Second-order hyperfine correction to H, D, and ${}^3$He energy levels

Abstract

The complete second-order hyperfine-interaction correction is calculated for centroid energy levels of H, D, and ${}^3$He atoms. For ${}^3$He, the corrections of $-2.075$~kHz and $-0.305$~kHz beyond the leading hyperfine-mixing contribution are obtained for the $2^{1}S$ and $2^{3}S$ states, respectively. These results shift the nuclear charge radii difference derived from the ${}^3$He -${}^4$He isotope shift and largely resolve the previously reported disagreement between the muonic and electronic helium determinations [van der Werf et al., arXiv:2306.02333 (2023); Schuhmann et al., arXiv:2305.11679 (2023)].

Comments: 11 pages