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arxiv: 1907.09827 · v1 · pith:YU77HA4Mnew · submitted 2019-07-23 · ✦ hep-lat

Parton distribution functions of Delta^+ on the lattice

Yahui Chai , Yuan Li , Shicheng Xia , Constantia Alexandrou , Krzysztof Cichy , Martha Constantinou , Xu Feng , Kyriakos Hadjiyiannakou
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Karl Jansen Giannis Koutsou Chuan Liu Aurora Scapellato Fernanda Steffens
This is my paper

Pith reviewed 2026-05-24 17:13 UTC · model grok-4.3

classification ✦ hep-lat
keywords parton distribution functionslattice QCDDelta baryonquasi-distribution functionslarge momentum effective theorytwisted mass fermionsmomentum smearing
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The pith

Renormalized matrix elements for the unpolarized quasi-distribution function of the Δ⁺ baryon are computed on the lattice using large momentum effective theory.

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

This paper presents results for renormalized matrix elements related to the unpolarized quasi-distribution function of the Δ⁺ baryon. It employs the large momentum effective theory on two ensembles of N_f=2+1+1 twisted mass fermions with pion masses of 250 MeV and 330 MeV. Momentum smearing is used to improve the overlap with the boosted state and reduce statistical errors in the correlation functions. The work extends the quasi-PDF approach to the Delta resonance. A sympathetic reader would care because it demonstrates the feasibility of accessing parton distributions in excited baryons on the lattice.

Core claim

We present results for renormalized matrix elements related to the unpolarized quasi-distribution function of the Δ⁺ baryon making use of the large momentum effective theory on two ensembles with pion masses of 250 MeV and 330 MeV, employing momentum smearing to significantly reduce statistical errors.

What carries the argument

Large momentum effective theory (LaMET) for extracting quasi-distribution functions from matrix elements of the boosted Δ⁺ baryon.

If this is right

  • The method allows for the determination of parton distribution functions inside the Δ⁺ baryon.
  • Results at two different pion masses enable assessment of pion mass dependence.
  • Momentum smearing technique proves effective in improving signal quality for three-point functions.
  • Renormalized matrix elements provide a basis for future extraction of light-cone PDFs for the Delta.

Where Pith is reading between the lines

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

  • This approach could be extended to other excited states or different quantum numbers to map out baryon structure more broadly.
  • If the quasi-PDFs converge to physical PDFs, it would validate LaMET for resonances and allow comparisons with nucleon PDFs to see excitation effects.
  • Future work might involve higher momenta or finer lattices to control systematics better.

Load-bearing premise

The large momentum effective theory applies reliably to the Δ⁺ on these ensembles and momentum smearing improves the signal without introducing uncontrolled systematic effects.

What would settle it

Observation that the matrix elements do not show the expected improvement in signal or fail to renormalize consistently with LaMET predictions at increasing boost momenta would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.09827 by Aurora Scapellato, Chuan Liu, Constantia Alexandrou, Fernanda Steffens, Giannis Koutsou, Karl Jansen, Krzysztof Cichy, Kyriakos Hadjiyiannakou, Martha Constantinou, Shicheng Xia, Xu Feng, Yahui Chai, Yuan Li.

Figure 1
Figure 1. Figure 1: The real (left) and imaginary (right) part of the ratio in eq. (3.5) yielding the isovector unpolar￾ized quasi-PDFs of the ∆ with boosted momentum Pz = 2π/L for different values of the source-sink time separation as a function of z/a. 5. Results at mπ = 250 MeV In this section we present results using a 323 ×64 lattice with lattice spacing a = 0.096 fm and pion mass mπ = 250 MeV. At this value of the pion … view at source ↗
Figure 2
Figure 2. Figure 2: , we show the bare matrix elements for momentum 4π/L with different stout smearing steps. As can be seen, the stout smearing increases the value of matrix elements. The results for different stout smearing converge as the smearing steps increase [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Real part (left) and imaginary (right) parts of the renormalization factor for different stout smearing steps. Fig.3 for different stout smearing steps. The renormalized matrix elements are shown in Fig.4 for momentum Pz = 4π/L ≈ 0.82 GeV. As expected, the renormalized matrix elements for different stout smearing steps are consistent, with stout smearing clearly reducing the errors in the renormal￾ization … view at source ↗
Figure 4
Figure 4. Figure 4: Renormalized matrix of quasi-PDF with boosted momentum 4π/L 6. Summary and outlook In this study, we perform a first and exploratory study of the renormalized matrix elements for the unpolarized isovector PDF of the ∆ +. The momentum boosts used are 0.41 GeV and 0.82 GeV for the mπ = 250 MeV ensemble, and 0.54 GeV for the mπ = 330 MeV ensemble. These values are rather small and the next step is therefore t… view at source ↗
read the original abstract

