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REVIEW 2 major objections 3 minor 40 references

This work reports the first lattice-QCD determination of the proton's momentum and angular-momentum decomposition at the continuum limit using only physical-mass ensembles, with total sums of 0.995(60)(29) and 0.507(43)(65).

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2026-08-01 08:05 UTC pith:YILMIRN6

load-bearing objection A credible continuum-limit momentum sum-rule result, but the spin decomposition hinges on unpublished axial charges; needs the companion papers before this is more than conditional. the 2 major comments →

arxiv 2607.21230 v1 pith:YILMIRN6 submitted 2026-07-23 hep-lat hep-exhep-phnucl-exnucl-th

Spin and momentum fraction carried by partons in the nucleon

Constantia Alexandrou (University of Cyprus & The Cyprus Institute) , Simone Bacchio (The Cyprus Institute) , Jacob Finkenrath (Wuppertal University) , Christos Iona (University of Cyprus & The Cyprus Institute) , Giannis Koutsou (The Cyprus Institute) , Christian Kummer , (University of Cyprus & Technical University of Berlin) , Yan Li (The Cyprus Institute)
show 2 more authors
Bhavna Prasad (The Cyprus Institute) Gregoris Spanoudes (University of Cyprus)
This is my paper
classification hep-lat hep-exhep-phnucl-exnucl-th PACS 12.38.Gc
keywords lattice QCDproton spinmomentum fractionangular momentumenergy-momentum tensorgravitational form factorscontinuum limitparton decomposition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper attempts to settle, from lattice QCD alone, how the proton's momentum and spin are split among quarks and gluons. It uses four gauge ensembles with up, down, strange, and charm quark masses set to their physical values, allowing the continuum limit to be taken directly at the physical pion mass. The paper reports that the quark and gluon pieces add to 0.995(60)(29) for momentum and 0.507(43)(65) for spin, matching the momentum and spin sum rules within uncertainties. If correct, this is the first complete flavor decomposition of the proton's momentum and angular momentum in the continuum limit without chiral extrapolation, and it locates about 40% of each sum in gluons and roughly 10% in sea quarks.

Core claim

The central claim is that the proton's momentum fraction and total angular momentum can be decomposed into quark (up, down, strange, charm) and gluon contributions directly from lattice QCD at the physical point, with all renormalization factors—including the quark-singlet and gluon mixing—computed non-perturbatively. The authors obtain total momentum fraction <x>_N = 0.995(60)(29) and total spin J_N = 0.507(43)(65), consistent with the sum rules <x>_q + <x>_g = 1 and J_q + J_g = 1/2. They further split J_q into intrinsic quark spin (1/2)ΔΣ_q, taken from axial charges of a companion paper, and orbital angular momentum L_q = J_q - (1/2)ΔΣ_q, yielding a decomposition in which the up quark carr

What carries the argument

The machinery is the nucleon matrix element of the QCD energy-momentum tensor, parameterized by gravitational form factors A20(Q^2) and B20(Q^2). In the forward limit, A20(0) is the momentum fraction <x>; the Ji relation J = (1/2)[A20(0) + B20(0)] gives the total angular momentum. The paper computes these matrix elements on four twisted-mass clover-improved ensembles at different lattice spacings, renormalizes the quark singlet and gluon operators non-perturbatively using a 2x2 mixing matrix, and extrapolates to the continuum using constant and linear fits in a^2 averaged with the Akaike information criterion.

Load-bearing premise

The spin-orbit split assumes the axial-charge values (1/2)ΔΣ_q taken from a companion paper listed as 'In preparation' are correct; if those are wrong, L_q and the spin decomposition change, although the total J_N from A20+B20 is unaffected.

What would settle it

Recompute L_q using independently determined axial charges; if the resulting L_q values move by more than the quoted errors, the decomposition is not yet robust. Alternatively, a future measurement of the proton's total angular momentum (e.g., through deeply virtual Compton scattering) that disagrees with J_N = 0.507(43)(65) would falsify the sum-rule claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The momentum sum rule is satisfied at the physical point without chiral extrapolation: <x>_N = 0.995(60)(29), so no missing momentum needs to be attributed to unknown sources.
  • The spin sum J_N = 0.507(43)(65) confirms the Ji spin sum rule, with gluons contributing about 42% and sea quarks about 10%, providing a first-principles answer to the proton spin puzzle.
  • The quark momentum fractions agree with global PDF analyses, and the lattice intrinsic quark spin has better precision than current polarized-PDF integrals, offering a benchmark for phenomenological fits.
  • The small but statistically non-zero strange and charm quark momentum fractions imply intrinsic heavy-quark contributions in the nucleon that global fits should accommodate.
  • The individual quark orbital angular momenta, extracted as L_q = J_q - (1/2)ΔΣ_q, are a new continuum-limit prediction that future GPD-based experiments could test.

