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REVIEW 4 major objections 6 minor 2 cited by

Proton Gravitational Structure and Mass Decomposition on the Light Front

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper computes the proton's quark and gluon gravitational form factors from light-front wave functions and finds that at 4 GeV^2 the proton mass is 31.5% quark energy, 34.7% gluon field energy, 11.3% quark condensate, and 22.5% trace…

desk verdict BLFQ quark+gluon GFFs and mass decomposition are new and mostly plausible, but the unstated model-to-4-GeV scale connection makes the central comparison hinge on an unverified premise. read the letter →

arxiv 2506.07554 v1 pith:FYJUJQP4 submitted 2025-06-09 hep-ph

Sreeraj Nair , Chandan Mondal , Siqi Xu , Xingbo Zhao , James P. Vary This is my paper · ORCID
classification hep-ph MSC 81V05 PACS 12.38.-t14.20.Dh
keywords gravitationalformfactorsprotonmassdecompositionlight-frontquantizationBLFQgluonstructureD-termtraceanomalymechanicalradius
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 tries to establish that the proton's gravitational form factors—the quantities that encode how mass, pressure, and shear are distributed among quarks and gluons—can be computed nonperturbatively from light-front quantization. Using a Hamiltonian with explicit quark and gluon degrees of freedom in two Fock sectors, the authors extract quark and gluon contributions separately and compare them with lattice QCD and experimental data. A sympathetic reader would care because gluon contributions to the proton's mechanical structure have been poorly constrained; if these results hold, they provide a direct, first-principles budget for the proton's mass and a prediction for its internal pressure and shear.

What carries the argument

The central object is the light-front wave function of the proton, obtained by diagonalizing a light-front quantized Hamiltonian that includes effective QCD interactions and a confining potential. The machinery is basis light-front quantization (BLFQ): the proton state is expanded in Fock sectors |qqq> and |qqqg>, with longitudinal momentum fractions discretized by a resolution K and transverse modes expanded in two-dimensional harmonic oscillator functions. The gravitational form factors A, B, C, and D are then expressed as overlaps of these wave functions, with the gluon contribution entering through the dynamical gluon in the |qqqg> sector; this is what allows the paper to quantify gluon effects on pressure, shear, and the mass budget.

What would settle it

A decisive test would be to compute the same GFFs at two different model scales, evolve them to 4 GeV² with the QCD evolution equations, and check whether the evolved A_q, A_g, D_q, D_g match the paper's values within the quoted uncertainties. Alternatively, a precise measurement of the gluon D-term from near-threshold J/ψ photoproduction at an electron-ion collider that yields D_g(0) outside −2.04 ± 0.41 would contradict the central claim.

Watch

Extended reading notes

Core claim

The central claim is that solving the light-front QCD Hamiltonian in the |qqq> and |qqqg> Fock sectors yields gravitational form factors that agree with lattice QCD and experimental extractions, and that the proton's mass decomposes into quark energy (31.5%), gluon field energy (34.7%), quark condensate (11.3%), and trace anomaly (22.5%) at μ² = 4 GeV². The authors report A_q(0) = 0.54(04), A_g(0) = 0.46(03), D_q(0) = −1.73(35), D_g(0) = −2.04(41), giving a total D-term D(0) = −3.77 ± 0.74, and they extract mechanical and mass radii of 0.73(5) fm and 0.79(4) fm. They also find that the gluon form factor C_g(Q²) is positive while C_q(Q²) is negative, with C(0) ≈ 0, consistent with the sum rule arising from EMT conservation.

Load-bearing premise

The results are quoted at μ² = 4 GeV², but the wave functions are solved at a low-resolution 'initial scale' and no evolution equations connect the two, so the implicit assumption is that the model scale is 4 GeV² or that DGLAP/ERBL evolution of the GFFs is negligible.

Editorial extensions

If this is right

  • If the computed gluon D-term D_g(0) = −2.04(41) is correct, near-threshold charmonium photoproduction experiments can test it directly through two-gluon exchange.
  • The mass decomposition at μ² = 4 GeV² provides a common scale at which lattice QCD results and experimental extractions can be compared, sharpening the long-standing puzzle of proton mass.
  • The separate quark and gluon mechanical and mass radii (0.78/0.76 fm and 0.70/0.82 fm) give concrete targets for future electron-ion collider measurements.
  • The finding that C_q < 0 and C_g > 0 with C(0) ≈ 0 confirms, in a nonperturbative calculation, the EMT sum rule that partial contributions conspire to conserve total energy-momentum.
  • The two-term internal-energy decomposition, where gluons contribute 55.7% of the proton mass, reinforces the picture that gluons dominate the proton's rest mass.

