Past exam of the particle physics course of the University of Oxford / 2022 / 3 / a

The ${\Delta }^{++}$ Delta baryon has electric charge $+2$ and valence quark content $uuu$. In its ground state the spatial wavefunction has orbital angular momentum $L=0$ and is symmetric. Its spin is $3/2$; the maximal projection state $u_{\uparrow }{u}_{\uparrow }{u}_{\uparrow }$ is symmetric in spin, and its quark flavour wavefunction is symmetric as well. Three identical fermions require an antisymmetric total wavefunction by the Pauli exclusion principle. A further internal degree of freedom must therefore supply the antisymmetry.

With three color charges, that factor is the normalized colour singletIt changes sign under exchange of any two quarks. The remaining symmetric factors then give the required antisymmetric total state. This argument motivates color charge independently of the already present quark flavour and spin labels.