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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2010
• Main Authors: Friot, S., Greynat, D.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2010
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/146502
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation / S. Friot, D. Greynat // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 15 назв. — англ.

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.