP-odd term F̃μνFμν introduced by the axial anomaly is a gauge-invariant renormalisable interaction which is also generated by instanton transitions between Yang–Mills vacua with different winding numbers. The Peceei–Quinn symmetry is discussed as a possible explanation why this term does not contribute to the QCD action.Less

The axial anomaly is derived both from the non-invariance of the path-integral measure under UA(1) transformations and calculations of specific triangle diagrams. It is demonstrated that the anomalous terms are cancelled in the electroweak sector of the standard model, if the electric charge of all fermions adds up to zero. The CP-odd term F̃μνFμν introduced by the axial anomaly is a gauge-invariant renormalisable interaction which is also generated by instanton transitions between Yang–Mills vacua with different winding numbers. The Peceei–Quinn symmetry is discussed as a possible explanation why this term does not contribute to the QCD action.