## Symmetry, Integrability and Geometry: Methods and Applications Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin–Barnes Representation

### BibTeX

@MISC{Friot_symmetry,integrability,

author = {Samuel Friot and David Greynat},

title = {Symmetry, Integrability and Geometry: Methods and Applications Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin–Barnes Representation},

year = {}

}

### OpenURL

### Abstract

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 φ 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. Key words: exactly and quasi-exactly solvable models; Mellin–Barnes representation; hyperasymptotics; resurgence; non-perturbative effects; field theories in lower dimensions 2010 Mathematics Subject Classification: 41A60; 30E15 1

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Citation Context ... a larger region of validity (a larger domain of definition in the complex expansion parameter) and a greater accuracy than conventional asymptotic expansions, appeared in the mathematical literature =-=[2]-=-. With them, a new asymptotic theory emerged: exponential asymptotics (or hyperasymptotics 3 ). These asymptotic objects (hyperasymptotic expansions) are very interesting since they correspond, for so... |

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Citation Context ...ourous approach which is completely justified at a later stage in the paper. Zero-dimensional φ 4 field theory has already been used many times to explain new theoretical approaches (see for instance =-=[5, 6]-=- and, more recently, [7]) and we will see that it leads here to non-trivial and interesting issues. We would like to add that although zero-dimensional φ 4 field theory cannot be considered, strictly ... |

5 |
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Citation Context ... 2k) (3!) k λ k = 1 +∞∑ √ 2π k=n ) Γ ( k + 1 4 ) ( ) 3 Γ k + 4 Γ(k + 1) (2.13) ( ) k −2λ . (2.14) 3 Our main tool is the so-called inverse factorial expansion which may be obtained from Barnes’ lemma =-=[10]-=- (see also [4, Chapter 2, Section 2.2]) Γ ( k + 1 ) ( ) 3 m−1 4 Γ k + ∑ 4 = (−1) Γ(k + 1) j AjΓ(k − j) + 1 ∫ c+m+i∞ ds f(s)Γ(k − s), (2.15) 2iπ j=0 c+m−i∞ where f(s) = Γ ( s + 1 ) ( ) 3 4 Γ s + 4 Γ(−s... |

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Citation Context ...ew perturbative terms is in general in very good agreement with “exact” experimental results gives a piece of evidence that perturbation theory is asymptotic to “something”. The question is: to what? =-=[1]-=-2 S. Friot and D. Greynat However, one has of course to keep in mind that when dealing with divergent asymptotic perturbative power expansions, there always remains a finite limit of precision beyond... |

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Citation Context ...eme proposed in this reference. One finds in this case Z(0)| λ= 1 = 0.965560480. 322 S. Friot and D. Greynat (and the smoothing of the Borel ambiguity), we would like to mention an interesting paper =-=[15]-=- which appeared a few days before the end of the writing of our manuscript and dealing, among others things, with a hyperasymptotic approach of instantons in topological string theory and c = 1 matrix... |

1 |
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Citation Context ...ourous approach which is completely justified at a later stage in the paper. Zero-dimensional φ 4 field theory has already been used many times to explain new theoretical approaches (see for instance =-=[5, 6]-=- and, more recently, [7]) and we will see that it leads here to non-trivial and interesting issues. We would like to add that although zero-dimensional φ 4 field theory cannot be considered, strictly ... |

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Citation Context ...mpletely justified at a later stage in the paper. Zero-dimensional φ 4 field theory has already been used many times to explain new theoretical approaches (see for instance [5, 6] and, more recently, =-=[7]-=-) and we will see that it leads here to non-trivial and interesting issues. We would like to add that although zero-dimensional φ 4 field theory cannot be considered, strictly speaking, as a realistic... |

1 |
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Citation Context ...ns) or in the study of the resummation of higher order corrections in quantum mechanical models and/or superconvergent quantum field theories that are considered in condensed matter physics (see e.g. =-=[8]-=-). The paper is organized as follows. In the introductive Section 2.1 where basic facts are recalled, the perturbative approach is detailed and, to fix ideas, numerical results are given for a particu... |

1 |
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Citation Context ... 2|λ| ]−i∞ dt ( 1 λ ) −t ( ) t 2 3 π Γ(t − j) sin(πt) , (2.47) which constitutes the optimally truncated hyperasymptotic expansion of Z(0) at first level, in the Berry–Howls optimal truncation scheme =-=[3, 13]-=-, that we call OTS1 in the following. In fact, in OTS1, equation (2.45) constitutes an asymptotic expansion which, being extracted from the exponentially suppressed remainder R0| 3 n= , allows us to c... |

1 |
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(Show Context)
Citation Context ...(−1) j Aj 1 2iπ ∫ 3 d+[ |λ| ]+i∞ d+[ 3 |λ| ]−i∞ dt ( ) −t 1 λ , (2.48) which constitutes the first level of the hyperasymptotic expansion of Z(0) in the Olde Daalhuis– Olver optimal truncation scheme =-=[14]-=- that we call OTS2 in the following. Comparing (2.44) and (2.48), one sees that the first level of the hyperasymptotic expansion in OTS2 therefore leads to an exponential improvement of the superasymp... |