Nonrelativistic meson masses from the Curci-Ferrari model

Anaclara Alvez; Florencia Ben\'itez; Marcela Pel\'aez; Nahuel Barrios

REVIEW 4 major objections 5 minor 110 references

Nonrelativistic meson masses from the Curci-Ferrari model

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A Yukawa-plus-Cornell potential from the Curci-Ferrari model, with a running gluon mass, reproduces charmonium, bottomonium, and B_c masses to about 70 MeV—and the fit prefers a nonzero gluon mass.

desk verdict A transparent first application of the Curci-Ferrari model to heavy-quark spectra; the claim that a nonzero gluon mass is preferred is plausible but not established, and the authors know it. read the letter →

arxiv 2509.04365 v1 pith:HCTDBJTY submitted 2025-09-04 hep-ph hep-lathep-th

Anaclara Alvez , Nahuel Barrios , Florencia Benítez , Marcela Peláez This is my paper

classification hep-phhep-lathep-th

keywords heavymesonspectraquarkoniumCurci-FerrarimodelgluonmassCornellpotentialYukawanonrelativisticSchrödingerequationcharmoniumandbottomonium

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper uses the Curci-Ferrari model, a renormalizable deformation of QCD in Landau gauge with an explicit gluon mass, as the basis of a nonrelativistic potential model for heavy quarkonia. From the tree-level one-gluon exchange amplitude it derives a Yukawa potential, adds a linear Cornell confining term, solves the Schrödinger equation for the n=1, ℓ=0,1 states, and adds spin-dependent corrections perturbatively. Fitting the five parameters to the measured charmonium, bottomonium, and B_c spectra, it reports agreement to about 70 MeV, with best-fit charm and bottom masses close to standard values and a string tension at the high end of typical ranges. The fit's minimum lies at a nonzero gluon mass (about 370 MeV at the charm scale and 312 MeV at the bottom scale), and the authors argue the massless case falls outside the region compatible with the data under their error estimate. A sympathetic reader would care because this connects a renormalizable infrared model of QCD to the observed heavy-meson spectrum and suggests the dynamical gluon mass is visible in hadronic masses.

What carries the argument

The central object is the Curci-Ferrari model in Landau gauge, a renormalizable extension of the Faddeev-Popov action with a gluon mass term; its tree-level gluon propagator is that of a massive vector field. The calculation's engine is the nonrelativistic reduction of the one-gluon exchange amplitude: applying Born's approximation to the tree-level quark-antiquark scattering amplitude with q0=0 yields the Hamiltonian, whose leading piece is the Yukawa potential plus a linear Cornell confining term. The Schrödinger equation for the dominant Hamiltonian is solved numerically, and the subleading spin-dependent terms are treated as first-order perturbations. The scale dependence enters through

What would settle it

A direct way to settle the claim is to compute the omitted O(λ²) term M2 in the mass expansion (38) explicitly, or to propagate the scan uncertainties through the error χ: if any parameter set with m→0 yields χ ≤ λ² = (g²N_c/16π²)² ≈ 0.0064 after that correction, the massless-gluon case is not excluded, and the paper's central preference for a massive gluon would fail.

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Extended reading notes

Core claim

The central claim is that the Curci-Ferrari model, treated as a nonrelativistic potential model, reproduces the low-lying spectra of charmonium, bottomonium, and B_c mesons, and that the experimental and lattice masses favor a nonvanishing gluon mass over the massless limit. The Hamiltonian is built from the Fourier transform of the one-massive-gluon exchange amplitude in the static q0=0 limit, giving a Yukawa term plus relativistic Breit-type corrections; a linear Cornell term supplies confinement. Solving the Schrödinger equation for the dominant kinetic-plus-Yukawa-plus-linear Hamiltonian and treating the remaining terms perturbatively, the authors scan the parameters (string tension, cha

Load-bearing premise

The claim that the data rule out a massless gluon rests on an order-of-magnitude estimate of the omitted next-order term (Eq. (39)) rather than on propagated parameter uncertainties, and the authors note that small parameter shifts move their contour plots; under a different or more conservative error model, the massless point could become acceptable.

