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

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and s calar phi (3) theory, from all nestings and chainings of a primitive self-e nergy subdivergence. Here we formulate the nonperturbative problems which t hese 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 scala r theory, at d = 6, the nonperturbative problem is more severe: we transfor m it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbatio n series to 500 loops in 7 minutes. Finally, we establish the propagator-co upling duality underlying these achievements making use of the Hopf structu re of Feynman diagrams. (C) 2001 Elsevier science B.V. All rights reserved.