Griffiths Introduction to Elementary Particles · Problem 9.6

Quantum Chromodynamics — David J. Griffiths

Physics — QFT · octet-color-factor

Problem

Verify that all members of the color octet (9.4) give the same color factor f=−16f = -\frac{1}{6} for the qqˉq\bar{q} interaction (Example 9.1).
Intuitive explanation

The color factor ff measures how strongly a quark-antiquark pair couples through gluon exchange, and it depends on which color state they occupy. Since the color octet is an irreducible representation of SU(3)SU(3), all eight gluon states must be related by gauge symmetry—so they should all produce the same coupling strength. You'll verify this by computing f=⟨Ta⊗Ta⟩f = \langle \mathbf{T}_a \otimes \mathbf{T}_a \rangle for the octet, using the Casimir operator and the fact that color charge operators act identically on all octet members up to basis rotations.

Intuitive explanation

The color factor ff measures how strongly a quark-antiquark pair couples through gluon exchange, and it depends on which color state they occupy. Since the color octet is an irreducible representation of SU(3)SU(3), all eight gluon states must be related by gauge symmetry—so they should all produce the same coupling strength. You'll verify this by computing f=⟨Ta⊗Ta⟩f = \langle \mathbf{T}_a \otimes \mathbf{T}_a \rangle for the octet, using the Casimir operator and the fact that color charge operators act identically on all octet members up to basis rotations.

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