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Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin–Barnes Representation
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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Using a method mixing Mellin–Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "$N$-point" functions for the simple case of zero-dimensional $\phi^4$ field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin–Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.
Keywords: exactly and quasi-exactly solvable models; Mellin–Barnes representation; hyperasymptotics; resurgence; non-perturbative effects; field theories in lower dimensions. 
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     author = {Samuel Friot and David Greynat},
     title = {Non-Perturbative {Asymptotic} {Improvement} of {Perturbation} {Theory} and {Mellin{\textendash}Barnes} {Representation}},
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}
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Samuel Friot; David Greynat. Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin–Barnes Representation. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a78/

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