# Why has the Higgs potential the form it has?

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

## 1 Answer

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