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REVIEW 3 major objections 3 minor 67 references

Momentum Distributions and Spatial Signatures of Proton Halos in the sd Shell

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper predicts that 26P shows the strongest proton-halo signatures of three sd-shell candidates, and that reliable identification requires combining momentum and spatial observables rather than relying on any single width.

desk verdict Competent, internally consistent predictions for three proton-halo candidates; the 26P > 27S > 31Ar ranking is plausible, but the quantitative FWHM values and the 27S/31Ar separation inherit a fixed Woods-Saxon geometry that is never tested. read the letter →

arxiv 2608.06524 v1 pith:2YY47DBY submitted 2026-08-06 nucl-th nucl-ex

Taslima S.C. Diba , Carlos A. Bertulani , Ronaldo V. Lobato This is my paper · ORCID
classification nucl-thnucl-ex PACS 25.40.-h21.10.Gv21.60.-n
keywords protonhaloquasifreeknockout(p2p)reactionslongitudinalmomentumdistribution2s1/2orbitalCoulombbarriersd-shellnucleiGlauber-eikonalmodel
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

This paper asks whether the proton-rich nuclei $^{26}$P, $^{27}$S, and $^{31}$Ar are proton halos, meaning their outermost proton spends a significant fraction of its time far outside the nuclear core. It argues that no single observable can settle this: a narrow longitudinal momentum distribution, a large root-mean-square radius, and a high probability of finding the proton outside the core must all point the same way. Using a Glauber-eikonal reaction model for quasifree $(p,2p)$ knockout at 450 MeV per nucleon, with Woods-Saxon wave functions tuned to measured proton separation energies, the paper predicts that $^{26}$P shows the strongest proton-halo signatures, $^{27}$S retains pronounced halo-like features, and $^{31}$Ar is more suppressed by its larger Coulomb barrier. If correct, the paper gives specific, testable predictions for upcoming $(p,2p)$ experiments and shows why proton halos should not be treated as neutron halos with the sign of the charge flipped.

What carries the argument

The machinery is the link between the asymptotic tail of a weakly bound proton wave function and the width of the longitudinal momentum distribution after knockout. The valence proton is placed in a $2s_{1/2}$ Woods-Saxon orbital—an $s$-wave orbital with no centrifugal barrier—whose central depth is adjusted to reproduce the experimental separation energy while the geometry ($r_0 = 1.2 A^{1/3}$ fm, $a = 0.6$ fm, $V_{so} = -20$ MeV) is fixed; the Glauber-eikonal reaction model of Ref. [45] converts that wave function into the residue momentum distribution. The complementary spatial diagnostics are the rms radius $r_{\mathrm{rms}} = [\int_0^\infty r^2 |u(r)|^2 dr]^{1/2}$ and the exterior probability $P(r > R_{\mathrm{core}}) = \int_{R_{\mathrm{core}}}^\infty |u(r)|^2 dr$, with $R_{\mathrm{core}} = 1.2 A^{1/3}$ fm, which together with the FWHM form the internally consistent halo signature.

What would settle it

Measure the longitudinal momentum distribution of the residual nucleus in $^{26}$P($p,2p$) at about 450 MeV/u on a hydrogen target with Gaussian momentum resolution of about 40 MeV/c. If the inclusive spectrum shows no component substantially narrower than the $1d_{5/2}$ line shape, while the neutron-removal mirror $^{26}$Na shows the expected narrow peak, the paper's halo ranking for $^{26}$P is falsified.

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Extended reading notes

Core claim

The central claim is that within the adopted single-particle-plus-reaction model, $^{26}$P exhibits the strongest proton-halo signatures, $^{27}$S retains pronounced halo-like features, and $^{31}$Ar is more confined, and that this ranking cannot be established from any single observable. The narrow longitudinal-momentum component produced by the weakly bound $2s_{1/2}$ orbital is the primary reaction signature, but it becomes interpretable only when combined with the single-particle rms radius and the exterior probability $P(r > R_{\mathrm{core}})$. The paper further shows that Coulomb confinement systematically broadens proton momentum distributions relative to neutrons with the same separation energy, so weak binding alone is insufficient to characterize a proton halo. The practical conclusion is that future $(p,2p)$ measurements should seek a narrow $2s_{1/2}$ peak in the inclusive momentum distribution and that a consistent set of momentum-space and coordinate-space indicators, not any single width, is the reliable halo criterion.

