Improved Bound-Electron <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math>-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 Zα. Here, Z is the nuclear charge number and α 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 Zα. We apply our theory to the recently measured g factor of the hydrogenlike ^{118}Sn^{49+} ion [J. Morgner et al., Nature (London) 622, 53 (2023)NATUAS0028-083610.1038/s41586-023-06453-2] 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.