Axial perturbations of black holes with primary scalar hair

We study axial perturbations of static black holes with primary hair in a family of degenerate higher-order scalar-tensor (DHOST) theories. These solutions possess a scalar charge, fully independent of the mass, leading to a continuous one-parameter deformation of the standard Schwarzschild black hole. Starting from these solutions, we also construct new black holes, solutions of other DHOST theories, obtained via disformal transformations of the metric. In particular, we investigate two specific types of disformal transformations: the first leading to a theory where gravitational waves propagate at the speed of light, the second to a Horndeski theory, where the equations of motion remain second order. The dynamics of axial perturbations can be formally related to the general relativistic equations of motion of axial perturbations in an effective metric. The causal structure of the effective metric differs from that of the background metric, leading to distinct gravitational and luminous horizons. Using a WKB approximation, we compute the quasi-normal modes for the Schr\"odinger-like equation associated with the effective metric outside the gravitational horizon.