Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron

We have carried out a new evaluation of the eighth-order contribution to the electron $g-2$ using FORTRAN codes generated by an automatic code generator gencode$N$. Comparison of the new result with the old one has revealed an inconsistency in the treatment of the infrared divergences in the latter. With this error corrected we now have two independent determinations of the eighth-order term. This leads to the revised value $1159652182.79(7.71)\times {10}^{-12}$ of the electron $g-2$, where the uncertainty comes mostly from that of the best non-QED value of the fine structure constant $\alpha$. The new value of $\alpha$ derived from the revised theory and the latest experiment is ${\alpha }^{-1}=137.035999084(51)[0.37\mathrm{ppb}]$, which is about 4.7 ppb smaller than the previous ${\alpha }^{-1}$.

Figure 1

Eighth-order Group V diagrams represented by 47 self-energy-like diagrams $M_{01}–M_{47}$.

Figure 2

(a) Self-energy-like diagram of $M_{14}$. (b) Vertex diagrams $6g5$ and $4l$ from which the vertex renormalization constants $L_{6g5}$ and $L_{4l}$, respectively, are derived.

Figure 3

Self-energy-like diagrams of $M_{16}$ and $M_{18}$. Feynman parameters assigned to the electron lines are $z_1\sim z_7$ and those to the photon lines are $z_a\sim z_d$.

Figure 4

Fourth-order $q$-type diagrams. (a) Self-energy-like diagrams $M_{4a}$ and $M_{4b}$. (b) Vertex diagrams $4c$, $4x$, $4s$, and $4l$. Their contributions to the magnetic moment are related to $M_{4a}=2M_{4c}+M_{4x}$, and $M_{4b}=2M_{4s}+M_{4l}$.

Figure 5

Sixth-order $q$-type self-energy-like diagrams $M_{6x}$, $x=a,\dots ,h$. The time-reversal diagrams of $M_{6d}$ and $M_{6g}$ are not shown here. The fermion lines of a diagram are named 1 to 5 from left to right. The vertex diagram obtained by inserting an external photon vertex into the fermion line $i$ of the self-energy diagram $6x$ is named $6xi$.