Loop-after-loop contribution to the second-order Lamb shift in hydrogenlike low-Z atoms

We present a numerical evaluation of the loop-after-loop contribution to the second-order self-energy for the ground state of hydrogenlike atoms with low nuclear charge numbers Z. The calculation is carried out in the Fried-Yennie gauge and without an expansion in Zα. Our calculation confirms the results of Mallampalli and Sapirstein and disagrees with the calculation by Goidenko and co-workers. A discrepancy between different calculations is investigated. An accurate fitting of the numerical results provides a detailed comparison with analytic calculations based on an expansion in the parameter Zα. We confirm the analytic results of order α²(Zα)⁵ but disagree with Karshenboim’s calculation of the α²(Zα)⁶ln³(Zα)^-2 contribution.