Hypernuclei with Neural Network Quantum States

Alessandro Lovato; Andrea Di Donna; Francesco Pederiva; Lorenzo Contessi

REVIEW 3 major objections 5 minor 3 cited by

Hypernuclei with Neural Network Quantum States

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper extends neural-network quantum state Monte Carlo to include Λ hyperons, computing hypernuclei up to 16ΛO and matching experimental separation energies within a few percent, including the proton-radius shrinkage of 7ΛLi.

desk verdict First NQS calculations with a Lambda: a genuine method extension, well benchmarked, but the few-percent predictive claim lacks an EFT truncation error and partially rests on fitted systems. read the letter →

arxiv 2507.16994 v1 pith:AGC5J5QO submitted 2025-07-22 nucl-th

Andrea Di Donna , Lorenzo Contessi , Alessandro Lovato , Francesco Pederiva This is my paper

classification nucl-th MSC 81V3581V7082B80 PACS 21.80.+a21.10.Dr02.70.Uu

keywords hypernucleineuralnetworkquantumstatesvariationalMonteCarlopionlesseffectivefieldtheoryLambdaseparationenergyprotonradiusshrinkageGaussianprocessPfaffian-Jastrow

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 show that an ab initio, many-body description of light and medium-mass hypernuclei is achievable with a machine-learning solver. It extends variational Monte Carlo with neural network quantum states (VMC-NQS), previously applied to nucleons, to treat a $\Lambda$ hyperon as a distinguishable particle alongside protons and neutrons, and couples this ansatz with an improved leading-order pionless effective field theory Hamiltonian. The low-energy constants of that Hamiltonian are fitted to two-body scattering data and to the binding and separation energies of a few light systems, with uncertainties propagated through Gaussian Process surrogates and Markov Chain Monte Carlo. Using this setup the authors compute ground-state energies, $\Lambda$ separation energies, single-particle densities, and radii for $s$- and $p$-shell single-$\Lambda$ hypernuclei up to ${}^{16}_{\Lambda}\mathrm{O}$. For hypernuclei not included in the fit, the predicted separation energies agree with experiment to within an average of 9%, and the calculation reproduces the experimentally observed shrinkage of the proton radius in ${}^{7}_{\Lambda}\mathrm{Li}$ relative to ${}^{6}\mathrm{Li}$.

What carries the argument

The machinery that carries the argument is the hypernuclear Pfaffian-Jastrow neural-network quantum state: a permutation-invariant Jastrow factor, now including nucleon–$\Lambda$ pairs, multiplied by a Pfaffian that antisymmetrizes only the nucleons and by a separate single-particle orbital for the $\Lambda$ hyperon, all built from message-passing backflow-transformed coordinates. This ansatz is paired with an improved leading-order pionless-EFT Hamiltonian whose two-body regulators and low-energy constants are fixed to reproduce $np$ and $p\Lambda$ scattering lengths and effective ranges, and whose NNN and NN$\Lambda$ three-body couplings are calibrated through Gaussian Process surrogates trained on high-accuracy few-body calculations, with posterior uncertainties sampled by Markov Chain Monte Carlo. Because the $\Lambda$ is never antisymmetrized into the nuclear wave function, the variational state can treat it as a distinguishable particle while backflow and Jastrow correlations still keep the nucleon–$\Lambda$ interactions fully coupled to the nuclear dynamics.

What would settle it

A concrete test: refit the NN$\Lambda$ three-body couplings with a second, larger regulator range (for example one 20% larger) while still reproducing the same $np$ and $p\Lambda$ scattering data and fitted $B_\Lambda$ values, then recompute the separation energies of unconstrained hypernuclei; if ${}^{9}_{\Lambda}\mathrm{Be}$, ${}^{11}_{\Lambda}\mathrm{B}$, ${}^{12}_{\Lambda}\mathrm{C}$, ${}^{13}_{\Lambda}\mathrm{C}$, and ${}^{15}_{\Lambda}\mathrm{N}$ shift by more than the few-percent agreement claimed, the apparent accuracy is tied to the particular range rather than to the EFT expansion. An independent cross-check would be to compute ${}^{40}_{\Lambda}\mathrm{Ca}$ with this Hamiltonian and compare its $B_\Lambda$ with a many-body calculation based on a chiral hypernuclear interaction, exposing whether the few-percent agreement persists at medium mass.

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Extended reading notes

Core claim

The central claim is that a single, deliberately simple Hamiltonian — contact interactions at leading order in pionless effective field theory, with finite ranges and couplings fixed by two-body scattering data and a handful of few-body energies — is sufficient, when solved with a flexible neural-network variational wave function, to reproduce the $\Lambda$-separation energies of hypernuclei up to ${}^{16}_{\Lambda}\mathrm{O}$ within a few percent of experiment and to capture the structural role of the hyperon. The key new element is the wave function: the Pfaffian-Jastrow ansatz is extended so that the $\Lambda$, being distinguishable from nucleons, enters through its own single-particle orbital rather than through antisymmetrization, and this ansatz reproduces exact few-body Stochastic Variational Method benchmarks in light hypernuclei. The authors further identify a two-regime pattern in hyperon-induced size changes: in the lightest systems the weakly bound $\Lambda$ forms a halo that pulls nucleons outward, whereas from ${}^{7}_{\Lambda}\mathrm{Li}$ onward the $\Lambda$ occupies the central $1s$ orbital and acts as a glue that contracts the core — the proton radius of ${}^{7}_{\Lambda}\mathrm{Li}$ is computed to shrink by about 13% relative to ${}^{6}\mathrm{Li}$ — an effect that fades to about a percent by ${}^{16}_{\Lambda}\mathrm{O}$.

