# A stochastic root finding approach: The Homotopy Analysis Method applied to Dyson-Schwinger Equations

## TL;DR

This paper introduces a stochastic root finding method using the Homotopy Analysis Method for Dyson-Schwinger equations, demonstrating improved convergence and applicability to high-dimensional, non-perturbative problems in quantum field theory.

## Contribution

The paper develops a novel stochastic summation algorithm based on rooted-tree diagrams, enhancing convergence and efficiency over existing diagrammatic Monte Carlo methods for solving Dyson-Schwinger equations.

## Key findings

- Superior convergence compared to bold diagrammatic Monte Carlo
- Applicable to high-dimensional integral equations
- Accesses non-perturbative regimes in quantum field theories

## Abstract

We present the construction and stochastic summation of rooted-tree diagrams, based on the expansion of a root finding algorithm applied to the Dyson-Schwinger equations (DSEs). The mathematical formulation shows superior convergence properties compared to the bold diagrammatic Monte Carlo approach and the developed algorithm allows one to tackle generic high-dimensional integral equations, to avoid the curse of dealing explicitly with high-dimensional objects and to access non-perturbative regimes. The sign problem remains the limiting factor, but it is not found to be worse than in other approaches. We illustrate the method for $\phi^4$ theory but note that it applies in principle to any model.