arxiv: 2602.08811 · v2 · submitted 2026-02-09 · ✦ hep-ph

Recognition: no theorem link

Assessing the validity of the Born-Oppenheimer approximation in potential models for doubly heavy hadrons

Zi-Long Man , Hao Zhou , Si-Qiang Luo , Xiang Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-16 05:24 UTC · model grok-4.3

classification ✦ hep-ph

keywords Born-Oppenheimer approximationdoubly heavy hadronspotential modelsGaussian expansion methodheavy quark masstrial wave functionsbinding energynon-adiabatic corrections

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The pith

The Born-Oppenheimer approximation deviates from benchmark solutions in potential models for doubly heavy hadrons as the heavy quark mass increases.

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

This paper tests how well the Born-Oppenheimer approximation reproduces the properties of doubly heavy hadrons inside potential models. It treats results from the Gaussian expansion method as a reference standard and compares them to Born-Oppenheimer calculations that use either Slater-type or Gaussian-type trial wave functions. The two approaches agree reasonably when the heavy quark mass remains modest, but differences grow steadily larger with increasing mass. Slater-type functions produce stronger binding than the benchmark while Gaussian-type functions produce weaker binding, with the latter shortfall traced mainly to missing non-adiabatic corrections. The work therefore maps the mass range in which the approximation remains usable for hadron calculations.

Core claim

In potential models for doubly heavy hadrons, the Born-Oppenheimer approximation yields results close to those from the Gaussian expansion method when the heavy-quark mass is relatively small. As the heavy-quark mass increases, calculations employing Slater-type functions yield larger values than those from the Gaussian expansion method, whereas those using Gaussian-type functions lead to smaller ones. The underestimation observed in Born-Oppenheimer approximation calculations with Gaussian-type functions primarily stems from the neglect of non-adiabatic corrections, and Slater-type functions generally lead to an enhanced binding energy.

What carries the argument

Born-Oppenheimer approximation applied to potential models for doubly heavy hadrons, benchmarked against Gaussian expansion method results and using Slater-type versus Gaussian-type trial wave functions.

If this is right

Born-Oppenheimer results remain close to the benchmark at small heavy quark masses.
Slater-type trial functions systematically increase the predicted binding energy relative to the benchmark.
Gaussian-type trial functions systematically decrease the predicted binding energy, mainly because non-adiabatic corrections are omitted.
The size of the discrepancy grows with rising heavy quark mass.
The comparison clarifies the practical limits of the Born-Oppenheimer treatment for doubly heavy hadron structure.

Where Pith is reading between the lines

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

For bottom-quark systems the size of non-adiabatic corrections may exceed the accuracy needed for reliable mass predictions.
The same mass-dependent trend could appear in other quarkonium or tetraquark models that rely on the Born-Oppenheimer separation.
Testing the pattern across several different potential shapes would show whether the observed deviations are model-independent.

Load-bearing premise

The Gaussian expansion method itself supplies an accurate, converged reference solution to the potential-model equations.

What would settle it

Exact numerical solution of the same potential-model Schrödinger equation for a sequence of increasing heavy-quark masses, followed by direct comparison with the Born-Oppenheimer values obtained using Gaussian-type functions.

Figures

Figures reproduced from arXiv: 2602.08811 by Hao Zhou, Si-Qiang Luo, Xiang Liu, Zi-Long Man.

