Abstract
We study $U(1)$ gauged gravitating compact $Q$-ball, $Q$-shell solutions in a nonlinear sigma model with the target space $\mathbb{C}P^N$. The models with odd integer $N$ and a special potential can be parameterized by $N$-th complex scalar fields and they support compact solutions. Implementing the $U(1)$ gauge field in the model, the behavior of the solutions become complicated than the global model. Especially, they exhibit branch, i.e., two independent solutions with same shooting parameter. The energy of the solutions in the first branch behaves as $E\sim Q^{5/6}$ for small $Q$, where $Q$ stands for the $U(1)$ Noether charge. For the large $Q$, it gradually deviates from the scaling $E\sim Q^{5/6}$ and, for the $Q$-shells it is $E\sim Q^{7/6}$, which forms the second branch. A coupling with gravity allows for harboring of the Schwarzschild black holes for the $Q$-shell solutions, forming the charged boson shells. The space-time then consist of a charged black hole in the interior of the shell, surrounded by a $Q$-shell, and the outside becomes a Reissner-Nordstr\"om space-time. These solutions inherit the scaling behavior of the flat space-time.
Highlights
A complex scalar field theory with some self-interactions has stationary soliton solutions called Q-balls [1,2,3,4]
The repulsive force by the electric interaction is effectively strong and the solution deviates from Q5=6 behavior
We have considered Uð1Þ gauged CPN nonlinear sigma model with a compact support in flat space-time and coupled with gravity
Summary
A complex scalar field theory with some self-interactions has stationary soliton solutions called Q-balls [1,2,3,4]. The compact boson stars and shells are extensively studied in [11,31,32,34] for the global models and in [35,36,37] for the Uð1Þ gauged model. For the boson shell configurations, one possibility is the case that the gravitating boson shells surround a flat Minkowski-like interior region r < Rin while the exterior region r > Rout is the exterior of a Reissner-Nordström solution Another and even more interesting possibility is the existence of the charged black hole in the interior region. The authors of [43,44] proposed boson star solutions for a collection of an arbitrary odd number N of complex scalar fields with an internal symmetry UðNÞ.
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