arxiv: 2605.02826 · v2 · submitted 2026-05-04 · ⚛️ nucl-th

Recognition: unknown

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

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:08 UTC · model grok-4.3

classification ⚛️ nucl-th

keywords three-body modelasymptotic normalization constantastrophysical S-factor8B nucleus7Be(p gamma)8Bdirect capturesolar neutrinoshyperspherical harmonics

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The pith

A three-body model of 8B gives the zero-energy S-factor for 7Be(p,γ)8B as 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 paper models the 8B and 8Li nuclei as three-body systems consisting of an alpha particle plus a helium-3 or tritium cluster plus a nucleon. It solves the Schrödinger equation with the hyperspherical Lagrange-mesh method and realistic two-body potentials to obtain binding energies, radii, and wave functions. The asymptotic normalization constants are extracted by matching the overlap of the three-body wave function with the two-body 7Be+p channel to its known asymptotic form. These ANC values are inserted into Baye's asymptotic theory to compute the astrophysical S-factor at zero energy. The resulting S17(0) value lies close to one recent solar model while exceeding the currently recommended average.

Core claim

Within the alpha + 3He + p three-body potential cluster model using the hyperspherical Lagrange-mesh method with literature two-body potentials, convergent binding energies and matter radii are obtained for the ground 2+ and excited 1+ states of 8B and 8Li at maximal hypermomenta Kmax=22 and 28. The ANC values for the 8B virtual transition are 0.211 fm^{-1/2} and 0.739 fm^{-1/2} in the spin-1 and spin-2 channels. Insertion of these constants into Baye's asymptotic theory produces the zero-energy astrophysical factor S17(0) = 22.492 ± 0.014 eV b for the 7Be(p,γ)8B direct capture process, with the spin-2 channel contributing 20.838 ± 0.014 eV b.

What carries the argument

The asymptotic normalization constants extracted from the overlap integrals of the three-body wave functions matched to their known exponential decay, which are then inserted into Baye's asymptotic theory for the radiative capture cross section.

If this is right

The spin-2 channel supplies the large majority of the total S17(0) value.
The computed S17(0) agrees with the input used in the BAR2M solar model while lying above the SF III recommended value.
The same framework supplies ANC estimates for the mirror nucleus 8Li of 0.220 fm^{-1/2} and 0.774 fm^{-1/2}.
Converged results require hypermomentum cutoffs up to Kmax = 28 for the excited 1+ state.

Where Pith is reading between the lines

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

A higher S17(0) would increase the predicted boron-8 solar neutrino flux in standard solar models.
The method can be applied directly to other low-energy capture reactions on light nuclei once suitable two-body potentials are chosen.
Differences with other S17(0) calculations may be traceable to the specific choice of two-body potentials or the treatment of channel coupling in the three-body wave functions.
Direct measurement of the ANC for 8B at low momentum transfer would provide an independent test of the model's wave-function tail.

Load-bearing premise

The three-body wave functions generated from literature two-body potentials, once matched to their asymptotic form, correctly reproduce the ANC values that determine the S-factor.

What would settle it

An experimental measurement of the asymptotic normalization constant for the 8B ground state that deviates by more than a few percent from 0.739 fm^{-1/2} in the dominant spin-2 channel.

Figures

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

**Figure 1.** Figure 1: FIG. 1. The Jacobi koordinates for the nuclei view at source ↗

**Figure 1.** Figure 1: FIG. 1. The Jacobi coordinates for the three-body system view at source ↗

**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: FIG. 2. Convergence of the ground (2 view at source ↗

**Figure 3.** Figure 3: FIG. 3. Convergence of the matter radii of the view at source ↗

**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 $\alpha-^3$He($^3$H), $\alpha$-N, and $^3$He($^3$H)-N 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. For the zero-energy astrophysical factor of the direct nuclear capture process $^7$Be(p,$\gamma)^8$B 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 most important contribution comes from the spin 2 channel 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.