Load-bearing premise

The load-bearing premise is that a fixed-range, leading-order pionless-EFT Hamiltonian, with regulators and couplings calibrated to two-body scattering and a handful of few-body energies, is accurate enough to predict unconstrained hypernuclear separation energies to within a few percent — a premise the paper cannot test by varying the cutoff, because the regulator range is frozen by the fit, so the EFT truncation error is never explicitly included in the quoted uncertainties.

Editorial extensions

If this is right

Hypernuclei not included in the fit, up to ${}^{16}_{\Lambda}\mathrm{O}$, have predicted $\Lambda$ separation energies within an average of 9% of experiment, so the Hamiltonian ansatz is predictive rather than merely interpolative.
The same solver and interaction can be applied to medium-mass hypernuclei such as ${}^{40}_{\Lambda}\mathrm{Ca}$ and ${}^{48}_{\Lambda}\mathrm{Ca}$, giving access to the isospin dependence of the N$\Lambda$ interaction; the paper states these systems have been measured and are within reach of the method's scaling.
In ${}^{16}_{\Lambda}\mathrm{O}$ the $\Lambda$ localizes in the central $1s$ orbital, meaning the NN$\Lambda$ interaction is probed near saturation density, the regime relevant to neutron-star cores and the hyperon puzzle.
The two-regime pattern in hyperon-induced radius changes — expansion in the lightest hypernuclei, contraction from ${}^{7}_{\Lambda}\mathrm{Li}$ onward, fading by ${}^{16}_{\Lambda}\mathrm{O}$ — is a systematic, mass-dependent prediction for how $\Lambda$ particles modify nuclear sizes.
Residual underbinding in $p$-shell systems is attributed to missing $p$-wave contributions, so adding leading $p$-wave contact terms to both NN and N$\Lambda$ interactions is the paper's stated route to removing the largest deviations.

Reading between the lines

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

The quoted uncertainties include Monte Carlo statistics and the spread of the NN$\Lambda$ parameter posterior, but not the EFT truncation error, because the fixed regulator range makes a cutoff-variation study impossible; the few-percent agreement therefore includes an unquantified model-error component.
The Gaussian Process plus MCMC calibration pipeline would transfer almost unchanged to chiral hypernuclear interactions, making the workflow itself a reusable tool for sensitivity analysis rather than a single-Hamiltonian study.
The sharp halo-to-glue transition between $A=5$ and $A=7$ implies a specific, testable signature: hypernuclear and parent-nucleus charge radii should cross somewhere in that mass range, an observable that would be accessible to hypernuclear laser spectroscopy if such measurements become feasible.
A direct check of the paper's attribution of the largest deviations would be to confirm that the same $p$-wave correction that fixes the parent nuclei (${}^{11}\mathrm{B}$, ${}^{11}\mathrm{C}$) also removes the excess underbinding in ${}^{12}_{\Lambda}\mathrm{B}$, ${}^{12}_{\Lambda}\mathrm{C}$, and ${}^{13}_{\Lambda}\mathrm{C}$, without requiring new hypernuclear physics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a genuine technical advance. The Pfaffian-Jastrow VMC-NQS method is extended to single-Lambda hypernuclei, with the Lambda treated as distinguishable, and the authors carefully benchmark against SVM for A <= 5, getting few-keV agreement. Pushing to 16_Lambda O and computing radii and densities, including the 7_Lambda Li proton shrinkage, is real new content. The GP-based LEC fitting is sensible, and the paper is honest about several limitations.

What it does well: the benchmark table is solid, the radii results are an independent prediction that goes beyond energy reproduction, and the explicit statement that the truncation error is not included (Section IV, after Fig. 5) shows the authors are not trying to hide the main caveat.

The soft spots are real, but they are the kind one expects in a first methods paper. First, there is no truncation-error estimate: the regulators are fixed by the fit, so cutoff variation is infeasible, and the few-percent claim carries no EFT error bar. Second, the abstract's "remarkably good agreement" leans on systems that are in the fit. The non-fitted B_Lambda values average below 9% deviation, but the p-shell cases 12_Lambda B, 12_Lambda C, and 13_Lambda C sit around 12%, larger than the combined statistical and model uncertainties. The authors attribute this to missing p-wave terms, which is plausible but not tested by a subleading-order calculation. Third, the p.Lambda scattering input is from 1968 and comes without uncertainties, and the 2.5% inflation of experimental errors on binding energies is ad hoc. Fourth, no code or data are released, which would help reproducibility but is not disqualifying.

The stress-test note is essentially correct: the few-percent predictive claim is not fully secured. But I would not call this a load-bearing flaw for the paper's main purpose. The method is credible, the benchmark proves it, and the physics conclusions are qualitative and reasonable. The next step should be quantified p-wave contributions or an explicit statement that the truncation error dominates the quoted uncertainties.

This is a paper for people working on ab initio hypernuclear structure, NQS methods, and the hyperon puzzle. It deserves a serious referee. My recommendation: send to review, require a clearer separation of fitted versus predicted results, and ask for a subleading-order estimate or an explicit bound on the truncation error.

