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For the purposes of this question, I'll limit the question to quantum electrodynamics, although this question could just as well be asked of electroweak theory and QCD.

For a particular order of perturbation theory, how do I know when I have drawn all the Feynman diagrams? For example, I have heard that in electron-electron scattering, at second order, we have t-channel and u-channel diagrams. But how did we know to draw these two channels - I would have stopped at just the t-channel diagram. And besides, if this is second order, what ever happened to first order diagrams of moller scattering?

One factoid I know about how many diagrams to draw is: in QED, a diagram that is third order in $\alpha$ has three vertices. Each vertex has one in-going and one out-going fermion and one virtual photon. A diagram that is fourth order in $\alpha$ has four vertices and so forth. I suppose I could use simple combinatorics to know when I've exhausted all the vertex combinations, but I don't know how to tell when two graphs are redundant, and there's still the t-channel and u-channel mystery as explained above.

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  • $\begingroup$ t-channel and u-channel diagrams are at first-order, too $\endgroup$
    – probably_someone
    May 3, 2020 at 2:34
  • $\begingroup$ Can you (or anyone else) please elaborate on that? The way I see it, all feynman diagrams that involve two-particle scattering should have both a t- and a u-channel diagram at all orders. But maybe I'm just imagining this. $\endgroup$
    – the_photon
    May 3, 2020 at 2:54
  • $\begingroup$ it's challenging problem, but there are techniques to solve it, i'm no expert but see e.g., inspirehep.net/literature/315611 it seems to be quite a computer-sciency topic $\endgroup$
    – innisfree
    May 3, 2020 at 2:58
  • $\begingroup$ There might be more modern descriptions in the FeynArts program manual, feynarts.de/FA3Guide.pdf $\endgroup$
    – innisfree
    May 3, 2020 at 2:59
  • $\begingroup$ Related: physics.stackexchange.com/q/304869/2451 $\endgroup$
    – Qmechanic ♦
    May 3, 2020 at 14:41

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It's a combinatorics question. You can break diagrams down into decision trees (start with a bunch of points, and then decide where to connect, line by line) and you've only written down all diagrams once you've hit every possible branch of the tree, meanwhile obeying constraints like "no duplicate diagrams" (these become symmetry factors) or disconnected diagrams (these don't contribute to scattering amplitudes), etc. I usually start with the external lines but that may be a matter of taste.

You can improve your intuition for it if you go through the derivation of where the diagrams come from, I used the path integral approach but you can probably get it through Wick contractions etc as well if that was your route. I did the derivation thoroughly for the hermitian scalar field because it was simplest. Each new field requires some adaptation, but not much.

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