Editorial extensions

If this is right

If the central claim holds, heavy-quark spectroscopy provides an independent, low-energy observable that constrains the gluon mass, complementing lattice and functional-method determinations.
The same Hamiltonian can be applied to higher radial and angular excitations or to other heavy-flavor combinations; the n=1, ℓ=0,1 restriction is a practical limit, not an obstacle of principle.
Because the best fit yields a string tension at the high end of the usual range while quark masses stay standard, the model implies that part of what other potential models call confinement is being absorbed by the massive-exchange term.
A nonzero gluon mass suppresses the short-distance Yukawa attraction relative to a massless Coulomb exchange; the successful fit of hyperfine and spin-orbit splittings means the Breit corrections are consistent with that suppression.
The paper's error criterion, χ ≲ λ², provides a template for judging when a parameter region is excluded; applying the same criterion to other observables could sharpen or overturn the massless-gluon exclusion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The massless-versus-massive comparison is only as strong as the λ² threshold: if a next-order calculation shows the omitted term M2 is suppressed relative to the measured masses, the massless point could move inside the allowed region, weakening the headline claim while leaving the spectrum fit intact.
A cleaner discriminator would be to fit the same data with the gluon mass fixed to zero and let the string tension, quark masses, and coupling float freely; if the massless fit reaches a comparable error, the data alone do not single out a gluon mass, and the preference would come from the model's running.
Because the running depends only on the ratio of scales, the choice of initialization scale is effectively arbitrary for the fit; testing different schemes would show how much of the preferred m(μc)=370 MeV is scheme-dependent.
The charmonium squared velocity estimate (roughly 0.26–0.32) is not deeply nonrelativistic, so a relativistic Bethe-Salpeter calculation in the Curci-Ferrari model, which the authors flag as future work, would be a sharper test of whether the gluon mass really improves the charmonium states.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a transparent, workmanlike first application of the Curci-Ferrari model to heavy-quark spectroscopy. The central claim that a nonzero gluon mass is preferred is not established by the evidence in the paper, but the authors themselves flag the main weakness, so the paper is honest rather than overreaching.

What's new: the potential is the standard Breit-Yukawa form, as they note, with a Cornell confining term. The new piece is using the CF running of g and m, fixed earlier to lattice correlators in [1], and fitting charmonium, bottomonium, and B_c simultaneously. That specific calculation hasn't appeared before. The fit works: largest discrepancy around 70 MeV, which is decent for a five-parameter fit to fourteen states. The derivation of the Hamiltonian is careful, and the appendices give enough detail to reproduce the expectation values.

Soft spots, in order of importance. First, the massless-gluon comparison is never actually computed. The scans start at m=0.02 GeV, and the claim that m=0 is "safely outside" the allowed region is an extrapolation. The stress-test note is right that H5 and H6 carry 1/m^2 factors; taking m→0 in the Hamiltonian before forming expectation values is not obviously regular. Second, the compatibility threshold χ≲λ^2 uses λ^2=(g^2 N_c/16π^2)^2≈0.0064, which assumes the omitted M2 term is of order the measured masses. That's a dimensional estimate, not an uncertainty propagation. The paper says so itself, in the same paragraph. Third, there is no statistical error model for the fit; χ is a relative error, and five parameters to fourteen states gives a lot of flexibility. The paper's own caveat that small parameter changes alter the contour plots undercuts the robustness of the conclusion.

That said, I don't think this is fatal. The paper's stated goal is exploratory—"first exploration" appears several times—and the conclusion is explicitly hedged. The value is in connecting the CF model to spectroscopy with a concrete calculation that can be improved. The two-step fitting procedure (fix M_b from bottomonium, then fit the rest globally) is reasonable.

Who it's for: people working on CF model applications or quarkonia potential models. It won't reshape the field, but it's a solid data point. I'd send it to peer review; a referee can ask for the m=0 limit to be done explicitly and for a better error model. For a reading group, it's a maybe—useful as an example of an honest phenomenological fit with a clear caveat.

