Eighth-order QED contribution to the anomalous magnetic moment of the muon

Abstract

We report a calculation of the eighth-order QED contribution to the muon anomalous magnetic moment $a_{\mu }^{\mathrm{}\left(8\right)\mathrm{}}$ coming from 469 Feynman diagrams, all of which contain electron loops of vacuum-polarization type and/or light-by-light scattering type. Our result is $126.92\left(41\right)\mathrm{}{\left(\frac{\alpha }{\pi }\right)}^4$. The error represents the estimated accuracy (90% confidence limit) of the required numerical integration. We also report an estimate of the tenth-order contribution to $a_{\mu }$. Combining these with the lower-order results and the latest theoretical value for the electron anomaly $a_e$, we find that the QED contribution to the muon anomaly is given by $a_{\mu }^{\mathrm{QED}}=1165846947\mathrm{}\left(46\right)\mathrm{}\left(28\right)\mathrm{}\times {10}^{-12\mathrm{}}$, where the first error is an estimate of theoretical uncertainty and the second reflects the measurement uncertainty in $\alpha$. Including the hadronic and electroweak contributions, the best theoretical prediction for $a_{\mu }$ available at present is $a_{\mu }^{\mathrm{theory}}=116591920\mathrm{}\left(191\right)\mathrm{}\times {10}^{-11\mathrm{}}$, where the error comes predominantly from the hadronic contribution.