Contrary to what was previously believed, two-loop radiative corrections to the $g$-factor of an electron bound in a hydrogen-like ion at $\mathcal{ O}\left(\alpha^2 (Z\alpha)^5\right)$ exhibit logarithmic enhancement. This previously unknown contribution is due to a long-distance light-by-light scattering amplitude. Taking an effective field theory approach, and using the Euler-Heisenberg Lagrangian, we find $\Delta g = -\left( \frac{ \alpha}{ \pi} \right)^2\left(Z\alpha\right)^5 \frac{ 56 \pi}{ 135}\ln Z\alpha$.
Categories: hep-ph physics.atom-ph