  It has recently been demonstrated that black holes with spatially regular
horizons can support external scalar fields (scalar hairy configurations) which
are non-minimally coupled to the Gauss-Bonnet invariant of the curved
spacetime. The composed black-hole-scalar-field system is characterized by a
critical existence line $\alpha=\alpha(\mu r_{\text{H}})$ which, for a given
mass of the supported scalar field, marks the threshold for the onset of the
spontaneous scalarization phenomenon [here $\{\alpha,\mu,r_{\text{H}}\}$ are
respectively the dimensionless non-minimal coupling parameter of the field
theory, the proper mass of the scalar field, and the horizon radius of the
central supporting black hole]. In the present paper we use analytical
techniques in order to explore the physical and mathematical properties of the
marginally-stable composed black-hole-linearized-scalar-field configurations in
the eikonal regime $\mu r_{\text{H}}\gg1$ of large field masses. In particular,
we derive a remarkably compact analytical formula for the critical
existence-line $\alpha=\alpha(\mu r_{\text{H}})$ of the system which separates
bare Schwarzschild black-hole spacetimes from composed hairy (scalarized)
black-hole-field configurations.
