Black hole hairs in scalar-tensor gravity and the lack thereof

Scalar-tensor theories are a natural alternative to general relativity, as they may provide an effective dark energy phenomenology on cosmological scales while passing local tests, but their black hole solutions are still poorly understood. Here, we generalize existing no-hair theorems for spherical black holes and specific theories in the scalar-tensor class. We show that shift symmetry prevents the appearance of scalar hairs in rotating (asymptotically flat, stationary and axisymmetric) black holes for all theories in the Horndeski/beyond Horndeski/DHOST classes, but for those with a coupling between the scalar and the Gauss--Bonnet invariant. Our proof also applies to higher dimensions. We also compute the values of the scalar hair charges if shift symmetry and asymptotic flatness are violated by a time growth of the scalar field at infinity, under suitable regularity conditions at the event horizon.