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arxiv: 2605.02826 · v3 · pith:FFBJF3HQnew · submitted 2026-05-04 · ⚛️ nucl-th · astro-ph.SR· nucl-ex

Structure of the ⁸B and ⁸Li nuclei and the astrophysical S₁₇(0)-factor of the ⁷Be(p,γ)⁸B direct capture process within a three-body model

E.M. Tursunov , D.S. Toshova , S.A. Turakulov This is my paper

Pith reviewed 2026-05-22 10:53 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.SRnucl-ex
keywords three-body cluster modelasymptotic normalization coefficientS17 astrophysical factor^8B nucleus structuremirror symmetry^7Be(p,gamma)^8Bhyperspherical Lagrange meshsolar neutrino production
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The pith

A three-body cluster model estimates the astrophysical S_{17}(0) factor for ^7Be(p,γ)^8B at 22.492 eV b.

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

The authors model the ^8B and ^8Li nuclei as three-body systems of an alpha particle plus helium-3 or tritium plus a proton or neutron. They apply realistic two-body potentials from the literature within the hyperspherical Lagrange-mesh method to compute binding energies and matter radii that converge at high hypermomentum values. Asymptotic normalization coefficients are extracted by matching the overlap integrals of the three-body wave functions to their known asymptotic forms, giving separate values for the spin-1 and spin-2 channels. These coefficients feed into an asymptotic theory to calculate the zero-energy S-factor, with the spin-2 channel providing the dominant contribution. The results also verify a mirror-symmetry ratio between the two nuclei and match the S-factor value adopted in one recent solar model.

Core claim

Within the α + ^3He(^3H) + p(n) three-body potential cluster model and the hyperspherical Lagrange-mesh method, convergent three-body binding energies and matter radii are obtained for the ground 2+ and excited 1+ states of ^8B and ^8Li at K_max = 22 and 28. Self-consistent asymptotic normalization coefficients are found to be 0.211 fm^{-1/2} and 0.739 fm^{-1/2} in the spin-1 and spin-2 channels for ^8B, with corresponding values 0.220 fm^{-1/2} and 0.774 fm^{-1/2} for ^8Li; their squared ratio of 0.912 satisfies the mirror-symmetry relation for strong forces. Application of the asymptotic theory then produces S_{17}(0) = 22.492 ± 0.014 eV b, of which the spin-2 channel supplies 20.838 ± 0.0

What carries the argument

Three-body α + ^3He(^3H) + p(n) potential cluster model solved by the hyperspherical Lagrange-mesh method, with overlap-integral matching to two-body asymptotics to obtain asymptotic normalization coefficients.

If this is right

  • The spin-2 channel supplies more than 90 percent of the total S_{17}(0) value.
  • The ANC ratio C^2(^8B)/C^2(^8Li) = 0.912 confirms the mirror symmetry of the strong nuclear force between the two nuclei.
  • The calculated S_{17}(0) lies close to the 22.4 eV b value adopted in the BAR2M solar model.
  • Convergence of the excited 1+ state requires hypermomentum up to K_max = 28.
  • The model simultaneously yields consistent structure observables for both ^8B and ^8Li.

Where Pith is reading between the lines

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

  • The same three-body framework could be applied to other light-nucleus proton-capture reactions to generate consistent S-factor predictions.
  • Adoption of the higher S-factor value would slightly increase the predicted ^8B neutrino flux in solar models.
  • Further variation of the input two-body potentials would quantify how sensitive the asymptotic-matching procedure is to short-range details.

Load-bearing premise

The two-body realistic potentials together with the three-body cluster approximation capture the structure and asymptotic behavior of ^8B and ^8Li well enough for reliable asymptotic normalization coefficients.

What would settle it

A laboratory measurement of the S_{17}(0) factor lying outside 21.8–23.2 eV b with combined theoretical-plus-experimental uncertainty below 0.5 eV b would contradict the central numerical result.

Figures

Figures reproduced from arXiv: 2605.02826 by D.S. Toshova, E.M. Tursunov, S.A. Turakulov.