Load-bearing premise

The load-bearing assumption is that the true many-body wave function of the removed proton has the same long-distance tail as the single-particle Woods-Saxon wave function, whose geometry is fixed and whose depth is adjusted only to match the measured separation energy; if that tail is different, the predicted momentum widths and the halo ranking shift.

Editorial extensions

If this is right

  • A $(p,2p)$ measurement on $^{26}$P at 450 MeV/u with momentum resolution near 40 MeV/c should reveal a narrow $2s_{1/2}$ component in the inclusive longitudinal momentum distribution, with a weaker but visible version for $^{27}$S.
  • Adding realistic $1d_{5/2}$ spectroscopic strength broadens but does not erase the narrow peak, so configuration mixing does not destroy the halo signature.
  • For equal separation energies, a larger core charge broadens the momentum distribution and reduces the exterior probability, so the ordering $^{26}$P, $^{27}$S, $^{31}$Ar reflects binding energy and Coulomb barrier jointly.
  • Mirror-pair calculations predict systematically broader proton-removal than neutron-removal distributions, quantified by a positive $\Delta\mathrm{FWHM}_{\mathrm{mirror}}$, isolating the Coulomb effect.
  • No single observable—width, radius, or exterior probability—identifies a proton halo reliably; the paper's criterion requires them to agree.

Reading between the lines

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

  • Beyond the paper, the combined criterion suggests a quantitative working definition of a proton halo—for example, a minimum exterior probability or a maximum FWHM for a given separation energy—that could be applied to other proton-rich candidates.
  • If a high-resolution $(p,2p)$ measurement of $^{26}$P fails to show the predicted narrow peak, the single-particle assumption that the many-body overlap has the same long-distance tail as the Woods-Saxon wave function would be the first element to revisit, rather than the reaction model.
  • The framework is naturally extendable to heavier proton-rich nuclei, where the Coulomb barrier grows further; the authors mention this direction but do not quantify it.
  • Exclusive measurements gating on the $1d_{5/2}$ excited-state residue would separate the two components and provide a sharper test of the FWHM and spectroscopic strengths assumed here.
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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 / 3 minor

Summary. The paper computes longitudinal momentum distributions, one-proton knockout cross sections, FWHM values, rms radii, and exterior probabilities for proton removal from 26P, 27S, and 31Ar in the Glauber-eikonal (p,2p) model of Ref. [45], using Woods-Saxon single-particle wave functions adjusted to AME2020 separation energies. It also studies Coulomb-barrier systematics, mirror pairs, spectroscopic mixtures from shell-model/VS-IMSRG inputs, detector-resolution convolution, and separation-energy sensitivity. The central claim is that proton-halo identification requires combining momentum-space and coordinate-space observables, and that within this model 26P has the strongest halo signatures, 27S retains pronounced halo features, and 31Ar shows progressive suppression.

Significance. If correct, the paper provides concrete, falsifiable predictions for future R3B (p,2p) measurements: a narrow 2s1/2 component in the longitudinal momentum distribution of 26P (weaker in 27S), with the FWHM ranking driven by separation energy and Coulomb barrier. The calculations are not fitted to the halo conclusion; the ranking follows from external AME2020 separation energies and VS-IMSRG spectroscopic factors plus a standard reaction model. The explicit convolution with experimental resolution and the S_p sensitivity study in Fig. 11 are useful. The main limitation is that the single-particle geometry and the rough 1d5/2 spectroscopic inputs are not varied, which weakens the demonstrated robustness of the ranking and of the inclusive narrow-component visibility.

