Vector boson star solutions with a quartic order self-interaction

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 ${\cal O}[\sqrt{\lambda}M_{pl}^3/\mu^2 \ln (\lambda M_{pl}^2/\mu^2)]$, which is different from that of the scalar BSs, ${\cal O}(\sqrt{\lambda_\phi}M_{pl}^3/\mu_\phi^2)$, where $\lambda_\phi$ and $\mu_\phi$ are the coupling constant and the mass of the complex scalar field.