Improvedα4term of the muon anomalous magnetic moment

We have completed the evaluation of all mass-dependent ${\alpha }^4$ QED contributions to the muon $g-2$, or $a_{\mu }$, in two or more different formulations. Their numerical values have been greatly improved by an extensive computer calculation. The new value of the dominant ${\alpha }^4$ term $A_2^{(8)}(m_{\mu }/m_e)$ is 132.6823 (72), which supersedes the old value 127.50 (41). The new value of the three-mass term $A_3^{(8)}(m_{\mu }/m_e,m_{\mu }/m_{\tau })$ is 0.0376 (1). The term $A_2^{(8)}(m_{\mu }/m_{\tau })$ is crudely estimated to be about 0.005 and may be ignored for now. The total QED contribution to $a_{\mu }$ is $116584719.58(0.02)(1.15)(0.85)\times {10}^{-11}$, where 0.02 and 1.15 are uncertainties in the ${\alpha }^4$ and ${\alpha }^5$ terms and 0.85 is from the uncertainty in $\alpha$ measured by atom interferometry. This raises the standard model prediction by $13.9\times {10}^{-11}$, or about 1/5 of the measurement uncertainty of $a_{\mu }$. It is within the noise of current uncertainty ($\sim 100\times {10}^{-11}$) in the estimated hadronic contributions to $a_{\mu }$.