Linear stability of black holes with static scalar hair in full
  Horndeski theories: generic instabilities and surviving models
  In full Horndeski theories, we show that the static and spherically symmetric
black hole (BH) solutions with a static scalar field $\phi$ whose kinetic term
$X$ is nonvanishing on the BH horizon are generically prone to ghost/Laplacian
instabilities. We then search for asymptotically flat hairy BH solutions with a
vanishing $X$ on the horizon free from ghost/Laplacian instabilities. We show
that models with regular coupling functions of $\phi$ and $X$ result in no-hair
Schwarzschild BHs in general. On the other hand, the presence of a coupling
between the scalar field and the Gauss-Bonnet (GB) term $R_{\rm GB}^2$, even
with the coexistence of other regular coupling functions, leads to the
realization of asymptotically flat hairy BH solutions without ghost/Laplacian
instabilities. Finally, we find that hairy BH solutions in power-law $F(R_{\rm
GB}^2)$ gravity are plagued by ghost instabilities. These results imply that
the GB coupling of the form $\xi(\phi)R_{\rm GB}^2$ plays a prominent role for
the existence of asymptotically flat hairy BH solutions free from
ghost/Laplacian instabilities.
