Theory for the Rydberg states of helium: Results for $2 \le n \le 35$ and comparison with experiment for the singlet and triplet $P$-states

TL;DR

This work advances high-precision helium theory by extending nonrelativistic $P$-state energies to all singlet and triplet states up to $n=35$ using a triple Hylleraas basis, achieving ~1 part in $10^{22}$ accuracy for the base energy and enabling 1 kHz-level final accuracies when including relativistic and QED corrections. It directly derives ionization energies for the $2\,^1S_0$ and $2\,^3S_1$ states by combining theory with experimental high-$n$ transitions, providing 11 independent measurements of the $2\,^3S_1$ ionization energy and a stringent test of quantum defect extrapolation (QDT) methods. The results show excellent agreement with QDT within ~17 kHz but reveal a 0.471(53) MHz discrepancy with the most complete recent theory, highlighting a 9$\sigma$ tension that challenges current fundamental-physics modeling in this two-electron system. The study also delivers detailed Breit-related matrix elements and demonstrates the viability of direct theory–experiment comparisons at very high precision for helium Rydberg states.

Abstract

High precision variational calculations in Hylleraas coordinates are presented for all singlet and triplet $P$-states of helium up to principal quantum number $n = 35$ with a uniform accuracy of 1 part in $10^{22}$ for the nonrelativistic energy. Mass polarization, relativistic and quantum electrodynamic effects are included to achieve a final accuracy of $\pm$1 kHz or better for the ionization energy of the Rydberg states of $^4$He in the range $24\le n \le 35$. The results are combined with 11 transition frequency measurements of Clausen et al. Phys. Rev. A 111, 012817 (2025) to obtain complementary measurements of the ionization energy of the $1s2s\;^3S_1$ state that do not depend on quantum defect extrapolations to the series limit. The result from the triplet spectrum yields an ionization energy of 1152 842 742.728(6) MHz, which agrees with but is larger than the experimental value by 14 $\pm$17 kHz. However, it confirms a much larger 9$σ$ discrepancy of $0.468\pm0.055$ MHz with the theoretical ionization energy of Patkóš et al. Phys. Rev. A 103, 042809 (2021). The results provide a test of the quantum defect extrapolation method at the level of $\pm$17 kHz. This revised version contains an additional table of spin-dependent matrix elements of the Breit interaction in the appendix for $24\le n\le 35$. (12 pages, 1 figure).

Figures (1)

• Figure 1: Ionization energy of the $2\,^3S_1$ state of $^4$He as determined by adding the calculated ionization frequency of the $n\,^3P_{\rm c}$ state to the measured $2\,^3S_1-n\,^3P_{\rm c}$ transition frequency Clausen2025, and similarly for the singlet case Clausen2021 together with the measured $2\,^1S_0-2\,^3S_1$ transition frequency Rengelink2018.