Logarithms ofαin QED Bound States from the Renormalization Group

The velocity renormalization group is used to determine $\ln \alpha$ contributions to QED bound state energies. The leading-order anomalous dimension for the potential gives the ${\alpha }^5\ln \alpha$ Lamb shift. The next-to-leading-order anomalous dimension determines the ${\alpha }^6\ln \alpha$, ${\alpha }^7{ln}^2\alpha$, and ${\alpha }^8{ln}^3\alpha$ corrections to the energy. These are used to obtain the ${\alpha }^8{ln}^3\alpha$ Lamb shift and ${\alpha }^7{ln}^2\alpha$ hyperfine splitting for hydrogen, muonium, and positronium, as well as the ${\alpha }^2\ln \alpha$ and ${\alpha }^3{ln}^2\alpha$ corrections to the ortho- and parapositronium lifetimes. This shows for the first time that these logarithms can be computed from the renormalization group.