Spinning $Q$-ball superradiance in $3+1D$

Recently, it was found that a $Q$-ball can amplify waves incident upon it, due to rotation in the internal space and the interaction of the two modes in the complex scalar field. While the spherically symmetric 3D case has been investigated previously, here we explore the 3D axisymmetric case, which is numerically much more challenging. The difficulty comes because a partial wave expansion is needed, and the different partial waves can not be separated, for either the background spinning $Q$-ball solution or the perturbative scattering on top of it. A relaxation method and a high dimensional shooting method are applied to compute the $Q$-ball solutions and the amplification factors respectively. We also classify the behavior of the amplification factors and we discuss their bounds and the superradiance criteria.