Spontaneous scalarization of Gauss-Bonnet black holes surrounded by massive scalar fields

Understand

For massless scalar fields, a relation $\Delta_{n}=\frac{\sqrt{3}}{2}\pi$ for $n\rightarrow \infty$ was observed in the scalar-Gauss-Bonnet theory.

• In the present paper, we extend the discussion by including a nonzero scalar field mass.
• For massive scalar fields, we show that the relation $\Delta_{n}=\frac{\sqrt{3}}{2}\pi$ for $n\rightarrow \infty$ still holds.
• We demonstrate this relation with both analytical and numerical methods.