# Colour-ordering formula of QCD amplitudes (tree-level)

I have been studying colour-ordered amplitudes and spinor helicity formalism for a while. It is now apparent to me that I do not fully understand the 'master' formula which allows us to relate the full amplitude to the partial ones by dressing them with appropriate colour factors (which are decomposed as traces of $$SU(N)$$ generators):

$$$$A(1 \ldots n) = g_s^{n-2} \sum_{\sigma\in S_{n-1}} \mathrm{Tr}[1\sigma(2 \ldots n)] \, \tilde{A}[1\sigma(2 \ldots n)],$$$$ where the sum runs over permutations of the $$n-1$$ elements while holding the first one fixed. (This formula might have additional factors of $$i$$ or $$\sqrt{2}$$ depending on the normalisation of generators.)

I understand the general ideas leading to this decomposition, namely the Fierz identity and the crossing-symmetry between the s-, t-, u-channels of Feynman diagrams, however I do not see how to prove it rigorously. I have looked in a multitude of books and papers, yet the discussion always concludes with 'one can see that the following is true'. Is there a source where the proof has been shown explicitly?

I think the most pedagogical explanation I have seen so far is in Matthew Schwartz's QFT and the Standard Model textbook. In Chaper 27.4, he derives the above formula explicitly for the case of 4-gluon scattering. However, I am still confused about a few things. Following the usual Feynman rules, one has the three usual s-, t-, u-channels (two 3-gluon vertices + propagator) and the 'contact term', i.e. the 4-gluon vertex. Through a clever choice of reference vectors for the polarisations, one can show that the contact term always vanishes, regardless of gluon helicities (see Scattering Ampltudes in Gauge Theory and Gravity by Elvang&Huang, Exercise. 2.22). So we are left with the 3 channels. Each of them gets a colour factor of the form $$f^{abe}f^{cde}$$ or similar, which can be decomposed into a sum of four traces, hence $$3 \times 4 = 12$$ terms. The claim is that this can be reduced to a sum of $$6$$ terms of the form shown in the RHS of the master formula. This number makes sense as, for $$n=4$$, $$S_3$$ contains $$6$$ permutations. However, I tried to show this explicitly and failed to simplify all $$12$$ terms to just $$6$$ with this particular ordering.

I also don't understand why the definition of colour-ordered amplitudes $$\tilde{A}$$ includes only diagrams with no crossed legs - essentially the u-channel drops out. Is that because the three colour structures are not independent, but related by Jacobi identity, so there is some cancellation occurring?

I would greatly appreciate it if someone could offer a detailed explanation of the process of producing the master formula. For reference, I'm aware of these related questions: Colour decomposition in QCD, Colour decomposition of $n-$gluon tree amplitude, however they only talk about the general idea of representing strings of structure constants as sums of traces, which I understand. It's the second part that I struggle with - how to combine these traces with individual amplitudes for s-, t-, u-, channels to produce the colour-ordered $$\tilde{A}$$ that does not depend on the u-channel.