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Parton Distribution Functions from Ioffe time pseudo-distributions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The first dynamical-fermion Ioffe-time pseudo-distribution calculation yields a nucleon valence PDF and its first two moments, with estimates of continuum and finite-volume systematics.

desk verdict First dynamical, two-spacing/two-volume pseudo-PDF calculation; honest and useful even though the x-shape rests on a single ansatz and on leading-twist assumptions that the paper itself flags. read the letter →

arxiv 1908.09771 v3 pith:ZUW4M27S submitted 2019-08-26 hep-lat hep-ph

Bálint Joó , Joseph Karpie , Kostas Orginos , Anatoly Radyushkin , David Richards , Savvas Zafeiropoulos This is my paper · ORCID
classification hep-lathep-ph PACS 12.38.Gc11.15.Ha14.20.Dh
keywords partondistributionfunctionslatticeQCDIoffetimepseudo-distributionsnucleonstructureoperatorproductexpansioncontinuumextrapolationfinite-volumeeffects
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 aims to show that the Ioffe-time pseudo-distribution method, previously tested in the quenched approximation, works with dynamical fermions and can produce a nucleon valence PDF together with an estimate of the main systematic errors. Using three ensembles with two lattice spacings and two volumes at a pion mass around 400 MeV, the authors form a double ratio that cancels the ultraviolet renormalization of the non-local operator, evolve the data at next-to-leading order, and extract both the $x$-dependence of the valence quark distribution and the first two of its moments. The two lattice spacings give an estimate of the continuum extrapolation error, and the two volumes give an estimate of finite-volume effects. The moments are found to be higher than the phenomenological values from an NLO global fit, consistent with the heavier-than-physical pion mass. If this is right, the pseudo-PDF approach is a viable path to PDFs from first principles with a controlled error budget.

What carries the argument

The central object is the reduced pseudo-ITD $$ \overline{\mathcal M}(\nu,$z^{2}$)=\frac{\mathcal $M^{0}$(\nu,$z^{2}$)/\mathcal $M^{0}$(\nu,0)}{\mathcal $M^{0}$(0,$z^{2}$)/\mathcal $M^{0}$(0,0)}, $$ where $\mathcal M^0(\nu,z^2)$ is the bare forward nucleon matrix element of $\bar\psi(z)\gamma_4 W(z;0)\psi(0)$ with $\nu=p\cdot z$. The double ratio cancels the Wilson-line renormalization constants and the quark-number normalization, and the result is renormalization-group invariant. The $z^2$ dependence is governed at one loop by the Altarelli–Parisi kernel $B(u)$, and the NLO matching to the $\overline{\rm MS}$ Ioffe-time distribution uses the kernel $K(u,z^2\mu^2,\alpha_s)$. The $x$-space PDF is recovered by fitting the matched real-part ITD to a valence-quark parameterization, while the moments $b_n(z^2)$ of the pseudo-PDF are matched to the PDF moments $a_n(\mu^2)$ through the multiplicative Wilson coefficients $K_n$.

What would settle it

Split the data by separation: fit the PDF and the moments using only the points with $z/a\le 4$ and only those with $z/a\ge 5$, after applying the same evolution and matching, and check whether the two results agree within statistical errors. A disagreement beyond errors would show that the separation dependence is not fully removed by the double ratio and that the simultaneous all-$z$ fit is contaminated.

Watch

Extended reading notes

Core claim

The paper claims that the reduced Ioffe-time pseudo-distribution—a double ratio of forward matrix elements of a non-local quark bilinear with a straight Wilson line—carries the same dominant short-distance information as the light-cone Ioffe-time distribution, with the ultraviolet and power divergences canceled by the ratio. On three dynamical ensembles the authors find that this reduced function is nearly independent of the separation $z^2$ at fixed Ioffe time $\nu$, so all data with $z/a$ up to 8 can be evolved with the one-loop kernel and matched at next-to-leading order to the $\overline{\rm MS}$ Ioffe-time distribution. They then fit the matched real part with a valence-quark parameterization to obtain the $x$-dependent PDF, and independently obtain the first two PDF moments—the average momentum fraction and the second moment—from the Taylor expansion of the same function using operator product expansion Wilson coefficients. The moments are systematically higher than those of the NLO global fit benchmark at this pion mass, as expected for $m_\pi\simeq 400$ MeV, and are consistent with one NNLO set within its errors.

