Letter
Open Access

Toward resolving the discrepancy in helium-3 and helium-4 nuclear charge radii

Xiao-Qiu Qi ^1,2, Pei-Pei Zhang², Zong-Chao Yan^3,2, Li-Yan Tang², Ai-Xi Chen¹, Ting-Yun Shi^2,*, and Zhen-Xiang Zhong ^4,2,†

¹Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
²State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
³Department of Physics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
⁴Center for Theoretical Physics, School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China

^*Contact author: tyshi@wipm.ac.cn
^†Contact author: zxzhong@hainanu.edu.cn

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Phys. Rev. Research 7, L022020 – Published 21 April, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L022020

Abstract

The discrepancy in the squared nuclear charge radius difference, $Δ R^{2}$ , between ${}^{3}{He}$ and ${}^{4}{He}$ , as determined from electronic and muonic atom energy levels, presents an unresolved puzzle. This paper shows that accounting for off-diagonal hyperfine mixing effects can substantially reduce this discrepancy. We find that hyperfine mixing with the $n^{3} S$ and $n^{1} S$ states ( $n > 2$ ) in ${}^{3}{He}$ introduces a correction of $- 1.37$ kHz to the isotope shift of the $2^{1} S - 2^{3} S$ transition, a factor of seven times larger than the current uncertainty. This correction modifies $Δ R^{2}$ by $- 0.0064 {fm}^{2}$ , shifting it from $1.0757 (15) {fm}^{2}$ to $1.0693 (15) {fm}^{2}$ , as initially reported by Werf et al. [arXiv:2306.02333]. This brings $Δ R^{2}$ closer to the value of $1.0636 (31) {fm}^{2}$ obtained from muonic helium $μ {He}^{+}$ by Schuhmann et al. [arXiv:2305.11679], narrowing the existing discrepancy from $3.6 σ$ to $1.7 σ$ . The adjusted value $Δ R^{2}$ also agrees well with the result of 1.069(3) ${fm}^{2}$ derived from the helium $2^{3} S - 2^{3} P$ transitions, as analyzed by Patkóš et al. [Phys. Rev. A 94, 052508 (2016)]. Our results provide key insights for resolving the discrepancy in $Δ R^{2}$ .

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References (49)