Where Pith is reading between the lines

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

  • Because the total J_N is built from A20 and B20 alone, while the spin-orbit split uses external axial charges, the decomposition is only as solid as those axial charges; a revision of the companion paper's values would change L_q but not J_N.
  • The same lattice machinery could be extended to other second Mellin moments, such as helicity and transversity, to complete the parton picture at the same physical-point continuum limit.
  • A natural testable extension is to compute the axial charges on the same four ensembles rather than importing them from a separate analysis, removing an external input and letting the spin-orbit decomposition stand alone.
  • If future global analyses tighten the small-x region of polarized PDFs, the lattice intrinsic-spin values here provide a cross-check that could distinguish genuine QCD sea polarization from model assumptions.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper reports a lattice QCD determination of the quark and gluon contributions to the nucleon momentum fraction and total angular momentum, using four Nf=2+1+1 twisted-mass ensembles at physical pion mass and lattice spacings from 0.049 to 0.079 fm, with a continuum extrapolation via AIC-weighted constant/linear fits. It quotes ⟨x⟩_N = 0.995(60)(29) and J_N = 0.507(43)(65), consistent with the momentum and spin sum rules. The quark intrinsic spin and orbital angular momentum are derived by combining the computed J_q with axial charges taken from a companion study (Ref. [24]); the paper claims to provide the first complete parton decomposition at physical masses with a continuum limit.

Significance. If correct, the results constitute a major step: a first-principles check of the momentum and spin sum rules at the physical point, with a flavor decomposition and continuum extrapolation. The internal consistency of the sums, the use of four lattice spacings, non-perturbative renormalization including quark-gluon mixing, and AIC-weighted model averaging are strengths. The cross-checks against phenomenological PDFs are useful. The main reservation is that the advertised spin decomposition depends on axial charges from an unpublished companion paper.

major comments (2)
  1. [Results and discussion, Table II] The central advertised result—a complete decomposition of the proton spin into intrinsic quark spin and orbital angular momentum—rests on the values of 1/2 ΔΣ_q taken from Ref. [24], which is cited as 'In preparation.' These values are not derived or shown in this manuscript. Since L_q = J_q − 1/2 ΔΣ_q (defined in the paragraph before Table II), any error or bias in the axial charges directly changes all L_q and the interpretation of the spin decomposition. This is load-bearing for the conclusion 'first quantitatively complete parton decomposition.' Please either include the axial-charge calculation and its full error budget here, or replace the reference with a publicly available, published analysis. In addition, state how the quoted L_q errors are propagated from J_q and ΔΣ_q, including whether correlations between the two are neglected.
  2. [Table II footnote; Conclusions] The error definitions are inconsistent. Table II states that the first error on ⟨x⟩ and J combines the statistical error with the AIC model-average error of the continuum extrapolation, while the Conclusions state that 'the first error is the statistical error.' This mislabel affects the interpretation of the central results. The footnote also gives only one error for 1/2 ΔΣ_q; the systematic uncertainty on L_q therefore does not include the systematic error of the axial-charge input. Please clarify the error definitions and, if possible, provide a complete error breakdown for the decomposed quantities.
minor comments (3)
  1. [Extraction section] Typo: 'solve it using the its singular value decomposition' should be 'solve it using its singular value decomposition'.
  2. [Table II footnote] Typo: 'taken form Ref. [24]' should be 'taken from Ref. [24]'.
  3. [Conclusions] The statement 'we also find a small non-zero contribution of the charm quark to the momentum fraction' is based on ⟨x⟩_c = 0.025(10)(7); even with errors added in quadrature this is only about a 2σ effect. Please soften to 'consistent with a small positive charm contribution' or provide a quantitative significance.

