Energy Landscape of d-Dimensional Q-balls

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 $V(\varphi {\varphi }^{†})={\sum }_{n=1}^3{a}_n(\varphi {\varphi }^{†}{)}^n$, 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 $Q_{\mathrm{m}\mathrm{i}\mathrm{n}}\sim \exp (d)$. We also briefly investigate the existence of $Q$-clouds in $d$ dimensions.