Running Coulomb potential and Lamb shift in QCD

The QCD $\beta$ function and the anomalous dimensions for the Coulomb potential and the static potential first differ at three loop order. We evaluate the three loop ultrasoft anomalous dimension for the Coulomb potential and give the complete three loop running. Using this result, we calculate the leading logarithmic Lamb shift for a heavy-quark–antiquark bound state, which includes all contributions to the binding energies of the form $m{\alpha }_s^4({\alpha }_s\ln {\alpha }_s{)}^k,$ $k>~0.$