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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.

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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.

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Weak SU(2) charge conservation rules out the cubic term--- the Higgs doublet can't make a cubic term which is gauge invariant. –  Ron Maimon Aug 10 '12 at 19:12
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