Black hole perturbations in vector-tensor theories: The odd-mode
  analysis
  In generalized Proca theories with vector-field derivative couplings, a bunch
of hairy black hole solutions have been derived on a static and spherically
symmetric background. In this paper, we formulate the odd-parity black hole
perturbations in generalized Proca theories by expanding the corresponding
action up to second order and investigate whether or not black holes with
vector hair suffer ghost or Laplacian instabilities. We show that the models
with cubic couplings $G_3(X)$, where $X=-A_{\mu}A^{\mu}/2$ with a vector field
$A_{\mu}$, do not provide any additional stability condition as in General
Relativity. On the other hand, the exact charged stealth Schwarzschild solution
with a nonvanishing longitudinal vector component $A_1$, which originates from
the coupling to the Einstein tensor $G^{\mu\nu}A_\mu A_\nu$ equivalent to the
quartic coupling $G_4(X)$ containing a linear function of $X$, is unstable in
the vicinity of the event horizon. The same instability problem also persists
for hairy black holes arising from general quartic power-law couplings $G_4(X)
\supset \beta_4 X^n$ with the nonvanishing $A_1$, while the other branch with
$A_1=0$ can be consistent with conditions for the absence of ghost and
Laplacian instabilities. We also discuss the case of other exact and numerical
black hole solutions associated with intrinsic vector-field derivative
couplings and show that there exists a wide range of parameter spaces in which
the solutions suffer neither ghost nor Laplacian instabilities against
odd-parity perturbations.