H. Gao and M. Vanderhaeghen, The proton charge radius, Rev. Mod. Phys. 94, 015002 (2022).
S. Scheidegger and F. Merkt, Precision-spectroscopic determination of the binding energy of a two-body quantum system: The hydrogen atom and the proton-size puzzle, Phys. Rev. Lett. 132, 113001 (2024).
A. D. Brandt, S. F. Cooper, C. Rasor, Z. Burkley, A. Matveev, and D. C. Yost, Measurement of the $2 S_{1 / 2} - 8 D_{5 / 2}$ transition in hydrogen, Phys. Rev. Lett. 128, 023001 (2022).
R. Pohl, F. Nez, L. M. P. Fernandes, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, M. Diepold, A. Giesen, A. L. Gouvea, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, P. Knowles, F. Kottmann, E.-O. L. Bigot, Y.-W. Liu et al., Laser spectroscopy of muonic deuterium, Science 353, 669 (2016).
Y. van der Werf, K. Steinebach, R. Jannin, H. Bethlem, and K. Eikema, The alpha and helion particle charge radius difference from spectroscopy of quantum-degenerate helium, arXiv:2306.02333.
R. J. Rengelink, Y. van der Werf, R. P. M. J. W. Notermans, R. Jannin, K. S. E. Eikema, M. D. Hoogerl, and W. Vassen, Precision spectroscopy of helium in a magic wavelength optical dipole trap, Nat. Phys. 14, 1132 (2018).
K. Schuhmann, L. M. P. Fernandes, F. Nez, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, T.-L. Chen, D. S. Covita, A. J. Dax, M. Diepold, B. Franke, S. Galtier, A. L. Gouvea, J. Götzfried, T. Graf, T. W. HÄnsch, M. Hildebrandt, P. Indelicato, L. Julien et al., The helion charge radius from laser spectroscopy of muonic helium-3 ions, arXiv:2305.11679.
J. J. Krauth, K. Schuhmann, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, T.-L. Chen, D. S. Covita, A. J. Dax, M. Diepold, L. M. P. Fernandes, B. Franke, S. Galtier, A. L. Gouvea, J. Götzfried, T. Graf, T. W. Hänsch, J. Hartmann, M. Hildebrandt, P. Indelicato et al., Measuring the $α$ -particle charge radius with muonic helium-4 ions, Nature (London) 589, 527 (2021).
S. S. Li Muli, T. R. Richardson, and S. Bacca, Revisiting the helium isotope-shift puzzle with improved uncertainties from nuclear structure corrections, Phys. Rev. Lett. 134, 032502 (2025).
P. C. Pastor, G. Giusfredi, P. D. Natale, G. Hagel, C. de Mauro, and M. Inguscio, Absolute frequency measurements of the $2^{3} S_{1} \to 2^{3} P_{0, 1, 2}$ atomic helium transitions around 1083 nm, Phys. Rev. Lett. 92, 023001 (2004).
P. C. Pastor, L. Consolino, G. Giusfredi, P. De Natale, M. Inguscio, V. A. Yerokhin, and K. Pachucki, Frequency metrology of helium around 1083 nm and determination of the nuclear charge radius, Phys. Rev. Lett. 108, 143001 (2012).
X.-Q. Qi, P.-P. Zhang, Z.-C. Yan, G. W. F. Drake, Z.-X. Zhong, T.-Y. Shi, S.-L. Chen, Y. Huang, H. Guan, and K.-L. Gao, Precision calculation of hyperfine structure and the Zemach radii of ${}^{6, 7}{Li}^{+}$ ions, Phys. Rev. Lett. 125, 183002 (2020).
H. Guan, S. Chen, X.-Q. Qi, S. Liang, W. Sun, P. Zhou, Y. Huang, P.-P. Zhang, Z.-X. Zhong, Z.-C. Yan, G. W. F. Drake, T.-Y. Shi, and K. Gao, Probing atomic and nuclear properties with precision spectroscopy of fine and hyperfine structures in the $^{7} {Li}^{+}$ ion, Phys. Rev. A 102, 030801(R) (2020).
X.-Q. Qi, P.-P. Zhang, Z.-C. Yan, T.-Y. Shi, G. W. F. Drake, A.-X. Chen, and Z.-X. Zhong, Theoretical hyperfine splittings of $^{7, 9} {Be}^{2 +}$ ions for future studies of nuclear properties, Phys. Rev. A 107, L010802 (2023).
W. Sun, P.-P. Zhang, P.-P. Zhou, S.-L. Chen, Z.-Q. Zhou, Y. Huang, X.-Q. Qi, Z.-C. Yan, T.-Y. Shi, G. W. F. Drake, Z.-X. Zhong, H. Guan, and K.-L. Gao, Measurement of hyperfine structure and the Zemach radius in ${}^{6}{Li}^{+}$ using optical Ramsey technique, Phys. Rev. Lett. 131, 103002 (2023).
Z.-T. Lu, P. Mueller, G. W. F. Drake, W. Nörtershäuser, S. C. Pieper, and Z.-C. Yan, Colloquium: Laser probing of neutron-rich nuclei in light atoms, Rev. Mod. Phys. 85, 1383 (2013).
R. van Rooij, J. S. Borbely, J. Simonet, M. D. Hoogerland, K. S. E. Eikema, R. A. Rozendaal, and W. Vassen, Frequency metrology in quantum degenerate helium: Direct measurement of the $2^{3} S_{1} \to 2^{1} S_{0}$ transition, Science 333, 196 (2011).
G. Clausen, P. Jansen, S. Scheidegger, J. A. Agner, H. Schmutz, and F. Merkt, Ionization energy of the metastable $2^{1} S_{0}$ state of ${}^{4}{He}$ from Rydberg-series extrapolation, Phys. Rev. Lett. 127, 093001 (2021).
G. Clausen, S. Scheidegger, J. A. Agner, H. Schmutz, and F. Merkt, Imaging-assisted single-photon doppler-free laser spectroscopy and the ionization energy of metastable triplet helium, Phys. Rev. Lett. 131, 103001 (2023).
Z.-C. Yan and G. W. F. Drake, High precision calculation of fine structure splittings in helium and He-like ions, Phys. Rev. Lett. 74, 4791 (1995).
