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REVIEW 3 major objections 7 minor 53 references

Symbolic regression turns a neural TMD fit into a closed 9-constant formula that still fits 482 Drell–Yan points.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 07:07 UTC pith:WH55B3WU

load-bearing objection Solid NN-to-analytic TMD compression with a usable 9-constant formula; the “genuine x–b_T correlation” claim is overstated relative to the ansatz. the 3 major comments →

arxiv 2607.24710 v1 pith:WH55B3WU submitted 2026-07-27 hep-ph nucl-thphysics.data-an

Symbolic Extraction of Non-Perturbative Transverse-Momentum-Dependent Distributions from Drell-Yan Data

Cole Granger , Alessandro Bacchetta , Valerio Bertone , Chiara Bissolotti , Matteo Cerutti , Marco Radici , Simone Rodini , Lorenzo Rossi
show 1 more author
Cristiano Fanelli
This is my paper
classification hep-ph nucl-thphysics.data-an
keywords TMD PDFsDrell-Yansymbolic regressionnon-perturbative QCDneural network parametrizationCollins-Soper kernelN³LL
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.

Non-perturbative pieces of transverse-momentum-dependent parton distributions are usually guessed as simple Gaussians or exponentials, or else left inside hard-to-read neural networks. This paper trains a deliberately factorized network on a global Drell–Yan dataset at N³LL accuracy, then runs symbolic regression on each factor so that compact analytic expressions replace the network weights. The chosen formula has nine free numbers, reaches χ²/ndf = 1.040 on 482 points, and keeps a mild cross term linking the longitudinal momentum fraction x to the transverse separation b_T even when a sparsity prior tries to drive that term to zero. The result is a ready-to-code analytical non-perturbative function whose structure was read off the data rather than postulated, and it shows that symbolic regression can turn flexible machine-learning fits into interpretable QCD parametrizations.

Core claim

A closed-form non-perturbative unpolarized quark TMD, obtained by symbolic regression on a factorized neural network trained directly on 482 Drell–Yan cross-section points at N³LL, achieves χ²/ndf = 1.040 with only nine free constants and retains a non-trivial x–b_T cross term despite an L1 prior that biases the cross term toward zero.

What carries the argument

The factorized enveloping ansatz R(x,b_T)=exp[h_x(x)h_b(b_T)+h_xb(x,b_T)] multiplied by a Collins–Soper factor exp[−g₂²(b_T)b_T² ln(ζ/Q₀²)], trained as four small networks against experimental χ² and then distilled component-wise by symbolic regression, with the final expression chosen from a Pareto front of total complexity versus experimental χ².

Load-bearing premise

The claim that the data themselves prefer this structure rests on an architectural template that forces factorization plus a small cross term and on a sparsity prior that already pushes the cross term toward zero; a different template could produce different components at similar fit quality.

What would settle it

Replace the factorized-plus-sparse-cross-term template with an unconstrained two-dimensional network or an alternate factorization, re-run the full symbolic pipeline on the same 482-point Drell–Yan set, and check whether any comparably compact formula still retains a statistically required x–b_T correlation and χ²/ndf near 1.

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

If this is right

  • Global TMD fit codes can drop in a nine-constant closed form instead of a neural network or an ad-hoc Gaussian.
  • The retained x–b_T cross term becomes a concrete, testable claim about non-perturbative quark correlations rather than a modeling choice.
  • The same train-then-symbolically-distill pipeline can be applied to polarized TMDs and to flavor-dependent extractions once SIDIS data are included.
  • Pareto selection on complexity versus experimental χ² supplies a reproducible route from flexible ML fits to analytically usable QCD functions.

