 Q-ball

For a complex scalar field with global $U(1)$ invariance and Lagrangian $\mathcal{L}=\frac{1}{2}|{\partial }_t\varphi {|}^2-\frac{1}{2}|\nabla \varphi {|}^2-U(|\varphi |)$, a Q-ball is a localized phase-rotating configuration with conserved Noether charge $Q=\int \mathrm{Im}(\bar{\varphi }{\varphi }_t)$. At fixed charge its real profile can minimizeSuch solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is ${\inf }_{f>0}2U(f)/f^2<U^{\prime \prime }(0)$.

A Q-ball is a theoretical concept in the field of particle physics and cosmology. It refers to a type of non-topological soliton, which is a stable, localized solution of field equations in certain scalar field theories. Q-balls can arise in models that involve scalar fields with a global U(1) symmetry and are characterized by a conserved charge, denoted as \(Q\).