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arxiv: 2604.11918 · v1 · submitted 2026-04-13 · ⚛️ nucl-th

Parameter-free deformation variables of the proxy-SU(3) symmetry in even-even atomic nuclei with Z=28-82, N=28-126

Dennis Bonatsos , V. K. B. Kota , Andriana Martinou , S. K. Peroulis , D. Petrellis , P. Vasileiou , T. J. Mertzimekis , N. Minkov This is my paper

Pith reviewed 2026-05-10 15:34 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords proxy-SU(3)deformation variablesbeta and gammaeven-even nucleihighest-weight irrepnuclear shapesshell modelPauli principle
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0 comments X

The pith

Proxy-SU(3) symmetry predicts nuclear deformation variables beta and gamma parameter-free for even-even nuclei with Z=28-82 and N=28-126.

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

The paper demonstrates that the proxy-SU(3) approximation restores SU(3) symmetry in the shell model beyond the sd shell and uses it to calculate the collective shape parameters beta and gamma for even-even nuclei. Selection of the highest-weight irreducible representation of SU(3), dictated by the Pauli principle and the short-range attractive nucleon-nucleon force, supplies the predictions without any adjustable parameters. Complete tables of these representations and the resulting beta and gamma values are supplied for the entire range from nickel to lead isotopes, with a few examples showing how the assignments illuminate specific trends in different parts of the nuclear chart. A sympathetic reader would care because the approach covers a large fraction of the chart of stable and near-stable nuclei using only group-theoretical rules.

Core claim

The proxy-SU(3) approximation predicts the collective deformation variables beta and gamma of even-even atomic nuclei in a parameter-free way, based on the most symmetric irreducible representation of SU(3) allowed by the Pauli principle and the short-range nature of the nucleon-nucleon interaction, which in group theoretical language is the highest weight irrep. In the few cases in which the hw irrep turns out to be completely symmetric and able to accommodate only the ground state band, the next hw irrep becomes indispensable. Complete tables of the hw and nhw irreps are given for all nuclei from Z=28, N=28 to Z=82, N=126, along with the corresponding parameter-free predictions for beta, 2

What carries the argument

The highest-weight (hw) irreducible representation of SU(3) under the proxy-SU(3) mapping, which selects the most symmetric allowed state to fix the ground-state deformation variables beta and gamma.

If this is right

  • The tabulated hw and nhw irreps directly yield beta and gamma values that can be compared to measured deformations throughout the given nuclear range.
  • When the hw irrep is fully symmetric, switching to the nhw irrep supplies the additional states needed for collective bands.
  • The assignments provide a group-theoretical explanation for the onset and variation of deformation in different mass regions.
  • The same parameter-free procedure applies uniformly from the lightest to the heaviest nuclei in the stated window.

Where Pith is reading between the lines

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

  • If the predictions hold, the dominant contribution to ground-state nuclear shapes in this range arises from symmetry selection rules rather than from details of the residual interaction.
  • The tables could be used as a starting point for calculations that later add configuration mixing to refine energies or transition rates.
  • Similar proxy mappings might be constructed for nuclei outside the stated Z and N limits provided the underlying harmonic-oscillator shell structure remains valid.

Load-bearing premise

The proxy-SU(3) mapping together with automatic choice of the highest-weight or next-highest-weight irrep is enough to fix ground-state beta and gamma accurately across the full range without corrections from the complete shell-model Hamiltonian or configuration mixing.

What would settle it

A precise experimental measurement of beta for any even-even nucleus in the Z=28-82, N=28-126 range that differs substantially from the tabulated prediction based on its hw or nhw irrep.

Figures

Figures reproduced from arXiv: 2604.11918 by Andriana Martinou, Dennis Bonatsos, D. Petrellis, N. Minkov, P. Vasileiou, S. K. Peroulis, T. J. Mertzimekis, V. K. B. Kota.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The parameter-free predictions for the collective [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The parameter-free predictions for the collective [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The parameter-free predictions for the collective [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
read the original abstract

The proxy-SU(3) approximation to the shell model, which restores the SU(3) symmetry of the 3-dimensional harmonic oscillator beyond the sd shell, predicts the collective deformation variables beta and gamma of even-even atomic nuclei in a parameter-free way, based on the most symmetric irreducible representation (irrep) of SU(3) allowed by the Pauli principle and the short-range nature of the nucleon-nucleon interaction, which in group theoretical language is the highest weight (hw) irrep. In the few cases in which the hw irrep turns out to be completely symmetric, thus being able to accommodate only the ground state band, the next hw (nhw) irrep becomes indispensable. In the present article complete tables of the hw and nhw irreps are given for all atomic nuclei ranging from Z=28, N=28 to Z=82, N=126, along with the corresponding parameter-free predictions for the deformation variables beta and gamma. A few examples using the tabulated results for providing microscopic insight for specific effects in various regions of the nuclear chart are also given.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript tabulates the highest-weight (hw) and next-highest-weight (nhw) SU(3) irreducible representations for all even-even nuclei with 28 ≤ Z ≤ 82 and 28 ≤ N ≤ 126 within the proxy-SU(3) approximation to the shell model. These irreps are selected by the Pauli principle and the short-range character of the nucleon-nucleon interaction; the corresponding (λ, μ) labels are then converted to parameter-free predictions for the collective deformation variables β and γ. A small number of illustrative examples are given to show how the tabulated values can furnish microscopic insight into specific nuclear phenomena.

Significance. If the central mapping holds, the work supplies a fully parameter-free, symmetry-based route to nuclear deformations over a wide swath of the chart, together with exhaustive tables that can be used directly by the community. The explicit construction from group labels and the absence of fitted parameters constitute a clear methodological strength, especially for regions where large-scale shell-model diagonalizations remain prohibitive.

major comments (2)
  1. [§4] §4 (examples): the claim that the hw/nhw irrep furnishes the ground-state β and γ rests on the assumption that configuration mixing and non-SU(3) terms in the realistic Hamiltonian do not shift the ground state away from this irrep. No quantitative comparison of the predicted β, γ values to experimental data or to full shell-model results is presented for any of the illustrated cases, leaving the size of the approximation error untested.
  2. [Tables 1–10] Tables 1–10 (hw/nhw tabulations): the conversion from (λ, μ) to β and γ is performed with a fixed, parameter-free formula, but the manuscript does not state the explicit expression used (or its reference) nor provide uncertainty estimates arising from the proxy-orbital mapping itself. This omission makes it impossible to judge how sensitive the tabulated deformations are to the details of the proxy construction.
minor comments (2)
  1. The notation for the deformation variables (β, γ) is introduced without a brief reminder of the standard Bohr-Mottelson definitions; a single sentence would improve readability for readers outside nuclear structure.
  2. A few table entries list the same (λ, μ) for both hw and nhw; a footnote clarifying when this occurs (completely symmetric irreps) would prevent confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the significance of our work and for the constructive comments. We respond point by point to the major comments below, indicating where revisions will be incorporated.

read point-by-point responses

  1. Referee: [§4] §4 (examples): the claim that the hw/nhw irrep furnishes the ground-state β and γ rests on the assumption that configuration mixing and non-SU(3) terms in the realistic Hamiltonian do not shift the ground state away from this irrep. No quantitative comparison of the predicted β, γ values to experimental data or to full shell-model results is presented for any of the illustrated cases, leaving the size of the approximation error untested.

    Authors: We acknowledge that the manuscript presents no quantitative comparisons of the predicted β and γ values to experimental data or full shell-model results in §4. The examples are intended to illustrate how the tabulated hw/nhw irreps can furnish microscopic insight into specific nuclear phenomena, rather than to serve as a systematic validation of the approximation. The underlying assumption—that the hw (or nhw) irrep dominates the ground state—is inherent to the proxy-SU(3) framework, which selects the most symmetric allowed irrep on the basis of the Pauli principle and the short-range character of the nucleon-nucleon interaction. We agree that an explicit discussion of the possible effects of configuration mixing and non-SU(3) terms would strengthen the presentation. In the revised manuscript we will expand the introductory paragraph of §4 to state these assumptions clearly and to note that the size of the approximation error is left for future quantitative studies, while emphasizing that the current work focuses on delivering the complete, parameter-free tabulation. revision: partial

  2. Referee: [Tables 1–10] Tables 1–10 (hw/nhw tabulations): the conversion from (λ, μ) to β and γ is performed with a fixed, parameter-free formula, but the manuscript does not state the explicit expression used (or its reference) nor provide uncertainty estimates arising from the proxy-orbital mapping itself. This omission makes it impossible to judge how sensitive the tabulated deformations are to the details of the proxy construction.

    Authors: We thank the referee for identifying this omission. The conversion from the SU(3) labels (λ, μ) to the deformation parameters β and γ employs the standard, parameter-free relations derived from the quadratic Casimir operator of SU(3) and the mapping to the collective quadrupole deformation (as introduced in the original proxy-SU(3) papers). We will add the explicit expressions together with the appropriate references in the revised manuscript, most naturally in the paragraph preceding the tables. With regard to uncertainty estimates, the proxy-orbital mapping is fixed by construction to restore the SU(3) symmetry of the harmonic oscillator as accurately as possible within each major shell; the scheme is therefore parameter-free by design. We will nevertheless insert a concise discussion of the robustness of the proxy construction, referencing earlier works that examined its accuracy, so that readers can assess the sensitivity of the tabulated values to the details of the mapping. revision: yes

Circularity Check

0 steps flagged

No circularity: beta/gamma predictions follow directly from hw/nhw irrep selection via fixed SU(3) mapping

full rationale

The derivation selects the highest-weight (or next-highest-weight) SU(3) irrep for each nucleus using the Pauli principle plus the short-range NN interaction rule, then converts the resulting (lambda, mu) labels to beta and gamma through the standard, parameter-free SU(3) deformation formulas. No step fits parameters to data or renames a fitted quantity as a prediction; the output is not equivalent to the input by construction. Self-citations to prior proxy-SU(3) papers exist but are not load-bearing for the central mapping, which remains independent and externally falsifiable against measured deformations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the proxy-SU(3) approximation itself and on the rule that selects the highest-weight irrep on the basis of the Pauli principle and the short-range attractive character of the nucleon-nucleon force; both are domain assumptions rather than derived results.

axioms (2)
  • domain assumption Proxy-SU(3) restores an effective SU(3) symmetry of the 3D harmonic oscillator beyond the sd shell
    Invoked in the first sentence of the abstract as the foundation of the method.
  • domain assumption The Pauli principle together with the short-range attractive nucleon-nucleon interaction selects the highest-weight irrep
    Explicitly stated as the basis for choosing the irrep that determines beta and gamma.

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