arxiv: 2605.14094 · v1 · pith:YNIKB75Pnew · submitted 2026-05-13 · ✦ hep-ph · hep-th

Approximate mass spectra of the heavy mesons under a Coulomb plus logarithmic spin-dependent potential function

E. Omugbe , E. P Inyang , K. Adegoke , E. M. Khokha This is my paper

Pith reviewed 2026-05-15 01:45 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords bottomoniumcharmoniummass spectraperturbation theoryCoulomb logarithmic potentialquarkoniaheavy mesons

0 comments

The pith

A Coulomb plus logarithmic potential treated with first-order perturbation theory reproduces experimental bottomonium masses with 0.24% average deviation.

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

This paper develops an approximate analytical method using perturbation theory to calculate the mass spectra of heavy mesons like bottomonium and charmonium under a potential that combines Coulomb and logarithmic terms with spin dependence. The approach aims to capture key features of quantum chromodynamics, such as asymptotic freedom at short distances and quark confinement at long distances. By fitting the potential parameters to experimental data, the calculated masses for bottomonium agree closely with observations, achieving a small average deviation that improves on prior models. The method is validated against numerical solutions, suggesting it can extend reliably to higher states.

Core claim

The energy equation derived to first-order corrections in perturbation theory for the Coulomb plus logarithmic spin-dependent potential, when parameters are fitted to Particle Data Group values, yields bottomonium masses with an absolute percentage average deviation of 0.24% and charmonium masses with 1.65%, both in good agreement with experiment.

What carries the argument

The Coulomb plus logarithmic spin-dependent potential, combined with first-order perturbation theory to obtain the energy levels for quarkonium systems.

If this is right

The potential successfully models both short-distance one-gluon exchange and long-distance confinement behaviors in QCD.
Bottomonium mass predictions improve upon several existing theoretical calculations.
Charmonium vector and pseudoscalar masses are obtained with comparable accuracy to competing models.
The approximation's reliability is confirmed by small errors compared to exact numerical solutions from the matrix Numerov method.
The approach can be extended to higher excited states of quarkonia.

Where Pith is reading between the lines

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

This framework may provide a simpler alternative to numerical methods for estimating spectra in similar potential models.
Improved accuracy for bottomonium suggests the potential form is particularly suitable for heavier quark systems.
Validation against Numerov method indicates the perturbation series converges quickly for these states.

Load-bearing premise

That first-order perturbation theory remains accurate for the chosen potential and that parameters fitted to low-lying states will continue to describe the system reliably when extended to higher excitations.

What would settle it

A measurement of a higher radial excitation mass in bottomonium that deviates by more than a few percent from the predicted value would indicate the breakdown of the first-order approximation or the potential form.

read the original abstract

In this paper, we presented an approximate analytical treatment of the Coulomb plus logarithmic potential using perturbation theory to investigate the mass spectra of bottomonium and charmonium mesons for the low-order quantum states. The derived energy equation, to first-order corrections, was employed to model the free potential parameters through fitting to experimental data of the Particle Data Group. The proposed potential successfully reproduces asymptotic freedom at short distances through one-gluon exchange interactions and quark confinement at large distances, which are the essential features of the strong interactions in Quantum chromodynamics theory. The calculated bottomonium masses exhibited excellent agreement with experimental values, yielding an absolute percentage average deviation (APAD) of 0.24%, which improves upon several previously reported theoretical results. Similarly, the vector and pseudoscalar charmonium masses were obtained with an APAD of 1.65%, demonstrating improved and comparable accuracy relative to existing competing theoretical calculations. Although our results were limited to first-order corrections to the energy spectra within the perturbation theory, the reliability of the approximation was validated by comparison with exact numerical solutions obtained using the matrix Numerov method. The small percentage errors obtained confirm the effectiveness of the phenomenological potential and perturbation approximation in describing quarkonia systems. The results suggest that the approach can be reliably extended to higher excited states.

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 0.24% APAD for bottomonium is obtained after fitting the two potential parameters directly to the PDG masses used in the comparison, so it measures fit quality more than predictive power.

read the letter

The main thing to know is that the small average deviations come from tuning the Coulomb and logarithmic coefficients to the same experimental masses that are later used to compute the APAD. Without a clear split between fitted states and out-of-sample ones, or any cross-validation, the numbers show how well the model can be adjusted rather than how well the potential form predicts new data. The Numerov check only confirms that first-order perturbation theory is accurate for those fitted states, not that the model generalizes. The paper applies standard first-order perturbation theory to a Coulomb-plus-logarithmic potential for low-lying charmonium and bottomonium states. They derive the energy formula, fit the two free parameters to PDG data, and report the resulting masses with direct comparisons to experiment and to other phenomenological calculations. They also run the matrix Numerov method on the same potential to show that the perturbative truncation error stays small. This is a straightforward, competent execution of an established approach in quarkonium phenomenology. The potential choice captures the expected short-distance Coulomb behavior and long-distance confinement, and the numerical validation step is honest and useful. The calculations are reproducible from the given formulas. The central limitation is the circularity in the validation. The paper does not state exactly which states entered the fit and which are true predictions, so the claim of improvement over prior work rests on a fitted result rather than an independent test. Extending the model to higher excitations is mentioned but not actually carried out or checked. This paper is for readers who track incremental variants of effective potentials for heavy mesons. It adds one more data point in that space but introduces no new method or first-principles insight. I would bring it to a reading group as a maybe, mainly to discuss how fitting protocols affect reported accuracy in this literature. I would not cite it unless I needed a specific comparison number. It deserves peer review because the technical steps are solid and the Numerov comparison adds value, even though the main claim would benefit from tighter language around the data usage.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an approximate analytical method based on first-order perturbation theory applied to a Coulomb plus logarithmic spin-dependent potential to compute the mass spectra of bottomonium and charmonium mesons for low-lying states. Free parameters in the potential are fitted to Particle Data Group (PDG) experimental masses, resulting in reported absolute percentage average deviations (APAD) of 0.24% for bottomonium and 1.65% for charmonium. The perturbation approximation is cross-checked against numerical solutions obtained via the matrix Numerov method.

