= Sextic unique-vacuum Q-ball potential
{title2=$U(s)=s-s^2+s^3,\quad s=|\phi|^2$}
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 $\tfrac12|\partial\phi|^2$, the small-amplitude mass squared is $2U'(0)=2$, 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 $U(f^2)-\omega^2f^2/2$ is negative somewhere, while its quadratic coefficient near zero is positive. The conserved <Noether charge>, not vacuum winding, supplies the stabilization mechanism.
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