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 ChernSimons-scalar theory. Here, we introduce two quadratic scalar couplings (phi(2)(1),phi(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 phi(1) is the Klein- Gordon equation with an effective mass, while the perturbation equation for phi(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 phi(1) perturbation, leading to scalarized black holes, while the black hole is stable against phi(2) and metric perturbations, implying no scalarized black holes.