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Zemach radii and nuclear structure effects in hyperfine splitting of Lithium

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Nuclear polarizability, not nuclear density shape, explains the 6Li Zemach-radius discrepancy.

desk verdict First ab initio 6,7Li Zemach radii and a neat SU(4) ratio rule, but the headline 6Li polarizability dominance rests on a closure energy fitted to the data it explains. read the letter →

arxiv 2509.01303 v1 pith:G3VFMW5F submitted 2025-09-01 nucl-th nucl-exphysics.atom-ph

Yilong Yang , Evgeny Epelbaum , Chen Ji , Pengwei Zhao This is my paper
classification nucl-thnucl-exphysics.atom-ph
keywords Zemachradiushyperfinesplittingnuclearpolarizabilitylithiumisotopeschiraleffectivefieldtheoryneural-networkvariationalMonteCarloSU(4)symmetryclosureapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to explain why the effective Zemach radius of 6Li extracted from hyperfine splitting is about 40% smaller than the elastic Zemach radius built from measured charge and magnetic radii, even though the two agree in 7Li. It argues that virtual nuclear excitations — nuclear polarizability — are negligible in 7Li but dominate in 6Li, and that this same mechanism accounts for similar discrepancies in 2H and 3He. To make the case, the authors compute both radii from first principles using a new neural-network variational Monte Carlo method with chiral nuclear forces, and they derive a compact closure-approximation expression that turns the polarizability correction into a ground-state expectation value. If the argument holds, isotope-dependent nuclear corrections to hyperfine splitting in light atoms are no longer mysterious: polarizability is the dominant and largely predictable source.

What carries the argument

The load-bearing object is the closure-approximation polarizability operator δrpol(ωbar), which rewrites the two-photon-exchange current using current conservation and replaces every virtual excitation energy ωN with a single state-independent average ωbar, turning δrpol into a ground-state expectation value built from single-nucleon charge and magnetic densities. On top of this, approximate SU(4) symmetry reduces the correction to a counting rule: only unpaired nucleons contribute, so δrpol vanishes for even-Z, odd-N nuclei, scales as 1/Z for odd-Z nuclei, and is enhanced by µp/(µp+µn)≈3 in odd-odd nuclei. The neural-network variational Monte Carlo wave function provides the ground states w

What would settle it

Perform an explicit sum over excited nuclear states, rather than the closure average, for 6Li and 7Li using the same chiral Hamiltonians and currents, and compare the resulting polarizability correction with the fitted-closure values (ωbar ≈ 23 and 29 MeV). If the explicit correction is much smaller, the claim that polarizability dominates in 6Li would fail.

Watch

Extended reading notes

Core claim

The central claim is that the 6Li mismatch between reffZ (from hyperfine splitting) and rZ (from elastic nuclear densities) is caused by nuclear polarizability, not by the Gaussian-shape assumption used to convert charge and magnetic rms radii into rZ. Using ab initio wave functions, the paper computes rZ directly from realistic densities and shows it agrees with the Gaussian-based elastic radius, ruling out the density-shape explanation. It then evaluates the polarizability correction δrpol in the closure approximation and demonstrates that it is large in 6Li and small in 7Li; adjusting a single closure energy reproduces the experimental effective Zemach radii. The paper further claims that

Load-bearing premise

The calculation replaces all the many possible excitation energies of the nucleus with one single average number, and that number is chosen to reproduce the very hyperfine measurements being explained.

