Pauli constraint on three-quark flavour multiplets

In a three-quark colour singlet, a symmetric spatial state requires symmetric spin-flavour symmetry. Completely symmetric decuplet flavour couples to symmetric spin

3/2

; mixed octet flavour couples to mixed spin

1/2

in the symmetric permutation combination. Antisymmetric singlet flavour cannot have a symmetric S-wave orbital state, since

⋀^{3} C^{2} = 0

forbids a completely antisymmetric state of three spin-

1/2

quarks. Mixed orbital and spin states can instead realize the required antisymmetry.

Table of contents
- Light-diquark spin-isospin correlation Pauli constraint on three-quark flavour multiplets
  - Baryons with one isospin-singlet quark Light-diquark spin-isospin correlation
- Symmetric spin-flavour SU6 representation Pauli constraint on three-quark flavour multiplets

Light-diquark spin-isospin correlation ( $(I, S_{u d}) = (0, 0) or (1, 1)$ )

 0  0

Two light quarks in a symmetric spatial state and the antisymmetric colour pair inside a three-quark colour singlet require symmetric spin-flavour symmetry. Their isospin singlet and spin singlet are both antisymmetric, while their respective triplets are symmetric. Thus the allowed pairs have

(I, S_{u d}) = (0, 0)

(1, 1)

; opposite exchange parities are excluded. Coupling an isospin-zero spin-one-half quark gives an

I = 0, J = 1/2

baryon or

I = 1, J = 1/2, 3/2

baryons.

Baryons with one isospin-singlet quark ( $Q (h [u d]) = q_{h} + 1/3$ )

 0  0

For an isospin singlet quark

h

of charge

q_{h}

, the antisymmetric light-flavour pair gives an

I = 0, J = 1/2

state with charge

q_{h} + 1/3

. The symmetric light-flavour pairs

uu, {u d}, dd

give

I = 1

triplets of spins

1/2

and

3/2

, with charges

q_{h} + 4/3, q_{h} + 1/3, q_{h} - 2/3

. These follow from light-diquark spin-isospin correlation and addition of constituent electric charges, with zero orbital angular momentum.

Symmetric spin-flavour SU6 representation ( $56 = (10, 4) \oplus (8, 2)$ )

 0  0

Three light quarks have a six-dimensional single-quark spin-flavour space

C^{3} \otimes C^{2}

. A symmetric spatial ground state and the antisymmetric three-quark colour singlet require the symmetric cube of that space. Its dimension is

(3 8) = 56

, and restriction to flavour

S U (3)

and spin

S U (2)

gives a baryon decuplet of spin

3/2

and a baryon octet of spin

1/2

. The two mixed-permutation multiplicity spaces pair once in the symmetric sector, explaining why the two flavour-octet copies in the unconstrained tensor cube do not represent two distinct ground-state octets.