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Kaon Gluon Parton Distribution and Momentum Fraction from 2+1+1 Lattice-QCD with High Statistics

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Pith's one-line read Kaon gluons carry about twice the pion's momentum fraction: $\langle x\rangle_g^{K}=0.557(18)(24)(56)$ at $\mu=2$ GeV, with ratio $1.92(18)$.

desk verdict A careful, high-statistics kaon gluon moment and PDF, but the headline ratio rests on an uncomputed quark-gluon mixing term. read the letter →

arxiv 2506.03002 v2 pith:FWMI3SCO submitted 2025-06-03 hep-lat

Alex NieMiera , William Good , Huey-Wen Lin This is my paper · ORCID
classification hep-lat PACS 12.38.-t
keywords kaongluonpartondistributionfunctionmomentumfractionlatticeQCDpseudo-PDFmethodreducedpseudo-Ioffe-timenonperturbativerenormalizationmesonstructure
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

This paper asks how much of the kaon's momentum is carried by gluons, using lattice QCD with 1,296,640 kaon-correlator measurements on one ensemble with lattice spacing about 0.12 fm and a 310 MeV pion mass. It reports a nonperturbatively renormalized gluon momentum fraction $\langle x\rangle_g^{\overline{\text{MS}},K}=0.557(18)_{\text{stat}}(24)_{\text{NPR}}(56)_{\text{mixing}}$ at $\mu=2$ GeV in the $\overline{\text{MS}}$ scheme. The central result is the kaon-to-pion ratio $\langle x\rangle_g^K/\langle x\rangle_g^\pi = 1.92(18)$: on the same ensemble, gluons carry about twice as much momentum in the kaon as in the pion. The paper also extracts the kaon gluon PDF through pseudo-PDF matching and finds its normalized shape agrees with a Dyson-Schwinger calculation for $x\gtrsim 0.4$. If the ratio holds up, it would mean the gluon momentum share is not the same for kaon and pion as earlier calculations suggested, and future electron-ion collider measurements have a concrete prediction to test.

What carries the argument

The argument is carried by the reduced pseudo–Ioffe-time distribution (RpITD), a double ratio of fitted ground-state gluon matrix elements $\mathcal{M}(\nu,z^2)=\frac{M(z\cdot P_z,z^2)/M(0\cdot P_z,0)}{M(z\cdot0,z^2)/M(0\cdot0,0)}$ that cancels renormalization factors and ultraviolet divergences. The gluon operator $F^{ti}(z)W(z,0)F^t_i(0)-F^{ij}(z)W(z,0)F_{ij}(0)$, with a gauge-invariant Wilson line $W$, produces the matrix elements; for the first moment a separate OPE gluon operator is used and related to $\langle x\rangle_g$ by a kinematic factor. The RpITD is matched to the light-cone gluon PDF through a convolution with the gluon-gluon kernel $R_{gg}$, using the model $x^\alpha(1-x)^\beta/B(\alpha+1,\beta+1)$ with Bayesian priors. A smearing study over hypercubic, Stout, and Wilson-flow steps leads to the choice of five steps of HYP smearing as the conservative balance between noise reduction and preservation of the physics. Quark-gluon mixing in the renormalization is not computed; it is replaced by a 10 percent systematic error.

What would settle it

Compute the quark disconnected diagrams and the $Z_{gq}\langle x\rangle_q$ mixing contribution on the same ensemble and renormalize with the full mixing matrix: if the resulting $\langle x\rangle_g^{\overline{\text{MS}},K}$ moves by more than 0.056, or if the kaon-to-pion ratio moves outside $1.92(18)$, the central claim is falsified. A physical-continuum extrapolation with full mixing that brings the kaon moment down to the earlier value near 0.42 would also falsify the factor-of-two conclusion.