Editorial extensions

If this is right

  • The discrepancy between effective and elastic Zemach radii in 6Li is resolved as a nuclear-polarizability effect, so the Gaussian density assumption in earlier extractions is not the source of the anomaly.
  • Hyperfine-structure measurements in light atoms can be used as probes of nuclear excitation spectra, with polarizability corrections estimated from ground-state wave functions alone through the closure formula.
  • The SU(4) ratio rule gives a predictive grid: among light isotopes, odd-odd nuclei should show roughly three times larger polarizability corrections than odd-even ones, with the correction decreasing as proton number grows.
  • The same unified framework reproduces the effective Zemach radii of 2H and 3He along with 6Li and 7Li, connecting previously separate anomalies in deuterium and helium to the lithium puzzle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the fitted closure energies (23 and 29 MeV for 6Li and 7Li) sit near known resonances, a testable extension is that ωbar tracks the energy-weighted centroid of the magnetic-dipole excitation spectrum, allowing it to be fixed by photoabsorption or inelastic-scattering data instead of by the hyperfine measurements themselves.
  • The residual 6% deviation in 3He, where the calculated reffZ exceeds the measured value while δrpol is always negative in closure, suggests a positive missing term of comparable size; computing two-body currents and relativistic corrections would provide a sharp test of the closure approximation at the few-percent level.
  • If the SU(4) ratio rule extends to beryllium, the effective Zemach radius of 9Be (odd-odd) should deviate from its elastic value in the opposite sense relative to 7Be (odd-even), a signature that precision Be+ hyperfine spectroscopy could look for.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript presents ab initio variational Monte Carlo calculations of elastic and effective Zemach radii for 6,7Li using a novel neural-network wave function, together with a closure-approximation treatment of nuclear polarizability contributions to hyperfine splitting. The elastic radii from realistic densities are found to closely match the values obtained from Gaussian-density assumptions, ruling out density-shape effects as the source of the reffZ/rZ discrepancy in 6Li. The central claim is that nuclear polarizability, computed via a state-independent closure energy ωbar, dominates in 6Li and accounts for the observed difference between effective and elastic Zemach radii, while being negligible in 7Li; the same mechanism is argued to explain trends in 2H and 3He, with an SU(4) enhancement factor for odd-odd nuclei. The manuscript includes validation of elastic radii against experiments and a cross-check of the closure energy for 2H against pionless EFT.

Significance. If the central claim holds, this is a notable step: it would establish nuclear polarizability as the dominant source of the long-standing 6Li effective-vs-elastic Zemach radius discrepancy, and it provides a concrete SU(4)-based pattern for light nuclei that is testable in other systems. The elastic Zemach radii themselves are a solid result: the VMC-NN densities are shown to reproduce experimental radii, and the direct comparison with Gaussian shapes is a useful resolution of a debated point. The closure-approximation expression relates polarizability to ground-state observables, and the 2H cross-check against pionless EFT gives some independent support for the method. The SU(4) ratio analysis is largely insensitive to the chosen ωbar and provides a robust qualitative explanation of the isotope trend. However, the absolute polarizability contribution in 6Li/7Li is not predicted, because the closure energies are fitted to the same HFS data the claim is supposed to explain; this undermines the headline statement and needs to be addressed before the paper can be accepted.

major comments (3)
  1. [Eqs. (9)–(11) and Fig. 4] The absolute polarizability contribution in 6Li and 7Li is not predicted: the closure energies ωbar=23 MeV and 29 MeV are adjusted to reproduce the experimental reffZ values. The agreement in Fig. 4 is therefore partly built in and cannot by itself establish that nuclear polarizability dominates the reffZ−rZ discrepancy in 6Li. The 2H cross-check (ωbar=14.4(4) MeV from pionless EFT versus 16 MeV from HFS) is valuable but does not validate the closure energy for A=6,7. Please provide an independent determination of ωbar (e.g., from the computed excitation spectrum or a sum-rule/energy-weighting calculation) or a quantified closure-error band, and state which conclusions are independent of the fitting.
  2. [Fig. 4, lower panel (3He)] The acknowledged ~6% deviation for 3He, which cannot be removed by varying ωbar because δrpol is always negative while experimental reffZ>rZ, signals missing contributions (the authors suggest two-body currents and relativistic corrections). Since the same formalism is used for lithium, the potential size of these omitted effects in 6Li is unquantified. This does not invalidate the SU(4) ratio trend, but it undercuts the absolute claim that polarizability, rather than omitted currents/relativistic effects, is responsible for the 6Li discrepancy. Please quantify or bound the analogous residual for 6Li, or temper the absolute claim accordingly.
  3. [Eq. (12) and Fig. 3] The SU(4)-limit formula predicts δrpol=0 for even-Z odd-N nuclei, which would make the 3He polarizability contribution zero, yet the manuscript treats 3He with a nonzero fitted ωbar=25 MeV and describes the observed reffZ−rZ discrepancy for 3He as smaller than for 6Li/2H but not zero. Please clarify how the full VMC calculation produces a nonzero 3He polarizability and how Fig. 3 represents 3He relative to the SU(4) prediction. As written, Eq. (12) appears inconsistent with the 3He discussion.
minor comments (4)
  1. [Eq. (9)] The text 'using (H−E0)/ωN ≡ 1' is informal: this is an identity only when acting on excited states with excitation energy ωN. Please reformulate to avoid implying an operator identity in the ground-state subspace.
  2. [Eqs. (10)–(11)] The derivation of the coordinate-space kernel f(ω,r) and the impulse-approximation current operators is relegated to the Supplemental Material. For a Letter, at least the definitions of f and the current operator conventions should be stated in the main text or in an appendix.
  3. [Fig. 3, right panel] The 'experimental' δrpol values are obtained by subtracting recoil and single-nucleon TPE contributions from reffZ−rZ, but the magnitudes and uncertainties of those subtraction terms are not shown. Please display them or list them so the error bars on the experimental δrpol can be assessed.
  4. [p. 4, closure-energy justification] The statement that the chosen ωbar values 'lie close to known resonances' is too vague. Provide a quantitative comparison (e.g., excitation-energy centroids) and explain why proximity to a resonance validates a state-independent closure approximation.

