Tenth-Order QED Lepton Anomalous Magnetic Moment --- Eighth-Order Vertices Containing a Second-Order Vacuum Polarization

This paper reports the evaluation of the tenth-order QED contribution to the lepton g-2 from the gauge-invariant set of 2072 Feynman diagrams, called Set IV, which are obtained by inserting a second-order lepton vacuum-polarization loop into 518 eighth-order vertex diagrams of four-photon exchange type. The numerical evaluation is carried out by the adaptive-iterative Monte-Carlo integration routine vegas using the fortran codes written by the automatic code-generating algorithm gencodeN. Some of the numerical results are confirmed by comparison with the values of corresponding integrals that have been obtained previously by a different method. The result for the mass-independent contribution of the Set IV to the electron g-2 is $-7.7296 (48) (\alpha/\pi)^5$. There is also a small mass-dependent contribution to the electron g-2 due to the muon loop: $-0.01136 (7) (\alpha/\pi)^5$. The contribution of the tau-lepton loop is $-0.0000937(104)(\alpha/\pi)^5$. The sum of all these contributions to the electron g-2 is $-7.7407(49) (\alpha/\pi)^5$. The same set of diagrams enables us to evaluate the contributions to the muon g-2 from the electron loop, muon loop, and tau-lepton loop. They add up to $-46.95 (17) (\alpha/\pi)^5$.