Tenth-Order Lepton Anomalous Magnetic Moment -- Second-Order Vertex   Containing Two Vacuum Polarization Subdiagrams, One Within the Other
  This paper reports the tenth-order QED contribution to the g-2 of electron and muon from two gauge-invariant sets, Set I(g) and Set I(h). In the case of electron g-2 Set I(g) consists of 9 Feynman diagrams which have a 4th-order vacuum-polarization loop containing another 4th-order vacuum-polarization loop. Set I(h) consists of 30 Feynman diagrams which have a proper 6th-order vacuum-polarization loop containing a second-order vacuum-polarization loop. The results of numerical integration, including mass-dependent terms containing one closed loop of muon, are 0.028 597 (4) (alpha/pi)^5 for Set I(g) and 0.001 685 (13) (alpha/pi)^5 for Set I(h), respectively. We also report the contributions of Set I(g) and Set I(h) to the muon anomaly. Diagrams included are those containing electron, muon, and tau-lepton loops. Their sums are 2.640 9 (4) (alpha/pi)^5 and -0.564 8 (11) (alpha/pi)^5, respectively. The sum of contributions of Sets I(g) and I(h) containing only electron loops are in fair agreement with the recently obtained asymptotic analytic results. 
T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio