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 gencode$N$. 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 $-46.95(17)(\alpha /\pi {)}^5$.

Figure 1

The eighth-order Group V diagrams. The solid line represents the electron propagating in a weak, constant, magnetic field.

Figure 2

The eighth-order diagram $M_{47}$ of Group V and the tenth-order diagram $M_{47,P2}$ of Set IV. The diagram $M_{47,P2}$ represents the sum of diagrams obtained by inserting a second-order vacuum-polarization loop into each of four photon lines of the eighth-order diagram $M_{47}$.