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Interpreting Parton Distributions with Shapley Values

T0 review · 2 major / 3 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Shapley values on PDF flavors reveal a hidden gluon blind spot at intermediate x and quantify which data constrain which partons.

desk verdict Clean, model-agnostic PDF diagnostic that recovers textbook lore and exposes a real intermediate-x gluon blind spot; the single-bump probe is a limitation but not a fatal one. read the letter →

arxiv 2607.02647 v2 pith:77JG3XLE submitted 2026-07-02 hep-ph hep-ex

Raphaël Bonnet-Guerrini , Stefano Carrazza , Stefano Forte , Eva Groenendijk , Vincenzo Piuri , Ramon Winterhalder This is my paper · ORCID
classification hep-phhep-ex
keywords partondistributionfunctionsShapleyvaluesexplainableAIgluonPDFsensitivityNNPDFMellinmomentshyperparameteroptimization
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

Parton distribution functions (PDFs) sit at the heart of every hadron-collider prediction, yet the map from experimental data to PDFs is nonlocal, scale-dependent and often opaque. This paper treats the eight PDF flavors as players in a cooperative game whose payoff is the χ² agreement between theory and data. By applying a calibrated Gaussian bump to each flavor in a chosen x-region, restoring sum rules, and averaging the exact Shapley value over all coalitions, the authors obtain a quantitative score of how much each flavor, in each x-bin, contributes to the fit quality. The scores recover textbook expectations about which processes constrain which PDFs, but they also expose a previously unnoticed dip in gluon sensitivity around x ≈ 0.07. That dip is traced to the vanishing eigenvalue of the singlet-gluon anomalous-dimension matrix at the momentum sum-rule moment N = 2 and appears in three independent global PDF sets. The same diagnostic can be used to redesign the data folds employed in neural-network hyperparameter optimisation.

What carries the argument

The exact Shapley value φ_j(x_µ) for flavor j at perturbation center x_µ, obtained by averaging the marginal change in χ² over all 128 coalitions of the eight flavors while a calibrated Gaussian bump is applied and sum rules are re-enforced.

What would settle it

Recompute the gluon Shapley profile after replacing the Gaussian bump with an alternative localized deformation (e.g., a Mellin-space moment shift or a different width) and check whether the dip at x ≈ 0.07 persists across NNPDF4.0, CT18 and MSHT20; its disappearance would falsify the claimed universality of the sensitivity loss.

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Extended reading notes

Core claim

Exact Shapley values computed on the eight PDF flavors, after a localized Gaussian perturbation of amplitude equal to the 68 % PDF uncertainty and with sum rules restored, quantitatively recover the known data-to-PDF map and reveal a universal local loss of sensitivity of the gluon PDF around x ≈ 0.07 that is invisible to ordinary uncertainty bands.

Load-bearing premise

That a single, symmetric Gaussian bump of fixed logarithmic width, applied simultaneously to every flavor inside a coalition and then averaged over positive and negative signs, faithfully probes the local sensitivity of the full nonlinear PDF-to-observable map.

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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

2 major / 3 minor

Summary. The paper introduces exact Shapley values as a model-agnostic diagnostic for parton distributions, treating the eight PDF flavors (in flavor or evolution basis) as players in a cooperative game whose value function is the experimental χ^{2}. A localized Gaussian perturbation of amplitude equal to the 68% PDF uncertainty, with sum rules restored after each coalition, is applied one x-region at a time; the resulting Shapley values φ_j(x_µ) quantify the marginal contribution of each flavor to the description of a chosen dataset. Applied to NNPDF4.0 (and cross-checked on CT18 and MSHT20), the method recovers the standard data-to-PDF map while revealing a previously unnoticed, universal dip in gluon sensitivity near x≈0.07. The dip is traced to the vanishing eigenvalue of the singlet-gluon anomalous-dimension matrix at N=2 (enforced by the momentum sum rule) and mapped to x-space via the saddle point of the Mellin inversion. Two applications are shown: a diagnostic of under-constrained regions relevant to BSM searches and Higgs production, and a proof-of-concept improvement of the K-fold design used in NNPDF hyperparameter optimization.

