# SU(3) octet scalar quartic interactions

1. Feb 18, 2016

### Lamia

Hi.
General question: Is there a fixed way to find all invariant tensor for a generic representation?

Example problem: Suppose you search for all indipendent quartic interactions of a scalar octet field $\phi^{a}$ in the adjoint representation of SU(3). They will be terms like

$L_{int}=\phi^{\dagger a}\phi^{b}\phi^{\dagger c}\phi^{d} T^{abcd}$,

where $T^{abcd}$ is an invariant tensor of $8 \otimes8 \otimes8 \otimes8$.

So, using $8 \otimes 8=1\oplus8\oplus27\oplus8\oplus10\oplus\bar{10}$, plus the fact that the adjoint is real, plus the fact that $N \otimes \bar{N}=1\oplus...$, plus the fact that an invariant tersor exists for each siglets in irreps reduction, one can conclude that there are seven $T^{abcd}$ invariant tensor of $8 \otimes8 \otimes8 \otimes8$. But what are they?

I'm trying to combine SU(3) main tensors, such as $\delta^{ab}$, generators of algebra, $d^{abc}$ completely simmetric tensor, etc.., but i would like to know if is there a more systematic approach.

2. Feb 23, 2016

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Feb 23, 2016

### Orodruin

Staff Emeritus
You should look into Young tableaux methods for determining the reduction of products of irreps into irreps. It becomes fairly straight forward for all SU(N) groups.