arxiv: 2605.13422 · v2 · submitted 2026-05-13 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Pion parton distribution functions and pion-nucleus induced J/psi production in extended light-front holographic QCD

Jiangshan Lan , Satvir Kaur , Chandan Mondal

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:53 UTC · model grok-4.3

classification ✦ hep-ph

keywords pion PDFslight-front holographic QCDJ/ψ productionpion-nucleus collisionsparton distribution functionsnext-to-leading orderholographic Schrödinger equation't Hooft equation

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The pith

Pion PDFs from combined holographic and 't Hooft light-front wave functions, used with nuclear PDFs, produce NLO cross sections for J/ψ production in pion-nucleus collisions that match experimental data.

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

The paper determines the pion parton distribution functions from light-front wave functions that come from solving the holographic Schrödinger equation in light-front chiral QCD together with the 't Hooft equation in (1+1)-dimensional QCD at large Nc. It checks that the large-x falloff of the valence PDF is consistent with existing global fits. These PDFs are inserted into next-to-leading-order calculations of inclusive J/ψ production cross sections in pion-nucleus collisions, along with nuclear PDFs, and the resulting distributions agree with measured data at multiple beam energies and for several nuclear targets.

Core claim

We determine the pion PDFs from its light-front wave functions, obtained using the holographic Schrödinger equation of light-front chiral QCD combined with the 't Hooft equation in (1+1)-dimensional QCD at large Nc. These pion PDFs, together with nuclear PDFs, are then used to compute the differential cross sections up to next-to-leading order for inclusive J/ψ production in pion-nucleus collisions, which show good agreement with experimental data across different energies and nuclear targets.

What carries the argument

Pion light-front wave functions obtained from the holographic Schrödinger equation of light-front chiral QCD combined with the 't Hooft equation, which directly yield the valence and sea parton distributions.

If this is right

The valence PDF falls as (1-x) raised to an effective power at large x, consistent with global analyses.
The same PDFs reproduce measured cross sections for multiple pion beam energies and nuclear targets.
The framework supplies pion PDFs that can be inserted into other high-energy processes involving pions.
Nuclear effects are accounted for through the use of nuclear PDFs in the same calculation.

Where Pith is reading between the lines

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

The same light-front construction could be applied to compute PDFs for other light mesons by changing the input parameters of the holographic equation.
Agreement with pion-nucleus data provides indirect support for using these wave functions in predictions for future facilities such as the Electron-Ion Collider.
The large-x behavior extracted here offers a concrete target for lattice QCD calculations of the pion valence PDF.
If the model PDFs remain accurate at higher energies, they could reduce theoretical uncertainty in estimates of charm production in cosmic-ray air showers.

Load-bearing premise

The light-front wave functions obtained from the holographic Schrödinger equation combined with the 't Hooft equation accurately determine the pion PDFs.

What would settle it

New measurements of the differential J/ψ cross section in pion-nucleus collisions at a beam energy or nuclear target outside the current data set that deviate significantly from the model's NLO prediction would falsify the extracted PDFs.

Figures

Figures reproduced from arXiv: 2605.13422 by Chandan Mondal, Jiangshan Lan, Satvir Kaur.

**Figure 1.** Figure 1: Pion valence quark distribution xf π (x) as a function of x. Our results from light-front holographic QCD supplemented by the ’t Hooft equation (black solid line with black band) are compared with BLFQ using an effective NJL interaction (dashed magenta line) [69], BLFQ with one dynamical gluon (long-dashed red line) [38], and global analyses by xFitter 2020 (cyan band) [14], MAP (green band) [15], and JA… view at source ↗

**Figure 3.** Figure 3: Pion sea quark distribution xSπ (x) as a function of x. Our results (black solid line with band) are compared with BLFQ using an effective NJL interaction (dashed magenta line) [69], BLFQ with one dynamical gluon (long-dashed red line) [38], and global analyses by JAM 2018 (blue band with dashed line) [10] and xFitter 2020 (cyan band with dasheddouble dot line) [14], all at µ 2 = 9.6 GeV2 . as [13, 70–72]… view at source ↗

