Scalar in adjoint representation

[MManuel Abad]

Hello, people.

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 \Phi, 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:

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

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.

 

[Bill_K]

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

There's only two invariants for SU(3), one of them quadratic, the other cubic. Consequently your first term should be discarded. Tr(Φ⁴) and Tr(Φ²)² are basically the same thing.

 

[ChrisVer]

Are they really the same thing?
Only in the case in which:

Suppose the matrix to have diagonal elements h_{ii}
They are the same if:
\sum_{i} h_{ii}^{4} = (\sum_{i}h_{ii}^{2})^{2}
Maybe this holds? why?

As for the cubic term, it's not so common to take in account cubic terms in the lagrangians because they are somehow destroying the symmetry under reflections:
Φ \rightarrow -Φ
In general if you want to break this symmetry, nothing stops you from taking this kind of terms into account.

 

[Bill_K]

The field Φ is said to be an SU(3) octet. That's Φ_i, i=1,...,8. There's one way to form an SU(3) invariant which is quadratic, namely δ_ijΦ_iΦ_j. That's the Casimir invariant, which could be written Tr(Φ²).

How do you form a cubic invariant? In SU(3) there's one way to do that, namely d_ijkΦ_iΦ_jΦ_k.

Ok, now try to form an SU(3) invariant which is quartic. Φ_iΦ_jΦ_kΦ_l times what? The only available things you can multiply this by are δ_ijδ_kl or δ_ikδ_jl or δ_ilδ_jk. They all lead to the same thing: just the Casimir invariant squared.

 

[MManuel Abad]

Ok, now try to form an SU(3) invariant which is quartic. Φ_iΦ_jΦ_kΦ_l times what? The only available things you can multiply this by are δ_ijδ_kl or δ_ikδ_jl or δ_ilδ_jk. They all lead to the same thing: just the Casimir invariant squared.

What about terms with coefficients d^{ijk} d^{klm}? Wouldn't they be able to produce \Phi^4 terms as well? Or does this contraction also have an expression in terms of \delta^{ij}'s? If that is the case, do you happen to know it?

 

[samalkhaiat]

Almost all useful relations of the algebraic structure of SU(N) are listed below:
<br /> [ T_{ a } , T_{ b } ] = i f_{ a b c } T_{ c } , \ \ \ \ \ (1)<br />
<br /> \{ T_{ a } , T_{ b } \} = \frac{ 1 }{ N } \delta_{ a b } I_{ N } + d_{ a b c } T_{ c } , \ \ \ (2)<br />
where I_{ N } is the N-dimensional unit matrix. The f_{ a b c } are antisymmetric and the d_{ a b c } symmetric under the interchange of any two indices.

Traces:

<br /> \mbox{ Tr } ( T_{ a } ) = 0, \ \ \mbox{ Tr } ( T_{ a } T_{ b } T_{ c } ) = \frac{ 1 }{ 4 } ( d_{ a b c } + i f_{ a b c } ) , \ \ \ (3)<br />
<br /> \mbox{ Tr } ( T_{ a } T_{ b } ) = \frac{ 1 }{ 2 } \delta_{ a b }, \ \ \mbox{ Tr } ( T_{ a } T_{ b } T_{ a } T_{ c } ) = - \frac{ 1 }{ 4 N } \delta_{ b c }, \ \ (4)<br />
<br /> 2 T_{ a } T_{ b } = \frac{ 1 }{ N } \delta_{ a b } I_{ N } + ( d_{ a b c } + i f_{ a b c } ) T_{ c }, \ \ \ (5)<br />
<br /> 2 ( T_{ a } )_{ i j } ( T_{ a } )_{ k l } = \delta_{ i l } \delta_{ j k } - \frac{ 1 }{ N } \delta_{ i j } \delta_{ k l } . \ \ \ \ (6)<br />

Useful matrices for the N^{ 2 } – 1 dimensional adjoint representation:

