Existence of stable asymptotically flat black holes beyond the scalar–Gauss–Bonnet class

Determine whether stable, asymptotically flat black holes with static scalar hair exist in GLPV theories beyond the scalar–Gauss–Bonnet class considered in the paper.

Background

The paper develops local no-ghost and high-frequency gradient-stability criteria for static, spherically symmetric black holes with radial scalar profiles in general quartic–quintic GLPV theories. Its near-horizon analysis establishes broad obstructions for branches with nonzero horizon scalar kinetic term, while its explicit stable constructions are confined to weakly coupled scalar–Gauss–Bonnet black holes with compatible beyond-Horndeski deformations and vanishing horizon scalar kinetic term.

The authors therefore leave unresolved whether other GLPV interactions can support stable, asymptotically flat black holes carrying static scalar hair. Resolving this question would require identifying candidate solutions outside the scalar–Gauss–Bonnet class and testing them using the complete perturbation framework, including local stability conditions and, ultimately, finite-frequency mode analyses where appropriate.