Where Pith is reading between the lines

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

  • If the mild x–b_T correlation survives replica and flavor tests, it may constrain the functional form of non-perturbative models used in lattice-informed or holographic TMD constructions.
  • The same skeleton-plus-symbolic pipeline is likely portable to other convolution-heavy observables (e.g., semi-inclusive DIS multiplicities or jet-broadening) where networks fit well but formulas are still guessed.
  • Because the large-b_T node in h_b lies outside the data-supported window, independent processes that probe larger b_T could falsify or reinforce the extrapolated shape without waiting for new DY data.

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

3 major / 7 minor

Summary. The manuscript presents a two-stage extraction of the non-perturbative unpolarized quark TMD f_NP from 482 Drell-Yan points (fixed-target, Tevatron, RHIC, LHC) at N3LL accuracy in NangaParbat. A small factorized neural network, R(x,b_T)=exp[h_x(x)h_b(b_T)+h_xb(x,b_T)] with an L1-sparsified, capacity-capped cross term, is trained directly against the full experimental covariance via a precomputed Ogata-quadrature "convolution skeleton"; PySR symbolic regression then distills each component, and the final closed form is selected from a combined complexity-chi2 Pareto front with a worst-subset minimax criterion. The selected complexity-31 formula (Eqs. 15-18) has 9 free constants and achieves chi2/ndf = 1.040 +/- 0.065 (parent NN: 0.941), with worst-subset chi2/Ns = 2.96. The abstract further claims that a non-trivial x-b_T cross term survives the sparsity prior, indicating a "mild but genuine" x-b_T correlation.

Significance. If the result holds, this is the first compact, closed-form f_NP obtained by fitting data rather than by postulating a functional template, and it is immediately re-implementable in existing global-fit codes. The methodology is carefully specified and substantially reproducible: training against the full experimental covariance via nuisance-parameter profiling identical to Ref. [15], subset-balanced loss, minimax checkpointing, component-wise PySR with combined experimental-chi2 Pareto selection, and a public interactive Pareto explorer. The compression from O(100) NN parameters to 9 analytic constants at a cost of only ~0.1 in chi2/ndf is a genuine and quantified achievement. The paper is also commendably candid about its own limitations (Sec. VI B): the enveloping ansatz is an architectural choice, flavor universality is assumed, and uncertainty quantification captures only seed variance. The weak point is that one of the two headline claims — the "genuine" x-b_T correlation — is not yet supported by the evidence presented, and the printed formula for g_2 appears to contain a transcription error.

