Quark Model - Baryons

Baryons

Since quarks are fermions, the spin-statistics theorem implies that the wavefunction of a baryon must be antisymmetric under exchange of any two quarks. This antisymmetric wavefunction is obtained by making it fully antisymmetric in colour and symmetric in flavour, spin and space put together. With three flavours, the decomposition in flavour is

.

The decuplet is symmetric in flavour, the singlet antisymmetric and the two octets have mixed symmetry. The space and spin parts of the states are thereby fixed once the orbital angular momentum is given.

It is sometimes useful to think of the basis states of quarks as the six states of three flavours and two spins per flavour. This approximate symmetry is called spin-flavour SU(6). In terms of this, the decomposition is

The 56 states with symmetric combination of spin and flavour decompose under flavour SU(3) into

where the superscript denotes the spin, S, of the baryon. Since these states are symmetric in spin and flavour, they should also be symmetric in space—a condition that is easily satisfied by making the orbital angular momentum L = 0. These are the ground state baryons. The S = 1⁄2 octet baryons are the two nucleons (p+, n0), the three Sigmas (Σ+, Σ0, Σ−), the two Xis (Ξ0, Ξ−), and the Lambda (Λ0). The S = 3⁄2 decuplet baryons are the four Deltas (Δ++, Δ+, Δ0, Δ−), three Sigmas (Σ∗+, Σ∗0, Σ∗−), two Xis (Ξ∗0, Ξ∗−), and the Omega (Ω−). Mixing of baryons, mass splittings within and between multiplets, and magnetic moments are some of the other questions that the model deals with.