Referee Report

3 major / 5 minor

Summary. The paper extends the variational Monte Carlo method with neural network quantum states (VMC-NQS), based on a Pfaffian-Jastrow ansatz with backflow, to single-Λ hypernuclei. The Hamiltonian is an improved leading-order pionless EFT with fixed finite-range regulators; the NN and NΛ two-body LECs and regulators are fitted to scattering data, and the NNN and NNΛ three-body LECs and regulators are fitted to binding and separation energies using Gaussian-process emulators of SVM and NQS energies followed by MCMC sampling. The NQS solver is benchmarked against SVM for A ≤ 5 hypernuclei (Table IV) with agreement at the few-keV level. The authors compute Λ separation energies for selected s- and p-shell hypernuclei up to 16ΛO, report an average discrepancy below 9% for systems not included in the NNΛ fit, and find a proton-radius shrinkage of δr ≈ −0.13 in 7ΛLi relative to 6Li, consistent with experiment.

Significance. The methodological advance is genuine and valuable: this is the first application of NQS to hypernuclear systems, it provides pure estimators for densities and radii, and it is benchmarked against SVM for light hypernuclei with excellent agreement. The Gaussian-process-plus-MCMC framework for propagating LEC uncertainties is a useful template. If the Hamiltonian truncation error were quantified, the approach would offer a scalable path to medium-mass hypernuclei. However, the paper's central few-percent predictive claim is conditional on the fixed-range leading-order Hamiltonian, and the largest p-shell deviations are about 12%, with no subleading-order estimate to show whether those deviations are a controlled EFT correction or a limitation of the model.

major comments (3)

[Section IV (text after Fig. 5 and Fig. 6)] The central few-percent predictive claim is not secured because the truncation error of the fixed-range improved-LO Hamiltonian is not estimated. The text explicitly states after Fig. 5 that a cutoff-variation study is not feasible and that truncation error is not included. The same Hamiltonian underbinds the p-shell parents 11B and 11C by about 9%, and the largest BΛ deviations in Fig. 6 (12ΛB, 12ΛC, 13ΛC) are about 12%, larger than the combined statistical and NNΛ model uncertainties. The paper attributes these deviations to missing p-wave terms, but it does not test this hypothesis with a subleading-order calculation, an estimate of omitted p-wave contributions, or any range-sensitivity study. As a result, the agreement for non-fitted p-shell hypernuclei is not yet distinguishable from an accidental outcome of the fitted central interaction.
[Section IV, Fig. 6] The statement that agreement "remains excellent—within theoretical and experimental uncertainties—even for the 4ΛHeS=1 and 4ΛHeS=0 states" is not supported by the numbers in Fig. 6. For 4ΛHeS=0, the NQS value is BΛ = 2.129 ± 0.042 MeV versus the experimental 2.347 ± 0.036 MeV, a discrepancy of 0.218 MeV, which is several times the combined uncertainty and the error bars do not overlap. This system is not included in the NNΛ fit, so it is a direct test of the model, and the discrepancy should be reported quantitatively and included in the accuracy assessment.
[Section IV and Section V (Conclusions)] Per-system accuracy for non-fitted p-shell hypernuclei is weaker than the paper's summary language. The abstract and conclusions say "few percent" and "average discrepancy below 9%", but individual non-fitted systems deviate by 12–13% (12ΛB ≈ −12.3%, 12ΛC ≈ −11.9%, 13ΛC ≈ −12.8%, 4ΛHe1 ≈ +13.1%), so the average is not representative of the worst cases. A table listing each system with its fitted/non-fitted status, BΛ, experimental value, and signed percentage deviation would make the predictive claim transparent and is needed before the summary statements can be assessed.

minor comments (5)

[Section V, Conclusions] There is a typo: "VNC-NQS" should read "VMC-NQS".
[Eq. (18)] In the definition of ψ(xi), the arguments of uψ and vψ are written as (xi, xj), but xj is not defined for a single-particle orbital; they should depend only on xi.
[Eq. (15)] The exponent for η(xi, xj) has an unbalanced parenthesis: "a tanh(uη(xi, xj)/a) + iπ vη(xi, xj))" contains one closing parenthesis too many.
[Fig. 1 caption] The caption contains a duplicated article: "Joint distributions of the the NNN potential". Similar small grammar issues appear elsewhere, e.g., "The overlaying best-fit values and corresponding energies, are indicated" in Fig. 2.
[Section II A] The Gaussian-process emulator is used to replace expensive few-body and NQS calculations in the fit, but no cross-validation or estimate of GP interpolation error is reported. Even if the GP uncertainty is small relative to experimental uncertainties, stating this explicitly would strengthen the reliability of the fitted LECs and posterior distributions.

Circularity Check

1 steps flagged · score 2.0 of 10

Only minor circularity: the NNΛ (and NNN) calibration systems are explicitly labeled as fitted, while the paper's independent predictive content — non-fitted hypernuclear separation energies and the 7ΛLi radius shrinkage — is benchmarked against external data and does not reduce to its inputs.

fitted input called prediction [Section II A (Eq. 8, Tables II–III) and Section IV (Figure 6)]
"Similarly, the low-energy constants and regulators of the NNΛ potential can be adjusted to reproduce the Λ separation energies ... for 3ΛH, 4ΛHS=0, 4ΛHS=1, and 5ΛHe. As in the nuclear case, we also include the separation energy of 16ΛO in the fit. ... As expected, the NQS calculations for the systems used to calibrate the NNΛ interaction closely reproduce the experimental data."

