Searching for non-minimally coupled scalar hairs

In this work we study the asymptotically flat, static, and spherically symmetric black-hole solutions of the theory described by the action $$S=\int d^{n}x\sqrt{-g}\{(1-\xi {\varphi }^2)R-g^{\mu \nu }{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}_{\nu }\varphi \},$$ 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 $\mathrm{\mid }\varphi (r)\mathrm{\mid }<1\mathrm{/}\sqrt{\xi }$ or $\mathrm{\mid }\varphi (r)\mathrm{\mid }>1\mathrm{/}\sqrt{\xi }$. For $0<\xi <{\xi }_c$, we prove the absence of scalar hairs such that $\mathrm{\mid }\varphi (r)\mathrm{\mid }<1\mathrm{/}\sqrt{\xi }$ or $\frac{1}{\xi }<{\varphi }^2(r)<\frac{{\xi }_c}{\xi ({\xi }_c-\xi )}$.