Precision Spectroscopy Reaffirms Gap Between Theory and Experiment

Aaron T. Bondy

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Precision Spectroscopy Reaffirms Gap Between Theory and Experiment

Aaron T. Bondy

Department of Physics and Astronomy, Drake University, Des Moines, IA

June 2, 2025• Physics 18, 110

New physics may explain discrepant values for the ionization energy of a metastable state of helium.

Figure caption — **Figure 1:** The newly determined ionization energy of a helium-3 atom in its 1s2s³S₁ triplet state is as distant from theoretical calculations as the previously determined ionization energy of a helium-4 atom in the same state. The x axes show the energy, measured in frequency units (MHz), relative to the nearest integral unit of the corresponding ionization energy.

In the search for new physics beyond the standard model of particle physics, a significant discrepancy between theory and experiment attracts attention, especially in a simple atomic system such as helium. Recently, evidence has appeared for a 9 $𝜎$ discrepancy in the ionization energy of the metastable triplet state of helium-4 (⁴He) [1, 2]. This stands out like a sore thumb in a field where theory and experiment are both highly accurate and normally in agreement. However, in assessing the validity of the discrepancy, there is always the possibility that something has been overlooked or miscalculated. Now Gloria Clausen and Frédéric Merkt of the Swiss Federal Institute of Technology (ETH) Zurich have released the results of their latest research [3] in a series of high-precision experiments [1, 4]. Their results (Fig. 1) indicate that the discrepancy remains, as does the possibility that the culprit is new physics.

To determine ionization energies with enough precision to confront theory, researchers measure the excitation energies of a series of Rydberg states. These are states whose principal quantum number n is about 20 or higher. An atom’s ionization energy is the energy required to remove an electron, that is, to reach a Rydberg state with n equal to infinity. Its value can be determined to high precision by robust extrapolation from the excitation energies of lower Rydberg states. In previous experiments, Clausen, Merkt, and their collaborators determined the absolute ionization energies of the metastable 1s2s³S₁ (triplet) and 1s2s¹S₀ (singlet) states of ⁴He with impressive uncertainties of $\pm$ 60 kHz (5.2 ppt) [1] and $\pm$ 32 kHz (33 ppt), respectively [4].

Clausen and Merkt’s latest experiment (Fig. 2) followed the same approach, but this time they applied it to the metastable triplet state of helium-3 (³He) [3]. Using a tunable ultraviolet laser, they excited the state to a Rydberg state with n in the range 27–55. They determined the excitation energies by measuring the minimum voltage needed to ionize the ³He atoms. Unlike ⁴He, ³He has nonzero nuclear spin, which introduces additional complications. Nevertheless, the results indicate that the difference in charge radii between the helion and the alpha particle (the nuclei of ³He and ⁴He, respectively) is consistent with all previous experiments that measured atomic structure properties—except, importantly, the ionization energy. The charge radii can be extracted from the measured isotope shift—provided the atomic structure contributions can be accurately calculated and subtracted. However, that route is blocked by the 9 $𝜎$ discrepancy between theory and experiment.

To reconcile with the experimental values, correspondingly accurate theoretical values are needed for both the n = 2 metastable triplet states and the high-lying Rydberg states. High-precision variational methods are used to construct explicitly correlated two-electron wave functions to solve the quantum-mechanical three-body problem with a point nucleus. The resulting wave functions are essentially exact for all practical purposes in the nonrelativistic limit. Energies accurate to 20 or more significant figures are readily obtained. The wave functions provide a solid foundation on which to build relativistic and quantum electrodynamic corrections, expressed as a power series in the fine-structure constant $𝛼$ , along with nuclear-size effects. In a monumental series of calculations, Krzysztof Pachucki of the University of Warsaw, Poland, and his co-workers evaluated all these terms to order $𝛼$ ⁴ Ry together with estimates of higher-order terms [2, 5]. Their calculation of the ionization energy of metastable triplet ionization energy in ³He disagrees with Clausen and Merkt’s experimental value by 482 $\pm$ 53 kHz (9 $𝜎$ ). The calculations included such effects as electron self-energy and vacuum polarization, which are well known from the spectrum of hydrogen.

Do theory and experiment also disagree for the ionization energies of the Rydberg states at the upper end of the measured excitation transition? The high-precision calculations that work well for the low-lying states typically lose accuracy with increasing n and are impractical for n as high as 24. My collaborators and I recently solved this problem by introducing three distinct distance scales for the inner and outer electrons [6]. As in the case for the lower-lying triplet state, the resulting wave functions provide a solid foundation on which to build the relativistic and quantum electrodynamic (QED) corrections. However, there is one important difference. All the relativistic and QED corrections decrease approximately in proportion to 1/n³ and are suppressed, for example, by a factor of 6 × 10^–4 in the n = 24 Rydberg state compared to the n = 2 state. A comparison of theory and experiment for the high-lying Rydberg states with n = 24 and higher should therefore be free of any QED inaccuracies that could be causing a discrepancy for the n= 2 triplet state. In fact, theory and experiment are in excellent agreement for the ionization energy of the n = 24 Rydberg state of ⁴He. Corresponding calculations for ³He yield similar agreement between theory and experiment [7]. Our findings therefore lay bare the 9 $𝜎$ discrepancy for the n= 2 triplet state and, we believe, boost confidence in the experimentally derived values.

The path to a final resolution of the discrepancy is unclear. The disagreement is present only on the triplet side [1, 3] of the spectrum, and not on the singlet side [4]. Assuming that all terms have been correctly determined through a long and complicated calculation [2, 5], and the experimental uncertainties have been correctly assessed, a possible implication is that there is an additional spin-dependent interaction that affects only the triplet part of the spectrum. Spin-dependent interactions are a hot topic in, among other things, spectroscopy-based searches for new physics at the low-energy frontier [8]. Additional work is clearly called for on both the theoretical and experimental sides.

References

G. Clausen et al., “Metrology in a two-electron atom: The ionization energy of metastable triplet helium 2³S₁,” Phys. Rev. A 111, 012817 (2025).
V. Patkóš et al., “Complete $𝛼^{7}$ m Lamb shift of helium triplet states,” Phys. Rev. A 103, 042809 (2021).
G. Clausen and F. Merkt, “Ionization energy of metastable ³He (2 ³S₁) and the alpha- and helion-particle charge-radius difference from precision spectroscopy of the np Rydberg series,” Phys. Rev. Lett. 134, 223001 (2025).
G. Clausen et al., “Ionization energy of the metastable 2 ¹S₀ state of ⁴He from Rydberg-series extrapolation,” Phys. Rev. Lett. 127, 093001 (2021).
K. Pachucki et al., “Testing fundamental interactions on the helium atom,” Phys. Rev. A 95, 062510 (2017).
A. T. Bondy et al., “Theory for the Rydberg states of helium: Comparison with experiment for the 1s24p¹P₁ state (n = 24),” Phys. Rev. A No. 111, L010803.
A. T. Bondy and G. W. F. Drake, (unpublished).
L. Cong et al., “Spin-dependent exotic interactions,” Rev. Mod. Phys. (to be published) arXiv:2408.15691.

About the Author

Aaron Bondy is a theoretical physicist at Drake University in Iowa, where he works as a postdoctoral researcher in Klaus Bartschat’s group. He earned his PhD from the University of Windsor in Canada under the supervision of Gordon Drake. His current work focuses on laser–atom interactions, including at ultrafast (attosecond) timescales, and on electron–atom collisions. He is also interested in atomic structure, having carried out high‑precision calculations that have been compared with experiments to stringently test QED—and to potentially reveal new physics beyond the standard model of particle physics.