Circularity Check

1 steps flagged · score 6.0 of 10

Closure-energy fit to the same HFS data makes the absolute polarizability-dominance claim for 6Li non-predictive; an independent ωbar is required.

  1. fitted input called prediction [Nuclear polarizability effects section, Fig. 4 discussion (page 4); Summary (page 5)]
    "The reffZ values from HFS are well reproduced by adjusting the closure energies, ¯ω = 23 MeV for 6Li and ¯ω = 29 MeV for 7Li. ... These results validate the closure approximation with a finite closure energy and motivate future efforts to explicitly take into account nuclear excitations, which are technically challenging, in order to further refine the nuclear polarizability contributions in 6,7Li."

    The nuclear polarizability contribution δrpol(¯ω) in Eqs. (10)–(11) is inversely proportional to the closure energy ¯ω. Choosing ¯ω to reproduce the experimental reffZ (two data points, two parameters) makes the calculated reffZ match the target by construction. The paper then presents this forced agreement as a 'validation' of the closure approximation and as evidence for the central claim that polarizability dominates in 6Li. The absolute magnitude of δrpol is thus a fitted input, not an ab initio prediction. The elastic rZ and the SU(4) ratio pattern are independent of this fit, but the quantitative dominance claim reduces to the fitted closure energies.

full rationale

This paper has a genuine ab initio component: the elastic Zemach radii rZ are computed from VMC-NN densities (Eq. 4) and agree with experiment; the SU(4) ratio predictions (Eq. 12, Fig. 3) are independent of the fitted closure energy and match the observed reffZ − rZ trend across 2H, 3He, 6Li, and 7Li. However, the absolute magnitude of the nuclear polarizability effect δrpol—the centerpiece of the claim that polarizability dominates in 6Li—is not predicted from first principles. It is computed in the closure approximation (Eqs. 10–11) with a state-independent excitation energy ¯ω that is adjusted to reproduce the experimental reffZ (¯ω = 23 MeV for 6Li, 29 MeV for 7Li). Because δrpol scales parametrically with 1/¯ω, choosing ¯ω to fit the two HFS data points forces the calculated reffZ to agree; the 'validation' of the closure approximation in Fig. 4 is therefore tautological for these nuclei. The only independent anchor is 2H, where pionless EFT gives ¯ω = 14.4(4) MeV close to the fitted 16 MeV; no such independent determination is provided for 6Li/7Li. The 3He residual (~6%, not removable by varying ¯ω) shows the framework has systematic omissions (two-body currents, relativistic corrections) whose size for 6Li is unquantified. Thus the central quantitative claim is partially circular: the SU(4) pattern is an independent prediction, but the absolute dominance of polarizability in 6Li reduces to a two-parameter fit. Score 6.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central numerical results rest on four closure energies fitted per nucleus, plus the closure approximation, impulse approximation, SU(4) symmetry, and the accuracy of VMC-NN densities. No new particles, forces, or conserved quantities are introduced. The closure energies are the clearest data-fitted inputs: without them, the calculated effective Zemach radii do not reproduce the HFS values.

