Q-ball (source code)
= Q-ball
{c}
{title2=$\phi(t,x)=e^{i\omega t}f(x)$}
{wiki}
For a <complex scalar field> with global $U(1)$ invariance and Lagrangian $\mathcal L=\tfrac12|\partial_t\phi|^2-\tfrac12|\nabla\phi|^2-U(|\phi|)$, a <Q-ball> is a localized phase-rotating configuration with conserved <Noether charge> $Q=\int\operatorname{Im}(\bar\phi\phi_t)$. At fixed charge its real profile can minimize
$$
E_Q[f]=\int\left(\frac12|\nabla f|^2+U(f)\right)+
\frac{Q^2}{2\int f^2}.
$$
Such solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is $\inf_{f>0}2U(f)/f^2<U''(0)$.
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