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Extended reading notes

Core claim

The paper's central claim is that, on a 2+1+1-flavor HISQ ensemble with $a\approx0.12$ fm and $M_\pi\approx310$ MeV, the nonperturbatively renormalized kaon gluon momentum fraction is $\langle x\rangle_g^{\overline{\text{MS}},K}=0.557(18)_{\text{stat}}(24)_{\text{NPR}}(56)_{\text{mixing}}$ at $\mu=2$ GeV. Taking the pion's value from the same ensemble, the paper finds $\langle x\rangle_g^K/\langle x\rangle_g^\pi=1.92(18)$, which is in tension with earlier lattice and Dyson-Schwinger results that put the two moments nearly equal. Using the pseudo-PDF method, the paper extracts the kaon gluon PDF and reports that its shape, divided by the first moment, agrees with the Dyson-Schwinger prediction for $x\gtrsim0.4$ and with the pion's shape within one standard deviation; the absolute distributions differ because of the factor-of-two momentum fraction. Higher normalized moments are $\langle x^2\rangle_g^K/\langle x\rangle_g^K=0.123(16)$ and $\langle x^3\rangle_g^K/\langle x\rangle_g^K=0.0277(46)$.

Load-bearing premise

The load-bearing premise is that neglecting the quark-gluon mixing term $Z_{gq}$ in the renormalization and absorbing it into a 10 percent systematic error is adequate; if the true mixing for the kaon exceeds 10 percent, the central value 0.557 and the kaon-to-pion ratio shift by more than the quoted mixing uncertainty.

Editorial extensions

If this is right

  • If the kaon-to-pion ratio $1.92(18)$ is confirmed, the kaon's gluon momentum fraction is about twice the pion's, so by the momentum sum rule the quark momentum fraction in the kaon is correspondingly smaller.
  • The extracted gluon PDF, matched at $\mu=2$ GeV, gives a lattice prediction for kaon gluon distributions that future electron-ion collider measurements can test directly for $x\gtrsim0.4$.
  • The high-statistics data reveal low-momentum contamination in the RpITD that was not visible at lower statistics, so future pseudo-PDF analyses should exclude or model the smallest $P_z$ points.
  • The result provides a single-ensemble data point toward a physical-continuum extrapolation of the kaon gluon moment; combining it with other ensembles would sharpen the comparison with earlier continuum-extrapolated lattice results.
  • The smearing study identifies five steps of HYP smearing as a conservative default for gluonic matrix elements, which can inform the design of future gluon-PDF calculations.

Reading between the lines

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

  • One consequence the paper does not pursue: if the factor-of-two ratio survives the physical-continuum limit, then SU(3) flavor breaking in the gluonic sector is much larger than in the quark sector, which would be a new constraint on models of emergent hadronic mass.
  • A direct test is to compute the quark disconnected diagrams and the full quark-gluon mixing term on the same ensemble; if the correction exceeds the assumed 10 percent, both the central moment and the kaon-to-pion ratio would move, possibly reconciling the result with earlier near-equality predictions.
  • The observation that the normalized kaon and pion gluon PDF shapes agree within 1$\sigma$, while the absolute fractions differ by a factor of two, suggests the $x$-dependence of gluon distributions in light pseudoscalar mesons may be approximately universal even when the total gluon momentum share is not; a physical-pion-mass comparison would test this.
  • The small-$P_z$ contamination identified in the RpITD data implies that previous lower-statistics pseudo-PDF results may have underestimated similar systematics; re-analyzing older ensembles with the same fit-window and momentum cuts would quantify that.
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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. The paper reports a high-statistics lattice-QCD calculation of the kaon gluon momentum fraction and gluon PDF on a single MILC 2+1+1 HISQ ensemble at a≈0.12 fm and Mπ≈310 MeV. Using 1,296,640 kaon-correlator measurements, the authors extract bare gluon matrix elements via two-state fits to three-point correlators, study the dependence on HYP, Stout, and Wilson smearing, and quote a nonperturbatively renormalized kaon gluon momentum fraction ⟨x⟩_g^{MS,K}=0.557(18)_stat(24)_NPR(56)_mixing at μ=2 GeV in the MS scheme. They also extract the kaon gluon PDF from reduced pseudo-Ioffe-time distributions (RpITDs) using pseudo-PDF matching, compare it with a DSE prediction and with a pion gluon PDF from the same ensemble, and report a kaon-to-pion gluon momentum fraction ratio of 1.92(18)_stat.

