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

E. M. Khokha; E. Omugbe; E. P Inyang; K. Adegoke

arxiv: 2605.14094 · v2 · 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 2026-05-19 16:55 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords bottomoniumcharmoniummass spectraperturbation theoryCoulomb potentiallogarithmic potentialquarkoniumheavy mesons

0 comments

The pith

A Coulomb plus logarithmic potential with first-order perturbation corrections reproduces bottomonium and charmonium masses to within 0.24% and 1.65% average deviation from experiment.

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

The paper develops an approximate analytical solution for the Schrödinger equation under a potential that combines a Coulomb term for short-distance one-gluon exchange with a logarithmic term for long-distance quark confinement. Parameters are fixed by fitting to known experimental masses, and the first-order energy corrections are then used to predict the spectra of low-lying vector and pseudoscalar states in bottomonium and charmonium. The resulting masses agree closely with Particle Data Group values, outperforming or matching several earlier theoretical calculations. Reliability is cross-checked by comparing the perturbative results against exact numerical solutions from the matrix Numerov method. The work shows that this simple phenomenological form captures the essential short- and long-distance features of QCD for heavy quarkonia.

Core claim

The energy equation obtained to first order in perturbation theory for the Coulomb-plus-logarithmic potential yields mass predictions for the low-order quantum states of bottomonium and charmonium that agree with experimental data to absolute percentage average deviations of 0.24% and 1.65%, respectively, while reproducing the asymptotic freedom and confinement properties required by QCD.

What carries the argument

The Coulomb plus logarithmic spin-dependent potential, inserted into the radial Schrödinger equation and treated with first-order perturbation theory to obtain an analytic energy formula.

If this is right

The same potential and perturbative formula can be applied to predict masses of additional excited states without refitting.
The approach offers a computationally lightweight alternative to full numerical solutions or lattice QCD for estimating heavy-meson spectra.
The small errors relative to exact Numerov results indicate that higher-order perturbative corrections are not required for these ground and low-lying states.
The potential form already encodes both the short-distance Coulomb behavior and the long-distance logarithmic rise demanded by QCD.

Where Pith is reading between the lines

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

If the same potential parameters also describe other heavy-quark systems such as B_c mesons, the model would gain broader phenomenological support.
Extending the comparison to decay widths or leptonic decay constants could test whether the wave functions obtained from this potential are realistic beyond the mass spectrum.
The success for low states suggests the method might serve as a quick initial estimate before investing in more computationally intensive approaches for newly discovered exotic states.

Load-bearing premise

That the first-order perturbation corrections remain sufficiently accurate for the chosen low-lying states when the potential parameters are adjusted to match the essential short- and long-distance features of QCD.

What would settle it

New experimental mass measurements for higher radial or orbital excitations of bottomonium or charmonium that lie outside the range used for parameter fitting; if those measured masses deviate systematically from the predicted values, the first-order approximation would be shown to break down.

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

Standard phenomenological fit of a Coulomb-plus-log potential to charmonium and bottomonium masses via first-order perturbation theory, with numerical checks but results driven by parameter tuning to PDG data.

read the letter

The main takeaway is that this paper takes a Coulomb term plus a logarithmic confining term with added spin dependence, applies first-order perturbation theory to get an analytic energy expression for low-lying heavy meson states, fits the strength parameters to experimental masses, and reports average deviations of 0.24% for bottomonium and 1.65% for charmonium. They also check the perturbative results against matrix Numerov numerics and find the approximation holds reasonably for the states considered. The potential is set up to reproduce short-distance one-gluon exchange and long-distance confinement, which are the usual QCD-inspired features in these models. That numerical validation step is the part that actually adds some technical value here, since it shows the first-order correction is not wildly off for the ground and first excited states. Similar combined potentials and perturbative treatments have appeared before, so the specific combination is an incremental extension rather than a sharp departure. The clear limitation is that the free parameters are adjusted directly to the PDG values for the states being calculated. The small APAD numbers therefore reflect the quality of that fit for the included states rather than independent predictions. Without a clear separation of fitted versus extrapolated states or a parameter count that is fully transparent, it is hard to judge how much the model improves on earlier work beyond the quoted percentages. This kind of calculation is mainly useful to people already working in quarkonia potential models who want a quick analytic handle on low states. A reader who needs practical mass estimates or wants to see how well perturbation theory performs on this potential form can extract something from it. The work is coherent enough on its own terms to go to a serious referee in a specialized journal, mainly to verify the algebra, the fitting procedure, and the direct comparisons to prior calculations. I would send it for review rather than desk-reject, but with the expectation that referees will press on the predictive content versus the tuning.

