Accuracy of calculations involvingα3vacuum-polarization diagrams: Muonic hydrogen Lamb shift and muong−2

The contribution of the ${\alpha }^3$ single electron-loop vacuum-polarization diagrams to the Lamb shift of 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 Padé approximation method. The high precision of these values offers an opportunity to examine the reliability of the Monte Carlo integration as well as that of the Padé method. Our examination covers both the muonic hydrogen atom and muon $g-2.$ We test 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 muonic hydrogen and also resolves the long-standing discrepancy between two previous evaluations of the muon $g-2$ value.