Significance. If the result withstands scrutiny, the kaon-to-pion ratio ⟨x⟩_g^K/⟨x⟩_g^π≈1.92 would be a notable finding, contradicting the near-equality suggested by ETMC and DSE and with implications for emergent-mass phenomenology. The paper has clear strengths: the statistics are very large for gluonic observables; the smearing study is systematic and informative; the ground-state matrix-element extractions are checked against variations in fit windows; the OPE-based moment is independent of the PDF model; and the error budget is reported transparently. The main caveat is that the central renormalized value and the PDF both rest on an uncomputed quark-gluon mixing term that is replaced by an ad hoc 10% systematic, and the ratio is quoted without propagating that uncertainty.

major comments (4)
  1. [III.A, Eq. (14)] The headline moment 0.557(18)_stat(24)_NPR(56)_mixing is obtained by setting Z_gq⟨x⟩_q^bare to zero and assigning a 10% systematic based on Refs. [18,20,31]. The paper itself cites those works as reporting mixing between 2% and 20%, so the quoted 56_mixing uncertainty is not an envelope of the cited range but a choice near its lower end. A 20% mixing contribution would shift the central value by roughly ±0.11, nearly twice the quoted mixing error. The authors should either compute Z_gq⟨x⟩_q^bare, or quote the central value with a conservative uncertainty derived from the full cited range, and should state explicitly that the current 10% is an assumption rather than a derived systematic.
  2. [III.A, ratio paragraph] The ratio ⟨x⟩_g^K/⟨x⟩_g^π = 1.92(18)_stat is quoted with only the statistical error. The kaon moment has additional 24_NPR and 56_mixing uncertainties, and the pion moment from Ref. [21] carries its own 13_NPR uncertainty; none of these are propagated into the ratio. Adding just the kaon mixing uncertainty in quadrature changes the ratio uncertainty to about 0.19, and including the NPR terms brings the total to roughly 0.29. The conclusion that the kaon gluon carries significantly more momentum than the pion is therefore not supported by the quoted error. The ratio must be accompanied by a full uncertainty budget, including the quadrature combination or, preferably, an estimate of the correlation between the two moments.
  3. [III.B, Eq. (15)] The pseudo-PDF matching used to extract the kaon gluon PDF drops the quark-gluon kernel R_gq, and the paper justifies this by citing previous nucleon and pion gluon PDF studies [23,24,29]. Those studies do not automatically cover the kaon, whose quark disconnected contributions can differ. Because the same neglected mixing term affects both the renormalized moment and the RpITD entering the fit, the comparisons with DSE for x≳0.4 and the higher-moment ratios are not independent of the mixing assumption. The authors should provide a quantitative estimate of the R_gq contribution to M(ν,z²), or propagate a corresponding systematic into the extracted PDF band.
  4. [III.B, fit-range paragraph] The fit excludes Pz=1 because those data cannot be properly described by the matching, yet no systematic uncertainty is assigned to this exclusion. The paper itself notes that small-Pz contamination has become visible only at the present statistics, which suggests that the remaining Pz∈[2,5] data may also carry a residual, smaller version of the same effect. At minimum, the authors should report a fit that includes Pz=1 with a modified treatment, or vary the minimum Pz in the fit and include the spread in the quoted PDF uncertainties.
minor comments (6)
  1. [Abstract and Sec. I] There are typos that should be corrected, including 'in in the MS scheme' in the abstract and 'koan' for 'kaon' in Sec. I.
  2. [Sec. II, Fig. 5 discussion] The text refers to 'WILSON2' although the labels used elsewhere are WILSON1 and WILSON3; also, the Wilson-flow labels should be explicitly defined (flow time in units of a²) since the text states Nsteps=100 for both chosen flow times.
  3. [III.B, Eq. (20)] Equation (20) as written displays ⟨x^n⟩_g on the left-hand side, but the expression B(α+n,β+1)/B(α+1,β+1) actually gives the normalized ratio ⟨x^n⟩_g/⟨x⟩_g. The notation should be corrected to avoid implying that the quoted numbers are absolute moments.
  4. [Fig. 8] The y-axis label appears as 'M( , z2)' with the Ioffe-time variable ν missing; this should be fixed for clarity.
  5. [III.B, PDF fit] The fitted values of α and β, along with the χ²/dof of the RpITD fits, are not reported. Reporting these values would greatly improve reproducibility and allow readers to judge the influence of the Bayesian priors.
  6. [Fig. 10 and surrounding text] The bands in Fig. 10 exclude the uncertainties in ⟨x⟩_g; this is stated in the caption but should be emphasized in the main text, and ideally a version with those uncertainties propagated should be provided so that the visual separation between pion and kaon curves is not overinterpreted.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor circularity: higher moments labeled 'predictions' are analytic functions of the fit parameters; the central momentum fraction is an independent OPE extraction.

