Gauged Wess-Zumino terms for a general coset space

TL;DR

This paper develops explicit, practical formulas for Wess-Zumino terms in effective theories of Nambu-Goldstone bosons for general coset spaces, including gauge couplings, across various spacetime dimensions.

Contribution

It provides the first comprehensive construction of Wess-Zumino terms for arbitrary coset spaces with gauge coupling in multiple dimensions, filling a gap in practical applications.

Findings

Explicit matrix expressions for WZ terms in 4D

Construction applicable to QCD-like theories

Includes coupling to gauge fields for general groups

Abstract

The low-energy physics of systems with spontaneously broken continuous symmetry is dominated by the ensuing Nambu-Goldstone bosons. It has been known for half a century how to construct invariant Lagrangian densities for the low-energy effective theory of Nambu-Goldstone bosons. Contributions, invariant only up to a surface term -- also known as the Wess-Zumino (WZ) terms -- are more subtle, and as a rule are topological in nature. Although WZ terms have been studied intensively in theoretically oriented literature, explicit expressions do not seem to be available in sufficient generality in a form suitable for practical applications. Here we construct the WZ terms in $d=1,2,3,4$ spacetime dimensions for an arbitrary compact, semisimple and simply connected symmetry group $G$ and its arbitrary connected unbroken subgroup $H$, provided that the $d$-th homotopy group of the coset space $G/H$ is trivial. Coupling to gauge fields for the whole group $G$ is included throughout the construction. We list a number of explicit matrix expressions for the WZ terms in four spacetime dimensions, including those for QCD-like theories, that is vector-like gauge theories with fermions in a complex, real or pseudoreal representation of the gauge group.