Hairy black holes by spontaneous symmetry breaking

We study hairy black hole solutions in Einstein(-Maxwell)-scalar-Gauss-Bonnet theory. The complex scalar coupling function includes quadratic and quartic terms, so the gravitational action has a U(1) symmetry. We argued that when the effective mass of the scalar field is at the critical value, the black holes without hairs transform into hairy black holes in a symmetry-broken vacuum via spontaneous symmetry breaking. These hairy black holes are stable under scalar perturbations, and the Goldstone bosons are trivial. Moreover, we found that the spontaneous symmetry breaking associated with local U(1) is unlikely to occur in this theory.