Tenth-order QED contribution to lepton anomalous magnetic moment: Fourth-order vertices containing sixth-order vacuum-polarization subdiagrams

This paper reports the tenth-order contributions to the $g-2$ of the electron $a_e$ and those of the muon $a_{\mu }$ from the gauge-invariant Set II(c), which consists of 36 Feynman diagrams, and Set II(d), which consists of 180 Feynman diagrams. Both sets are obtained by insertion of sixth-order vacuum-polarization diagrams in the fourth-order anomalous magnetic moment. The mass-independent contributions from Set II(c) and Set II(d) are $-0.116489(32)(\alpha /\pi {)}^5$ and $-0.24300(29)(\alpha /\pi {)}^5$, respectively. The leading contributions to $a_{\mu }$, which involve electron loops only, are $-3.88827(90)(\alpha /\pi {)}^5$ and $0.4972(65)(\alpha /\pi {)}^5$ for Set II(c) and Set II(d), respectively. The total contributions of the electron, muon, and tau-lepton loops to $a_e$ are $-0.116874(32)(\alpha /\pi {)}^5$ for the Set II(c), and $-0.24310(29)(\alpha /\pi {)}^5$ for the Set II(d), respectively. The contributions of the electron, muon, and tau-lepton loop to $a_{\mu }$ are $-5.5594(11)(\alpha /\pi {)}^5$ for the Set II(c) and $0.2465(65)(\alpha /\pi {)}^5$ for the Set II(d), respectively.

Figure 1

Typical diagrams of the tenth-order Set II(c) and Set II(d).

Figure 2

Sixth-order vacuum-polarization diagrams consisting of two fermion loops.

Figure 3

Self-energy-like diagrams of fourth order.

Figure 4

Sixth-order vacuum-polarization diagrams consisting of a single fermion loop.

Figure 5

Fourth-order vacuum-polarization diagrams consisting of one lepton loop.