In theoretical physics, Q-ball is a type of non-topological soliton. A soliton is a localized field configuration that is stable—it cannot spread out and dissipate. In the case of a non-topological soliton, the stability is guaranteed by a conserved charge: the soliton has lower energy per unit charge than any other configuration (in physics, charge is often represented by the letter "Q", and the soliton is spherically symmetric, hence the name).
Intuitive explanation A Q-ball arises in a theory of bosonic particles when there is an attraction between the particles. Loosely speaking, the Q-ball is a finite-sized "blob" containing a large number of particles. The blob is stable against fission into smaller blobs and against "evaporation" via emission of individual particles, because, due to the attractive interaction, the blob is the lowest-energy configuration of that number of particles. (This is analogous to the fact that nickel-62 is the most stable nucleus because it is the most stable configuration of neutrons and protons. However, nickel-62 is not a Q-ball, in part because neutrons and protons are fermions, not bosons.) For there to be a Q-ball, the number of particles must be conserved (i.e. the particle number is a conserved "charge", so the particles are described by a complex-valued field ϕ {\displaystyle \phi } ), and the interaction potential V ( ϕ ) {\displaystyle V(\phi )} of the particles must have a negative (attractive) term. For non-interacting particles, the potential would be just a mass term V free ( ϕ ) = m 2 | ϕ | 2 {\displaystyle V_{\text{free}}(\phi )=m^{2}|\phi |^{2}} , and there would be no Q-ball. But if one adds an attractive − λ | ϕ | 4 {\displaystyle -\lambda |\phi |^{4}} term (and positive higher powers of ϕ {\displaystyle \phi } to ensure that the potential has a lower bound), then there are values of ϕ {\displaystyle \phi } where V ( ϕ ) < V free ( ϕ ) {\displaystyle V(\phi )<V_{\text{free}}(\phi )} , i.e. the energy of these field values is less than the energy of a free field. This corresponds to saying that one can create blobs of non-zero field (i.e. clusters of many particles) whose energy is lower than the same number of individual particles far apart. Those blobs are therefore stable against evaporation into individual particles.
Construction In its simplest form, a Q-ball is constructed in a field theory of a complex scalar field ϕ {\displaystyle \phi } , in which Lagrangian is invariant under a global U ( 1 ) {\displaystyle U(1)} symmetry. The Q-ball solution is a state that minimizes energy while keeping the charge Q associated with the global U ( 1 ) {\displaystyle U(1)} symmetry constant. A particularly transparent way of finding this solution is via the method of Lagrange multipliers. In particular, in three spatial dimensions we must minimize the functional
E ω = E + ω [ Q − 1 2 i ∫ d 3 x ( ϕ ∗ ∂ t ϕ − ϕ ∂ t ϕ ∗ ) ] , {\displaystyle E_{\omega }=E+\omega \left[Q-{\frac {1}{2i}}\int d^{3}\,x(\phi ^{*}\partial _{t}\phi -\phi \partial _{t}\phi ^{*})\right],}
where the energy is defined as
E = ∫ d 3 x [ 1 2 ϕ ˙ 2 + 1 2 | ∇ ϕ | 2 + U ( ϕ , ϕ ∗ ) ] , {\displaystyle E=\int d^{3}\,x\left[{\frac {1}{2}}{\dot {\phi }}^{2}+{\frac {1}{2}}|\nabla \phi |^{2}+U(\phi ,\phi ^{*})\right],}
and ω {\displaystyle \omega } is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional E ω {\displaystyle E_{\omega }} as
… excerpt ends here. Continue reading the full article.
