Gaussian vs. Real Wavefunction of Nuclear Clusters and Hypernuclei

Elena Bratkovskaya; Jiaxing Zhao; Joerg AICHELIN

arxiv: 2606.11008 · v1 · pith:4T4VHFKGnew · submitted 2026-06-09 · ⚛️ nucl-th · hep-ph

Gaussian vs. Real Wavefunction of Nuclear Clusters and Hypernuclei

Jiaxing Zhao , Joerg Aichelin , Elena Bratkovskaya This is my paper

Pith reviewed 2026-06-27 11:17 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph

keywords nuclear clusterswave functionsGaussian ansatzhypernucleiA=4 clustersSchrödinger equationproduction yieldsspatial distributions

0 comments

The pith

Realistic wave functions of nuclear clusters and hypernuclei are significantly broader and non-Gaussian than Gaussian forms matched only on rms radius.

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

The paper solves the Schrödinger equation for realistic N-body wave functions of A=4 clusters and compares them directly to Gaussian approximations that share the same root-mean-square radius. The microscopic solutions spread farther and display complex non-Gaussian shapes. This structural difference is then linked to production dynamics through a phenomenological two-body interaction, offering one route to close the gap between calculated and measured yields of these clusters.

Core claim

Realistic N-body wave functions obtained from solutions of the Schrödinger equation exhibit significantly broader spatial distributions and pronounced non-Gaussian structures compared to Gaussian ansatze constrained to the same rms radius. Investigation of production channels for A=4 clusters with a phenomenological two-body interaction supplies a mechanism that may reduce the underestimation of cluster yields in theoretical models relative to experiment.

What carries the argument

Direct comparison of microscopic N-body wave functions from the Schrödinger equation against Gaussian ansatze fixed solely by rms radius.

If this is right

Models relying on Gaussian wave functions will continue to underpredict A=4 cluster production yields unless the non-Gaussian tails are incorporated.
The same broadening applies to hypernuclear wave functions, altering predicted binding and decay properties.
Phenomenological two-body interactions can be tuned to reproduce observed cluster yields once the realistic wave-function shapes are used.
Transport simulations that embed these microscopic wave functions should show improved agreement with measured cluster multiplicities in heavy-ion collisions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Replacing Gaussian ansatze with tabulated microscopic wave functions in coalescence or transport codes would provide a direct test of whether the yield discrepancy shrinks.
The non-Gaussian tails may also affect electromagnetic observables such as form factors or transition rates in light nuclei.

Load-bearing premise

Matching the Gaussian form only to the rms radius creates a fair benchmark, and the chosen two-body interaction adequately represents the production process for A=4 clusters.

What would settle it

Direct experimental measurement of the spatial extent or two-particle correlation functions for A=4 clusters that deviates from the broader non-Gaussian distributions predicted by the microscopic wave functions.

Figures

Figures reproduced from arXiv: 2606.11008 by Elena Bratkovskaya, Jiaxing Zhao, Joerg AICHELIN.

**Figure 1.** Figure 1: The radial probability distributions Pκ(ρ) ≡ |Rκ(ρ)| 2ρ 3N−4 of S -wave d, t, 3He, 3 Λ H, 4He, 4 Λ He, 4 Λ H, 5 Λ He, and 5 ΛΛHe are shown as red solid lines. The corresponding Gaussian wave-function probabilities are shown as black dashed lines for comparison. nucleon–nucleon interactions and the Usmani potential for hyperon–nucleon (Y–N) interactions. To facilitate the solution of the many-body problem,… view at source ↗

**Figure 2.** Figure 2: The interaction potential of p − p from Argonne-18 and t − p by fitting the mass of the 4He (left panel). The radial distribution P(r) of S -wave 4He from 4body system p − p − n − n and 2body system t − p (right panel). where L α n generalized (associated) Laguerre polynomial. The ground state gives, Φ0(ρ, Ω) = [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

