# Making the most general lagrangian of triplet scalar fields under $SU(3)$ symmetry

I need to write the most general renormalized lagrangian under those conditions:

1. The symmetry is global $$SU(3)_G$$

2. There is 3 triplets of scalar field $$\phi(3)_i$$ , $$i=1,2,3$$

3. There are no fermion fields.

I use Einsteins notation $$\mathcal L=(\partial_{\mu}\phi_i)^2-m_i^2\phi^{\dagger}_{i}\phi_i-\eta(some- combination -of -three -fields)-\lambda_i( \phi^{\dagger}_{i}\phi_i)^2$$

So as you can see in the lagrangian I dont know how to build the trilinear term - I know it must be a singlet, but how do I create singlet out of 3 triplets? The second question is does there is more quadratic and quartic elements exist in the lagrangian then of how I wrote this?

• Presumably in 4D? – Qmechanic Aug 7 at 10:13
• Of course due to Lorentz invariance and Renormalizability – Daniel Vainshtein Aug 7 at 10:16
• Is there any symmetry acting on the $i$ index? Do you consider redefinitions of the $\phi_i$? What about $\phi_1^\dagger \phi_2$? – Toffomat Aug 7 at 10:30
• That's just it: $3\times 3 \times 3$ contains a singlet, but it is antisymmetric, so it's good for fermions but not scalars (unless you give those a flavor permitting antisymmetrization...). – Cosmas Zachos Aug 7 at 14:12
• @CosmasZachos I don't see the ambiguity here, doesn't the parenthesis in $\phi (3)$ imply that the scalar fields sits in the triplet representation and he says there are 3 triplets which means the system he is referring to has, as you mention $3\times 3\times 2$ dof. – Sanjay M Aug 8 at 19:45

Yeah , it is a bit tricky to find the third order term. Essentially, it is pretty clear that we need use the epsilon tensor. Once you figure that out , it might seem that the antisymmetric tensor will give a zero term as mentioned by Cosmos Zachos , but we need to note that we have three different flavours in the model. Hence, we could have a term of the form , $$\epsilon^{abc}\phi_{1a}\phi_{2b}\phi_{3c}\sim \det[\Phi]$$ where the indices $$a,b,c$$ are the $$SU(3)$$ indices.
Note that this could only be done since we have at least 3 flavours. Also, if you want to give the most general renormalizable Lagrangian, you should in general allow varied coupling constants instead of a single $$\lambda$$ and also varied masses for the three flavours
• What is $\Phi$ ? – Daniel Vainshtein Aug 9 at 8:52
• It's the matrix with the three columns (or rows) being $\phi_1,\phi_2,\phi_3$ – Sanjay M Aug 9 at 12:25