free parameters (4)
  • closure energy omega_bar for 6Li = 23 MeV
    Adjusted to reproduce the HFS-derived effective Zemach radius of 6Li; value is said to lie near known resonances but is not independently calculated.
  • closure energy omega_bar for 7Li = 29 MeV
    Adjusted to reproduce the HFS-derived effective Zemach radius of 7Li.
  • closure energy omega_bar for 2H = 16 MeV (ab initio alternative 14.4(4) MeV from pionless EFT)
    Adjusted to reproduce the experimental effective Zemach radius; an independent pionless-EFT determination is also given.
  • closure energy omega_bar for 3He = 25 MeV
    Adjusted to reproduce the experimental effective Zemach radius; a 6 percent residual remains and is attributed to omitted two-body currents.
assumptions (4)
  • ad hoc to paper Closure approximation: all nuclear excitation energies omega_N in the two-photon-exchange sum are replaced by one state-independent average omega_bar.
    Used to turn Eq. (8) into the ground-state matrix element Eq. (10); omega_bar is then chosen per nucleus to reproduce experimental HFS, so the approximation is not independently justified for 6Li and 7Li.
  • domain assumption Current conservation [rho(r), H] = -i div j(r) and impulse-approximation one-body charge/magnetic density operators.
    Needed to rewrite Eq. (8) as Eq. (9) and to obtain Eq. (10); two-body meson-exchange currents are omitted, which the authors invoke to explain the 3He residual.
  • domain assumption Approximate SU(4) spin-isospin symmetry of nuclear forces and saturation of paired nucleon spins.
    Underlies Eq. (12) and the factor mu_p/(mu_p+mu_n) of about 3; the quantitative 'universal' ratios depend on this approximate symmetry.
  • domain assumption VMC-NN ground-state wave functions are accurate enough for charge and magnetic density distributions.
    Figure 1 validates energies against GFMC and AFDMC, but densities are not directly benchmarked; the entire calculation of rZ and delta_rpol starts from these wave functions.

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Cite this review

Pith. "Pith review of Zemach radii and nuclear structure effects in hyperfine splitting of Lithium." pith.science (2026). https://pith.science/paper/G3VFMW5F

@misc{pith2026250901303,
  author       = {Pith},
  title        = {Pith review of: Zemach radii and nuclear structure effects in hyperfine splitting of Lithium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3VFMW5F}},
  note         = {Machine review of arXiv:2509.01303}
}
abstract

Nuclear structure effects are essential for describing hyperfine splittings from high-precision atomic spectroscopy measurements. These effects are often parametrized by the effective or elastic Zemach radii, with their difference poorly understood. We solve the longstanding discrepancy between the effective and elastic Zemach radii in ${}^6$Li and ${}^7$Li by performing \emph{ab initio} nuclear structure calculations that take into account nuclear polarizability effects. Our results demonstrate that nuclear polarizability effects, negligible in ${}^7$Li, dominate in ${}^6$Li and explain the observed significant deviation between the effective and elastic Zemach radii. Furthermore, we show that the ratios between the nuclear polarizability contributions in different nuclei are universal in the limit of closure and SU(4) symmetry of nuclear forces. In particular, the nuclear polarizability contribution in an odd-odd nucleus is enhanced by a factor of $\mu_p/(\mu_p+\mu_n)\simeq 3$, with $\mu_{n,p}$ denoting the nucleon magnetic moments, compared to its odd-$A$ isotopes. The same mechanism also explains the Zemach radius deviations observed in ${}^2$H and ${}^3$He. These findings establish nuclear polarizability as the dominant source of isotope-dependent nuclear corrections to hyperfine splitting in light atoms.

Figures

Figures reproduced from arXiv: 2509.01303 by the authors.

Figure 1
Figure 1. FIG. 1. Ground-state energies of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The charge and magnetic density distribution of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left panel: Ratios of nuclear polarizability con [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the effective Zemach radii [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Reference graph

Works this paper leans on

72 extracted references · 69 canonical work pages · cited by 1 Pith paper

  1. [1]

    Yan and G

    Z.-C. Yan and G. W. F. Drake, High precision calcula- tion of fine structure splittings in helium and he-like ions, Phys. Rev. Lett. 74, 4791 (1995)

    work page 1995

  2. [2]

    ( 3), are parametrized by neural networks [37]

    and FLS in Eq. ( 3), are parametrized by neural networks [37]. The neural-network wave function is optimized via VMC calculations that minimize the energy expectation value using the stochastic reconfiguration method [31, 38]. We employ several nuclear Hamiltonians that in- clude two- and three-nucleon interactions: the phe- nomenological A V8′+UIX′ [39] a...

    work page

  3. [3]

    experimental

    as δrpol = 3i 16µZ ∫ d3rd3r′ ∫∑ N ̸=0 f (ωN , |r′ − r|) ωN × ⟨0| [∇ · j(r), ((r − r′) × j(r′))z] |0⟩. (9) We use the closure approximation that replaces ωN with a state-independent averaged value ¯ ω, summing up the excited states implicitly via ∫ ∑ N ̸=0 = ˆ 1 − |0⟩⟨0| in Eq. ( 9), and makes δrpol a ground-state observable with averaged intermediate-stat...

