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