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The potential for the Higgs field is 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, which may lead to differences in the derived particle masses.

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I always thought is was more "urban sombrero" shaped. – Jiminion Mar 4 '15 at 15:50

In four dimensions, 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 (e.g. symmetry under $\phi \mapsto -\phi$) rules 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
Regarding the first sentence, you'd have to say in $d=4$. – JamalS Mar 4 '15 at 14:37
@JamalS: I edited my post accordingly. – Arnold Neumaier Mar 4 '15 at 15:48
What do you mean by worst? – Jiminion Mar 4 '15 at 15:51
It means that it is the higher power that works : $\phi^{5}$ would not be renormalizable – agemO Mar 4 '15 at 15:55

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