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.

pdf (374 Kb)   ps (481 Kb)   tex (142 Kb)

References

1. Fredenhagen K., Rehren K.-H., Seiler E., Quantum field theory: where we are, in Approaches to Fundamental Physics, Lecture Notes in Phys., Vol. 721, Springer, Berlin, 2007, 61-87, hep-th/0603155.
2. Olver F.W.J., Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral, SIAM J. Math. Anal. 22 (1991), 1460-1474.
3. Berry M.V., Howls C.J., Hyperasymptotics, Proc. Roy. Soc. London Ser. A 430 (1990), no. 1880, 653-668.
4. Paris R.B., Kaminski D., Asymptotics and Mellin-Barnes integrals, Encyclopedia of Mathematics and its Applications, Vol. 85, Cambridge University Press, Cambridge, 2001.
5. Zinn-Justin J., Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation, Phys. Rep. 70 (1981), 109-167.
6. Häußling R., Quantum field theory on a discrete space and noncommutative geometry, Ann. Physics 299 (2002), 1-77, hep-th/0109161.
7. Rivasseau V., Constructive field theory in zero dimension, Adv. Math. Phys. 2009 (2009), 180159, 12 pages, arXiv:0906.3524.
8. Nalimov M.Yu., Sergeev V.A., Sladkoff L., Borel resummation of the ε-expansion of the dynamic exponent z in the model A of the φ⁴(O(n)) theory, Theoret. and Math. Phys. 159 (2009), 499-508.
9. Dingle R.B., Asymptotic expansions: their derivation and interpretation, Academic Press, London - New York, 1973.
10. Olver F.W.J., On an asymptotic expansion of a ratio of Gamma functions, Proc. Roy. Irish Acad. Sect. A 95 (1995), 5-9.
11. Beneke M., Jamin M., α_s and the τ hadronic width: fixed-order, contour-improved and higher-order perturbation theory, J. High Energy Phys. 2008 (2008), no. 9, 044, 42 pages, arXiv:0806.3156.
12. Abramowitz M., Stegun I. (Editors), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Stegun Dover Publications, Inc., New York 1966.
13. Berry M.V., Howls C.J., Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London Ser. A 434 (1991), no. 1892, 657-675.
14. Olde Daalhuis A.B., Olver F.W.J., Hyperasymptotic solutions of second-order linear differential equations. I, Methods Appl. Anal. 2 (1995), 173-197.
15. Pasquetti S., Schiappa R., Borel and Stokes nonperturbative phenomena in topological string theory and c=1 matrix models, Ann. Henri Poincaré 11 (2010), 351-431, arXiv:0907.4082.