
SIGMA 6 (2010), 079, 23 pages arXiv:0907.5593
http://dx.doi.org/10.3842/SIGMA.2010.079
NonPerturbative Asymptotic Improvement of Perturbation Theory and MellinBarnes Representation
Samuel Friot ^{a, b} and David Greynat ^{c}
^{a)} Univ ParisSud, Institut de Physique Nucléaire,
UMR 8608, Orsay, F91405, France
^{b)} CNRS, Orsay, F91405, France
^{c)} Institut de Física Altes Energies,
Universitat Autònoma de Barcelona, E08193 Bellaterra, Barcelona, Spain
Received June 09, 2010, in final form September 30, 2010; Published online October 07, 2010
Abstract
Using a method mixing MellinBarnes 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 ''Npoint'' functions for the simple case of zerodimensional φ^{4} field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven nonperturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a nonperturbative 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 MellinBarnes 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 quasiexactly solvable models; MellinBarnes representation; hyperasymptotics; resurgence; nonperturbative effects; field theories in lower dimensions.
pdf (374 Kb)
ps (481 Kb)
tex (142 Kb)
References
 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, 6187,
hepth/0603155.
 Olver F.W.J.,
Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral,
SIAM J. Math. Anal. 22 (1991), 14601474.
 Berry M.V., Howls C.J.,
Hyperasymptotics,
Proc. Roy. Soc. London Ser. A 430 (1990), no. 1880, 653668.
 Paris R.B., Kaminski D.,
Asymptotics and MellinBarnes integrals,
Encyclopedia of Mathematics and its Applications, Vol. 85, Cambridge University Press, Cambridge, 2001.
 ZinnJustin J.,
Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation,
Phys. Rep. 70 (1981), 109167.
 Häußling R.,
Quantum field theory on a discrete space and noncommutative geometry,
Ann. Physics 299 (2002), 177,
hepth/0109161.
 Rivasseau V.,
Constructive field theory in zero dimension,
Adv. Math. Phys. 2009 (2009), 180159, 12 pages,
arXiv:0906.3524.
 Nalimov M.Yu., Sergeev V.A., Sladkoff L.,
Borel resummation of the εexpansion of the dynamic exponent z in the model A of the φ^{4}(O(n)) theory,
Theoret. and Math. Phys. 159 (2009), 499508.
 Dingle R.B.,
Asymptotic expansions: their derivation and interpretation,
Academic Press, London  New York, 1973.
 Olver F.W.J.,
On an asymptotic expansion of a ratio of Gamma functions,
Proc. Roy. Irish Acad. Sect. A 95 (1995), 59.
 Beneke M., Jamin M.,
α_{s} and the τ hadronic width: fixedorder, contourimproved and higherorder perturbation theory,
J. High Energy Phys. 2008 (2008), no. 9, 044, 42 pages,
arXiv:0806.3156.
 Abramowitz M., Stegun I. (Editors),
Handbook of mathematical functions, with formulas, graphs, and mathematical tables,
Stegun Dover Publications, Inc., New York 1966.
 Berry M.V., Howls C.J.,
Hyperasymptotics for integrals with saddles,
Proc. Roy. Soc. London Ser. A 434 (1991), no. 1892, 657675.
 Olde Daalhuis A.B., Olver F.W.J.,
Hyperasymptotic solutions of secondorder linear differential equations. I,
Methods Appl. Anal. 2 (1995), 173197.
 Pasquetti S., Schiappa R.,
Borel and Stokes nonperturbative phenomena in topological string theory and c=1 matrix models,
Ann. Henri Poincaré 11 (2010), 351431,
arXiv:0907.4082.

