 Solution

A physical baryon is a colour singlet. Its three-quark colour factor is antisymmetric, proportional to ${ϵ}_{abc}|abc⟩$. In the hypothetical scalar-quark theory, the quarks obey Bose-Einstein statistics, so the complete three-particle state must be symmetric under exchange. At $L=0$ the spatial factor is symmetric, and three zero spins supply no nontrivial spin factor. The flavour factor must therefore be antisymmetric to cancel the antisymmetry of the colour factor.

There is exactly one completely antisymmetric state made from three flavour basis states: the flavour singlet of three scalar quarks,Equivalently, $\dim {\bigwedge }^3{\mathbb{C}}^3=1$. A repeated flavour gives zero under complete antisymmetrization. Only one ground-state flavour singlet, containing $uds$ and having $J^P=0^{+}$, remains. Keeping fermionic exchange antisymmetry while changing the spins to zero would contradict the assumed Spin-statistics theorem. The standard antisymmetric colour factor and the ordinary-quark flavour decomposition are documented in the quark-model review.^[ref]