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

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ3 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.