Introduction: Precision spectroscopy's advancing accuracy offers powerful opportunities to test Quantum Electrodynamics (QED) and search for physics beyond the Standard Model. As experimental techniques improve, measurements of atomic energy levels and bound-particle g-factors now challenge the accuracy of fundamental QED calculations. This growing precision creates a critical need to refine computational methods for key radiative corrections, particularly the bound-electron self-energy—a dominant contribution to the Lamb shift.
Methods: The modern approach for renormalization and calculation of the bound-electron self-energy diagram relies on a potential expansion, where the electron propagator is expanded perturbatively in a Dyson series of nuclear interaction. A key challenge is the slow convergence of the resulting infinite partial-wave expansion for the many-potential contribution, typically computed in coordinate space. The recently proposed Sapirstein–Cheng scheme accelerates this convergence by employing a difference method, calculating an auxiliary term in both momentum and coordinate space, which yields a faster-converging series for extrapolation.
Results: In this work, we present a further optimization of this method. Based on a detailed analysis of the asymptotic behavior of the partial-wave series, we introduce a new parameter into the subtraction term. Since the asymptotic behavior of both the many-potential term and the difference series decays as 1/k², this parameter can be tuned to cancel the leading asymptotic term completely. This optimization enhances the efficiency of the numerical extrapolation, improving the final accuracy of the self-energy correction calculation by an additional one to two orders of magnitude.
Conclusions: By accelerating the convergence of the partial-wave expansion, the developed method significantly increases the precision of self-energy calculations for hydrogen-like systems. This advancement is crucial for matching the accuracy of ongoing high-precision spectroscopic experiments and provides a more robust computational framework for evaluating higher-order Feynman diagrams, thereby enabling more sensitive searches for New Physics beyond the Standard Model.
