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I am reading Quantum Field Theory in a Nutshell by A.Zee. In Chapter 5 Section 6, Under the subtitle A nonperturbative phenomenon, He commented

"That the mass of the kink comes out inversely proportional to the coupling is a clear sign that field theorists could have done perturbation theory till they were blue in the face without ever discovering the kink."

I don't quite get the meaning of the sentence, why this is a clear sign? Why does it imply non-perturbative effects? What is the non-perturbative effects he was referring to?

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More generally, if one is doing perturbation theory in some coupling constant $g$, the result is a power series in $g$, i.e. non-negative powers of $g$. To get negative/inverse powers of $g$, one must include non-perturbative effects.

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