Scalarization of Taub-NUT Black Holes in Extended scalar-tensor-Gauss-Bonnet Theory

Recently, scalarization of Schwarzschild black hole are extensively studied. In this work, we explore the scalarization of Taub-NUT black hole. The theory we consider is the extended scalar-tensor-Gauss-Bonnet theory, which admits Ricci-flat Taub-NUT black hole as a solution. An analysis of probe scalar field is carried out to identify the mass parameter and NUT parameter (m,n) where the hairy black holes start to emerge. Then, we use shooting method to construct the scalarized Taub-NUT black hole numerically. Being different from the Schwarzschild case, there exists two branches of new hairy black holes which are smoothly connected to each other. We calculate the entropy of scalarized black holes and compare it with the entropy of scalar-free Taub-NUT black holes, it turns out that the entropy of the new hairy black holes are larger than that of scalar-free black holes. A novel phenomena emerges in this system that the entropy of the black holes at the bifurcation point is constant for positive mass parameter. We then conjecture a maximal entropy bound for all the scalarized black hole whose mass parameter at the bifurcation point is greater than zero.