  1. fitted input called prediction [Section III.B, Eq. (20) and surrounding text]
    "In addition to the kaon gluon momentum fraction, we make predictions for its normalized higher moments using the jackknife ensemble of fitted parameters (α, β) obtained from fitting the RpITD data. Given the model PDF defined in Eq. 16, the nth moment can be written analytically in terms of beta functions: ⟨xn⟩g = B(α + n, β + 1)/B(α + 1, β + 1)."

    The normalized higher moments are exact closed-form functions of α and β, which are the parameters obtained from minimizing χ² against the same RpITD lattice data (Eqs. 16–17). Therefore ⟨x²⟩/⟨x⟩ = 0.123(16) and ⟨x³⟩/⟨x⟩ = 0.0277(46) are propagated fit outputs, not independent lattice predictions. The subsequent comparison with DSE — 'Our findings deviate from these DSE results by approximately 2.1σ... and 1.3σ...' — is a comparison of a transformed fit result to DSE rather than a fresh first-principles prediction. The headline first moment is unaffected because it comes from the OPE operator (Eqs. 5, 13–14), but calling the higher moments 'predictions' overstates their independence.

full rationale

The central claim, the kaon gluon momentum fraction ⟨x⟩_g^{MS,K} = 0.557(18)_stat(24)_NPR(56)_mixing, is derived from the bare matrix element of the OPE operator (Eq. 5, Eq. 13) divided by the nonperturbative renormalization constant Z_gg^{MS} = 1.512(65) from Ref. [19]. This is a direct lattice measurement of the gluon moment and is independent of the PDF model, the priors, and the RpITD fit. Reusing the published renormalization constant from the same group is standard practice and carries its own quoted NPR error, so it is not circular. The PDF shape is extracted by fitting the model of Eq. 16 to the RpITD data with Bayesian priors; comparing that fitted shape with DSE is a legitimate, falsifiable result rather than a definitional reduction. The only clear circularity is the labeling of the higher moments in Eq. 20 as 'predictions': they are analytic functions of the α, β fitted to the same data, so the quoted moment ratios and the DSE deviation are transformed fit outputs. The neglected quark-gluon mixing (Z_gq in Eq. 14 and R_gq in Eq. 15) is an external systematic assumption with a 10% error based on prior works; this is an uncertainty estimate, not a derivation that reduces to its own inputs, though it does warrant a correctness-risk note that the 10% may not cover the cited 20% upper end.

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

The central claim rests on standard lattice-QCD tools plus several hand-set choices: the smearing level, the PDF model and its priors, the Pz = 1 exclusion, and the 10% mixing uncertainty. None of these is independently verified in the paper, so the ledger is dominated by domain assumptions rather than free numerical parameters.