major comments (3)
  1. [Sec. V A, Eq. (18); Sec. VI A(d)] Eq. (18) as printed, g_2(b_T) = 0.169 + 0.0328 sqrt(b_T/s) + 0.636, is inconsistent with the paper's own quoted values. It gives g_2(0) = 0.805, whereas Sec. V A and Sec. VI A(d) quote g_2(0) ≃ 0.195, g_2(2.5 GeV^-1) ≃ 0.227, and g_2(5 GeV^-1) ≃ 0.247. The quoted values are instead reproduced by g_2 = 0.169 + 0.0328 sqrt(b_T/s + 0.636): at b_T=0, 0.169+0.0328*sqrt(0.636)=0.195; at b_T=2.5, 0.227; at b_T=5, 0.247. The constant 0.636 therefore almost certainly belongs inside the square root. Since the closed-form formula is the central deliverable of the paper, this must be corrected and the corrected expression re-verified against the reported chi2/ndf = 1.040 before publication.
  2. [Abstract; Sec. IV A; Sec. V A, Eq. (17); Sec. VI A(c)] The claim of a 'mild but genuine x-b_T correlation' is not established for the component that actually carries the correlation. In the selected h_xb = (b_T/s)[0.766 - 0.0567(b_T/s - x)^2], the dominant 0.766*b_T/s piece is x-independent. Because h_b(b_T) enters the exponent of Eq. (9) only multiplied by h_x(x), an additive b_T-only term has no place in the factorized backbone and is structurally forced into h_xb; its retention under the L1 prior therefore cannot be read as evidence of x-b_T correlation. The genuinely x-dependent part is only the 0.0567 coefficient: at the paper's own reference point (x=0.1, b_T=1 GeV^-1, Sec. VI A(c)) it contributes 0.046 of the 0.720 cross-term total (~6%). The manuscript never tests whether this term is statistically required: there is no strictly-separable (h_xb ≡ 0) baseline, no ablation deleting the 0.0567 term, and no lambda_L1 scan showing the par
  3. [Sec. IV A-B; Sec. VI B] Related to the previous point but distinct: the paper's framing of the extraction as 'discovery' should be reconciled with its own Sec. VI B admission that the smallness of h_xb is 'in part imposed rather than purely data-driven' (L1 prior plus four-neuron cap plus subset balancing). Given that the unconstrained 2D fit yielded no candidate with chi2/Ndat < 1.3 (Sec. IV A), the reader cannot tell whether the retained component structure reflects the data or the prior. A minimal control — e.g., reporting how the extracted h_xb and chi2 vary as lambda_L1 is scanned over one to two orders of magnitude around 10^-4 — would substantially strengthen the robustness claim and is within the existing pipeline's scope.
minor comments (7)
  1. [Eq. (14) vs Eq. (8)] Eq. (14) writes the Collins-Soper factor as -(g_2^(k)(b))^2 b_T^2 ln(zeta/Q_0^2), whereas Eq. (8) has -(g_2/2) b_T^2 ln(zeta/Q_0^2) with g_2 not squared. Please make the convention consistent, or state explicitly if the symbolic g_2 is defined as the square root of the conventional one.
  2. [Sec. VI A(d)] Units: g_2 is quoted in GeV ('g_2 ≃ 0.195 GeV'), but for the exponent in Eq. (8) to be dimensionless with b_T in GeV^-1, g_2 must carry units of GeV^2 (as in the MAP convention). Please check throughout.
  3. [Sec. IV D] The argument that the O(100)-to-9 compression is 'incompatible with a network that has fit noise' is heuristic; overfitting diagnostics would be strengthened by quoting the chi2 on a simple held-out split (even if not used for training) or by a replica-level comparison. This does not block publication given the small architecture, but a sentence acknowledging the limitation of the compression argument would help.
  4. [Sec. V, Fig. 1] Fig. 1 is very small and its axis tick labels are hard to read; the complexity-31 selection point should be explicitly marked on the figure rather than described only in the text.
  5. [Sec. V B, Table II] Table II aggregates experiments (e.g., 'Fermilab 233'), but the worst-subset discussion (E772 8<Q<9 GeV at chi2/Ns ≃ 2.96) refers to subsets not shown in the table. A short supplementary table of the five worst subsets, or a pointer to the exact view in the Pareto explorer, would make the minimax criterion verifiable.
  6. [Introduction, Sec. II] The phrase 'first analytical formula for f_NP obtained directly from data rather than postulated' is somewhat at odds with Sec. VI B, where the enveloping structure itself is acknowledged to be postulated. Suggest softening to 'discovered within a fixed enveloping ansatz'.
  7. [Throughout] Typographical: 'EXPERIMENT AL DA T A' and 'RESUL TS' section headings (OCR artifacts); 'the Pareto-knee analysis ... act as' (subject-verb agreement, Sec. IV D); stray comma after 'constants,' at the end of Sec. V A.

Circularity Check

1 steps flagged

Main extraction is ordinary data-driven fitting, not circular; only the abstract’s ‘genuine x–b_T correlation’ gloss mildly over-reads a residual channel defined to absorb non-product terms.

