Hairy black hole solutions in U(1) gauge-invariant scalar–vector–tensor theories

In U(1) gauge-invariant scalar–vector–tensor theories with second-order equations of motion, we study the properties of black holes (BH) on a static and spherically symmetric background. In shift-symmetric theories invariant under the shift of scalar ϕ→ϕ+c, we show the existence of new hairy BH solutions where a cubic-order scalar–vector interaction gives rise to a scalar hair manifesting itself around the event horizon. In the presence of a quartic-order interaction besides the cubic coupling, there are also regular BH solutions endowed with scalar and vector hairs