A Simple approach for precision calculation of Bethe logarithm

TL;DR

This paper introduces a simplified, highly precise method for calculating the Bethe logarithm, reducing computational complexity and achieving 49-digit accuracy for hydrogen's ground state.

Contribution

The authors present a novel approach that simplifies the calculation of Bethe logarithm using specific operators and dimensional regularization, enhancing precision and efficiency.

Findings

Achieved 49 significant digits for hydrogen ground state Bethe logarithm.

Reduced complexity in pseudostate construction for numerical evaluation.

Applicable to multi-electron systems with improved simplicity and efficiency.

Abstract

In this article we propose a simple approach for the precision calculation of Bethe logarithm. The leading contributions are obtained using specific operators, while the remaining terms are eliminated by adjusting the parameter $\lambda$. Through the use of dimensional regularization, singular divergences are algebraically canceled. Compared to the standard form of Bethe logarithm, our approach significantly reduces the complexity of constructing pseudostates in numerical evaluations. Using this approach we obtain a very highly precise result of Bethe logarithm for the ground state of the hydrogen, achieving 49 significant digits. And for multi-electron systems this approach appears simplicity and efficiency as well.