Black hole minimum size and scalar charge in shift-symmetric theories

It is known that, for shift-symmetric scalars, only a linear coupling with the Gauss-Bonnet invariant can introduce black hole hair. Such hairy black holes have a minimum mass, determined by the coupling of this interaction, and a scalar charge that is uniquely determined by their mass and spin for a fixed value of that coupling. Here we explore how additional shift-symmetric interactions affect the structure of the black hole, the value of the minimum mass, and the scalar charge.