Four-loop Dyson-Schwinger-Johnson anatomy

Dyson-Schwinger equations are used to evaluate the 4-loop anomalous dimensi ons of quenched QED in terms of finite, scheme-independent, 3-loop integral s. Three of the results serve as strong checks of terms from scheme-depende nt 4-loop QCD calculations. The fourth, for the anomalous dimension of <(ps i)over bar>sigma(mu v)psi was previously known only to 2-loop order. The 4- loop beta function has 24 unambiguous terms. Two of the simplest give beta( 4) = 1/3(22 - 160) = -46. The rational, zeta(3) and zeta(5) parts of the ot her 22 miraculously sum to zero. Vertex anomalous dimensions have 40 terms, with no dramatic cancellations. Our methods come from work by the late Ken neth Johnson, done more than 30 years ago. They are entirely free of the su btractions and infrared rearrangements of later methods. (C) 1999 Elsevier Science B.V. All rights reserved.