Coupled vector Gauss-Bonnet theories and hairy black holes

We study vector-tensor theories in which a 4-dimensional vector field $A_{\mu}$ is coupled to a vector quantity ${\cal J}^{\mu}$, which is expressed in terms of $A_{\mu}$ and a metric tensor $g_{\mu \nu}$. The divergence of ${\cal J}^{\mu}$ is equivalent to a Gauss-Bonnet (GB) term. We show that an interacting Lagrangian of the form $f(X)A_{\mu}{\cal J}^{\mu}$, where $f$ is an arbitrary function of $X=-(1/2)A_{\mu}A^{\mu}$, belongs to a scheme of beyond generalized Proca theories. For $f(X)=\alpha={\rm constant}$, this interacting Lagrangian reduces to a particular class of generalized Proca theories. We apply the latter coupling to a static and spherically symmetric vacuum configuration by incorporating the Einstein-Hilbert term, Maxwell scalar, and vector mass term $\eta X$ ($\eta$ is a constant). Under an expansion of the small coupling constant $\alpha$ with $\eta \neq 0$, we derive hairy black hole solutions endowed with nonvanishing temporal and radial vector field profiles. The asymptotic properties of solutions around the horizon and at spatial infinity are different from those of hairy black holes present in scalar-GB theories. We also show that black hole solutions without the vector mass term, i.e., $\eta=0$, are prone to ghost instability of odd-parity perturbations.