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

For massless scalar fields, a relation Δn=32π for n→∞ 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 Δn=32π for n→∞ 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.