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The calculation for the electron positron annihilation to 2 photons amplitude takes up 5 pages in $\LaTeX$, and the answer is astonishingly simple

$$\mathcal{M}=-4g_e^2.$$

Even Griffiths notes this quite sarcastically in his book. Is there a deeper reason that the machinery of QED makes this simple process so hard to calculate? Does it have something to do with gauge fixing and the complications that it brings? Is there a slick way to come to this simple answer by utilizing certain tricks and symmetries?

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    $\begingroup$ Isn’t this the ultra-relativistic limit? And isn’t it averaged over helicities? I’m fairly sure that it isn’t the general amplitude. $\endgroup$ – G. Smith May 30 at 21:35
  • $\begingroup$ annihilation to what? That is surely not the full answer, but UR and/or massless limit for sure $\endgroup$ – Giorgio Busoni May 30 at 21:45
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    $\begingroup$ This has to be annihilation to two photons. $\endgroup$ – G. Smith May 30 at 22:03
  • $\begingroup$ Sorry to everyone. This is average over all helicities and tree level amplitude for $e^+ e^-\rightarrow \gamma \gamma$. $\endgroup$ – gsuer May 31 at 6:48
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    $\begingroup$ I think it is also the ultrarelativistic limit. $\endgroup$ – G. Smith Jun 1 at 0:30

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