
SIGMA 6 (2010), 079, 23 pages arXiv:0907.5593
https://doi.org/10.3842/SIGMA.2010.079
NonPerturbative Asymptotic Improvement of Perturbation Theory and MellinBarnes Representation
Samuel Friot ^{a, b} and David Greynat ^{c}
^{a)} Univ ParisSud, Institut de Physique Nucléaire,
UMR 8608, Orsay, F91405, France
^{b)} CNRS, Orsay, F91405, France
^{c)} Institut de Física Altes Energies,
Universitat Autònoma de Barcelona, E08193 Bellaterra, Barcelona, Spain
Received June 09, 2010, in final form September 30, 2010; Published online October 07, 2010
Abstract
Using a method mixing MellinBarnes 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 ''Npoint'' functions for the simple case of zerodimensional φ^{4} field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven nonperturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a nonperturbative 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 MellinBarnes 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.
Key words:
exactly and quasiexactly solvable models; MellinBarnes representation; hyperasymptotics; resurgence; nonperturbative effects; field theories in lower dimensions.
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