Abstract
We study compact gravitating $Q$-ball, $Q$-shell solutions in a sigma model with the target space $\mathbb{C}P^N$. Models with odd integer $N$ and suitable potential can be parameterized by $N$-th complex scalar fields and they support compact solutions. A coupling with gravity allows for harboring of the Schwarzschild black holes for the $Q$-shell solutions. The energy of the solutions behaves as $E\sim |Q|^{5/6}$, where $Q$ stands for the $U(1)$ Noether charge, for both the gravitating and the black hole solutions.Notable difference from the solutions of the flat space is that upper bound of $|Q|$ appears when the coupling with gravity is stronger. The maximal value of $|Q|$ quickly reduces for larger coupling constant. It may give us a useful hint of how a star forms its shape with a certain finite number of particles.
Highlights
Q-balls are stationary soliton solutions of certain complex scalar field theories with self-interactions [1,2,3,4]
The energy of the solutions behaves as E ∼ jQj5=6, where Q stands for the Uð1Þ Noether charge, for both the gravitating and the black hole solutions
We have considered the family of CPN nonlinear sigma models coupled with gravity which possess solutions with compact support
Summary
Q-balls are stationary soliton solutions of certain complex scalar field theories with self-interactions [1,2,3,4]. Such configurations of fields are called Q-shells Such shell solutions have no restrictions on upper bound for jQj. The authors claim that the energy of compact Q-balls scales as ∼Q5=6 and of Q-shells for large Q as ∼Q7=6. 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 an Reissner-Nordström solution Another and even more interesting possibility is the existence of the charged black hole in the interior region. In the complex signum-Gordon model with local symmetry, Q-balls exist due to presence of the gauge field, whereas in the case of the CPN model, they appear as the result of self-interactions between scalar fields. We consider model containing complex scalar fields coupled to gravity and obtain the compact Qball and Q-shell solutions.
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