Lets say one has a gauge group for example SU(n) or SO(n) and has a scalar field which belongs to a certain representation (m-ranked tensor). If one wants to write down the invariant potential for the scalar field, for lower dimensional representations (fundamental/adjoint/2-ranked symmetric/anti-symmetric), it is not difficult and studied in detail in the paper by L.F.Li http://journals.aps.org/prd/abstract/10.1103/PhysRevD.9.1723 .

On the contrary, if one has a product group, very little is mentioned on the above paper and calculating the invariant potential seems quite difficult, specially, constructing terms that are linearly independent of each other.

The question is: how to construct linearly independent invariant scalar potential for a representation under a product group. is there any general method to find out which are the terms that are linearly independent and the terms that are not linearly independent? any good reference exist that explains such techniques?

As for example, lets consider a product group $$ SU(2)_{L} \times SU(2)_{R} \times SU(4)_{C} $$ and a scalar field in the representation ${\phi_{\alpha \dot{\beta}}}_{,\;i}^{\;\;j} =(2,2,15)$
where, $\alpha,\dot{\beta}=1,2$ are the indices of $SU(2)_{L}$ and $SU(2)_{R}$ respectively and i,j=1,2,3,4 are the indices of $SU(4)_{C}$.

Lets try to construct quartic,$\phi^{4}$ terms. To make things simple, lets impose a $Z_{2}$ symmetry and restricts terms like $\phi \phi \phi \phi$ and only terms allowed are of the form $\phi^{\dagger} \phi \phi^{\dagger} \phi$. The field $\phi$ is assumed to be complex field. According to group theory 14 such quartic terms are possible. But by tensor construction method one can actually construct more than 14 terms. How to find out the linearly independent 14 terms out of them?



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