Fast summation of divergent series and resurgent transseries from Meijer-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>G</mml:mi></mml:math> approximants

Abstract

We develop a resummation approach based on Meijer-$G$ functions and apply it to approximate the Borel sum of divergent series and the Borel-\'Ecalle sum of resurgent transseries in quantum mechanics and quantum field theory (QFT). The proposed method is shown to vastly outperform the conventional Borel-Pad\'e and Borel-Pad\'e-\'Ecalle summation methods. The resulting Meijer-$G$ approximants are easily parametrized by means of a hypergeometric ansatz and can be thought of as a generalization to arbitrary order of the Borel-hypergeometric method [Mera et al., Phys. Rev. Lett. 115, 143001 (2015)]. Here we demonstrate the accuracy of this technique in various examples from quantum mechanics and QFT, traditionally employed as benchmark models for resummation, such as zero-dimensional ${\ensuremath{\phi}}^{4}$ theory; the quartic anharmonic oscillator; the calculation of critical exponents for the $N$-vector model; ${\ensuremath{\phi}}^{4}$ with degenerate minima; self-interacting QFT in zero dimensions; and the summation of one- and two-instanton contributions in the quantum-mechanical double-well problem.