    work page

  4. [4]

    We demonstrate below that varying ¯ ω per nucleus resolves this deviation

    The SU(4) estimates, assuming a uniform ¯ ω value across all considered nuclei, deviate slightly from experiments. We demonstrate below that varying ¯ ω per nucleus resolves this deviation. The full VMC-NN calculations of reff Z for 6,7Li are de- picted in Fig. 4, in comparison with the values extracted from atomic HFS measurements [9–11, 50, 59]. The reff ...

    work page

  5. [5]

    For reff Z , closure energies of ¯ω = 16 MeV for 2H and ¯ω = 25 MeV for 3He are adopted

    The predicted rZ show excellent agreements with the values from electron-scattering experiments. For reff Z , closure energies of ¯ω = 16 MeV for 2H and ¯ω = 25 MeV for 3He are adopted. In the case of 2H, the HFS value is accurately reproduced, while for 3He, the calculated re- sult deviates slightly (by about 6%) from the experimen- tal value. This small ...

    work page

  6. [6]

    Pachucki, V

    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)

    2024

  7. [7]

    Yan and G

    Z.-C. Yan and G. W. F. Drake, Relativistic and QED Energies in Lithium, Phys. Rev. Lett. 81, 774 (1998)

    work page 1998

  8. [8]

    To go beyond 2H, cer- tain approximations have to be adopted to make numer- ical calculations tractable

    sums over all excited states in the TPE process, currently feasible only for the two-body nucleus 2H [21]. To go beyond 2H, cer- tain approximations have to be adopted to make numer- ical calculations tractable. The previous study for A ≤ 3 systems [23, 24] utilizes the Low-term formalism [47], which takes the limit ωN → 0 such that f (ωN , r) → r. Howeve...

    work page

Show all 72 references
  1. [9]

    Z.-T. Lu, P. Mueller, G. W. F. Drake, W. N¨ ortersh¨ auser, S. C. Pieper, and Z.-C. Yan, Colloquium: Laser probing of neutron-rich nuclei in light atoms, Rev. Mod. Phys. 85, 1383 (2013)

    2013

  2. [10]

    Pohl et al

    R. Pohl et al. (The CREMA Collaboration), Laser spec- troscopy of muonic deuterium, Science 353, 669 (2016) , https://www.science.org/doi/pdf/10.1126/science.aaf2468

    2016 doi

  3. [11]

    J. J. Krauth et al., Measuring the α-particle charge radius with muonic helium-4 ions, Nature 589, 527 (2021)

    2021

  4. [12]

    T. E. Chupp, P. Fierlinger, M. J. Ramsey- Musolf, and J. T. Singh, Electric dipole mo- ments of atoms, molecules, nuclei, and particles, Rev. Mod. Phys. 91, 015001 (2019)

    2019

  5. [13]

    A. C. Zemach, Proton structure and the hyperfine shift in hydrogen, Phys. Rev. 104, 1771 (1956)

    1956

  6. [14]

    Puchalski and K

    M. Puchalski and K. Pachucki, Ground State Hyper- fine Splitting in 6,7Li Atoms and the Nuclear Structure, Phys. Rev. Lett. 111, 243001 (2013)

    2013

  7. [15]

    Qi, P.-P

    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 Hy- perfine Structure and the Zemach Radii of 6,7Li + Ions, Phys. Rev. Lett. 125, 183002 (2020)

    2020

  8. [16]

    Sun, P.-P

    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 6Li+ Using Optical Ramsey Tech- nique, Phys. Rev. Lett. 131, 1030...

    2023

  9. [17]

    V. A. Yerokhin, Hyperfine structure of Li and Be +, Phys. Rev. A 78, 012513 (2008)

    2008

  10. [18]

    C. Ji, N. Nevo Dinur, S. Bacca, and N. Barnea, Nu- clear Polarization Corrections to the µ4He+ Lamb Shift, Phys. Rev. Lett. 111, 143402 (2013)

    2013

  11. [19]

    Hernandez, C

    O. Hernandez, C. Ji, S. Bacca, N. Nevo Dinur, and N. Barnea, Improved estimates of the nuclear structure corrections in µd, Phys. Lett. B 736, 344 (2014)

    2014

  12. [20]

    Nevo Dinur, C

    N. Nevo Dinur, C. Ji, S. Bacca, and N. Barnea, Nuclear structure corrections to the Lamb shift in µ3He+ and µ3H, Phys. Lett. B 755, 380 (2016)

