Convergence acceleration method for precision calculation of the Bethe logarithm

Abstract In this article we propose a simple approach for the precision calculation of the Bethe logarithm. In this approach, the Bethe logarithm is recast into a more flexible form that includes a free parameter λ . The leading contributions are obtained from specific operators, while the remaining terms are eliminated by adjusting λ . Through the use of dimensional regularization, singular divergences are algebraically canceled. Our approach notably reduces the complexity of Bethe logarithm calculations. Using this approach, we obtain a highly precise result for the Bethe logarithm for the ground state of hydrogen, achieving 49 significant digits. In the case of helium, this approach retains its simplicity and efficiency.