K. Pachucki, $α^{4} R$ corrections to singlet states of helium, Phys. Rev. A 74, 022512 (2006).
K. Pachucki, Helium energy levels including $m α^{6}$ corrections, Phys. Rev. A 74, 062510 (2006).
V. A. Yerokhin and K. Pachucki, Theoretical energies of low-lying states of light helium-like ions, Phys. Rev. A 81, 022507 (2010).
V. Patkóš, V. A. Yerokhin, and K. Pachucki, Complete quantum electrodynamic $α^{6} m$ correction to energy levels of light atoms, Phys. Rev. A 100, 042510 (2019).
V. Patkóš, V. A. Yerokhin, and K. Pachucki, Complete $α^{7} m$ Lamb shift of helium triplet states, Phys. Rev. A 103, 042809 (2021).
V. Patkóš, V. A. Yerokhin, and K. Pachucki, Radiative $α^{7} m$ QED contribution to the helium Lamb shift, Phys. Rev. A 103, 012803 (2021).
V. Patkóš, V. A. Yerokhin, and K. Pachucki, Higher-order QED corrections to the hyperfine splitting in ${}^{3}{He}$ , Phys. Rev. Lett. 131, 183001 (2023).
A. T. Bondy, G. W. F. Drake, C. McLeod, E. M. R. Petrimoulx, X.-Q. Qi, and Z.-X. Zhong, Theory for the Rydberg states of helium: Comparison with experiment for the $1 s 24 p^{1} p_{1}$ state ( $n = 24$ ), Phys. Rev. A 111, L010803 (2025).
K. Pachucki and V. A. Yerokhin, Theory of the helium isotope shift, J. Phys. Chem. Ref. Data 44, 031206 (2015).
V. Patkóš, V. A. Yerokhin, and K. Pachucki, Higher-order recoil corrections for triplet states of the helium atom, Phys. Rev. A 94, 052508 (2016).
V. Patkóš, V. A. Yerokhin, and K. Pachucki, Higher-order recoil corrections for singlet states of the helium atom, Phys. Rev. A 95, 012508 (2017).
D. Shiner, R. Dixson, and V. Vedantham, Three-nucleon charge radius: A precise laser determination using ${}^{3}{He}$ , Phys. Rev. Lett. 74, 3553 (1995).
X. Zheng, Y. R. Sun, J.-J. Chen, W. Jiang, K. Pachucki, and S.-M. Hu, Measurement of the frequency of the $2 {}^{3}S - 2 {}^{3}P$ transition of ${}^{4}{He}$ , Phys. Rev. Lett. 119, 263002 (2017).
Y.-J. Huang, Y.-C. Guan, J.-L. Peng, J.-T. Shy, and L.-B. Wang, Precision laser spectroscopy of the $2^{1} S_{0} - 3^{1} D_{2}$ two-photon transition in ${}^{3}{He}$ , Phys. Rev. A 101, 062507 (2020).
C. R. Ottermann, G. Köbschall, K. Maurer, K. Röhrich, C. Schmitt, and V. Walther, Elastic electron scattering from ${}^{3}{He}$ and ${}^{4}{He}$ , Nucl. Phys. A 436, 688 (1985).
I. Sick, Precise root-mean-square radius of ${}^{4}{He}$ , Phys. Rev. C 77, 041302(R) (2008).
I. Sick, Zemach moments of ${}^{3}{He}$ and ${}^{4}{He}$ , Phys. Rev. C 90, 064002 (2014).
X.-Q. Qi, P.-P. Zhang, Z.-C. Yan, L.-Y. Tang, A.-X. Chen, T.-Y. Shi, and Z.-X. Zhong, Revised $^{3}$ He nuclear charge radius due to electronic hyperfine mixing, arXiv:2409.09279.
K. Pachucki, V. Patkóš, and V. A. Yerokhin, Second-order hyperfine correction to H, D, and ${}^{3}{He}$ energy levels, Phys. Rev. A 110, 062806 (2024).
G. Clausen, K. Gamlin, J. A. Agner, H. Schmutz, and F. Merkt, Metrology in a two-electron atom: The ionization energy of metastable triplet helium $2^{3} S_{1}$ , Phys. Rev. A 111, 012817 (2025).
R. M. Potvliege, Spectroscopy of light atoms and bounds on physics beyond the standard model, arXiv:2412.16315.
K. Pachucki, V. Patkóš, and V. A. Yerokhin, Recoil nuclear size corrections in hydrogenic systems, arXiv:2502.09455.
A. Wienczek, K. Pachucki, M. Puchalski, V. Patkóš, and V. A. Yerokhin, Quantum-electrodynamic corrections to the $1 s 3 d$ states of the helium atom, Phys. Rev. A 99, 052505 (2019).
See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.L022020 for details of formulas and numerical results.
K. Pachucki, V. A. Yerokhin, and P. Cancio Pastor, Quantum electrodynamic calculation of the hyperfine structure of ${}^{3}{He}$ , Phys. Rev. A 85, 042517 (2012).
D. C. Morton, Q. X. Wu, and G. W. F. Drake, Energy levels for the stable isotopes of atomic helium ( ${}^{4}{He}$ I and ${}^{3}{He}$ I), Can. J. Phys. 84, 83 (2006).
E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, Codata recommended values of the fundamental physical constants: 2018, Rev. Mod. Phys. 93, 025010 (2021).
J.-L. Wen, J.-D. Tang, Y.-N. Lv, Y. R. Sun, C.-L. Zou, J.-F. Dong, and S.-M. Hu, Postselection shifts the transition frequency of helium in an atomic beam, Sci. Adv. 11, eadu9796 (2025).
K. Pachucki, V. Lensky, F. Hagelstein, S. S. Li Muli, S. Bacca, and R. Pohl, Comprehensive theory of the Lamb shift in light muonic atoms, Rev. Mod. Phys. 96, 015001 (2024).

Phys. Rev. Research 7, L022020– Published 21 April, 2025

Vol. 7, Iss. 2 — April - June 2025

Received 31 August 2024
Accepted 25 March 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L022020

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Toward resolving the discrepancy in helium-3 and helium-4 nuclear charge radii

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