    2016

  13. [21]

    C. Ji, S. Bacca, N. Barnea, O. J. Hernandez, and N. Nevo Dinur, Ab initio calculation of nuclear-structure corrections in muonic atoms, J. Phys. G 45, 093002 (2018)

    2018

  14. [22]

    Pachucki, Nuclear structure corrections in muonic deuterium, Phys

    K. Pachucki, Nuclear structure corrections in muonic deuterium, Phys. Rev. Lett. 106, 193007 (2011)

    2011

  15. [23]

    S. S. Li Muli, T. R. Richardson, and S. Bacca, Revisiting the helium isotope-shift puzzle with im- proved uncertainties from nuclear structure corrections, Phys. Rev. Lett. 134, 032502 (2025)

    2025

  16. [24]

    I. B. Khriplovich and A. I. Milstein, Corrections to deu- terium hyperfine structure due to deuteron excitations, J. Exp. Theor. Phys. 98, 181 (2004)

    2004

  17. [25]

    Kalinowski, K

    M. Kalinowski, K. Pachucki, and V. A. Yerokhin, Nuclear-structure corrections to the hyperfine splitting in muonic deuterium, Phys. Rev. A 98, 062513 (2018)

    2018

  18. [26]

    C. Ji, X. Zhang, and L. Platter, Nuclear Structure Ef- fects on Hyperfine Splittings in Ordinary and Muonic Deuterium, Phys. Rev. Lett. 133, 042502 (2024)

    2024

  19. [27]

    Bonilla, T

    J. Bonilla, T. R. Richardson, S. Bacca, C. Ji, and L. Platter, Improved nuclear-structure corrections to the hyperfine sp litting of electronic (2025), arXiv:2508.18776 [nucl-th]

    2025 arXiv

  20. [28]

    Friar and G

    J. Friar and G. Payne, The nuclear physics of hyperfine structure in hydrogenic atoms, Phys. Lett. B 618, 68 (2005)

    2005

  21. [29]

    J. L. Friar and G. L. Payne, Nuclear corrections to hyperfine structure in light hydrogenic atoms, Phys. Rev. C 72, 014002 (2005)

    2005

  22. [30]

    Carlson, S

    J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Quantum Monte Carlo methods for nuclear physics, Rev. Mod. Phys. 87, 1067 (2015)

    2015

  23. [31]

    Hornik, M

    K. Hornik, M. Stinchcombe, and H. White, Multi- layer feedforward networks are universal approximators, Neural Networks 2, 359 (1989)

    1989

  24. [32]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017)

    2017

  25. [33]

    Adams, G

    C. Adams, G. Carleo, A. Lovato, and N. Rocco, Vari- ational Monte Carlo Calculations of A ≤ 4 Nuclei with an Artificial Neural-Network Correlator Ansatz, Phys. Rev. Lett. 127, 022502 (2021)

    2021

  26. [34]

    Yang and P

    Y. Yang and P. Zhao, A consistent description of the relativistic effects and three-body interactions in atomic nuclei, Phys. Lett. B 835, 137587 (2022)

    2022

  27. [35]

    Lovato, C

    A. Lovato, C. Adams, G. Carleo, and N. Rocco, Hidden-nucleons neural-network quantum states for the nuclear many-body problem, Phys. Rev. Research 4, 043178 (2022)

    2022

  28. [36]

    Y. L. Yang and P. W. Zhao, Deep-neural-network ap- proach to solving the ab initio nuclear structure problem, Phys. Rev. C 107, 034320 (2023)

    2023

  29. [37]

    B. Fore, J. M. Kim, G. Carleo, M. Hjorth- Jensen, A. Lovato, and M. Piarulli, Dilute neu- tron star matter from neural-network quantum states, Phys. Rev. Res. 5, 033062 (2023)

    2023

  30. [38]

    Gnech, B

    A. Gnech, B. Fore, A. J. Tropiano, and A. Lo- vato, Distilling the essential elements of nu- clear binding via neural-network quantum states, Phys. Rev. Lett. 133, 142501 (2024)

    2024

  31. [39]

    Yang and P.-W

    Y.-L. Yang and P.-W. Zhao, Reconciling light nuclei and nuclear matter: Relativistic ab initio calculations, Chinese Phys. Lett. 42, 051201 (2025)

    2025

  32. [40]

