Tenth-Order Lepton Anomalous Magnetic Moment—Sixth-Order Vertices Containing Vacuum-Polarization Subdiagrams

This paper reports the values of contributions to the electron $g-2$ from 300 Feynman diagrams of the gauge-invariant Set III(a) and 450 Feynman diagrams of the gauge-invariant Set III(b). The evaluation is carried out in two versions. Version A is to start from the sixth-order magnetic anomaly $M_6$ obtained in the previous work. The mass-independent contributions of Set III(a) and Set III(b) are 2.1275(2) and 3.3271(6) in units of $(\alpha /\pi {)}^5$, respectively. Version B is based on the recently developed automatic code generation scheme. This method yields 2.1271(3) and 3.3271(8) in units of $(\alpha /\pi {)}^5$, respectively. They are in excellent agreement with the results of the first method within the uncertainties of numerical integration. Combining these results as statistically independent we obtain the best values, 2.1273(2), and 3.3271(5) times $(\alpha /\pi {)}^5$, for the mass-independent contributions of the Set III(a) and Set III(b), respectively. We have also evaluated mass-dependent contributions of diagrams containing muon and/or tau-particle loop. Including them the total contribution of Set III(a) is 2.1349(2) and that of Set III(b) is 3.3299(5) in units of $(\alpha /\pi {)}^5$. The total contributions to the muon $g-2$ of various leptonic vacuum-polarization loops of Set III(a) and Set III(b) are 112.418(32) and 15.407(5) in units of $(\alpha /\pi {)}^5$, respectively.

Figure 1

The sixth-order $q$-type diagrams. The solid line represents the electron in a constant magnetic field. The time-reversal diagrams of $M_{6D}$ and $M_{6G}$ are omitted for simplicity.

Figure 2

Typical diagrams of Set III.

Figure 3

Typical tenth-order diagrams of Set III(a) obtained by insertion of two second-order vacuum-polarization loops ${\Pi }_2$ in lepton diagrams of the three-photon-exchange type. The $\mathrm{subset}\mathrm{III}(a_d)$ consists of diagrams in which ${\Pi }_2$ are inserted in different photon lines, while the $\mathrm{subset}\mathrm{III}(a_s)$ consists of diagrams in which ${\Pi }_2$ are inserted in the same photon line. There are 150 diagrams in each subset.

Figure 4

Typical tenth-order diagrams of Set III(b) obtained by insertion of a fourth-order vacuum-polarization loop ${\Pi }_4$ in lepton diagrams of the three-photon-exchange type. Altogether, there are 450 diagrams of this type.