Revised and improved value of the QED tenth-order electron anomalous magnetic moment

In order to improve the theoretical prediction of the electron anomalous magnetic moment $a_e$ we have carried out a new numerical evaluation of the 389 integrals of Set V, which represent 6,354 Feynman vertex diagrams without lepton loops. During this work, we found that one of the integrals, called $X024$, was given a wrong value in the previous calculation due to an incorrect assignment of integration variables. The correction of this error causes a shift of $-1.26$ to the Set V contribution, and hence to the tenth-order universal (i.e., mass-independent) term $A_1^{(10)}$. The previous evaluation of all other 388 integrals is free from errors and consistent with the new evaluation. Combining the new and the old (excluding $X024$) calculations statistically, we obtain $7.606(192)(\alpha /\pi {)}^5$ as the best estimate of the Set V contribution. Including the contribution of the diagrams with fermion loops, the improved tenth-order universal term becomes $A_1^{(10)}=6.675(192)$. Adding hadronic and electroweak contributions leads to the theoretical prediction $a_e(\text{theory})=1159652182.032(720)\times {10}^{-12}$. From this and the best measurement of $a_e$, we obtain the inverse fine-structure constant ${\alpha }^{-1}(a_e)=137.0359991491(331)$. The theoretical prediction of the muon anomalous magnetic moment is also affected by the update of QED contribution and the new value of $\alpha$, but the shift is much smaller than the theoretical uncertainty.

Figure 1

Self-energy-like diagram $X024(abcbddecea)$. The straight and wavy lines represent fermion and photon propagators, respectively. Indices assigned to the fermion lines are $1,2,\dots ,9$ from left to right, and those to the photon lines are $a,b,\dots ,e$. The nine vertex diagrams related to this self-energy-like diagram are obtained by inserting an external photon vertex in each of the nine fermion lines.