A Note on Wess-Zumino Terms and Discrete Symmetries

TL;DR

The paper investigates conditions under which Wess-Zumino terms vanish in sigma models, showing that vectorlike models can have vanishing terms due to symmetry, while chiral models cannot, based on an index theorem.

Contribution

It demonstrates a symmetry-based criterion for the vanishing of Wess-Zumino terms in vectorlike sigma models and proves an associated index theorem from the SU(N)×SU(N) algebra.

Findings

Vectorlike sigma models can have vanishing Wess-Zumino terms due to symmetry.

Chiral sigma models cannot have vanishing Wess-Zumino terms for symmetry reasons.

An index theorem for the axialvector coupling matrix is proven from the SU(N)×SU(N) algebra.

Abstract

Sigma models in which the integer coefficient of the Wess-Zumino term vanishes are easy to construct. This is the case if all flavor symmetries are vectorlike. We show that there is a subset of SU(N)XSU(N) vectorlike sigma models in which the Wess-Zumino term vanishes for reasons of symmetry as well. However, there is no chiral sigma model in which the Wess-Zumino term vanishes for reasons of symmetry. This can be understood in the sigma model basis as a consequence of an index theorem for the axialvector coupling matrix. We prove this index theorem directly from the SU(N)XSU(N) algebra.