# When can colour charge indices be equated?

I'm currently studying QFT and QCD for the first time and I have a question about the colour charge indices given below. I was asked the following question:

(a) Derive the Feynman rule for the 3-gluon vertex $$V\left[A_{\mu_1}^{a_1}\left(p_1\right), A_{\mu_2}^{a_2}\left(p_2\right), A_{\mu_3}^{a_3}\left(p_3\right)\right]$$ from $$\mathcal{L}_{\mathrm{QCD}}=-\frac{1}{4} F_{\mu \nu}^a F^{a, \mu \nu}+\ldots, \quad F_{\mu \nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a-g_{\mathrm{s}} f^{a b c} A_\mu^b A_\nu^c.$$

I was given the correct working but I'm confused about the below section of it, specifically how the colour indices d and e seem to become b and c when collecting the terms. I'm sure it's probably straightforward and I'm missing something obvious but any help would be appreciated.

\begin{aligned} & i L_{Q C D}=-\frac{i}{4} F_{\mu v}^a F^{a, \mu v}+\ldots \\ & =-\frac{i}{4}\left(\partial_\mu A_\nu^a-\partial_\nu A_\mu^a-g_s f^{a b c} A_\mu^b A_\nu^c\right)\left(\partial^\mu A^{a, \nu}-\partial^v A^{a, \mu}-g_s f^{a d e} A^{d, \mu} A^{e, v}\right) \\ & \end{aligned}

Collecting terms with AAA: \begin{aligned} i L_{Q C D}= & -i\left(\partial_\mu A_\nu^a-\partial_\nu A_\mu^a\right)\left(-g_s f^{a d e} A^{d, \mu} A^{e, \nu}\right) \\ & -\frac{i}{4}\left(-g_s f^{a b c} A_\mu^b A_v^c\right)\left(\partial^\mu A^{a, v}-\partial^v A^{a, \mu}\right)+\ldots \\ = & \frac{i g_s}{2}\left(\partial_\mu A_v^a-\partial_v A_\mu^a\right) f^{a b c} A^{b, \mu} A^{c, v}+\ldots \end{aligned}

• Since $X^{abcbc}=X^{dede}$, $X^{abcbc}+X^{adede}=2X^{abcbc}$. (I'm taking $X^{afghi}:=f^{afg}A^{h,\,\mu}A^{i,\,\nu}$, hiding Greek indices.)
– J.G.
Dec 29, 2023 at 11:43