Non-Abelian vertex 3-gauge-boson I am trying to understand how the vertices depicted in page 507 of Peskin and Schroeder come about. I understand that vertex where we have 1 gauge boson and two fermions but I'm confused on the indices used on the 3-vertex and 4-vertex interaction.
Then in (16.8) they write down one of the possible combinations which would be
$$-igf^{abc}(-ik^\nu)g^{\mu\rho}.\tag{16.8}$$
I understand the prefactor and that the derivative part gives the $(-ik)$ term (if the momentum flows in) but how do we chose the metric indices?
 A: *

*Briefly speaking, the Feynman rule in position space for the cubic gluon vertex (16.8) follows from the cubic gluon field term in the Lagrangian density (times $i$)
$$\begin{align} i{\cal L}_3~=~&-igf^{abc}(\partial_{\kappa}A^a_{\lambda}) A^{\kappa b}A^{\lambda c}\cr~=~&-igf^{abc}g^{\mu\rho}(\partial^{\nu}A^a_{\mu}) A^b_{\nu}A^c_{\rho},\end{align} \tag{16.6}$$


*The derivative $\partial^{\nu}$ becomes $-ik^{\nu}$ by Fourier transformation:
$$\begin{align} iS_3~=~& i\int\!d^4x~{\cal L}_3
~\stackrel{(16.6)}{=}~\ldots\cr
~=~&-igf^{abc}g^{\mu\rho}\int\! \frac{d^4k}{(2\pi)^4}\int\! \frac{d^4p}{(2\pi)^4}\int\! \frac{d^4q}{(2\pi)^4} \cr 
&\times (2\pi)^4\delta^4(k\!+\!p\!+\!q) (-ik^{\nu})\widetilde{A}^a_{\mu}(k)\widetilde{A}^b_{\nu}(p)\widetilde{A}^c_{\rho}(q).\end{align} $$
From the Dirac delta function we learn that each vertex obeys momentum conservation.


*The full cubic gluon vertex Feynman rule in momentum space involves a symmetrization of the 3 legs, i.e. it has 3!=6 terms, cf. Fig. 16.1.
References:

*

*M.E. Peskin & D.V. Schroeder, An Intro to QFT. 1995; p. 507.