We compare realistic $N$-body wave functions obtained from solutions of the Schr\"odinger equation with Gaussian ans\"atze constrained to the same rms radius. The microscopic wave functions exhibit significantly broader spatial distributions, revealing pronounced non-Gaussian structures. In addition, we investigate possible production channels for $A=4$ clusters using a phenomenological two-body interaction. This study provides a potential mechanism that may help alleviate the underestimation of $A=4$ cluster yields in theoretical models compared to experimental data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The rms-radius-only Gaussian benchmark does not test the forms actually used in production models, so the claimed difference may not fix the A=4 yield problem.

read the letter

The main thing to know is that the paper compares realistic N-body wave functions to Gaussians fixed only by rms radius and finds the microscopic ones broader and non-Gaussian, then floats a phenomenological two-body interaction as a possible fix for underpredicted A=4 yields. That comparison choice is the load-bearing step.

The work does the standard thing of solving the Schrödinger equation for light clusters and hypernuclei and plotting the resulting spatial distributions against a simple Gaussian. It is useful to see the non-Gaussian tails made explicit, and the link to the known yield mismatch in collision models is a reasonable motivation. The title includes hypernuclei, so the comparison presumably covers those cases as well.

The soft spot is exactly the one the stress-test note flags. Production calculations usually optimize or fit their Gaussian (or Gaussian-like) wave functions to binding energies, separation energies, or electromagnetic data, not rms radius alone. Without showing that the difference survives against the actual ansatz used in the yield models, the broader distributions do not demonstrate a mechanism that would raise the calculated yields. The two-body interaction part is even thinner: it is introduced without any check that it reproduces known A=4 observables or that two-body dynamics dominate over medium or multi-body effects. The abstract supplies no numbers, errors, or figures, so the size of the effect remains unquantified here.

This is for people already working on coalescence or transport models for clusters in heavy-ion collisions. A reader who needs a concrete example of non-Gaussian structure in A=4 systems might pull one figure, but the paper does not introduce new formalisms or falsifiable predictions.

I would send it to peer review. The yield discrepancy is a standing issue in the subfield, and the wave-function comparison is at least a concrete step even if the argument needs tightening on the benchmark and validation.

Referee Report

2 major / 1 minor

Summary. The manuscript compares realistic N-body wave functions obtained from solutions of the Schrödinger equation with Gaussian ansatze constrained to the same rms radius for nuclear clusters and hypernuclei. It reports that the microscopic wave functions exhibit significantly broader spatial distributions and pronounced non-Gaussian structures. The study also investigates possible production channels for A=4 clusters using a phenomenological two-body interaction, providing a potential mechanism to alleviate the underestimation of A=4 cluster yields in theoretical models compared to experimental data.

Significance. If the central comparison and production mechanism hold after addressing the benchmark issues, the work could help explain and correct underpredictions of light cluster yields in nuclear collision models by highlighting limitations of Gaussian wave function approximations. The explicit contrast between microscopic solutions and simple ansatze addresses a practical modeling choice in the field.

major comments (2)

[Abstract] Abstract: the central claim that microscopic wave functions exhibit significantly broader, non-Gaussian distributions (and thus a mechanism for A=4 yields) rests on comparing to a Gaussian constrained solely by rms radius. Production calculations typically employ Gaussians optimized to binding energies, separation energies, or electromagnetic observables rather than rms radius alone; without demonstrating that the reported difference persists against such optimized forms, the implication for yields is not established.
[Production investigation] Production investigation (abstract): the phenomenological two-body interaction is introduced to investigate production without showing that it reproduces known A=4 observables or that two-body dynamics dominate over multi-body or medium effects. With free parameters of the interaction listed among the adjustable quantities, the claimed mechanism risks reducing to a fit rather than an independent prediction.

minor comments (1)

The abstract supplies no equations, error estimates, or numerical data, which limits assessment of the quantitative support for the non-Gaussian claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific comments on the abstract and production investigation. We respond to each point below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that microscopic wave functions exhibit significantly broader, non-Gaussian distributions (and thus a mechanism for A=4 yields) rests on comparing to a Gaussian constrained solely by rms radius. Production calculations typically employ Gaussians optimized to binding energies, separation energies, or electromagnetic observables rather than rms radius alone; without demonstrating that the reported difference persists against such optimized forms, the implication for yields is not established.

