Gauss-Bonnet black holes with a massive scalar field

In the present paper, we consider the extended scalar-tensor Gauss-Bonnet gravity with a massive scalar field. We prove numerically the existence of Gauss-Bonnet black holes for three different forms of the coupling function including the case of spontaneous scalarization. We have performed a systematic study of the black hole characteristics such as the area of the horizon, the entropy, and the temperature for these coupling functions and compared them to the Schwarzschild solutions. The introduction of scalar field mass leads to a suppression of the scalar field, and the increase of this mass brings the black holes closer to the Schwarzschild case. For linear and exponential coupling, a nonzero scalar field mass expands the domain of existence of black hole solutions. Larger deviations from the Schwarzschild solution are observed only for small masses, and these differences decrease with the increase of the scalar field mass. In the case of a coupling function which leads to scalarization, the scalar field mass has a significant influence on the bifurcation points where the scalarized black holes branch out of the Schwarzschild solution. The largest deviation from the case with a massless scalar field are observed for black hole masses close to the bifurcation point.