Improved $α^4$ Term of the Muon Anomalous Magnetic Moment

Abstract

We have completed the evaluation of all mass-dependent $α^4$ QED contributions to the muon $g-2$, or $a_μ$, in two or more different formulations. Their numerical values have been greatly improved by an extensive computer calculation. The new value of the dominant $α^4$ term $A_2^{(8)} (m_μ/ 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_μ/ m_e, m_μ/ m_τ) $ is 0.0376 (1). The term $A_2^{(8)} (m_μ/ m_τ) $ is crudely estimated to be about 0.005 and may be ignored for now. The total QED contribution to $a_μ$ is $116 584 719.58 (0.02)(1.15)(0.85) \times 10^{-11}$, where 0.02 and 1.15 are uncertainties in the $α^4$ and $α^5$ terms and 0.85 is from the uncertainty in $α$ 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 $a_μ$. It is within the noise of current uncertainty ($\sim 100 \times 10^{-11}$) in the estimated hadronic contributions to $a_μ$.