Circularity Check

0 steps flagged

No circularity: momentum and spin fractions are computed from matrix elements and summed; the spin-orbit decomposition uses an external, independent axial-charge input.

full rationale

The paper's central derivation computes quark and gluon momentum fractions and total angular momentum directly from lattice matrix elements of the energy-momentum tensor. The momentum sum rule and spin sum rule are not imposed as constraints; the paper states 'we check the momentum and spin sums by computing ab initio all components.' The continuum extrapolation, renormalization, and A20/B20 extraction are standard and not defined in terms of the final sum-rule results. The spin decomposition uses L_q = J_q − 1/2 ΔΣ_q with ΔΣ_q taken from Ref. [24], a companion paper by the same collaboration that is 'In preparation.' This is a genuine external-input dependency and a limitation for auditability, but it is not circular: the axial charges are a separate observable (nucleon axial charges), not the paper's target result, and the relation used is a definitional decomposition, not a fitted parameter called a prediction. Similarly, Refs. [20] and [11] supply methodology and previous results, not the target claims. Therefore no circular step can be exhibited; the derivation does not reduce to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim rests entirely on established lattice-QCD machinery and standard sum-rule identities. No new particles, forces, or conserved quantities are introduced. The main external inputs are the lattice spacings from Refs. [17,18], the nonperturbative renormalization details from companion Ref. [20], and the axial charges from Ref. [24]; the latter two are not fully public, which limits independent verification.

axioms (6)
  • standard math The QCD energy-momentum tensor can be decomposed into quark and gluon parts whose forward matrix elements give momentum fractions via A20(0) (Eq. 3) and total angular momentum via J=1/2(A20+B20) (Eq. 4).
    Standard decomposition of the traceless EMT; Ji's spin sum rule.
  • standard math Nucleon matrix elements of the EMT are parametrized by the gravitational form factors A20, B20, C20 (Eq. 2).
    Lorentz structure and spinor parameterization; standard in the field.
  • domain assumption The twisted-mass clover action is at maximal twist, so physical matrix elements are automatically O(a) improved without additional operator improvement.
    Relies on Refs. [15,16] and underlies the continuum extrapolation.
  • domain assumption The four ensembles with m_pi between 136.5 and 140.8 MeV are treated as being at the physical pion mass, so the continuum limit can be taken in a^2 without a chiral extrapolation.
    A modeling choice; the paper does not correct for the residual pion-mass scatter.
  • domain assumption The nonperturbative RI'-MOM renormalization conditions (including the 2x2 mixing matrix in Eq. 5 and suppression of gauge-noninvariant mixing) are sufficient to yield the physical MS matrix elements at 2 GeV.
    Details are deferred to Ref. [20]; incorrect renormalization conditions would shift the momentum and spin fractions.
  • domain assumption The intrinsic quark spin 1/2 DeltaSigma_q equals half the axial charge g_A^q from Ref. [24] and those values are correct.
    The spin/orbital decomposition uses these in-preparation results; no derivation is given in this paper.

pith-pipeline@v1.3.0-alltime-deepseek · 9811 in / 21898 out tokens · 228172 ms · 2026-08-01T08:05:43.311829+00:00 · methodology

0 comments
read the original abstract

We determine the momentum fraction and angular momentum carried by quarks and gluons in the proton in lattice QCD. We use four ensembles simulated with up, down, strange and charm quarks with their masses tuned to their physical values. These ensembles have similar physical volume and different lattice spacings allowing us to take the continuum limit directly at the physical pion mass point. We extract the quark and gluon momentum fractions and total angular momentum in the continuum limit as well as the intrinsic quark spin and orbital angular momentum contributions to the proton spin. We find the total momentum fraction $\langle x_N \rangle= 0.995(60)(29)$ and the total spin $J_N = 0.507(43)(65)$, showing that both the momentum and spin sum rules are satisfied. We compare our results to those extracted from phenomenological analyses.

Figures

Figures reproduced from arXiv: 2607.21230 by Bhavna Prasad (The Cyprus Institute), Christian Kummer, Christos Iona (University of Cyprus & The Cyprus Institute), Constantia Alexandrou (University of Cyprus & The Cyprus Institute), Giannis Koutsou (The Cyprus Institute), Gregoris Spanoudes (University of Cyprus), Jacob Finkenrath (Wuppertal University), Simone Bacchio (The Cyprus Institute), (University of Cyprus & Technical University of Berlin), Yan Li (The Cyprus Institute).

Figure 2
Figure 2. Figure 2: FIG. 2. Results on the proton angular momentum fractions [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Continuum extrapolation of the proton momen [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Continuum-extrapolated quark and gluon momen [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Continuum-extrapolated results for the proton total [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Continuum extrapolated results for the intrinsic [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of our results for quark and gluon momentum fractions [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of our results for the quark intrinsic spin contributions, ∆Σ [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗

discussion (0)

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Reference graph

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