Searching for nonminimally coupled scalar hairs

We study the asymptotically flat, static, and spherically symmetric black-hole solutions of the theory described by the action S={∫} dⁿˣ{}-g{(1-{ξ}φ²)R-g^μν{{{∂}}}_{{{μ}}}φ∂_{{ν}}}, with n3 and arbitrary ξ. We demonstrate the absence of scalar hairs for ξ0. For ξ{{ξ}}c$=(n-2)/4(n-1), we show that there is no scalar hair obeying |{φ}(r)|1/{}{ξ} or |{φ}(r)|{}1/{}{ξ}. For 0ξc, we prove the absence of scalar hairs such that |{φ}(r)|1/{}{ξ} or 1/{ξ}φ²(r)ξc/{ξ}(ξc-{ξ}). {} 1996 The American Physical Society.