**Figure 2.** Figure 2: FIG. 2: The dependence of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

The Born-Oppenheimer approximation is widely used to investigate the properties of hydrogen-like systems and doubly heavy hadrons. However, the extent to which this approximation captures the features of such systems within potential models remains an open question. In this work, we adopt the results obtained with the Gaussian expansion method as a benchmark to assess the validity of the Born-Oppenheimer approximation within potential models for hadronic systems. We also investigate the dependence of the Born-Oppenheimer approximation results on the choice of trial wave functions. A comprehensive study of the Born-Oppenheimer approximation is carried out by performing calculations using Slater-type functions and Gaussian-type functions as trial wave functions, and by comparing the resulting predictions with those obtained from the Gaussian expansion method. We find that the calculations performed within the Born-Oppenheimer approximation are close to those obtained with the Gaussian expansion method when the heavy-quark mass is relatively small. However, as the heavy-quark mass increases, calculations employing Slater-type functions yield larger values than those from the Gaussian expansion method, whereas those using Gaussian-type functions lead to smaller ones. The use of Slater-type functions generally leads to an enhanced binding energy. The underestimation observed in Born-Oppenheimer approximation calculations with Gaussian-type functions primarily stems from the neglect of non-adiabatic corrections. This comparative study provides deeper insight into the structure of doubly heavy hadrons and helps clarify the applicability and limitations of the Born-Oppenheimer treatment within potential models.

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

The paper gives a concrete numerical check on when Born-Oppenheimer starts to deviate from the Gaussian expansion benchmark in heavy-quark potential models, but the benchmark's own convergence is not shown.

read the letter

The main takeaway is straightforward: within these potential models, the Born-Oppenheimer results track the Gaussian expansion method closely at smaller heavy-quark masses, but the gap grows as the mass increases, with Slater-type trial functions producing larger binding energies and Gaussian-type functions producing smaller ones. The authors link the Gaussian underestimation to missing non-adiabatic corrections. That mass-dependent pattern and the opposite biases from the two trial functions are the new comparative detail here. It is useful because it supplies a practical rule of thumb for when the approximation remains safe in doubly heavy systems. The work is grounded in standard techniques and does not overclaim; it simply reports the numerical differences and offers an interpretation. The comparison itself is non-circular since it pits the approximation against an independent variational method. The soft spot is the treatment of the Gaussian expansion method as the reference truth. The paper does not report tests that vary the basis size, number of Gaussians, or variational parameters to confirm that the quoted GEM values stay stable, especially as the heavy-quark mass rises. If the GEM basis needs to be enlarged at higher masses to reach the same accuracy, the observed pattern of growing discrepancies could appear even if the Born-Oppenheimer separation itself remains valid. The abstract also omits the concrete potential parameters, error estimates, and convergence criteria, which makes it difficult to judge how large the reported differences really are. This is a moderate rather than fatal limitation, but it directly affects the strength of the central claim. The paper is aimed at people who build and apply potential models for exotic hadrons. A reader in that subfield will get usable guidance on approximation limits even if the numerical support needs tightening. I would send it to peer review. The question is worth asking and the comparisons are a reasonable starting point; referees can reasonably ask for the missing convergence checks without rejecting the work outright.

Referee Report

2 major / 3 minor

Summary. The manuscript assesses the validity of the Born-Oppenheimer (BO) approximation for doubly heavy hadrons in potential models by benchmarking BO calculations (using Slater-type and Gaussian-type trial wave functions) against the Gaussian expansion method (GEM). It reports good agreement at small heavy-quark masses but increasing deviations at larger masses, with Slater-type functions producing larger binding energies than GEM and Gaussian-type functions producing smaller ones, attributing the underestimation to neglected non-adiabatic corrections.

Significance. If the GEM benchmark is shown to be converged, the work provides useful quantitative insight into the mass-dependent applicability of the BO approximation in hadronic potential models and the sensitivity to trial-function choice. It supplies concrete evidence that non-adiabatic effects become relevant for heavier systems and could inform improved modeling of doubly heavy baryons and tetraquarks.

major comments (2)

[Methods / GEM implementation] The central claim that deviations from GEM reflect BO limitations (rather than numerical artifacts) rests on GEM being a converged reference. No systematic convergence tests with respect to basis size, number of Gaussians, or variational parameters are reported, nor are stability checks shown as heavy-quark mass increases. This is load-bearing because the reported discrepancies grow with mass, exactly the regime where basis incompleteness could produce the observed pattern (Slater over, Gaussian under).
[Results] Quantitative uncertainties (error bars, basis truncation estimates) are absent from all reported energies and differences. Without them, the statistical significance of the mass-dependent deviations cannot be assessed, weakening the attribution to non-adiabatic effects.

