$Z_N$-balls: Solitons from $Z_N$-symmetric scalar field theory

TL;DR

This paper explores the existence and properties of $Z_N$-balls, which are finite-energy, static scalar field configurations with $Z_N$-symmetry, constructed from models inspired by finite-temperature Yang-Mills theories.

Contribution

It introduces explicit solutions for $Z_N$-balls in a scalar field theory with $Z_N$-symmetry, analyzing their existence for various N and their node structures.

Findings

$Z_N$-balls exist for N=3,4,6,8,10.

Only N odd solutions are node-free; even N solutions can have radial nodes.

Solutions are modeled after effective theories related to SU(N) Yang-Mills.

Abstract

We discuss the conditions under which static, finite-energy, configurations of a complex scalar field $\phi$ with constant phase and spherically symmetric norm exist in a potential of the form $V(\phi^*\phi, \phi^N+\phi^{*N})$ with $N\in\mathbb{N}$ and $N\geq2$, i.e. a potential with a $Z_N$-symmetry. Such configurations are called $Z_N$-balls. We build explicit solutions in $(3+1)$-dimensions from a model mimicking effective field theories based on the Polyakov loop in finite-temperature SU($N$) Yang-Mills theory. We find $Z_N$-balls for $N=$3, 4, 6, 8, 10 and show that only static solutions with zero radial node exist for $N$ odd, while solutions with radial nodes may exist for $N$ even.