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 $S = \int d^nx\sqrt{-g} \left\{\left(1-ξϕ^2 \right)R - g^{μν}\partial_μϕ\partial_νϕ\right\},$ with $n>3$ and arbitrary $ξ$. We demonstrate the absence of scalar hairs for $ξ<0$. For $ξ>ξ_c=\frac{n-2}{4(n-1)}$, we show that there is no scalar hair obeying $|ϕ(r)| < 1/\sqrtξ$ or $|ϕ(r)| > 1/\sqrtξ$. For $0<ξ<ξ_c$, we prove the absence of scalar hairs such that $|ϕ(r)| < 1/\sqrtξ$ or $\frac{1}ξ < ϕ^2(r) < \frac{ξ_c}{ξ(ξ_c-ξ)}$.

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