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

For massless scalar fields, a relation $Δ_{n}=\frac{\sqrt{3}}{2}π$ 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 $Δ_{n}=\frac{\sqrt{3}}{2}π$ for $n\rightarrow \infty$ still holds. We demonstrate this relation with both analytical and numerical methods. The analytical analysis implies that this relation may be a very universal behavior.