Virtual Delbr\"uck scattering and the Lamb shift in light hydrogen-like atoms

TL;DR

This paper investigates the light-by-light contribution to hydrogen atom energy levels, revealing a new correction related to Delbrück scattering, and evaluates its impact on various states using an effective field theory approach.

Contribution

It introduces an additional correction to the Lamb shift from Delbrück scattering, expanding the understanding of light-by-light effects in hydrogen-like atoms.

Findings

New correction larger than previous light-by-light terms

Effective potential evaluated for states with non-zero orbital momentum

Asymptotic behavior analyzed for muonic atoms

Abstract

We return to the problem of evaluation of the light-by-light contribution to the energy levels of the hydrogen atom. We find an additional contribution directly related to the Delbr\"uck scattering amplitude. The new correction is larger than the previously included light-by-light terms at order~$\alpha^2 (Z\alpha)^6\ln(Z\alpha)\, m_e$. We consider the effective potential in position space using an effective field theory approach and evaluate light-by-light corrections to the energy levels of states with non-zero orbital momentum as well as to the weighted difference of $s$ states. We also determine the large distance asymptotic behaviour of the effective potential induced by the light-by-light scattering in muonic atoms.