Q-balls in Maxwell-Chern-Simons theory

We examine the energetics of Q-balls in Maxwell-Chern-Simons theory in two space dimensions. Whereas gauged Q-balls are unallowed in this dimension in the absence of a Chern-Simons term due to a divergent electromagnetic energy, the addition of a Chern-Simons term introduces a gauge field mass and renders finite the otherwise-divergent electromagnetic energy of the Q-ball. Similar to the case of gauged Q-balls, Maxwell-Chern-Simons Q-balls have a maximal charge. The properties of these solitons are studied as a function of the parameters of the model considered, using a numerical technique known as relaxation. The results are compared to expectations based on qualitative arguments. A class of non-topological solitons (see [1] for a comprehensive review) dubbed Q-balls were examined some time ago by Coleman [2]. These objects owe their existence to a conserved global charge. Under certain circumstances, a localized configuration of charge Q can be created which has a lower energy than the “naive” lowest-energy configuration, namely, Q widely-separated ordinary particles (each of unit charge) at zero momentum. This latter state obviously has energy Qm, where m is the mass of the quanta of the theory. If another configuration of charge Q can be constructed whose energy is lower, then that state cannot decay into ordinary matter: either it is stable or some other “non-naive” configuration of the same charge and still lower energy is stable. Coleman studied the three-dimensional case with a charged scalar field φ. The Q-ball is a spherically symmetric configuration where |φ| is nonzero inside a core region and tends to zero as r → ∞; its phase varies linearly in time. A mechanical analogy permits a clean demonstration of the necessary conditions which must be satisfied by the potential in order for Q-balls to exist. It is particularly easy to analyze the case of large charge, since then the surface energy can be neglected compared to the volume energy. Among Coleman’s conclusions is the fact that there is no upper limit to the charge of a Q-ball (if they exist in the first place); furthermore, the interior of a sufficiently large Q-ball is homogeneous. The case of small Q-balls was analyzed by Kusenko [3], who (along with many others) also proposed possible astrophysical signatures ofQ-balls (see [4] and references therein). Possible applications in condensed matter physics have been analyzed in [5, 6, 7]. A number of other variations have been studied since, including non-abelian Q-balls [8, 9], gauged Q-balls [10, 11], Q-stars [12], Q-balls in other dimensions [13, 14], higherdimensional Q-objects [15, 16], spinning Q-balls [17, 18], and so on. Of particular interest here is the paper of Lee, et al. [10], who discussed the case of gauged Q-balls in three dimensions, using a combination of analytical and numerical techniques. They argued that when the charge exceeds a critical value the Q-ball’s energy exceeds Qm, so the Q-ball is at best metastable. This is intuitively reasonable, since a ball of electric charge will have a Coulomb energy which grows roughly as the square of the charge, so eventually the Q-ball will be unable to compete with ordinary matter. On the other hand, as the charge decreases the Q-ball gets smaller and smaller; surface effects become important and eventually destabilize the Q-ball. These two observations indicate that there may or may not be a range of charges for which Q-balls exist, depending on under what circumstances each effect becomes significant. Another result of their analysis is that the core of a large gauged Q-ball is not homogeneous, essentially because the charge repels itself, and the electromagnetic energy is reduced by having the charge migrate to the surface of the Q-ball.