In this work we study the asymptotically flat, static, and spherically symmetric black-hole solutions of the theory described by the action $$S = \int d^nx\sqrt{ -g} \left\{ \left(1-\xi\phi^2 \right)R - g^{ \mu\nu}\partial_\mu\phi\partial_\nu\phi\right\},$$ with $n>3$ and arbitrary $\xi$. We demonstrate the absence of scalar hairs for $\xi<0$. For $\xi>\xi_c=\frac{ n-2}{ 4(n-1)}$, we show that there is no scalar hair obeying $|\phi(r)| < 1/\sqrt{ \xi}$ or $|\phi(r)| > 1/\sqrt{ \xi}$. For $0<\xi<\xi_c$, we prove the absence of scalar hairs such that $|\phi(r)| < 1/\sqrt{ \xi}$ or $\frac{ 1}{ \xi} < \phi^2(r) < \frac{ \xi_c}{ \xi(\xi_c-\xi)}$.
Categories: gr-qc