Scrutinizing black hole stability in cubic vector Galileon theories

In a subclass of generalized Proca theories where a cubic vector Galileon term breaks the $U(1)$ gauge invariance, it is known that there are static and spherically symmetric black hole (BH) solutions endowed with nonvanishing temporal and longitudinal vector components. Such hairy BHs are present for a vanishing vector-field mass ($m=0$) with a non-zero cubic Galileon coupling $\beta_3$. We study the linear stability of those hairy BHs by considering even-parity perturbations in the eikonal limit. In the angular direction, we show that one of the three dynamical perturbations has a nontrivial squared propagation speed $c_{\Omega,1}^2$, while the other two dynamical modes are luminal. We could detect two different unstable behaviors of perturbations in all the parameter spaces of hairy asymptotically flat BH solutions we searched for. In the first case, an angular Laplacian instability on the horizon is induced by negative $c_{\Omega,1}^2$. For the second case, it is possible to avoid this horizon instability, but in such cases, the positivity of $c_{\Omega,1}^2$ is violated at large distances. Hence these hairy BHs are generally prone to Laplacian instabilities along the angular direction in some regions outside the horizon. Moreover, we also encounter a pathological behavior of the radial propagation speeds $c_r$ possessing two different values of $c_r^2$ for one of the dynamical perturbations. Introducing the vector-field mass $m$ to cubic vector Galileons, however, we show that the resulting no-hair Schwarzschild BH solution satisfies all the linear stability conditions in the small-scale limit, with luminal propagation speeds of three dynamical even-parity perturbations.