Charged black holes in non-linear Q-clouds with O(3) symmetry

We construct charged soliton solutions around spherical charged black holes with no angular momentum in asymptotically flat spacetime. These solutions are non-linear generalizations of charged scalar clouds, dubbed Q-clouds, and they do not contradict the non-existence theorem for free (linear) scalar clouds around charged black holes. These solutions are the first examples of O(3) solutions for Q-clouds around a non-extremal and non-rotating BH in the Abelian gauge theory. We show that a solution exists with an infinitely short cloud in the limit of extremal black holes. We discuss the evolution of Q-cloud in a system with fixed total charge and describe how the existence of Q-clouds is related to the weak-gravity conjecture. The reason that the no-hair theorem by Mayo and Bekenstein cannot be applied to the massive scalar field is also discussed.