Two types of boson stars in $U(1)$ gauged $3+1$-dimensional $O(3)$ sigma-model

TL;DR

The paper investigates self-gravitating boson stars in an $O(3)$ nonlinear sigma-model with a symmetry-breaking potential, highlighting two distinct families: type I (emerging from vacuum as the frequency $ω$ drops below the mass threshold) and type II (originating from static solitons at $ω=0$). Using a spherically symmetric ansatz and numerical solutions of the coupled Einstein–Maxwell–scalar system, the authors map out the domains of existence in terms of the gravitational coupling $α$, gauge coupling $e$, and potential parameter $β$, revealing bifurcations where the two spirals detach and reconnect into a connected branch spanning $0≤ω≤μ$. The gauged case modifies these patterns via electrostatic repulsion, allowing or suppressing certain branches (e.g., type I disappearing at large $e$) and sometimes yielding Q-ball–like limits as $ω→1$. These results expand the landscape of boson-star solutions and motivate future work on rotating configurations and hairy black holes within the same model, with potential observational implications.

Abstract

We investigate boson stars in an $O(3)$ scalar field theory with a symmetry-breaking potential. By constructing numerically spherically symmetric solutions, we demonstrate that the model gives rise to a rich set of field configurations. The negative coupling constant of the scalar self-interactions allows for two types of boson stars: Type I solutions represent the usual boson stars, that emerge from the vacuum as the boson frequency is decreased below the boson mass, whereas type II boson stars emerge from a set of static soliton solutions. Depending on the strengths of the gravitational and the electric coupling constants, both types or only one type is present. At a critical set of coupling constants, both types undergo a bifurcation. There the spirals of both types disconnect from their branches and reconnect with each other, while the remaining branches of both types also reconnect with each other.

Figures (13)

• Figure 1: Spherically symmetric ungauged ($e=0$) $O(3)$ boson stars: The ADM mass $M$ in units of $8\pi$ (upper plot), the central values of the scalar component $\phi_3(0)$,(bottom left) and the metric component $-g_{00}(0)$ (bottom right) vs the frequency $\omega$ are plotted for some set of values of the gravitational coupling $\alpha$ and $\beta=0.5$. The solid and dashed lines correspond to solutions of types I and II, respectively.
• Figure 2: Spherically symmetric ungauged ($e=0$) $O(3)$ boson stars: The ADM mass $M$ of the type I boson stars in units of $8\pi$ vs the frequency $\omega$ for $\alpha=0.6$ and $\beta=0.5$ (upper plot) and illustrative radial profiles of the scalar component $\phi_3$ (bottom left), and the metric component $-g_{00}$ (bottom right) of the solutions, indicated by dots on the upper plot.
• Figure 3: Spherically symmetric ungauged ($e=0$) $O(3)$ boson stars: The ADM mass of the type II boson stars in units of $8\pi$ vs the frequency $\omega$ for $\alpha=0.6$ and $\beta=0.5$ (upper plot) and illustrative radial profiles of the scalar component $\phi_3$ (bottom left) and the metric component $-g_{00}$ (bottom right) of the solutions, indicated by dots on the upper plot.
• Figure 4: Bifurcation of two types of spherically symmetric ungauged ($e=0$) $O(3)$ boson stars: The ADM mass $M$ in units of $8\pi$ (upper plot), the central values of the scalar component $\phi_3(0)$,(bottom left) and the metric component $-g_{00}(0)$ (bottom right) vs the frequency $\omega$ are plotted for some set of values of the gravitational coupling $\alpha$ and $\beta=0.5$. The solid and dashed lines correspond to solutions of types I and II, respectively.
• Figure 5: Evolution of the detached double spirals of ungauged ($e=0$) $O(3)$ boson stars in an isolated domain: The ADM mass $M$ in units of $8\pi$ (upper plot), the central values of the scalar component $\phi_3(0)$,(bottom left) and the metric component $-g_{00}(0)$ (bottom right) vs the frequency $\omega$ are plotted for some set of values of the gravitational coupling $\alpha$ and $\beta=0.5$.