The NNΛ LECs and regulators are fitted to the very BΛ values of 3ΛH, 4ΛH(S=0,1), 5ΛHe, and 16ΛO listed in Table III; therefore their reproduction is guaranteed by construction once the GP/MCMC fit converges, not a prediction. The same holds for 3H, 3He, 4He, and 16O, which constrain the NNN force. However, the paper transparently identifies these as calibration systems and bases its genuine predictive claim on the separately stated 'average discrepancy below 9%' for hypernuclei not included in the fit, plus the independent radius comparison. Thus the circularity is real but partial and explicitly acknowledged.

full rationale

The only concrete reduction-by-construction is the agreement for the calibration systems: the NNΛ potential is fitted to the experimental BΛ values of 3ΛH, 4ΛH(S=0,1), 5ΛHe, and 16ΛO, so restating those as 'predicted binding energies show remarkably good agreement' is partly circular. The paper is candid about this ('As expected, the NQS calculations for the systems used to calibrate the NNΛ interaction closely reproduce the experimental data'). The independent content is the non-fitted hypernuclear separation energies (4ΛHe, 7ΛLi, 9ΛBe, 11ΛB, 12ΛB, 12ΛC, 13ΛC, 15ΛN) and the proton-radius shrinkage in 7ΛLi, neither of which enters the Hamiltonian fit; agreement with experiment there is a genuine external benchmark rather than a consequence of the fitting procedure. The fixed-range improved pionless-EFT Hamiltonian is anchored to scattering data and to few/nuclear-body binding energies, with the resulting truncation error explicitly not quantified in Section IV; that is a model-accuracy limitation, not circularity. The strategy of fixing regulators to effective ranges is inherited from Refs. [46,54,55], some of which share authors with this paper, but it is not presented as a uniqueness theorem and is here tested against non-fit data. No load-bearing self-citation chain, ansatz-smuggling, or renaming of known results was found. Overall circularity is minor and confined to inputs that the paper itself labels as calibration.

Assumptions & free parameters 15 free parameters · 5 assumptions · 0 invented entities

The calculation relies on a simplified interaction whose parameters are mostly fitted to the very observables later compared with experiment. The structurally new content is the NQS solver and the non-fitted hypernuclear observables, especially the 7ΛLi radius shrinkage. No new physical entities are postulated.

free parameters (15)

C^{10}_λ (NN 1S0 LEC) = -31.0633 MeV
Fitted with the variable-phase method to reproduce the np scattering length and effective range in the 1S0 channel. All nuclear and hypernuclear results inherit this calibration.
λ^{10} (NN 1S0 regulator) = 1.10117 fm^-1
Fixed to reproduce the effective range in the np 1S0 channel rather than taken to infinity, which is the improved-LO scheme used throughout.
C^{01}_λ (NN 3S1 LEC) = -68.3747 MeV
Fitted to the np scattering length and effective range in the 3S1 channel, and it reproduces the deuteron binding energy.
λ^{01} (NN 3S1 regulator) = 1.30512 fm^-1
Fixed to the np 3S1 effective range.
tilde{C}^{1/2 0}_λ (NΛ 1S0 LEC) = -33.5417 MeV
Fitted to the 1968 Λp scattering length and effective range in the 1S0 channel, whose uncertainties are not listed in Table I.
tilde{λ}^{1/2 0} (NΛ 1S0 regulator) = 1.54720 fm^-1
Fixed to the Λp 1S0 effective range.
tilde{C}^{1/2 1}_λ (NΛ 3S1 LEC) = -25.3115 MeV
Fitted to the 1968 Λp scattering length and effective range in the 3S1 channel.
tilde{λ}^{1/2 1} (NΛ 3S1 regulator) = 1.41379 fm^-1
Fixed to the Λp 3S1 effective range.
Dλ (NNN LEC) = 25.54 (+14.34/-8.56) MeV
Fitted to the ground-state energies of 3H, 3He, 4He, and 16O, with experimental errors inflated to 2.5%.
λ (NNN regulator) = 1.93 (+0.17/-0.15) fm^-1
Determined in the same fit; including 16O is essential to break the degeneracy in the (λ, Dλ) plane.
tilde{D}^{0,1/2}_λ (NNΛ LEC) = -1.35 (+1.09/-1.13) MeV
Fitted to BΛ of 3ΛH, 4ΛHS=0, 4ΛHS=1, 5ΛHe, and 16ΛO via a Gaussian-process surrogate and MCMC.
tilde{D}^{1,1/2}_λ (NNΛ LEC) = 12.28 (+2.50/-2.27) MeV
Fitted to the same Lambda separation energies and strongly constrained by the 4ΛH and 5ΛHe data.
tilde{D}^{0,3/2}_λ (NNΛ LEC) = 6.45 (+1.00/-1.02) MeV
Fitted to the same Lambda separation energies and required by the spin-3/2 channel.
tilde{λ} (NNΛ regulator) = 1.494 (+0.033/-0.031) fm^-1
Determined in the NNΛ fit; including 16ΛO is needed to constrain the range.
Experimental error inflation factor = 0.025
Chosen by hand to stabilize the fitting algorithm; it changes the posterior widths reported in Figures 1 to 4.

assumptions (5)

domain assumption Non-relativistic Schrodinger dynamics with the Hamiltonian of Eq. (1), including only two- and three-body contact interactions regularized by Gaussian regulators.
Pion exchanges are not explicit; the entire many-body calculation rests on this interaction form.
ad hoc to paper Fixed finite-range regulators define the improved leading-order pionless EFT; no lambda-to-infinity limit or renormalization-group invariance is enforced.
Sections II and IV; this prevents cutoff-variation studies, so the truncation error is not computed.
domain assumption The NΛ system has a single shallow virtual state in each s-wave channel, and the Lambda is distinguishable from nucleons with isospin tz=0.
Used in Section II for the NΛ potential and in Section III for the product ansatz of Eq. (19).
ad hoc to paper The Gaussian-process emulator with rational-quadratic plus Matern-3/2 kernels and uniform priors accurately interpolates SVM and NQS energies over the fitted parameter ranges.
Section II A; if the GP is biased, the MCMC posteriors and all downstream energies inherit the bias.
domain assumption The Pfaffian-Jastrow plus backflow ansatz is expressive enough to represent the relevant ground states of nuclei and hypernuclei up to A=16.
Validated for A<=5 against SVM in Table IV and for nuclei in prior work, but assumed for p-shell hypernuclei up to 16ΛO.