specific steps
  1. self definitional [Abstract; Sec. IV A Eqs. (8)–(9); Sec. VI A(c); Eq. (17)]
    "A non-trivial x–b_T cross term is retained even under a sparsity prior that biases it toward zero, indicating a mild but genuine correlation between the longitudinal momentum fraction and the transverse momentum. ... where the cross term h_xb(x, b_T), which captures x–b_T correlations, is included throughout the training. ... h_xb(x, b_T) = b_T/s (0.766 − 0.0567 (b_T/s − x)^2)"

    h_xb is defined as the additive residual outside the product h_x(x)h_b(b_T). Any pure-b_T piece whose b_T shape is not proportional to h_b cannot live in the factorized backbone and is forced into h_xb even with no x dependence. The selected h_xb is dominated by the x-independent 0.766 b_T/s term; only the small 0.0567(b_T/s−x)^2 piece is true mixing. Inferring ‘genuine x–b_T correlation’ solely from ‘h_xb≠0 under L1’ therefore partly restates the architectural definition of the residual channel rather than an independent correlation test (no h_xb≡0 baseline or term-ablation is shown).

full rationale

The paper does not claim a first-principles QCD derivation of f_NP. It trains a constrained factorized NN on DY cross sections, distills components with PySR, and selects a Pareto formula by the same experimental χ²—standard phenomenological compression. Constants in Eqs. (15)–(18) are fit parameters, not independent predictions of the spectra they describe; that is not Eq.X≡Eq.Y circularity. Self-citations to MAP/NangaParbat supply the shared dataset, evolution skeleton, and NN benchmark, not a load-bearing uniqueness theorem that forces the symbolic form. The sole mild issue is interpretive: h_xb is introduced as the channel that ‘captures x–b_T correlations,’ yet under R=exp[h_x h_b+h_xb] it also absorbs any pure-b_T shape that is not proportional to h_b, so non-zero h_xb under L1 does not by itself establish genuine x–b_T correlation (and Sec. VI B already concedes smallness is partly prior-imposed). That weakens one sentence of physical gloss; it does not collapse the derivation of the nine-constant formula or the reported χ²/ndf. Score 2.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 2 invented entities

The central claim rests on standard TMD factorization and N³LL evolution as implemented in NangaParbat, on a specific factorized exponential ansatz for R, on DY data under q_T/Q<0.2 cuts, and on nine fitted constants selected via symbolic regression and Pareto rules. No new particles or forces are introduced; the ‘invented’ objects are architectural (factorized networks, sparsity-stabilized cross term, complexity-31 selection criterion).

