α(Zα)5m finite-nuclear-size contribution to the energy levels in light muonic atoms

Abstract

We present a theoretical consideration of the radiative two-photon finite-nuclear-size contributions to the Lamb shift in a light muonic atom. They are of the order of $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{5}m$ and may depend differently on $m{R}_{N}$. The contributions are due to the electronic and muonic vacuum polarization and the muon self-energy. The consideration is done within the external field approximation. We also found the leading logarithmic finite-nuclear-size contributions in the next order. One of them is of the order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{6}{ln}^{2}(Z\ensuremath{\alpha})\phantom{\rule{0.16em}{0ex}}{(m{R}_{N})}^{2}m$ and the other is of the order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{5}{ln}^{2}({m}_{e}{R}_{N})\phantom{\rule{0.16em}{0ex}}{(m{R}_{N})}^{3}m$. Special attention is paid to higher-order effects in muonic hydrogen and the related applications of various parametrizations of the proton electric form factors. We also consider the general expressions in coordinate space for the finite-size radiative contributions to the Lamb shift in two-body muonic atoms. The numerical results are present for the atoms with $Z\ensuremath{\le}10$.