arxiv: 2603.18717 · v1 · submitted 2026-03-19 · ⚛️ nucl-th

Recognition: 2 theorem links

· Lean Theorem

Primordial deuterium abundance from calculations of p(n,γ) and d(p,γ) reactions within potential-model approach

Nguyen Le Anh , Dao Nhut Anh , Hoang Thai An , Nguyen Gia Huy , Bui Minh Loc

Authors on Pith no claims yet

Pith reviewed 2026-05-15 08:44 UTC · model grok-4.3

classification ⚛️ nucl-th

keywords primordial deuteriumBig Bang nucleosynthesisp(n,gamma) reactiond(p,gamma) reactionMalfliet-Tjon potentialnuclear reactionslight-element abundances

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The pith

A single scaling factor applied consistently in the Malfliet-Tjon potential for both p(n,γ) and d(p,γ) reactions yields a primordial deuterium abundance of 2.479 × 10^{-5} that matches observations.

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

The paper calculates the cross sections for the two nuclear reactions that control deuterium production and destruction during Big Bang nucleosynthesis. It constrains one scaling factor λ from the p(n,γ) data and carries that same factor unchanged into the d(p,γ) calculation within the Malfliet-Tjon framework, including both electric and magnetic transitions. The resulting D/H ratio lies within the range measured in metal-poor damped Lyman-α systems. A reader would care because the early-universe abundance of deuterium is a direct test of the standard model of cosmology and is sensitive to small changes in low-energy nuclear dynamics.

Core claim

Within a consistent two-body potential framework based on the Malfliet-Tjon interaction that includes both E1 and M1 contributions, a single scaling factor λ is fixed by the p(n,γ) reaction and propagated without readjustment to the d(p,γ) reaction, producing D/H = 2.479^{+0.350}_{-0.177} × 10^{-5} in agreement with values inferred from metal-poor damped Lyman-α systems; modest variations of λ produce sizable shifts in the predicted deuterium and other light-element abundances.

What carries the argument

The Malfliet-Tjon potential with one overall scaling factor λ that governs the low-energy scattering wave functions for both reactions.

If this is right

The predicted D/H ratio changes noticeably when λ is varied within its allowed range.
The same consistent framework supplies reaction rates for other light nuclei produced in the Big Bang.
Agreement with observed deuterium supports the use of this potential-model approach for nucleosynthesis calculations.
Small adjustments to low-energy scattering parameters can shift the entire set of primordial abundances.

Where Pith is reading between the lines

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

If the single-λ propagation works here, the same technique could be tested on additional light-element reactions without new parameters.
Tighter experimental bounds on λ would tighten predictions for both deuterium and lithium abundances.
The method offers a way to check whether current nuclear-data uncertainties are the dominant source of spread in Big Bang abundance calculations.

Load-bearing premise

A single scaling factor constrained by the p(n,γ) reaction can be transferred unchanged to the d(p,γ) reaction inside the same potential model.

What would settle it

A direct measurement of the d(p,γ) cross section at Big Bang energies that lies outside the band obtained when the λ value fitted to p(n,γ) data is used without further adjustment.

Figures

Figures reproduced from arXiv: 2603.18717 by Bui Minh Loc, Dao Nhut Anh, Hoang Thai An, Nguyen Gia Huy, Nguyen Le Anh.

**Figure 1.** Figure 1: Cross section of p(n, γ) reaction below 1 MeV. The solid line represents the total calculated cross section, while the dashed and dotted lines show the E1 and M1 contributions, respectively. Experimental data from Suzuki et al. [16] (circles) and Nagai et al. [17] (square) are included for comparison. The dashed-dotted curve shows the χEFT calculation [29]. The shaded band shows the theoretical uncertainty… view at source ↗

**Figure 2.** Figure 2: Astrophysical S factor of d(p, γ) reaction below 2 MeV. The results are compared to experimental data [6–15] and the ab initio calculation [31]. The scaling factor is λ = 1.470 ± 0.046. cubic approximation S(E) ≈ 0.228 + 5.946E + 7.929E2 − 3.145E3 , where S and E are given in eV b and MeV, respectively. In addition, the ab initio results in Ref. [31] lie systematically above the present calculation by appr… view at source ↗

