Exact Solutions Of Dyson Schwinger Equations For Iterated One Loop Integrals And Propagator Coupling Duality英文电子书.pdf

MZ–TH/00–28 hep-th/0012146 Nov 20 2000 Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling 0 0 duality 0 2 c e D. J. Broadhurst1) and D. Kreimer2) D 7 1 1 v 6 4 1 2 1 0 0 Abstract The Hopf algebra of undecorated rooted trees has tamed the combinatorics / of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ3 h t- theory, from all nestings and chainings of a primitive self-energy subdivergence. Here p we formulate the nonperturbative problems which these resummations approximate. For e h Yukawa theory, at spacetime dimension d = 4, we obtain an integrodifferential Dyson- : v Schwinger equation and solve it parametrically in terms of the complementary error func- i X tion. For the scalar theory, at d = 6, the nonperturbative problem is more severe; we r transform it to a nonlinear fourth-order differential equation. After intensive use of sym- a bolic 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. 1) D.Broadhurst@open.ac.uk; http://physics.open.ac.uk/dbroadhu Physics Dept, Open University, Milton Keynes MK7 6AA, UK 2 ) Dirk.Kreimer@uni-mainz.de; http://dipmza.physik.uni-mainz.de/kreimer/homepage.html Heisenberg Fellow, Physics Dept, Univ. Mainz, 55099 Mainz, Germany 1 Introdu