Reading between the lines

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

  • The paper's numbers are quoted at μ² = 4 GeV², but the Hamiltonian is solved at a low-resolution scale; if proper QCD evolution were applied between scales, the percentages and radii could shift, so a natural extension is to compute GFFs at two model scales and evolve them to test the assumption.
  • The same BLFQ wave functions could be used to predict gluon GPDs and thus make contact with deeply virtual Compton scattering observables, offering a testable extension of the framework.
  • Including a |qqqgg> Fock sector would test whether the gluon contributions reported here are stable against additional gluonic degrees of freedom, which would strengthen or challenge the mass-decomposition result.
  • Since B(Q²) ≈ 0 implies J ≈ A/2, a measurement of the gluon spin contribution J_g through high-energy scattering would provide an independent check of the wave functions used here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper presents a basis light-front quantization (BLFQ) calculation of the quark and gluon gravitational form factors (GFFs) of the proton. The proton wave function is obtained from an effective light-front QCD Hamiltonian in the |qqq> and |qqqg> Fock sectors with a phenomenological confinement term in the three-quark sector, and the GFFs A_i, B_i, C_i, D_i are extracted from overlaps of these wave functions. The authors report A_q(0)=0.54(04), A_g(0)=0.46(03), D_q(0)=-1.73(35), D_g(0)=-2.04(41), C_q(0)=-0.11(02), C_g(0)=0.11(02), mechanical and mass radii 0.73(5) fm and 0.79(4) fm, and a Ji mass decomposition at mu^2=4 GeV^2 of 31.5% quark energy, 34.7% gluon field energy, 11.3% quark mass, and 22.5% trace anomaly. These are compared with lattice QCD, JLab, and vector-meson photoproduction extractions.

Significance. If correct, the computation would provide a nonperturbative, relativistic estimate of the gluon contribution to proton mechanical properties, a quantity that is currently poorly constrained. The explicit inclusion of the |qqqg> Fock sector, the separate treatment of the C_i form factors with the C(0)=0 sum rule, and the large set of falsifiable predictions (D-term, radii, mass fractions) are strengths. The calculation is not circular: the mass decomposition follows from the computed a=A_q(0) and b=A_q(0)+4C_q(0) through Ji's formulas. However, the absence of a scale-evolution procedure and the normalization-rescaled gluon comparison presently leave the central quantitative claims unverified.

major comments (4)
  1. [Proton wave functions from light-front QCD Hamiltonian; Gravitational form factors; Fig. 1] The central results are quoted at mu^2=4 GeV^2, but the Hamiltonian is diagonalized at a low-resolution model scale (Nmax=9, K=16.5, with parameters fixed to low-energy observables). No DGLAP/ERBL evolution or scale-setting argument connects the model scale to mu^2=4 GeV^2. Since A_i(0), D_i(0), C_i(0), the slopes entering Eq. (10), and hence the mass-decomposition percentages and radii are all scale-dependent, the agreement with lattice at 4 GeV^2 is not established as stated. Please either provide the evolution of the GFFs from the model scale to 4 GeV^2, or re-label all results as model-scale predictions and compare at a matching scale, or state and justify why the model scale is 4 GeV^2.
  2. [Fig. 1 caption] The statement that the shaded band represents a 10% uncertainty originating from the initial scale is not supported by any derivation or sensitivity study in the text or Supplemental Material. If this is an estimate from varying the model scale, the variation and its effect on A_i(Q^2), D_i(Q^2), and the derived percentages should be shown; otherwise the quoted uncertainties on A_q(0), A_g(0), D(0), and the mass decomposition are incomplete.
  3. [Fig. 1 caption and A_g(0) discussion] The caption says all A_g(Q^2) curves, including the authors', are rescaled to A_g(0)=0.414, while the text reports A_g(0)=0.46(03). Rescaling removes the normalization from the comparison, so the plotted 'agreement' with lattice does not test the predicted A_g(0). Please show both the unnormalized curves and the rescaling, and discuss the 0.46 vs 0.414 difference explicitly.
  4. [Confinement in |qqqg>; after Eq. (2)] The omission of explicit confinement in the |qqqg> sector is justified by the claim that the restricted transverse basis and the massive gluon capture the essential effects, but no quantitative test is provided. Since the gluon GFFs and the 34.7% gluon energy contribution are central results, a sensitivity check (e.g., varying the basis truncation or adding a confining term in the four-particle sector) is needed to show that the omitted term is not dominating the gluon observables.
minor comments (6)
  1. [Throughout] The text uses 'initial scale', 'model scale', and 'low-resolution scales' interchangeably; please define one convention and use it consistently.
  2. [Fig. 5 and two-term decomposition] The two-term decomposition reports a gluon contribution of 55.7%, whereas using U_g=A_g(0)+C_g(0)=0.46+0.11=0.57 gives 57%; the valence/sea split should be defined and the arithmetic reconciled.
  3. [Supplemental Material, Sec. III] Both dipole and tripole fits are tabulated, but the text does not state which fit was used to generate the pressure/shear distributions and the radii; please specify.
  4. [Abstract and Fig. 5] In Ji's four-term scheme, M_m is the quark mass (or sigma-term) contribution; calling it 'quark condensate' is imprecise and should be corrected.
  5. [Eq. (7) and eigenvalue equation] There are small notational/formatting issues: Eq. (7) uses the symbol N both as the Fock-sector label and in the integration measure, and the eigenvalue equation is typeset with P+P- without an explicit operator symbol; please clean these up.
  6. [Comparison with JLab D_q(Q^2)] The comparison with JLab D_q(Q^2) [11,32] should clarify that the experimental extraction involves model assumptions; the current wording 'agrees with JLab data' is stronger than warranted.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the GFFs and mass decomposition are calculated from light-front wave functions, not fitted to the quoted final values, and the results are benchmarked against external lattice and experimental data.