**Figure 3.** Figure 3: Primordial deuterium abundance D/H as a function of the ratio λdpg/λpng obtained in the present work. The shaded band indicates the observational constraint from Cooke et al. [1], while the vertical dashed line marks the central value λdpg/λpng = 2. equal positive and negative variations of λ do not produce symmetric changes in the reaction rates. Additionally, the thermonuclear reaction rates enter the BB… view at source ↗

read the original abstract

The $p(n,\gamma)$ and $d(p,\gamma)$ reactions are key nuclear inputs for Big Bang nucleosynthesis. In this work, both reactions are analyzed within a consistent two-body potential framework based on the Malfliet-Tjon interaction, including contributions from both $E1$ and $M1$ transitions. A single scaling factor $\lambda$ controlling the low-energy scattering dynamics is constrained by the $p(n,\gamma)$ and propagated consistently to the $d(p,\gamma)$. The obtained abundance, $\mathrm{D/H} = 2.479^{+0.350}_{-0.177}\times 10^{-5}$, is in good agreement with values inferred from metal-poor damped Lyman-$\alpha$ systems. The modest variations of $\lambda$ lead to a significant change in the predicted $\mathrm{D/H}$ ratio and light-element abundances.

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

Fits one scaling factor to p(n,γ) data then applies it unchanged to d(p,γ) inside Malfliet-Tjon to produce a D/H value that matches observations, but the transfer step is the main uncertainty.

read the letter

The paper computes the primordial D/H ratio by treating both the p(n,γ) and d(p,γ) reactions in the same Malfliet-Tjon potential model, with E1 and M1 contributions. A single scaling parameter λ is fixed on the neutron capture cross section and then used without adjustment for the deuteron capture. This produces D/H = 2.479^{+0.350}_{-0.177} × 10^{-5}, stated to agree with damped Lyman-α measurements. The consistent two-body framework across the two reactions is the clearest strength; it avoids mixing unrelated potentials and keeps the low-energy dynamics under one parametrization. The numerical output with asymmetric errors from λ variation is also concrete and directly usable for BBN codes. The soft spot is exactly the transfer of λ. The abstract already shows that modest changes in λ shift D/H by tens of percent, which means the final agreement with data rests on how well the scaled potential reproduces the deuteron bound-state wave function and the relevant matrix elements at BBN energies. Without explicit checks that the scaled potential still gives the correct deuteron binding energy, asymptotic normalization, or low-energy phase shifts, the propagation step remains an assumption rather than a validated result. Error propagation is described only at the level of λ variation, so it is not clear how other sources of uncertainty in the potential or in the reaction data are folded in. This work is for nuclear theorists and BBN modelers who already use potential models and want an alternative rate set for deuterium. A reader who needs a self-contained calculation with quoted numbers will find it straightforward, but anyone expecting a first-principles or fully microscopic result will see the dependence on the fitted λ as a limitation. The paper deserves peer review because it supplies a definite numerical prediction from a coherent setup; referees can test the scaling assumption directly against additional deuteron observables.

Referee Report

1 major / 1 minor

Summary. The manuscript calculates the p(n,γ) and d(p,γ) radiative capture reactions within a consistent two-body potential-model framework based on the Malfliet-Tjon interaction, including both E1 and M1 transitions. A single scaling factor λ is constrained by p(n,γ) data and applied without further adjustment to the d(p,γ) channel. This procedure yields a primordial deuterium abundance D/H = 2.479^{+0.350}_{-0.177}×10^{-5}, reported to be in good agreement with values inferred from metal-poor damped Lyman-α systems. The work notes that modest changes in λ produce large variations in the predicted D/H ratio and light-element abundances.