Desk Editor's Note private letter to a colleague

This three-body calculation yields S17(0) = 22.5 eV b with channel details, but the reported uncertainty understates the model dependence on the fixed potentials.

read the letter

The paper gives a new numerical value for the zero-energy S-factor of the 7Be(p,gamma)8B reaction at 22.492 eV b, obtained from asymptotic normalization coefficients in a three-body model. This sits near the input used in the BAR2M solar model but above the SF III average. The calculation uses the hyperspherical Lagrange-mesh method with fixed realistic two-body potentials taken from earlier papers. Convergence is shown for the binding energies and matter radii at reasonably high hypermomentum cutoffs. The ANCs for both 8B and 8Li are extracted by matching the three-body overlap integrals to their known asymptotic behavior, and the S-factor is then computed channel by channel using Baye's formula. The spin-2 contribution dominates, as expected. This approach is straightforward and the results are presented clearly with a channel breakdown. The agreement with one solar model and the difference from another is noted without overclaiming. The soft spot is the error estimate. The ±0.014 eV b is tied to the stability of the numerical matching at the chosen Kmax values. No tests are reported that change the depths or ranges of the input potentials or that compare to alternative parametrizations. Since the S-factor depends quadratically on the ANCs, any shift from those choices would scale directly into the quoted number. That systematic piece is left unquantified, which is common in this class of models but still limits how much weight the central value can carry on its own. The paper is useful for nuclear astrophysicists who track the 8B neutrino flux and need theoretical cross-section inputs to compare against experiment. It is a careful application of an established technique rather than a new method, so the main value is the specific numbers and the convergence data. I think it deserves peer review. A referee can check the technical implementation and suggest whether a short sensitivity section would strengthen the uncertainty discussion. The work is coherent and the numerics appear solid.

Referee Report

2 major / 2 minor

Summary. The manuscript applies a three-body α + ³He + p (and mirror) cluster model within the hyperspherical Lagrange-mesh method to the ground (2⁺) and excited (1⁺) states of ⁸B and ⁸Li. Literature two-body potentials are used without adjustment; convergence of binding energies and radii is reported at K_max = 22 (ground) and 28 (excited). Asymptotic normalization coefficients are extracted by explicit matching of the three-body overlap integrals to their known asymptotic forms, giving channel-specific values 0.211 and 0.739 fm^{-1/2} for ⁸B (spin 1 and 2). These ANCs are inserted into Baye’s asymptotic formula to obtain S_{17}(0) = 22.492 ± 0.014 eV b, with the spin-2 channel contributing 20.838 ± 0.014 eV b.

Significance. If the ANCs prove robust, the calculation supplies an independent, channel-resolved theoretical value for S_{17}(0) that lies close to the input used in the BAR2M solar model and within the broader SF II uncertainty band. Explicit convergence checks at high K_max and direct matching of the overlap integrals to asymptotics are positive features, as is the clean separation of spin-1 and spin-2 contributions. The result could usefully inform solar-neutrino analyses once model-dependence is quantified.

major comments (2)

[Abstract and S-factor results] Abstract and the section describing the S_{17}(0) evaluation: the quoted uncertainty ±0.014 eV b is stated to reflect only the numerical stability of the ANC-matching procedure. Because S_{17}(0) ∝ ANC² in each channel and the two-body potentials are taken unchanged from the literature, any systematic shift arising from reasonable variations in potential depths, ranges, or alternative parametrizations would propagate directly into the central value. No such sensitivity tests are reported, so the reported precision does not capture the dominant theoretical uncertainty.
[ANC extraction and convergence tests] Section on ANC extraction and convergence: while K_max convergence is demonstrated for the three-body binding energies and radii, the manuscript does not show that the extracted ANCs themselves have stabilized with respect to further increases in K_max or with respect to the choice of the three-body Lagrange-mesh parameters. Because the final S_{17}(0) is obtained solely from these ANCs via Baye’s formula, residual truncation error in the ANCs remains unquantified.

minor comments (2)

[Abstract] Abstract: 'it's asymptotics' should read 'its asymptotics'.
[Discussion] The manuscript compares the new S_{17}(0) only to the SF II and SF III compilations and to the BAR2M value. A compact table listing additional recent theoretical and experimental determinations (with references) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and indicate the revisions made to strengthen the presentation of uncertainties and convergence.

read point-by-point responses

Referee: Abstract and the section describing the S_{17}(0) evaluation: the quoted uncertainty ±0.014 eV b is stated to reflect only the numerical stability of the ANC-matching procedure. Because S_{17}(0) ∝ ANC² in each channel and the two-body potentials are taken unchanged from the literature, any systematic shift arising from reasonable variations in potential depths, ranges, or alternative parametrizations would propagate directly into the central value. No such sensitivity tests are reported, so the reported precision does not capture the dominant theoretical uncertainty.

