I want to calculate the cross section associated to the $\mathrm{e}^++\mathrm{e}^-\to \mathrm{q}+\bar{\mathrm{q}}+\mathrm{g}$ process as in the Feynman diagram
(source) and the other one in which the gluon is "emitted" from the antiquark leg. I'm neglecting any contribution from the weak interaction.
I'm not sure how to associate the colour factors to the outgoing quarks and gluon. My idea is to treat the colour indices of three particles as completely uncorrelated, so that when I compute the unpolarized cross section I have to sum over the three colours of $\mathrm{q}$, the three colours of $\bar{\mathrm{q}}$ and the eight degrees of freedom of $\mathrm{g}$, separately. Is this the correct procedure? The book I'm studying this on, QCD and collider physics by R. K. Ellis, doesn't seem to specify how to do it.
