We consider the fourth-order contribution of the one-bubble diagram to $a=\frac{\mathrm{}(g-2\mathrm{})\mathrm{}}{2}$ of the charged leptons for arbitrary mass ratio $k=\frac{m_1}{m_2}$. We verify that our recent formulas for $a_1\mathrm{}\left(k\right)=a\left(k\right)\mathrm{}$, $k<\mathrm{}1\mathrm{}$, and $a_2\mathrm{}\left(k\right)=a\left(k\right)\mathrm{}$, $k>\mathrm{}1\mathrm{}$, reduce smoothly to $a(k=1\mathrm{})=\frac{119}{36}-\frac{{\pi }^2}{3}$ from both sides, thus verifying the correctness of our formulas for $a_1$ and $a_2$. We also give very accurate results for all combinations of $m_1$, $m_2=m_e, m_{\mu }, \mathrm{and} m_{\tau }$, where the accuracy is limited only by the present experimental accuracy of the masses $m_e$, $m_{\mu }$, and $m_{\tau }$. We consider a new more accurate measurement of the mass of the $\tau$ lepton and compute its effect on the QED contribution to $a_{\mu }$ and $a_{\tau }$. The results for $a_{\tau }^{\mathrm{QED}}$ and $a_{\tau }$ are unchanged. Our result for $a_{\mu }^{\mathrm{QED}}$ is changed by 3.3(5)×${10}^{-12\mathrm{}}$, but $a_{\mu }^{\mathrm{th}}$ remains unchanged.

• Received 14 July 1992

DOI:https://doi.org/10.1103/PhysRevD.47.1723