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

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.

On convergent series representations of Mellin-Barnes integrals

“…The first one is the toy integral Z(0) corresponding to the zero-dimensional version of the vacuum-to-vacuum generating functional of λ φ 4 theory, used in [1] for the exposition of a method allowing to obtain non-perturbative asymptotic improvements directly from divergent perturbative expansions:…”

“…In [1], the quantity of interest (and starting point of the non-perturbative asymptotic analysis) was the divergent expansion (3), therefore it was not mentioned that one may also close the contour of (2) to the right to get the series…”

“…arXiv:1107.0328v1 [math-ph] 1 Jul 2011 1 We will see that this property is in fact not restricted to the case of double series.…”

“…They converge in different regions of values of the parameters, and it is not obvious to obtain them. For twofold integrals we present a method which allows to derive straightforwardly and systematically:(a) different sets of poles which correspond to different convergent double series representations of a given integral, (b) the regions of convergence of all these series (without an a priori full knowledge of their general term), (c) the general term of each series (this may be performed, if necessary, once the relevant domain of convergence has been found).This systematic procedure is illustrated with some integrals which appear, among others, in the calculation of the two-loop hexagon Wilson loop in N = 4 SYM theory.Mellin-Barnes integrals of higher dimension are also considered.arXiv:1107.0328v1 [math-ph] 1 Jul 2011 1 We will see that this property is in fact not restricted to the case of double series.…”