No-Go Theorems for Hairy Black Holes in Scalar- or Vector-Tensor-Gauss-Bonnet Theory

In this paper, we show a no-go theorem for static spherically symmetric black holes with vector hair in Einstein-$\Lambda$-Vector-Tensor-Gauss-Bonnet theory where complex field non-minimally couples Gauss-Bonnet invariant. For purpose, expand metric functions and radial of around the event horizon, substitute expansions into equations motion. Demanding that motion are satisfied each order, vanishes on horizon. Moreover, when horizon is degenerated, it also implied outside