Calculating a particular amplitude with Feynman diagrams

Ringo Hendrix · Oct 9, 2019

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

Handwritten notes with three expressions: the initial electron-positron state |i> built from creation operators acting on the vacuum, the final two-photon state |f>, and the free electron and positron field expansions for Psi and Psi-bar as sums over spin with plane-wave terms.

$Handwritten photon field expansion A^mu = sum over lambda of integral d^3p over sqrt((2pi)^3 sqrt(2E_p)) times epsilon_lambda^mu times a_p e^{-ipx} + a_p^dagger e^{ipx}, followed by a handwritten amplitude A = sqrt(16 E E E E) times <0| a a T e^{ig integral} |0>.$

$Handwritten perturbation expansion: first line ig integral dx of <0| a_p2 a_q2 T{psi-bar(x) gamma^i A_i(x) psi(x) b_p2 c_p2 |0>; second line (ig)^2/2! double integral dx dy of <0| a a T{psi-bar gamma A psi-bar gamma A psi b c} |0>.$

PeterDonis · Oct 9, 2019

Ringo Hendrix said:

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 · Oct 9, 2019

Ringo Hendrix said:

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).

Calculating a particular amplitude with Feynman diagrams

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