Yang-Mills Feynman rules

Question

Good morning/evening.

In Peskin & Schroeder chapter 16 on gauge invariance, the gauge boson self interaction vertex rules are given. For three gauge bosons, the relevant interaction term in the Lagrangian is

$$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)} $$

I have rewritten this term using the total asymmetry of the structure constants:

$$ \mathcal{L}_{YM} \supset \dfrac{g}{6} f^{ijk} \left[ A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \left(\partial^{\mu} A^{\nu}{}^{(i)} - \partial^{\nu} A^{\mu}{}^{(i)}\right) + A_{\mu}{}^{(i)} A_{\nu}{}^{(k)} \left(\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)}\right) + A_{\mu}{}^{(k)} A_{\nu}{}^{(i)} \left(\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)}\right)\right] $$

I know that the derivative of the field will make the momenta appear in the expression. The problem is that I do not understand which gauge field momentum appears from which derivative, and how to go from this expression to the answer, which is

$$ g f^{abc} \left[ g^{\mu \nu} (k-p)^\rho + g^{\nu \rho} (p-q)^\mu + g^{\rho \mu} (q-k)^\nu \right] $$

where the momenta and indices are taken according to the attached diagram.

Thanks in advance for any clarification!

Answer 1

To see which momentum appears, its useful to fix some configuration as Peskin & Schroeder suggest. After writing out the Lagrangian, the term linear in g is \begin{align} -igf^{abc}\partial_\mu A_\nu^aA^{b\,\mu}A^{c\,\nu} \end{align} This term appears (after some index renaming) four times and cancels the factor $\frac{1}{4}$ from the Lagrangian. To get the vertex factor, pull down the indices with the metric. Then the vertex factor for the configuration $A_\mu^a(k)$, $A_\nu^b(p)$ and $A_\rho^c(q)$ is \begin{align} -gk^\nu f^{abc}\eta^{\mu\rho} \end{align} as prefactor of $A_\mu^a(k)A_\nu^b(p)A_\rho^c(q)$. Now you have to sum all possible permutations of this expression, which are 6 in this case. You can do this for example by switching $A$'s and renaming Minkowski and gauge indices to have the same index order as in the first considered term. As an example: \begin{align} -g(k^\nu\eta^{\mu\rho}f^{abc}A_\mu^aA_\nu^bA_\rho^c+k^\rho\eta^{\mu\nu}f^{acb}A_\mu^aA_\rho^cA_\nu^b) \end{align} It is important here to keep track of the gauge fields, since it is always the momentum of the first A in the expression that gives the momentum Switching now $A_\nu^b$ in the above expression to be in front means that it will contribute with p. Now bringing all structure constants into some chosen order, e.g. $f^{abc}$, will give you the signs that appear in the resulting formula