Load-bearing premise

The calculation assumes that after the double ratio cancels the ultraviolet divergences, the remaining dependence on the quark-antiquark separation is small enough to be described by the one-loop perturbative evolution, so that data at all separations up to eight lattice spacings can be fit together; if non-perturbative short-distance effects at the largest separations are large enough to produce a systematic drift, the extracted PDF and the higher moments would be biased.

Editorial extensions

If this is right

  • If the central claim holds, future pseudo-PDF computations can report a continuum and finite-volume error budget from two lattice spacings and two volumes, rather than quoting only statistical errors.
  • The near $z^2$-independence of the reduced pseudo-ITD justifies fitting all separations simultaneously with a single PDF Ansatz, turning an ill-posed Fourier inversion into a stable fit.
  • The first two PDF moments can be computed from the same correlation functions without an inverse Fourier transform, providing a direct low-cost check against global fits.
  • The combination of momentum smearing and Feynman–Hellmann extraction extends the usable Ioffe-time range to $\nu\simeq 9$, which is what makes the higher moments and the large-$x$ region accessible at all.

Reading between the lines

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

  • Editorial inference: the near-$z^2$ independence suggests that the double-ratio subtraction absorbs most higher-twist contamination at the tested separations; if that survives at finer lattices, large-$z$ data, the only points sensitive to higher moments, can be used without per-separation higher-twist fits.
  • Editorial inference: the gap between the lattice moments and the NLO global-fit values at $m_\pi\simeq 400$ MeV gives a quantitative anchor for the pion-mass dependence; repeating the calculation at lower pion masses would show whether the moments approach the physical values linearly in $m_\pi^2$ or with a different rate.
  • Editorial inference: the same double-ratio plus NLO-matching pipeline could be transferred to the pion PDF or to sea-quark and gluon distributions, using the imaginary component of the pseudo-ITD that isolates the $q+\bar q$ combination; a pion benchmark would give an independent test of the leading-twist assumption.
  • Editorial inference: because the Feynman–Hellmann extraction reaches its plateau at earlier Euclidean times, this setup may reduce the cost of physical-pion-mass simulations where excited-state contamination is the dominant bottleneck.
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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

3 major / 5 minor

Summary. This manuscript presents a lattice QCD calculation of the unpolarized nucleon valence PDF using the Ioffe-time pseudo-distribution (pseudo-ITD) formalism. The authors use three 2+1-flavor Wilson-clover ensembles with lattice spacings a = 0.127 fm and a = 0.094 fm and two volumes, all at a pion mass of roughly 400 MeV. Matrix elements are extracted with a Feynman-Hellmann method and momentum smearing, and the reduced pseudo-ITD is formed through a double ratio that cancels the Wilson-line renormalization constants. The paper extracts the first two PDF moments from a Taylor/Vandermonde analysis, fits the Ioffe-time distribution to a four-parameter PDF ansatz, and studies discretization and finite-volume effects by comparing the two lattice spacings and two volumes. The central claims are that this is the first dynamical-fermion pseudo-PDF calculation, that the continuum and finite-volume systematics can be estimated with these ensembles, and that the resulting x-dependence and moments can be compared with global fits.

Significance. If the analysis is robust, this paper is a valuable step for the pseudo-PDF program: it demonstrates the method with dynamical fermions, uses momentum smearing and Feynman-Hellmann extraction to control excited states, and addresses two sources of systematic uncertainty (lattice spacing and volume) that were not both considered in earlier pseudo-PDF work. The paper is transparent: bare matrix elements are tabulated for all ensembles, jackknife errors are propagated, and the authors explicitly refrain from claiming a continuum-extrapolated PDF. The renormalization-group-invariant double ratio and the NLO matching kernel provide a clean route from the lattice matrix element to the MS PDF. The main risks are the reliance on leading-twist dominance at large separations and the dependence of the extracted x-dependence on a single model ansatz, both of which are acknowledged in the text but not fully quantified.