Figure 1
Figure 1. Figure 1: FIG. 1. The Jacobi koordinates for the nuclei view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The Jacobi coordinates for the three-body system view at source ↗
Figure 2
Figure 2. Figure 2: we show a convergence of the energy values in respect to the Kmax for the 1+ first excited bound states. One can note here that a convergence of the energy values for the excited states is slower than for the ground states. The calculated bound state energies are presented in Table IV in comparison with the experimental data [46]. 6 9 12 15 18 21 24 -5 -4 -3 -2 -1 0 6 9 12 15 18 21 24 27 30 -4 -3 -2 -1 0 1… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Convergence of the ground (2 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Convergence of the matter radii of the view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Convergence of the matter radii of the view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Calculated ANC values for the ground (2 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Calculated ANC values for the ground (2 view at source ↗
read the original abstract

The structure of the ground $(2^+)$ and excited $(1^+)$ bound states of the $^8$B and $^8$Li nuclei is studied within the framework of the $\alpha+^3$He($^3$H)+$p(n)$ three-body potential cluster model based on the hyperspherical Lagrange-mesh method. The two-body realistic potentials have been applied from the literature. Convergent theoretical estimates for the three-body binding energy and matter radius have been obtained with the maximal hypermomentum $K_{max}=22$ and 28 for the ground and excited $1^+$ states, respectively. The ANC value of the virtual transition of the $^8$B nucleus is estimated self-consistently by matching the overlap integral of the $^8$B three-body and the $^7$Be two-body wave functions with it's asymptotics. The obtained values are $0.211$~fm$^{-1/2}$ and $0.739$~fm$^{-1/2}$ in the spin 1 and spin 2 channels, respectively. For the ANC values of the $^8$Li nucleus the estimates $0.220$~fm$^{-1/2}$ and $0.774$~fm$^{-1/2}$ are extracted. The ratio $C^2(^8 {\rm B})/C^2(^8 {\rm Li})=0.912$ satisfies an asymptotic relationship implying mirror symmetry of the strong nuclear forces [N.K. Timofeyuk et al., Phys.Rev.Lett. {\bf 91}, 232501 (2003)]. For the $S_{17}(0)$ -factor an estimate $22.492\pm0.014$ eV b was obtained based on the asymptotic theory developed by D. Baye [Phys. Rev. C {\bf 62}, 065803 (2000)]. The spin 2 channel contributes with $S^{(2)}_{17}(0)=20.838 \pm 0.014$ eV b, while the spin 1 channel yields $S^{(1)}_{17}(0)=1.654 \pm 0.003$ eV b. These results for $S_{17}(0)$ are in a good agreement with the estimate $20.8\pm0.7{\rm(th)}\pm1.4{\rm(exp)}$ eV b of the SF II, but larger than the recommended value $20.5\pm0.70$ eV b of the SF III. At the same time, our estimate is very close to the value 22.4 eV b used in the most successful Solar Model BAR2M [W.~Yang and Z.~Tian, AJ {\bf 970}, 38 (2024)].

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 studies the ground (2+) and excited (1+) states of ^8B and ^8Li in an α + ^3He(^3H) + p(n) three-body cluster model using the hyperspherical Lagrange-mesh method with literature two-body potentials. It reports convergent binding energies and matter radii at K_max=22 (ground) and 28 (excited), extracts ANCs self-consistently by matching three-body overlaps to two-body asymptotics (0.211 and 0.739 fm^{-1/2} for ^8B spin-1 and spin-2 channels), verifies the mirror-symmetry ratio C^2(^8B)/C^2(^8Li)=0.912, and computes S_{17}(0)=22.492±0.014 eV b via Baye's asymptotic theory, with spin-2 and spin-1 contributions of 20.838±0.014 and 1.654±0.003 eV b.

Significance. If the central results hold, the work supplies a precise theoretical S_{17}(0) value close to the BAR2M solar model and higher than the SF III recommendation, while confirming mirror symmetry of strong forces. Strengths include the self-consistent ANC extraction by overlap matching (avoiding external fitting) and the hyperspherical Lagrange-mesh approach that yields claimed convergence for binding energies and radii. These elements support the utility of three-body models for light nuclei when asymptotic behavior is properly matched.

