Four-loop Dyson-Schwinger-Johnson anatomy

Dyson-Schwinger equations are used to evaluate the 4-loop anomalous dimensions of quenched QED in terms of finite, scheme-independent, 3-loop integrals. Three of the results serve as strong checks of terms from scheme-dependent 4-loop QCD calculations. The fourth, for the anomalous dimension of $\bar\psi\sigma_{\mu\nu}\psi$, was previously known only to 2-loop order. The 4-loop beta function has 24 unambiguous terms. Two of the simplest give $\beta_4=\frac13(22-160)=-46$. The rational, $\zeta_3$ and $\zeta_5$ parts of the other 22 miraculously sum to zero. Vertex anomalous dimensions have 40 terms, with no dramatic cancellations. Our methods come from work by the late Kenneth Johnson, done more than 30 years ago. They are entirely free of the subtractions and infrared rearrangements of later methods