The relation between the QED charge renormalized in MSbar and on-shell   schemes at four loops, the QED on-shell beta-function at five loops and   asymptotic contributions to the muon anomaly at five and six loops
  In this paper we compute the four-loop corrections to the QED photon self-energy Pi(Q^2) in the two limits of q=0 and Q^2->infinity. These results are used to explicitly construct the conversion relations between the QED charge renormalized in on-shell(OS) and MSbar scheme. Using these relations and results of Baikov et al. [1] we construct the momentum dependent part of Pi(Q^2,m,alpha) at large Q^2 at five loops in both MSbar and OS schemes. As a direct consequence we arrive at the full result for the QED beta-function in the OS scheme at five loops. These results are applied, in turn, to analytically evaluate a class of asymptotic contributions to the muon anomaly at five and six loops. 
P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn, C. Sturm
hep-ph