Scalar interactions after complete SU2 breaking ($m_h^2=\lambda v^2$)

In unitary gauge, $\varphi =(0,(v+h)/\sqrt{2}{)}^T$ gives $\frac{g^2}{8}(v+h{)}^2{A}_{\mu }^a{A}^{a\mu }$. With potential $\frac{\lambda }{2}({\varphi }^{†}\varphi -v^2/2{)}^2$, the physical scalar has $m_h^2=\lambda v^2$ and interactions $-\lambda v{h}^3/2-\lambda h^4/8$. This normalization matters when extracting both masses and identical-leg Feynman vertices.