 Sextic unique-vacuum Q-ball potential

This nonnegative potential has its only vacuum at $s=0$, since $U(s)=s[(s-1/2{)}^2+3/4]$. For the complex kinetic normalization $\frac{1}{2}|\partial \varphi {|}^2$, the small-amplitude mass squared is $2U^{\prime }(0)=2$, whereas ${\min }_{s>0}2U(s)/s=3/2$. The radial Q-ball equation has a positive localized profile for frequencies $3/2<{\omega }^2<2$, for example in three spatial dimensions. It is a standard non-topological soliton potential: the effective profile potential $U(f^2)-{\omega }^2{f}^2/2$ is negative somewhere, while its quadratic coefficient near zero is positive. The conserved Noether charge, not vacuum winding, supplies the stabilization mechanism.