Referee Report

2 major / 2 minor

Summary. The paper develops an approximate analytical treatment of the Coulomb plus logarithmic spin-dependent potential using first-order perturbation theory to compute the mass spectra of bottomonium and charmonium for low-order quantum states. Potential parameters are fitted to PDG experimental data, producing APAD values of 0.24% for bottomonium and 1.65% for charmonium that are presented as excellent agreement improving on prior theoretical results; the first-order approximation is validated by comparison to matrix Numerov numerical solutions.

Significance. If the central results hold after clarification, the work offers a computationally convenient phenomenological tool that incorporates QCD-inspired short-distance Coulomb and long-distance logarithmic features for heavy-meson spectroscopy. The explicit numerical validation against the Numerov method provides a useful internal consistency check for the perturbation approach on low-lying states. The reported APAD improvement is a potential strength if demonstrated with transparent fitting details, but the overall significance remains that of a standard fitted potential model rather than a parameter-free derivation.

major comments (2)

Abstract and results sections: the reported APAD of 0.24% for bottomonium (and 1.65% for charmonium) is obtained after fitting the Coulomb and logarithmic strength coefficients to the PDG masses of the states whose spectra are calculated. This makes the low deviations expected by construction for the fitted states rather than independent predictions; the manuscript must explicitly identify the fitted versus any predicted states and state the precise number of free parameters to support the claim of improvement over previous calculations.
Perturbation and validation section: while comparison to the matrix Numerov method is used to confirm reliability of the first-order corrections, the manuscript provides no quantitative table of percentage errors per state (e.g., 1S, 2S, 1P). This detail is needed to assess whether the perturbation remains accurate once parameters are fixed by the fit, which is load-bearing for extending the method to higher states as suggested in the abstract.

minor comments (2)

Notation: ensure the symbols for the Coulomb strength, logarithmic strength, and any spin-dependent coefficients are defined once and used consistently in all equations and tables.
Tables: add a column or footnote in the mass-comparison tables indicating which states entered the parameter fit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our phenomenological approach. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: Abstract and results sections: the reported APAD of 0.24% for bottomonium (and 1.65% for charmonium) is obtained after fitting the Coulomb and logarithmic strength coefficients to the PDG masses of the states whose spectra are calculated. This makes the low deviations expected by construction for the fitted states rather than independent predictions; the manuscript must explicitly identify the fitted versus any predicted states and state the precise number of free parameters to support the claim of improvement over previous calculations.

Authors: We agree that the reported APAD values measure the goodness of fit to the PDG masses for the low-lying states included in our analysis, rather than serving as independent predictions. In the revised manuscript we will explicitly state the precise number of free parameters (the overall strength coefficients of the Coulomb and logarithmic terms, with quark masses taken from standard values and not refitted) and list the specific states (e.g., 1S, 2S, 1P for both charmonium and bottomonium) that entered the fit. This clarification will allow readers to compare our results on an equal footing with other fitted potential models in the literature. revision: yes
Referee: Perturbation and validation section: while comparison to the matrix Numerov method is used to confirm reliability of the first-order corrections, the manuscript provides no quantitative table of percentage errors per state (e.g., 1S, 2S, 1P). This detail is needed to assess whether the perturbation remains accurate once parameters are fixed by the fit, which is load-bearing for extending the method to higher states as suggested in the abstract.

Authors: We accept this observation. The revised manuscript will include a new table that reports the relative percentage error between our first-order perturbative masses and the corresponding matrix Numerov numerical solutions for each individual state (1S, 2S, 1P, etc.). This will provide a transparent, state-by-state assessment of the approximation’s accuracy after the parameters have been fixed by the fit. revision: yes

Circularity Check

1 steps flagged

Mass spectra agreement achieved by fitting free potential parameters to PDG experimental data

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%, which improves upon several previously reported theoretical results. Similarly, the vector and pseudoscalar charmonium masses were obtained with an APAD of 1.65%"

The free parameters are fitted directly to the PDG experimental masses for the low-order states; the energy equation is then used to recompute those same masses, so the low APAD values measure the quality of the fit by construction rather than constituting independent predictions from first principles.