major comments (3)
  1. [§II.B and §V.B] The central ranking 26P > 27S > 31Ar and the quantitative FWHM values are obtained with a single fixed Woods-Saxon geometry (r0 = 1.2 A^{1/3} fm, a = 0.6 fm, V_so = −20 MeV), with only the central depth adjusted to the separation energy. Because the asymptotic normalization of the valence wave function is set by this geometry, the predicted widths and the ordering are not demonstrated to be robust to the choice of single-particle potential. The 27S (S_p = 0.87 MeV) versus 31Ar (S_p = 0.67 MeV) ordering involves a cancellation between weaker binding and stronger Coulomb confinement; Fig. 11 varies S_p only and does not test r0, a, V_so, or the charge radius used in the Coulomb potential. I request a sensitivity study (e.g., ±10–20% variations of r0 and a, or a comparison with an alternative geometry) with the resulting FWHM values and halo ranking quoted as uncertainty bands.
  2. [§II.B, Table I, and §IV.A] The inclusive momentum distributions and the conclusion that the narrow 2s1/2 component survives configuration mixing rest on the 1d5/2 spectroscopic strengths of Table I, which the text itself labels as 'rough estimates,' and on the assumed E_x = 2 MeV. No uncertainty or alternative mixture is considered. Since a smaller C^2S(2s1/2) relative to the broad 1d5/2 background would make the narrow peak harder to identify after the σ_M = 40–80 MeV/c convolution, the robustness of the inclusive narrow-component claim requires a sensitivity study varying the 2s1/2 spectroscopic factor and the 1d5/2 admixture; otherwise the prediction is underdetermined by the adopted input.
  3. [§III.C] The text at the end of the P(r > R_core) discussion states that the sensitivity of this quantity to reasonable variations of R_core is 'discussed below,' but no such discussion appears in Sections IV or V. Since Eq. (4) defines the exterior probability using an adopted R_core = 1.2 A^{1/3} fm, the paper should either provide the promised sensitivity analysis (e.g., varying R_core by ±0.2 fm or using a channel-radius prescription) or revise the text to remove the unfulfilled promise.
minor comments (3)
  1. [§III.C] Using the term 'independent observables' for r_rms, P(r > R_core), and FWHM is an overstatement because all three are computed from the same single-particle wave function; agreement among them is a model-consistency check rather than an independent empirical test. Please rephrase.
  2. [Title/Abstract] The title contains 'thesdShell' with a missing space; it should read 'the sd Shell' (the same issue appears in the PDF heading).
  3. [Data Availability] The data availability statement says the data are not publicly available because depositing them is 'not technically feasible' and/or 'prohibitive'; for calculations of this scale, providing a small companion file with the tabulated FWHM values, adopted C^2S values, and the plotted distribution curves would improve reproducibility and is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the computed observables follow from external separation energies, independent spectroscopic inputs, and a reaction model that is not introduced for this paper.

full rationale

The paper's derivation chain is not circular at the level of its central predictions. The bound-state wave functions are generated by solving the Schrödinger equation in a Woods-Saxon potential whose central depth is adjusted to reproduce experimental proton separation energies from AME2020 (external inputs), with fixed geometry and spin-orbit parameters stated directly in Sec. II B. The spectroscopic factors for the inclusive distributions are taken from VS-IMSRG and shell-model studies (Refs. [28,29,64–66]) or are explicitly labeled as rough estimates, not fitted to the halo observables being predicted. The reaction model of Aumann et al. [45] is co-authored by one of the present authors, but it predates this work, is applied uniformly to all nuclei and orbitals, and has been used by several independent groups (Refs. [47–59]); no uniqueness theorem or ansatz is imported from the authors' prior work to force the conclusion. The correlation between P(r>Rcore), r_rms, and FWHM is an internal consistency of the same single-particle wave function rather than an independent confirmation, and the paper appropriately frames its conclusions as holding 'within the present model.' The ordering 26P > 27S > 31Ar is not trivially identical to the input separation energies because 31Ar has a smaller S_p than 27S yet is predicted to be more confined; the ordering emerges from the calculation. The fixed Woods-Saxon geometry and rough 1d5/2 strengths are genuine model-sensitivity concerns, but they are limitations of the assumed structure model, not circular reductions of the predictions to their inputs.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard nuclear-structure and reaction-model assumptions rather than new physics entities. The main free parameters are the geometric choices of the Woods-Saxon well, the assumed E_x = 2 MeV for d5/2 strength, and the inferred spectroscopic factors; the halo ranking itself is driven by the external separation energies and Coulomb barriers, which are not fitted.

