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I’ve seen people say that if the coupling constant is large, we can’t trust Feynman Diagrams (like in the case of QCD). The logic is that high couplings describe bound states, whereas Feynman Diagrams describe scattering states, but this argument is not completely satisfactory, since it doesn’t prove that the perturbation series for a large coupling will give an inherently wrong result.

So why can’t we use Feynman Diagrams for a large coupling?

Qmechanic
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    How is the perturbation series supposed to be useful when the higher terms get increasingly more relevant? – ACuriousMind Aug 28 '22 at 13:17
  • @ACuriousMind Higher order terms already become relevant even when the coupling is small (see Dyson’s argument for an asymptotic series) –  Aug 30 '22 at 01:07
  • That's not a counter-argument, the number of useful terms in an asymptotic series is proportional to the inverse of the coupling constant (hence larger coupling constants mean useless asymptotic series), see e.g. https://physics.stackexchange.com/a/62916/50583. (And if your question is specifically about the asymptotic nature of the perturbation series, please include that directly in your question) – ACuriousMind Aug 30 '22 at 07:41

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Feynman diagrams are a graphical interpretation of some perturbation series. For those to be defined, you have to consider a parameter to be small, usually the coupling. In a large coupling context there are several approaches possible. You can e.g. still consider the coupling small, and perform high order expansions in the Feynman diagrams, followed by ressumations techniques (keywords: Padé/Borel resumation techniques) to extrapolate your results for high coupling (this is usually very hard). Or you can choose another parameter to be the perturbative one (keyword: large N expansions). Or lastly, abandon Feynman diagrams and use your favorite self-consistent / non pertubative technique.

Simon Metayer
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  • You’re begging the question. Why must the coupling be small for the diagrams to be defined? –  Aug 30 '22 at 01:05
  • As I stated in my first sentence, Feynman diagrams are just a graphical way to represent each term of the perturbative (asymptotic) series obtained by expanding the Dyson equations, e.g. at small coupling. If you expand in another parameter, e.g. in 1/N, N being the number of flavor, you will obtain a different series, an thus different diagrams. I insist, these are a graphical way to represent an hugly series. You can perform the computations without drawing diagrams, just expand in series your equations. – Simon Metayer Aug 30 '22 at 09:43