Complete two-loop correction to the bound-electrongfactor

Within a systematic approach based on dimensionally regularized nonrelativistic quantum electrodynamics, we derive a complete result for the two-loop correction to order ${(\alpha ∕\pi )}^2{\left(Z\alpha \right)}^4$ for the $g$ factor of an electron bound in an $nS$ state of a hydrogenlike ion. The results obtained significantly improve the accuracy of the theoretical predictions for the hydrogenlike carbon and oxygen ions and influence the value of the electron mass inferred from $g$-factor measurements.