Authors: We agree that the uncertainty ±0.014 eV b represents only the numerical stability of the ANC-matching procedure and does not include systematic effects from variations in the two-body potentials. In the revised manuscript, we will explicitly state this in the abstract and results section. We will also add a discussion noting that the potentials are taken from established literature sources without adjustment, and that our S_{17}(0) value aligns closely with the SF II recommendation and the BAR2M solar model. A comprehensive sensitivity analysis to potential parameters is beyond the scope of the current work but will be considered in future studies. revision: partial
Referee: Section on ANC extraction and convergence: while K_max convergence is demonstrated for the three-body binding energies and radii, the manuscript does not show that the extracted ANCs themselves have stabilized with respect to further increases in K_max or with respect to the choice of the three-body Lagrange-mesh parameters. Because the final S_{17}(0) is obtained solely from these ANCs via Baye’s formula, residual truncation error in the ANCs remains unquantified.

Authors: We acknowledge that explicit convergence of the ANCs with K_max and Lagrange-mesh parameters was not demonstrated, although the binding energies and radii converge at the reported K_max values. In the revised version, we will include new figures or tables displaying the ANC values as a function of K_max for both channels, confirming stabilization at K_max = 22 (ground state) and 28 (excited state). We will also discuss the insensitivity to the Lagrange-mesh parameters in the converged regime, thereby quantifying the truncation error in the ANCs. revision: yes

Circularity Check

0 steps flagged

No circularity: S17(0) derived from external potentials and independent asymptotic formula

full rationale

The derivation proceeds as: literature two-body potentials (α-³He, α-N, ³He-N) → hyperspherical Lagrange-mesh three-body wave functions at finite K_max → numerical overlap integral matched to known asymptotic form → ANC values → insertion into Baye's external formula (Phys. Rev. C 62, 065803) to obtain S17(0). No parameter is adjusted to the S17 datum, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the quoted ±0.014 error is stated to arise only from matching precision. The central result is therefore a genuine model prediction, not a renaming or re-derivation of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on literature two-body potentials and the validity of the three-body cluster picture; no new free parameters are introduced beyond the numerical cutoff Kmax, and no new particles or forces are postulated.

axioms (2)

domain assumption The three-body potential cluster model with the chosen literature potentials accurately represents the low-energy structure of 8B and 8Li.
Invoked throughout the framework description in the abstract.
standard math The hyperspherical Lagrange-mesh expansion converges for Kmax = 22 (ground) and 28 (excited).
Stated as the basis for the reported binding energies and radii.

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Reference graph

Works this paper leans on

67 extracted references · 1 canonical work pages · 1 internal anchor

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739 fm − 1/ 2 in the spin 1 and spin 2 channels, respectively

211 fm − 1/ 2 and CI=2 = 0 . 739 fm − 1/ 2 in the spin 1 and spin 2 channels, respectively. For the ANC values of the 8Li→ 7Li+n virtual transition the estimates CI=1 = 0 . 220 fm − 1/ 2 and CI=2 =
[2]

For the zero-energy astrophysical factor of the direct nuclear capture process 7Be(p,γ)8B an estimate S17(0) = 22

774 fm − 1/ 2 are extracted. For the zero-energy astrophysical factor of the direct nuclear capture process 7Be(p,γ)8B an estimate S17(0) = 22 . 492 ± 0. 014 eV b was obtained based on the asymptotic theory developed by D. Baye [Phys. Rev. C 62, 065803 (2000)]. It was found that the most important contribution comes from the spin 2 channel with S(2) 17 (0...

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for this quantity. The ANC value can also be cal- culated within purely realistic theoretical models such as the microscopic three-body cluster model [10–12], the ab-initio no-core shell model/resonating group method (NCSM/RGM) [13], and Skyrme Hartree-Fock theory [14], as well as halo eﬀective ﬁeld theory (EFT) [15, 16] at next-to-leading order. The ANC ...

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instead of <r 2 3He >1/ 2 as for the second particle. In Fig. 3 we examine a convergence of the calculated matter radii for the ground states of the 8B and 8Li nuclei as a function of Kmax, which varies from 4 up to 22. Within the developed three-body model the matter rms radius of the 8B one-proton halo nucleus is overestimated by about 5%. At the same t...
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