Black holes in Gauss-Bonnet and Chern-Simons-scalar theory

We carry out the stability analysis of the Schwarzschild black hole in Gauss-Bonnet and Chern-Simons-scalar theory. Here, we introduce two quadratic scalar couplings (${\varphi }_1^2,{\varphi }_2^2$) to Gauss-Bonnet and Chern-Simons terms, where the former term is parity-even, while the latter one is parity-odd. The perturbation equation for the scalar ${\varphi }_1$ is the Klein-Gordon equation with an effective mass, while the perturbation equation for ${\varphi }_2$ is coupled to the parity-odd metric perturbation, providing a system of two coupled equations. It turns out that the Schwarzschild black hole is unstable against ${\varphi }_1$ perturbation, leading to scalarized black holes, while the black hole is stable against ${\varphi }_2$ and metric perturbations, implying no scalarized black holes.