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Pith. "Pith review of Hypernuclei with Neural Network Quantum States." pith.science (2026). https://pith.science/paper/AGC5J5QO

@misc{pith2026250716994,
  author       = {Pith},
  title        = {Pith review of: Hypernuclei with Neural Network Quantum States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGC5J5QO}},
  note         = {Machine review of arXiv:2507.16994}
}

abstract

Leveraging complementary machine-learning-based approaches, we compute properties of $s$- and $p$-shell $\Lambda$ hypernuclei - including binding energies, single-particle densities, and radii - starting from the individual interactions among their constituents. These interactions are modeled using an improved leading-order pionless effective field theory expansion, with coefficients determined via a Gaussian Process framework anchored on virtually exact few-body techniques. We solve the many-body Schr\"odinger equation using a variational Monte Carlo method based on neural network quantum states, extending it for the first time to include $\Lambda$ particles alongside protons and neutrons. The predicted binding energies show remarkably good agreement with experimental results, given the simplicity of the input Hamiltonian. We also confirm the experimentally observed shrinkage of the proton radius in $^7_\Lambda$Li compared to its parent nucleus, $^6$Li. This work paves the way for an ab initio description of medium-mass and heavy hypernuclei, as well as for understanding the onset of strange degrees of freedom in the core of neutron stars.

Figures

Figures reproduced from arXiv: 2507.16994 by the authors.

**Figure 2.** Joint posterior distributions of nuclear binding energies (MeV) obtained by varying the NNN potential [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Joint distributions of the LECs (in MeV) and regulator (in fm [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Same as Figure 2 but for the hypernuclear separation energies (in MeV). [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Binding energy per nucleon for selected nuclei [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: illustrates the Λ-separation energies of selected hypernuclei up to 16 Λ O. The NQS calculations are benchmarked against experimental values from the Hypernuclear Database [62] (blue), as well as recent theoretical predictions [40, 41, 82] (green, purple, and red re…

**Figure 7.** Figure 7: Neutron (orange squares), proton (green triangles), and Λ hyperon (blue circles) single-particle radial density distributions for 5 ΛHe (upper panel), 7 ΛLi (middle panel), and 16 Λ O (lower panel). The protons densities of the corresponding parent nuclei are also show…

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Meson-Nucleus Bound States with Neural-Network Quantum States
hep-ph 2026-06 unverdicted novelty 7.0 of 10

Neural-network quantum states applied to HAL QCD meson-nucleon potentials predict bound states for phi at A>=2, J/psi at A>=4, and eta_c at A>=6, with binding energies from tens of MeV to sub-MeV scales.
Medium-mass nuclei with neural quantum states
nucl-th 2026-07 conditional novelty 6.5 of 10

Pfaffian-Jastrow neural quantum states yield ground-state energies and charge radii for nuclei up to A=58, with weak p-wave terms reducing average energy error to ~3% while revealing Hamiltonian sensitivity and A^3 scaling.
Machine learning the single-$\Lambda$ hypernuclei with neural-network quantum states
nucl-th 2025-08 unverdicted novelty 6.0 of 10

Neural-network quantum states with new spin and isospin treatments compute light hypernuclei spectra at claimed high accuracy and benchmark pionless effective field theory Hamiltonians.

Reference graph

Works this paper leans on

87 extracted references · 34 canonical work pages · cited by 3 Pith papers

[1]

Epelbaum, H.-W

E. Epelbaum, H.-W. Hammer, and U.-G. Meissner, Rev. Mod. Phys.81, 1773 (2009), arXiv:0811.1338 [nucl-th]

arXiv 2009
[2]

Machleidt and D

R. Machleidt and D. R. Entem, Phys. Rept. 503, 1 (2011), arXiv:1105.2919 [nucl-th]

arXiv 2011
[3]

H. W. Hammer, S. König, and U. van Kolck, Rev. Mod. Phys. 92, 025004 (2020), arXiv:1906.12122 [nucl-th]

arXiv 2020
[4]

Svensson, A

I. Svensson, A. Ekström, and C. Forssén, Phys. Rev. C 105, 014004 (2022), arXiv:2110.04011 [nucl-th]

arXiv 2022
[5]

J. M. Bub, M. Piarulli, R. J. Furnstahl, S. Pastore, and D. R. Phillips, Phys. Rev. C111, 034005 (2025), arXiv:2408.02480 [nucl-th]

arXiv 2025
[6]

Hagen, T

G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, Rept. Prog. Phys. 77, 096302 (2014), arXiv:1312.7872 [nucl-th]

arXiv 2014
[7]

Hergert, S

H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tsukiyama, Phys. Rept. 621, 165 (2016), arXiv:1512.06956 [nucl-th]

arXiv 2016
[8]

W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004), arXiv:nucl-th/0402034

arXiv 2004

Show all 87 references

[9]

V. Soma, C. Barbieri, and T. Duguet, Phys. Rev. C87, 011303 (2013), arXiv:1208.2472 [nucl-th]

2013 arXiv
[10]

J.Carlson, S.Gandolfi, F.Pederiva, S.C.Pieper, R.Schi- avilla, K. E. Schmidt, and R. B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015), arXiv:1412.3081 [nucl-th]

2015 arXiv
[11]

