 Vector Higgs field

In the SO(3) vector Higgs model, a real scalar field $\Phi \in {\mathbb{R}}^3$ transforms as $\Phi \mapsto R\Phi$. Its gauge covariant derivative is $D_{\mu }\Phi ={\partial }_{\mu }\Phi +g{\mathbf{A}}_{\mu }\times \Phi$ for generators acting by the cross product.