Black holes in vector-tensor theories
  We study static and spherically symmetric black hole (BH) solutions in
second-order generalized Proca theories with nonminimal vector field derivative
couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann
tensor. We find concrete Lagrangians which give rise to exact BH solutions by
imposing two conditions of the two identical metric components and the constant
norm of the vector field. These exact solutions are described by either
Reissner-Nordstr\"{o}m (RN), stealth Schwarzschild, or extremal RN solutions
with a non-trivial longitudinal mode of the vector field. We then numerically
construct BH solutions without imposing these conditions. For cubic and quartic
Lagrangians with power-law couplings which encompass vector Galileons as the
specific cases, we show the existence of BH solutions with the difference
between two non-trivial metric components. The quintic-order power-law
couplings do not give rise to non-trivial BH solutions regular throughout the
horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead
to BH solutions with a secondary hair. For all the solutions, the vector field
is regular at least at the future or past horizon. The deviation from General
Relativity induced by the Proca hair can be potentially tested by future
measurements of gravitational waves in the nonlinear regime of gravity.