Significance. If the results hold, the work supplies a quantitative, methodology-independent tool that can be applied to any existing PDF set without re-fitting. Exact evaluation over the 128 coalitions (rather than the additive SHAP approximation) correctly retains flavor correlations and sum-rule constraints, a clear technical advance over prior XAI proposals in the field. The recovery of textbook expectations places those expectations on a numerical footing, while the gluon dip—confirmed on three independent global fits and given a clean QCD explanation—is a genuine new observation with direct phenomenological consequences for gluon-fusion Higgs and possible new-physics reabsorption. The fold-optimization example further demonstrates immediate practical utility for the NNPDF pipeline. These strengths make the paper a useful addition to the PDF-diagnostics literature.

major comments (2)
  1. [Sec. 3.2, Eqs. (21-23)] Sec. 3.2, Eqs. (21–23): the central claim that the observed gluon dip is a genuine local loss of sensitivity rests on a highly restrictive probe (identical-sign Gaussian bump of fixed logarithmic width w applied simultaneously to every flavor inside a coalition, then averaged over + and - signs, one x-bin at a time). Because the momentum sum rule and the coupled singlet-gluon evolution are non-local, this coalition geometry can cancel or amplify the marginal contribution of the gluon relative to a pure single-flavor, single-x probe. The paper itself notes the combinatorial barrier to a fuller feature space, yet provides no robustness checks (variation of w, single-flavor-only games, or opposite-sign coalitions). Without such tests the dip could be an artifact of the chosen probe rather than a property of the PDF-to-observable map.
  2. [Appendix B, Table 9] Appendix B and Table 9: the iterative removal of any dataset for which v_Di(S)>10^5 is an ad-hoc regularization whose threshold is a free parameter. The discarded points (chiefly fixed-target DY ratios) change with PDF set and with x_µ; it is not shown how much the gluon Shapley profile (especially the depth and location of the dip) moves when the threshold is varied or when those datasets are retained with a milder cut. Because the dip is the paper’s principal new result, its stability under this pruning must be quantified.
minor comments (3)
  1. [Figs. 4-5] Fig. 4 and Fig. 5: the symlog color scale and the vertical offsets used for readability make quantitative comparison of the size of φ_j across x_µ difficult; a linear inset or a table of peak values would help.
  2. [Sec. 4.3] Sec. 4.3: the broader secondary dip near x~10^{-2} (visible for CT18 and, more weakly, for the other sets) is noted but not discussed; a short remark on its possible relation to the gluon-fusion Higgs region would strengthen the phenomenological claims.
  3. [Sec. 3.2] The Gaussian width w is fixed once and for all to the bin spacing of the 50-point scan; a one-sentence statement of the numerical value and a brief stability check would remove an unnecessary free parameter from the method.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the method treats any pre-existing PDF set as an external black-box input and evaluates exact Shapley values from the definition of the cooperative game; nothing is forced by construction or by self-citation.

full rationale

The paper’s derivation chain is self-contained and non-circular. A previously determined PDF set (NNPDF4.0, or CT18/MSHT20) is loaded as an immutable external input; no information about the fitting procedure that produced it is ever used (explicitly stated in Sec. 3 and Fig. 1). Players are the eight PDF flavors; the value function is the ordinary experimental χ² evaluated after a controlled, sum-rule-restoring Gaussian perturbation (Eqs. 21–23). The Shapley value is then the exact combinatorial average of marginal contributions (Eq. 18). This is a definitional computation, not a prediction that re-uses a fitted parameter. The observed gluon dip at xµ≈0.07 is an output of that computation; its subsequent physical explanation (vanishing eigenvalue of the anomalous-dimension matrix at N=2 mapped to x-space by the saddle point of the Mellin inversion) relies only on textbook QCD evolution and is independently verified on two external PDF sets. Self-citations to NNPDF papers are ordinary methodological background and are not load-bearing for the central claims. The restrictive single-bump probe is an acknowledged modeling choice (end of Sec. 3.2), not a circular reduction. Consequently the strongest claim—that the method recovers known data–PDF maps and reveals a universal local loss of gluon sensitivity—does not reduce to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper rests on standard QCD factorization, DGLAP evolution and sum rules, plus the classical definition of the Shapley value. The only free choices are the concrete form and width of the probe perturbation and the numerical cut used to discard pathological data sets; no new dynamical entities are postulated.

free parameters (3)
  • Gaussian width w
    Fixed by hand to the logarithmic spacing of the 50-point xµ grid (w = log10(0.9/10^{-4})/50); controls the locality of the probe.
  • perturbation amplitude
    Set equal to the upper/lower 68 % PDF uncertainty at the bump center; a modeling choice that normalizes the Shapley value to the existing uncertainty band.
  • pathology threshold 10^5
    Ad-hoc cut on χ² used to iteratively discard data sets that produce vanishing cross-section ratios (Appendix B).
assumptions (3)
  • domain assumption QCD factorization theorem and DGLAP evolution equations hold to the order used in the theory predictions
    Invoked throughout Sec. 2.1; the entire PDF-to-observable map rests on them.
  • domain assumption Momentum and valence sum rules must be restored after every coalition perturbation
    Enforced by rescaling normalization constants Ai (Sec. 3.2); without them the coalitions would be unphysical.
  • standard math Shapley value formula (Eq. 18) with combinatorial weights correctly attributes marginal contributions
    Taken from classical cooperative game theory; used without modification.