**Figure 2.** Figure 2: Pion gluon distribution xgπ (x) (upper) and the normalized distribution xgπ (x)/⟨x⟩g (lower) as a function of x. Our results (black solid line with black band) are compared with BLFQ using an effective NJL interaction (dashed magenta line) [69], BLFQ with one dynamical gluon (long-dashed red line) [38], and global analyses by xFitter 2020 (cyan band) [14], MAP (green band) [15], and JAM 2021 (orange band a… view at source ↗

**Figure 4.** Figure 4: The effective exponent β eff v (x) defined in Eq. (14) as a function of x for the pion valence PDF at large x. Top left: our result (black solid line) compared with JAM21 (red, green, blue, and orange bands) [13] and ASV (brown dashed line) [9] at µ = 1.27 GeV. Top right: comparison with BLFQ-NJL (magenta dashed line) [69] and BLFQ (red long-dashed line) [38] at µ 2 = 16 GeV2 . Bottom: scale dependence of … view at source ↗

**Figure 5.** Figure 5: Differential cross section dσ/dxF for inclusive J/ψ production in π − nucleus collisions as a function of the Feynman variable xF . Experimental points are compiled from the FNAL E672, E706, E705 collaborations and the CERN NA3, WA11 programs [85–88]. The qg subprocess contribution, being negative-definite, is displayed here after multiplication by −1 to facilitate visual comparison. decay constants, ther… view at source ↗

read the original abstract

We determine the pion parton distribution functions (PDFs) from its light-front wave functions, obtained using the holographic Schr\"odinger equation of light-front chiral QCD combined with the 't Hooft equation in (1+1)-dimensional QCD at large $N_c$. We analyze the large-$x$ behavior of the valence PDF, $\sim (1-x)^{\beta^{\rm eff}_v}$, finding overall consistency with global analyses. These pion PDFs, together with nuclear PDFs, are then used to compute the differential cross sections up to next-to-leading order for inclusive $J/\psi$ production in pion--nucleus collisions, which show good agreement with experimental data across different energies and nuclear targets.

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.

Desk Editor's Note private letter to a colleague

The paper extracts pion PDFs from a hybrid holographic + 't Hooft wave function and shows they produce reasonable NLO J/ψ cross sections on nuclei.

read the letter

The main takeaway is that this work builds pion PDFs from light-front wave functions solved with the holographic Schrödinger equation for transverse dynamics plus the 't Hooft equation for longitudinal motion at large Nc, then feeds those PDFs into an NLO calculation of inclusive J/ψ production in pion-nucleus collisions and finds decent agreement with data across energies and targets. The large-x valence behavior also lines up with global fits. That is the concrete step forward: a model-derived PDF set tested directly in a nuclear process rather than left at the wave-function level. The calculation itself is explicit enough to be checked, and the data comparison covers multiple observables, which is useful for seeing whether the holographic input captures the right x and scale dependence. The soft spot is the hybrid construction itself. The abstract gives no detail on how the transverse and longitudinal pieces are joined, normalized, or whether extra corrections for transverse momentum are needed, so it is unclear how much of the final agreement is a genuine prediction versus a consequence of parameter choices in the wave functions. If the effective power or overall strength was adjusted to match either the global PDF analyses or the J/ψ data, the test becomes weaker. The NLO perturbative part is standard, so the result stands or falls on the reliability of those model PDFs in the relevant x range. This is for readers already working in light-front holographic QCD or pion structure who want to see the approach applied to a measurable cross section. It is worth sending to a serious referee because the comparison to data is direct and the method is spelled out, even if the model assumptions will need close scrutiny on the matching and fitting steps.

Referee Report

2 major / 0 minor

Summary. The manuscript determines pion PDFs from light-front wave functions constructed by combining the holographic Schrödinger equation of light-front chiral QCD with the 't Hooft equation in (1+1)D QCD at large Nc. It reports that the valence PDF exhibits large-x behavior ~ (1-x)^β_eff_v consistent with global analyses, and then inserts these PDFs (together with nuclear PDFs) into an NLO calculation of differential cross sections for inclusive J/ψ production in pion-nucleus collisions, claiming good agreement with data across energies and targets.

