This paper reports the Feynman-parametric representation of the vacuum-polarization function consisting of 105 Feynman diagrams of the eighth order, and its contribution to the gauge-invariant set called Set I(i) of the tenth-order lepton anomalous magnetic moment. Numerical evaluation of this set is carried out using FORTRAN codes generated by an automatic code generation system gencodevpN developed specifically for this purpose. The contribution of diagrams containing an electron loop to the electron $g-2$ is $0.01747(11)(\alpha /\pi {)}^5$. The contribution of diagrams containing a muon loop is $0.00000167(3)(\alpha /\pi {)}^5$. The contribution of a tau-lepton loop is negligible at present. The sum of all these terms is $0.01747(11)(\alpha /\pi {)}^5$. The contribution of diagrams containing an electron loop to the muon $g-2$ is $0.0871(59)(\alpha /\pi {)}^5$. This is to be compared with the unpublished asymptotic analytic result $(0.25237+O(m_e/m_{\mu }))(\alpha /\pi {)}^5$. The contribution of a tau-lepton loop to $a_{\mu }$ is $0.000237(1)(\alpha /\pi {)}^5$. The total contribution to $a_{\mu }$, the sum of these terms and the mass-independent term, is $0.1048(59)(\alpha /\pi {)}^5$.

• Received 12 January 2011

DOI:https://doi.org/10.1103/PhysRevD.83.053003