Contribution to the eighth-order anomalous magnetic moment of the muon

Abstract

The eighth-order contribution to the muon anomaly from second-order vacuum polarization insertions into photon-photon scattering diagrams is accurately determined. The result is $a_{\mu }^{\mathrm{}\left(8\right)\mathrm{}}\left(\gamma \gamma \right)=(117.4\mathrm{}\pm 0.5){\left(\frac{\alpha }{\pi }\right)}^4$. The coefficients of the ${\ln }^2\left(\frac{m_{\mu }}{m_e}\right)$ and $\ln \left(\frac{m_{\mu }}{m_e}\right)$ terms are also evaluated. The coefficient of ${\ln }^2\left(\frac{m_{\mu }}{m_e}\right)$ is found to be just one half the value expected from naive application of renormalization-group methods, and it is shown how this arises.