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Two-loop self-energy corrections to the bound-electron $g$ factor areinvestigated theoretically to all orders in the nuclear binding strengthparameter $Z\alpha$. The separation of divergences is performed by dimensionalregularization, and the contributing diagrams are regrouped into specificcategories to yield finite results. We evaluate numerically the loop-after-loopterms, and the remaining diagrams by treating the Coulomb interaction in theelectron propagators up to first order. The results show that such two-loopterms are mandatory to take into account for projected near-future stringenttests of quantum electrodynamics and for the determination of fundamentalconstants through the $g$ factor.

Theory of the two-loop self-energy correction to the $\boldsymbol g$ factor in non-perturbative Coulomb fields