J. E. Lynn, I. Tews, S. Gandolfi, and A. Lovato, Ann. Rev. Nucl. Part. Sci.69, 279 (2019), arXiv:1901.04868 [nucl-th]

2019 arXiv
[12]

Danysz and J

M. Danysz and J. Pniewski, The London, Edinburgh, and Dublin Philosophical Maga- zine and Journal of Science 44, 348 (1953), https://doi.org/10.1080/14786440308520318

1953 doi
[13]

Mayet al., Phys

M. Mayet al., Phys. Rev. Lett.51, 2085 (1983)

1983
[14]

Tanidaet al., Phys

K. Tanidaet al., Phys. Rev. Lett.86, 1982 (2001)

2001
[15]

Akikawaet al., Phys

H. Akikawaet al., Phys. Rev. Lett.88, 082501 (2002)

2002
[16]

Hashimoto and H

O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006)

2006
[17]

T. O. Yamamotoet al. (J-PARC E13), Phys. Rev. Lett. 115, 222501 (2015), arXiv:1508.00376 [nucl-ex]

2015 arXiv
[18]

Gogami et al

T. Gogami et al. , Phys. Rev. C 93, 034314 (2016), arXiv:1511.04801 [nucl-ex]

2016 arXiv
[19]

A. Gal, E. V. Hungerford, and D. J. Millener, Rev. Mod. Phys. 88, 035004 (2016), arXiv:1605.00557 [nucl-th]

2016 arXiv
[20]

Bombaci, JPS Conf

I. Bombaci, JPS Conf. Proc. 17, 101002 (2017), arXiv:1601.05339 [nucl-th]

2017 arXiv
[21]

K. A. Maslov, E. E. Kolomeitsev, and D. N. Voskre- sensky, Phys. Lett. B748, 369 (2015), arXiv:1504.02915 [astro-ph.HE]

2015 arXiv
[22]

Demorest, T

P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature 467, 1081 (2010), arXiv:1010.5788 [astro-ph.HE]

2010 arXiv
[23]

Antoniadis et al

J. Antoniadis et al. , Science 340, 6131 (2013), arXiv:1304.6875 [astro-ph.HE]

2013 arXiv
[24]

H. T. Cromartieet al. (NANOGrav), Nature Astron.4, 72 (2019), arXiv:1904.06759 [astro-ph.HE]

2019 arXiv
[25]

Fonseca et al., Astrophys

E. Fonseca et al., Astrophys. J. Lett.915, L12 (2021), arXiv:2104.00880 [astro-ph.HE]

2021 arXiv
[26]

Lonardoni, A

D. Lonardoni, A. Lovato, S. Gandolfi, and F. Pederiva, Phys. Rev. Lett. 114, 092301 (2015), arXiv:1407.4448 [nucl-th]

2015 arXiv
[27]

J.Haidenbauer, U.G.Meißner, N.Kaiser, andW.Weise, Eur. Phys. J. A53, 121 (2017), arXiv:1612.03758 [nucl- th]

2017 arXiv
[28]

Nogga, H

A. Nogga, H. Kamada, and W. Gloeckle, Phys. Rev. Lett. 88, 172501 (2002), arXiv:nucl-th/0112060

2002 arXiv
[29]

Hiyama, M

E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Phys. Rev. C 65, 011301 (2002), arXiv:nucl-th/0106070

2002 arXiv
[30]

Nemura, Y

H. Nemura, Y. Akaishi, and Y. Suzuki, Phys. Rev. Lett. 89, 142504 (2002), arXiv:nucl-th/0203013

2002 arXiv
[31]

I. N. Filikhin and A. Gal, Phys. Rev. Lett.89, 172502 (2002), arXiv:nucl-th/0209003

2002 arXiv
[32]

Garcilazo, A

H. Garcilazo, A. Valcarce, and T. Fernandez-Carames, Phys. Rev. C76, 034001 (2007), arXiv:0708.0199 [hep- ph]

2007 arXiv
[33]

Contessi, N

L. Contessi, N. Barnea, and A. Gal, Phys. Rev. Lett. 121, 102502 (2018), arXiv:1805.04302 [nucl-th]

2018 arXiv
[34]

Lonardoni, S

D. Lonardoni, S. Gandolfi, and F. Pederiva, Phys. Rev. C 87, 041303 (2013), arXiv:1301.7472 [nucl-th]

2013 arXiv
[35]

Lonardoni, F

D. Lonardoni, F. Pederiva, and S. Gandolfi, Phys. Rev. C 89, 014314 (2014), arXiv:1312.3844 [nucl-th]

2014 arXiv
[36]

Wirth, D

R. Wirth, D. Gazda, P. Navrátil, A. Calci, J. Langham- mer, and R. Roth, Phys. Rev. Lett.113, 192502 (2014), arXiv:1403.3067 [nucl-th]

2014 arXiv
[37]

Wirth and R

R. Wirth and R. Roth, Phys. Rev. Lett. 117, 182501 (2016), arXiv:1605.08677 [nucl-th]

2016 arXiv
[38]

Wirth and R

R. Wirth and R. Roth, Phys. Lett. B779, 336 (2018), arXiv:1710.04880 [nucl-th]

2018 arXiv
[39]

Wirth and R

R. Wirth and R. Roth, Phys. Rev. C100, 044313 (2019), arXiv:1902.03324 [nucl-th]

2019 arXiv
[40]

Knöll and R

M. Knöll and R. Roth, Phys. Lett. B846, 138258 (2023), arXiv:2307.11577 [nucl-th]

2023 arXiv
[41]

