Physical charged-vector Lagrangian for an adjoint SU2 Higgs model ($m_{W^{\pm }}=g_{c}v,m_{\eta }^2=\lambda v^2$)

For a real triplet potential $\lambda (|\varphi {|}^2-v^2{)}^2/8$ and canonical gauge coupling $g_c$, unitary gauge gives one neutral Higgs mode, two oppositely charged massive vectors and one massless Abelian vector. The scalar kinetic contribution is $\frac{1}{2}(\partial \eta {)}^2+g_c^2(v+\eta {)}^2{W}^{+}{W}^{-}$, and its potential is $\lambda v^2{\eta }^2/2+\lambda v{\eta }^3/2+\lambda {\eta }^4/8$. This potential convention is distinct from using $\lambda (|\varphi {|}^2-v^2{)}^2/4$.

For a real triplet with potential $\lambda ({\Phi }^2-v^2{)}^2/8$, unitary gauge gives $\Phi =(0,0,v+h)$ and scalar interaction terms $-\lambda v{h}^3/2-\lambda h^4/8+2e^{2}vh{W}^{+}{W}^{-}+e^2{h}^2{W}^{+}{W}^{-}$. Functional differentiation, including identical-field multiplicities, gives vertices $-3i\lambda v$, $-3i\lambda$, $2i{e}^{2}v{g}_{\mu \nu }$ and $2i{e}^2{g}_{\mu \nu }$ respectively. The physical scalar is neutral under the residual U(1), so there are no $hAA$ or $hhAA$ vertices.