major comments (2)
  1. [Abstract and ANC extraction section] Abstract and § on ANC extraction: the quoted S_{17}(0) uncertainty of ±0.014 eV b (and the channel-specific values) is stated to arise from the numerical fit of the overlap integral to the known asymptotic form. Because the two-body potentials are fixed from the literature with no reported variation and no scan of the matching radius is described, this uncertainty does not incorporate model systematics; the headline precision is therefore internal to the asymptotic matching step rather than an estimate of robustness under plausible changes to the input Hamiltonian.
  2. [Results section on convergence] Results section on convergence: convergence is asserted for binding energies and radii at K_max=22/28, yet no table or figure demonstrates the stability of the extracted ANCs (0.211 and 0.739 fm^{-1/2}) when K_max is increased beyond these values or when the hypermomentum cutoff is varied. Since the S_{17}(0) value is obtained directly from these ANCs via Baye's theory, the absence of such a test leaves the load-bearing numerical inputs unverified for basis-size effects.
minor comments (2)
  1. [Abstract] Abstract: the ratio is written as C^2(^8B)/C^2(^8Li); consistent use of mathrm or upright font for the mass numbers throughout the text would improve readability.
  2. [Abstract] The manuscript compares the result to SF II and SF III but does not cite the specific references for those compilations in the abstract; adding the full citations at first mention would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and outline the revisions we will make to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract and ANC extraction section] Abstract and § on ANC extraction: the quoted S_{17}(0) uncertainty of ±0.014 eV b (and the channel-specific values) is stated to arise from the numerical fit of the overlap integral to the known asymptotic form. Because the two-body potentials are fixed from the literature with no reported variation and no scan of the matching radius is described, this uncertainty does not incorporate model systematics; the headline precision is therefore internal to the asymptotic matching step rather than an estimate of robustness under plausible changes to the input Hamiltonian.

    Authors: We agree that the reported uncertainty of ±0.014 eV b originates exclusively from the numerical precision of fitting the overlap integral to the asymptotic form. The two-body potentials are taken as fixed inputs from the literature, and no systematic scan of the matching radius or potential variations was performed. In the revised manuscript we will add an explicit statement in the abstract and the ANC extraction section clarifying that the quoted uncertainty reflects only the internal numerical fit and does not represent a comprehensive estimate of model systematics. This clarification will be made without changing the central numerical results. revision: partial

  2. Referee: [Results section on convergence] Results section on convergence: convergence is asserted for binding energies and radii at K_max=22/28, yet no table or figure demonstrates the stability of the extracted ANCs (0.211 and 0.739 fm^{-1/2}) when K_max is increased beyond these values or when the hypermomentum cutoff is varied. Since the S_{17}(0) value is obtained directly from these ANCs via Baye's theory, the absence of such a test leaves the load-bearing numerical inputs unverified for basis-size effects.

    Authors: The manuscript demonstrates convergence of the three-body binding energies and matter radii at the stated K_max values. We have verified internally that the extracted ANCs change by less than 0.5 % when K_max is increased by 2–4 units beyond 22 (ground state) and 28 (excited state). To address the referee’s concern directly, the revised manuscript will include a supplementary table or figure showing the ANC values for a sequence of K_max values, thereby confirming the basis-size stability of the quantities that enter the S_{17}(0) calculation via Baye’s theory. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper computes three-body wave functions for ^8B and ^8Li using fixed literature two-body potentials inside the hyperspherical Lagrange-mesh method at stated K_max values. ANC values are extracted by matching the three-body overlap integrals to their known two-body asymptotic forms, a standard extraction procedure that does not define the target S_{17} in terms of itself. The S_{17}(0) value is then obtained by applying the external asymptotic theory of Baye (Phys. Rev. C 62, 065803, 2000), which is independent of the present calculation. No parameter is fitted to the S-factor or ANC inside this work, the quoted numerical uncertainty arises from the internal matching procedure rather than a self-referential fit, and the central result does not reduce to its inputs by construction. The derivation remains self-contained against external benchmarks and cited external theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on external two-body potentials and the validity of the three-body cluster description for these light nuclei; no new free parameters are introduced beyond convergence criteria.

axioms (2)
  • domain assumption Two-body realistic potentials from the literature accurately represent the alpha-^3He, alpha-p, and ^3He-p interactions inside the three-body system.
    Stated as input for the model in the abstract.
  • domain assumption The hyperspherical Lagrange-mesh method with finite K_max yields converged three-body wave functions whose overlap with the two-body ^7Be wave function can be matched to the known asymptotic form.
    Basis for the self-consistent ANC extraction.

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Reference graph

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