Nonrelativistic QED Approach to the Bound-Electron $g$ Factor

Within a systematic approach based on nonrelativistic quantum electrodynamics, we derive the one-loop self-energy correction of order $\alpha (Z\alpha {)}^4$ to the bound-electron $g$ factor. In combination with numerical data, this analytic result improves theoretical predictions for the self-energy correction for carbon and oxygen by an order of magnitude. Basing on one-loop calculations, we obtain the logarithmic two-loop contribution of order ${\alpha }^2(Z\alpha {)}^4\ln [(Z\alpha {)}^{-2}]$ and the dominant part of the corresponding constant term. The results obtained improve the accuracy of the theoretical predictions for the $1S$ bound-electron $g$ factor and influence the value of the electron mass determined from $g$-factor measurements.