Bethe logarithm for the helium atom

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 are 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.