The relation between the QED charge renormalized in MS¯ and on-shell schemes at four loops, the QED on-shell β-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 Π(Q2)Π(Q2) in the two limits of q=0q=0 and Q2→∞Q2→∞. These results are used to explicitly construct the conversion relations between the QED charge renormalized in on-shell (OS) and MS¯ scheme. Using these relations and results of Baikov et al. [1] we construct the momentum dependent part of Π(Q2,m,α)Π(Q2,m,α) at large Q2Q2 at five loops in both MS¯ and OS schemes. As a direct consequence we arrive at the full result for the QED β-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.