Significance. Should the results prove robust after addressing the fitting concerns, the paper would offer a straightforward phenomenological potential that incorporates essential QCD features of asymptotic freedom at short range and confinement at long range. The small PT errors validated numerically support the use of this approximation for these systems, and the accuracy claims surpass some existing models. However, the primary value lies in demonstrating fit quality rather than strong predictive tests of the potential form.

major comments (2)

[§4 (Results)] The reported APAD of 0.24% for bottomonium masses is obtained by fitting the Coulomb and logarithmic coefficients directly to the same PDG experimental values used for the comparison. Without an explicit statement of the fitting dataset versus any out-of-sample states, or a cross-validation procedure, this metric largely reflects the optimization rather than independent agreement with experiment.
[§3 (Perturbation Theory)] While the Numerov method comparison confirms small truncation errors for the fitted low-lying states, the manuscript does not provide evidence that the first-order PT remains accurate when the fitted parameters are applied to higher excitations, as suggested for future extension in the abstract.

minor comments (2)

[Abstract] The abstract claims the potential 'successfully reproduces' the features of QCD, but this reproduction occurs after parameter fitting to data; a brief clarification on this point would improve clarity.
The definition and exact formula for the absolute percentage average deviation (APAD) should be provided explicitly in the main text for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and careful review of our manuscript. We address each major point below and will incorporate revisions to improve clarity on the fitting procedure and the scope of the perturbation theory validation.

read point-by-point responses

Referee: [§4 (Results)] The reported APAD of 0.24% for bottomonium masses is obtained by fitting the Coulomb and logarithmic coefficients directly to the same PDG experimental values used for the comparison. Without an explicit statement of the fitting dataset versus any out-of-sample states, or a cross-validation procedure, this metric largely reflects the optimization rather than independent agreement with experiment.

Authors: We acknowledge that the potential parameters were fitted directly to the PDG masses of the low-lying states included in the study, and the reported APAD values therefore measure the quality of this fit rather than independent predictive power. This fitting approach is standard in phenomenological quarkonia models to determine the strength of the Coulomb and logarithmic terms. To address the concern, we will revise Section 4 to explicitly list the specific states used for fitting (e.g., the ground and first few excited states for both bottomonium and charmonium) and clarify that the APAD reflects agreement within the fitted dataset. We will also note that tests on additional out-of-sample states or cross-validation could be pursued in extensions of the work. revision: yes
Referee: [§3 (Perturbation Theory)] While the Numerov method comparison confirms small truncation errors for the fitted low-lying states, the manuscript does not provide evidence that the first-order PT remains accurate when the fitted parameters are applied to higher excitations, as suggested for future extension in the abstract.

Authors: We agree that the Numerov validation was performed only for the low-lying fitted states and does not constitute evidence for the accuracy of first-order perturbation theory at higher excitations. The abstract's phrasing regarding reliable extension is therefore not fully supported by the presented results. We will revise the abstract and the final section to remove or qualify the suggestion of extension, stating instead that the small errors observed for low-lying states indicate the approximation is effective in the regime studied, while accuracy for higher excitations would require separate numerical checks. revision: yes

Circularity Check

1 steps flagged

Parameters fitted to PDG masses produce the reported 0.24% APAD by construction

specific steps

fitted input called prediction [Abstract]
"The derived energy equation, to first-order corrections, was employed to model the free potential parameters through fitting to experimental data of the Particle Data Group. The calculated bottomonium masses exhibited excellent agreement with experimental values, yielding an absolute percentage average deviation (APAD) of 0.24%"

Parameters are adjusted to reproduce PDG masses for low-lying states; the same masses are then used to evaluate the APAD between PT-derived masses and experiment. The reported deviation therefore follows directly from the fitting procedure rather than constituting an independent prediction of the potential form.

full rationale

The central claim of excellent agreement (0.24% APAD for bottomonium) rests on fitting the free parameters of the Coulomb+log potential to the same PDG experimental masses that are later used to compute the deviation. The energy formula is obtained via first-order PT, but the fit step makes the subsequent comparison to experiment a measure of fit quality rather than an independent test. No explicit train/test split or out-of-sample states are identified, and the Numerov comparison only validates the PT truncation for the already-fitted states. This matches the fitted_input_called_prediction pattern exactly.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on fitting several free parameters in the potential to experimental masses and on the assumption that first-order perturbation suffices for the low states examined.

free parameters (1)

Coulomb and logarithmic potential coefficients
Strength parameters of the Coulomb term and the logarithmic term are adjusted by fitting to PDG experimental masses.

axioms (1)

domain assumption First-order perturbation theory is adequate for the energy corrections in the chosen potential
The energy equation is derived only to first-order corrections using perturbation theory.

pith-pipeline@v0.9.0 · 5543 in / 1299 out tokens · 59803 ms · 2026-05-15T01:45:09.522706+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 unclear

?

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

The derived energy equation, to first-order corrections, was employed to model the free potential parameters through fitting to experimental data of the Particle Data Group.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

The proposed potential successfully reproduces asymptotic freedom... and quark confinement...

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

Works this paper leans on

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