major comments (3)
  1. [§4.1–§4.2] The simultaneous fit of all z/a separations in §4.2 is in tension with the statement in §4.1 that the higher Taylor coefficients, which control the large-ν behavior, are only constrained by the largest-z data, and that "Since these data potentially have significant higher twist corrections, these results should be considered questionable." The large-ν points in the PDF fit of §4.2 are precisely the large-z points, so the extracted x-dependence and higher moments are directly sensitive to the same higher-twist contamination that the paper flags as questionable. The observed z² independence of the reduced pseudo-ITD is empirical support, but the double ratio only partially cancels O(z² Λ²_QCD) effects and is not a proof. I request a systematic stability test: repeat the PDF fit excluding points with z/a > 5 and z/a > 4, and/or include an explicit higher-twist term such as M(ν,z²) = M_LT(ν) + z² H(ν), and report whether the valence PDF and moments change within errors.
  2. [§4.2, Figs. 17–19] The central result—the x-dependence of the valence PDF—is never displayed: Figs. 17–19 show only the fitted Ioffe-time distribution Q(ν) compared with the lattice data, and the fitted parameters a, b, c, d of Eq. (4.4) are not reported for any ensemble. As a result, the reader cannot quantitatively assess the extracted x-space PDF or its uncertainty, nor can the moments be reproduced from the fit. Please include a plot of x f_v(x) (or f_v(x)) with a jackknife error band for each ensemble, and give a table of the fitted parameters with their correlations.
  3. [§4.2, Eq. (4.4)] The x-dependence is obtained from a single four-parameter ansatz, Eq. (4.4), taken from CJ/MSTW global fits. The paper acknowledges that the choice of parameterization introduces model-dependent bias, but it does not quantify that bias. Since the x-dependence is one of the paper's main claims, the analysis should include at least one alternative functional form (e.g., P(x)=1+c x+d x², or a form without the √x term in Eq. (4.4)) and propagate the difference between the resulting PDFs as a systematic uncertainty. The quoted χ²/d.o.f. values (2.0–2.5) are not small enough to argue that the data alone select this particular form.
minor comments (5)
  1. [Abstract and §4.3–4.4] The abstract states that "continuum limit and infinite volume extrapolation systematic errors of the PDF are considered," but the body explicitly declines to produce a continuum-extrapolated PDF and studies these effects on the reduced pseudo-ITD rather than on the final PDF or moments; please rephrase to avoid overstating the scope.
  2. [§3.5, text near Eq. (3.21)] The statement that the double ratio is exactly unity at ν=0 "with no possible higher twist effects and lattice spacing errors" is true by construction, but it may be clearer to note explicitly that this is a normalization condition that fixes the zeroth moment rather than a physical cancellation.
  3. [§3.7, Eq. (3.24)] The polynomial interpolation uses degree six (real) and degree five (imaginary) for data reaching ν ≈ 9.4; the paper says a cubic spline was checked, but does not report the numerical comparison. A sentence quantifying the difference between the two interpolations would help the reader judge the systematic error introduced here.
  4. [§4.1, Fig. 16] The residual z² dependence of the matched MS moments is said to be negligible, but no numerical χ² or p-value for a constant fit is given; a quantitative statement would strengthen the justification for combining all z² in the PDF fit.
  5. [§4.4, Eq. (4.9)] The finite-volume ansatz M_Inf = M_Latt + C_L exp(-m(L-z)) is presented as an assumption, and the conclusions note that the functional form still needs to be determined. In Fig. 21 the finite-volume difference is shown, but not divided by the combined statistical error; please add a quantitative estimate of the significance of the volume effect.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported moments and x-dependence are not forced by the ν=0 normalization or by the comparison to global fits.

full rationale

The derivation chain is self-contained against external benchmarks, and no load-bearing step reduces to its own inputs by construction. The reduced pseudo-ITD double ratio of Eq. (3.21) is normalized to unity at ν=0, which sets the zeroth moment/sum rule by construction, but the paper's reported first and second moments come from the c1 and c2 coefficients of the imaginary and real polynomial fits in Eq. (3.24), and those coefficients are not fixed by the normalization. The NLO matching kernel in Eq. (2.13) is cited to three independent derivations [59-61], including groups with no author overlap, and it is not fitted to the lattice data; the comparison with CJ15, MSTW, and NNPDF is a comparison of independently computed lattice moments to published global fits, not a fit to those fits. The decision to fit all z simultaneously in Section 4.2 rests on the empirically observed z2-independence of the reduced ITD in Section 3.6, which is an assumption with higher-twist risk at large z, but an unproven assumption is a correctness concern rather than a circular reduction. Self-citations to the pseudo-PDF framework appear, but the paper re-derives the cancellation of renormalization constants and the OPE moment matching in the text, and the framework is also corroborated by non-overlapping external derivations, so those citations are not load-bearing in a circular sense. No specific equation is shown to equal another by construction, and no fitted parameter is renamed as a prediction; therefore no circular step is exhibited.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The result rests on eight modeling and domain assumptions and five groups of fitted parameters; no invented entities. The most consequential input not paid for upstream is leading-twist dominance of the reduced pseudo-ITD at the z values used, followed by the four-parameter PDF Ansatz and the assumed finite-volume exponential form.

