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

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 $Z\alpha .$ 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 $Z\alpha .$ We confirm the analytic results of order ${\alpha }^2(Z\alpha {)}^5$ but disagree with Karshenboim’s calculation of the ${\alpha }^2\left(Z\alpha {)}^6{\ln }^3\right(Z\alpha {)}^{-2}$ contribution.