    Y. Yang, E. Epelbaum, J. Meng, L. Meng, and P. Zhao, Chiral symmetry and peripheral neutron- α scattering (2025), arXiv:2502.09961 [nucl-th]

    2025

  33. [41]

    P. Wen, A. Gezerlis, and J. W. Holt, Neural-network correlation functions for light nuclei wit h chiral two- (2025), arXiv:2505.11442 [nucl-th]

    2025 arXiv

  34. [42]

    See Supplemental Material at [URL] for details on VMC- NN calculations and the derivations on the coordinate- 7 space expression for the nuclear polarizability contribu- tions to HFS within the closure approximation

  35. [43]

    Sorella, Wave function optimization in the variational Monte Carlo method, Phys

    S. Sorella, Wave function optimization in the variational Monte Carlo method, Phys. Rev. B 71, 241103 (2005)

    2005

  36. [44]

    B. S. Pudliner, V. R. Pandharipande, J. Carl- son, S. C. Pieper, and R. B. Wiringa, Quantum Monte Carlo calculations of nuclei with A < 7, Phys. Rev. C 56, 1720 (1997)

    1997

  37. [45]

    Gezerlis, I

    A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi, K. Hebeler, A. Nogga, and A. Schwenk, Quantum Monte Carlo Calculations with Chiral Effective Field Theory In- teractions, Phys. Rev. Lett. 111, 032501 (2013)

    2013

  38. [46]

    J. E. Lynn, I. Tews, J. Carlson, S. Gandolfi, A. Gezerlis, K. E. Schmidt, and A. Schwenk, Chiral Three-Nucleon Interactions in Light Nuclei, Neutron- α Scattering, and Neutron Matter, Phys. Rev. Lett. 116, 062501 (2016)

    2016

  39. [47]

    J. E. Lynn, I. Tews, J. Carlson, S. Gandolfi, A. Gezerlis, K. E. Schmidt, and A. Schwenk, Quantum monte carlo calculations of light nuclei with local chiral two- and three-nucleon interactions, Phys. Rev. C 96, 054007 (2017)

    2017

  40. [48]

    R. B. Wiringa, S. C. Pieper, J. Carlson, and V. R. Pand- haripande, Quantum Monte Carlo calculations of A = 8 nuclei, Phys. Rev. C 62, 014001 (2000)

    2000

  41. [49]

    Q. N. Usmani, K. Anwar, and N. Abdullah, Further im- provement in the variational many-body wave functions for light nuclei, Phys. Rev. C 86, 034323 (2012)

    2012

  42. [50]

    Lonardoni, S

    D. Lonardoni, S. Gandolfi, J. E. Lynn, C. Petrie, J. Carlson, K. E. Schmidt, and A. Schwenk, Auxil- iary field diffusion Monte Carlo calculations of light and medium-mass nuclei with local chiral interactions, Phys. Rev. C 97, 044318 (2018)

    2018

  43. [51]

    Fermi, ¨ uber die magnetischen momente der atomk- erne, Zeitschrift f¨ ur Physik60, 320 (1930)

    E. Fermi, ¨ uber die magnetischen momente der atomk- erne, Zeitschrift f¨ ur Physik60, 320 (1930)

    1930

  44. [52]

    Low, On the Effects of Internal Nuclear Mo- tion on the Hyperfine Structure of Deuterium, Phys

    F. Low, On the Effects of Internal Nuclear Mo- tion on the Hyperfine Structure of Deuterium, Phys. Rev. 77, 361 (1950)

    1950

  45. [53]

    J. J. Kelly, Simple parametrization of nucleon form fac- tors, Phys. Rev. C 70, 068202 (2004)

    2004

  46. [54]

    Wigner, On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei, Phys

    E. Wigner, On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei, Phys. Rev. 51, 106 (1937)

    1937

  47. [55]

    Pachucki, V

    K. Pachucki, V. Patk´ oˇ s, and V. A. Yerokhin, Hyperfine splitting in 6,7Li+, Phys. Rev. A 108, 052802 (2023)

    2023

  48. [56]

    J. J. Krauth, M. Diepold, B. Franke, A. An- tognini, F. Kottmann, and R. Pohl, The- ory of the n=2 levels in muonic deuterium, Ann. Phys. (Amsterdam) 366, 168 (2016)

    2016

  49. [57]

    Patk´ oˇ s, V

    V. Patk´ oˇ s, V. A. Yerokhin, and K. Pachucki, Nu- clear polarizability effects in 3He+ hyperfine splitting, Phys. Rev. A 107, 052802 (2023)

