Higgs potential

Question

The potential for the Higgs field is standard a quartic one (Mexican hat). Is this done for simplicity or are there fundamental reasons for this choice? I can imagine further contributions to this potential without altering the essentials. But this may lead to differences in the derived particle masses.

Answer 1

The quartic is the worst interaction between scalar fields that is renormalizable (by power counting arguments).

Cubics would also do but only in combination with quartics as a pure cubic potential is not bounded below. A further discrete symmetry assumption ules out the cubic term.