H. Le, J. Haidenbauer, U.-G. Meißner, and A. Nogga, Phys. Rev. Lett.134, 072502 (2025), arXiv:2409.18577 [nucl-th]

2025 arXiv
[42]

Adams, G

C. Adams, G. Carleo, A. Lovato, and N. Rocco, Phys. Rev. Lett. 127, 022502 (2021), arXiv:2007.14282 [nucl- th]

2021 arXiv
[43]

Lovato, C

A. Lovato, C. Adams, G. Carleo, and N. Rocco, Phys. Rev. Res.4, 043178 (2022), arXiv:2206.10021 [nucl-th]

2022 arXiv
[44]

7,148 (2024), arXiv:2305.08831 [cond-mat.quant-gas]

J.Kim, G.Pescia, B.Fore, J.Nys, G.Carleo, S.Gandolfi, M.Hjorth-Jensen, andA.Lovato,Commun.Phys. 7,148 (2024), arXiv:2305.08831 [cond-mat.quant-gas]

2024 arXiv
[45]

B. Fore, J. Kim, M. Hjorth-Jensen, and A. Lovato, Com- mun. Phys.8, 108 (2025), arXiv:2407.21207 [nucl-th]. 15

2025 arXiv
[46]

Schiavilla, L

R. Schiavilla, L. Girlanda, A. Gnech, A. Kievsky, A. Lo- vato, L. E. Marcucci, M. Piarulli, and M. Viviani, Phys. Rev. C103, 054003 (2021), arXiv:2102.02327 [nucl-th]

2021 arXiv
[47]

Auerbach, J

N. Auerbach, J. Hüfner, A. K. Kerman, and C. M. Shakin, Reviews of Modern Physics44, 48 (1972)

1972
[48]

R. W. Hackenburg, Phys. Rev. C73, 044002 (2006)

2006
[49]

Alexander, U

G. Alexander, U. Karshon, A. Shapira, G. Yekutieli, R. Engelmann, H. Filthuth, and W. Lughofer, Phys. Rev. 173, 1452 (1968)

1968
[50]

P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52, 339 (2002), arXiv:nucl-th/0203055

2002 arXiv
[51]

Stetcu, B

I. Stetcu, B. R. Barrett, and U. van Kolck, Phys. Lett. B 653, 358 (2007), arXiv:nucl-th/0609023

2007 arXiv
[52]

Contessi, A

L. Contessi, A. Lovato, F. Pederiva, A. Roggero, J. Kirscher, and U. van Kolck, Phys. Lett. B772, 839 (2017), arXiv:1701.06516 [nucl-th]

2017 arXiv
[53]

Bansal, S

A. Bansal, S. Binder, A. Ekström, G. Hagen, G. R. Jansen, and T. Papenbrock, Phys. Rev. C 98, 054301 (2018), arXiv:1712.10246 [nucl-th]

2018 arXiv
[54]

B.-N. Lu, N. Li, S. Elhatisari, D. Lee, E. Epelbaum, and U.-G. Meißner, Phys. Lett. B797, 134863 (2019), arXiv:1812.10928 [nucl-th]

2019 arXiv
[55]

Gnech, B

A. Gnech, B. Fore, A. J. Tropiano, and A. Lovato, Phys. Rev. Lett. 133, 142501 (2024), arXiv:2308.16266 [nucl- th]

2024 arXiv
[56]

Contessi, M

L. Contessi, M. Schäfer, and U. van Kolck, Phys. Rev. A 109, 022814 (2024), arXiv:2310.15760 [physics.atm-clus]

2024 arXiv
[57]

Contessi, M

L. Contessi, M. Pavon Valderrama, and U. van Kolck, Phys. Lett. B 856, 138903 (2024), arXiv:2403.16596 [nucl-th]

2024 arXiv
[58]

L.Contessi, M.Schäfer, A.Gnech, A.Lovato, andU.van Kolck, (2025), arXiv:2505.09299 [nucl-th]

2025 arXiv
[59]

Calogero, Il Nuovo Cimento27, 261 (1963)

F. Calogero, Il Nuovo Cimento27, 261 (1963)

1963
[60]

Live chart of nuclides,

International Atomic Energy Agency (IAEA), “Live chart of nuclides,”https://www-nds.iaea.org/relnsd/ vcharthtml/VChartHTML.html
[61]

Ekström and L

A. Ekström and L. Platter, Phys. Lett. B860, 139207 (2025), arXiv:2409.08197 [nucl-th]

2025 arXiv
[62]

Chart of Hypernu- clides — Hypernuclear Structure and Decay Data,

P. Eckert and P. Achenbach, “Chart of Hypernu- clides — Hypernuclear Structure and Decay Data,” https://hypernuclei.kph.uni-mainz.de/ (2023)

2023
[63]

C. E. Rasmussen and C. K. I. Williams,Gaussian Pro- cesses for Machine Learning , 3rd ed., Adaptive Compu- tation and Machine Learning (MIT Press, Cambridge, Mass., 2008)

2008
[64]

Suzuki and K

Y. Suzuki and K. Varga, Stochastic Variational Ap- proach to Quantum Mechanical Few-Body Problems, Lec- ture Notes in Physics New Series M, Monographs No. 54 (Springer, Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo, 1998)

1998
[65]

Duvenaud,Automatic model construction with Gaus- sian processes, Ph.D

D. Duvenaud,Automatic model construction with Gaus- sian processes, Ph.D. thesis, University of Cambridge (2014)

2014
[66]

D. J. C. MacKay, in Maximum Entropy and Bayesian Methods, edited by G. R. Heidbreder (Springer Nether- lands, Dordrecht, 1996) pp. 221–234

