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

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

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-dimensio...

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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
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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.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-146502
record_format dspace
spelling Friot, S.
Greynat, D.
2019-02-09T19:28:19Z
2019-02-09T19:28:19Z
2010
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.
1815-0659
2010 Mathematics Subject Classification: 41A60; 30E15
DOI:10.3842/SIGMA.2010.079
https://nasplib.isofts.kiev.ua/handle/123456789/146502
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.
We would like to thank Santi Peris and Eduardo de Rafael for their comments, as well as one of the referees for his suggestions to improve the manuscript. D.G. acknowledges financial support from CICYT-FEDER-FPA2008-01430, and the Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042).
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
spellingShingle Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
Friot, S.
Greynat, D.
title_short Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
title_full Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
title_fullStr Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
title_full_unstemmed Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation
title_sort non-perturbative asymptotic improvement of perturbation theory and mellin-barnes representation
author Friot, S.
Greynat, D.
author_facet Friot, S.
Greynat, D.
publishDate 2010
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description 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.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/146502
citation_txt 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.
work_keys_str_mv AT friots nonperturbativeasymptoticimprovementofperturbationtheoryandmellinbarnesrepresentation
AT greynatd nonperturbativeasymptoticimprovementofperturbationtheoryandmellinbarnesrepresentation
first_indexed 2025-12-07T16:29:29Z
last_indexed 2025-12-07T16:29:29Z
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