Exact Black Hole Solutions with a Conformally Coupled Scalar Field and Dynamic Ricci Curvature in $f(R)$ Gravity Theories

We report exact black hole solutions in asymptotically flat or (A)dS four-dimensional spacetime with a conformally coupled self-interacting scalar field in $f(R)$ gravity. We first consider the asymptotically flat model $f(R) = R -2\alpha \sqrt{R}$ and derive an exact black hole solution. Then, we consider the asymptotically (A)dS model $f(R) =R -2 \Lambda -2 \alpha \sqrt{R-4 \Lambda }$ and derive an exact black hole solution. In both cases the modified gravity parameter $\alpha$, which has the dimension of the inverse mass, cannot be set to zero and the self-interacting potential is determined from the Klein-Gordon equation, preserving the conformal invariance. The thermodynamics of the solutions is also studied.