A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge

TL;DR

This paper introduces a simple cutoff method for Feynman diagram series that improves convergence, outperforming traditional techniques like Pade and Borel summation, especially for non-Borel summable series.

Contribution

The authors propose a novel cutoff approach to ensure convergence of asymptotic Feynman series, applicable to complex quantum field problems and non-Borel summable series.

Findings

Modified series converge exponentially close to exact values.

Outperforms Pade and Borel methods for certain lambda values.

Applicable to non-Borel summable series like double-well potentials.

Abstract

We show that for two non-trivial lambda phi ^4 problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Pade's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff.