Gauged Wess-Zumino terms for a general coset space

Tomáš Brauner,Helena Kolešová

doi:10.1016/j.nuclphysb.2019.114676

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.

Highlights

Spontaneous breaking of a continuous global symmetry is a fascinating phenomenon that dramatically affects the physics in the quantum realm
The case of most physical interest is the cohomology of degree 5
We introduce the notation (TL,A, TR,B) for the generators of the GL × GR symmetry group, where both entries in the parentheses run over the generators of G

Summary

Introduction

Spontaneous breaking of a continuous global symmetry is a fascinating phenomenon that dramatically affects the physics in the quantum realm. Another, related consequence is the fact that spontaneously broken symmetry is not realized by unitary operators on the Hilbert space of physical states of the system. Transformations induced by the spontaneously broken generators of the symmetry group G are realized nontrivially even on the space of fields, through nonlinear functions of the NG fields. The latter feature presents a challenge to model building. The action of the lowenergy effective theory, formulated in terms of the NG degrees of freedom, must preserve the symmetry properties of the underlying microscopic theory [1, 2]. It is sufficient that the latter changes upon a symmetry transformation by a surface term, in which case we speak of a quasi-invariant Lagrangian

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