full rationale

The derivation chain is: solve the light-front Hamiltonian eigenvalue problem with the QCD plus confinement interactions of Eqs. (1) and (2), obtain the LFWFs, compute the quark and gluon GFFs from the overlaps in Eq. (7), and then obtain radii and mass decomposition from Eqs. (10) and the Ji formulas with a = A_q(0) and b = A_q(0) + 4 C_q(0). The Hamiltonian parameters are fixed to reproduce the proton mass and electromagnetic properties, as stated before Eq. (9) and attributed to prior BLFQ work [42]. The quantities quoted as predictions--A_q(0) = 0.54(04), A_g(0) = 0.46(03), D_q(0) = -1.73(35), D_g(0) = -2.04(41), and the mass percentages 31.5/34.7/11.3/22.5--are not among the fitted inputs; they are outcomes of wavefunction overlaps and standard sum rules. The paper independently checks quark and gluon GFFs against lattice QCD, JLab data, and phenomenological extractions [29,30,32,49,11,24,50,52], which provides external grounding not reducible to the model inputs. The self-citations to earlier BLFQ papers describe the framework and Hamiltonian but do not import the GFF or mass-decomposition results themselves, so the self-citation is not load-bearing. The main caveat is the scale identification: the Hamiltonian is solved at a low-resolution initial scale while results are quoted at mu^2 = 4 GeV^2, with no explicit DGLAP/ERBL evolution supplied. That is an unverified modeling assumption and a correctness risk, not circularity, because the quoted outputs are not defined in terms of that scale by construction. Similarly, omitting explicit confinement in the |qqqg> sector is a stated limitation rather than a circular step. Overall, the central derivation is self-contained against external benchmarks, with only minor self-citation and methodological reliance. Score 2 reflects these minor issues, not circular reasoning.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

No genuinely new physical entities are introduced: the massive gluon and effective vertex mass are model parameters, not new particles or forces. The central calculation rests on six fitted Hamiltonian parameters, basis choices, and an unstated scale-evolution assumption. These do not make the claim circular, but they mark the result as a model prediction rather than a direct first-principles computation.