Referee Report

4 major / 5 minor

Summary. The paper develops a nonrelativistic potential model for heavy quarkonia starting from the Curci-Ferrari (CF) model in Landau gauge. From the one-massive-gluon-exchange Born amplitude, the authors derive a Hamiltonian containing a Yukawa potential plus relativistic and spin-dependent corrections, add a Cornell linear confining term, solve the Schrödinger equation numerically for n=1, ℓ=0,1 states, and treat the remaining terms perturbatively. Using one-loop UV beta functions for the gauge coupling and gluon mass taken from earlier CF-model work, and evaluating parameters at the scale μ=M/2, they scan five parameters (string tension b, quark masses Mc and Mb, coupling g(μc), gluon mass m(μc)) and fit 14 states of charmonium, bottomonium, and Bc mesons. The reported fit agrees with data at the level of tens of MeV, with a largest discrepancy around 70 MeV, and the authors conclude that a nonvanishing gluon mass is preferred over the massless case because a vanishing m lies outside the region χ≲λ², where λ=g²Nc/(16π²). The paper is exploratory and explicitly flags several limitations of its own analysis.

Significance. If the central preference claim were robust, this work would meaningfully extend the Curci-Ferrari model to a new observable—heavy quarkonium spectroscopy—and would support the massive-gluon picture with a transparent, mostly analytic Hamiltonian. The paper has real strengths: the derivation from the Born amplitude is explicit, the perturbative corrections are listed and evaluated, the parameter scans are described, and the authors clearly acknowledge that additional studies are needed. However, the headline claim that the massless gluon case is excluded is not yet established. The massless limit is never actually computed, the χ<λ² criterion is a dimensional estimate without propagated uncertainties, and the fit is presented without a statistical error model. These issues are load-bearing for the paper’s main conclusion, so the manuscript requires major revision rather than being ready for acceptance.

major comments (4)

[Sec. VI, Eqs. (38)-(39) and Fig. 2] The central conclusion that a vanishing gluon mass is excluded is not supported by the presented evidence. The scans for the full fit start at m(μc)=0.02 GeV with step 0.01 GeV; no m=0 point is computed. The threshold χ≲λ² is obtained by assuming, in Eq. (39), that the omitted second-order term M2 is of the same order as the experimental masses; this is a dimensional estimate, not an uncertainty propagation, and the global minimum value of χ for the combined fit is never quoted. The text itself concedes that 'relatively small changes in the values of the parameters can lead to modifications on the contour plots.' A direct m=0 run, a more conservative error band, and at least a sensitivity study over scan grids are needed before one can say the massless case is 'safely outside' the allowed region.
[Sec. III, Eqs. (14)-(15)] The m→0 limit is not taken in the Hamiltonian. H5 and H6 carry explicit 1/m² factors; individually they are singular as m→0. For m=0, H6 reduces to -H5, so the combination H5+H6 is regular only through a cancellation between inverse-power terms. The paper does not perform this analytic limit before forming expectation values, and the numerical evaluation of T5 and T6 in Appendix B also uses m in denominators. Thus the massless-gluon case is not actually evaluated, even approximately, and the claim that the data disfavors m=0 rests on an extrapolation. The authors should either compute the m=0 limit of H2-H9 (or of the combined T5+T6) and solve the corresponding Schrödinger equation, or clearly restrict the claim to the scanned m>0 region.
[Sec. VI, Tables I-III] The fit is presented as a five-parameter point estimate of 14 masses with no statistical model. No uncertainties are given for the fitted parameters, the global χ minimum for the combined fit is not reported, and the parenthetical uncertainties on the calculated masses in Tables I-III are never defined. The quality of the agreement is therefore difficult to assess. In particular, the bottomonium residuals are much larger than the experimental errors (ηb differs by about 70 MeV), and the text after Fig. 3 explicitly states that the error of the procedure is underestimated for bottomonium. This weakens the claim of 'very good agreement' and makes the χ contours hard to interpret as confidence statements.
[Sec. VII and Abstract] The abstract and Section VII state that a nonvanishing gluon mass 'provides a better description' and that the data favor massive gluons, but the body of the paper concludes only that further studies are needed before reaching a robust conclusion. Given the two issues above, the strong wording in the abstract overstates what the analysis establishes. The abstract should be softened to match the conditional nature of the result, e.g., 'suggests' with the explicit caveat that the massless case has not been directly computed.