    2023

  50. [58]

    Friar and I

    J. Friar and I. Sick, Zemach moments for hydrogen and deuterium, Phys. Lett. B 579, 285 (2004)

    2004

  51. [59]

    Sick, Zemach moments of 3He and 4He, Phys

    I. Sick, Zemach moments of 3He and 4He, Phys. Rev. C 90, 064002 (2014)

    2014

  52. [60]

    Antognini, F

    A. Antognini, F. Hagelstein, and V. Pascalutsa, The proton structure in and out of muonic hydrogen, Ann. Rev. Nucl. Part. Sci. 72, 389 (2022)

    2022

  53. [61]

    Antognini, Y.-H

    A. Antognini, Y.-H. Lin, and U.-G. Meißner, Preci- sion calculation of the recoil–finite-size correction for t he hyperfine splitting in muonic and electronic hydrogen, Phys. Lett. B 835, 137575 (2022)

    2022

  54. [62]

    Tomalak, Two-photon exchange correction to the lamb shift and hyperfine splitting of s levels, Eur

    O. Tomalak, Two-photon exchange correction to the lamb shift and hyperfine splitting of s levels, Eur. Phys. J. A 55, 64 (2019)

    2019

  55. [63]

    Tomalak, Two-photon exchange on the neutron and the hyperfine splitting, Phys

    O. Tomalak, Two-photon exchange on the neutron and the hyperfine splitting, Phys. Rev. D 99, 056018 (2019)

    2019

  56. [64]

    R. Li, Y. Wu, Y. Rui, B. Li, Y. Jiang, L. Ma, and H. Wu, Absolute Frequency Measurement of 6Li D Lines with khz-Level Uncertainty, Phys. Rev. Lett. 124, 063002 (2020)

    2020

  57. [65]

    Yamagata, S

    T. Yamagata, S. Nakayama, H. Akimune, M. Fujiwara, K. Fushimi, M. B. Greenfield, K. Hara, K. Y. Hara, H. Hashimoto, K. Ichihara, K. Kawase, M. Kinoshita, Y. Matsui, K. Nakanishi, M. Tanaka, H. Utsunomiya, and M. Yosoi, Excitations of the α cluster in A = 6 and 7 nuclei, Phys. R...

    2004

  58. [66]

    Nevo Dinur, O

    N. Nevo Dinur, O. J. Hernandez, S. Bacca, N. Barnea, C. Ji, S. Pastore, M. Piarulli, and R. B. Wiringa, Zemach moments and radii of 2,3H and 3,4He, Phys. Rev. C 99, 034004 (2019)

    2019

  59. [67]

    Pastore, R

    S. Pastore, R. Schiavilla, and J. L. Goity, Electromag- netic two-body currents of one- and two-pion range, Phys. Rev. C 78, 064002 (2008)

    2008

  60. [68]

    Pastore, L

    S. Pastore, L. Girlanda, R. Schiavilla, and M. Vi- viani, Two-nucleon electromagnetic charge operator in chiral effective field theory ( χEFT) up to one loop, Phys. Rev. C 84, 024001 (2011)

    2011

  61. [69]

    K¨ olling, E

    S. K¨ olling, E. Epelbaum, H. Krebs, and U. G. Meißner, Two-pion exchange electromagnetic current in chiral ef- fective field theory using the method of unitary transfor- mation, Phys. Rev. C 80, 045502 (2009)

    2009

  62. [70]

    K¨ olling, E

    S. K¨ olling, E. Epelbaum, H. Krebs, and U.-G. Meißner, Two-nucleon electromagnetic current in chiral effective field theory: One-pion exchange and short-range contri- butions, Phys. Rev. C 84, 054008 (2011)

    2011

  63. [71]

    Qi, P.-P

    X.-Q. Qi, P.-P. Zhang, Z.-C. Yan, T.-Y. Shi, G. W. F. Drake, A.-X. Chen, and Z.-X. Zhong, Theoretical hyper- fine splittings of 7,9Be2+ ions for future studies of nuclear properties, Phys. Rev. A 107, L010802 (2023)

    2023

  64. [72]

    Dickopf, B

    S. Dickopf, B. Sikora, A. Kaiser, M. M¨ uller, S. Ulmer, V. A. Yerokhin, Z. Harman, C. H. Keitel, A. Mooser, and K. Blaum, Precision spectroscopy on 9Be overcomes lim- itations from nuclear structure, Nature 632, 757 (2024)

    2024

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