Electrically charged black hole with scalar hair

An electrically charged black hole solution with scalar hair in four dimensions is presented. The self-interacting scalar field is real and it is minimally coupled to gravity and electromagnetism. The event horizon is a surface of negative constant curvature and the asymptotic region is locally an AdS spacetime. The asymptotic falloff of the fields is slower than the standard one. The scalar field is regular everywhere except at the origin and is supported by the presence of electric charge which is bounded from above by the AdS radius. In turn, the presence of the real scalar field smooths the electromagnetic potential everywhere. Regardless the value of the electric charge, the black hole is massless and has a fixed temperature. The entropy follows the usual area law. It is shown that there is a nonvanishing probability for the decay of the hairy black hole into a charged black hole without scalar field. Furthermore, it is found that an extremal black hole without scalar field is likely to undergo a spontaneous dressing up with a nontrivial scalar field, provided the electric charge is below a critical value.