The Hopf algebra of undecorated rooted trees has tamed the combinatorics of
perturbative contributions, to anomalous dimensions in Yukawa theory and scalar
Ď•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.Comment: 20p, 2 epsf fi