free parameters (5)
  • PDF Ansatz shape parameters a, b, c, d (Eq. 4.4) = Not reported; four per ensemble fit to ITD
    These four parameters regulate the ill-posed Fourier inversion and determine the extracted x dependence of the valence PDF.
  • Sixth-degree interpolation coefficients c1,c3,c5 (Im) and c2,c4,c6 (Re) in Eq. 3.24 = Not reported; fit per z^2 per ensemble
    Used to interpolate the reduced pseudo-ITD for the convolution integrals in evolution and matching; consistency with cubic spline is checked.
  • Excited-state fit parameters A_p^(j)(z^2), B_p^(j)(z^2) and gap Delta_p in Eq. 3.19 = Not reported
    Simultaneous fits to effective matrix elements extract the ground-state reduced ITD; their values are data-dependent.
  • Polynomial coefficients in Eq. 4.6 (labeled a,b,c,d) for discretization and finite-volume comparison = Not reported
    Fit to averaged reduced pseudo-ITD to estimate O(a) and O(a^2) and volume effects; coefficients not quoted.
  • Momentum smearing parameter zeta = 0.0, 1.5, 3.0, and 4.0 depending on ensemble and momentum range
    Chosen by maximizing signal-to-noise of two-point functions; not a physics fit, but affects which momenta have usable precision.
assumptions (8)
  • domain assumption The non-local Wilson-line operator is multiplicatively renormalizable and its renormalization constants cancel in the double ratio Eq. (3.21).
    Sections 2.2 and 3.5; relies on ref. [58] and on the RGI nature of the reduced pseudo-ITD.
  • domain assumption The OPE of the reduced pseudo-ITD is dominated by leading twist with NLO Wilson coefficients K_n (Eq. 2.18), with O(z^2 Lambda_QCD^2) terms small enough to ignore.
    Sections 2.4 and 4.2; load-bearing for fitting all z^2 data simultaneously. Empirical z^2 independence supports it but does not prove it.
  • domain assumption The NLO matching kernel K in Eq. (2.13) from refs. [59-61] correctly connects the renormalization-group-invariant pseudo-ITD to the MS ITD at mu=2 GeV.
    Section 2.3; perturbative corrections beyond O(alpha_s) and non-perturbative evolution are neglected.
  • domain assumption The pseudo-PDF has canonical support x in [-1,1] and the reduced ITD has a finite continuum limit.
    Section 2 and refs. [32,34]; justifies the Fourier transform and lattice ratio.
  • domain assumption The four-parameter valence PDF Ansatz of Eq. (4.4) is an adequate prior for the ill-posed inverse problem.
    Section 4.2; the paper acknowledges model-dependent bias; alternative parameterizations are not tested.
  • domain assumption The large-time behavior of the Feynman-Hellmann effective matrix element is given by Eq. (3.11) with exponentially suppressed excited states parameterized by Eq. (3.19).
    Sections 3.2-3.4; relies on refs. [66,67].
  • ad hoc to paper Finite-volume effects follow M_Inf = M_Latt + C_L exp(-m(L-z)) (Eq. 4.9).
    Section 4.4; assumed by analogy to ref. [74], and unknown powers of (L-z) are neglected.
  • domain assumption The lattice ensembles are valid discretizations of QCD with parameters listed in Tab. 1, with scale set by w0 from ref. [65].
    Section 3; configurations come from unpublished JLab/W&M generation [64].

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Cite this review

Pith. "Pith review of Parton Distribution Functions from Ioffe time pseudo-distributions." pith.science (2026). https://pith.science/paper/ZUW4M27S

@misc{pith2026190809771,
  author       = {Pith},
  title        = {Pith review of: Parton Distribution Functions from Ioffe time pseudo-distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZUW4M27S}},
  note         = {Machine review of arXiv:1908.09771}
}
abstract

In this paper, we present a detailed study of the unpolarized nucleon parton distribution function (PDF) employing the approach of parton pseudo-distribution functions. We perform a systematic analysis using three lattice ensembles at two volumes, with lattice spacings $a=$ 0.127 fm and $a=$ 0.094 fm, for a pion mass of roughly 400 MeV. With two lattice spacings and two volumes, both continuum limit and infinite volume extrapolation systematic errors of the PDF are estimated. In addition to the $x$ dependence of the PDF, we compute their first two moments and compare them with the pertinent phenomenological determinations.

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Forward citations

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