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

For massless scalar fields, a relation Delta(n) = root 3/2 pi for n -> 8 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) = root 3/2 pi for n -> 8 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. (C) 2020 The Author. Published by Elsevier B.V.