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.