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The asymptotically flat, spherical, electro-vacuum black holes (BHs) areshown to support static, spherical configurations of a gauged,self-interacting, scalar field, minimally coupled to the geometry. Consideringa $Q$-ball type potential for the scalar field, we dub these configurations$Q$-clouds, in the test field approximation. The clouds exist under a resonancecondition, at the threshold of (charged) superradiance. This is similar to thestationary clouds supported by Kerr BHs, which exist for a synchronisationcondition, at the threshold of (rotational) superradiance. In contrast with therotating case, however, $Q$-clouds require the scalar field to be massive andself-interacting; no similar clouds exist for massive but free scalar fields.First, considering a decoupling limit, we construct $Q$-clouds aroundSchwarzschild and Reissner-Nordstr\"om BHs, showing there is always a mass gap.Then, we make the $Q$-clouds backreact, and construct fully non-linearsolutions of the Einstein-Maxwell-gauged scalar system describing spherical,charged BHs with resonant, scalar $Q$-hair. Amongst other properties, weobserve there is non-uniqueness of charged BHs in this model and the $Q$-hairyBHs can be entropically preferred over Reissner-Nordstr\"om, for the samecharge to mass ratio; some $Q$-hairy BH solutions can be overcharged. We alsodiscuss how some well known no-hair theorems in the literature, applying toelectro-vacuum plus minimally coupled scalar fields, are circumvented by thisnew type of BHs.

Spherical electro-vacuum black holes with resonant, scalar $Q$-hair