# Permitted bound states in extended QCD color group $SU(4)$

I'm very much curious about the possible quark combinations to form bound states in $$SU(4)$$ QCD. We already know Wilson has proven that only colorless combinations can arise in $$SU(3)$$ of these formats: $$q\bar{q}$$, $$qqq$$, tetra quarks and penta quarks.
I guess in $$SU(4)$$ one can only find $$q\bar{q}$$ and $$qqqq$$ and combinations of these two. Am I right?

The argument is still the same: A hadron has to be a color singlet $${\bf 1}$$ under the $$SU(N)_C$$ color gauge group, due to color confinement.

• A single quark $$q$$ transforms in the fundamental representation $${\bf N}$$ of $$SU(N)_C$$, and is hence not allowed since $$N>1$$.

• In a meson, the quark-antiquark-pair $$q\bar{q}$$ belongs to $${\bf N}\otimes\bar{\bf N}\cong{\bf 1}\oplus({\bf N^2\!-\!1})_{\rm Adj}$$, which contains a singlet $${\bf 1}$$, and is hence allowed.

• In a baryon, the $$N$$ quarks $$\overbrace{qq\ldots q}^{N\text{ times}}$$ form a totally antisymmetric representation $$\wedge^N {\bf N}\cong {\bf 1}$$ of $$SU(N)_C$$, which is isomorphic to a singlet $${\bf 1}$$, and is hence allowed.

As one can see the number of quarks minus the number of anti-quarks should be divisible by $$N$$. For $$N=3$$, see also my related Phys.SE answer here.

In fact yes, a simple way to get all possible states is by the following argument: quarks belong to the fundamental representation of the color group, and to them one can assign the upper index $$q^{i}$$, the antiquarks belong to the antifundamental representation, and to them we assign the lower index $$\bar{q}_i$$. Also there is a Levi-Civita symbol $$\varepsilon^{i_1 \ldots i_N}$$ ($$N$$ being the number of colors). The asymptotic states in QCD are color singlets $$-$$ so to get all possible states, you should contract all indices with each other in some way.

As for quark-antiquark pair you form a singlet by: $$\bar{q}_i q^{i} \,,$$Which means taking the trace in color space. And baryons (antibaryons) correspond to: $$\varepsilon_{i_1 \ldots i_N} q^{i_1} \ldots q^{i_N} \,,\qquad \varepsilon^{i_1 \ldots i_N} \bar{q}_{i_1} \ldots \bar{q}_{i_N} \,.$$