Nuclear recoil corrections to the Lamb shift of hydrogen and light hydrogen-like ions

Accurate calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms are presented. Numerical results are reported for the $ns$ states with $n\le 5$ and for the $2p_{1\mathrm{/}2}$ and $2p_{3\mathrm{/}2}$ states. The calculations are performed to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter $Z\alpha$ (where $Z$ is the nuclear charge number and $\alpha$ is the fine structure constant). The obtained results provide accurate predictions for the higher-order remainder beyond the known $Z\alpha$-expansion terms. In the case of hydrogen, the remainder was found to be much larger than anticipated. This result resolves the previously reported disagreement between the numerical all-order and the analytical $Z\alpha$-expansion approaches for the nuclear recoil effect in the hydrogen Lamb shift.