Accuracy of Calculations Involving $\alpha^3$ Vacuum-Polarization   Diagrams: Muonic Hydrogen Lamb Shift and Muon $g-2$
  The contribution of the $\alpha^3$ single electron-loop vacuum-polarization diagrams to the Lamb shift of the muonic hydrogen has been evaluated recently by two independent methods. One uses the exact parametric representation of the vacuum-polarization function while the other relies on the Pad\'{e} approximation method. High precision of these values offers an opportunity to examine the reliability of the Monte-Carlo integration as well as that of the Pad\'{e} method. Our examination covers both muonic hydrogen atom and muon $g-2$. We tested them further for the cases involving two-loop vacuum polarization, where an exact analytic result is known. Our analysis justifies the result for the Lamb shift of the muonic hydrogen and also resolves the long-standing discrepancy between two previous evaluations of the muon $g-2$ value. 
T. Kinoshita and M. Nio