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 nine Feynman diagrams which have a fourth-order vacuum-polarization loop containing another fourth-order vacuum-polarization loop. Set I(h) consists of 30 Feynman diagrams which have a proper sixth-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) (α/π)⁵ for Set I(g) and 0.001 685 (13) (α/π)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)(α/π)⁵ and -0.564 8 (11)(α/π)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.