Why does the standard model have a higgs boson quadratic and a cuartic term but it does not have a cubic term? is there any problem if it happens to have a cubic term?
But that's after we had shifted the field by its VEV, if I'm not mistaken. That occurs even for:
[tex]
\mathcal{L}_{4} = \frac{\lambda}{2} \, \left( \vert \varphi \vert^2 - v^2 \right)^2
[/tex]
if you make the subst:
[tex]
\varphi = (v + \rho) \, e^{i \, \chi}
[/tex]