Authors: The comparison in the manuscript is performed with a Gaussian ansatz fixed solely by the rms radius, as stated. We agree that this does not match the optimization procedures (binding energies, separation energies, or electromagnetic observables) commonly used in production calculations. The manuscript contains no such additional comparisons, so the implication that the observed difference provides a mechanism for A=4 yields is not demonstrated. We will revise the abstract to remove the reference to alleviating underestimation of yields and add a clarifying sentence in the discussion noting the limited scope of the rms-only constraint. revision: yes
Referee: [Production investigation] Production investigation (abstract): the phenomenological two-body interaction is introduced to investigate production without showing that it reproduces known A=4 observables or that two-body dynamics dominate over multi-body or medium effects. With free parameters of the interaction listed among the adjustable quantities, the claimed mechanism risks reducing to a fit rather than an independent prediction.

Authors: The production study is presented as an exploratory illustration using a simple phenomenological two-body interaction. The manuscript does not demonstrate reproduction of A=4 observables, dominance of two-body dynamics, or independence from parameter adjustment. We will revise the abstract and the relevant section to state explicitly that the calculation is illustrative and phenomenological, without claiming predictive power or dominance over multi-body or medium effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper compares N-body Schrödinger solutions to Gaussian ansätze matched solely on rms radius and reports broader distributions; this is a direct, non-reductive comparison with no self-definition or fitted-input-as-prediction structure. The phenomenological two-body interaction is introduced for investigating A=4 production channels without any quoted indication that its parameters are adjusted to the target yields or that the result reduces to the input by construction. No self-citation load-bearing steps, uniqueness theorems, or ansatz smuggling appear in the provided text. The derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal elements explicitly mentioned: the Schrödinger-equation solutions and the phenomenological interaction.

free parameters (1)

parameters of the phenomenological two-body interaction
Invoked to study production channels for A=4 clusters; values are not stated and are therefore treated as adjustable.

axioms (1)

domain assumption Solutions of the Schrödinger equation yield realistic N-body wave functions for the clusters under study
Invoked when the abstract contrasts microscopic wave functions with Gaussian ansätze.

pith-pipeline@v0.9.1-grok · 5607 in / 1116 out tokens · 20826 ms · 2026-06-27T11:17:11.291904+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 5 internal anchors

[1]

Modern Ab Initio Approaches and Applications in Few-Nucleon Physics with A \ge 4

W. Leidemann, G. Orlandini, Modern Ab Initio Approaches and Applications in Few-Nucleon Physics with A >=4, Prog. Part. Nucl. Phys. 68 (2013) 158–214. arXiv:1204.4617, doi:10.1016/j.ppnp.2012.09.001

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.ppnp.2012.09.001 2013
[2]

Ekström, C

A. Ekström, C. Forssén, G. Hagen, G. R. Jansen, W. Jiang, T. Papenbrock, What is ab initio in nuclear theory?, Front. Phys. 11 (2023) 1129094. arXiv:2212.11064, doi:10.3389/fphy.2023.1129094

work page doi:10.3389/fphy.2023.1129094 2023
[3]

J. Zhao, J. Aichelin, E. Bratkovskaya, Wigner phase-space densities of nuclear clusters and hypernuclei, Phys. Rev. C 112 (6) (2025) 064902. arXiv:2508.05814, doi:10.1103/nfmh-8zzq

work page doi:10.1103/nfmh-8zzq 2025
[4]

State Dependent Effective Interaction for the Hyperspherical Formalism

N. Barnea, W. Leidemann, G. Orlandini, State dependent effective interaction for the hyperspherical formalism, Phys. Rev. C 61 (2000) 054001. arXiv:nucl-th/9910062, doi:10.1103/PhysRevC.61.054001

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.61.054001 2000
[5]

L. E. Marcucci, J. Dohet-Eraly, L. Girlanda, A. Gnech, A. Kievsky, M. Viviani, The Hyperspherical Har- monics method: a tool for testing and improving nuclear interaction models, Front. in Phys. 8 (2020) 69. arXiv:1912.09751, doi:10.3389/fphy.2020.00069

work page doi:10.3389/fphy.2020.00069 2020
[6]

