Improved bound-electron g-factor theory through complete two-loop QED calculations

The two-loop self-energy correction to the bound-electron $g$-factor in hydrogenlike ions is investigated, taking into account the electron-nucleus interaction exactly. This all-order calculation is required to improve the total theoretical uncertainty of the $g$-factor, which is limited by the fact that two-loop self-energy corrections have only been calculated so far in the form of an expansion in $Z\alpha$. Here, $Z$ is the nuclear charge number and $\alpha$ is the fine-structure constant. In this work, we report calculations of the last missing parts of the total two-loop self-energy correction, exactly in $Z\alpha$. We apply our theory to the recently measured $g$-factor of the hydrogenlike $^{118}$Sn$^{49+}$ ion [J. Morgner et al., Nature 622, 53 (2023)] and, with a factor of 8, improve the accuracy of its state-of-the-art theoretical value by almost one order of magnitude, enabling more detailed tests of quantum electrodynamics and new physics in strong fields.