<br /> ( F_{ a } )_{ b c } = - i f_{ a b c }, \ \ ( D_{ a } )_{ b c } = d_{ a b c } . \ \ \ (7)<br />

The Jacobi identities:

f_{ a b e } f_{ e c d } + f_{ c b e } f_{ a e d } + f_{ d b e } f_{ a c e } = 0 , \ \ \ \ (8a)
f_{ a b e } d_{ e c d } + f_{ c b e } d_{ a e d } + f_{ d b e } d_{ a c e } = 0 , \ \ \ (8b)
or equivalently
[ F_{ a } , F_{ b } ] = i f_{ a b c } F_{ c } , \ \ \ \ (9a)
[ F_{ a } , D_{ b } ] = i f_{ a b c } D_{ c } . \ \ \ \ (9b)

Contractions and more traces:

<br /> f_{ a b e } f_{ c d e } = \frac{ 2 }{ N } ( \delta_{ a c } \delta_{ b d } - \delta_{ a d } \delta_{ b c } ) + ( d_{ a c e } d_{ b d e } - d_{ b c e } d_{ a d e } ), \ \ (10)<br />
f_{ a b b } = 0, \ \ \Rightarrow \ \ \mbox{ Tr } ( F_{ a } ) = 0 , \ \ (11a)
d_{ a b b } = 0, \ \ \Rightarrow \ \ \mbox{ Tr } ( D_{ a } ) = 0 , \ \ (11b)
f_{ a c d } f_{ b c d } = N \delta_{ a b }, \Rightarrow \mbox{ Tr } ( F_{ a } F_{ b } ) = N \delta_{ a b } , \ \ (12a)
F_{ a } F_{ a } = N I_{ N^{ 2 } - 1 } , \ \ \ (12b)
<br /> f_{ a c d } d_{ b c d } = 0 , \ \mbox{ Tr } ( F_{ a } D_{ b } ) = 0 , \ \ F_{ a } D_{ a } = 0 , \ \ \ (13)<br />
d_{ a c d } d_{ b c d } = \frac{ N^{ 2 } - 4 }{ N } \delta_{ a b }, \ \ \ (14a)
\mbox{ Tr } ( D_{ a } D_{ b } ) = \frac{ N^{ 2 } - 4 }{ N } \delta_{ a b } \ \ (14b)
D_{ a } D_{ a } = \frac{ N^{ 2 } - 4 }{ N } I_{ N^{ 2 } - 1 } , \ \ \ \ (14c)
\mbox{ Tr } ( F_{ a } F_{ b } F_{ c } ) = \frac{ i N }{ 2 } f_{ a b c } \ \ \ (15a)
\mbox{ Tr } ( D_{ a } F_{ b } F_{ c } ) = \frac{ N }{ 2 } d_{ a b c } , \ \ \ (15b)
\mbox{ Tr } ( D_{ a } D_{ b } F_{ c } ) = i \frac{ N^{ 2 } - 4 }{ 2 N } f_{ a b c } , \ \ (15c)
\mbox{ Tr } ( D_{ a } D_{ b } D_{ c } ) = \frac{ N^{ 2 } - 12 }{ 2 N } d_{ a b c } , \ \ (15d)

And finally my favourite

\mbox{ Tr } ( F_{ a } F_{ b } F_{ a } F_{ c } ) = \frac{ N^{ 2 } }{ 2 } \delta_{ b c } . \ \ \ (16)

Sam

 

[Bill_K]

SU(3) is a rank two Lie group, and so we know there can only be two independent Casimir invariants. So it's not profitable to spend time looking for a third one. Any other invariant we form is guaranteed to be algebraically dependent on those two.

What about terms with coefficients d^{ijk} d^{klm}? Wouldn't they be able to produce \Phi^4 terms as well? Or does this contraction also have an expression in terms of \delta^{ij}'s? If that is the case, do you happen to know it?

See Eq. (3.10) of this paper.

 

[MManuel Abad]

Wooow! Thank you so much all, this was so complete!