Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation

TL;DR

This paper presents an exact, unbiased solution to the Dyson-Schwinger equation in 2D $\

Contribution

It introduces a novel approach using homotopy analysis and Monte Carlo sampling to solve Dyson-Schwinger equations for quantum field theories.

Findings

Series solution can be asymptotic or convergent near phase transition

Method applicable to fermionic theories

Provides a new framework for solving integro-differential equations

Abstract

We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for $\phi^4$ theory in 2D. It is based on an exact treatment of the functional derivative $\partial \Gamma / \partial G$ of the 4-point vertex function $\Gamma$ with respect to the 2-point correlation function $G$ within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around $G=0$ in a HAM control parameter $c_0G$, or even a convergent one up to the phase transition point if shifts in $G$ can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving integro-differential equations.