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 $λ(A^μ\bar{A}_μ)^2/4$ and the mass term $μ\bar{A}^μA_μ/2$, where $A_μ$ is the complex vector field and ${\bar A}_μ$ is the complex conjugate of $A_μ$, and $λ$ and $μ$ 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 $λ=0$, the maximal ADM mass and Noether charge increase for $λ>0$ and decrease for $λ<0$. 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 $λ>0$ it can be expressed by the simple analytic expression. For a sufficiently large positive coupling $Λ:=M_{pl}^2λ/(8πμ^2) \gg 1$, the maximal ADM mass and Noether charge of the vector BSs are obtained from the critical central amplitude and of ${\cal O}[\sqrtλM_{pl}^3/μ^2 \ln (λM_{pl}^2/μ^2)]$, which is different from that of the scalar BSs, ${\cal O}(\sqrt{λ_ϕ}M_{pl}^3/μ_ϕ^2)$, where $λ_ϕ$ and $μ_ϕ$ are the coupling constant and the mass of the complex scalar field.