minor comments (5)

[Sec. IV, after Eq. (22)] The text lists ℓ = 0, 1/2, 1, 3/2, 2... as angular momentum eigenvalues of spherical harmonics. Orbital angular momentum in a central potential takes integer values; the half-integer entries appear to be a typo or confusion with total spin. This should be corrected.
[Fig. 2 caption] The caption does not identify the axes or the fixed parameter values for both panels unambiguously. It should state which parameters are plotted on each axis and list all parameter values used for the contours, including the gluon mass and coupling grids.
[Sec. VI] The global fit minimum χ is not given numerically, only the bottomonium-specific χb=0.0004895. Since the contours are central to the paper’s argument, the authors should quote the minimum χ for the full 14-state fit and the corresponding parameter values.
[Tables I-III] The uncertainties shown in parentheses for the calculated masses are not defined. If they come from parameter scan ranges or numerical error, this should be explained; otherwise remove them or replace with a clearly defined error estimate.
[Introduction, paragraph 3] The sentence 'preceding the confirmation through numerical simulations, such solutions had already been observed' is grammatically awkward. Rephrase for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the meson agreement is explicitly a fit, and the self-cited running couplings are external lattice-calibrated inputs.

full rationale

The paper does not present the meson spectra as a prediction: the abstract and Sec. VI state that the experimental/lattice masses are fitted, e.g. "By studying the parameter space of the model we fit the experimental mass spectrum" and "To fit the data we minimized the error χ". Agreement after a five-parameter fit is therefore a fit-quality statement, not a claimed derivation of the data, and the paper is transparent about this. The running of g and m is imported from Ref. [1], which shares an author, but those beta functions were determined in prior work from lattice correlation functions, not from meson masses; hence this self-citation is independent support rather than a circular input. The massless-vs-massive conclusion is based on a scan over m starting at 0.02 GeV together with a heuristic dimensional threshold λ² in Eqs. (38)-(39); the massless point is not directly computed and the threshold is an order-of-magnitude estimate. These are legitimate limitations or statistical overclaims, but they are not cases where a conclusion reduces to its own input by definition or where a fitted parameter is renamed a prediction. The Hamiltonian is derived from the tree-level one-gluon amplitude plus a Cornell confining term, and the perturbative corrections are computed from that Hamiltonian without assuming the target spectra. No circular step satisfying the quoted-reduction criterion is present.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

All five parameters (b, Mc, Mb, g0, m0) are fitted to the very spectra that the paper then compares with, so the reported agreement is expected. The running of g and m is imported from the one-loop CF beta functions of [1] (overlapping authorship), which were previously calibrated to lattice propagators. The Cornell string tension and the choice of renormalization scale mu = M/2 are ad hoc model inputs.

free parameters (5)

b (string tension) = 0.26 GeV^2 (global fit); 0.33 GeV^2 in bottomonium-only fit
Controls the linear confinement term Vb=br in Eq. (20); scanned 0.02-0.5 GeV^2.
Mc (charm quark mass) = 1.20 GeV
Charm quark mass, scanned 1.1-1.3 GeV.
Mb (bottom quark mass) = 4.575 GeV
Bottom quark mass; extracted from bottomonium-only fit then fixed in global fit; scanned 4.0-6.5 GeV.
g(mu_c) (gauge coupling) = 2.05 at mu_c=600 MeV
Gauge coupling at the charm scale; scanned 1.4-2.3; runs via Eq. (35).
m(mu_c) (gluon mass) = 370 MeV at mu_c=600 MeV
Gluon mass at the charm scale; scanned 0.02-0.8 GeV; runs via Eq. (36).

