Black hole scalarisation from the breakdown of scale-invariance

Electro-vacuum black holes are scale-invariant; their energy-momentum tensor is traceless. Quantum corrections of various sorts, however, can often produce a trace anomaly and a breakdown of scale-invariance. The (quantum-corrected) black hole solutions of the corresponding gravitational effective field theory (EFT) have a non-vanishing Ricci scalar. Then, the presence of a scalar field with the standard non-minimal coupling $\xi \phi^2 R$ naturally triggers a spontaneous scalarisation of the corresponding black holes. This scalarisation phenomenon occurs for an (infinite) discrete set of $\xi$. We illustrate the occurrence of this phenomenon for two examples of static, spherically symmetric, asymptotically flat black hole solution of EFTs. In one example the trace anomaly comes from the matter sector -- a novel, closed form, generalisation of the Reissner-Nordstr\"om solution with an $F^4$ correction -- whereas in the other example it comes from the geometry sector -- a noncommutative geometry generalization of the Schwarzschild black hole. For comparison, we also consider the scalarisation of a black hole surrounded by (non-conformally invariant) classical matter (Einstein-Maxwell-dilaton black holes). We find that the scalarised solutions are, generically, entropically favoured.