We present results for renormalized matrix elements related to the unpolarized quasi-distribution function of the $\Delta^+$ baryon making use of the large momentum effective theory. Two ensembles of $N_f=2+1+1$ twisted mass fermions with a clover term and pion masses of 250 MeV and 330 MeV are analyzed. We employ momentum smearing to improve the overlap with the boosted $\Delta$ state significantly reducing in this way the statistical error of both two- and three-point functions.

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.

Referee Report

2 major / 0 minor

Summary. The manuscript presents results for renormalized matrix elements related to the unpolarized quasi-distribution function of the Δ⁺ baryon, computed via the large momentum effective theory (LaMET) on two Nf=2+1+1 twisted-mass fermion ensembles (pion masses 250 MeV and 330 MeV). Momentum smearing is used to improve overlap with the boosted Δ state and thereby reduce statistical errors in the two- and three-point correlation functions.

Significance. If the results and underlying assumptions hold, the work would constitute the first lattice calculation of quasi-PDF matrix elements for a baryon resonance. This extends the LaMET framework beyond the nucleon and supplies a practical demonstration of momentum smearing for three-point functions involving excited states. Such results could serve as a benchmark for future resonance PDF studies once matching and renormalization procedures are fully validated.

major comments (2)
  1. [Method description / Abstract] The central claim that LaMET applies reliably to the Δ⁺ resonance and that momentum smearing introduces no uncontrolled bias rests on untested assumptions. The method description provides no quantitative checks (smearing-parameter variation, comparison of Δ versus nucleon matrix elements on the same ensembles, or tests of residual excited-state contamination in the boosted frame) that would be required to establish control over these systematics.
  2. [Abstract] No numerical values, error budgets, or figures for the renormalized matrix elements themselves are referenced in the abstract or summary statements, preventing direct assessment of whether the reported reduction in statistical error is statistically significant or physically meaningful.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Method description / Abstract] The central claim that LaMET applies reliably to the Δ⁺ resonance and that momentum smearing introduces no uncontrolled bias rests on untested assumptions. The method description provides no quantitative checks (smearing-parameter variation, comparison of Δ versus nucleon matrix elements on the same ensembles, or tests of residual excited-state contamination in the boosted frame) that would be required to establish control over these systematics.

    Authors: The referee correctly identifies that the manuscript does not contain the quantitative checks listed. This work is the first computation of quasi-PDF matrix elements for a resonance, and the primary aim was to demonstrate the feasibility of the calculation on the given ensembles using momentum smearing. We will revise the text to state the assumptions more explicitly and to note that systematic validation (including the suggested checks) lies beyond the scope of the present study and is left for future work. revision: partial

  2. Referee: [Abstract] No numerical values, error budgets, or figures for the renormalized matrix elements themselves are referenced in the abstract or summary statements, preventing direct assessment of whether the reported reduction in statistical error is statistically significant or physically meaningful.

    Authors: We agree that the abstract would be improved by referencing concrete results. In the revised manuscript we will update the abstract to include brief mention of representative numerical values for the renormalized matrix elements and to point to the figures that display the error reduction. revision: yes

Circularity Check

0 steps flagged

No circularity: direct lattice results with no self-referential derivations

full rationale

The manuscript reports numerical matrix elements computed on two twisted-mass ensembles using standard LaMET matching and momentum smearing; no equations or claims reduce a derived quantity to a fitted parameter or prior self-citation by construction. All central outputs are direct simulation results, rendering the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5654 in / 946 out tokens · 19209 ms · 2026-05-24T17:13:23.080501+00:00 · methodology

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Reference graph

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