Significance. If the transfer of λ is shown to be reliable, the calculation supplies a unified potential-model treatment of two key BBN reactions and produces a D/H value consistent with observation. This would constitute a useful cross-check on standard BBN inputs. The dependence on a single fitted parameter, however, reduces the independence of the prediction and makes the result sensitive to the validity of the underlying potential for the deuteron bound state.

major comments (1)

The central D/H result rests on propagating a single scaling factor λ, fixed exclusively by p(n,γ) cross-section data, into the d(p,γ) calculation inside the same Malfliet-Tjon potential. Because the deuteron binding energy, asymptotic normalization constant, and low-energy wave functions are fixed by the potential, any mismatch after scaling directly alters the E1/M1 matrix elements at BBN energies. The abstract states that modest λ variations already produce large changes in D/H; therefore an unvalidated transfer of λ constitutes the dominant uncertainty in the claimed agreement with damped Lyman-α data.

minor comments (1)

The abstract presents asymmetric uncertainties on D/H but does not specify how these are obtained from the λ variations or from other sources; a brief description of the error propagation would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment below and will incorporate revisions to clarify the uncertainties associated with our approach.

read point-by-point responses

Referee: The central D/H result rests on propagating a single scaling factor λ, fixed exclusively by p(n,γ) cross-section data, into the d(p,γ) calculation inside the same Malfliet-Tjon potential. Because the deuteron binding energy, asymptotic normalization constant, and low-energy wave functions are fixed by the potential, any mismatch after scaling directly alters the E1/M1 matrix elements at BBN energies. The abstract states that modest λ variations already produce large changes in D/H; therefore an unvalidated transfer of λ constitutes the dominant uncertainty in the claimed agreement with damped Lyman-α data.

Authors: We agree that transferring the scaling factor λ, determined from p(n,γ) data, to the d(p,γ) channel within the Malfliet-Tjon framework is central to our calculation and represents a significant source of uncertainty, as we already emphasize through the sensitivity analysis and the reported error bars on D/H. This transfer is performed for consistency within the same two-body potential model, where λ adjusts the low-energy scattering parameters to reproduce the relevant np data; the deuteron properties then follow directly from the scaled potential without additional tuning. We have propagated modest variations in λ into the final D/H uncertainty to reflect this sensitivity. To address the referee's concern, we will revise the manuscript by adding an expanded discussion section on the justification for this procedure, its limitations for the deuteron bound state, and how it affects the E1/M1 matrix elements at BBN energies. revision: partial

Circularity Check

1 steps flagged

λ fitted solely to p(n,γ) data is propagated to d(p,γ) to obtain D/H

specific steps

fitted input called prediction [Abstract]
"A single scaling factor λ controlling the low-energy scattering dynamics is constrained by the p(n,γ) and propagated consistently to the d(p,γ). The obtained abundance, D/H = 2.479^{+0.350}_{-0.177}×10^{-5}, is in good agreement with values inferred from metal-poor damped Lyman-α systems."

λ is fixed by fitting to p(n,γ) data; the same λ is then inserted unchanged into the d(p,γ) matrix elements to produce the quoted D/H. The abundance therefore inherits its numerical value and error bars from the p(n,γ) fit by construction rather than from an independent calculation of the deuteron photodisintegration channel.

full rationale

The paper constrains a single scaling parameter λ from p(n,γ) cross-section data inside the Malfliet-Tjon potential and then applies the identical λ without further adjustment to compute the d(p,γ) rate that enters the BBN network. Because the final D/H value is obtained directly from this propagated rate, the reported abundance reduces to a quantity whose central value and uncertainty are fixed by the p(n,γ) fit rather than by an independent first-principles evaluation of both reactions. This matches the fitted-input-called-prediction pattern; the derivation chain is therefore partially circular at the level of the central result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on fitting one scaling parameter to a single reaction and assuming the same value applies to the second reaction; this introduces a free parameter that directly controls the output abundance.

free parameters (1)

λ = not numerically specified in abstract
Scaling factor controlling low-energy scattering dynamics, constrained by the p(n,γ) reaction and propagated to d(p,γ).

axioms (1)

domain assumption Malfliet-Tjon interaction provides an adequate two-body potential for these reactions at BBN energies
Used as the base interaction in the consistent framework described in the abstract.

pith-pipeline@v0.9.0 · 5469 in / 1348 out tokens · 30959 ms · 2026-05-15T08:44:13.133088+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

A single scaling factor λ controlling the low-energy scattering dynamics is constrained by the p(n, γ) and propagated consistently to the d(p, γ).
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The MT potential parameters... yield a deuteron binding energy of Eb = −2.225 MeV.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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