J. Zhao, S. Shi, P. Zhuang, Fully-heavy tetraquarks in a strongly interacting medium, Phys. Rev. D 102 (11) (2020) 114001. arXiv:2009.10319, doi:10.1103/PhysRevD.102.114001

work page doi:10.1103/physrevd.102.114001 2020
[7]

Y . H. Leung, Y . Zhou, N. Herrmann, Data-guided coalescence model for production of light nuclei and hypernuclei in relativistic heavy-ion collisions at sNN=3–200GeV, Phys. Rev. C 113 (3) (2026) 034912. arXiv:2510.06758, doi:10.1103/9m2b-99x4

work page doi:10.1103/9m2b-99x4 2026
[8]

Mahlein, L

M. Mahlein, L. Barioglio, F. Bellini, L. Fabbietti, C. Pinto, B. Singh, S. Tripathy, A realistic coalescence model for deuteron production, Eur. Phys. J. C 83 (9) (2023) 804. arXiv:2302.12696, doi:10.1140/epjc/s10052-023- 11972-3

work page doi:10.1140/epjc/s10052-023- 2023
[9]

Pi-mesonic decay of the hypertriton

H. Kamada, J. Golak, K. Miyagawa, H. Witala, W. Gloeckle, Pi mesonic decay of the hypertriton, Phys. Rev. C 57 (1998) 1595–1603. arXiv:nucl-th/9709035, doi:10.1103/PhysRevC.57.1595

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.57.1595 1998
[10]

A. Gal, E. V . Hungerford, D. J. Millener, Strangeness in nuclear physics, Rev. Mod. Phys. 88 (3) (2016) 035004. arXiv:1605.00557, doi:10.1103/RevModPhys.88.035004

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/revmodphys.88.035004 2016
[11]

Lambda-Lambda interaction from relativistic heavy-ion collisions

K. Morita, T. Furumoto, A. Ohnishi,ΛΛinteraction from relativistic heavy-ion collisions, Phys. Rev. C 91 (2) (2015) 024916. arXiv:1408.6682, doi:10.1103/PhysRevC.91.024916

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.91.024916 2015
[12]

Acharya, et al., Measurement of (anti)alpha production in central Pb–Pb collisions at sNN=5.02 TeV, Phys

S. Acharya, et al., Measurement of (anti)alpha production in central Pb–Pb collisions at sNN=5.02 TeV, Phys. Lett. B 858 (2024) 138943. arXiv:2311.11758, doi:10.1016/j.physletb.2024.138943

work page doi:10.1016/j.physletb.2024.138943 2024
[13]

Abdulhamid, et al., Production of protons and light nuclei in Au+Au collisions at sNN=3 GeV with the STAR detector, Phys

M. Abdulhamid, et al., Production of protons and light nuclei in Au+Au collisions at sNN=3 GeV with the STAR detector, Phys. Rev. C 110 (5) (2024) 054911. arXiv:2311.11020, doi:10.1103/PhysRevC.110.054911. 4

work page doi:10.1103/physrevc.110.054911 2024

[1] [1]

Modern Ab Initio Approaches and Applications in Few-Nucleon Physics with A \ge 4

W. Leidemann, G. Orlandini, Modern Ab Initio Approaches and Applications in Few-Nucleon Physics with A >=4, Prog. Part. Nucl. Phys. 68 (2013) 158–214. arXiv:1204.4617, doi:10.1016/j.ppnp.2012.09.001

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.ppnp.2012.09.001 2013

[2] [2]

Ekström, C

A. Ekström, C. Forssén, G. Hagen, G. R. Jansen, W. Jiang, T. Papenbrock, What is ab initio in nuclear theory?, Front. Phys. 11 (2023) 1129094. arXiv:2212.11064, doi:10.3389/fphy.2023.1129094

work page doi:10.3389/fphy.2023.1129094 2023

[3] [3]

