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!

Moritz · Answer

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^{murho}
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^nueta^{murho}f^{abc}A_mu^aA_nu^bA_rho^c+k^rhoeta^{munu}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

Yang-Mills Feynman rules

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