free parameters (5)
  • alpha (PDF shape parameter) = not quoted in text
    Fitted to the reduced pseudo-ITD data via the model in Eq. 16; controls the small-x behavior and determines all reported higher moments through Eq. 20.
  • beta (PDF shape parameter) = not quoted in text
    Fitted together with alpha; controls large-x suppression; also determines the higher moments reported in Sec. III.B.
  • prior centers and widths for alpha, beta = alpha_bar = -0.5, beta_bar = 5, sigma_alpha = sigma_beta = 5
    Hand-chosen Gaussian priors in Eq. 19 that guide the fit toward phenomenological expectations; they directly influence the extracted PDF shape and hence the moment ratios.
  • mixing systematic (10%) = 0.10
    Ad hoc uncertainty assigned because Z_gq is not computed; justified by a 2 to 20 percent range from previous studies [18,20,31].
  • HYP5 smearing steps = 5
    Smearing amount selected after comparing HYP/Stout/Wilson schemes in Sec. II; no smearing-systematic is propagated into the final uncertainty.
assumptions (6)
  • domain assumption The nonperturbative renormalization factor Z_MS_gg = 1.512(65) from Ref. [19] applies to this kaon calculation.
    Taken from MSULat's earlier nucleon gluon work, not recomputed or cross-checked on this ensemble.
  • domain assumption The pseudo-PDF matching relation Eq. 15 with only the gluon-gluon kernel Rgg describes the kaon RpITD data.
    The quark-gluon mixing kernel Rgq is dropped; the text cites previous gluon PDF studies claiming the effect is smaller than statistical errors, but it is not quantified here.
  • domain assumption The double ratio Eq. 12 is free of significant higher-twist or kinematic contamination for the included Pz = 2 to 5 data.
    Pz = 1 data are excluded because they cannot be properly described by the matching; the text notes possible small-Pz contamination persists at low nu, so the boundary at Pz = 2 is an assumption.
  • domain assumption A single ensemble at a = 0.1207 fm and M_pi = 310 MeV is representative enough for the quoted kaon gluon moment and PDF comparison.
    No continuum or physical-mass extrapolation is performed; the paper acknowledges this in the conclusions but still presents the number as the main result.
  • ad hoc to paper The Gaussian priors in Eq. 19 reflect the physically expected range for the PDF parameters.
    The centers and widths are chosen by hand, and the paper does not test the sensitivity of the final PDF to these choices.
  • domain assumption Two-state fits to the two- and three-point correlators in Eqs. 9 and 10 capture excited-state contamination.
    This is a standard modeling choice; the paper checks fit-window stability in Figs. 1 to 3, which supports but does not prove the assumption.

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Cite this review

Pith. "Pith review of Kaon Gluon Parton Distribution and Momentum Fraction from 2+1+1 Lattice-QCD with High Statistics." pith.science (2026). https://pith.science/paper/FWMI3SCO

@misc{pith2026250603002,
  author       = {Pith},
  title        = {Pith review of: Kaon Gluon Parton Distribution and Momentum Fraction from 2+1+1 Lattice-QCD with High Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWMI3SCO}},
  note         = {Machine review of arXiv:2506.03002}
}
abstract

We present a high-statistics lattice-QCD determination of the kaon gluon parton distribution function and gluon momentum fraction. We use clover valence fermion action to take 1,296,640 kaon-correlator measurements on a HISQ ensemble with $a \approx 0.12$~fm and 310-MeV pion mass, generated by the MILC collaboration. A detailed investigation into the impact of gauge-link smearing on the gluonic matrix elements indicates that five steps of hypercubic smearing offer an effective balance between signal quality and preservation of long-distance physics. We report a nonperturbatively renormalized kaon gluon momentum fraction of $\langle x \rangle_g^{\overline{\text{MS}}, K} = 0.557(18)_\text{stat}(24)_\text{NPR}(56)_\text{mixing}$ at $\mu = 2$ GeV in in the $\overline{\text{MS}}$ scheme. Using reduced pseudo-ITD matrix elements and pseudo-PDF matching, we extract the kaon gluon PDF and compare with the prediction from the Dyson-Schwinger equation and with the pion PDF obtained from the same ensemble.

Figures

Figures reproduced from arXiv: 2506.03002 by the authors.

Figure 1
Figure 1. FIG. 1: Example ratio plots used to extract the ground-state matrix element with the gluon operator [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Ratio plots and source-sink separation dependencies for stout smearing with 10 steps (top row) and 20 steps (bottom [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Ratio plots and source-sink separation dependencies for Wilson smearing with 1 steps (top row) and 3 steps (bottom [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the fitted ground state matrix ele [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Ratio plots and source-sink separation dependencies for HYP5 smearing using the gluon operator [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The bare gluon momentum fraction [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Kaon (left) and pion (right) RpITD for the a12m310 ensemble. The bands are the gluon PDF fits from simultaneously [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: (Left) The kaon gluon PDF [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The kaon (green) and pion (red) gluon PDF [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

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