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I'm studying (as an exercise) the breaking of an SU(3) gauge group to SU(2) x U(1) via a Higgs mechanism. The scalar responsible for the breaking is [itex]\Phi[/itex], who transforms under the adjoint representation of SU(3) (an octet). First of all I want to construct the most general potential for this scalar. So far I've got:

[itex]V= a Tr(\Phi^{4}) + b (Tr(\Phi^2))^2 + c Tr(\Phi^3) + d Tr(\Phi^2) [/itex].

Is the cubic term supposed to be there or is there a reason for it to disappear? I've been looking in the literature for examples of this kind of terms in potentials but I couldn't find any. I know that it would destabilize the vacuum, but there are quartic terms and therefore there are bound states and thus the vacuum is stable. If I want to minimize this potential my answer would depend on the sign of "c", wouldn't it?

Of course, I'm taking "a" and "b" positive while "d" is negative.

Cheers and thanks a lot.

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# Scalar in adjoint representation

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