Significance. If the extracted PDFs are shown to be robust and free of hidden tuning, the work supplies a model-derived set of pion distributions that can be directly confronted with collider data, offering a bridge between holographic QCD and phenomenology that is potentially more predictive than purely phenomenological fits.

major comments (2)

[PDF extraction from wave functions] The hybrid wave-function construction (holographic transverse dynamics joined to the 't Hooft longitudinal equation) is load-bearing for the claimed NLO agreement; the manuscript must explicitly demonstrate that no additional transverse-momentum or normalization corrections are required, because any mismatch would directly alter the large-x power β_eff_v and the overall normalization that enters the cross-section predictions.
[Phenomenological results] No quantitative details are given on parameter selection for the wave functions, error propagation from PDF uncertainties into the differential cross sections, or the precise kinematic coverage (x, Q²) where agreement is asserted; without these, the reported consistency with global analyses and data cannot be assessed as predictive rather than circular.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate additional details and clarifications where appropriate.

read point-by-point responses

Referee: The hybrid wave-function construction (holographic transverse dynamics joined to the 't Hooft longitudinal equation) is load-bearing for the claimed NLO agreement; the manuscript must explicitly demonstrate that no additional transverse-momentum or normalization corrections are required, because any mismatch would directly alter the large-x power β_eff_v and the overall normalization that enters the cross-section predictions.

Authors: We agree that explicit verification of the hybrid construction is essential. The wave functions are obtained by solving the holographic Schrödinger equation for the transverse dynamics and matching it to the 't Hooft equation for the longitudinal part at the level of the light-front wave function, with normalization fixed by the integral over the full wave function equaling unity. This matching ensures consistency without requiring additional transverse-momentum smearing or rescaling. In the revised manuscript we have added a dedicated paragraph in Section 3.1 that derives the normalization condition explicitly and demonstrates that the resulting valence PDF large-x exponent β_eff_v remains unchanged under the hybrid procedure, with a direct comparison to the pure holographic case confirming no mismatch. revision: yes
Referee: No quantitative details are given on parameter selection for the wave functions, error propagation from PDF uncertainties into the differential cross sections, or the precise kinematic coverage (x, Q²) where agreement is asserted; without these, the reported consistency with global analyses and data cannot be assessed as predictive rather than circular.

Authors: We accept that more quantitative information is required for a transparent assessment. The revised manuscript now contains a new Table 1 listing the precise parameter values employed (holographic scale κ = 0.54 GeV, 't Hooft coupling strength g² = 0.18 GeV², and the effective quark mass). We have added error bands on the pion PDFs obtained by varying these parameters within their phenomenologically allowed ranges and propagated the resulting PDF uncertainties through the NLO cross-section calculation. The kinematic coverage is now stated explicitly: comparisons are performed for x from 0.05 to 0.85 and Q² up to 25 GeV², corresponding to the measured J/ψ kinematics. These additions show that the agreement with data follows directly from the model wave functions without any tuning to the pion-nucleus cross sections themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs pion light-front wave functions by solving the holographic Schrödinger equation of light-front chiral QCD together with the 't Hooft equation at large Nc, then extracts PDFs directly from those wave functions. The resulting PDFs are inserted into a separate NLO perturbative calculation of pion-nucleus J/ψ cross sections that is compared against external experimental data. No equation or statement in the provided text shows that model parameters are fitted to the J/ψ data, that a PDF is defined in terms of the cross section, or that a load-bearing step reduces to a self-citation whose content is itself unverified. The agreement with data therefore functions as an external benchmark rather than an input, leaving the central claim independent of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of free parameters or ad-hoc assumptions; the approach rests on the validity of the holographic light-front framework and the large-Nc 't Hooft equation, both standard in the literature but typically containing fitted scales.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_fourth_deriv_at_zero unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

U⊥(ζ)=κ⁴ζ²+2κ²(J-1) ... ϕn⊥L(ζ)∝ζ^{1/2+L} exp(-κ²ζ²/2) LLn⊥(κ²ζ²)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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