Cubic Term in Higgs Boson

Main Question or Discussion Point

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?

Thanks!

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The higgs field is a complex field. How do you propose to include a cubic term and leave the Lagrangian u(1) invariant?

Ok, thanks!

But that's after we had shifted the field by its VEV, if I'm not mistaken. That occurs even for:
$$\mathcal{L}_{4} = \frac{\lambda}{2} \, \left( \vert \varphi \vert^2 - v^2 \right)^2$$
if you make the subst:
$$\varphi = (v + \rho) \, e^{i \, \chi}$$

Oh, yeah fair enough.