Q-ball

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 ${\displaystyle \phi }$), and the interaction potential ${\displaystyle V(\phi )}$ of the particles must have a negative (attractive) term. For non-interacting particles, the potential would be just a mass term ${\displaystyle V_{\rm {free}}(\phi )=m^{2}|\phi |^{2}}$, and there would be no Q-ball. But if one adds an attractive ${\displaystyle -\lambda |\phi |^{4}}$ term (and positive higher powers of ${\displaystyle \phi }$ to ensure that the potential has a lower bound) then there are values of ${\displaystyle \phi }$ where ${\displaystyle V(\phi )<V_{\rm {free}}(\phi )}$, 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.

In its simplest form, a Q-ball is constructed in a field theory of a complex scalar field ${\displaystyle \phi }$, in which Lagrangian is invariant under a global ${\displaystyle U(1)}$ symmetry. The Q-ball solution is a state which minimizes energy while keeping the charge Q associated with the global ${\displaystyle U(1)}$ 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

${\displaystyle E_{\omega }=E+\omega \left[Q-{\frac {1}{2i}}\int d^{3}x(\phi ^{*}\partial _{t}\phi -\phi \partial _{t}\phi ^{*})\right],}$

where the energy is defined as

${\displaystyle E=\int d^{3}x\left[{\frac {1}{2}}{\dot {\phi }}^{2}+{\frac {1}{2}}|\nabla \phi |^{2}+U(\phi ,\phi ^{*})\right],}$

and ${\displaystyle \omega }$ is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional ${\displaystyle E_{\omega }}$ as

${\displaystyle E_{\omega }=\int d^{3}x\left[{\frac {1}{2}}|{\dot {\phi }}-i\omega \phi |^{2}+{\frac {1}{2}}|\nabla \phi |^{2}+{\hat {U}}_{\omega }(\phi ,\phi ^{*})\right]}$

where ${\displaystyle {\hat {U}}_{\omega }=U-{\frac {1}{2}}\omega ^{2}\phi ^{2}}$. Since the first term in the functional is now positive, minimization of this terms implies

${\displaystyle \phi ({\vec {r}},t)=\phi _{0}({\vec {r}})e^{i\omega t}.}$

We therefore interpret the Lagrange multiplier ${\displaystyle \omega }$ as the frequency of oscillation of the field within the Q-ball.

The theory contains Q-ball solutions if there are any values of ${\displaystyle \phi ^{*}\phi }$ at which the potential is less than ${\displaystyle m^{2}\phi ^{*}\phi }$. In this case, a volume of space with the field at that value can have an energy per unit charge that is less than ${\displaystyle m}$, meaning that it cannot decay into a gas of individual particles. Such a region is a Q-ball. If it is large enough, its interior is uniform, and is called "Q-matter". (For a review see Lee et al. (1992).[1]

Configurations of a charged scalar field that are classically stable (stable against small perturbations) were constructed by Rosen in 1968.[4] Stable configurations of multiple scalar fields were studied by Friedberg, Lee, and Sirlin in 1976.[5] The name "Q-ball" and the proof of quantum-mechanical stability (stability against tunnelling to lower energy configurations) come from Sidney Coleman.[2]

It has been theorized that dark matter might consist of Q-balls (Frieman et al.. 1988,[6] Kusenko et al.. 1997[7]) and that Q-balls might play a role in baryogenesis, i.e. the origin of the matter that fills the universe (Dodelson et al.. 1990,[8] Enqvist et al.. 1997[9]). Interest in Q-balls was stimulated by the suggestion that they arise generically in supersymmetric field theories (Kusenko 1997[10]), so if nature really is fundamentally supersymmetric then Q-balls might have been created in the early universe, and still exist in the cosmos today.

• In the movie Sunshine, the Sun is undergoing a premature death. The movie's science adviser, scientist Brian Cox, proposed "infection" with a Q-ball as the mechanism for this death, but this is mentioned only in the commentary tracks and not in the movie itself.
• In the fictional universe of Orion's Arm, Q-balls are one of the speculated sources for the large amounts of antimatter used by certain groups.

• Cosmic anarchists, by Hazel Muir. A popular account of the proposal of Alexander Kusenko.