Virtual light-by-light scattering and the $g$ factor of a bound electron

The contribution of the light-by-light diagram to the $g$ factor of an electron and muon bound in a Coulomb field is obtained. For an electron in a ground state, our results are in good agreement with the results of other authors obtained numerically for large $Z$. For relatively small $Z$ our results have essentially higher accuracy as compared to the previous ones. For muonic atoms, the contribution is obtained with a high accuracy in the whole region of $Z$.

Figure 1

The diagram corresponding to the magnetic-loop contribution to the $g$ factor of a bound electron and first nonvanishing terms of expansion of this loop with respect to the Coulomb field. The double line denotes the electron propagator and the wave function in a Coulomb field, the dashed line with the cross denotes the Coulomb field, the wavy line with the square denotes the external homogeneous magnetic field, and the internal wavy line corresponds to the photon propagator.

Figure 2

The ratio $F\left(x\right)∕F\left(0\right)$ as a function of $x=q∕m$.

Figure 3

The correction $\Delta g$ for $1S_{1∕2}$. Solid curve: the exact result. Dashed curve: $\Delta g_0+\Delta g_1$. Dotted curve: $\Delta g_{nr}$.