full rationale

The paper derives an approximate energy equation via first-order perturbation theory for the Coulomb-plus-logarithmic potential, which is an independent analytical step. However, the load-bearing claim of 'excellent agreement' (APAD 0.24% bottomonium, 1.65% charmonium) and improvement over prior work is obtained after explicitly fitting the free parameters to the PDG masses of the states under study. This reduces the reported accuracy to a measure of fit quality for the included low-order states rather than an independent prediction. The Numerov numerical check validates the perturbation approximation but does not remove the fitting dependence. This matches the 'fitted input called prediction' pattern with partial circularity on the central validation claim; the derivation chain itself is not fully self-referential.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a small number of fitted potential strengths and the domain assumption that the chosen functional form simultaneously encodes asymptotic freedom and confinement; no new entities are postulated.

free parameters (1)

Coulomb and logarithmic strength coefficients
These are adjusted by fitting the derived energy equation to selected PDG masses so that the model reproduces the reported APAD values.

axioms (1)

domain assumption The Coulomb plus logarithmic spin-dependent potential captures the essential short-distance and long-distance features of QCD.
Invoked in the abstract to justify the potential choice before any fitting or calculation.

pith-pipeline@v0.9.0 · 5774 in / 1385 out tokens · 49376 ms · 2026-05-19T16:55:42.484675+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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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/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

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

V(r) = -A/r + B ln(r) + spin-dependent terms

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.
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Reference graph

Works this paper leans on

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[1] [1]

Spectroscopy and annihilation decay widths of charmonium and bottomonium excitations,

M. A. Christas. and R. Dhir, “Spectroscopy and annihilation decay widths of charmonium and bottomonium excitations,” Journal of Physics G: Nuclear and Particle Physics 51, no. 11 (2024): 115004, https://doi.org/10.1088/1361-6471/ad7bd5

work page doi:10.1088/1361-6471/ad7bd5 2024

[2] [2]

Mass spectra, Regge trajectories and decay properties of heavy -flavour mesons,

R. Mistry, M. Shah, and A. Majethiya, “Mass spectra, Regge trajectories and decay properties of heavy -flavour mesons,” Revista Mexicana de Física 70, no. 1 (2024): 010801 https://doi.org/10.31349/revmexfis.70.010801

work page doi:10.31349/revmexfis.70.010801 2024

[3] [3]

Mass Spectroscopy of Charmonium Using a Screened Potential,

M. Sreelakshmi and A. Ranjan, “Mass Spectroscopy of Charmonium Using a Screened Potential,” International Journal of Theoretical Physics 64, (2025): 58, https://doi.org/10.1007/s10773-025-05924-8

work page doi:10.1007/s10773-025-05924-8 2025

[4] [4]

Time-translation invariance symmetry break- ing hidden by finite-scale singularities

R. Garg, K. K. Vishwakarma, and A. Upadhyay, “Bottomonia in quark –antiquark confining potential,” Physica Scripta 99, no. 9 (2024): 095301, https://doi.org/10.1088/1402- 4896/ad680f

work page doi:10.1088/1402- 2024

[5] [5]

Review of Particle Physics,

C. A. Bokade, and A. Bhaghyesh, “Predictions for bottomonium from a relativistic screened potential model,” Chinese Physics C 49, no. 7 (2025): 073 102, https://doi.org/10.1088/1674- 1137/adc084

work page doi:10.1088/1674- 2025

[6] [6]

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M. A. Sultan, W. Hao, E. S. Swanson, and L. Chang, “Bottomonium meson spectrum with quenched and unquenched quark models,” The European Physical Journal A 61, no. (2025): 137, https://doi.org/10.1140/epja/s10050-025-01607-4

work page doi:10.1140/epja/s10050-025-01607-4 2025

[7] [7]

Thermophysical properties and mass spectra of meson systems via the Nikiforov –Uvarov method,

R. Horchani, O. Al Kharusi, A. N. Ikot, and F. Ahmed, “Thermophysical properties and mass spectra of meson systems via the Nikiforov –Uvarov method,” Progress of Theoretical and Experimental Physics 2024, no. 12 (2024): 123A02, https://doi.org/10.1093/ptep/ptae157

work page doi:10.1093/ptep/ptae157 2024

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