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

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 $V({\phi }^{\ast }\phi ,{\phi }^N+{\phi }^{\ast N})$ with $N\in \mathbb{N}$ and $N\ge 2$, 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.