Tenth-order QED contribution to the lepton anomalous magnetic moment: Sixth-order vertices containing an internal light-by-light-scattering subdiagram

This paper reports the tenth-order QED contribution to the lepton $g-2$ from the gauge-invariant set, called Set III(c), which consists of 390 Feynman vertex diagrams containing an internal fourth-order light-by-light-scattering subdiagram. The mass-independent contribution of Set III(c) to the electron $g-2$ ($a_e$) is 4.9210 (103) in units of $(\alpha /\pi {)}^5$. The mass-dependent contributions to $a_e$ from diagrams containing a muon loop is 0.00370 (37) $(\alpha /\pi {)}^5$. The tau-lepton loop contribution is negligible at present. Altogether the contribution of Set III(c) to $a_e$ is 4.9247 (104) $(\alpha /\pi {)}^5$. We have also evaluated the contribution of the closed electron loop to the muon $g-2$ ($a_{\mu }$). The result is 7.435 (134) $(\alpha /\pi {)}^5$. The contribution of the tau-lepton loop to $a_{\mu }$ is 0.1999 (28) $(\alpha /\pi {)}^5$. The total contribution of various leptonic loops (electron, muon, and tau-lepton) of Set III(c) to $a_{\mu }$ is 12.556 (135)$(\alpha /\pi {)}^5$.

Figure 1

Tenth-order self-energy-like diagrams in which lepton lines propagate in the magnetic field. They represent 390 vertex diagrams of Set III(c). Assignment of Feynman parameters $z_1,z_2,\dots ,z_9$ to the lepton lines and $z_a,z_b,\dots ,z_e$ to the photon lines is indicated in the figure $A1$, $B1$, and $C1$.

Figure 2

Vertex diagrams of the sixth order containing an $l\mathrm{\text{−}}l$ subdiagram. The diagram $6LL(3)$ and $6LL(4)$ are identical with each other because of the time-reversal symmetry. There are six vertex diagrams of this type, taking into account of two directions in which the closed fermion loop can take.

Figure 3

Various diagrams of the eighth order needed for renormalization. $8LL\alpha (\alpha =E,F,G,H,I)$ is denoted as $LL\alpha$ in Ref. 35. A vertex diagram $8LL\alpha (i)(i=5,6,7)$ is obtained by inserting an external vertex into a fermion line $i$ of the diagram $8LL\alpha$. It is denoted as $LL\alpha (i)$ in Ref. 36.

Figure 4

The eighth-order self-energy-like diagrams $8LLJ$, $8LLK$, and $8LLL$, containing a light-by-light scattering loop internally. Open lepton lines propagate in the weak magnetic field. They represent 18 vertex diagrams in total. The vertex diagram $8LLJ(1)$ is a part of the diagram $8LLJ$ in which the magnetic vertex is attached only to the fermion line 1 of the self-energy diagram $8LLJ$.