# Ambiguity in Asymptotic Perturbative Series and Instantons

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I know there are a number of questions about the asymptoticity of perturbative series and about instantons on StackExchange (e.g. Instantons and Non Perturbative Amplitudes in Gravity from user566, Instantons and Borel Resummation asked by felix, and How can an asymptotic expansion give an extremely accurate predication, as in QED? asked by yonni). Reading them was helpful but left me with two short questions:

1) What is meant by "ambiguity" in this context? Several posters use the term in alluding to the problems in asymptotic series. Does it have a technical meaning here?

2) How can we see that the instantons would "correct" the series in the full theory?

Perhaps the only thing to do is to read these notes ("Instantons and large N" by Marino) (which I plan on doing) but I was wondering if someone could give a quick answer for #1 and perhaps a clever or intuitive way of making #2 plausible.

This post imported from StackExchange Physics at 2014-08-07 15:39 (UCT), posted by SE-user gn0m0n
asked Jun 30, 2014
Great question!! The quick and dirty way to see what is going on is to consider what in your notes is called the 'toy integral', so I would read sections 2.3 and 4.2 first. Note that eqn 2.32 is an instanton solution in the toy model. The ambiguity is the branch cut talked about in 2.3. You can also see what is going on in eqn 2.38: the perturbation series doesn't converge because of factorial growth of the size of the terms in the series. Also note that in QFT instantons are not the only nonperturbative features that perturbation theory misses, there are also renormalons.

This post imported from StackExchange Physics at 2014-08-07 15:39 (UCT), posted by SE-user Andrew
@Andrew thanks, I will look at those sections ASAP

This post imported from StackExchange Physics at 2014-08-07 15:39 (UCT), posted by SE-user gn0m0n
Some explanations are given here and Prof. Carl Bender's Mathematical Physics course, which exactly deals with asymptotic series, might be helpful.

This post imported from StackExchange Physics at 2014-08-07 15:39 (UCT), posted by SE-user Dilaton

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