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I am trying to calculate the 2-loop correction for a basic vertex in QED

enter image description here

So I need to figure out what diagrams I need to consider. My literature says (without much explanation), that there are 7 contributing diagrams (including pictures). But naively it is possible to draw many more 2-loop diagrams with two electron-legs and one photon leg. Most of them are excluded by simple rules such as

  1. All legs should be amputated.
  2. The diagram should be connected (i.e. no vacuum bubbles).

But what about the diagram

enter image description here

? It seems to obey all rules, but yet it is not listed as one of the contributing diagrams. Why is that? What rule am I missing?

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1 Answer 1

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The rule you're missing is Furry's theorem. There's another diagram where the internal electron loop has the arrows reversed, related to the original diagram by charge conjugation, and the two should exactly cancel. This generally happens whenever you have a fermion loop with an odd number of photons attached.

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  • $\begingroup$ Ah, I can't believe I didn't notice this! The last two diagrams of my answer cancel off each other. Silly, silly me. $\endgroup$
    – AccidentalFourierTransform
    Commented May 4, 2017 at 16:40
  • $\begingroup$ Thank you so much! This actually helped me to understand QED! Our lecturer defined this theorem too mathematically(a green function of odd number of photons gives zero - never explained what green function, that it is related to loop green functions), even very briefly, just "by the way", and did not show any proof("it is obvious" :) ). So I actually had no idea what it meant. You saved me hours! $\endgroup$
    – user74200
    Commented Feb 18, 2018 at 21:29
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    $\begingroup$ @user74200 No problem! For what it’s worth, a Green’s function in this context is nothing but a kind of correlation function, which is really what you’ve been computing the whole time. I recommend looking at Timo Weigand’s lecture notes for this kind of vocabulary. $\endgroup$
    – knzhou
    Commented Feb 18, 2018 at 22:56

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