free parameters (6)
  • Woods-Saxon radius parameter r0 = 1.2 A^{1/3} fm
    Standard geometry chosen, not fitted to these nuclei; sets the nuclear surface position and affects the asymptotic wave function.
  • Woods-Saxon diffuseness a = 0.6 fm
    Chosen standard value; controls the surface thickness and the tail of the bound-state wave function.
  • Spin-orbit strength V_so = -20 MeV
    Assumed value; influences the 1d5/2 binding and wave function but not the 2s1/2 halo ranking.
  • Excitation energy E_x for 1d5/2 strength = 2 MeV
    Ad hoc assumption; places the lumped 1d5/2 removal strength at Sp + 2 MeV, which affects the inclusive momentum distribution width.
  • Summed 1d5/2 spectroscopic strengths = 5.2 (26P), 5.5 (27S), 5.6 (31Ar)
    Called 'rough estimates' in Sec. II B, inferred from VS-IMSRG studies; used in the inclusive mixture and affect cross sections.
  • Core radius R_core for P(r > Rcore) = 1.2 A^{1/3} fm
    Adopted definition for the exterior probability; the numerical P(r > Rcore) values depend on this choice.
assumptions (5)
  • domain assumption The valence proton is described by a single-particle Woods-Saxon wave function with fixed geometry; only the central depth is adjusted to reproduce the separation energy.
    Sec. II B. This is the core modeling assumption; the predicted FWHM and radii depend on the chosen geometry.
  • domain assumption The Glauber-eikonal factorization and sudden approximation are valid at 450 MeV/u for (p,2p), with the residue as a spectator.
    Sec. II A, following Ref. [45]. Standard for intermediate-energy knockout but an approximation.
  • standard math The inclusive cross section is an incoherent sum over final states weighted by spectroscopic factors.
    Eq. (2); assumes that distinct final states do not interfere.
  • domain assumption Core densities are taken from Skyrme-Hartree-Fock calculations with the SLy4 functional.
    Sec. II A; affects the eikonal S-matrices and absorption probabilities.
  • domain assumption Spectroscopic factors and occupancies from VS-IMSRG and shell-model studies in Refs. [28,29,64-66] are reliable; some are 'rough estimates'.
    Sec. II B and Table I; central to the inclusive mixture results.

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Cite this review

Pith. "Pith review of Momentum Distributions and Spatial Signatures of Proton Halos in the sd Shell." pith.science (2026). https://pith.science/paper/2YY47DBY

@misc{pith2026260806524,
  author       = {Pith},
  title        = {Pith review of: Momentum Distributions and Spatial Signatures of Proton Halos in the sd Shell},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YY47DBY}},
  note         = {Machine review of arXiv:2608.06524}
}
abstract

We perform a theoretical study of intermediate-energy quasifree one-proton knockout reactions on proton targets. Single-particle wave functions constrained by the experimental proton separation energies are employed to calculate longitudinal momentum distributions, one-proton removal cross sections, and full momentum-space profiles for $^{26}$P, $^{27}$S and $^{31}$Ar nuclei. To establish robust criteria to identify proton halos, the analysis is extended beyond the traditional momentum-width approach by investigating the spatial extension of the valence proton through root-mean-square radii and the probability that the proton resides outside the core nucleus, $P(r>R_{\rm core})$. We also examine Coulomb-barrier systematics, mirror-nucleus comparisons, realistic spectroscopic mixtures, finite experimental momentum resolution, and uncertainties associated with the proton separation energy. Our calculations indicate that proton-halo structure cannot be identified reliably from a single observable. A consistent interpretation emerges only when momentum distributions, spatial observables, Coulomb effects, and many-body structure are considered simultaneously. Within the present model, $^{26}$P exhibits the strongest proton-halo signatures, while $^{27}$S retains pronounced halo-like features despite its larger Coulomb barrier. The more strongly confined $^{31}$Ar provides a useful comparison and illustrates the progressive suppression of halo observables with increasing binding and core charge.

Figures

Figures reproduced from arXiv: 2608.06524 by the authors.

Figure 2
Figure 2. FIG. 2. Calculated root-mean-square (rms) radii of the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Longitudinal momentum distributions for one-proton [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Calculated probability for the valence proton to lie [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Calculated full widths at half maximum (FWHM) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Calculated probability for the valence proton to [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Calculated longitudinal momentum distributions for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Difference between the calculated longitudinal mo [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Calculated longitudinal momentum distributions [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Calculated longitudinal momentum full widths [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

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