  We investigate scalar field perturbations of the hairy black holes involved
with spontaneous symmetry breaking of the global U(1) symmetry in
Einstein-scalar-Gauss-Bonnet theory for asymptotically flat spacetimes. We
consider the mechanism that black holes without hairs become unstable at the
critical point of the coupling constant and undergo a phase transition to hairy
black holes in the symmetry-broken phase driven by spontaneous symmetry
breaking. This transition occurs near the black hole horizon due to the
diminishing influence of the Gauss-Bonnet term at infinity. To examine such
process, we introduce a scalar field perturbation on the newly formed
background spacetime. We solve the linearized perturbation equation using
Green's function method. We begin by solving the Green's function,
incorporating the branch cut contribution. This allows us to analytically
investigate the late-time behavior of the perturbation at both spatial and null
infinity. We found that the late-time behavior only differs from the
Schwarzschild black hole by a mass term. We then proceed to calculate the
quasinormal modes (QNMs) numerically, which arise from the presence of poles in
the Green's function. Our primary interest lies in utilizing QNMs to
investigate the stability of the black hole solutions both the symmetric and
symmetry-broken phases. Consistent with the prior study, our analysis shows
that hairy black holes in the symmetric phase become unstable when the
quadratic coupling constant exceeds a critical value for a fixed value of the
quartic coupling constant. In contrast, hairy black holes in the
symmetry-broken phase are always stable at the critical value. These numerical
results provide strong evidence for a dynamical process that unstable black
holes without hairs transition into stable hairy black holes in the
symmetry-broken phase through the spontaneous symmetry breaking.
