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In QFT, the perturbation series of some amplitudes diverge. As far as I know, perturbation series assume the solution is an analytic function of the small parameter. However this assumption can be wrong. Does that hint that the amplitudes in the consideration are not analytic functions of small parameters? i.e non-perturbative effects exist?

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The thing I want to ask is if non-perturbative effects are the cause of divergent perturbation series in QFT, which also includes QED.

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In some sense, the answer for your edited question is yes. We can read the non-perturbative effects from the divergent perturbation series by using resummation methods called Borel transformation. (e.g. see the link in the above comment or this link).

Also, the important point is each singular point of the Borel transformation denotes the divergent structure of perturbation series, and such singular point can be related to the non-perturbative effect. In this sense, the divergence of purturbative series is related to non-perturbative effects. Such a correspondence is called as “resurgence structure”. (e.g. see this review or more mathematical bases are given in “J. Ecalle, Les Fonctions Resurgentes(Publ. Math. Orsay)”. In addition to this, the introduction of this article is really educational.)

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