The Tenth-Order QED Contribution to the Lepton g-2: Evaluation of Dominant ${\alpha }^5$ Terms of Muon g-2

The QED contribution to the anomalous magnetic moments of electron and muon are known very precisely up to the order ${\alpha }^4$. However, the knowledge of ${\alpha }^5$ term will also be required when the precision of measurement improves further. This paper reports the first systematic attempt to evaluate the ${\alpha }^5$ term. Feynman diagrams contributing to this term can be classified into six gauge-invariant sets which can be subdivided further into 32 gauge-invariant subsets. Thus far we have numerically evaluated all integrals of 17 gauge-invariant subsets which contain light-by-light-scattering subdiagrams and/or vacuum-polarization subdiagrams. They cover most of leading terms of muon $g-2$ and lead to a preliminary result 663(20) $(\alpha \mathrm{/}\pi {)}^5$, which is 8.5 times more precise than the old estimate.