Calculating a particular amplitude with Feynman diagrams

[Ringo Hendrix]

TL;DR Summary

This isn’t homework, I’m self taught. Pardon me for using pictures instead of typing out equations, I have no clue how to do that. So please don’t downvote this for that :-) I’m just really desperate to learn this.

So I’m trying to compute the probability amplitude of an electron with momentum p1 and a positron with momentum p2 annihilating into a photons with momenta q1 and q2.

My question is how do you use Feynman diagrams to calculate the first and second order expansions (seen in the third image)? I wouldn’t mind a step-by-step with minimal assumption similar to the pictures I posted. Any help would be greatly appreciated. Again I’m sorry about the pictures, but I really am unsure how to type out the equations but I hope and assume my writing is legible enough.

What I have so far

 

[PeterDonis]

I really am unsure how to type out the equations

You do it using LaTeX:

https://www.physicsforums.com/help/latexhelp/
Please take the time to learn it and use it to type your equations. The issue is not just legibility but being able to quote parts of your post in responses; we can't do that if all the equations are in images. That makes it much harder for people to help you.

 

[strangerep]

This isn’t homework, I’m self taught. [...] I’m just really desperate to learn this.

But,... presumably you are learning from QFT textbook(s), yes? If so, which one(s)? If not, go get a copy of (e.g.,) Peskin & Schroeder.

My question is how do you use Feynman diagrams to calculate the first and second order expansions (seen in the third image)? [...] I wouldn’t mind a step-by-step with minimal assumption [...]

Any respectable QFT textbook should do that. Best to try and learn from the textbook first, then come back here to clarify any details or steps that you don't follow, or can't figure out (but if you do this, be sure to cite page+eqn numbers in the book).