Bethe logarithm for the11Sand21Sstates of helium

Abstract

We present variational calculations of the Bethe logarithm for the $1{}^{1}S$ and $2{}^{1}S$ states of helium. The approach is based on the explicit second-order perturbation formula and closely follows the method of Schwartz. The final values are $\mathrm{ln}[{K}_{0}(1{}^{1}S)/(1\mathrm{Ry})]=4.3701579(5)$ and $\mathrm{ln}[{K}_{0}(2{}^{1}S)/(1\mathrm{Ry})]=4.3664091(7)$. The latter result reduces the difference between theoretical and experimental values for the ionization potential of the $2{}^{1}S$ state to $0.15$ MHz.