Convergence-acceleration approach to partial-wave expansion of two-electron self-energy contributions to the Lamb shift

Methods of bound-state QED that treat the self-energy contributions to the Lamb shift within the partial-wave expansion usually face the problem of slow convergence of the latter. Inspired by an approach formulated by Sapirstein and Cheng [J. Sapirstein and K. T. Cheng, Phys. Rev. A 108, 042804 (2023)], we propose a modification of the standard procedure for calculating the contributions of two-electron self-energy diagrams. The performance of the method is studied by evaluating the corresponding corrections to the binding energies of He-like ions and by comparing the obtained results with the state-of-the-art values available in the literature: our calculations involving a much smaller number of partial waves show an improvement in accuracy.