where B α α β β = ∂ ∂ α α ϕ ϕ β β − − ∂ ∂ β β ϕ ϕ α α is the (coordinate independent) exterior derivative of ϕ ϕ α α , , μ μ is the mass of the vector field, ω ω characterizes the strength of the coupling between the fifth force and matter, and V ϕ ϕ is a self-interaction potential.