free parameters (7)
  • h_x coefficients (−0.854, −1.45) = −0.854 − 1.45 x
    Linear coefficients in the symbolic h_x(x) fixed by regression to the trained network / experimental χ².
  • h_b coefficients (0.919, −0.368) = 0.919, −0.368
    Coefficients in h_b(b_T)= (b_T/s)(0.919 − 0.368√(b_T/s)).
  • h_xb coefficients (0.766, −0.0567) = 0.766, −0.0567
    Coefficients of the cross term retained on the Pareto front.
  • g_2 coefficients (0.169, 0.0328, 0.636) = 0.169 + 0.0328√(b_T/s) + 0.636 (as written)
    Three constants in the mildly b_T-dependent Collins–Soper nonperturbative coupling.
  • λ_L1, λ_L2, λ_bal, subset weights w_s = λ_L1=1e-4; λ_L2=1e-4; λ_bal=5; w_s=3 (E772 5–9 GeV)
    Hand-chosen regularization and balancing hyperparameters that shape the trained NN before distillation (λ_L1=10^{-4}, λ_L2=10^{-4}, λ_bal=5, w_s=3 on selected E772 bins).
  • b_cut_T = 2.5 GeV^{-1} and b_max, b_min in b* = b_cut_T=2.5 GeV^{-1}; b_max=2e^{-γ_E} GeV^{-1}
    Kinematic cutoffs defining the fitted and evolved domain; not data-derived constants of f_NP but choices that bound the extraction.
  • Pareto complexity threshold (complexity 31 selection) = complexity 31
    Discrete selection rule (worst-subset χ²/N_s<3 and knee at complexity 31) chooses which symbolic candidate is ‘the’ formula.
axioms (6)
  • domain assumption TMD factorization of the DY cross section for q_T ≪ Q with the standard hard factor, charges, and Fourier–Bessel convolution (Eq. 1).
    Load-bearing QCD factorization assumption; analysis restricted to q_T/Q<0.2.
  • domain assumption N³LL TMD evolution with b* prescription to avoid the Landau pole and f_NP→1 as b_T→0 (Eqs. 4–7).
    Standard CSS/TMD evolution setup shared with MAP literature; defines how f_NP enters observables.
  • ad hoc to paper R(x,b_T)=exp[h_x(x)h_b(b_T)+h_xb(x,b_T)] with architectural boundaries h_b(0)=h_xb(x,0)=0 and flavor-universal f_NP.
    Enveloping ansatz chosen for symbolic tractability; paper admits alternative templates may give different components.
  • ad hoc to paper L1 sparsity prior and four-neuron cap on h_xb plus subset-balanced χ² make a small cross term the preferred residual correlation.
    Stabilizers bias toward separability; interpretation of ‘genuine’ correlation depends on them (Sec. IV A, VI B).
  • domain assumption Experimental covariance matrices and analytic nuisance profiling as in the reference MAPNN fit correctly represent uncertainties.
    χ² definition identical to Ref. [15]; goodness-of-fit claims inherit that treatment.
  • ad hoc to paper PySR search with chosen operators, parsimony, and b_T<2.5 GeV^{-1} grid adequately explores the expression space for each component.
    Symbolic completeness is not proved; Pareto front is relative to the search configuration (Sec. IV E–F).
invented entities (2)
  • Factorized NN components (h_x, h_b, h_xb, g_2) as the distillation interface no independent evidence
    purpose: Reduce 2D symbolic regression to 1D pieces plus a small cross term so compact formulas exist.
    Not a physical particle but an architectural entity that defines what can be ‘discovered’; independent evidence only via fit quality to DY data.
  • Complexity-31 Pareto-selected symbolic f_NP formula no independent evidence
    purpose: Provide a portable closed-form nonperturbative function with 9 constants.
    The specific algebraic form is the paper’s product; falsifiable by refitting independent processes (SIDIS, etc.), which is proposed but not done here.

pith-pipeline@v1.2.0-grok45-kimik3 · 22147 in / 4620 out tokens · 89457 ms · 2026-07-31T07:07:21.741240+00:00 · methodology

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read the original abstract

We present an analytical parametrization of the non-perturbative transverse-momentum-dependent (TMD) parton distribution function of unpolarized quarks, extracted from Drell-Yan data using a combination of neural-network fitting and symbolic regression. A factorized neural network is trained directly against experimental cross-section data from fixed-target, Tevatron, RHIC, and LHC experiments at next-to-next-to-next-to-leading logarithmic accuracy, and symbolic regression is subsequently applied to each network component to discover compact analytical expressions. The final formula is selected from a Pareto front in the space of expression complexity and experimental $\chi^2$, yielding a closed-form non-perturbative function with 9 free numerical constants that achieves $\chi^2/\mathrm{ndf}=1.040$ over 482 data points. A non-trivial $x$-$b_T$ cross term is retained even under a sparsity prior that biases it toward zero, indicating a mild but genuine correlation between the longitudinal momentum fraction and the transverse momentum. This work demonstrates that symbolic regression is a viable tool for bridging flexible machine-learning fits and interpretable analytical TMD parametrizations, and opens a systematic path toward data-driven discovery of specific features of non-perturbative QCD.

Figures

Figures reproduced from arXiv: 2607.24710 by Alessandro Bacchetta, Chiara Bissolotti, Cole Granger, Cristiano Fanelli, Lorenzo Rossi, Marco Radici, Matteo Cerutti, Simone Rodini, Valerio Bertone.

Figure 1
Figure 1. Figure 1: FIG. 1. The Pareto front: candidate symbolic parametriza [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The non-perturbative [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Drell-Yan cross section as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

discussion (0)

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Reference graph

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