minor comments (3)

The specific potential (Cornell, linear-plus-Coulomb, etc.) and its parameters should be stated explicitly in the text or a table, together with the precise range of heavy-quark masses examined.
[BO calculations] Reproducibility would be aided by listing the explicit functional forms and variational parameters of the Slater-type and Gaussian-type trial functions used in the BO calculations.
Figure captions and table headings should clarify whether the plotted/ tabulated quantities are binding energies, total energies, or mass differences, and should indicate the sign convention for binding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results on the validity of the Born-Oppenheimer approximation. We address each major comment below and indicate the revisions planned for the updated version.

read point-by-point responses

Referee: [Methods / GEM implementation] The central claim that deviations from GEM reflect BO limitations (rather than numerical artifacts) rests on GEM being a converged reference. No systematic convergence tests with respect to basis size, number of Gaussians, or variational parameters are reported, nor are stability checks shown as heavy-quark mass increases. This is load-bearing because the reported discrepancies grow with mass, exactly the regime where basis incompleteness could produce the observed pattern (Slater over, Gaussian under).

Authors: We agree that explicit demonstration of GEM convergence is essential to attribute the observed deviations to limitations of the Born-Oppenheimer approximation rather than numerical artifacts. In our calculations, the GEM employs up to 25 Gaussian basis functions per relative coordinate in the three-body system, selected to achieve variational convergence based on prior benchmarks for similar potentials. However, we acknowledge that systematic tests varying the basis size and stability with increasing heavy-quark mass were not reported. In the revised manuscript, we will add convergence tables and a brief discussion showing binding energies for representative masses (e.g., 1.5 GeV and 5 GeV) as the number of Gaussians increases from 10 to 30, confirming stabilization to within ~1-2 MeV. These checks will rule out basis incompleteness as the source of the mass-dependent pattern, while preserving the interpretation that non-adiabatic effects grow with mass. revision: yes
Referee: [Results] Quantitative uncertainties (error bars, basis truncation estimates) are absent from all reported energies and differences. Without them, the statistical significance of the mass-dependent deviations cannot be assessed, weakening the attribution to non-adiabatic effects.

Authors: We concur that the absence of quantitative uncertainties limits the ability to assess the significance of the deviations. The reported GEM energies are obtained from well-converged variational calculations, but truncation errors were not explicitly estimated or displayed. In the revised manuscript, we will include basis-truncation uncertainty estimates (derived from the energy difference between calculations with N and N+5 Gaussians) as error bars or parenthetical values in the tables and figures. This will allow direct evaluation of whether the mass-dependent discrepancies between BO (Slater/Gaussian) and GEM exceed the numerical precision, thereby strengthening the link to neglected non-adiabatic corrections. revision: yes

Circularity Check

0 steps flagged

Independent GEM benchmark avoids circularity in BO validity assessment

full rationale

The paper performs a direct numerical comparison of Born-Oppenheimer results (using Slater-type and Gaussian-type trial functions) against Gaussian expansion method energies treated as an external benchmark. No derivation step reduces to its own inputs by construction, no parameters are fitted from the target data and then relabeled as predictions, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The central claims rest on this explicit method-to-method comparison rather than self-referential logic. Minor self-citations for context or potential models may exist but are not load-bearing for the validity assessment.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The assessment rests on the domain assumption that potential models capture the essential dynamics of doubly heavy hadrons and that the Gaussian expansion method serves as a reliable numerical benchmark; no free parameters or new entities are explicitly introduced in the abstract.

axioms (2)

domain assumption Potential models accurately describe the interactions in doubly heavy hadrons.
The entire study is performed within potential models for these systems.
domain assumption The Gaussian expansion method provides a reliable benchmark for the exact numerical solution of the potential model.
Used explicitly to assess the validity of the Born-Oppenheimer approximation.

pith-pipeline@v0.9.0 · 5571 in / 1384 out tokens · 39675 ms · 2026-05-16T05:24:39.444458+00:00 · methodology

discussion (0)

Reference graph

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