free parameters (9)
  • m_u = 0.31 GeV
    Up quark mass parameter; tuned with the other Hamiltonian parameters to reproduce the proton mass and electromagnetic properties (Ref. [42]).
  • m_d = 0.25 GeV
    Down quark mass parameter; fitted as part of the same parameter set.
  • m_g = 0.50 GeV
    Phenomenological gluon mass introduced to account for nonperturbative effects; appears in the kinetic term of the gluon field.
  • kappa = 0.54 GeV
    Confinement strength in the leading Fock sector confinement potential; fitted to hadronic properties.
  • m_f = 1.80 GeV
    Effective vertex mass parameterizing nonperturbative contributions to quark renormalization; fitted with the other parameters.
  • g_c = 2.40
    Effective QCD coupling in the light-front Hamiltonian; fitted to proton mass and electromagnetic form factors.
  • HO scale b = 0.70 GeV
    Two-dimensional harmonic oscillator basis scale; chosen by hand, controls transverse resolution.
  • UV cutoff b_inst = 3.00 GeV
    Cutoff for the instantaneous interaction; chosen by hand.
  • Basis truncations Nmax and K = Nmax = 9, K = 16.5
    Truncation parameters defining the finite basis; no convergence study with respect to these cutoffs is shown.
assumptions (5)
  • domain assumption The effective light-front Hamiltonian P^- = P^-_QCD + P^-_I, with one dynamical massive gluon and no explicit confinement in the |qqqg> sector, is a sufficient approximation to QCD for computing GFFs.
    Invoked in the Hamiltonian section; the model is not full QCD and the gluon mass and missing gluon-sector confinement are phenomenological.
  • ad hoc to paper The model's low-resolution initial scale can be identified with mu^2 = 4 GeV^2, or that evolution between the two scales is negligible.
    Results are presented at 4 GeV^2, but the wave functions are solved at the 'initial scale' with no evolution equations given.
  • domain assumption The basis truncations Nmax = 9 and K = 16.5 are sufficient for convergence of the GFFs.
    The calculation uses a single truncation setting; no extrapolation or convergence study is shown.
  • standard math Ji's mass decomposition formulas with gamma_m = -0.15 for n_f = 3 correctly connect A_q(0) and C_q(0) to the four mass terms.
    The formulas are taken from the cited literature and treated as known results.
  • standard math The EMT parametrization and light-front overlap formulas in Eqs. (4)-(8) are correct for extracting A, B, D, and C.
    These are standard definitions and methods in the GFF literature.

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Cite this review

Pith. "Pith review of Proton Gravitational Structure and Mass Decomposition on the Light Front." pith.science (2026). https://pith.science/paper/FYJUJQP4

@misc{pith2026250607554,
  author       = {Pith},
  title        = {Pith review of: Proton Gravitational Structure and Mass Decomposition on the Light Front},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYJUJQP4}},
  note         = {Machine review of arXiv:2506.07554}
}
abstract

Gravitational form factors (GFFs) of hadrons encode essential information about the internal distributions of mass, spin, pressure, and shear among their quark and gluon constituents. We compute the quark and gluon GFFs of the proton using a fully relativistic, nonperturbative framework based on a light-front quantized Hamiltonian with quantum chromodynamics (QCD) input. This allows us to quantify the impact of a dynamical gluon on the proton's mechanical properties, such as pressure and shear distributions. Our predictions agree well with recent lattice QCD results and experimental extractions. We also determine the proton's mass and mechanical radii and address the long-standing puzzle of its mass decomposition. At the scale $\mu^2 = 4~\mathrm{GeV}^2$, we find that quark energy, gluon field energy, the quark condensate, and the QCD trace anomaly contribute $31.5\%$, $34.7\%$, $11.3\%$, and $22.5\%$, respectively, which are consistent with lattice QCD findings.

Figures

Figures reproduced from arXiv: 2506.07554 by the authors.

Figure 1
Figure 1. presents the proton GFFs A(Q2 ) and D(Q2 ) at µ 2 = 4 GeV2 , showing separate quark and gluon con￾tributions. We compare our results with lattice QCD and experimental extractions based on various phenomeno￾logical approaches. Our D(Q2 ) and its quark and gluon components agree well with lattice QCD [30], while A(Q2 ), particularly the quark contribution, shows some deviation at high Q2 [30, 49]. The quark form facto… view at source ↗
Figure 2
Figure 2. presents GFF C(Q2 ), highlighting opposite signs for quark (Cq < 0) and gluon (Cg > 0). The sum rule C(Q2 ) = 0 is approximately satisfied at Q2 = 0: C(0) ≈ 0, with Cq(0) = −0.11(02), Cg(0) = 0.11(02). Mechanical densities and radii.—Defined through the EMT and by analogy with classical mechanical systems, the GFF D(Q2 ) provides insight into the internal pres￾sure and shear force distributions in the proton [6]. Th… view at source ↗
Figure 4
Figure 4. FIG. 4. Proton mechanical and mass radii compared with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Proton mass decomposition following Ji’s four-term [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 2
Figure 2. Figure 2: presents the gravitational form factors (GFFs) B(Q2 ) and J(Q2 ) as functions of Q2 , separately indicating the quark and gluon contributions. At Q2 = 0, we obtain the values Jq(0) = 0.27(02), Jg(0) = 0.22(02), J(0) = 0.50(01); Bq(0) = 0.0087(15), Bg(0) = −0.0087(13), …
Figure 1
Figure 1. Figure 1: compares our calculated total proton D￾term, D(0) = −3.77 ± 0.74, with lattice QCD and other theoretical predictions. Most results are consistent within our uncertainty, particularly the lattice QCD results from the dipole fit [1], which align closely with our calculat…

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Forward citations

Cited by 2 Pith papers

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.