assumptions (6)

domain assumption The Curci-Ferrari Lagrangian (Eq. 1) with a bare gluon mass is a valid phenomenological starting point for infrared QCD.
Sec. II, last paragraph: the gluon mass term 'has not been derived from first principles' and the model is treated as a phenomenological tool.
domain assumption The one-loop ultraviolet beta functions in Eqs. (33)-(34), taken from [1], govern the running of g and m in the 3-10 GeV range.
Sec. V: the leading UV contribution to one-loop results is used, with Nf=4 active flavors.
ad hoc to paper Confinement is modeled by adding the Cornell linear potential br by hand to the one-gluon-exchange Hamiltonian.
Sec. IV, Eq. (20): 'To take into account confinement, we can add... This is the Cornell potential.'
standard math The static approximation q0=0 for the exchanged gluon momentum is valid for bound states.
Sec. III: choice q0=0, justified by the independence of bound-state poles.
standard math The corrections H1-H9 can be treated as first-order perturbations in the basis of HS eigenstates.
Sec. IV, Eq. (25): E1 = <n0|HP|n0>, with HP the sum of higher-order terms.
ad hoc to paper The running parameters g(mu) and m(mu) are evaluated at the scale mu = M/2 for each meson family.
Sec. VI: parameters are evaluated at the reduced mass scale mu_b = M_b/2 and mu_c = M_c/2; footnote shows any alpha scaling gives the same results.

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Cite this review

Pith. "Pith review of Nonrelativistic meson masses from the Curci-Ferrari model." pith.science (2026). https://pith.science/paper/HCTDBJTY

@misc{pith2026250904365,
  author       = {Pith},
  title        = {Pith review of: Nonrelativistic meson masses from the Curci-Ferrari model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCTDBJTY}},
  note         = {Machine review of arXiv:2509.04365}
}

read the original abstract

We study the mass spectrum of nonrelativistic mesons composed of charm and bottom quarks within the framework of the Curci-Ferrari model in the Landau gauge, focusing on the influence of the gluon mass on our results. We derive the Hamiltonian from the scattering amplitude of a single massive gluon exchange. By incorporating a confining Cornell potential we solve the Schr\"odinger equation for the dominant terms of the Hamiltonian, which include the kinetic energy and a Yukawa-type potential. Corrections to the energy are then introduced perturbatively. By studying the parameter space of the model we fit the experimental mass spectrum of charmonium, bottomonium and charm-bottom mesons. From the five parameters of our approach, we allow the gluon mass and the gauge coupling to run with the energy scale according to the ultraviolet one-loop renormalization flows, computed in [1]. Our results show very good agreement with the data, suggesting that a nonvanishing gluon mass provides a better description of the spectrum of heavy mesons than the massless case.

Figures

Figures reproduced from arXiv: 2509.04365 by the authors.

**Figure 2.** FIG. 2: Contour lines of the error [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: Comparison between the experimental/lattice data and the CF results for the spectrum of charmonium (top), [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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= p 2M1 w′† 1 1 − p′2 1 8M 2 1 −w′† 1 σ1·p′ 1 2M1 , (5) are the Dirac spinors associated with quarks, and v(p′

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= p 2M2   σ2·p′ 2 2M2 w′ 2 1 − p′ 2 2 8M 2 2 w′2   , ¯v(p2) = p 2M2 w† 2 σ2·p2 2M2 −w† 2 1 − p2 2 8M 2 2 , (6) are those associated with antiquarks. Due to momentum conservation the initial and final quantities are related by p′ 1 = p1 − q and p′ 2 = p2 + q. There are several ways to determine the component q0 of the gluon momentum. One of them is to ...

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6 It is clear from Eqs

We performed the following scans: for the string tension, b, the same scan as the one implemented for the bottomonium spectrum; for the charm mass, Mc, a scan from 1 .1 GeV to 1 .3 GeV, with a step size of dMc = 0.05 GeV; for the coupling, g(µc), a scan from 1.4 to 2.3, with a step size of dg = 0.05; for the gluon mass, 5 Of course, Mc depends only on the...

2023

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