No static regular black holes in Einstein-complex-scalar-Gauss-Bonnet gravity

Understand

In this brief report, we investigate the existence of 4-dimensional static spherically symmetric black holes (BHs) in the Einstein-complex-scalar-Gauss-Bonnet (EcsGB) gravity with an arbitrary potential $V(\phi)$ and a coupling $f(\phi)$ between the scalar field $\phi$ and the Gauss-Bonnet (GB) term.

• We find that static regular BH solutions with complex scalar hairs do not exist.
• This conclusion does not depend on the coupling between the GB term and the scalar field, nor on the scalar potential $V(\phi)$ and the presence of a cosmological constant $\Lambda$ (which can be either positive or negative), as longer as the scalar field remains complex and is regular across the horizon.