# Bethe logarithm for the helium atom

## TL;DR

This paper provides highly precise calculations of the Bethe logarithm for helium atom states, including mass dependence and asymptotic expansions, enabling accurate results for various atomic states and nuclear masses.

## Contribution

It introduces a method to compute the Bethe logarithm with high precision and derives mass-dependent coefficients for finite nuclear mass effects.

## Key findings

- Achieved 12-14 digit precision for helium Bethe logarithm
- Derived coefficients for mass dependence expansion
- Provided accurate approximations for Rydberg states

## Abstract

The Bethe logarithm for a large set of states of the helium atom is calculated with a precision of 12-14 significant digits. The numerical data is obtained for the case of infinite mass of a nucleus. Then we study the mass dependence and provide coefficients of the $m_e/M$ expansion, which allows us to calculate accurate values for the Bethe logarithm for any finite mass. An asymptotic expansion for the Rydberg states is analyzed and a high-quality numerical approximation is found, which ensures 7-8 digit accuracy for the $S$, $P$, and $D$ states of the helium atom.