# Colour decomposition in QCD

I am looking to compute the matrix element for the process gg -> u ubar at leading order. It is straightforward to calculate the non-colour part of the usual s, t and u channels. I will call these amplitudes A_s, A_t and A_u (resp.) However I am struggling to apply the logic of 'colour decomposition' to simplify the remainder of the calculation. Based on references such as http://arxiv.org/pdf/hep-ph/0209271v2.pdf and, http://arxiv.org/pdf/1010.0748v3.pdf I understand that there are two colour flow diagrams; One which looks like the s-channel and one which looks like the t-channel (I am not 100% why there isn't also one which looks like the u-channel) - these diagrams are shown in figure 6 of the first reference. I am also unsure of how to construct the 'primitive amplitudes' associated with these colour flow terms. I have calculated the matrix element using MadGraph 5 (outputting in to c++) and I can see that the three amplitudes are combined in the following way:

$\mathcal{J}_1 = -i\mathcal{A}_s+\mathcal{A}_u$

$\mathcal{J}_2 = i\mathcal{A}_s+\mathcal{A}_t$

These amplitudes are then combined with various factors of 3, 16 and -2 to yield the full matrix element. If anyone could explain a bit more about the idea of colour decomposition, and in particular how the combinations above are formed (and then further combined) that would be a great help. I can provide more details about the Madgraph calculations and/or how to reproduce if required. Cheers, Jack

this paper might help.$^1$ It's written pedagogically and hence is easier to read. It goes on to discuss a lot more than just color decomposition too.
$^1$ Scattering Amplitudes, Henriette Elvang, Yu-tin Huang.