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

Abstract

The contribution of the α 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 Padé 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 Padé 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.

Improvedα4term of the muon anomalous magnetic moment

“…The 1998 results of numerical integration of ∆M 2,P 6i are listed in Table V. From the numerical values in Tables IV and V we obtain the value reported previously [40]:…”

“…However, it seems to be small compared with the quoted uncertainty [40]. In principle it is possible to prove or disprove it by more numerical work.…”

“…However, the discrepancy with (35) was still non-negligible. Finally, the problem was traced to round-off errors caused by insufficient number of effective digits in real*8 arithmetic in carrying out renormalization by numerical means [40]. This was resolved by going over to the real*16 arithmetic.…”

“…This will be called digit-deficiency errors. Various techniques had to be introduced to alleviate this problem [40]. See Appendix B for details.…”

Tenth-order QED contribution to lepton anomalous magnetic moment: Fourth-order vertices containing sixth-order vacuum-polarization subdiagrams

“…4) is not known, although the Padé approximant is known to provide a good approximation [30][31][32]. We follow here primarily the approach [29] which utilizes the vacuum-polarization function Π (6) itself, instead of its spectral function.…”