Energy Landscape of d-Dimensional Q-balls

TL;DR

This paper explores the properties and energy estimates of $Q$-balls in multiple spatial dimensions, providing analytical and numerical insights, and introduces a criterion to classify their size based on charge and dimensionality.

Contribution

It generalizes virial relations and energy estimates for $Q$-balls in arbitrary dimensions with specific potentials, and offers a classification criterion based on charge and dimension.

Findings

Analytical energy estimates match numerical results.

Minimum charge scales exponentially with dimension.

A criterion to distinguish large and small $Q$-balls is proposed.

Abstract

We investigate the properties of $Q$-balls in $d$ spatial dimensions. First, a generalized virial relation for these objects is obtained. We then focus on potentials $V(\phi\phi^{\dagger})= \sum_{n=1}^{3} a_n(\phi\phi^{\dagger})^n$, where $a_n$ is a constant and $n$ is an integer, obtaining variational estimates for their energies for arbitrary charge $Q$. These analytical estimates are contrasted with numerical results and their accuracy evaluated. Based on the results, we offer a simple criterion to classify ``large'' and ``small'' $d$-dimensional $Q$-balls for this class of potentials. A minimum charge is then computed and its dependence on spatial dimensionality is shown to scale as $Q_{\rm min} \sim \exp(d)$. We also briefly investigate the existence of $Q$-clouds in $d$ dimensions.