Two-loop QED corrections with closed fermion loops for the bound-electron $g$ factor

Two-loop QED corrections with closed fermion loops are calculated for the $1s$ bound-electron $g$ factor. Calculations are performed to all orders in the nuclear binding strength parameter $Z\alpha$ (where $Z$ is the nuclear charge and $\alpha$ is the fine-structure constant) except for the closed fermion loop, which is treated within the free-loop (Uehling) approximation in some cases. Comparison with previous $Z\alpha$-expansion calculations is made and the higher-order remainder of order ${\alpha }^2{\left(Z\alpha \right)}^5$ and higher is separated out from the numerical results.

Figure 1

The combined self-energy and vacuum-polarization correction to the bound-electron $g$ factor with vacuum-polarization insertion into the photon line, referred to as the S(VP)E correction. Double lines represent an electron propagating in the binding nuclear field, wave lines denote virtual photon, and the wave line terminated by a cross denotes interaction with an external magnetic field.

Figure 2

The S(VP)E correction to the $1s$ bound-electron $g$ factor. The left graph presents our numerical all-order results [dots and solid line (red)], together with the $Z\alpha$-expansion result [dashed line (black)], in terms of $\delta g=\Delta g/({\alpha }^2/{\pi }^2)$. The right graph shows the corresponding results for the higher-order remainder $G\left(Z\alpha \right)$.

Figure 3

The combined self-energy and vacuum-polarization correction to the bound-electron $g$ factor with vacuum-polarization insertion into the electron line, referred to as the SEVP correction.

Figure 4

SEVP correction to the $1s$ bound-electron $g$ factor. The left graph presents our numerical all-order results [dots and solid line (red)], together with the $Z\alpha$-expansion result [dashed line (black)], in terms of $\delta g=\Delta g/\left[({\alpha }^2/{\pi }^2){\left(Z\alpha \right)}^4\right]$. The right graph shows the results for the higher-order remainder $G\left(Z\alpha \right)$.

Figure 5

The two-loop vacuum-polarization correction to the bound-electron $g$ factor (the VPVP correction).

Figure 6

VPVP correction to the $1s$ bound-electron $g$ factor. The left graph presents our numerical all-order results [dots and solid line (red)], together with the $Z\alpha$-expansion results [dashed line (black)], in terms of $\delta g=\Delta g/\left[({\alpha }^2/{\pi }^2){\left(Z\alpha \right)}^4\right]$. The right graph shows the corresponding results for the higher-order remainder $G\left(Z\alpha \right)$.