# Photon-Gluon annihilation in QCD

I am starting to learn about QCD, and I wanted to calculate the squared matrix elements for photon-gluon annihilation into a quark and an anti-quark. However, I am having trouble writing down the correct matrix elements at tree level and computing the spin and color sum of the averaged matrix element.

I found this FeynCalc webpage, but I did not find it extremely helpful. I would really appreciate any help or resources.

The specific process I am interested in is $$\gamma(p_1)+g(p_{2})\rightarrow q(p_{3})+\bar{q}(p_{4}).$$ So the relevant Feynman rules and diagrams are:

I believe that I have made a mistake with the arrows on the first (left) diagram; I think they should be in the opposite direction. Thus the matrix elements for the first (left) diagram and the second (right) diagram are: $$M_{1} = -ie^2Q\left(v(p_{4})T^b \gamma ^{\alpha}\left(\frac{\not{p_{1}}-\not{p_{4}}+m}{(p_{4}-p_{1})^2-m^2}\right)\gamma^{\beta}\bar{u}(p_{3})\epsilon_{\alpha}(p_{1})\epsilon_{\beta}(p_{2})\right)$$ and $$M_{2} = -ie^2Q\left(\bar{u}(p_{3}) \gamma ^{\mu}\left(\frac{\not{p_{3}}-\not{p_{1}}+m}{(p_{3}-p_{1})^2-m^2}\right)T^a\gamma^{\nu}v(p_{4})\epsilon_{\mu}(p_{1})\epsilon_{\nu}(p_{2})\right)$$ However, I am still not sure that I have applied the rules correctly.

• Please show us what you have tried thus far, and where you are having a conceptual problem. thanks. Jul 18, 2021 at 8:26
• @joseph h is it fine to post the Feynman diagrams written by hand?
– Joel
Jul 18, 2021 at 8:35
• Sure. But if you are having problems with the math, post that here too using latex. Cheers. Jul 18, 2021 at 8:37
• @joseph h I have added more details to my question. Hope this clears things up. Thanks
– Joel
Jul 18, 2021 at 9:17
• very good. thanks. Jul 18, 2021 at 9:18

The fermion line on the left diagram should be reversed, since the fermion is outgoing. Then one should trace the fermion line backwards, so you have $$\bar{u}(p_3) ... v(p_4)$$ on both diagrams. I assume you know that the spinors are vectors in the Dirac space in which the gamma matrices live, so this expression is a scalar in this space, $$\bar{u}$$ is a row vector and $$v$$ is a column vector.
Then you should of course note that photon and gluon couplings are not equal! So you better use different symbol for them, conventionally $$e$$ for em coupling and $$g$$ for the strong coupling.