Vector boson star solutions with a quartic order self-interaction

Abstract

We investigate boson star (BS) solutions in the Einstein-Proca theory with the quartic order self-interaction of the vector field $\lambda (A^{\mu }{\bar{A}}_{\mu }{)}^2/4$ and the mass term $\mu {\bar{A}}^{\mu }{A}_{\mu }/2$, where $A_{\mu }$ is the complex vector field and ${\bar{A}}_{\mu }$ is the complex conjugate of $A_{\mu }$, and $\lambda$ and $\mu$ are the coupling constant and the mass of the vector field, respectively. The vector BSs are characterized by the two conserved quantities, the Arnowitt-Deser-Misner (ADM) mass and the Noether charge associated with the global $U(1)$ symmetry. We show that in comparison with the case without the self-interaction $\lambda =0$, the maximal ADM mass and Noether charge increase for $\lambda >0$ and decrease for $\lambda <0$. We also show that there exists the critical central amplitude of the temporal component of the vector field above which there is no vector BS solution, and for $\lambda >0$ it can be expressed by the simple analytic expression. For a sufficiently large positive coupling $\Lambda :=M_{pl}^2\lambda /(8\pi {\mu }^2)\gg 1$, the maximal ADM mass and Noether charge of the vector BSs are obtained from the critical central amplitude and of $\mathcal{O}[\sqrt{\lambda }{M}_{pl}^3/{\mu }^2\ln (\lambda M_{pl}^2/{\mu }^2)]$, which is different from that of the scalar BSs, $\mathcal{O}(\sqrt{{\lambda }_{\varphi }}{M}_{pl}^3/{\mu }_{\varphi }^2)$, where ${\lambda }_{\varphi }$ and ${\mu }_{\varphi }$ are the coupling constant and the mass of the complex scalar field.

10 pages

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