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Pith. "Pith review of Interpreting Parton Distributions with Shapley Values." pith.science (2026). https://pith.science/paper/77JG3XLE

@misc{pith2026260702647,
  author       = {Pith},
  title        = {Pith review of: Interpreting Parton Distributions with Shapley Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/77JG3XLE}},
  note         = {Machine review of arXiv:2607.02647}
}
abstract

We show that Shapley values can be used to trace how individual parton distributions (PDFs) shape the theory predictions for high-energy observables computed from them. This provides a tool for assessing the impact of data on PDFs when determining them, and the impact of PDF uncertainties when using the PDFs to compute collider observables. The Shapley value is computed by treating the regression of PDFs from data as a cooperative game. The PDFs are the players, and the reward is the likelihood ($\chi^2$) that characterizes the agreement between data and the predictions obtained from a given PDF, with theory and methodology held fixed. The method is agnostic to the way PDFs have been determined in the first place: for PDFs determined with a black-box AI model it may be used in order to explain the behavior of the model, and for PDFs determined using a fixed parametrization it may be used in order to expose the features and potential limitations of the parametrization. We find that the method recovers known expectations about which data constrain which PDFs in a global fit, while placing them on a more quantitative footing. We demonstrate its effectiveness in two ways. We uncover an unexpected loss of sensitivity of the gluon PDF at intermediate $x$, with potential implications for BSM searches and the gluon fusion Higgs cross section. We also show that the method can be used to improve the hyperparameter optimization procedure currently used by the NNPDF collaboration.

Figures

Figures reproduced from arXiv: 2607.02647 by the authors.

Figure 1
Figure 1. Schematic overview of the Shapley value analysis. A previously determined PDF set is loaded as input. For a fixed x region and coalition S, perturbed PDFs f (S) are propagated through the theory pipeline to obtain predictions that are compared to some dataset D through a suitable χ 2 log-likelihood. Shapley values ϕj quantify the contribution of each flavor for a chosen x region and dataset D. 9 [PITH_FULL_IMAGE:fi… view at source ↗
Figure 2
Figure 2. Calibrated perturbation Eq. (21) centered at xµ = 10−3 in the flavor basis, with amplitude normalized to 1σ at Q0 = 1.65 GeV and w = 0.3. the perturbation. As mentioned, a common way of representing PDF uncertainties, specifically used by NNPDF, is to deliver PDF sets as sets of replicas f k such that the central expected PDF is the ensemble mean and the uncertainty and correlation are found from the ensemble covari… view at source ↗
Figure 3
Figure 3. Shapley value obtained setting each PDF to zero in turn according to Eq. (25) in the flavor basis (top) and evolution basis (bottom). The dashed horizontal line corresponds to the average over flavors. evolution kernels in Eq. (2) is such that the behavior of the initial condition fi(x, Q2 0 ) in some x ∼ x0 region mostly affects the behavior of the evolved PDFs fi(x, Q2 ) for x < x0 when the evolved scale is greate… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Scan of flavor-basis (top) and evolution-basis (bottom) Shapley values as a function of perturbation center xµ for NNPDF4.0, computed for the central PDF set. The value ϕj (xµ) is shown for all PDF flavors j and for 50 values of xµ, equally spaced on a logarithmic scal…
Figure 5
Figure 5. Figure 5: The median Shapley value ϕ˜ j (xµ) over replicas, with an uncertainty band given by the central 68% interval [ϕ 16 j , ϕ84 j ] (16th–84th percentiles) of the replica distribution, plotted with a symlog scale on the y axis for six different values of the perturbation ce…
Figure 6
Figure 6. Figure 6: The Shapley values ϕj (0.5) in the flavor basis (same as shown in [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Same as [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: The momentum fraction x corresponding to the value N0 of the saddle point of the Mellin inversion integral for the gluon distribution, evaluated using NNPDF4.0 PDFs. the partonic cross section at NLO, while it already drives the scale dependence at LO through the coup…
Figure 12
Figure 12. Figure 12: Left: the gluon Shapley value computed from the central NNPDF4.0 PDF (same as shown in [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: The Shapley value computed using central PDFs (same as in [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

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