Neutron stars with a generalized Proca hair and spontaneous vectorization

In a class of generalized Proca theories, we study the existence of neutron star solutions with a nonvanishing temporal component of the vector field $A_\mu$ approaching 0 toward spatial infinity, as they may be the endpoints of tachyonic instabilities of neutron star solutions in general relativity with $A_{\mu}=0$. Such a phenomenon is called spontaneous vectorization, which is analogous to spontaneous scalarization in scalar-tensor theories with nonminimal couplings to the curvature or matter. For the nonminimal coupling $\beta X R$, where $\beta$ is a coupling constant and $X=-A_{\mu}A^{\mu}/2$, we show that there exist both 0-node and 1-node vector-field solutions, irrespective of the choice of the equations of state of nuclear matter. The 0-node solution, which is present only for $\beta=-{\cal O}(0.1)$, may be induced by some nonlinear effects such as the selected choice of initial conditions. The 1-node solution exists for $\beta=-{\cal O}(1)$, which suddenly emerges above a critical central density of star and approaches the general relativistic branch with the increasing central density. We compute the mass $M$ and radius $r_s$ of neutron stars for some realistic equations of state and show that the $M$-$r_s$ relations of 0-node and 1-node solutions exhibit notable difference from those of scalarized solutions in scalar-tensor theories. Finally, we discuss the possible endpoints of tachyonic instabilities.