A method for resummation of perturbative series based on the stochastic solution of Schwinger-Dyson equations

TL;DR

This paper introduces a stochastic numerical method for resumming perturbative series by solving Schwinger-Dyson equations, effectively estimating series coefficients and applying Borel resummation, demonstrated on scalar field theories.

Contribution

It presents a novel stochastic approach to resummation of perturbative series using Schwinger-Dyson equations, incorporating Borel resummation and sampling open Feynman diagrams with arbitrary external legs.

Findings

Confirmed triviality of scalar field theory in four and five dimensions.

Demonstrated instability of the trivial fixed point in three dimensions.

Validated the method by studying scale dependence of the renormalized coupling.

Abstract

We propose a numerical method for resummation of perturbative series, which is based on the stochastic perturbative solution of Schwinger-Dyson equations. The method stochastically estimates the coefficients of perturbative series, and incorporates Borel resummation in a natural way. Similarly to the "worm" algorithm, the method samples open Feynman diagrams, but with an arbitrary number of external legs. As a test of our numerical algorithm, we study the scale dependence of the renormalized coupling constant in a theory of one-component scalar field with quartic interaction. We confirm the triviality of this theory in four and five space-time dimensions, and the instability of the trivial fixed point in three dimensions.