Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 079, 23 pages      arXiv:0907.5593      https://doi.org/10.3842/SIGMA.2010.079

Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

Samuel Friot a, b and David Greynat c
a) Univ Paris-Sud, Institut de Physique Nucléaire, UMR 8608, Orsay, F-91405, France
b) CNRS, Orsay, F-91405, France
c) Institut de Física Altes Energies, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain

Received June 09, 2010, in final form September 30, 2010; Published online October 07, 2010

Abstract
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 φ4 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.

Key words: exactly and quasi-exactly solvable models; Mellin-Barnes representation; hyperasymptotics; resurgence; non-perturbative effects; field theories in lower dimensions.

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