The Quark Model — Question 6

Question 6

The Omega baryon uses color antisymmetry to satisfy the Pauli principle

The ground-state Omega minus has valence sss, spatial orbital angular momentum zero and spin 3/2. Assume its spatial, flavor and spin factors are symmetric under exchanging any two quarks.

1. Find the required exchange parity of its color factor.

2. Normalize a color wavefunction containing the six permutations of rgb with alternating signs, and find the amplitude of any one term.

Question 6 - Solution

The Omega baryon uses color antisymmetry to satisfy the Pauli principle

(a) Method and calculation

$$P_{12}\Psi_{\mathrm{total}}=-\Psi_{\mathrm{total}},\quad P_{12}(\psi_x\psi_f\psi_s)=+\psi_x\psi_f\psi_s.$$$$\boxed{P_{12}\psi_{\mathrm{color}}=-\psi_{\mathrm{color}}}.$$

(b) Interpretation and further calculation

$$\psi_{\mathrm{color}}=A(rgb-rbg+gbr-grb+brg-bgr).$$$$6|A|^2=1\Rightarrow\boxed{|A|=1/\sqrt6,\quad |A|^2=1/6}.$$

Physical check

The antisymmetric color singlet resolves the apparent conflict between aligned spins and three identical strange quarks.