Linear perturbations of dyonic black holes in the lowest-order $U(1)$ gauge-invariant scalar-vector-tensor theories

We study linear perturbations on top of the static and spherically symmetric background of dyonic black hole solutions endowed with electric and magnetic charges, as well as a scalar hair, in the lowest-order $U(1)$ gauge-invariant scalar-vector-tensor theories. The presence of magnetic charges in the background solutions gives rise to a mixing between the odd-parity and even-parity sectors of perturbations, which makes it impossible to analyze each sector separately. Thus, we expand the action up to second order in both odd-parity and even-parity perturbations and derive the general conditions for the absence of ghosts and Laplacian instabilities. We apply these general conditions to extended Einstein-Maxwell theories, which encompass a wide variety of concrete models from the literature known to have dyonic black hole solutions with the scalar hair, and examine their stabilities. Our general framework for studying stability conditions and dynamics of perturbations can be applied to a wide variety of theories, including nonlinear electrodynamics coupled to a scalar field, as well as to calculations of black hole quasinormal modes.