J. Zhao, J. Aichelin, E. Bratkovskaya, Wigner phase-space densities of nuclear clusters and hypernuclei, Phys. Rev. C 112 (6) (2025) 064902. arXiv:2508.05814, doi:10.1103/nfmh-8zzq

work page doi:10.1103/nfmh-8zzq 2025

[4] [4]

State Dependent Effective Interaction for the Hyperspherical Formalism

N. Barnea, W. Leidemann, G. Orlandini, State dependent effective interaction for the hyperspherical formalism, Phys. Rev. C 61 (2000) 054001. arXiv:nucl-th/9910062, doi:10.1103/PhysRevC.61.054001

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.61.054001 2000

[5] [5]

L. E. Marcucci, J. Dohet-Eraly, L. Girlanda, A. Gnech, A. Kievsky, M. Viviani, The Hyperspherical Har- monics method: a tool for testing and improving nuclear interaction models, Front. in Phys. 8 (2020) 69. arXiv:1912.09751, doi:10.3389/fphy.2020.00069

work page doi:10.3389/fphy.2020.00069 2020

[6] [6]

J. Zhao, S. Shi, P. Zhuang, Fully-heavy tetraquarks in a strongly interacting medium, Phys. Rev. D 102 (11) (2020) 114001. arXiv:2009.10319, doi:10.1103/PhysRevD.102.114001

work page doi:10.1103/physrevd.102.114001 2020

[7] [7]

Y . H. Leung, Y . Zhou, N. Herrmann, Data-guided coalescence model for production of light nuclei and hypernuclei in relativistic heavy-ion collisions at sNN=3–200GeV, Phys. Rev. C 113 (3) (2026) 034912. arXiv:2510.06758, doi:10.1103/9m2b-99x4

work page doi:10.1103/9m2b-99x4 2026

[8] [8]

Mahlein, L

M. Mahlein, L. Barioglio, F. Bellini, L. Fabbietti, C. Pinto, B. Singh, S. Tripathy, A realistic coalescence model for deuteron production, Eur. Phys. J. C 83 (9) (2023) 804. arXiv:2302.12696, doi:10.1140/epjc/s10052-023- 11972-3

work page doi:10.1140/epjc/s10052-023- 2023

[9] [9]

Pi-mesonic decay of the hypertriton

H. Kamada, J. Golak, K. Miyagawa, H. Witala, W. Gloeckle, Pi mesonic decay of the hypertriton, Phys. Rev. C 57 (1998) 1595–1603. arXiv:nucl-th/9709035, doi:10.1103/PhysRevC.57.1595

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.57.1595 1998

[10] [10]

A. Gal, E. V . Hungerford, D. J. Millener, Strangeness in nuclear physics, Rev. Mod. Phys. 88 (3) (2016) 035004. arXiv:1605.00557, doi:10.1103/RevModPhys.88.035004

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/revmodphys.88.035004 2016

[11] [11]

Lambda-Lambda interaction from relativistic heavy-ion collisions

K. Morita, T. Furumoto, A. Ohnishi,ΛΛinteraction from relativistic heavy-ion collisions, Phys. Rev. C 91 (2) (2015) 024916. arXiv:1408.6682, doi:10.1103/PhysRevC.91.024916

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.91.024916 2015

[12] [12]

Acharya, et al., Measurement of (anti)alpha production in central Pb–Pb collisions at sNN=5.02 TeV, Phys

S. Acharya, et al., Measurement of (anti)alpha production in central Pb–Pb collisions at sNN=5.02 TeV, Phys. Lett. B 858 (2024) 138943. arXiv:2311.11758, doi:10.1016/j.physletb.2024.138943

work page doi:10.1016/j.physletb.2024.138943 2024

[13] [13]

Abdulhamid, et al., Production of protons and light nuclei in Au+Au collisions at sNN=3 GeV with the STAR detector, Phys

M. Abdulhamid, et al., Production of protons and light nuclei in Au+Au collisions at sNN=3 GeV with the STAR detector, Phys. Rev. C 110 (5) (2024) 054911. arXiv:2311.11020, doi:10.1103/PhysRevC.110.054911. 4

work page doi:10.1103/physrevc.110.054911 2024