Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2000-12-17
Nucl.Phys. B600 (2001) 403-422
Physics
High Energy Physics
High Energy Physics - Theory
20p, 2 epsf fig
Scientific paper
10.1016/S0550-3213(01)00071-2
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.
Broadhurst David J.
Kreimer Dirk
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