1996
[67]

ALICE Collaboration, Physical Review Letters 131, 102302 (2023), arXiv:2209.07360 [nucl-ex]

2023 arXiv
[68]

Gpy: A gaussian processes frame- work,

SheffieldML, “Gpy: A gaussian processes frame- work,” https://github.com/SheffieldML/GPy (2024), accessed: 2024-10-10

2024
[69]

D.Foreman-Mackey, D.W.Hogg, D.Lang, andJ.Good- man, Publications of the Astronomical Society of the Pa- cific 125, 306 (2013), arXiv:1202.3665 [astro-ph]

2013 arXiv
[70]

Ekström, G

A. Ekström, G. R. Jansen, K. A. Wendt, G. Hagen, T. Papenbrock, B. D. Carlsson, C. Forssén, M. Hjorth- Jensen, P. Navrátil, and W. Nazarewicz, Phys. Rev. C 91, 051301 (2015), [Erratum: Phys.Rev.C 109, 059901 (2024)], arXiv:1502.04682 [nucl-th]

2015 arXiv
[71]

Bukov, M

M. Bukov, M. Schmitt, and M. Dupont, SciPost Physics 10, 147 (2021), arXiv:2011.11214 [physics.comp-ph]

2021 arXiv
[72]

Zaheer, S

M. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. Salakhutdinov, and A. Smola, arXiv e-prints , arXiv:1703.06114 (2017), arXiv:1703.06114 [cs.LG]

2017 arXiv
[73]

Wagstaff, F

E. Wagstaff, F. B. Fuchs, M. Engelcke, I. Posner, and M. Osborne, arXiv e-prints , arXiv:1901.09006 (2019), arXiv:1901.09006 [cs.LG]

2019 arXiv
[74]

R. P. Feynman and M. Cohen, Phys. Rev.102, 1189 (1956)

1956
[75]

Luo and B

D. Luo and B. K. Clark, Phys. Rev. Lett.122, 226401 (2019)

2019
[76]

Hermann, Z

J. Hermann, Z. Schätzle, and F. Noé, Nature Chem.12, 891 (2020)

2020
[77]

D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes, Phys. Rev. Res.2, 033429 (2020)

2020
[78]

Pescia, J

G. Pescia, J. Nys, J. Kim, A. Lovato, and G. Car- leo, Phys. Rev. B110, 035108 (2024), arXiv:2305.07240 [quant-ph]

2024 arXiv
[79]

Massella, F

P. Massella, F. Barranco, D. Lonardoni, A. Lovato, F. Pederiva, and E. Vigezzi, J. Phys. G 47, 035105 (2020), arXiv:1808.00518 [nucl-th]

2020 arXiv
[80]

Gattobigio, A

M. Gattobigio, A. Kievsky, and M. Viviani, Phys. Rev. C 100, 034004 (2019), arXiv:1903.08900 [nucl-th]

2019 arXiv
[81]

König, Eur

S. König, Eur. Phys. J. A 56, 113 (2020), arXiv:1910.12627 [nucl-th]

2020 arXiv
[82]

Hildenbrand, S

F. Hildenbrand, S. Elhatisari, Z. Ren, and U.- G. Meißner, Eur. Phys. J. A 60, 215 (2024), arXiv:2406.17638 [nucl-th]

2024 arXiv
[83]

Hiyama, M

E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Phys. Rev. C53, 2075 (1996)

1996
[84]

Hiyama, M

E. Hiyama, M. Kamimura, K. Miyazaki, and T. Motoba, Phys. Rev. C59, 2351 (1999)

1999
[85]

J. M. Yao, Z. P. Li, K. Hagino, M. T. Win, Y. Zhang, and J. Meng, Nucl. Phys. A 868-869, 12 (2011), arXiv:1104.3200 [nucl-th]

2011 arXiv
[86]

Lonardoni and F

D. Lonardoni and F. Pederiva, (2017), arXiv:1711.07521 [nucl-th]

2017 arXiv
[87]

Piarulli et al., Phys

M. Piarulli et al., Phys. Rev. Lett.120, 052503 (2018), arXiv:1707.02883 [nucl-th]

2018 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Epelbaum, H.-W

E. Epelbaum, H.-W. Hammer, and U.-G. Meissner, Rev. Mod. Phys.81, 1773 (2009), arXiv:0811.1338 [nucl-th]

arXiv 2009

[2] [2]

Machleidt and D

R. Machleidt and D. R. Entem, Phys. Rept. 503, 1 (2011), arXiv:1105.2919 [nucl-th]

arXiv 2011

[3] [3]

H. W. Hammer, S. König, and U. van Kolck, Rev. Mod. Phys. 92, 025004 (2020), arXiv:1906.12122 [nucl-th]

arXiv 2020

[4] [4]

Svensson, A

I. Svensson, A. Ekström, and C. Forssén, Phys. Rev. C 105, 014004 (2022), arXiv:2110.04011 [nucl-th]

arXiv 2022

[5] [5]

J. M. Bub, M. Piarulli, R. J. Furnstahl, S. Pastore, and D. R. Phillips, Phys. Rev. C111, 034005 (2025), arXiv:2408.02480 [nucl-th]

arXiv 2025

[6] [6]

Hagen, T

G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, Rept. Prog. Phys. 77, 096302 (2014), arXiv:1312.7872 [nucl-th]

arXiv 2014

[7] [7]

Hergert, S

H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tsukiyama, Phys. Rept. 621, 165 (2016), arXiv:1512.06956 [nucl-th]

arXiv 2016

[8] [8]

W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004), arXiv:nucl-th/0402034

arXiv 2004