Compact, charged boson stars and shells in the $\mathbb{C}{P}^N$ gravitating nonlinear sigma model

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. By implementing the $U(1)$ gauge field in the model, the behavior of the solutions becomes complicated than the global model. Especially, they have branches, i.e., two independent solutions with the 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 consists of a charged black hole in the interior of the shell, surrounded by a $Q$-shell, and the outside becomes a Reissner-Nordström space-time. These solutions inherit the scaling behavior of the flat space-time.

Figure 1

The gauged $Q$-ball solution for the $\mathbb{C}{P}^1$ case. Top left: the matter profile function $f(r)$ of the first branch. Top right: The gauge function $b(r)$ of the first branch. Bottom left: The matter profile function $f(r)$ of the second branch. Bottom right: the gauge function $b(r)$ of the second branch. Solutions of the first branch are plotted with the bold line and those of the second branch are plotted with the dot-dashed line. Solutions of the vacuum equations are depicted with the dashed line. Distinct curves correspond with different values of the shooting parameter $b_0$.

Figure 2

The gauged $Q$-shell solution for the $\mathbb{C}{P}^{11}$ case. Top left: the matter profile function $f(r)$ of the first branch. Top right: the gauge function $b(r)$ of the first branch. Bottom left: the matter profile function $f(r)$ of the second branch. Bottom right: the gauge function $b(r)$ of the second branch. Solutions of the first branch are plotted with the bold line and those of the second branch are plotted with the dot-dashed line. Solutions of the vacuum equations are depicted with the dashed line. Distinct curves correspond with different values of the shooting parameter $b_0$.

Figure 3

A relation between shooting parameter $b_0$ and frequency $\omega$ for $\mathbb{C}{P}^1$ (blue) and $\mathbb{C}{P}^{11}$ (green). The dots correspond to solutions with different $\omega$ of the first branch and the triangles to the second branch.

Figure 4

Left: the relation between $E^{-1/5}$ and $Q^{-1/6}$ for several gauged solutions. Right: the same as the left one but the plot is enlarged in the region of high $Q$, $E$. The dots correspond to solutions of the first branch. The triangles correspond to solutions of the second branch.

Figure 5

The relation between $E$ and $Q$. Left: $\mathbb{C}{P}^1$. Right: $\mathbb{C}{P}^{11}$. The gray dashed lines indicate $E={\beta }^{-5}Q$, where $\beta$ is the gradient of $E^{-1/5}-Q^{-1/6}$ in Fig. 4. We numerically fit the data of the solutions in the second branch with $E=\eta Q^{\alpha }+\xi$. The resultant functions are shown as the red line. For $\mathbb{C}{P}^1$, $\eta =1.12327$, $\xi =24.7417$, $\alpha =1.27389$. For $\mathbb{C}{P}^{11}$, $\eta =30.0064$, $\xi =68.5474$, $\alpha =1.16711\sim 7/6$.

Figure 6

The gauged gravitating $Q$-ball solution for the $\mathbb{C}{P}^1$. The parameter $b_0$ is fixed as $b_0=0.7627$. Top left: the matter profile function $f(r)$. Top right: the gauge function $b(r)$. Bottom left: the metric function $A(r)$. Bottom right: the metric function $C(r)$. Solutions of the first branch are plotted with the bold line and those of the second branch are plotted with the dot-dashed line. Solutions of the vacuum equations are depicted with the dashed line. Distinct curves correspond with different values of the coupling constant $\alpha$.

Figure 7

The gauged gravitating $Q$-shell solution for the $\mathbb{C}{P}^{11}$. The parameter $b_0$ is fixed as $b_0=1.657$. Top left: the matter profile function $f(r)$. Top right: the gauge function $b(r)$. Bottom left: the metric function $A(r)$. Bottom right: the metric function $C(r)$. Solutions of the first branch are plotted with the bold line and those of the second branch are plotted with the dot-dashed line. Solutions of the vacuum equations are depicted with the dashed line. Distinct curves correspond with different values of the coupling constant $\alpha$.

Figure 8

Left: the relation between $E^{-1/5}$ and $Q^{-1/6}$ for several gauged solutions. Right: the same as the left one but the plot is enlarged in region of high $Q$, $E$. The parameter $\alpha =0.001$. The dots correspond to solutions with $\omega$ of the first branch. The triangles correspond to solutions of the second branch.

Figure 9

The harbor solution for $\mathbb{C}{P}^{11}$ for a charged black hole. The parameters $b_0=1.657$ and $\alpha =0.001$. Top left: the matter profile function $f(r)$. Top right: the gauge function $b(r)$. Bottom left: the metric function $A(r)$. Bottom right: the metric function $C(r)$. Solutions of the first branch are plotted with the bold line and those of the second branch are plotted with the dot-dashed line. Solutions of the vacuum equations are depicted with the dotted line. The radius of the event horizon is chosen as $r_{\mathrm{H}}=1.420$.

Figure 10

Left: the relation between $E^{-1/5}$ and $Q^{-1/6}$ for harbor solutions of $\mathbb{C}{P}^{11}$. The dots with the same color indicate the solutions which differ only by the value of a horizon radius $r_{\mathrm{H}}$. The dots correspond to the first branch. Right: same as the left but the plots for $b_0=1.657$, 2.0 are enlarged. The dots correspond to the first branch and the triangles to the second branch. The parameters $\alpha =0.001$.

Figure 11

The charge density: (the Noether charge $Q$)/(the volume of the $Q$-shell $V$) for the several quantities of the model. Left top: for the shooting parameter $b_0$. Right top: for the frequency $\omega$. Bottom: the charge $Q$. The dots correspond to the first branch and the triangles to the second branch.