- #1

takunitoche

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- TL;DR Summary
- I can't seem to derive the vertex rules from the Yang-Mills lagrangian. I struggle to properly identify the origin of each momentum and the indices associated.

Hello everyone,

I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. 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:

$$\mathcal{L}_{YM} \supset \dfrac{g}{6} f^{ijk} \left[ A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} (\partial^{\mu} A^{\nu}{}^{(i)} - \partial^{\nu} A^{\mu}{}^{(i)}) + A_{\mu}{}^{(i)} A_{\nu}{}^{(k)} (\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)}) + A_{\mu}{}^{(k)} A_{\nu}{}^{(i)} (\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)})\right] $$Now consider the following diagram (from Peskin & Schroeder, section 16.1): View attachment 250501

This is where I'm stuck: 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.

Huge thanks for any help you might bring!

I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. 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:

$$\mathcal{L}_{YM} \supset \dfrac{g}{6} f^{ijk} \left[ A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} (\partial^{\mu} A^{\nu}{}^{(i)} - \partial^{\nu} A^{\mu}{}^{(i)}) + A_{\mu}{}^{(i)} A_{\nu}{}^{(k)} (\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)}) + A_{\mu}{}^{(k)} A_{\nu}{}^{(i)} (\partial^{\mu} A^{\nu}{}^{(j)} - \partial^{\nu} A^{\mu}{}^{(j)})\right] $$Now consider the following diagram (from Peskin & Schroeder, section 16.1): View attachment 250501

This is where I'm stuck: 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.

Huge thanks for any help you might bring!