Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality

TL;DR

This paper derives exact nonperturbative solutions to Dyson-Schwinger equations in Yukawa and scalar theories, revealing a propagator-coupling duality and enabling high-order perturbation series computation.

Contribution

It formulates and solves nonperturbative Dyson-Schwinger equations for specific quantum field theories, introducing a duality and an efficient algorithm for high-order expansions.

Findings

Exact parametric solutions using special functions

Extension of perturbation series to 500 loops in minutes

Identification of propagator-coupling duality

Abstract

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar $\phi^3$ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension $d=4$, we obtain an integrodifferential Dyson-Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at $d=6$, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator-coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.