Binding two-loop vacuum-polarization corrections to the bound-electrongfactor

Abstract

We commence the evaluation of the one- and two-loop binding corrections to the $g$ factor for an electron in a hydrogenlike system of order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{5}$ and consider diagrams with closed fermion loops. The one-loop vacuum-polarization correction is rederived and confirmed. For the two-loop vacuum-polarization correction, due to a specific gauge-invariant set of diagrams with closed fermion loops, we find a correction $\ensuremath{\delta}g=7.442{(\ensuremath{\alpha}/\ensuremath{\pi})}^{2}{(Z\ensuremath{\alpha})}^{5}$. Based on the numerical trend of the coefficients inferred from the gauge-invariant subset, we obtain a numerically large tentative estimate for the complete two-loop binding correction to the $g$ factor (sum of self-energy and vacuum polarization).